19  Moderation

In progress.

You may not recognize it, but you have much experience with moderation. Quite simply, a moderation is interaction. For example, we may test the efficacy of a new drug versus a placebo on ‘happiness’. We measure scores before taking the drug/placebo, and measure scores after some time of taking the drug/placebo. In this example we have two independent variables (time and drug) and one dependent variable (happiness). An interaction would indicate that the association between an IV and a DV is depenent on some other IV. That is, the change in time on happiness depends on whether you got the drug or the placebo. These figure represent potential interactions. Note that the lines are not parallel.

In ANOVAs you dealt with interactions between qualitative or categorical variables. Like the above example, we tested interactions between sex, gender, drug group, etc. These were cateogircal variables with mutually exclusive levels.

However, we can model interactions for continuous variables as well. We can test interactions between different types of variables:

Knowledge Check

Explain what moderation is.

Moderation or interactions occur when the effect of one IV on the DV depends on the level of a different IV.

19.1 Our Research

We are working with a new health company and been tasked with testing the association between average daily protein intake (DPI; measured in grams) and lean muscle mass (LMM; the amount of muscle tissue measured in pounds) in a group of individuals. The company is also interested in the association between LMM and gym activity (gym goers versus gym abstainers).

19.2 Our Model

Our initial regression model is defined as follow:

\(y_{lmm}=b_o+b_{dpi}(x_{i1})+b_{gym}(x_{i2})+e_i\)

So someone’s LMM is a function of their DPI and gym activity. Note that this model will only give us main effects. Let’s run a regression model using our data collected from 100 individuals (50 gym goers and 50 non-gym goers).

ID Gym DPI LMM
67 No Gym 22 80
29 Gym 23 93
42 Gym 28 74
74 No Gym 30 65
96 No Gym 33 59
9 Gym 36 120
34 Gym 38 71
87 No Gym 41 69
5 Gym 42 75
49 Gym 42 124
24 Gym 44 99
41 Gym 45 106
22 Gym 46 101
53 No Gym 48 52
80 No Gym 48 71
26 Gym 49 100
11 Gym 50 134
31 Gym 51 115
75 No Gym 51 83
7 Gym 52 120
23 Gym 52 79
30 Gym 52 117
58 No Gym 52 99
14 Gym 53 86
92 No Gym 53 56
3 Gym 54 113
36 Gym 54 118
78 No Gym 54 40
13 Gym 55 145
40 Gym 56 142
47 Gym 56 110
64 No Gym 56 57
94 No Gym 56 72
79 No Gym 57 50
82 No Gym 59 66
68 No Gym 60 52
15 Gym 62 137
77 No Gym 62 91
65 No Gym 63 49
32 Gym 64 106
48 Gym 64 154
71 No Gym 64 55
73 No Gym 64 42
56 No Gym 65 85
89 No Gym 65 58
4 Gym 66 134
63 No Gym 66 72
6 Gym 67 151
69 No Gym 67 67
76 No Gym 67 51
91 No Gym 67 74
28 Gym 68 128
66 No Gym 68 18
35 Gym 69 135
1 Gym 70 137
18 Gym 70 144
21 Gym 70 126
27 Gym 70 114
59 No Gym 70 78
55 No Gym 71 59
85 No Gym 71 63
98 No Gym 71 70
100 No Gym 71 24
72 No Gym 72 78
60 No Gym 73 58
37 Gym 75 170
86 No Gym 76 84
8 Gym 77 137
43 Gym 78 132
83 No Gym 79 54
52 No Gym 80 83
99 No Gym 80 82
93 No Gym 81 67
2 Gym 82 178
12 Gym 82 125
19 Gym 83 152
45 Gym 84 175
70 No Gym 84 79
39 Gym 85 161
33 Gym 86 139
51 No Gym 86 65
84 No Gym 86 30
46 Gym 87 171
57 No Gym 87 73
61 No Gym 87 95
62 No Gym 88 79
10 Gym 89 174
44 Gym 93 161
90 No Gym 94 86
97 No Gym 95 50
50 Gym 97 153
38 Gym 101 156
81 No Gym 104 74
25 Gym 105 185
88 No Gym 106 40
16 Gym 107 205
20 Gym 108 169
95 No Gym 111 62
54 No Gym 113 44
17 Gym 114 199

19.3 Our Results

The results of our model is as follows:

Observations 100
Dependent variable LMM
Type OLS linear regression
F(2,97) 140.23
R² 0.74
Adj. R² 0.74
Est. 2.5% 97.5% t val. p VIF
(Intercept) 17.72 1.55 33.89 2.18 0.03 NA
DPI 0.67 0.45 0.88 6.18 0.00 1.00
GymGym 70.45 61.66 79.23 15.91 0.00 1.00
Standard errors: OLS

…or with some additional details from the apaTables() package:



Regression results using LMM as the criterion
 

   Predictor       b       b_95%_CI sr2 sr2_95%_CI             Fit
 (Intercept)  17.72*  [1.55, 33.89]                               
         DPI  0.67**   [0.45, 0.88] .10 [.03, .17]                
      GymGym 70.45** [61.66, 79.23] .67 [.55, .79]                
                                                       R2 = .743**
                                                   95% CI[.65,.80]
                                                                  

Note. A significant b-weight indicates the semi-partial correlation is also significant.
b represents unstandardized regression weights. 
sr2 represents the semi-partial correlation squared.
Square brackets are used to enclose the lower and upper limits of a confidence interval.
* indicates p < .05. ** indicates p < .01.

Let’s also plot the data to look at it’s functional form:

After looking at the plot, do you notice anything? Look closely at the potential relationship between DPI and LLM; does it differ for gym goers versus non-gym goers?

Let’s work out the equation for gym goers and non-gym goers separately using the results of the regression. Recall that this is our model:

\(y_{lmm}=b_o+b_{dpi}(x_{i1})+b_{gym}(x_{i2})+e_i\)

And these are our results:



Regression results using LMM as the criterion
 

   Predictor       b       b_95%_CI sr2 sr2_95%_CI             Fit
 (Intercept)  17.72*  [1.55, 33.89]                               
         DPI  0.67**   [0.45, 0.88] .10 [.03, .17]                
      GymGym 70.45** [61.66, 79.23] .67 [.55, .79]                
                                                       R2 = .743**
                                                   95% CI[.65,.80]
                                                                  

Note. A significant b-weight indicates the semi-partial correlation is also significant.
b represents unstandardized regression weights. 
sr2 represents the semi-partial correlation squared.
Square brackets are used to enclose the lower and upper limits of a confidence interval.
* indicates p < .05. ** indicates p < .01.

The equation for non-gym goers is:

  • Non-gym Goers (\(x_2=0\))
    • \(y_{lmm}=17.72+0.67(x_{i1})+ 70.45(0)+e_i\)
    • \(y_{lmm}=17.72+0.67(x_{i1})+e_i\)

While the equation of gym goers is:

  • Gym Goers (\(x_2=1\))
    • \(y_{lmm}=17.72+0.67(x_{i1})+70.45(1)+e_i\)
    • \(y_{lmm}=88.17+0.67(x_{i1})+e_i\)

Let’s plot those lines on our previous figure:

So, when we inspect the lines of best fit for each level of our gym variable, we notice something strange happening. The points on the no gym line seem to be above the line for lower DPI, but below the line for the higher DPI. The reverse trend may be evident in the ‘gym’ line. Thus, if we allowed these lines to have different slopes, perhaps they would fit better. Modelling an interaction (or moderation) allows us to do this.

19.4 Moderation

But first…

19.4.1 Multicollinearity

Remember that an assumption of multiple regression is the independence of our independent variables. To REDUCE multicollinearity, we must center our continuous predictors. You can center predictors by calculating the mean of a variable and then subtracting the mean from each individual’s score.

For example, imagine we measure confidence in five people and get these scores:

Name Confidence
Jessica 17
Haley 8
Letyraial 13
Alec 10
Zainab 10

The mean of the confidence scores is 11.6. Thus, we will subtract each score by 11.6 to get the mean centered variable.

Name Confidence Confidence Centered
Jessica 17 5.4
Haley 8 -3.6
Letyraial 13 1.4
Alec 10 -1.6
Zainab 10 -1.6

Let’s see how using centered predictors impacts the correlations between variables and reduces multicollinearity. Consider two different variables, x1 and x2, that are correlated:

    x1  x2
x1 1.0 0.2
x2 0.2 1.0

You can see they correlated, \(r=.20\). Let’s multiply them together to create an interaction term (x3, which is \(x_3=x_1\times x_2\)) and see how correlated each predictor is with the interaction term.

             x1      x2     x1_x2
x1    1.0000000 0.20000 0.9108277
x2    0.2000000 1.00000 0.5719200
x1_x2 0.9108277 0.57192 1.0000000

Now, let’s center x1 and x2 and create a new interaction term.

            x1         x2      x1_x2
x1    1.000000 0.20000000 0.21388102
x2    0.200000 1.00000000 0.06792511
x1_x2 0.213881 0.06792511 1.00000000

Notice how the correlations between the interaction term and each original term is reduced, or attenuated. For example, the correlation between x2 and the interaction term, x1_x2, was about \(r=.572\). The correlation between the centered x2 and the interaction term was \(r=.068\). Also notice that the correlation between centered and uncentered independent variables (not the interaction term) does not change!

So, for our main DPI and LMM example, we would center all continuous independent variables: DPI. We would model an interaction by creating a new variable (\(x_3\)=interaction) that is the product of the other variables (\(x_1=DPI\), \(x_2=gym\)) - \(x_3=(x_1)(x_2)\)

Let’s run our new model with centered interaction terms.

Observations 100
Dependent variable LMM
Type OLS linear regression
F(3,96) 166.84
R² 0.84
Adj. R² 0.83
Est. 2.5% 97.5% t val. p VIF
(Intercept) 64.27 59.32 69.21 25.80 0.00 NA
DPI_Centered -0.06 -0.31 0.20 -0.44 0.66 2.25
GymGym 70.27 63.28 77.27 19.95 0.00 1.00
DPI_Centered:GymGym 1.31 0.97 1.66 7.57 0.00 2.24
Standard errors: OLS

Let’s work out the equations again using the new model:

  • No Gym (\(x_2=0\))
    • \(y_{lmm}=64.27-0.06(x_{i1cent})+70.27(x_{i2})+1.31(x_{i1cent})(x_{i2})+e_i\)
    • \(y_{lmm}=64.27-0.06(x_{i1cent})+70.27(0)+1.33(x_{i1cent})(0)+e_i\)
    • \(y_{lmm}=64.27-0.06(x_{i1cent})+e_i\)
  • Gym (\(x_2=1\))
    • \(y_{lmm}=64.27-0.06(x_{i1cent})+70.27(x_{i2})+1.31(x_{i1cent})(x_{i2})+e_i\)
    • \(y_{lmm}=64.27-0.06(x_{i1cent})+70.27(1)+1.31(x_{i1cent})(1)+e_i\)
    • \(y_{lmm}=(64.27+70.27)-(0.06(x_{i1cent})+1.31(x_{i1cent}))+e_i\)
    • \(y_{lmm}=134.54+1.25(x_{i1cent})+e_i\)

And a new visualization of our plot:

Notice how the new lines, with each group having their own intercept and slope seem to fit better. This is corroborated by the formal analysis, which resulted in the interaction being statistically significant (\(p<.001\)).

There is also a package in R that allows for 3D models of interactions. Have a look (you can interact with this figure) and try to understand what’s happening (note that the line of best fit becomes a ‘plane’ of best fit):

19.4.2 Continuous x Continuous Variable Interaction

We often will deal with multiple continuous variables that may interact. Fortunately, the process is similar. However, we will now need to center all predictors in the interaction.

Let’s stick to a similar example. Assume we want to determine if LMM regresses on DPI. But, we think that average hours in the gym per week will interact with protein intake to predict lean muscle mass. So, our previous categorical predictor of being a gym goer versus not is not a continuous predictor of average hours in the gym per week. So, we reach out to people and ask to now consider the average gym hours per week.

Our specific hypotheses are as follow:

19.5 Our Hypotheses

  1. DPI will predict LMM
  • \(H1: \beta_{1}\ne0\)
  1. Average gym hours will predict LMM
  • \(H2: \beta_{2}\ne0\)
  1. There will be an interaction between DPI and gym hours on LMM. Specifically, the relationship between DPI and LMM will be stronger for those who average more gym hours per week.
  • \(H3: \beta_{3}\ne0\)

19.5.1 Our Data

ID DPI Gym LMM
31 27 5 198
38 29 3 165
24 36 4 183
1 39 5 192
39 39 4 157
7 42 1 141
26 42 7 309
9 45 7 337
22 45 6 289
28 45 3 201
40 45 5 219
5 47 3 139
34 47 3 213
49 47 6 228
35 48 6 257
36 48 3 198
3 49 4 241
33 49 2 185
4 50 3 200
27 50 6 240
16 52 4 268
23 52 5 315
42 52 5 213
12 53 7 294
18 54 6 329
44 54 10 402
6 55 7 393
21 55 4 176
41 55 6 295
48 55 6 360
19 56 2 178
14 57 2 192
25 57 6 271
32 57 6 260
13 58 7 319
8 60 4 285
10 63 3 192
17 63 5 298
2 64 5 255
15 66 4 271
20 66 5 358
43 66 1 191
29 70 5 347
46 70 5 323
37 72 6 399
30 73 4 246
11 74 -1 5
45 76 7 396
50 76 6 298
47 98 5 430

19.6 Our Results

Our analysis results in the following:

Observations 50
Dependent variable LMM
Type OLS linear regression
F(3,46) 67.30
R² 0.81
Adj. R² 0.80
Est. 2.5% 97.5% t val. p VIF
(Intercept) 256.62 246.04 267.20 48.82 0.00 NA
DPI_Centered 2.75 1.92 3.57 6.69 0.00 1.01
Gym_Centered 31.73 26.14 37.33 11.41 0.00 1.04
DPI_Centered:Gym_Centered 0.70 0.22 1.18 2.93 0.01 1.05
Standard errors: OLS

Remember, the intercept here is for when all other variables are 0. However, we have centered our variables, so 0 carries a different meaning. Because we mean-centered, a score of 0 on a mean-centered variable is equal to the mean. So, the intercept in these results reflect the expected LMM score for an individual with an average DPI and Gym hours. We could expect someone who consumes the mean amount of protein and who goes to the gym an average amount of time to be 256.6237554.

Let’s visualize the new interaction.

Or in 3D:

19.7 Our Write up

Let’s write up the results of this last model.

We regressed individual’s lean muscle mass (LMM) onto their daily protein intake (DPI), average hours or gym per week, and the interaction between DPI and Gym hours. The results suggest that DPI was a statistically significant predictor and accounted for 18% of the variance in LMM, \(b=2.75, p<.001, sr^2=.18, 95\%CI[.06, .30]\). Gym hours was a statistically significant predictor of and accounted for an addition 53% of the variance in LMM, \(b=31.73, p<.001, sr^2=.53, 95\%CI[.33, .72]\). Finally, the interaction between DPI and Gym hours was statistically significant and accounted for an addition 3% of the variance in LMM, \(b=0.70, p=.01, sr^2=.03, 95\%CI[-.01, .08]\).

19.8 Simple Slopes

Simple slopes analysis tests whether the slope (coefficient) of one predictor (i.e., one IV) differs from 0 at given levels or values of the moderator (i.e., another IV). Although we can you any level of value on the moderator, the typically convention if to test at -1SD, 0, and +1SD on the moderator. Thus, if a variable has a mean of 20 and SD of 10, then our simple slopes analysis will test if the slope for the IV and DV is statistically significant for the values 10, 20, and 30 on the moderator.

These analyses can be used to provide additional information about a potential interaction. Let’s use the data from our last example and run a simple slopes analysis.

SIMPLE SLOPES ANALYSIS 

Slope of DPI when Gym = 2.715778 (- 1 SD): 

  Est.   S.E.   t val.      p
------ ------ -------- ------
  1.39   0.59     2.36   0.02

Slope of DPI when Gym = 4.660000 (Mean): 

  Est.   S.E.   t val.      p
------ ------ -------- ------
  2.75   0.41     6.69   0.00

Slope of DPI when Gym = 6.604222 (+ 1 SD): 

  Est.   S.E.   t val.      p
------ ------ -------- ------
  4.11   0.65     6.31   0.00

As is seen from the output, we are given a separate analysis for each value of the simple slopes analysis. In this specific example, the relationship between DPI and LMM was statistically significant at all tested levels (-1SD, mean, and +1SD) of gym hours.

19.9 Practice