Suppose I have a data set of 200 controls (group A; has no memory problems) and 100 cases (group B; has memory problems). And I'm looking at the relationship between memory and cognitive test score (low = good memory; high = bad memory). When I perform a 2 sample t-test, I find that group B scores significantly higher on the cognitive tests than group A. This is expected, as group B has poorer memory than its counterpart.
When I perform logistic regression, (I wanted to look at how cognitive test scores can predict whether person is in group A or group B), the coefficient of the test score is negative, i.e., the odds ratio is < 1. This would suggest that for a 1 unit increase in cognitive test score, the risk of having bad memory decreases. Why is this? Shouldn't it be for a 1 unit increase in cognitive test score, the risk of having bad memory increases? Because higher test scores are associated with worse memory.
Below is a plot of my data. You can see that the cases have worse memory (higher memory rating). But it seems that (as suggested by the best fit line) that the controls scored higher on the test. Although the t-test results suggest that the cases score worse on the test. So why is there a discrepancy between the linear regression best fit line and what the t-test result tells me?
I guess my biggest question is: t-test tells me group B (which has worse memory) scored higher on the cognitive test (This makes sense because higher score is indicative of worse memory). logistic regression tells me that higher score on the test decreases the odds of having bad memory (and when I look at the plot with the best fit lines below, it seems to make sense...but I'm not sure if I can relate logistic regression and linear regression like this). Why is there a reversal/discrepancy between the t-test and logistic regression results?