How to interpret the value of an interaction coefficient between two effects coded binary predictors?

Question

Let's imagine an experiment where people look at pictures and then rate their mood after each one from 1 (sad) to 7 (happy). The pictures can either be happy or sad (IV1), and participants are either doing the experiment on a high resolution or a low resolution monitor (IV2). (This is not my real experiment, just an example.)

These two predictors are effects coded and the data are analyzed at the trial level with mixed effects linear regressions.

Let's say the coefficient for picture type is +1. I think this means that on average, mood after happy pictures is rated two points higher than sad pictures (remember it is effects coded).

Let's say the coefficient for screen type is +.5: people are on average 1 point happier when doing the experiment on high resolution monitors.

Can you help me intuitively understand the interaction coefficient? Let's say, for example, I expect the effect of picture type to be a three point difference on the high resolution monitor, and a one point difference on the low resolution monitor. What interaction coefficient would that translate to? I think that I know how to work this out with dummy coding, but I am at a loss now with effects coding.

Answer 1

With dummy coding, $\{0,1\}$, for each IV, the 4 cells would take on the values

• sad/low: $\mu$
• sad/high: $\mu+\alpha$
• happy/low: $\mu+\beta$
• happy/high: $\mu + \alpha + \beta + \gamma$

So, the intercept of the model is just the average for the sad/low group, the effect for being in the high group is $\alpha$, the effect for being in the happy group is $\beta$, the the interaction (the deviation from the addition model) is $\gamma$. This can be interpreted either as the amount of "error" in the additive model for the 4th cell, or as the change in the "slope" as you move from one value of one of the IV to the other value. (Symmetry makes choice for the IV irrelevant in this case, but there usually is a better contextual choice for explanatory purposes with real data sets.)

With effects coding, $\{-1,1\}$ (assuming a balanced design), the 4 cells would take on the following values

• sad/low: $\mu - \alpha - \beta + \gamma$
• sad/high: $\mu + \alpha - \beta - \gamma$
• happy/low: $\mu -\alpha + \beta - \gamma$
• happy/high: $\mu +\alpha + \beta + \gamma$

In this case, $\mu$ is not the sad/low group, but the overall (grand) average. Thus, $\alpha$ is the half the difference of the measured effect between low and high (added to one and subtracted from the other). The $\beta$ is the same but for sad/happy. This leaves the interaction effect to be half the distance that the "slope" would shift if the additive model doesn't fully capture the nature of the data.