The command boxtid might be helpful in checking for linearity in the effect of any continuous x-variable(s) on the log odds of the y-variable. Note that this can also provide some clues as to why the link test produced such poor results. 

This command requires that you install a user-written package first. So, if you have not installed it already, type: 

ssc install boxtid

More information help boxtid

Practical example

Let us apply boxtid to our multiple regression model: 

boxtid logistic earlyret bmi sex ib1.educ if pop_logistic==1

The test of non-linearity for our continuous variable bmi is statistically significant (p=0.010), suggesting that the assumption of a linear effect is violated. Although we will not explore this further here, we could consider transformations of this variable (e.g. through categorisation, or log transformation). For example, we could categorise bmi into underweight, normal weight, overweight, and obesity.