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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?

enter image description here

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  • $\begingroup$ Maybe I'm misunderstanding, but it seems that your 'Case' cases have higher 'Memory' ratings than your control group? Wouldn't this mean that higher 'Memory' rating means worse memory? Didn't you also say that higher Test scores are worse? Wouldn't it then be logical that higher 'Memory' rating (worse memory) correlates to higher 'Test' scores (worse test results)? $\endgroup$ – cfh Apr 23 '15 at 10:12
  • $\begingroup$ Yes, higher memory rating should correlate with higher test scores. You are exactly right. And my t-test results DO show that group B (higher memory rating) score significantly higher than group A (lower memory rating). But in the plot the best fit line shows that the controls scored higher on the test than the cases? $\endgroup$ – Adrian Apr 23 '15 at 10:15
  • $\begingroup$ I don't think you can conclude that from the two regression lines. All the lines tell you is the influence of the Memory value on the Test score. In both groups, higher Memory values lead to higher Test scores, as expected, and the slopes are very similar. Control cases might score marginally worse on average given the same Memory rating, but this may well be due to noise. What's unclear to me about this data is why some control cases (who ostensibly have good memory) have way worse Memory ratings than the test cases. $\endgroup$ – cfh Apr 23 '15 at 10:34
  • $\begingroup$ How do you code variable group? It seems to me that you are predicting P(Y = A), in that case the coefficient should be negative. To check the relationship between group and score, you can cut the variable score into four groups (according to quartiles) and look at the contingency table - group, score. You can calculate row odds and see if the relationship is as expected or not. $\endgroup$ – Fimba Apr 23 '15 at 10:59
  • $\begingroup$ Do you have other variables or only "Memory" and "Score" to predict "Group" with logistic regression? $\endgroup$ – lanenok Apr 23 '15 at 11:00
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Since you have 6 variables, you may be facing the multicollinearity.

If the variables are highly correlated, your interpretation is never unambiguous. For example, the main risk of belonging to group B can be explained by another variable that is highly correlated with "Score" and the "Score" itself may only give small correction to this. Please check/post your correlation matrix or, even better, a matrix of scatterplots.

The highest correlation value 0.71 and VIF 2.6 are not so bad. But think about the interpretation.

When you write "for a 1 unit increase in cognitive test score" and look only at the coefficient $c_{Score}$, you implicitly presuppose that all other variables stay the same.

In your case, let us say "A" is 0.71 correlated with the "Score" and the coefficient $c_A$ is large positive, while $c_{Score}$ is small negative. Then for a 1 unit increase in "Score" you also have 0.71-unit increase in "A". The total effect is $0.71c_A + c_{Score}$, which is positive, the risk is increasing. The same logic for each correlated (and anticorrelated) variable. If "B" is anticorrelated with "Score" and $c_B$ is large negative, the total effect is again positive.

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