# Main effects flip sign when adding interaction term in ordinal logistic regression

I am running an ordinal logistic regression.

Dependent Variable: policy score (0-3).

Independent Variables: all continuous scale (GDP, corruption perception, total number of mines)

All Independent Variables have a positive correlation with the dependent variable.

When I run the regression without interactions, all covariates are positive.

When I add an interaction, GDP*corruption is positive, but their seperate effects become negative..

Can someone explain why this is happening?

Also, there's no multicolinearity (VIF < 1).

• plot your data! eg gdp on x axis and fitted values from main effects and interactions model. I think you will stop worrying once you see this. Apr 18 '19 at 8:08

First, as AlexK pointed out, this flipping of signs can happen because of the nature of multiple regressions. The coefficient in a multiple regression gives you the marginal effect of that variable with all other terms in the model held constant.

Let's look at interaction effects, but in a highly simplified 2x2 design. The principle can of course be extended to continuous variables, it is just a lot easier to think about it this way. Imagine your data set looked like this, with the means in the cells and the marginal means around it.

                   GDP
Low   High
Corruption    Low   10      8   9
High   8     20  14
9     14


As you can see, both GDP and Corruption have a positive main effect, because in both "High" groups, means are higher. The model would be:

y = b0 + b1*GDP + b2*Corr (with both IV either 0 or 1).

Estimating coefficients would lead to both b1 and b2 being positive, due to the high mean when both are 1. The model has no chance but to estimate coefficients in a way that fit the data best, and with this model, it will estimate two positive coefficients.

20 = b0 + b1*1 + b2*1
8 = b0 + b1*0 + b2*1
8 = b0 + b1*1 + b2*0
10 = b0 + b1*0 + b2*0


Let's include the interaction term (GDP*Corr), which will actually be 0 in all but the first equation.

20 = b0 + b1*1 + b2*1 + b3*1*1
8 = b0 + b1*0 + b2*1 + b3*0*1
8 = b0 + b1*1 + b2*0 + b3*1*0
10 = b0 + b1*0 + b2*0 + b3*0*0


Now the model has a chance to account for the "diagonal" effect of both IVs being "High" together. The high mean of 20 can now be accounted for by b3. The model can estimate marginal effects holding the interaction effect constant and show that the remaining (=marginal) effects reflected by coefficients b1 and b2 are actually negative.

I had to add: The interpretation of something like this can become tricky very quickly (imagine three-way interactions). If you have a significant interaction, be very careful how you interpret it and any remaining main effects!

This is due to the change in interpretation of coefficients for the main effects (GDP and corruption perception, in your case) when the interaction term is added.

Let's put aside the nature of your dependent variable (DV). When there is no interaction term, the coefficient on the GDP variable gives us the average change in DV for a one-unit increase in GDP, holding other covariates constant regardless of what the values of other covariates are. (And same thing is true for the corruption variable.) If you are familiar with calculus, this can be stated as the marginal effect of (or the partial derivative of DV with respect to) GDP is just the value of the coefficient on GDP.

But when you add the interaction term, a one-unit increase in GDP will not produce an average change in DV equal to the value of the coefficient on GDP. The marginal effect of GDP on DV now varies by the level of corruption variable. More precisely, the marginal effect of GDP will be equal to the value of the coefficient on GDP plus the value of the coefficient on the interaction term multiplied by the value of corruption variable. So the only case when the level of corruption variable will not affect the marginal effect of GDP on DV is when its value is 0, and that's what the coefficient on GDP tells you in the model with the interaction term: what the marginal effect of GDP on DV is when the level of corruption perception is 0. In your case, that marginal effect is negative.

If you have not already done so, you may want to center the main effect variables before creating the interaction term, as it may make your results more meaningful and easier to interpret. Centering means subtracting the mean of each variable from the individual values. If you do that, then the coefficient on GDP will give you the marginal effect of GDP on DV for the average level of corruption perception. And same for the marginal effect of corruption perception. Moreover, the interpretation of the intercept/constant term in the regression (if you include it) will be the expected value of DV for average levels of GDP and corruption perception.