formulating moderation hypothesis

The Three Most Common Types of Hypotheses

In this post, I discuss three of the most common hypotheses in psychology research, and what statistics are often used to test them.

• Post author By sean
• Post date September 28, 2013
• 37 Comments on The Three Most Common Types of Hypotheses

Simple main effects (i.e., X leads to Y) are usually not going to get you published. Main effects can be exciting in the early stages of research to show the existence of a new effect, but as a field matures the types of questions that scientists are trying to answer tend to become more nuanced and specific.  In this post, I’ll briefly describe the three most common kinds of hypotheses that expand upon simple main effects – at least, the most common ones I’ve seen in my research career in psychology – as well as providing some resources to help you learn about how to test these hypotheses using statistics.

Incremental Validity

“Can X predict Y over and above other important predictors?”

This is probably the simplest of the three hypotheses I propose. Basically, you attempt to rule out potential confounding variables by controlling for them in your analysis.  We do this because (in many cases) our predictor variables are correlated with each other. This is undesirable from a statistical perspective, but is common with real data. The idea is that we want to see if X can predict unique variance in Y over and above the other variables you include.

In terms of analysis, you are probably going to use some variation of multiple regression or partial correlations.  For example, in my own work I’ve shown in the past that friendship intimacy as coded from autobiographical narratives can predict concern for the next generation over and above numerous other variables, such as optimism, depression, and relationship status ( Mackinnon et al., 2011 ).

“Under what conditions does X lead to Y?”

Of the three techniques I describe, moderation is probably the most tricky to understand.  Essentially, it proposes that the size of a relationship between two variables changes depending upon the value of a third variable, known as a “moderator.”  For example, in the diagram below you might find a simple main effect that is moderated by sex. That is, the relationship is stronger for women than for men:

With moderation, it is important to note that the moderating variable can be a category (e.g., sex) or it can be a continuous variable (e.g., scores on a personality questionnaire).  When a moderator is continuous, usually you’re making statements like: “As the value of the moderator increases, the relationship between X and Y also increases.”

“Does X predict M, which in turn predicts Y?”

We might know that X leads to Y, but a mediation hypothesis proposes a mediating, or intervening variable. That is, X leads to M, which in turn leads to Y.  In the diagram below I use a different way of visually representing things consistent with how people typically report things when using path analysis.

I use mediation a lot in my own research. For example, I’ve published data suggesting the relationship between perfectionism and depression is mediated by relationship conflict ( Mackinnon et al., 2012 ). That is, perfectionism leads to increased conflict, which in turn leads to heightened depression. Another way of saying this is that perfectionism has an indirect effect on depression through conflict.

Helpful links to get you started testing these hypotheses

Depending on the nature of your data, there are multiple ways to address each of these hypotheses using statistics. They can also be combined together (e.g., mediated moderation). Nonetheless, a core understanding of these three hypotheses and how to analyze them using statistics is essential for any researcher in the social or health sciences.  Below are a few links that might help you get started:

Are you a little rusty with multiple regression? The basics of this technique are required for most common tests of these hypotheses. You might check out this guide as a helpful resource:

https://statistics.laerd.com/spss-tutorials/multiple-regression-using-spss-statistics.php

David Kenny’s Mediation Website provides an excellent overview of mediation and moderation for the beginner.

http://davidakenny.net/cm/mediate.htm

http://davidakenny.net/cm/moderation.htm

Preacher and Haye’s INDIRECT Macro is a great, easy way to implement mediation in SPSS software, and their MODPROBE macro is a useful tool for testing moderation.

http://afhayes.com/spss-sas-and-mplus-macros-and-code.html

If you want to graph the results of your moderation analyses, the excel calculators provided on Jeremy Dawson’s webpage are fantastic, easy-to-use tools:

http://www.jeremydawson.co.uk/slopes.htm

• Tags mediation , moderation , regression , tutorial

37 replies on “The Three Most Common Types of Hypotheses”

I want to see clearly the three types of hypothesis

Thanks for your information. I really like this

Thank you so much, writing up my masters project now and wasn’t sure whether one of my variables was mediating or moderating….Much clearer now.

Thank you for simplified presentation. It is clearer to me now than ever before.

Thank you. Concise and clear

hello there

I would like to ask about mediation relationship: If I have three variables( X-M-Y)how many hypotheses should I write down? Should I have 2 or 3? In other words, should I have hypotheses for the mediating relationship? What about questions and objectives? Should be 3? Thank you.

Hi Osama. It’s really a stylistic thing. You could write it out as 3 separate hypotheses (X -> Y; X -> M; M -> Y) or you could just write out one mediation hypotheses “X will have an indirect effect on Y through M.” Usually, I’d write just the 1 because it conserves space, but either would be appropriate.

Hi Sean, according to the three steps model (Dudley, Benuzillo and Carrico, 2004; Pardo and Román, 2013)., we can test hypothesis of mediator variable in three steps: (X -> Y; X -> M; X and M -> Y). Then, we must use the Sobel test to make sure that the effect is significant after using the mediator variable.

Yes, but this is older advice. Best practice now is to calculate an indirect effect and use bootstrapping, rather than the causal steps approach and the more out-dated Sobel test. I’d recommend reading Hayes (2018) book for more info:

Hayes, A. F. (2018). Introduction to mediation, moderation, and conditional process analysis: A regression-based approach (2nd ed). Guilford Publications.

Hi! It’s been really helpful but I still don’t know how to formulate the hypothesis with my mediating variable.

I have one dependent variable DV which is formed by DV1 and DV2, then I have MV (mediating variable), and then 2 independent variables IV1, and IV2.

How many hypothesis should I write? I hope you can help me 🙂

Thank you so much!!

If I’m understanding you correctly, I guess 2 mediation hypotheses:

IV1 –> Med –> DV1&2 IV2 –> Med –> DV1&2

Thank you so much for your quick answer! ^^

Could you help me formulate my research question? English is not my mother language and I have trouble choosing the right words. My x = psychopathy y = aggression m = deficis in emotion recognition

thank you in advance

I have mediator and moderator how should I make my hypothesis

Can you have a negative partial effect? IV – M – DV. That is my M will have negative effect on the DV – e.g Social media usage (M) will partial negative mediate the relationship between father status (IV) and social connectedness (DV)?

Thanks in advance

Hi Ashley. Yes, this is possible, but often it means you have a condition known as “inconsistent mediation” which isn’t usually desirable. See this entry on David Kenny’s page:

Or look up “inconsistent mediation” in this reference:

MacKinnon, D. P., Fairchild, A. J., & Fritz, M. S. (2007). Mediation analysis. Annual Review of Psychology, 58, 593-614.

This is very interesting presentation. i love it.

This is very interesting and educative. I love it.

Hello, you mentioned that for the moderator, it changes the relationship between iv and dv depending on its strength. How would one describe a situation where if the iv is high iv and dv relationship is opposite from when iv is low. And then a 3rd variable maybe the moderator increases dv when iv is low and decreases dv when iv is high.

This isn’t problematic for moderation. Moderation just proposes that the magnitude of the relationship changes as levels of the moderator changes. If the sign flips, probably the original relationship was small. Sometimes people call this a “cross-over” effect, but really, it’s nothing special and can happen in any moderation analysis.

i want to use an independent variable as moderator after this i will have 3 independent variable and 1 dependent variable…. my confusion is do i need to have some past evidence of the X variable moderate the relationship of Y independent variable and Z dependent variable.

Dear Sean It is really helpful as my research model will use mediation. Because I still face difficulty in developing hyphothesis, can you give examples ? Thank you

Hi! is it possible to have all three pathways negative? My regression analysis showed significant negative relationships between x to y, x to m and m to y.

Hi, I have 1 independent variable, 1 dependent variable and 4 mediating variable May I know how many hypothesis should I develop?

Hello I have 4 IV , 1 mediating Variable and 1 DV

My model says that 4 IVs when mediated by 1MV leads to 1 Dv

Pls tell me how to set the hypothesis for mediation

Hi I have 4 IVs ,2 Mediating Variables , 1DV and 3 Outcomes (criterion variables).

Pls can u tell me how many hypotheses to set.

Thankyou in advance

I am in fact happy to read this webpage posts which carries tons of useful information, thanks for providing such data.

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what if the hypothesis and moderator significant in regrestion and insgificant in moderation?

Thank you so much!! Your slide on the mediator variable let me understand!

Very informative material. The author has used very clear language and I would recommend this for any student of research/

Hi Sean, thanks for the nice material. I have a question: for the second type of hypothesis, you state “That is, the relationship is stronger for men than for women”. Based on the illustration, wouldn’t the opposite be true?

Yes, your right! I updated the post to fix the typo, thank you!

I have 3 independent variable one mediator and 2 dependant variable how many hypothesis I have 2 write?

Sounds like 6 mediation hypotheses total:

X1 -> M -> Y1 X2 -> M -> Y1 X3 -> M -> Y1 X1 -> M -> Y2 X2 -> M -> Y2 X3 -> M -> Y2

Clear explanation! Thanks!

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Statistics: Data analysis and modelling

Chapter 6 moderation and mediation.

In this chapter, we will focus on two ways in which one predictor variable may affect the relation between another predictor variable and the dependent variable. Moderation means the strength of the relation (in terms of the slope) of a predictor variable is determined by the value of another predictor variable. For instance, while physical attractiveness is generally positively related to mating success, for very rich people, physical attractiveness may not be so important. This is also called an interaction between the two predictor variables. Mediation is a different way in which two predictors affect a dependent variable. It is best thought of as a causal chain , where one predictor variable determines the value of another predictor variable, which then in turn determines the value of the dependent variable. the difference between moderation and mediation is illustrated in Figure 6.1 .

Figure 6.1: Graphical depiction of the difference between moderation and mediation. Moderation means that the effect of a predictor ( \(X_1\) ) on the dependent variable ( \(Y\) ) depends on the value of another predictor ( \(X_2\) ). Mediation means that a predictor ( \(X_1\) ) affects the dependent variable ( \(Y\) ) indirectly, through its relation to another predictor ( \(X_2\) ) which is directly related to the dependent variable.

6.1 Moderation

6.1.1 physical attractiveness and intelligence in speed dating.

Fisman, Iyengar, Kamenica, & Simonson ( 2006 ) conducted a large scale experiment 15 on dating behaviour. They placed their participants in a speed dating context, where they were randomly matched with a number of potential partners (between 5 and 20) and could converse for four minutes. As part of the study, after each meeting, participants rated how much they liked their speed dating partners, as well as more specifically on their attractiveness, sincerity, intelligence, fun, and ambition. We will focus in particular on ratings of physical attractiveness, fun, and intelligence, and how these are related to the general liking of a person. Ratings were given on a 10-point scale, from 1 (“awful”) to 10 (“great”). A multiple regression analysis predicting general liking from attractiveness, fun, and intelligence (Table 6.1 ) shows that all three predictors have a significant and positive relation with general liking.

6.1.2 Conditional slopes

If we were to model the relation between overall liking and physical attractiveness and intelligence, we might use a multiple regression model such as: 16 \[\texttt{like}_i = \beta_0 + \beta_{\texttt{attr}} \times \texttt{attr}_i + \beta_\texttt{intel} \times \texttt{intel}_i + \epsilon_i \quad \quad \epsilon_i \sim \mathbf{Normal}(0,\sigma_\epsilon)\] which is estimated as \[\texttt{like}_i = -0.0733 + 0.527 \times \texttt{attr}_i + 0.392 \times \texttt{intel}_i + \hat{\epsilon}_i \quad \quad \hat{\epsilon}_i \sim \mathbf{Normal}(0, 1.25)\] The estimates indicate a positive relation to liking of both attractiveness and intelligence. Note that the values of the slopes are different from those in Table 6.1 . The reason for this is that the model in the Table also includes fun as a predictor. Because the slopes reflect unique effects , these depend on all predictors included in the model. When there is dependence between the predictors (i.e. there is multicollinearity) both the estimates of the slopes and the corresponding significance tests will vary when you add or remove predictors from the model.

In the model above, a relative lack in physical attractiveness can be overcome by high intelligence, because in the end, the general liking of someone depends on the sum of both attractiveness and intelligence (each “scaled” by their corresponding slope). For example, someone with an attractiveness rating of \(\texttt{attr}_i = 8\) and an intelligence rating of \(\texttt{intel}_i = 2\) would be expected to be liked as much as a partner as someone with an attractiveness rating of \(\texttt{attr}_i = 3.538\) and an intelligence rating of \(\texttt{intel}_i = 8\) : \[\begin{aligned} \texttt{like}_i &= -0.073 + 0.527 \times 8 + 0.392 \times 2 = 4.924 \\ \texttt{like}_i &= -0.073 + 0.527 \times 3.538 + 0.392 \times 8 = 4.924 \end{aligned}\]

But what if for those lucky people who are very physically attractive, their intelligence doesn’t matter that much , or even at all ? And what if, for those lucky people who are very intelligent, their physical attractiveness doesn’t really matter much or at all? In other words, what if the more attractive people are, the less intelligence determines how much other people like them as a potential partner, and conversely, the more intelligent people are, the less attractiveness determines how much others like them as a potential partner? This implies that the effect of attractiveness on liking depends on intelligence, and that the effect of intelligence on liking depends on attractiveness. Such dependence is not captured by the multiple regression model above. While a relative lack of intelligence might be overcome by a relative abundance of attractiveness, for any level of intelligence, the additional effect of attractiveness is the same (i.e., an increase in attractiveness by one unit will always result in an increase of the predicted liking of 0.527).

Let’s define \(\beta_{\texttt{attr}|\texttt{intel}_i}\) as the slope of \(\texttt{attr}\) conditional on the value of \(\texttt{intel}_i\) . That is, we allow the slope of \(\texttt{attr}\) to vary as a function of \(\texttt{intel}\) . Similarly, we can define \(\beta_{\texttt{intel}|\texttt{attr}_i}\) as the slope of \(\texttt{intel}\) conditional on the value of \(\texttt{attr}\) . Our regression model can then be written as: \[\begin{equation} \texttt{like}_i = \beta_0 + \beta_{\texttt{attr}|\texttt{intel}_i} \times \texttt{attr}_i + \beta_{\texttt{intel} | \texttt{attr}_i} \times \texttt{intel}_i + \epsilon_i \tag{6.1} \end{equation}\] That’s a good start, but what would the value of \(\beta_{\texttt{attr}|\texttt{intel}_i}\) be? Estimating the slope of \(\texttt{attr}\) for each value of \(\texttt{intel}\) by fitting regression models to each subset of data with a particular value of \(\texttt{intel}\) is not really doable. We’d need lots and lots of data, and furthermore, we wouldn’t also be able to simultaneously estimate the value of \(\beta_{\texttt{intel} | \texttt{attr}_i}\) . We need to supply some structure to \(\beta_{\texttt{attr}|\texttt{intel}_i}\) to allow us to estimate its value without overcomplicating things.

6.1.3 Modeling slopes with linear models

One idea is to define \(\beta_{\texttt{attr}|\texttt{intel}_i}\) with a linear model: \[\beta_{\texttt{attr}|\texttt{intel}_i} = \beta_{\texttt{attr},0} + \beta_{\texttt{attr},1} \times \texttt{intel}_i\] This is just like a simple linear regression model, but now the “dependent variable” is the slope of \(\texttt{attr}\) . Defined in this way, the slope of \(\texttt{attr}\) is \(\beta_{\texttt{attr},0}\) when \(\texttt{intel}_i = 0\) , and for every one-unit increase in \(\texttt{intel}_i\) , the slope of \(\texttt{attr}\) increases (or decreases) by \(\beta_{\texttt{attr},1}\) . For example, let’s assume \(\beta_{\texttt{attr},0} = 1\) and \(\beta_{\texttt{attr},1} = 0.5\) . For someone with an intelligence rating of \(\texttt{intel}_i = 0\) , the slope of \(\texttt{attr}\) is \[\beta_{\texttt{attr}|\texttt{intel}_i} = 1 + 0.5 \times 0 = 1\] For someone with an intelligence rating of \(\texttt{intel}_i = 1\) , the slope of \(\texttt{attr}\) is \[\beta_{\texttt{attr}|\texttt{intel}_i} = 1 + 0.5 \times 1 = 1.5\] For someone with an intelligence rating of \(\texttt{intel}_i = 2\) , the slope of \(\texttt{attr}\) is \[\beta_{\texttt{attr}|\texttt{intel}_i} = 1 + 0.5 \times 2 = 2\] As you can see, for every increase in intelligence rating by 1 point, the slope of \(\texttt{attr}\) increases by 0.5. In such a model, there will be values of \(\texttt{intel}\) which result in a negative slope of \(\texttt{attr}\) . For instance, for \(\texttt{intel}_i = -4\) , the slope of \(\texttt{attr}\) is \[\beta_{\texttt{attr}|\texttt{intel}_i} = 1 + 0.5 \times (-4) = - 1\]

We can define the slope of \(\texttt{intel}\) in a similar manner as \[\beta_{\texttt{intel}|\texttt{attr}_i} = \beta_{\texttt{intel},0} + \beta_{\texttt{intel},1} \times \texttt{attr}_i\] When we plug these definitions into Equation (6.1) , we get \[\begin{aligned} \texttt{like}_i &= \beta_0 + (\beta_{\texttt{attr},0} + \beta_{\texttt{attr},1} \times \texttt{intel}_i) \times \texttt{attr}_i + (\beta_{\texttt{intel},0} + \beta_{\texttt{intel},1} \times \texttt{attr}_i) \times \texttt{intel}_i + \epsilon_i \\ &= \beta_0 + \beta_{\texttt{attr},0} \times \texttt{attr}_i + \beta_{\texttt{intel},0} \times \texttt{intel}_i + (\beta_{\texttt{attr},1} + \beta_{\texttt{intel},1}) \times (\texttt{attr}_i \times \texttt{intel}_i) + \epsilon_i \end{aligned}\]

Looking carefully at this formula, you can recognize a multiple regression model with three predictors: \(\texttt{attr}\) , \(\texttt{intel}\) , and a new predictor \(\texttt{attr}_i \times \texttt{intel}_i\) , which is computed as the product of these two variables. While it is thus related to both variables, we can treat this product as just another predictor in the model. The slope of this new predictor is the sum of two terms, \(\beta_{\texttt{attr},1} + \beta_{\texttt{intel},1}\) . Although we have defined these as different things (i.e. as the effect of \(\texttt{intel}\) on the slope of \(\texttt{attr}\) , and the effect of \(\texttt{attr}\) on the slope of \(\texttt{intel}\) , respectively), their value can not be estimated uniquely. We can only estimate their summed value. That means that moderation in regression is “symmetric”, in the sense that each predictor determines the slope of the other one. We can not say that it is just intelligence that determines the effect of attraction on liking, nor can we say that it is just attraction that determines the effect of intelligence on liking. The two variables interact and each determine the other’s effect on the dependent variable.

With that in mind, we can simplify the notation of the resulting model somewhat, by renaming the slopes of the two predictors to \(\beta_{\texttt{attr}} = \beta_{\texttt{attr},0}\) and \(\beta_{\texttt{intel}} = \beta_{\texttt{intel},0}\) , and using a single parameter for the sum \(\beta_{\texttt{attr} \times \texttt{intel}} = \beta_{\texttt{attr},1} + \beta_{\texttt{intel},1}\) :

\[\begin{equation} \texttt{like}_i = \beta_0 + \beta_{\texttt{attr}} \times \texttt{attr}_i + \beta_{\texttt{intel}} \times \texttt{intel}_i + \beta_{\texttt{attr} \times \texttt{intel}} \times (\texttt{attr} \times \texttt{intel})_i + \epsilon_i \end{equation}\]

Estimating this model gives \[\texttt{like}_i = -0.791 + 0.657 \times \texttt{attr}_i + 0.488 \times \texttt{intel}_i - 0.0171 \times \texttt{(attr}\times\texttt{intel)}_i + \hat{\epsilon}_i \] The estimate of the slope of the interaction, \(\hat{\beta}_{\texttt{attr} \times \texttt{intel}} = -0.017\) , is negative. That means that the higher the value of \(\texttt{intel}\) , the less steep the regression line relating \(\texttt{attr}\) to \(\texttt{like}\) . At the same time, the higher the value of \(\texttt{attr}\) , the less steep the regression line relating \(\texttt{intel}\) to \(\texttt{like}\) . You can interpret this as meaning that for more intelligent people, physical attractiveness is less of a defining factor in their liking by a potential partner. And for more attractive people, intelligence is less important.

A graphical view of this model, and the earlier one without moderation, is provided in Figure 6.2 . The plot on the left represents the model which does not allow for interaction. You can see that, for different values of intelligence, the model predicts parallel regression lines for the relation between attractiveness and liking. While intelligence affects the intercept of these regression lines, it does not affect the slope. In the plot on the right – although subtle – you can see that the regression lines are not parallel. This is a model with an interaction between intelligence and attractiveness. For different values of intelligence, the model predicts a linear relation between attractiveness and liking, but crucially, intelligence determines both the intercept and slope of these lines.

Figure 6.2: Liking as a function of attractiveness (intelligence) for different levels of intelligence (attractiveness), either without moderation or with moderation of the slope of attraciveness by intelligence. Note that the actual values of liking, attractiveness, and intelligence, are whole numbers (ratings on a scale between 1 and 10). For visualization purposes, the values have been randomly jittered by adding a Normal-distributed displacement term.

Note that we have constructed this model by simply including a new predictor in the model, which is computed by multiplying the values of \(\texttt{attr}\) and \(\texttt{intel}\) . While including such an “interaction predictor” has important implications for the resulting relations between \(\texttt{attr}\) and \(\texttt{like}\) for different values of \(\texttt{intel}\) , as well as the relations between \(\texttt{intel}\) and \(\texttt{like}\) for different values of \(\texttt{attr}\) , the model itself is just like any other regression model. Thus, parameter estimation and inference are exactly the same as before. Table 6.2 shows the results of comparing the full MODEL G (with three predictors) to different versions of MODEL R, where in each we fix one of the parameters to 0. As you can see, these comparisons indicate that we can reject the null hypothesis \(H_0\) : \(\beta_0 = 0\) , as well as \(H_0\) : \(\beta_{\texttt{attr}} = 0\) and \(H_0\) : \(\beta_{\texttt{intel}} = 0\) . However, as the p-value is above the conventional significance level of \(\alpha=.05\) , we would not reject the null hypothesis \(H_0\) : \(\beta_{\texttt{attr} \times \texttt{intel}} = 0\) . That implies that, in the context of this model, there is not sufficient evidence that there is an interaction. That may seem a little disappointing. We’ve done a lot of work to construct a model where we allow the effect of attractiveness to depend on intelligence, and vice versa. And now the hypothesis test indicates that there is no evidence that this moderation is present. As we will see later, there is evidence of this moderation when we also include \(\texttt{fun}\) in the model. I have left this predictor out of the model for now to keep things as simple as possible.

6.1.4 Simple slopes and centering

It is very important to realise that in a model with interactions, there is no single slope for any of the predictors involved in an interaction, that is particularly meaningful in principle. An interaction means that the slope of one predictor varies as a function of another predictor. Depending on which value of that other predictor you focus on, the slope of the predictor can be positive, negative, or zero. Let’s consider the model we estimated again: \[\texttt{like}_i = -0.791 + 0.657 \times \texttt{attr}_i + 0.488 \times \texttt{intel}_i - 0.0171 \times \texttt{(attr}\times\texttt{intel)}_i + \hat{\epsilon}_i \] If we fill in a particular value for intelligence, say \(\texttt{intel} = 1\) , we can write this as

\[\begin{aligned} \texttt{intel}_i &= -0.791 + 0.657 \times \texttt{attr}_i + 0.488 \times 1 -0.017 \times (\texttt{attr} \times 1)_i + \epsilon_i \\ &= (-0.791 + 0.488) + (0.657 -0.017) \times \texttt{attr}_i + \epsilon_i \\ &= -0.303 + 0.64 \times \texttt{attr}_i + \epsilon_i \end{aligned}\]

If we pick a different value, say \(\texttt{intel} = 10\) , the the model becomes \[\begin{aligned} \texttt{intel}_i &= -0.791 + 0.657 \times \texttt{attr}_i + 0.488 \times 10 -0.017 \times (\texttt{attr} \times 10)_i + \epsilon_i \\ &= (-0.791 + 0.488 \times 10) + (0.657 -0.017\times 10) \times \texttt{attr}_i + \epsilon_i \\ &= 4.09 + 0.486 \times \texttt{attr}_i + \epsilon_i \end{aligned}\] This shows that the higher the value of intelligence, the lower the slope of \(\texttt{attr}\) becomes. If you’d pick \(\texttt{intel} = 38.337\) , the slope would be exactly equal to 0. 17 Because there is not just a single value of the slope, testing whether “the” slope of \(\texttt{attr}\) is equal to 0 doesn’t really make sense, because there is no single value to represent “the” slope. What, then, does \(\hat{\beta}_\texttt{attr} = 0.657\) represent? Well, it is the (estimated) slope of \(\texttt{attr}\) when \(\texttt{intel}_i = 0\) . Similarly, \(\hat{\beta}_\texttt{intel} = 0.488\) is the estimated slope of \(\texttt{intel}\) when \(\texttt{attr}_i = 0\)

A significance test of the null hypothesis \(H_0\) : \(\beta_\texttt{attr} = 0\) is thus a test whether, when \(\texttt{intel} = 0\) , the slope of \(\texttt{attr}\) is 0. This test is easy enough to perform, but is it interesting to know whether liking is related to attractiveness for people who’s intelligence was rated as 0? Perhaps not. For one thing, the ratings were on a scale from 1 to 10, so no one could actually receive a rating of 0. Because the slope depends on \(\texttt{intel}\) and we know that for some value of \(\texttt{intel}\) , the slope of \(\texttt{attr}\) will equal 0, the hypothesis test will not be significant for some values of \(\texttt{intel}\) , and will be significant for others. At which value of \(\texttt{intel}\) we might want to perform such a test is up to us, but the result seems somewhat arbitrary.

That said, we might be interested in assessing whether there is an effect of \(\texttt{attr}\) for particular values of \(\texttt{intel}\) . For instance, whether, for someone with an average intelligence rating, their physical attractiveness matters for how much someone likes them as a potential partner. We can obtain this test by centering the predictors. Centering is basically just subtracting the sample mean of each value of a variable. So for example, we can center \(\texttt{attr}\) as follows: \[\texttt{attr_cent}_i = \texttt{attr}_i - \overline{\texttt{attr}}\] Centering does not affect the relation between variables. You can view it as a simple relabelling of the values, where the value which was the sample mean is now \(\texttt{attr_cent}_i = \overline{\texttt{attr}} - \overline{\texttt{attr}} = 0\) , all values below the mean are now negative, and values above the mean are now positive. The important part of this is that the centered predictor is 0 where the original predictor was at the sample mean. In a model with centered predictors \[\begin{align} \texttt{like}_i =& \beta_0 + \beta_{\texttt{attr_cent}} \times \texttt{attr_cent}_i + \beta_{\texttt{intel_cent}} \times \texttt{intel_cent}_i \\ &+ \beta_{\texttt{attr_cent} \times \texttt{intel_cent}} \times (\texttt{attr_cent} \times \texttt{intel_cent})_i + \epsilon_i \end{align}\] the slope \(\beta_{\texttt{attr_cent}}\) is, as usual, the slope of \(\texttt{attr_cent}\) whenever \(\texttt{intel_cent}_i = 0\) . We know that \(\texttt{intel_cent}_i = 0\) when \(\texttt{intel}_i = \overline{\texttt{intel}}\) . Hence, \(\beta_{\texttt{attr_cent}}\) is the slope of \(\texttt{attr}\) when \(\texttt{intel} = \overline{\texttt{intel}}\) , i.e. it represents the effect of \(\texttt{attr}\) for those with an average intelligence ratings.

Figure 6.3 shows the resulting model after centering both attractiveness and intelligence. When you compare this to the corresponding plot in Figure 6.2 , you can see that the only real difference is in the labels for the x-axis and the scale for intelligence. In all other respects, the uncentered and centered models predict the same relations between attractiveness and liking, and the models provide an equally good account, providing the same prediction errors.

Figure 6.3: Liking as a function of centered attractiveness for different levels of (centered) intelligence in a model including an interaction between attractiveness and intelligence. Note that the actual values of liking, attractiveness, and intelligence, are whole numbers (ratings on a scale between 1 and 10). For visualization purposes, the values have been randomly jittered by adding a Normal-distributed displacement term.

The results of all model comparisons after centering are given in Table 6.3 . A first important thing to notice is that centering does not affect the estimate and test of the interaction term . The slope of the interaction predictor reflects the increase in the slope relating \(\texttt{attr}\) to \(\texttt{like}\) for every one-unit increase in \(\texttt{intel}\) . Such changes to the steepness of the relation between \(\texttt{attr}\) and \(\texttt{like}\) should not – and are not – affected by changing the 0-point of the predictors through centering. A second thing to notice is that centering changes the estimates and test of the “simple slopes” and intercept . In the centered model, the simple slope \(\hat{\beta}_\texttt{attr_cent}\) reflects the effect of \(\texttt{attr}\) on \(\texttt{like}\) for cases with an average rating on \(\texttt{intel}\) . In Figure 6.3 , this is (approximately) the regression line in the middle. In the uncentered model, the simple slope \(\hat{\beta}_\texttt{attr}\) reflects the effect of \(\texttt{attr}\) on \(\texttt{like}\) for cases with \(\texttt{intel} = 0\) . In the top right plot in Figure 6.2 , this is (approximately) the lower regression line. This latter regression line is quite far removed from most of the data, because there are no cases with an intelligence rating of 0. The regression line for people with an average intelligence rating lies much more “within the cloud of data points”, and reflects the model predictions for many more cases in the data. As a result, the reduction in the SSE that can be attributed to the simple slope is much higher in the centered model (Table 6.3 ) than the uncentered one (Table 6.2 ). This results in a much higher \(F\) statistic. You can also think of this as follows: because there are hardly any cases with an intelligence rating close to 0, estimating the effect of attractiveness on liking for these cases is rather difficult and unreliable. Estimating the effect of attractiveness on liking for cases with an average intelligence rating is much more reliable, because there are many more cases with a close-to-average intelligence rating.

6.1.5 Don’t forget about fun! A model with multiple interactions

Up to now, we have looked at a model with two predictors, attractiveness and intelligence, and have allowed for an interaction between these. To simplify the discussion a little, we have not included \(\texttt{fun}\) in the model. It is relatively straightforward to extend this idea to multiple predictors. For instance, it might also be the case that the effect of \(\texttt{fun}\) is moderated by \(\texttt{intel}\) . To investigate this, we can estimate the following regression model:

\[\begin{aligned} \texttt{like}_i =& \beta_0 + \beta_{\texttt{attr}} \times \texttt{attr}_i + \beta_{\texttt{intel}} \times \texttt{intel}_i + \beta_{\texttt{fun}} \times \texttt{fun}_i \\ &+ \beta_{\texttt{attr} \times \texttt{intel}} \times (\texttt{attr} \times \texttt{intel})_i + \beta_{\texttt{fun} \times \texttt{intel}} \times (\texttt{fun} \times \texttt{intel})_i + \epsilon_i \end{aligned}\]

The results, having centered all predictors, are given in Table 6.4 . As you can see there, the simple slopes of \(\texttt{attr}\) , \(\texttt{intel}\) , and \(\texttt{fun}\) are all positive. Each of these represents the effect of that predictor when the other predictors have the value 0. Because the predictors are centered, that means that e.g. the slope of \(\texttt{attr}\) reflects the effect of attractiveness for people with an average rating on intelligence and fun. As before, the estimated interaction between \(\texttt{attr}\) and \(\texttt{intel}\) is negative, indicating that attractiveness has less of an effect on liking for those seen as more intelligent, and that intelligence has less of an effect for those seen as more attractive. The hypothesis test of this effect is now also significant, indicating that we have reliable evidence for this moderation. This shows that by including more predictors in a model, it is possible to increase the reliability of the estimates for other predictors. There is also a significant interaction between \(\texttt{fun}\) and \(\texttt{intel}.\) The estimated interaction is positive here. This indicates that fun has more of an effect on liking for those seen as more intelligent, and that intelligence has more of an effect for those seen as more fun. Perhaps you can think of a reason why intelligence appears to lessen the effect of attractiveness, but appears to strengthen the effect of fun…

6.2 Mediation

6.2.1 legacy motives and pro-environmental behaviours.

Zaval, Markowitz, & Weber ( 2015 ) investigated whether there is a relation between individuals’ motivation to leave a positive legacy in the world, and their pro-environmental behaviours and intentions. The authors reasoned that long time horizons and social distance are key psychological barriers to pro-environmental action, particularly regarding climate change. But if people with a legacy motivation put more emphasis on future others than those without such motivation, they may also be motivated to behave more pro-environmentally in order to benefit those future others. In a pilot study, they recruited a diverse sample of 245 U.S. participants through Amazon’s Mechanical Turk. Participants answered three sets of questions: one assessing individual differences in legacy motives, one assessing their beliefs about climate change, and one assessing their willingness to take pro-environmental action. Following these sets of questions, participants were told they would be entered into a lottery to win a $10 bonus. They were then given the option to donate part (between $0 and $10) of their bonus to an environmental cause (Trees for the Future). This last measure was meant to test whether people actually act on any intention to act pro-environmentally.

For ease of analysis, the three sets of questions measuring legacy motive, belief about the reality of climate change, and intention to take pro-environmental action, were transformed into three overall scores by computing the average over the items in each set. After eliminating participants who did not answer all questions, we have data from \(n = 237\) participants. Figure 6.4 depicts the pairwise relations between the four variables. As can be seen, all variables are significantly correlated. The relation is most obvious for \(\texttt{belief}\) and \(\texttt{intention}\) . Looking at the histogram of \(\texttt{donation}\) , you can see that although all whole amounts between $0 and $10 have been chosen at least once, it looks like three values were particularly popular, namely $0, $5, and to a lesser extent $10. This results in what looks like a tri-modal distribution. This is not necessarily an issue when modelling \(\texttt{donation}\) with a regression model, as the assumptions in a regression model concern the prediction errors , and not the dependent variable itself.

Figure 6.4: Pairwise plots for legacy motives, climate change belief, intention for pro-environmental action, and donations.

According to the Theory of Planned Behavior ( Ajzen, 1991 ) , attitudes and norms shape a person’s behavioural intentions, which in turn result in behaviour itself. In the context of the present example, that could mean that legacy motive and climate change beliefs do not directly determine whether someone behaves in a pro-environmental way. Rather, these factors shape a person’s intentions towards pro-environmental behaviour, which in turn may actually lead to said pro-environmental behaviour. This is an example of an assumed causal chain , where legacy motive (partly) determines behavioural intention, and intention determines behaviour. Mediation analysis is aimed at detecting an indirect effect of a predictor (e.g.  \(\texttt{legacy}\) ) on the dependent variable (e.g.  \(\texttt{donation}\) ), via another variable called the mediator (e.g.  \(\texttt{intention}\) ), which is the middle variable in the causal chain.

6.2.2 Causal steps

A traditional method to assess mediation is the so-called causal steps approach ( Baron & Kenny, 1986 ) . The basic idea behind the causal steps approach is as follows: if there is a causal chain from predictor ( \(X\) ) to mediator ( \(M\) ) to dependent variable ( \(Y\) ), then, ignoring the mediator for the moment, we should be able to see a relation between the predictor and dependent variable. This relation reflects the indirect effect of the predictor on the dependent variable. We should also be able to detect an effect of the predictor on the mediator, as well as an effect of the mediator on the dependent variable. Crucially, if there is a true causal chain, then the predictor should not offer any additional predictive power over the mediator. Because the effect of the predictor is assumed to go only “through” the mediator, once we know the value of the mediator, this should be all we need to predict the dependent variable. In more fancy statistical terms, this means that conditional on the mediator, the dependent variable is independent of the predictor, i.e.  \(p(Y \mid M, X) = p(Y \mid M)\) . In the context of a multiple regression model, we could say that in a model where we predict \(Y\) from \(M\) , the predictor \(X\) would not have a unique effect on \(Y\) (i.e. its slope would equal \(\beta_X = 0\) ).

The causal steps (Figure 6.5 ) approach involves assessing a pattern of significant relations in three different regression models. The first model is a simple regression model where we predict \(Y\) from \(X\) . In this model, we should find evidence for a relation between \(X\) and \(Y\) , meaning that we can reject the null hypothesis that the slope of \(X\) on \(Y\) (referred to here as \(\beta_X = c\) ) equals 0. The second model is a simple regression model where we predict \(M\) from \(X\) . In this model, we should find evidence for a relation between \(X\) and \(M\) , meaning that we can reject the null hypothesis that the slope of \(X\) on \(M\) (referred to here as \(\beta_X = a\) here) equals 0. The third model is a multiple regression model where we predict \(Y\) from both \(M\) and \(X\) . In this model, we should find evidence for a unique relation between \(M\) and \(Y\) , meaning that we can reject the null hypothesis that the slope of \(M\) on \(Y\) (referred to here as \(\beta_M = b\) here) equals 0. Controlling for the effect of \(M\) on \(Y\) , in a true causal chain, there should no longer be evidence for a relation between \(X\) and \(Y\) (as any relation between \(X\) and \(Y\) is captured through \(M\) ). Hence, we should not be able to reject the null hypothesis that the slope of \(X\) on \(Y\) in this model (referred to here as \(\beta_X = c\) ’, to distinguish it from the relation between \(X\) and \(Y\) in the first model, which was labelled as \(c\) ) equals 0. If this is so, then we speak of full mediation . When there is still evidence of a unique relation between \(X\) and \(Y\) in the model that includes \(M\) , but the relation is reduced (i.e.  \(|c'| < |c|\) ), we speak of partial mediation .

Figure 6.5: Assessing mediation with the causal steps approach involves testing parameters of three models. MODEL 1 is a simple regression model predicting \(Y\) from \(X\) and the slope of \(X\) ( \(c\) ) should be significant MODEL 2 is a simple regression model predicting \(M\) from \(X\) and the slope of \(X\) ( \(a\) ) should be significant. MODEL 3 is a multiple regression model predicting \(Y\) from both \(X\) and \(M\) . The slope of \(M\) ( \(b\) ) should be significant. The slope of \(X\) ( \(c\) ’) should not be significant (“full” mediation) or be substantially smaller in absolute value (“partial” mediation).

6.2.2.1 Testing mediation of legacy motive by intention with the causal steps approach

Let’s see how the causal steps approach works in practice by assessing whether the relation between \(\texttt{legacy}\) on \(\texttt{donation}\) is mediated by \(\texttt{intention}\) .

In MODEL 1 (Table 6.5 ), we assess the relation between \(\texttt{legacy}\) and \(\texttt{donation}\) . In this model, we find a significant and positive relation between legacy motives and donations, such that people with stronger legacy motives donate more of their potential bonus to a pro-environmental cause. The question is now whether this is a direct effect of legacy motive, or an indirect effect “via” behavioural intent.

In MODEL 2 (Table 6.6 ), we assess the relation between \(\texttt{legacy}\) and \(\texttt{intention}\) . In this model, we find a significant and positive relation between legacy motives and intention to act pro-environmentally, such that people with stronger legacy motives have a stronger intention to act pro-environmentally.

In MODEL 3 (Table 6.7 ), we assess the relation between \(\texttt{legacy}\) , \(\texttt{intention}\) , and \(\texttt{donation}\) . In this model, we find a significant and positive relation between intention to act pro-environmentally and donation to a pro-environmental cause, such that people with stronger intentions donate more. We also find evidence of a unique and positive effect of legacy motive on donation, such that people with stronger legacy motives donate more. Because there is still evidence of an effect of legacy motive on donations, after controlling for the effect of behavioural intent, we would not conclude that the effect of legacy motive is fully mediated by intent. When you compare the slope of \(\texttt{legacy}\) in MODEL 3 to that in MODEL 1, you can however see that the (absolute) value is smaller. Hence, when controlling for the effect of behavioural intent, a one-unit increase in \(\texttt{legacy}\) is estimated to increase the amount of donation less then in a model where \(\texttt{intention}\) is not taken into account.

In conclusion, the causal steps approach indicates that the effect of legacy motive of pro-environmental action (donations) is partially mediated by pro-environmental behavioural intentions. There is a residual direct effect of legacy motive on donations that is not captured by behavioural intentions.

6.2.3 Estimating the mediated effect

One potential problem with the causal steps approach is that it is based on a pattern of significance in four hypothesis tests (one for each parameter \(a\) , \(b\) , \(c\) , and \(c'\) ). This can result in a rather low power of the procedure ( MacKinnon, Fairchild, & Fritz, 2007 ) , which seems to be particularly related to the requirement of a significant \(c\) (the direct effect of \(X\) on \(Y\) in the model without the mediator).

An alternative to the causal steps approach is to estimate the mediated (indirect) effect of the predictor on the dependent variable directly. Algebraically, this mediated effect can be worked out as ( MacKinnon et al., 2007 ) :

\[\begin{equation} \text{mediated effect} = a \times b \end{equation}\]

The rationale behind this is reasonably straightforward. The slope \(a\) reflects the increase in the mediator \(M\) for every one-unit increase in the predictor \(X\) . The slope \(b\) reflects the increase in the dependent variable \(Y\) for every one unit increase in the mediator. So a one-unit increase in \(X\) implies an increase in \(M\) by \(a\) units, which in turn implies an increase in \(Y\) of \(a \times b\) units. Hence, the mediated effect can be expressed as \(a \times b\) .

In a single mediator model such as the one looked at here, the mediated effect \(a \times b\) turns out to be equal to \(c - c'\) , i.e. the difference between the direct effect of \(X\) on \(Y\) in a model without the mediator, and the unique direct effect of \(X\) on \(Y\) in a model which includes the mediator.

To test whether the mediated effect differs from 0, we can try to work out the sampling distribution of the estimated effect \(\hat{a} \times \hat{b}\) , under the null-hypothesis that in reality, \(a \times b = 0\) . Note that this null hypothesis can be true when \(a = 0\) , \(b = 0\) , or both \(a = b = 0\) . In the so-called Sobel-Aroian test, this sampling distribution is assumed to be Normal. However, it has been found that this assumption is often inaccurate. As there is no method to derive an accurate sampling distribution analytically, modern procedures rely on simulation. There are different ways to do this, but we’ll focus on one, namely the nonparametric bootstrap approach ( Preacher & Hayes, 2008 ) . This involves generating a large number (e.g.  \(>1000\) ) of simulated datasets by randomly sampling \(n\) cases with replacement from the original dataset. This means that any given case (i.e. a row in the dataset) can occur 0, 1, 2, times in a simulated dataset. For each simulated dataset, we can estimate \(\hat{a} \times \hat{b}\) by fitting the two corresponding regression models. The variance in these estimates over the different datasets forms an estimate of the variance of the sampling distribution. A 95% confidence interval can then also be computed through by determining the 2.5 and 97.5 percentiles. Because just the original data is used, there is no direct assumption made about the distribution of the variables, apart from that the original data is a representative sample from the Data Generating Process. Applying this procedure (with 1000 simulated datasets) provides a 95% confidence interval for \(a \times b\) of \([0.104, 0.446]\) . As this interval does not contain the value 0, we reject the null hypothesis that the mediated effect of \(\texttt{legacy}\) on \(\texttt{donation}\) “via” \(\texttt{intention}\) equals 0.

Note that in solely focusing on the mediated effect, we do not address the issue of total vs partial mediation. Using our simulated datasets, we can however also compute a bootstrap confidence interval for \(c'\) . For the present set of simulations, the 95% confidence interval for \(c'\) is \([0.189, 0.807]\) . As this interval does not contain the value 0, we reject the null hypothesis that the unique direct effect of \(\texttt{legacy}\) on \(\texttt{donation}\) equals 0. This thus provides a similar conclusion to the causal steps approach.

Here, we analyse only a subset of their data. ↩︎

Note that I’m using more descriptive labels here. If you prefer the more abstract version, then you can replace \(Y_i = \texttt{like}_i\) , \(\beta_1 = \beta_{\texttt{attr}}\) , \(X_{1,i} = \texttt{attr}_i\) . \(\beta_2 = \beta_{\texttt{intel}}\) , \(X_{2,i} = \texttt{intel}_i\) . ↩︎

The value for which the slope is 0 is easily worked out as \(\frac{\hat{\beta}_\texttt{attr}}{- \hat{\beta}_{\texttt{attr} \times \texttt{intel}}}\) . ↩︎

• Chapter 1: Introduction
• Chapter 2: Indexing
• Chapter 3: Loops & Logicals
• Chapter 4: Apply Family
• Chapter 5: Plyr Package
• Chapter 6: Vectorizing
• Chapter 7: Sample & Replicate
• Chapter 8: Melting & Casting
• Chapter 9: Tidyr Package
• Chapter 10: GGPlot1: Basics
• Chapter 11: GGPlot2: Bars & Boxes
• Chapter 12: Linear & Multiple
• Chapter 13: Ploting Interactions
• Chapter 14: Moderation/Mediation
• Chapter 15: Moderated-Mediation
• Chapter 16: MultiLevel Models
• Chapter 17: Mixed Models
• Chapter 18: Mixed Assumptions Testing
• Chapter 19: Logistic & Poisson
• Chapter 20: Between-Subjects
• Chapter 21: Within- & Mixed-Subjects
• Chapter 22: Correlations
• Chapter 23: ARIMA
• Chapter 24: Decision Trees
• Chapter 25: Signal Detection
• Chapter 26: Intro to Shiny
• Chapter 27: ANOVA Variance
• Download Rmd

Chapter 14: Mediation and Moderation

Alyssa blair, 1 what are mediation and moderation.

Mediation analysis tests a hypothetical causal chain where one variable X affects a second variable M and, in turn, that variable affects a third variable Y. Mediators describe the how or why of a (typically well-established) relationship between two other variables and are sometimes called intermediary variables since they often describe the process through which an effect occurs. This is also sometimes called an indirect effect. For instance, people with higher incomes tend to live longer but this effect is explained by the mediating influence of having access to better health care.

In R, this kind of analysis may be conducted in two ways: Baron & Kenny’s (1986) 4-step indirect effect method and the more recent mediation package (Tingley, Yamamoto, Hirose, Keele, & Imai, 2014). The Baron & Kelly method is among the original methods for testing for mediation but tends to have low statistical power. It is covered in this chapter because it provides a very clear approach to establishing relationships between variables and is still occassionally requested by reviewers. However, the mediation package method is highly recommended as a more flexible and statistically powerful approach.

Moderation analysis also allows you to test for the influence of a third variable, Z, on the relationship between variables X and Y. Rather than testing a causal link between these other variables, moderation tests for when or under what conditions an effect occurs. Moderators can stength, weaken, or reverse the nature of a relationship. For example, academic self-efficacy (confidence in own’s ability to do well in school) moderates the relationship between task importance and the amount of test anxiety a student feels (Nie, Lau, & Liau, 2011). Specifically, students with high self-efficacy experience less anxiety on important tests than students with low self-efficacy while all students feel relatively low anxiety for less important tests. Self-efficacy is considered a moderator in this case because it interacts with task importance, creating a different effect on test anxiety at different levels of task importance.

In general (and thus in R), moderation can be tested by interacting variables of interest (moderator with IV) and plotting the simple slopes of the interaction, if present. A variety of packages also include functions for testing moderation but as the underlying statistical approaches are the same, only the “by hand” approach is covered in detail in here.

Finally, this chapter will cover these basic mediation and moderation techniques only. For more complicated techniques, such as multiple mediation, moderated mediation, or mediated moderation please see the mediation package’s full documentation.

1.1 Getting Started

If necessary, review the Chapter on regression. Regression test assumptions may be tested with gvlma . You may load all the libraries below or load them as you go along. Review the help section of any packages you may be unfamiliar with ?(packagename).

2 Mediation Analyses

Mediation tests whether the effects of X (the independent variable) on Y (the dependent variable) operate through a third variable, M (the mediator). In this way, mediators explain the causal relationship between two variables or “how” the relationship works, making it a very popular method in psychological research.

Both mediation and moderation assume that there is little to no measurement error in the mediator/moderator variable and that the DV did not CAUSE the mediator/moderator. If mediator error is likely to be high, researchers should collect multiple indicators of the construct and use SEM to estimate latent variables. The safest ways to make sure your mediator is not caused by your DV are to experimentally manipulate the variable or collect the measurement of your mediator before you introduce your IV.

Total Effect Model.

Basic Mediation Model.

c = the total effect of X on Y c = c’ + ab c’= the direct effect of X on Y after controlling for M; c’=c-ab ab= indirect effect of X on Y

The above shows the standard mediation model. Perfect mediation occurs when the effect of X on Y decreases to 0 with M in the model. Partial mediation occurs when the effect of X on Y decreases by a nontrivial amount (the actual amount is up for debate) with M in the model.

2.1 Example Mediation Data

Set an appropriate working directory and generate the following data set.

In this example we’ll say we are interested in whether the number of hours since dawn (X) affect the subjective ratings of wakefulness (Y) 100 graduate students through the consumption of coffee (M).

Note that we are intentionally creating a mediation effect here (because statistics is always more fun if we have something to find) and we do so below by creating M so that it is related to X and Y so that it is related to M. This creates the causal chain for our analysis to parse.

2.2 Method 1: Baron & Kenny

This is the original 4-step method used to describe a mediation effect. Steps 1 and 2 use basic linear regression while steps 3 and 4 use multiple regression. For help with regression, see Chapter 10.

The Steps: 1. Estimate the relationship between X on Y (hours since dawn on degree of wakefulness) -Path “c” must be significantly different from 0; must have a total effect between the IV & DV

Estimate the relationship between X on M (hours since dawn on coffee consumption) -Path “a” must be significantly different from 0; IV and mediator must be related.

Estimate the relationship between M on Y controlling for X (coffee consumption on wakefulness, controlling for hours since dawn) -Path “b” must be significantly different from 0; mediator and DV must be related. -The effect of X on Y decreases with the inclusion of M in the model

Estimate the relationship between Y on X controlling for M (wakefulness on hours since dawn, controlling for coffee consumption) -Should be non-significant and nearly 0.

2.3 Interpreting Barron & Kenny Results

Here we find that our total effect model shows a significant positive relationship between hours since dawn (X) and wakefulness (Y). Our Path A model shows that hours since down (X) is also positively related to coffee consumption (M). Our Path B model then shows that coffee consumption (M) positively predicts wakefulness (Y) when controlling for hours since dawn (X). Finally, wakefulness (Y) does not predict hours since dawn (X) when controlling for coffee consumption (M).

Since the relationship between hours since dawn and wakefulness is no longer significant when controlling for coffee consumption, this suggests that coffee consumption does in fact mediate this relationship. However, this method alone does not allow for a formal test of the indirect effect so we don’t know if the change in this relationship is truly meaningful.

There are two primary methods for formally testing the significance of the indirect test: the Sobel test & bootstrapping (covered under the mediatation method).

The Sobel Test uses a specialized t-test to determine if there is a significant reduction in the effect of X on Y when M is present. Using the sobel function of the multilevel package will show provide you with three of the basic models we ran before (Mod1 = Total Effect; Mod2 = Path B; and Mod3 = Path A) as well as an estimate of the indirect effect, the standard error of that effect, and the z-value for that effect. You can either use this value to calculate your p-value or run the mediation.test function from the bda package to receive a p-value for this estimate.

In this case, we can now confirm that the relationship between hours since dawn and feelings of wakefulness are significantly mediated by the consumption of coffee (z’ = 3.84, p < .001).

However, the Sobel Test is largely considered an outdated method since it assumes that the indirect effect (ab) is normally distributed and tends to only have adequate power with large sample sizes. Thus, again, it is highly recommended to use the mediation bootstrapping method instead.

2.4 Method 2: The Mediation Pacakge Method

This package uses the more recent bootstrapping method of Preacher & Hayes (2004) to address the power limitations of the Sobel Test. This method computes the point estimate of the indirect effect (ab) over a large number of random sample (typically 1000) so it does not assume that the data are normally distributed and is especially more suitable for small sample sizes than the Barron & Kenny method.

To run the mediate function, we will again need a model of our IV (hours since dawn), predicting our mediator (coffee consumption) like our Path A model above. We will also need a model of the direct effect of our IV (hours since dawn) on our DV (wakefulness), when controlling for our mediator (coffee consumption). When can then use mediate to repeatedly simulate a comparsion between these models and to test the signifcance of the indirect effect of coffee consumption.

2.5 Interpreting Mediation Results

The mediate function gives us our Average Causal Mediation Effects (ACME), our Average Direct Effects (ADE), our combined indirect and direct effects (Total Effect), and the ratio of these estimates (Prop. Mediated). The ACME here is the indirect effect of M (total effect - direct effect) and thus this value tells us if our mediation effect is significant.

In this case, our fitMed model again shows a signifcant affect of coffee consumption on the relationship between hours since dawn and feelings of wakefulness, (ACME = .28, p < .001) with no direct effect of hours since dawn (ADE = -0.11, p = .27) and significant total effect ( p < .05).

We can then bootstrap this comparison to verify this result in fitMedBoot and again find a significant mediation effect (ACME = .28, p < .001) and no direct effect of hours since dawn (ADE = -0.11, p = .27). However, with increased power, this analysis no longer shows a significant total effect ( p = .08).

3 Moderation Analyses

Moderation tests whether a variable (Z) affects the direction and/or strength of the relation between an IV (X) and a DV (Y). In other words, moderation tests for interactions that affect WHEN relationships between variables occur. Moderators are conceptually different from mediators (when versus how/why) but some variables may be a moderator or a mediator depending on your question. See the mediation package documentation for ways of testing more complicated mediated moderation/moderated mediation relationships.

Like mediation, moderation assumes that there is little to no measurement error in the moderator variable and that the DV did not CAUSE the moderator. If moderator error is likely to be high, researchers should collect multiple indicators of the construct and use SEM to estimate latent variables. The safest ways to make sure your moderator is not caused by your DV are to experimentally manipulate the variable or collect the measurement of your moderator before you introduce your IV.

Basic Moderation Model.

3.1 Example Moderation Data

In this example we’ll say we are interested in whether the relationship between the number of hours of sleep (X) a graduate student receives and the attention that they pay to this tutorial (Y) is influenced by their consumption of coffee (Z). Here we create the moderation effect by making our DV (Y) the product of levels of the IV (X) and our moderator (Z).

3.2 Moderation Analysis

Moderation can be tested by looking for significant interactions between the moderating variable (Z) and the IV (X). Notably, it is important to mean center both your moderator and your IV to reduce multicolinearity and make interpretation easier. Centering can be done using the scale function, which subtracts the mean of a variable from each value in that variable. For more information on the use of centering, see ?scale and any number of statistical textbooks that cover regression (we recommend Cohen, 2008).

A number of packages in R can also be used to conduct and plot moderation analyses, including the moderate.lm function of the QuantPsyc package and the pequod package. However, it is simple to do this “by hand” using traditional multiple regression, as shown here, and the underlying analysis (interacting the moderator and the IV) in these packages is identical to this approach. The rockchalk package used here is one of many graphing and plotting packages available in R and was chosen because it was especially designed for use with regression analyses (unlike the more general graphing options described in Chapters 8 & 9).

3.3 Interpreting Moderation Results

Results are presented similar to regular multiple regression results (see Chapter 10). Since we have significant interactions in this model, there is no need to interpret the separate main effects of either our IV or our moderator.

Our by hand model shows a significant interaction between hours slept and coffee consumption on attention paid to this tutorial (b = .23, SE = .04, p < .001). However, we’ll need to unpack this interaction visually to get a better idea of what this means.

The rockchalk function will automatically plot the simple slopes (1 SD above and 1 SD below the mean) of the moderating effect. This figure shows that those who drank less coffee (the black line) paid more attention with the more sleep that they got last night but paid less attention overall that average (the red line). Those who drank more coffee (the green line) paid more when they slept more as well and paid more attention than average. The difference in the slopes for those who drank more or less coffee shows that coffee consumption moderates the relationship between hours of sleep and attention paid.

4 References and Further Reading

Baron, R., & Kenny, D. (1986). The moderator-mediator variable distinction in social psychological research: Conceptual, strategic, and statistical considerations. Journal of Personality and Social Psychology, 51, 1173-1182.

Cohen, B. H. (2008). Explaining psychological statistics. John Wiley & Sons.

Imai, K., Keele, L., & Tingley, D. (2010). A general approach to causal mediation analysis. Psychological methods, 15(4), 309.

MacKinnon, D. P., Lockwood, C. M., Hoffman, J. M., West, S. G., & Sheets, V. (2002). A comparison of methods to test mediation and other intervening variable effects. Psychological methods, 7(1), 83.

Nie, Y., Lau, S., & Liau, A. K. (2011). Role of academic self-efficacy in moderating the relation between task importance and test anxiety. Learning and Individual Differences, 21(6), 736-741.

Tingley, D., Yamamoto, T., Hirose, K., Keele, L., & Imai, K. (2014). Mediation: R package for causal mediation analysis.

Recoding Introduction to Mediation, Moderation, and Conditional Process Analysis

7 fundamentals of moderation analysis.

The effect of \(X\) on some variable \(Y\) is moderated by \(W\) if its size, sign, or strength depends on or can be predicted by \(W\) . In that case, \(W\) is said to be a moderator of \(X\) ’s effect on \(Y\) , or that \(W\) and \(X\) interact in their influence on \(Y\) . Identifying a moderator of an effect helps to establish the boundary conditions of that effect or the circumstances, stimuli, or type of people for which the effect is large versus small, present versus absent, positive versus negative, and so forth. (p. 220, emphasis in the original)

7.1 Conditional and unconditional effects

If \(X\) ’s effect on \(Y\) is moderated by another variable in the model, that means \(X\) ’s effect depends on that other variable. But this model constrains \(X\) ’s effect to be unconditional on \(W\) , meaning that it is invariant across all values of \(W\) . (p. 224)

Hayes’s Table 7.1 and the related Figure 7.2 showcase this well. You might reproduce the data for both like this.

In previous chapters, we alteried our plot themes using either built-in settings from ggplot2 or extra themes from the ggthemes package. When we wanted to change them further, we did so with extra theme() arguments. One, it’s nice to be know how to make one’s own custom theme and, two, it’d be handy to condense our code a bit. Here we’ll take a few cues from the Building a New Theme section from Peng, Kross, and Anderson’s Mastering Software Development in R . We’ll make our own theme, theme_07 , by saving a handful of augmentations from the default theme_gray() theme.

This chapter’s color palette comes from the dutchmasters package , which was itself based of Vermeer’s The Little Street . To get a quick glance at the full palette, we’ll also use viz_palette() , a convenience function from the ochRe package .

With our new theme_07 in hand, we’re ready to make our version of Figure 7.2.

We borrowed geom_dl() form the directlabels package , which made it easy to insert the “W = \(i\) ” labels to the right of the lines.

I played around with the annotation in Figure 7.4 for a while and it was just way more trouble than it was worth. If you’re ever inspired to work it out, please share your code . I’m moving on.

7.1.1 Eliminating the constraint of unconditionality.

We can write the generic moderation model as

\[Y = i_Y + f(W) X + b_2 W + e_Y,\]

where \(f(W)\) is a function of \(W\) . Consider a simple function of the form \(f(W) = b_1 + b_3 W\) . This function of \(W\) looks like a simple linear regression model where \(b_1\) is the constant and \(b_3\) is the regression coefficient for \(W\) , except that rather than estimating some consequent variable from \(W\) , it is a model of the effect of \(X\) on \(Y\) . (p. 226)

If we use that definition of \(f(W)\) , we can update our equation to

\[\begin{align*} Y & = i_Y + (b_1 + b_3 W) X + b_2 W + e_Y, \text{or} \\ Y & = i_Y + b_1 X + b_2 W + b_3 XW + e_Y. \end{align*}\]

Hayes called this the simple linear moderation model . Out of this equation we can extract \(\theta_{X \rightarrow Y}\) , the conditional effect of \(X\) on \(Y\) , which takes the form

\[\theta_{X \rightarrow Y} = b_1 + b_3 W.\]

This is what Figure 7.2 was all about.

7.1.2 Symmetry in moderation.

It turns out that, mathematically, there’s no difference in speaking about \(X\) moderating \(W\) than speaking about \(W\) moderating \(X\) . These are all just capital letters standing in for variables and perhaps you’ll recall from grade school that \(a + b + ab\) is just the same as \(b + a + ba\) . Thus we can rearrange the simple moderation equations above to

\[Y = i_Y + b_1 X + (b_2 + b_3 X) W + e_Y\]

and speak instead about the conditional effect of \(W\) on \(Y\) ,

\[\theta_{w \rightarrow Y} = b_2 + b_3 X.\]

7.1.3 Interpretation of the regression coefficients.

With all this wacky stuff going on, we should clarify some of our terms.

• \(i_Y\) is still the expected value for \(Y\) with both \(X = 0\) and \(W = 0\) .
• \(b_1\) is the conditional effect of \(X\) on \(Y\) when \(W = 0\) .
• \(b_2\) is the conditional effect of \(W\) on \(Y\) when \(X = 0\) .

It turns out \(b_3\) requires more words, which we’ll provide in the next subsection.

7.1.4 The importance of \(b_3\) wen asking about mediation.

The simple moderation model allows \(X\) ’s effect on \(Y\) to be a linear function of \(W\) . Of course, allowing that effect to depend on \(W\) doesn’t mean that it actually does in reality. In most any sample of data, [the point estimate (i.e., posterior mean or median) for] \(b_3\) will be different from zero even when \(X\) ’s effect on \(Y\) is independent of \(W\) . Of interest when [modeling] a moderation hypothesis is not just allowing \(X\) ’s effect to be contingent on \(W\) , but also determining whether \(b_3\) deviates too far from zero than would be expected given that \(b_3\) , like any statistic, is subject to sampling variance. (p. 231)

7.2 An example: Climate change disasters and humanitarianism

Here we load a couple necessary packages, load Chapman and Lickel (2016) data, and take a glimpse() .

Here is how to get the ungrouped mean and \(SD\) values for justify and skeptic , as presented in Table 7.3.

And here we get the same summary values, this time grouped by frame .

Let’s open brms .

Now fit the simple univariable model.

The ‘Estimate’ (i.e., posterior mean) of the model intercept is the expected justify value for when frame is 0. The ‘Estimate’ for frame is the expected difference when frame is a 1. If all you care about is the posterior mean, you could execute

which matches up nicely with the equation on page 233. But this wouldn’t be very Bayesian of us. It’d be more satisfying if we had an expression of the uncertainty in the value. For that, we’ll follow our usual practice of extracting the posterior samples, making nicely-named vectors, and summarizing a bit.

Hayes referenced a \(t\) -test and accompanying \(p\) -value in the lower part of page 233. We, of course, aren’t going to do that. But we do have the 95% intervals in our print() output, above, which we can also look at with the brms::posterior_interval() function.

And we can always plot.

We’ll use the update() function to hastily fit model7.2 and model7.3 .

Note our use of the frame:skeptic syntax in model7.3 . With that syntax we didn’t need to make an interaction variable in the data by hand. The brms package just handled it for us. An alternative syntax would have been frame*skeptic . But if you really wanted to make the interaction variable by hand, you’d do this.

Once you have interaction_variable in the data, you’d specify a model formula within the brm() function like formula = justify ~ 1 + frame + skeptic + interaction_variable . I’m not going to do that, here, but you can play around yourself if so inclined.

Here are the quick and dirty coefficient summaries for our two new models.

Just focusing on our primary model, model7.3 , here’s another way to look at the coefficients.

By default, the brms::stanplot() function returns coefficient plots which depict the parameters of a model by their posterior means (i.e., dots), 50% intervals (i.e., thick horizontal lines), and 95% intervals (i.e., thin horizontal lines). As stanplot() returns a ggplot2 object, one can customize the theme and so on.

We’ll extract the \(R^2\) iterations in the usual way once for each model, and then combine them for a plot.

Here’s the \(\Delta R^2\) distribution for model7.3 minus model7.2 .

In addition to the \(R^2\) , one can use information criteria to compare the models. Here we’ll use the LOO to compare all three.

The LOO point estimate for both multivariable models were clearly lower than that for model7.1 . The point estimate for the moderation model, model7.3 , was within the double-digit range lower than that for model7.2 , which typically suggests better fit. But notice how wide the standard error was. There’s a lot of uncertainty, there. Hopefully this isn’t surprising. Our \(R^2\) difference was small and uncertain, too. We can also compare them with AIC-type model weighting, which you can learn more about starting at this point in this lecture or this related vignette for the loo package . Here we’ll keep things simple and weight with the LOO.

The model_weights() results put almost all the relative weight on model7.3 . This doesn’t mean model7.3 is the “true model” or anything like that. It just suggests that it’s the better of the three with respect to the data.

Here are the results of the equations in the second half of page 237.

7.2.1 Estimation using PROCESS brms.

Similar to what Hayes advertised with PROCESS, with our formula = justify ~ 1 + frame + skeptic + frame:skeptic code in model7.3 , we didn’t need to hard code an interaction variable into the data. brms handled that for us.

7.2.2 Interpreting the regression coefficients.

When you add an interaction term into a model, such as \(x_1 \cdot x_2\) , this is sometimes called a higher order term . The terms that made up the higher order term– \(x_1\) and \(x_2\) , in this case–are correspondingly called the lower order terms . As Hayes pointed out, these lower order terms are conditional effects. Interpret them with care.

7.2.3 Variable scaling and the interpretation of \(b_1\) and \(b_3\) .

Making the mean-centered version of our \(W\) variable, skeptic , is a simple mutate() operation. We’ll just call it skeptic_c .

And here’s how we might fit the model.

Here are the summaries of our fixed effects.

To practice, frame and skeptic_c are lower order terms and, as such, they are conditional effects. The higher order term is frame:skeptic_c . Anyway, here are the \(R^2\) distributions for model7.3 and model7.4 . They’re the same within simulation variance.

If you’re bothered by the differences resulting from sampling variation, you might increase the number of HMC iterations from the 2000-per-chain default. Doing so might look something like this.

Before we fit model7.5 , we’ll recode frame to a -.5/.5 metric and name it frame_.5 .

Time to fit model5 .

Our posterior summaries match up nicely with the output in Hayes’s Table 7.4.

Here’s a summary of the Bayesian \(R^2\) .

7.3 Visualizing moderation

A regression model with the product to two antecedent variables in an abstract mathematical representation of one’s data that can be harder to interpret than a model without such a produce. As described earlier, the coefficients for \(X\) and \(W\) are conditional effects that may not have any substantive interpretation, and the coefficient for \(XW\) is interpreted as a difference between differences that can be hard to make sense of without more information. (pp. 223–224)

This is why we plot. To get quick plots for the interaction effect in brms , you might use the conditional_effects() function.

By default, conditional_effects() will show three levels of the variable on the right side of the interaction term. The formula in model7.3 was justify ~ frame + skeptic + frame:skeptic , with frame:skeptic as the interaction term and skeptic making up the right hand side of the term. The three levels of skeptic in the plot, above, are the mean \(\pm 1\) standard deviation. See the brms reference manual for details on the conditional_effects() function.

On page 244, Hayes discussed using the 16th, 50th, and 84th percentiles for the moderator variable. We can compute those with quantile() .

The first two columns in Hayes’s Table 7.5 contain the values he combined with the point estimates of his model to get the \(\hat Y\) column. The way we’ll push those values through model7.3 ’s posterior is with brms::fitted() . As a preparatory step, we’ll put the predictor values in a data object, nd .

Now we’ve go our nd , we’ll get our posterior estimates for \(Y\) with fitted() .

When using the default summary = TRUE settings in fitted() , the function returns posterior means, \(SD\) s and 95% intervals for \(Y\) based on each row in the nd data we specified in the newdata = nd argument. You don’t have to name your newdata nd or anything like that; it’s just my convention.

Here’ a quick plot of what those values imply.

That plot is okay, but we can do better.

In order to plot the model-implied effects across the full range of skeptic values presented in Figure 7.7, you need to change the range of those values in the nd data. Also, although the effect is subtle in the above example, 95% intervals often follow a bowtie shape. In order to insure the contours of that shape are smooth, it’s often helpful to specify 30 or so evenly-spaced values in the variable on the x-axis, skeptic in this case. We’ll employ the seq() function for that and specify length.out = 30 . In addition, we add a few other flourishes to make our plot more closely resemble the one in the text.

Here’s our Figure 7.7.

Do you see that subtle bowtie shape?

7.4 Probing an interaction

As with other modeling contexts, there the results from our moderation models, like depicted in the last plot, carry uncertainty with them.

To deal with the uncertainty, it is common to follow up a test of interaction with a set of additional inferential tests to establish where in the distribution of the moderator \(X\) has an effect on \(Y\) that is different from zero and where it does not. This exercise is commonly known as “probing” an interaction, like you might squeeze an avocado or a mango in the produce section of the grocery store to assess its ripeness. The goal is to ascertain where in the distribution of the moderator \(X\) is related to \(Y\) and where it is not in an attempt to better discern the substantive interpretation of the interaction. In this section [we will cover] two approaches to probing an interaction. (p. 249)

As in earlier sections and chapters, our approach will largely follow Hayes’s, but we will not be emphasizing statistical tests, \(p\) -values, and so on. You’ll see.

7.4.1 The pick-a-point approach.

The pick-a-point approach ( Rogosa, 1980 ; Bauer & Curran, 2005 ), sometimes called an analysis of simple slopes or a spotlight analysis , is perhaps the most popular approach to probing an interaction and is described in most discussions of multiple regression with interactions (e.g., Aiken & West, 1991 ; Cohen et al., 2003 ; Darlington & Hayes, 2017 ; Hayes, 2005 ; Jaccard & Turrisi, 2003 ; Spiller, Fitzsimons, Lynch, & McClelland, 2013 ). This procedure involves selecting a value or values of the moderator \(W\) , calculating the conditional effect of \(X\) on \(Y (\theta_{X \rightarrow Y})\) at that value or values, and then conducting an inferential test or generating a confidence interval. (p. 249, emphasis in the original)

Though we will not be using inferential tests, we will use various ways of expressing the uncertainty in our Bayesian models, such as with 95% credible intervals and other ways of summarizing the posterior.

7.4.1.1 The pick-a-point approach implimented by regression centering working directly with the posterior.

Yes, if you wanted to use the regression centering approach, you could do that in brms . Just center the necessary variables in the way Hayes described in the text, refit the model, and summarize() . I suspect this would be particularly approachable for someone new to R and to the ins and outs of data wrangling. But I’m going leave that as an exercise for the interested reader.

Now that we’ve already got a posterior for our model, we can just either algebraically manipulate the vectors yielded by posterior_samples() or push predictions through fitted() . To give a sense, we’ll start off with the 16 th percentile for skeptic . Recall we can get that with the quantile() function.

Now we just need to feed that value and different values of frame into the posterior samples of the model coefficients. We then create a difference score for the model-implied estimates given frame is either 0 or 1 and then plot that difference .

Note how nicely that distribution corresponds to the output in the lower left corner of Hayes’s Figure 7.8. If we wanted the values for other values of skeptic (e.g., 2.8 and 5.2 as in the text), we’d just rinse, wash, and repeat. A nice quality of this method is it requires you to work explicitly with the model formula. But it’s also clunky if you want to do this over many values. The fitted() function offers an alternative approach.

Recall how the default fitted() settings are to return summaries of a model’s \(Y\) -variable given values of the predictor variables. In the previous section we put our preferred frame and skeptic values into a data object named nd and used the newdata argument to push those values through fitted() . Buy default, this yielded the typical posterior means, \(SD\) s, and 95% intervals for the predictions. However, if one sets summary = F , the output will differ. First. Let’s revisit what nd looks like.

Here’s what happens when we use summary = F .

With summary = F , fitted() returned a matrix of 4000 rows (i.e., one for each posterior draw) and 6 vectors (i.e., one for each row in our nd data). So now instead of summary information, we have a full expression of the uncertainty in terms of 4000 draws. If you prefer working within the tidyverse and plotting with ggplot2 , matrices aren’t the most useful data type. Let’s wrangle a bit.

Now we have our draws in a nice structure, we’re ready to plot.

And if you prefered summary information instead of plots, you might just use tidybayes::median_qi() .

7.4.2 The Johnson-Neyman technique.

The JN technique generalizes this approach over many values of \(W\) (i.e., skeptic in this example) in order to get a sense of the trend and summarize regions of the trend in terms of \(p\) -value thresholds. Since we’re emphasizing modeling and deemphasizing null-hypothesis testing in this project, I’ll show a Bayesian version of the approach without the \(p\) -values.

7.4.2.1 Implementation in PROCESS brms.

Since Figure 7.9 had skeptic values ranging from 1 to 6 with ticks on the 0.5s, we’ll use a similar approach for our version. We will display posterior samples with fitted() for skeptic values ranging from .5 to 6.5, one for each 0.5—13 in total. But since we have two levels of frame (i.e., 0 and 1), that really gives us 26. And we don’t just want 26 summaries; we want full posterior distributions for each of those 26.

We’ve got a lot of moving parts in the code, below. To help make sure everything adds up, we’ll save several important values as R objects.

[Note. I got the atop() trick for the label for the y-axis from Drew Steen’s answer to this stackoverflow question .]

This isn’t quite our version of Figure 7.9, but I’m hoping it’ll add some pedagogical value for what we’re doing. Since we specified summary = F within fitted() , we got full posterior distributions for each of our 26 conditions. Because Figure 7.9 is all about differences between each frame pair across the various values of skeptic , we needed to make a difference score for each pair; this is what we did with the last mutate() line before the plot code. This initial version of the plot shows the full posterior distribution for each difference score. The posteriors are depicted with violin plots, which are density plots set on their side and symmetrically reflected as if by a mirror to give a pleasing leaf- or violin-like shape (though beware ). The light dots and vertical lines are the posterior medians and 95% intervals for each.

Going from left to right, it appears we have a clearly emerging trend. We can more simply express the trend by summarizing each posterior with medians and 95% intervals.

Notice how the contour boundaries of the 95% intervals are a little clunky. That’s because our bowtie-shape is based on only 13 x-axis values. If you wanted a smoother shape, you’d specify more skeptic values in the data object you feed into fitted() ’s newdata argument. For linear effects, 30 or so usually does it.

Anyway, I got the values for the two vertical lines directly out of the text. It’s not clear to me how one might elegantly determine those values within the paradigm we’ve been using. But that leads to an important digression. The two vertical lines are quite \(p\) -value centric. They are an attempt to separate the x-axis into areas where the difference trend either is or is not statistically-significantly different from zero. That is, we’re dichotomizing–or “trichotomizing”, depending on how you look at it–a continuous phenomenon. This is somewhat at odds with the sensibilities of the Bayesians associated with Stan and brms (e.g., here ).

On page 259, Hayes wrote:

Although the JN technique eliminates the need to select arbitrary values of \(W\) when probing an interaction, it does not eliminate your need to keep your brain turned into the task and thinking critically about the answer the method gives you.

I think this is valuable advice, particularly when working within the Bayesian paradigm. Our version of Figure 7.9 gives some interesting insights into the moderation model, model7.3 . I’m just not so sure I’d want to encourage people to interpret a continuous phenomenon by heuristically dividing it into discrete regions.

7.5 The difference between testing for moderation and probing it

This is another section where the NHST-type paradigm contrasts with many within the contemporary Bayesian paradigm. E.g., Hayes opened the section with: “We test for evidence of moderation when we want to know whether the relationship between \(X\) and \(Y\) varies systematically as a function of a proposed moderator \(W\) ”. His use of “whether” suggests we are talking about a binary answer–either there is an effect or there isn’t. But, as Gelman argued , the default presumption in social science [and warning, I’m a psychologist and thus biased towards thinking in terms of social science] is that treatment effects–and more generally, causal effects–vary across contexts 3 . As such, asking “whether” there’s a difference or an interaction effect isn’t really the right question. Rather, we should presume variation at the outset and ask instead what the magnitude of that variation is and how much accounting for it matters for our given purposes. If the variation–read interaction effect –is tiny and of little theoretical interest, perhaps we might just ignore it and not include it in the model. Alternatively, if the variation is large or of theoretical interest, we might should include it in the model regardless of statistical significance.

Another way into this topic is posterior predictive checking. We’ve already done a bit of this in previous chapters. The basic idea, recall, is that better models should give us a better sense of the patterns in the data. In the plot below, we continue to show the interaction effect with two regression lines, but this time we separate them into their own panels by frame . In addition, we add the original data which we also separate and color code by frame .

When we separate out the data this way, it really does appear that when frame == 1 , the justify values do increase as the skeptic values increase, but not so much when frame == 0 . We can use the same plotting approach, but this time with the results from the non-interaction multivariable model, model7.2 .

This time when we allowed the intercept but not the slope to vary by frame , it appears the regression lines are missing part of the story. They look okay, but it appears that the red line on the left is sloping up to quickly and that the cream line on the right isn’t sloping steeply enough. We have missed an insight.

Now imagine scenarios in which the differences by frame are more or less pronounced. Imagine those scenarios fall along a continuum. It’s not so much that you can say with certainty where on such a continuous an interaction effect would exist or not, but rather, such a continuum suggests it would appear more or less important, of greater or smaller magnitude. It’s not that the effect exists or is non-zero. It’s that it is orderly enough and of a large enough magnitude, and perhaps of theoretical interest, that it appears to matter in terms of explaining the data.

And none of this is to serve as a harsh criticism of Andrew Hayes . His text is a fine effort to teach mediation and moderation from a frequentist OLS perspective. I have benefited tremendously from his work. Yet I’d also like to connect his work to some other sensibilities.

Building further, consider this sentence from the text:

Rather, probing moderation involves ascertaining whether the conditional effect of \(X\) on \(Y\) is different from zero at certain specified values of \(W\) (if using the pick-a-point approach) or exploring where in the distribution of \(W\) the conditional effect of \(X\) on \(Y\) transitions between statistically significant and non-significant (if using the Johnson-Neyman technique). (pp. 259–260)

From an NHST/frequentist perspective, this makes clear sense. But we’re dealing with an entire posterior distribution. Consider again a figure from above.

With the pick pick-a-point approach one could fixate on whether zero was a credible value within the posterior, given a particular skeptic value. And yet zero is just one point in the parameter space. One might also focus on the whole shapes of the posteriors of these three skeptic values. You could focus on where the most credible values (i.e., those at and around their peaks) are on the number line (i.e., the effect sizes) and you could also focus on the relative widths of the distributions (i.e., the precision with which the effect sizes are estimated). These sensibilities can apply to the JN technique, as well. Sure, we might be interested in how credible zero is. But there’s a lot more to notice, too.

Now consider a modified version of our JN technique plot, from above.

This time we emphasized the shape of the posterior with stacked semitransparent 10, 20, 30, 40, 50, 60, 70, 80, 90, and 99% intervals. We also deemphasized the central tendency–our analogue to the OLS point estimate–by removing the median line. Yes, one could focus on where the 95% intervals cross zero. And yes one could request we emphasize central tendency. But such focuses miss a lot of information about the shape–the entire smooth, seamless distribution of credible values.

I suppose you could consider this our version of Figure 7.10.

7.6 Artificial categorization and subgroups

There are multiple ways to dichotomize the data by skeptic . A quick simple way is to use if_else() to make a skeptic_hi dummy.

With our dummy in hand, we’re ready to fit the two models.

Behold the coefficient summaries.

You can use fitted() to get the posterior means and other summaries for the two frame groups, by model.

Do note that though brms ‘Est.Error’ is the posterior \(SD\) for the coefficient, it is not the same thing as descriptive statistic \(SD\) of a subset of the data. Thus, although our ‘Estimates’ correspond nicely to the mean values Hayes reported in the middle of page 264, his \(SD\) s will not match up with our ‘Est.Error’ values, and nor should they.

Anyway, our results don’t yield \(t\) -tests. But you don’t need those anyway. We’re working within the Bayesian regression paradigm! But if you’re really interested in the sub-model-implied differences between the two levels of frame by skeptic_hi subgroup, all you need is the frame coefficient of model7.6 and model7.7 . Here we’ll use bind_rows() to combine their posterior samples and then plot.

As within the frequentist paradigm, please don’t mean split as a Bayesian. When possible, use all available data and use the regression formula to model theoretically-meaningful variables in your analyses.

Hayes, A. F. (2018). Introduction to mediation, moderation, and conditional process analysis: A regression-based approach. (2nd ed.). New York, NY, US: The Guilford Press.

Session info

If you’re an experimental psychologist, check out the great paper by Bolger, Zee, Rossignac-Milon, and Hassin, Causal processes in psychology are heterogeneous . The rest of you social scientists aren’t off the hook. Check out Ellen Hamaker’s excellent book chapter, Why researchers should think “within-person”: A paradigmatic rationale . Both works suggest researchers might do well to switch out their fixed-effects models for multilevel models. Does this mean the methods we’ve been covering are of no use? No. But don’t stop here, friends. Keep learning! ↩

Passion Driven Statistics

14 moderation.

Please watch the Chapter 14 Video below.

In statistics and regression analysis, moderation occurs when the relationship between two variables depends on a third variable. The third variable is referred to as the moderator variable or simply the moderator . The effect of a moderating variable is characterized statistically as an interaction; that is, a categorical (e.g., sex, race, class) or quantitative (e.g., level of reward) variable that affects the direction and/or strength of the relation between dependent and independent variables. Specifically within a correlational analysis framework, a moderator is a third variable that affects the zero-order correlation between two other variables, or the value of the slope of the dependent variable on the independent variable. In analysis of variance (ANOVA) terms, a basic moderator effect can be represented as an interaction between a focal independent variable and a factor that specifies the appropriate conditions for its operation. 9

I have hypotheses about the association between smoking quantity and nicotine dependence for individuals with and without depression (the moderator). For example, for those with depression, any amount of smoking may indicate substantial risk for nicotine dependence (i.e., at both low and high levels of daily smoking), while among those without depression, smoking quantity might be expected to be more clearly associated with likelihood of experiencing nicotine dependence (i.e., the more one smokes, the more likely they are to be nicotine dependent). In other words, I am hypothesizing a non-significant association between smoking and nicotine dependence for individuals with depression and a significant, positive association between smoking and nicotine dependence for individuals without depression.

To test this, I can run two ANOVA tests, one examining the association between nicotine dependence (categorical) and level of smoking (quantitative) for those with depression and one examining the association between nicotine dependence (categorical) and level of smoking (quantitative) for those without depression.

The results show a significant association between smoking and nicotine dependence such that the greater the smoking, the higher the rate of nicotine dependence among those individuals with and without depression. In this example, we would say that depression does not moderate the relationship between smoking and nicotine dependence. In other words, the relationship between smoking and nicotine dependence is consistent for those with and without depression. The interaction between TobaccoDependence and MajorDerpression is not significant (p-value = \(0.206836\) ).

I have a similar question regarding alcohol dependence. Specifically, I believe that the association between smoking quantity and nicotine dependence is different for individuals with and without alcohol dependence (the potential moderator). For those individuals with alcohol dependence, I believe that smoking and nicotine dependence will not be associated (i.e there will be high rates nicotine dependence at low, moderate and high levels of smoking), while among those without alcohol dependence, smoking quantity will be significantly associated with the likelihood of experiencing nicotine dependence (i.e., the more one smokes, the more likely he/she is to be nicotine dependent). In other words, I am hypothesizing a non-significant association between smoking and nicotine dependence for individuals with alcohol dependence and a significant,positive association between smoking and nicotine dependence for individuals without alcohol dependence.

To test this, I run two ANOVA tests, one examining the association between smoking and nicotine dependence for those with alcohol dependence and one examining the association between smoking and nicotine dependence for those without alcohol dependence.

The results show that there is a significant association between smoking and nicotine dependence but, as I hypothesized, sized, only for those without alcohol dependence. That is, for those without alcohol dependence, nicotine dependence is positively associated with level of smoking. In contrast, for those with alcohol dependence, the association between smoking and nicotine dependence is non-significant (statistically similar rates of nicotine dependence at every level of smoking). Because the relationship between the explanatory variable (smoking) and the response variable (nicotine dependence) is different based on the presence or absence of our third variable (alcohol dependence), we would say that alcohol dependence moderates the relationship between nicotine dependence and smoking.

https://en.wikipedia.org/wiki/Moderation_(statistics) ↩

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• How to Write a Strong Hypothesis | Guide & Examples

How to Write a Strong Hypothesis | Guide & Examples

Published on 6 May 2022 by Shona McCombes .

A hypothesis is a statement that can be tested by scientific research. If you want to test a relationship between two or more variables, you need to write hypotheses before you start your experiment or data collection.

Table of contents

What is a hypothesis, developing a hypothesis (with example), hypothesis examples, frequently asked questions about writing hypotheses.

A hypothesis states your predictions about what your research will find. It is a tentative answer to your research question that has not yet been tested. For some research projects, you might have to write several hypotheses that address different aspects of your research question.

A hypothesis is not just a guess – it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations, and statistical analysis of data).

Variables in hypotheses

Hypotheses propose a relationship between two or more variables . An independent variable is something the researcher changes or controls. A dependent variable is something the researcher observes and measures.

In this example, the independent variable is exposure to the sun – the assumed cause . The dependent variable is the level of happiness – the assumed effect .

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Step 1: ask a question.

Writing a hypothesis begins with a research question that you want to answer. The question should be focused, specific, and researchable within the constraints of your project.

Step 2: Do some preliminary research

Your initial answer to the question should be based on what is already known about the topic. Look for theories and previous studies to help you form educated assumptions about what your research will find.

At this stage, you might construct a conceptual framework to identify which variables you will study and what you think the relationships are between them. Sometimes, you’ll have to operationalise more complex constructs.

Step 3: Formulate your hypothesis

Now you should have some idea of what you expect to find. Write your initial answer to the question in a clear, concise sentence.

Step 4: Refine your hypothesis

You need to make sure your hypothesis is specific and testable. There are various ways of phrasing a hypothesis, but all the terms you use should have clear definitions, and the hypothesis should contain:

• The relevant variables
• The specific group being studied
• The predicted outcome of the experiment or analysis

Step 5: Phrase your hypothesis in three ways

To identify the variables, you can write a simple prediction in if … then form. The first part of the sentence states the independent variable and the second part states the dependent variable.

In academic research, hypotheses are more commonly phrased in terms of correlations or effects, where you directly state the predicted relationship between variables.

If you are comparing two groups, the hypothesis can state what difference you expect to find between them.

Step 6. Write a null hypothesis

If your research involves statistical hypothesis testing , you will also have to write a null hypothesis. The null hypothesis is the default position that there is no association between the variables. The null hypothesis is written as H 0 , while the alternative hypothesis is H 1 or H a .

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is used by scientists to test specific predictions, called hypotheses , by calculating how likely it is that a pattern or relationship between variables could have arisen by chance.

A hypothesis is not just a guess. It should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations, and statistical analysis of data).

A research hypothesis is your proposed answer to your research question. The research hypothesis usually includes an explanation (‘ x affects y because …’).

A statistical hypothesis, on the other hand, is a mathematical statement about a population parameter. Statistical hypotheses always come in pairs: the null and alternative hypotheses. In a well-designed study , the statistical hypotheses correspond logically to the research hypothesis.

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McCombes, S. (2022, May 06). How to Write a Strong Hypothesis | Guide & Examples. Scribbr. Retrieved 12 August 2024, from https://www.scribbr.co.uk/research-methods/hypothesis-writing/

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Moderation Analysis in SPSS

Discover Moderation Analysis in SPSS ! Learn how to perform, understand SPSS output , and report results in APA style. Check out this simple, easy-to-follow guide below for a quick read!

Struggling with Moderation Analysis in SPSS? We’re here to help . We offer comprehensive assistance to students , covering assignments , dissertations , research, and more. Request Quote Now !

Introduction

Moderation analysis is a valuable tool in research, allowing researchers to understand how the relationship between two variables changes depending on a third variable, known as the moderator. This analysis is crucial for gaining insights into complex relationships and identifying conditions under which certain effects occur. As the field of data analysis grows, the ability to perform and interpret moderation analysis becomes increasingly important.

Using SPSS, a widely-used statistical software, can simplify the process of conducting moderation analysis. This blog post aims to provide a comprehensive guide on performing moderation analysis in SPSS. We will cover the fundamental concepts, differentiate between mediation and moderation, and outline the steps and assumptions involved in testing moderation. Additionally, we will explore practical examples, interpret SPSS output, and provide guidance on reporting results in APA format.

PS: This post explains the traditional regression method in SPSS for moderation analysis. If you prefer to use the Hayes PROCESS Macro, please visit our guide on “ Moderation Analysis with Hayes PROCESS Macro in SPSS .”

What is Moderation Analysis?

Moderation analysis examines how the relationship between an independent variable (X) and a dependent variable (Y) changes as a function of a third variable, called the moderator (M). The moderator can either strengthen, weaken, or reverse the effect of the independent variable on the dependent variable. By including a moderator, researchers can capture more nuanced relationships and better understand the conditions under which certain effects are stronger or weaker.

This type of analysis is particularly useful in social sciences, where the impact of one variable on another often depends on additional contextual factors. For instance, the effect of stress on performance might vary depending on levels of social support. This helps researchers identify these cofounding effects, providing deeper insights into the dynamics of the studied relationships.

What are Steps in Testing Moderation?

• Center the Moderator and Independent Variable: Mean-center the independent variable and the moderator to reduce multicollinearity and simplify the interpretation of the interaction term.
• Create Interaction Term: Multiply the centered independent variable and the centered moderator to create an interaction term.
• Run Regression Analysis: Enter the independent variable, moderator, and interaction term into a multiple regression model predicting the dependent variable.
• Plot Interaction: Plot the interaction to visualise how the relationship between the independent variable and the dependent variable changes at different levels of the moderator.

Which is the Method better: Using Hayes PROCESS Macro or Traditional Regression for Moderation Analysis?

Choosing between Hayes PROCESS Macro and traditional regression for moderation analysis depends on your research needs and statistical expertise. The Hayes PROCESS Macro offers a user-friendly interface, automating many steps of the moderation analysis and providing bootstrap confidence intervals for the interaction effects. This method reduces human error and enhances result reliability, making it a preferred choice for those who seek convenience and precision.

In contrast, traditional regression requires manual computation of interaction terms and more steps in the analysis process. While it offers flexibility and a deeper understanding of the moderation process, it demands a higher level of statistical knowledge. The regression might be better suited for researchers who prefer customising their analyses and exploring the underlying data in more detail. Both methods have their advantages, and the choice ultimately depends on the research context and the user’s familiarity with statistical tools.

In this blog, we will give details about regression for moderation analysis, but you can visit the Hayes PROCESS post to see details of the method.

What are the Assumptions of Moderation Analysis?

• Linearity: The relationships between the independent variable, moderator, and dependent variable must be linear.
• Independence of Errors: The error terms in the regression equations should be independent of each other.
• No Multicollinearity: The independent variable, moderator, and their interaction term should not be highly correlated with each other.
• Homoscedasticity: The variance of the error terms should be constant across all levels of the independent variable and the moderator.
• Normality: The residuals of the regression equations should be normally distributed.
• Measurement without Error: The variables involved in the moderation analysis should be measured accurately without error.

What is the Hypothesis of Moderation Analysis?

The primary hypothesis in moderation analysis posits that the strength or direction of the relationship between an independent variable (X) and a dependent variable (Y) depends on the level of a third variable, the moderator (M).

• H0 (The null hypothesis): The interaction term does not significantly predict the dependent variable (meaning there is no moderation effect.)
• H1 (The alternative hypothesis): the interaction term significantly predicts the dependent variable. (indicating the presence of a moderation effect.)

Testing these hypotheses involves examining the interaction term in the regression model to determine if the moderation effect is statistically significant.

An Example of Moderation Analysis

Consider a study examining the impact of work stress (X) on job performance (Y) and how this relationship is moderated by social support (M). The hypothesis posits that the negative effect of work stress on job performance will be weaker for employees with high social support compared to those with low social support. To test this, researchers would first mean-center the variables of work stress and social support.

Next, researchers would create an interaction term by multiplying the centered work stress and social support variables. By entering work stress, social support, and the interaction term into a regression model predicting job performance, researchers can assess the main effects and the interaction effect. If the interaction term is significant, it indicates that social support moderates the relationship between work stress and job performance.

How to Perform Moderation Analysis in SPSS

Step by Step: Running Moderation Analysis in SPSS Statistics

Let’s embark on a step-by-step guide on performing the Moderation Analysis using SPSS

– Open your dataset in SPSS, ensuring it includes the independent variable (X), dependent variable (Y), and moderator (M).

Center the Variables

– Compute the mean of the independent variable and the moderator, then subtract these means from their respective variables to create centered variables.

Create Interaction Term

– Multiply the centered independent variable by the centered moderator to create an interaction term.

Run Regression Analysis

– Navigate to ` Analyze > Regression > Linear `.

– Enter the dependent variable (Y) into the Dependent box.

– Move the centered independent variable (X), centered moderator (M), then click Next ” for block 2 enter the interaction term into the Independent box.

– Click OK to run the regression analysis.

Interpret the Output

– Examine the coefficients table to assess the significance of the independent variable, moderator, and interaction term.

– Significant interaction term indicates moderation.

Note: Conducting Moderation Analysis in SPSS provides a robust foundation for understanding the key features of your data. Always ensure that you consult the documentation corresponding to your SPSS version, as steps might slightly differ based on the software version in use. This guide is tailored for SPSS version 25 , and for any variations, it’s recommended to refer to the software’s documentation for accurate and updated instructions.

SPSS Output for Moderation Analysis

Spss output 1, spss output 2.

How to Interpret SPSS Output of Moderation Analysis

When interpreting the SPSS output of your moderation analysis, focus on three key tables: Model Summary, ANOVA, and Coefficients.

Model Summary Table:

• R: This represents the correlation between the observed and predicted values of the dependent variable. Higher values indicate a stronger relationship.
• R Square (R²): This value indicates the proportion of variance in the dependent variable explained by the independent, moderator, and interaction variables. An R² value closer to 1 suggests a better fit.
• Adjusted R Square: Adjusts the R² value for the number of predictors in the model. This value is useful for comparing models with different numbers of predictors.

ANOVA Table:

• F-Statistic: This tests the overall significance of the model. A significant F-value (p < 0.05) indicates that the model significantly predicts the dependent variable.
• (p-value): If the p-value is less than 0.05, the model is considered statistically significant, meaning the independent and mediator variables together significantly predict the dependent variable.

Coefficients Table:

• Unstandardized Coefficients (B): Coefficient of variable
• Constant (Intercept): The expected value of the dependent variable when all predictors are zero.
• Standardized Coefficients (Beta): These coefficients are useful for comparing the relative strength of each predictor in the model.
• t-Statistic and Sig. (p-value): Indicates whether each predictor is significantly contributing to the model. If the p-value is less than 0.05, the predictor is considered statistically significant.

By focusing on these tables, you can effectively interpret the results of your mediation analysis in SPSS, identifying the direct and indirect effects, as well as the overall model significance.

How to Report Results of Moderation Analysis in APA

Reporting the results of moderation analysis in APA (American Psychological Association) format requires a structured presentation. Here’s a step-by-step guide in list format:

• Introduction : Briefly describe the purpose of the moderation analysis and the variables involved.
• Descriptive Statistics : Report the means and standard deviations of the independent variable, moderator, and dependent variable.
• Main Effects : Provide the regression coefficients, standard errors, and p-values for the independent variable and moderator.
• Interaction Effect : Report the regression coefficient, standard error, and p-value for the interaction term.
• Model Summary : Include R² and adjusted R² values to indicate the model fit.
• Significance Tests : Present the results of the F-test and the significance levels for the overall model.
• Plot Interaction : Include a plot illustrating the interaction effect, showing how the relationship between the independent variable and the dependent variable changes at different levels of the moderator.
• Figures and Tables : Provide tables and figures to visually represent the statistical results and interaction effects.
• Conclusion : Summarise the key results and suggest directions for future research.

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Conceptualizing, Organizing, and Positing Moderation in Communication Research

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R Lance Holbert, Esul Park, Conceptualizing, Organizing, and Positing Moderation in Communication Research, Communication Theory , Volume 30, Issue 3, August 2020, Pages 227–246, https://doi.org/10.1093/ct/qtz006

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Meta-theoretical focus is given to how communication researchers are approaching and hypothesizing moderation. A moderation typology is offered and an evaluation of the field’s common practices for positing moderation reveals an inability to discern between three overarching classifications (Contributory, Contingent, Cleaved). A content analysis of eight communication journals reveals moderation hypotheses lacking a level of precision that can best aid the field’s knowledge generation. In addition, vague hypothesizing is leaving communication researchers vulnerable to the commitment of Type III error (i.e., correctly rejecting a null hypothesis for the wrong reason). Recommendations are provided in an effort to improve the field’s conceptualization and presentation of moderation.

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• How to Write a Strong Hypothesis | Steps & Examples

How to Write a Strong Hypothesis | Steps & Examples

Published on May 6, 2022 by Shona McCombes . Revised on November 20, 2023.

A hypothesis is a statement that can be tested by scientific research. If you want to test a relationship between two or more variables, you need to write hypotheses before you start your experiment or data collection .

Example: Hypothesis

Daily apple consumption leads to fewer doctor’s visits.

Table of contents

What is a hypothesis, developing a hypothesis (with example), hypothesis examples, other interesting articles, frequently asked questions about writing hypotheses.

A hypothesis states your predictions about what your research will find. It is a tentative answer to your research question that has not yet been tested. For some research projects, you might have to write several hypotheses that address different aspects of your research question.

A hypothesis is not just a guess – it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations and statistical analysis of data).

Variables in hypotheses

Hypotheses propose a relationship between two or more types of variables .

• An independent variable is something the researcher changes or controls.
• A dependent variable is something the researcher observes and measures.

If there are any control variables , extraneous variables , or confounding variables , be sure to jot those down as you go to minimize the chances that research bias  will affect your results.

In this example, the independent variable is exposure to the sun – the assumed cause . The dependent variable is the level of happiness – the assumed effect .

Prevent plagiarism. Run a free check.

Step 1. ask a question.

Writing a hypothesis begins with a research question that you want to answer. The question should be focused, specific, and researchable within the constraints of your project.

Step 2. Do some preliminary research

Your initial answer to the question should be based on what is already known about the topic. Look for theories and previous studies to help you form educated assumptions about what your research will find.

At this stage, you might construct a conceptual framework to ensure that you’re embarking on a relevant topic . This can also help you identify which variables you will study and what you think the relationships are between them. Sometimes, you’ll have to operationalize more complex constructs.

Step 3. Formulate your hypothesis

Now you should have some idea of what you expect to find. Write your initial answer to the question in a clear, concise sentence.

4. Refine your hypothesis

You need to make sure your hypothesis is specific and testable. There are various ways of phrasing a hypothesis, but all the terms you use should have clear definitions, and the hypothesis should contain:

• The relevant variables
• The specific group being studied
• The predicted outcome of the experiment or analysis

5. Phrase your hypothesis in three ways

To identify the variables, you can write a simple prediction in  if…then form. The first part of the sentence states the independent variable and the second part states the dependent variable.

In academic research, hypotheses are more commonly phrased in terms of correlations or effects, where you directly state the predicted relationship between variables.

If you are comparing two groups, the hypothesis can state what difference you expect to find between them.

6. Write a null hypothesis

If your research involves statistical hypothesis testing , you will also have to write a null hypothesis . The null hypothesis is the default position that there is no association between the variables. The null hypothesis is written as H 0 , while the alternative hypothesis is H 1 or H a .

• H 0 : The number of lectures attended by first-year students has no effect on their final exam scores.
• H 1 : The number of lectures attended by first-year students has a positive effect on their final exam scores.

If you want to know more about the research process , methodology , research bias , or statistics , make sure to check out some of our other articles with explanations and examples.

• Sampling methods
• Simple random sampling
• Stratified sampling
• Cluster sampling
• Likert scales
• Reproducibility

 Statistics

• Null hypothesis
• Statistical power
• Probability distribution
• Effect size
• Poisson distribution

Research bias

• Optimism bias
• Cognitive bias
• Implicit bias
• Hawthorne effect
• Anchoring bias
• Explicit bias

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A hypothesis is not just a guess — it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations and statistical analysis of data).

Null and alternative hypotheses are used in statistical hypothesis testing . The null hypothesis of a test always predicts no effect or no relationship between variables, while the alternative hypothesis states your research prediction of an effect or relationship.

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is used by scientists to test specific predictions, called hypotheses , by calculating how likely it is that a pattern or relationship between variables could have arisen by chance.

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McCombes, S. (2023, November 20). How to Write a Strong Hypothesis | Steps & Examples. Scribbr. Retrieved August 16, 2024, from https://www.scribbr.com/methodology/hypothesis/

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• v.36(50); 2021 Dec 27

Formulating Hypotheses for Different Study Designs

Durga prasanna misra.

1 Department of Clinical Immunology and Rheumatology, Sanjay Gandhi Postgraduate Institute of Medical Sciences, Lucknow, India.

Armen Yuri Gasparyan

2 Departments of Rheumatology and Research and Development, Dudley Group NHS Foundation Trust (Teaching Trust of the University of Birmingham, UK), Russells Hall Hospital, Dudley, UK.

Olena Zimba

3 Department of Internal Medicine #2, Danylo Halytsky Lviv National Medical University, Lviv, Ukraine.

Marlen Yessirkepov

4 Department of Biology and Biochemistry, South Kazakhstan Medical Academy, Shymkent, Kazakhstan.

Vikas Agarwal

George d. kitas.

5 Centre for Epidemiology versus Arthritis, University of Manchester, Manchester, UK.

Generating a testable working hypothesis is the first step towards conducting original research. Such research may prove or disprove the proposed hypothesis. Case reports, case series, online surveys and other observational studies, clinical trials, and narrative reviews help to generate hypotheses. Observational and interventional studies help to test hypotheses. A good hypothesis is usually based on previous evidence-based reports. Hypotheses without evidence-based justification and a priori ideas are not received favourably by the scientific community. Original research to test a hypothesis should be carefully planned to ensure appropriate methodology and adequate statistical power. While hypotheses can challenge conventional thinking and may be controversial, they should not be destructive. A hypothesis should be tested by ethically sound experiments with meaningful ethical and clinical implications. The coronavirus disease 2019 pandemic has brought into sharp focus numerous hypotheses, some of which were proven (e.g. effectiveness of corticosteroids in those with hypoxia) while others were disproven (e.g. ineffectiveness of hydroxychloroquine and ivermectin).

Graphical Abstract

DEFINING WORKING AND STANDALONE SCIENTIFIC HYPOTHESES

Science is the systematized description of natural truths and facts. Routine observations of existing life phenomena lead to the creative thinking and generation of ideas about mechanisms of such phenomena and related human interventions. Such ideas presented in a structured format can be viewed as hypotheses. After generating a hypothesis, it is necessary to test it to prove its validity. Thus, hypothesis can be defined as a proposed mechanism of a naturally occurring event or a proposed outcome of an intervention. 1 , 2

Hypothesis testing requires choosing the most appropriate methodology and adequately powering statistically the study to be able to “prove” or “disprove” it within predetermined and widely accepted levels of certainty. This entails sample size calculation that often takes into account previously published observations and pilot studies. 2 , 3 In the era of digitization, hypothesis generation and testing may benefit from the availability of numerous platforms for data dissemination, social networking, and expert validation. Related expert evaluations may reveal strengths and limitations of proposed ideas at early stages of post-publication promotion, preventing the implementation of unsupported controversial points. 4

Thus, hypothesis generation is an important initial step in the research workflow, reflecting accumulating evidence and experts' stance. In this article, we overview the genesis and importance of scientific hypotheses and their relevance in the era of the coronavirus disease 2019 (COVID-19) pandemic.

DO WE NEED HYPOTHESES FOR ALL STUDY DESIGNS?

Broadly, research can be categorized as primary or secondary. In the context of medicine, primary research may include real-life observations of disease presentations and outcomes. Single case descriptions, which often lead to new ideas and hypotheses, serve as important starting points or justifications for case series and cohort studies. The importance of case descriptions is particularly evident in the context of the COVID-19 pandemic when unique, educational case reports have heralded a new era in clinical medicine. 5

Case series serve similar purpose to single case reports, but are based on a slightly larger quantum of information. Observational studies, including online surveys, describe the existing phenomena at a larger scale, often involving various control groups. Observational studies include variable-scale epidemiological investigations at different time points. Interventional studies detail the results of therapeutic interventions.

Secondary research is based on already published literature and does not directly involve human or animal subjects. Review articles are generated by secondary research. These could be systematic reviews which follow methods akin to primary research but with the unit of study being published papers rather than humans or animals. Systematic reviews have a rigid structure with a mandatory search strategy encompassing multiple databases, systematic screening of search results against pre-defined inclusion and exclusion criteria, critical appraisal of study quality and an optional component of collating results across studies quantitatively to derive summary estimates (meta-analysis). 6 Narrative reviews, on the other hand, have a more flexible structure. Systematic literature searches to minimise bias in selection of articles are highly recommended but not mandatory. 7 Narrative reviews are influenced by the authors' viewpoint who may preferentially analyse selected sets of articles. 8

In relation to primary research, case studies and case series are generally not driven by a working hypothesis. Rather, they serve as a basis to generate a hypothesis. Observational or interventional studies should have a hypothesis for choosing research design and sample size. The results of observational and interventional studies further lead to the generation of new hypotheses, testing of which forms the basis of future studies. Review articles, on the other hand, may not be hypothesis-driven, but form fertile ground to generate future hypotheses for evaluation. Fig. 1 summarizes which type of studies are hypothesis-driven and which lead on to hypothesis generation.

STANDARDS OF WORKING AND SCIENTIFIC HYPOTHESES

A review of the published literature did not enable the identification of clearly defined standards for working and scientific hypotheses. It is essential to distinguish influential versus not influential hypotheses, evidence-based hypotheses versus a priori statements and ideas, ethical versus unethical, or potentially harmful ideas. The following points are proposed for consideration while generating working and scientific hypotheses. 1 , 2 Table 1 summarizes these points.

Evidence-based data

A scientific hypothesis should have a sound basis on previously published literature as well as the scientist's observations. Randomly generated (a priori) hypotheses are unlikely to be proven. A thorough literature search should form the basis of a hypothesis based on published evidence. 7

Unless a scientific hypothesis can be tested, it can neither be proven nor be disproven. Therefore, a scientific hypothesis should be amenable to testing with the available technologies and the present understanding of science.

Supported by pilot studies

If a hypothesis is based purely on a novel observation by the scientist in question, it should be grounded on some preliminary studies to support it. For example, if a drug that targets a specific cell population is hypothesized to be useful in a particular disease setting, then there must be some preliminary evidence that the specific cell population plays a role in driving that disease process.

Testable by ethical studies

The hypothesis should be testable by experiments that are ethically acceptable. 9 For example, a hypothesis that parachutes reduce mortality from falls from an airplane cannot be tested using a randomized controlled trial. 10 This is because it is obvious that all those jumping from a flying plane without a parachute would likely die. Similarly, the hypothesis that smoking tobacco causes lung cancer cannot be tested by a clinical trial that makes people take up smoking (since there is considerable evidence for the health hazards associated with smoking). Instead, long-term observational studies comparing outcomes in those who smoke and those who do not, as was performed in the landmark epidemiological case control study by Doll and Hill, 11 are more ethical and practical.

Balance between scientific temper and controversy

Novel findings, including novel hypotheses, particularly those that challenge established norms, are bound to face resistance for their wider acceptance. Such resistance is inevitable until the time such findings are proven with appropriate scientific rigor. However, hypotheses that generate controversy are generally unwelcome. For example, at the time the pandemic of human immunodeficiency virus (HIV) and AIDS was taking foot, there were numerous deniers that refused to believe that HIV caused AIDS. 12 , 13 Similarly, at a time when climate change is causing catastrophic changes to weather patterns worldwide, denial that climate change is occurring and consequent attempts to block climate change are certainly unwelcome. 14 The denialism and misinformation during the COVID-19 pandemic, including unfortunate examples of vaccine hesitancy, are more recent examples of controversial hypotheses not backed by science. 15 , 16 An example of a controversial hypothesis that was a revolutionary scientific breakthrough was the hypothesis put forth by Warren and Marshall that Helicobacter pylori causes peptic ulcers. Initially, the hypothesis that a microorganism could cause gastritis and gastric ulcers faced immense resistance. When the scientists that proposed the hypothesis themselves ingested H. pylori to induce gastritis in themselves, only then could they convince the wider world about their hypothesis. Such was the impact of the hypothesis was that Barry Marshall and Robin Warren were awarded the Nobel Prize in Physiology or Medicine in 2005 for this discovery. 17 , 18

DISTINGUISHING THE MOST INFLUENTIAL HYPOTHESES

Influential hypotheses are those that have stood the test of time. An archetype of an influential hypothesis is that proposed by Edward Jenner in the eighteenth century that cowpox infection protects against smallpox. While this observation had been reported for nearly a century before this time, it had not been suitably tested and publicised until Jenner conducted his experiments on a young boy by demonstrating protection against smallpox after inoculation with cowpox. 19 These experiments were the basis for widespread smallpox immunization strategies worldwide in the 20th century which resulted in the elimination of smallpox as a human disease today. 20

Other influential hypotheses are those which have been read and cited widely. An example of this is the hygiene hypothesis proposing an inverse relationship between infections in early life and allergies or autoimmunity in adulthood. An analysis reported that this hypothesis had been cited more than 3,000 times on Scopus. 1

LESSONS LEARNED FROM HYPOTHESES AMIDST THE COVID-19 PANDEMIC

The COVID-19 pandemic devastated the world like no other in recent memory. During this period, various hypotheses emerged, understandably so considering the public health emergency situation with innumerable deaths and suffering for humanity. Within weeks of the first reports of COVID-19, aberrant immune system activation was identified as a key driver of organ dysfunction and mortality in this disease. 21 Consequently, numerous drugs that suppress the immune system or abrogate the activation of the immune system were hypothesized to have a role in COVID-19. 22 One of the earliest drugs hypothesized to have a benefit was hydroxychloroquine. Hydroxychloroquine was proposed to interfere with Toll-like receptor activation and consequently ameliorate the aberrant immune system activation leading to pathology in COVID-19. 22 The drug was also hypothesized to have a prophylactic role in preventing infection or disease severity in COVID-19. It was also touted as a wonder drug for the disease by many prominent international figures. However, later studies which were well-designed randomized controlled trials failed to demonstrate any benefit of hydroxychloroquine in COVID-19. 23 , 24 , 25 , 26 Subsequently, azithromycin 27 , 28 and ivermectin 29 were hypothesized as potential therapies for COVID-19, but were not supported by evidence from randomized controlled trials. The role of vitamin D in preventing disease severity was also proposed, but has not been proven definitively until now. 30 , 31 On the other hand, randomized controlled trials identified the evidence supporting dexamethasone 32 and interleukin-6 pathway blockade with tocilizumab as effective therapies for COVID-19 in specific situations such as at the onset of hypoxia. 33 , 34 Clues towards the apparent effectiveness of various drugs against severe acute respiratory syndrome coronavirus 2 in vitro but their ineffectiveness in vivo have recently been identified. Many of these drugs are weak, lipophilic bases and some others induce phospholipidosis which results in apparent in vitro effectiveness due to non-specific off-target effects that are not replicated inside living systems. 35 , 36

Another hypothesis proposed was the association of the routine policy of vaccination with Bacillus Calmette-Guerin (BCG) with lower deaths due to COVID-19. This hypothesis emerged in the middle of 2020 when COVID-19 was still taking foot in many parts of the world. 37 , 38 Subsequently, many countries which had lower deaths at that time point went on to have higher numbers of mortality, comparable to other areas of the world. Furthermore, the hypothesis that BCG vaccination reduced COVID-19 mortality was a classic example of ecological fallacy. Associations between population level events (ecological studies; in this case, BCG vaccination and COVID-19 mortality) cannot be directly extrapolated to the individual level. Furthermore, such associations cannot per se be attributed as causal in nature, and can only serve to generate hypotheses that need to be tested at the individual level. 39

IS TRADITIONAL PEER REVIEW EFFICIENT FOR EVALUATION OF WORKING AND SCIENTIFIC HYPOTHESES?

Traditionally, publication after peer review has been considered the gold standard before any new idea finds acceptability amongst the scientific community. Getting a work (including a working or scientific hypothesis) reviewed by experts in the field before experiments are conducted to prove or disprove it helps to refine the idea further as well as improve the experiments planned to test the hypothesis. 40 A route towards this has been the emergence of journals dedicated to publishing hypotheses such as the Central Asian Journal of Medical Hypotheses and Ethics. 41 Another means of publishing hypotheses is through registered research protocols detailing the background, hypothesis, and methodology of a particular study. If such protocols are published after peer review, then the journal commits to publishing the completed study irrespective of whether the study hypothesis is proven or disproven. 42 In the post-pandemic world, online research methods such as online surveys powered via social media channels such as Twitter and Instagram might serve as critical tools to generate as well as to preliminarily test the appropriateness of hypotheses for further evaluation. 43 , 44

Some radical hypotheses might be difficult to publish after traditional peer review. These hypotheses might only be acceptable by the scientific community after they are tested in research studies. Preprints might be a way to disseminate such controversial and ground-breaking hypotheses. 45 However, scientists might prefer to keep their hypotheses confidential for the fear of plagiarism of ideas, avoiding online posting and publishing until they have tested the hypotheses.

SUGGESTIONS ON GENERATING AND PUBLISHING HYPOTHESES

Publication of hypotheses is important, however, a balance is required between scientific temper and controversy. Journal editors and reviewers might keep in mind these specific points, summarized in Table 2 and detailed hereafter, while judging the merit of hypotheses for publication. Keeping in mind the ethical principle of primum non nocere, a hypothesis should be published only if it is testable in a manner that is ethically appropriate. 46 Such hypotheses should be grounded in reality and lend themselves to further testing to either prove or disprove them. It must be considered that subsequent experiments to prove or disprove a hypothesis have an equal chance of failing or succeeding, akin to tossing a coin. A pre-conceived belief that a hypothesis is unlikely to be proven correct should not form the basis of rejection of such a hypothesis for publication. In this context, hypotheses generated after a thorough literature search to identify knowledge gaps or based on concrete clinical observations on a considerable number of patients (as opposed to random observations on a few patients) are more likely to be acceptable for publication by peer-reviewed journals. Also, hypotheses should be considered for publication or rejection based on their implications for science at large rather than whether the subsequent experiments to test them end up with results in favour of or against the original hypothesis.

Hypotheses form an important part of the scientific literature. The COVID-19 pandemic has reiterated the importance and relevance of hypotheses for dealing with public health emergencies and highlighted the need for evidence-based and ethical hypotheses. A good hypothesis is testable in a relevant study design, backed by preliminary evidence, and has positive ethical and clinical implications. General medical journals might consider publishing hypotheses as a specific article type to enable more rapid advancement of science.

Disclosure: The authors have no potential conflicts of interest to disclose.

Author Contributions:

• Data curation: Gasparyan AY, Misra DP, Zimba O, Yessirkepov M, Agarwal V, Kitas GD.