## How to test for moderation? (part 2)

If you have not read my previous post ‘How to test for moderation? (part 1)’ I recommend that you do it. In part 2 we are going to address how to test for moderation when both predictor (X) and moderator (M) are continuous (scenario #2), when X is continuous and M is categorical (scenario #3) and when X is categorical and M is continuous (scenario #4).

The first step under scenario #2 is to center both predictor and moderator. The model will look like:

Y=b0 +b1Xc +b2Mc +b3XcMc + e

where Xc and Mc are the centered versions of X and M. Remember that centering a variable is simply subtract its mean. A significant coefficient for the interaction term will indicate moderation. The way to look at how the effect of X varies across the values of M, unless there are some theoretical levels of the M for which you want to measure the effect of X, it is commonly done by using the mean, the mean plus one standard error and the mean minus one standard error. Computing the slope or marginal effect of X at each of these 3 values of M and plotting the 3 lines is an easy way of visually representing moderation.

Please, keep in mind that we are assuming that the moderator changes the relationship between X and Y in a linear way. Other assumptions could be that there is a quadratic relationship or a threshold, a value of M above which the effect of X on Y changes.

Under scenario #3, X is continuous, and M can have two or more levels.

Let’s assume that we are interested in testing whether the effect of job satisfaction (X) on job performance (Y) is moderated by type of contract (M).

Assuming that M has 3 levels: full-time permanent, part-time permanent and contingent, as we did under scenario #1, we will need to dummy code M by creating variables M1 (1=full-time permanent, 0=else) and M2 (1= part-time permanent, 0=else). Note that this coding leaves contingent workers as the reference category.

The equation model will look like this:

Y=b0 +b1X +b2M1 +b3M2 +b4XM1 + b5XM2 + e

In our example b1 represents the effect of job satisfaction among contingent workers, b4 the change in the effect of job satisfaction from contingent to full-time permanent employees, also known as the difference in slopes, and b5 represents the different in slopes between part-time and contingent employees. Testing moderation is equivalent to testing whether b4 and b5 are jointly equal to zero or equivalently whether the change in F from adding the two interaction terms is equal to zero. Rejection of the null of these tests indicates that there is moderation.

If we want to represent the moderation effect graphically we will need to estimate the slope for X at each of the 3 levels of M, which can be done from the model coefficients or by estimating the marginal effect of X in the categories of M.

Finally, the last of the four described scenarios, scenario #4, and probably the less common, is when we have a continuous moderator categorical predictor. Imagine that we are interested in testing whether salary moderates the relationship between marital status (living alone vs living with a partner) and health as measured by a health index. As in scenario # 2, we will center the moderator and we will have as many product terms as the levels of the predictor minus one. In our example, since we have 2 levels for marital status we will only have a product term as shown in the equation below:

Y=b0 +b1X +b2Mc +b3McX + e

Testing for moderation in this context is equivalent to testing whether the coefficient corresponding to the product term is significantly different from zero.