The sensitivity and specificity statistics reported in logistic regression are not really all that helpful. As I recall, programs like SPSS that report this statistic use a cut-off based on whether the predicted probability is above or below 0.5. If the probability of one of the two levels in your logistic regression model is much larger than the other, such a cut-off may not make a lot of sense. It also may not make a lot of sense if the costs of misclassification are much different (as another person has already noted). You should not trust a computer's choice here because the computer knows nothing about the context of the problem.
You can manipulate the cutoff so that specificty is very high and sensitivity is very low. You can also manipulate the cutoff so that specificity is very low and sensitivity is very high. So the disparity that you see with the default cutoff of 0.5 is pretty much meaningless.
Personally, I think it is very bad for programs to report sensitivity and specificity for a logistic regression model. It is almost never helpful and it is almost always confusing. Also, if you had to choose a value, 0.5 is probably one of the worst values for a cut-off. For highly skewed situations (situations where one level has a lot more data than another), it leads to really extreme values. A more logical choice would be to set the cutoff at the probability of the first level (usually the level coded as 0). That would be equivalent to a prior probability equal to the probabilities observed in your data and an assumption that the costs of misclassification are the same in either direction. But I wouldn't hold my breath waiting for this to happen. Every package develops a legacy of users over the years (and decades in the case of SPSS) who are so used to seeing things one way that they get upset when things change. Plus the developers of these programs would much rather roll out new features than to tweak old features.
So just ignore sensitivity and specificity entirely.
The area under the curve, however, is a more useful statistic. When it is close to 1, you have a good model and when it is close to 0.5, you have a bad model. Some people treat this statistic as a goodness of fit statistic, much like R-squared in linear regression. I think the analogy is a bit strained, but even so, it is a worthwhile statistic to report.
With a single continuous variable, the area under the curve has an interesting physical interpretation. It represents P[X > Y] where X is a vector of the continuous variable associated with one of the levels of your dependent variable and Y is a vector of the continuous variable associated with the other level of your dependent variable. When this probability is large, it shows that small values of your independent variable are very likely to be associated with one level of your dependent variable and large values of your independent variable are very likely to be associated with the other level. This is a good thing.
On the other hand if P[X >Y] is close to 0.5 it means that small and large values of your independent variable are just as likely to be associated with either level of your dependent variable.
One minor point to keep things precise. The area under the curve is sometimes P[X < Y] rather than P[X > Y]. It depends on the sign of the coefficient in your model.
So is 0.75 a good value? Well, it's larger than 0.5, but not very close to 1.0. Rather than compare this in a vacuum, consider this value to value observed in other models and for other data sets. If I had to describe what 0.75 means, I'd say "good, but not great." But without knowing more about your model and the results of competing models, I can't say a whole lot more.
I hope this helps.
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Stephen Simon
Independent Statistical Consultant
P. Mean Consulting
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