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I ran a logistic regression with only 1 independent variable and came with an 18.59 odds ratio with a c statistic of .50.

After adding 4 variables that independent variable odds ratio decreased from 18.59 to 9.9 and the c statistic of the model increased to .68.

Does this mean that if I reported just the odds ratio of single variables that they would be misleading since the c statistic is only .50? Is it better to report the odds ratio of 9.9 instead of 18.59 since the model has a higher c statistic even though the odds ratio is lower?

I've seen papers where they have reported only odds ratio for all independent variables independently and then others where they have run a logistic regression. Just trying to figure out what one is better.

Thank you!

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  • $\begingroup$ I suggest you report both adjusted and unadjusted. $\endgroup$ – mdewey May 11 '17 at 15:14
  • $\begingroup$ Would the adjusted be odds ratios found in the model and unadjusted be through the cross tabs? $\endgroup$ – Mike May 11 '17 at 16:04
  • $\begingroup$ Yes. By the way that would be thought a very high odds ratio in the field that I work in (health) but your mileage may vary. $\endgroup$ – mdewey May 11 '17 at 16:24
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It depends on the inference you want to make. If you want to predict or describe, the independent logistic regressions are fine. If you want to explain or make a causal inference, the independent regressions are majorly problematic. Decreasing odds ratios with each additional predictor means your predictors are correlated with each other and have the same relationship with the outcome. The c-statistic increases because your model becomes "better" with each additional predictor you include.

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  • $\begingroup$ I ran a correlation of all the ivs and the highest correlation between the variable that obtained a 18.59 then a 9.9 odds ratio and another variable was .06. Is this a strong enough correlation to create that? I want to predict what variables will lead do a greater liklihood of donations. $\endgroup$ – Mike May 11 '17 at 15:35
  • $\begingroup$ This answer is mostly right, but logistic regression functions slightly differently from OLS. Making a long story short, the coefficients of existing predictors will change even if the new predictors are uncorrelated to them, simply by virtue of how the total variance in the model is constrained. The total variance in Y is a mix of the observed and unobserved factors, and the unobserved ones are constrained by the logistic link function to be constant, so any increase in the explanatory power of the observed part will force the scale of the coefficients to change, changing their magnitudes. $\endgroup$ – Kenji May 12 '17 at 21:20

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