Statistics - (Interaction|Synergy) effect

In a multiple regression, is assumed that the effect on the target of increasing one unit of one predictor (is independent|has no influence) on the other predictor.

If this is not the case, sharing a value on two predictors, may have more effect than allocating the entire value to only one.

In marketing, this is known as a synergy effect, and in statistics it is referred to as an interaction effect.

The interaction effect is when the effect of one independent variable depends on the other.

The effect of one IV depends on the other IV (the simple effects of one IV change across the levels of the other IV)

This is exactly like a moderation effect in multiple regression.

This effect is called:

If the simple effects change as a function of the other IV, then you've got an interaction.

3 - Example

example of an interaction in a 2 X 3 factorial ANOVA:

• the effect of A is significant (but different in magnitude) at each level of B
• the effect of B is significant at one level of A but not at the other level of A
• the effect B is significant (but different in direction) at each level of A

example of no interaction in a 2 X 3 factorial ANOVA:

• the effect of A is significant (and the same magnitude) at each level of B

4 - Modelling interactions

4.1 - between quantitative variables

Introduction of a new variable: the (product|interaction) term.

$X_{1i} . X_{2i}$$$\begin{array}{ll} Y_i & = & B_0 + B_1 X_{1i} + B_2 X_{2i} + B_3 ( X_{1i} . X_{2i} ) + \epsilon_i \\ & = & B_0 + (B_1 + B_3 X_{2i}) X_{1i} + B_2 X_{2i} + \epsilon_i \\ & = & B_0 + B_1 X_{1i} + (B_2 + B_3 X_{1i}) X_{2i} + \epsilon_i \end{array}$$

In the second and third formula form, we got only two variables but we can see that one is dependant of the other.

Interpretation:

• for one unit increase in $X_2$

, there is

$(B_1 + B_3 X_{2i})$

increase in

$X_1$
• for one unit increase in $X_1$

, there is

$(B_2 + B_3 X_{1i})$

increase in

$X_2$

If the p-value for the interaction term is extremely low, it indicates that there is strong evidence that

$H_A : B_ 3 \neq 0$

.

If R2 for the interaction model has augmented, it means that more variability has been explained by the interaction term.

4.2 - Hierarchy

Sometimes an interaction term has a very small p-value, but the associated main effects do not (ie

$X_1 \text{and} X_2$

)

The hierarchy principle:

If we include an interaction in a model, we should also include the main effects, even if the p-values associated with their coefficients are not significant.

The rationale for this principle is that interactions are hard to interpret in a model without main effects (their meaning is changed).

Specifically, the interaction terms also contain main effects, if the model has no main effect terms.

4.3 - between qualitative and quantitative variables

Without an interaction term, the model takes the form:

$$\begin{array}{ll} Y_i & \approx & B_0 + B_1 X_i + \left\{ \begin{array}{ll} B_2 & \text{ if the ith person is a male} \\ 0 & \text{ if the ith person is a female} \end{array}\right. \\ & = &B_1 X_i + \left\{ \begin{array}{ll} B_0 + B_2 & \text{ if the ith person is a male} \\ B_0 & \text{ if the ith person is a female} \end{array}\right. \end{array}$$

By grouping the items (Second line formula), we can think of this as having a common slope but a different intercept depending on whether the person is a male or female.

By putting an interaction term, we're going to get both a different intercept and a different slope. The model takes the form:

If you're not a male, the term is 0, which means you get the baseline intercept and slope.

$$\begin{array}{ll} Y_i & \approx & B_0 + B_1 X_i + \left\{ \begin{array}{ll} B_2 + B_3 X_i & \text{ if the ith person is a male} \\ 0 & \text{ if the ith person is a female} \end{array}\right. \\ & = & \left\{ \begin{array}{ll} (B_0 + B_2) + (B_1 + B_3) X_i & \text{ if the ith person is a male} \\ B_0 + B_1 X_i & \text{ if the ith person is a female} \end{array}\right. \end{array}$$