# Interpretation of GLM Odds?

I have data on student graduation (1 = graduated, 0 = not graduated), as well as student gender and whether or not they were diagnosed with a disorder (categorical variable where it can be none, mild, moderate, or severe).

Using R studio to analyze data and get Odds Ratios:

outputx<- svyglm(GRADUATE~GENDER+DISORDER,  family = quasibinomial, data=SCHOOL, design = SWSCHOOL, maxit=100)
summary(outputx)
odds.ratio(outputx)


This provides the following:

                OR     2.5%    97.5%    P
(intercept)    0.01     0.007  0.0317   < 2.2e-16
GENDERFemale   4.31     2.08   16.61    <001269
DISORDERMild   0.84     0.33    2.26    0.604
DISORDERMed    1.18     0.954   3.18    0.085
DISORDERSev    0.48     0.234   0.91    0.048


I don't understand how the odds relate to the reference group for the independent variables.

Is it saying people who graduate are 4.31 more likely to be female compared to people who did not graduate?

Or is it saying people who are female are 4.31 more likely to graduate compared to males?

Ditto for the disorder variable - what does the 0.48 mean in relation to the reference group for "no disorder" here? It seems like it has no interpretation.

It's saying that the odds of females graduating is 4.31 higher than the odds of males, holding constant disorder. That doesn't mean females are 4.31 times more likely to graduate, but it does mean females are more likely to graduate. You can think of it as that among females, the ratio of the number who graduate to the number who don't graduate is 4.31 times higher than it is among males.

Odds ratios are notoriously challenging to interpret, and, given that all your variables are categorical, you could fit another model, like a binomial model with an identity link (just by setting family = quasibiomial(link = "identity") and start = rep(.15, 5)), which would allow you to interpret the coefficients as the difference in the probability of graduating.

For DISORDERSev, the value of 0.48 means that the odds of graduating for those with severe disorder is 0.48 the odds for those with no disorder, indicating that those with severe disorder are less likely to graduate than those with no disorder. The reference category is no disorder.

Typically, odds ratios are interpreted as scalars that represent the increased/decreased likelihood of association to the response variable. Your first interpretation is correct, with a slight modification: holding all other factors constant, the odds of females graduating is 4.31 more times that of males.

In other words, it depends on the way the model is specified.

It's harder to assess the odds ratios for insignificant covariates (in your case - mild and severe). However, for the severe condition, given that p < 0.05, we could say that the odds of students with severe disabilities graduating is only 0.48 times, or approximately half that of the odds of students without disabilities graduating.