Adding interaction terms to a regression model can greatly expand understanding of the relationships among the variables in the model and allows more hypotheses to be tested.

The example from Interpreting Regression Coefficients was a model of the height of a shrub (Height) based on the amount of bacteria in the soil (Bacteria) and whether the shrub is located in partial or full sun (Sun). Height is measured in cm, Bacteria is measured in thousand per ml of soil, and Sun = 0 if the plant is in partial sun and Sun = 1 if the plant is in full sun. The regression equation was estimated as follows:

Height = 42 + 2.3*Bacteria + 11*Sun

It would be useful to add an interaction term to the model if we wanted to test the hypothesis that the relationship between the amount of bacteria in the soil on the height of the shrub was different in full sun than in partial sun.

One possibility is that in full sun, plants with more bacteria in the soil tend to be taller, whereas in partial sun, plants with more bacteria in the soil are shorter. Another possibility is that plants with more bacteria in the soil tend to be taller in both full and partial sun, but that the relationship is much more dramatic in full than in partial sun.

The presence of a significant interaction indicates that the effect of one predictor variable on the response variable is different at different values of the other predictor variable. It is tested by adding a term to the model in which the two predictor variables are multiplied. The regression equation will look like this:

Height = B0 + B1*Bacteria + B2*Sun + B3*Bacteria*Sun

Adding an interaction term to a model **drastically changes the interpretation of all of the coefficients**. If there were no interaction term, B1 would be interpreted as the unique effect of Bacteria on Height. But the interaction means that the effect of Bacteria on Height is different for different values of Sun. So the unique effect of Bacteria on Height is not limited to B1, but also depends on the values of B3 and Sun. The unique effect of Bacteria is represented by everything that is multiplied by Bacteria in the model: B1 + B3*Sun. B1 is now interpreted as the unique effect of Bacteria on Height only when Sun = 0.

In our example, once we add the interaction term, our model looks like:

Height = 35 + 4.2*Bacteria + 9*Sun + 3.2*Bacteria*Sun

Adding the interaction term changed the values of B1 and B2. The effect of Bacteria on Height is now 4.2 + 3.2*Sun. For plants in partial sun, Sun = 0, so the effect of Bacteria is 4.2 + 3.2*0 = 4.2. So for two plants in partial sun, a plant with 1000 more bacteria/ml in the soil would be expected to be 4.2 cm taller than a plant with less bacteria.

For plants in full sun, however, the effect of Bacteria is 4.2 + 3.2*1 = 7.4. So for two plants in full sun, a plant with 1000 more bacteria/ml in the soil would be expected to be 7.4 cm taller than a plant with less bacteria.

Because of the interaction, the effect of having more bacteria in the soil is different if a plant is in full or partial sun. Another way of saying this is that the slopes of the regression lines between height and bacteria count are different for the different categories of sun. B3 indicates how different those slopes are.

Interpreting B2 is more difficult. B2 is the effect of Sun when Bacteria = 0. Since Bacteria is a continuous variable, it is unlikely that it equals 0 often, if ever, so B2 can be virtually meaningless by itself. Instead, it is more useful to understand the effect of Sun, but again, this can be difficult. The effect of Sun is B2 + B3*Bacteria, which is different at every one of the infinite values of Bacteria. For that reason, often the only way to get an intuitive understanding of the effect of Sun is to plug a few values of Bacteria into the equation to see how Height, the response variable, changes.

{ 14 comments… read them below or add one }

hi Karen,

Goad approach for regression interaction but can this model be extended to quantile regression?

Hello,

In your example there are only 2 independent variables, so the interaction is obvious, being the product of those 2 independent variables (x1*x2).

What if you had 3 variables, but you are not sure which and which had interaction (although you suspect there is some)? Would you add to the table of values the columns X1X2X3, X1X3, X2X3?

Thanks.

You should mention the opportunity to de-mean bacteria, so the interpretation of bacteria = 0 would actually make sense.

How can we interpret the odds ratio for the following case:

Suppose we are modeling the dependency of birth-weight (dependent variable; normal/ Low) on birth term (independent variable; premature/ fullmature).

Variable B S.E. Wald P Exp(B)

Birth Term(premature): -3.099 1.079 8.248 .004 .045

For example:

Height = 35 + 4.2*Bacteria + 9*Sun + 3.2*Bacteria*Sun

What is the 9 for sun and 3,2 for bacteria*sun are significant, but 4,2 for bacteria is not significant. How do I have to interpret that?

It wouldn’t change the interpretation. It just means that when there isn’t sun (the reference condition), there is no significant effect of bacteria.

How would this apply in a logistic regression model? I am trying to calculate the OR of main effects and significant interactions.

Thank you

hello,

please help me interpret this regression equation with interaction variables.

stock price = 2 + 0.3*book values + 0.4*earnings + 0.36*book values*earnings*firm size + 0.1* firm size.

how does “0.36*book values*earnings*firm size” differ from ” 0.1* firm size ” in explaining the dependent variable?

Could you please explain how can I test the interaction using ‘likelihood ratio test’ and wald test?

Hi Karen,

thank you for this posting, which explains very well how to interpret interactions in linear models. I wonder whether it is similar in generalized linear models? Would I simply use the same estimates (log odds) and the same formulae shown above to see the effect of a moderating covariate on another predictor? And if yes, then I could simply transform the result (in your example, the value of “Height”) into odds ration via exp() or calculate probabilities? If so, I could easily plot “probability curves” of interaction terms (one curve for main effect, one curve for moderating interaction on main effect) in logistic regressions…

Best, Daniel

Hi Daniel,

The general approach is similar in generalized linear models. They of course, are a little trickier because odds ratios, for example, are ratios, not differences. See this: http://www.theanalysisfactor.com/complicated-models-with-tricky-effects/

And yes, you can plot predicted probabilites to see the interaction effect–it makes interpretation of the interaction much, much easier.

Hi, You interpreted interaction term with an example very nice, thank you. But how do you interpret a quadratic term in a nonlinear regression? Thank you very much in advance.

Hi a, just in case you still need to know (or for anyone else reading), i’m no expert but I believe this is how it works:

If the “Sun” variable was zero, then every unit of bacteria would affect the height by 4.2 units. (just b2)

if Sun was one, then every unit of bacteria would affect the height by (4.2-4.2) = 0 units. That is, b2+b3*sun (where sun is 1). The result can be interpreted as follows: when there is (full) sun, bacteria has no statistically significant effect on height.

The “interaction” identifies how the relationships between the two independent variables change their relationships with the dependent variable.

Cheers, James

Hi Karen,

how would you interpret it, say, if your coefficient for the interaction effect Bacteria*Sun were negative, such that it “cancelled out” the main effect in a sense? E.g., if the coefficient for Bacteria is 4.2, and for Bacteria*Sun it is -4.2? If these coefficients were both significant, that just seems to make no sense to me.

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