Interpret Logistic Regression with Interactions using R

Question

I ran the following logistic regression in R where the dependent variable (Success) is 1 if the student graduates and 0 if the student does not graduate:

    Call:
    glm(formula = Success ~ Core.GPA + Lab.Yr.Taken + Frn.Lang.Yr.Taken + 
SAT.Converted + Incoming.Test, family = binomial, data = df)

    Deviance Residuals: 
        Min       1Q   Median       3Q      Max  
    -2.7123  -1.2211   0.6501   0.8495   1.3955  

    Coefficients:
                        Estimate Std. Error z value Pr(>|z|)    
    (Intercept)       -4.094e+00  6.007e-01  -6.816 9.39e-12 ***
    Core.GPA           1.024e+00  9.916e-02  10.329  < 2e-16 ***
    Lab.Yr.Taken       1.220e-01  2.904e-02   4.200 2.66e-05 ***
    Frn.Lang.Yr.Taken  6.335e-02  3.115e-02   2.034    0.042 *  
    SAT.Converted      9.867e-05  3.759e-04   0.262    0.793    
    Incoming.Test      4.018e-02  3.332e-03  12.056  < 2e-16 ***
    ---
    Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

    (Dispersion parameter for binomial family taken to be 1)

Null deviance: 7022.5  on 5947  degrees of freedom
    Residual deviance: 6573.9  on 5942  degrees of freedom
      (1108 observations deleted due to missingness)
    AIC: 6585.9

    Number of Fisher Scoring iterations: 4

Here are the Odds Ratio for that model:

  (Intercept)          Core.GPA      Lab.Yr.Taken Frn.Lang.Yr.Taken 
   0.01666638        2.78504399        1.12973489        1.06540330 
SAT.Converted     Incoming.Test 
   1.00009867        1.04099486 

My interpretation here (using Incoming.Test as an example) is: "Odds of success increase by 4.1% for each additional one unit increase in Incoming Test Credit" (1.04099486-1) x 100

However, I'd like to know if being a first generation student impacts these odds ratio and if so how? (i.e., does the likelihood that a student graduates based on any of my independent variables differ for a first gen and non first gen student?) To determine this, I ran the model again and set First Generation as an interaction variable with all variables in model.

Call:
glm(formula = Success ~ (Core.GPA + Lab.Yr.Taken + Frn.Lang.Yr.Taken + 
SAT.Converted + Incoming.Test) * firstGen, family = binomial, 
data = df)

Deviance Residuals: 
Min       1Q   Median       3Q      Max  
-2.6380  -1.1061   0.6362   0.8186   1.8288  

Coefficients:
                                Estimate  Std. Error z value Pr(>|z|)    
(Intercept)                   -3.5094889  0.6705484  -5.234 1.66e-07 ***
Core.GPA                       1.0698918  0.1087680   9.836  < 2e-16 ***
Lab.Yr.Taken                   0.0971983  0.0316549   3.071  0.00214 ** 
Frn.Lang.Yr.Taken              0.0377991  0.0338970   1.115  0.26480    
SAT.Converted                 -0.0001896  0.0004232  -0.448  0.65418    
Incoming.Test                  0.0358615  0.0035690  10.048  < 2e-16 ***
firstGenYes                   -1.7697353  1.6092509  -1.100  0.27145    
Core.GPA:firstGenYes           0.3528697  0.2899914   1.217  0.22367    
Lab.Yr.Taken:firstGenYes       0.0928141  0.0811118   1.144  0.25251    
Frn.Lang.Yr.Taken:firstGenYes  0.1247308  0.0900435   1.385  0.16598    
SAT.Converted:firstGenYes     -0.0009749  0.0010075  -0.968  0.33319    
Incoming.Test:firstGenYes      0.0296594  0.0106095   2.796  0.00518 ** 
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for binomial family taken to be 1)

Null deviance: 7022.5  on 5947  degrees of freedom
Residual deviance: 6483.5  on 5936  degrees of freedom
  (1108 observations deleted due to missingness)
AIC: 6507.5

Number of Fisher Scoring iterations: 4

and odds ratio:

              (Intercept)                      Core.GPA 
                0.0299122                     2.9150640 
             Lab.Yr.Taken             Frn.Lang.Yr.Taken 
                1.1020789                     1.0385226 
            SAT.Converted                 Incoming.Test 
                0.9998104                     1.0365123 
              firstGenYes          Core.GPA:firstGenYes 
                0.1703781                     1.4231457 
 Lab.Yr.Taken:firstGenYes Frn.Lang.Yr.Taken:firstGenYes 
                1.0972578                     1.1328434 
SAT.Converted:firstGenYes     Incoming.Test:firstGenYes 
                0.9990255                     1.0301036 

What I'm hoping to learn from these models is:

1. For Non-First Gen students, the odds of success increase/decrease by _% for each additional one unit increase in Incoming test Credit.

My assumption on this so far is: The odds ratio for Incoming.Test (1.0365123) represents the odds ratio for NON-first Generation students (aka, for non-first gen students odds of success increase by 3% for each additional increase in incoming test credits).

2. Same question as above, but for First Gen students.

??? Is a calculation needed to determine this?

3. What is the difference in odds between first gen and non first gen students

My assumption so far: The odds ratio for IncomingTestxFirstGen (1.03010136) represents the difference in odds between first gen and non first gen???

4. Is First Generation status significant or non significant when considering odds of success based on Incoming test credits?

My assumption: Yes, the difference is significant because IncomingTestxFirstGen is significant.

5. What does the "firstGenYes" odds ratio in the second model indicate/represent?

???

I would appreciate any feedback regarding my initial interpretation attempts! I have reviewed questions similar to this that have already been asked, but have not came across any explanation that fully encapsulates what I’m after. Thank you.

Answer 1

All of these questions are easily answered by understanding who the reference person is. Let's examine a slightly simpler model, perhaps Success ~ Incoming.Test*firstGen. Mathematically, the model is

$$\operatorname{logit}(p) = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_1 x_2$$

Here, let incoming test be $x_1$ and firstGen be $x_2$.

The reference group would be a non-first generation student with an incoming test score of 0. That corresponds to just the intercept $\beta_0$.

A non first generation student who scores a 1 on the incoming test would have a log odds of success equal to $\beta_0 + \beta_1$. A non first generation student who scores a 2 on the incoming test would have a log odds of success equal to $\beta_0 + 2\beta_1$.

A first generation student who scored a 1 on the incoming test would have a log odds of success equal to $\beta_0 + \beta_1 + \beta_2 + \beta_3$. A first generation student who scored a 2 on the incoming test would have a log odds of success equal to $\beta_0 + 2\beta_1 + \beta_2 + 2\beta_3$.

Now, let's compare how a change in incoming test changes the log odds for both groups. First, the non first generation students. The difference in log odds between non first generation students who differ by 1 point on the incoming test is $\beta_1$. The odds ratio is then $\exp(\beta_1)$.

The difference in log odds between first generation students who differ by 1 point on the incoming test would be $\beta_1 + \beta_3$. The odds ratio would then be $\exp(\beta_3)\exp(\beta_1)$.

This strategy applies to your first two questions.

As for the remaining questions, I can answer with a little more brevity.

3. Because you include interactions, the difference between first gen and non first gen students depends on the values of the other variables.

4. This is an illposed question. Because you include interactions, I think a more salient question is "Does the model which includes these interactions explain the variation in the data more than I would expect under chance?". You can address this with a likelihood ratio test between your two models.

5. firstGenYes is the additive effect of being a 1st generation student when the other (interacting) variables are all exactly 0. Were you to mean center the continuous variables, then the interpretation would simply be the additive effect of being a first generation student.