I have data $y$ which is the rate of success in $n$ trials. I also have covariates $X$ that I want to regress against $y$ to understand the relationship between them. I tried 2 different approaches. The first is to take the log of $y$ and apply OLS regression. The second is to apply logistic regression to $y$ directly.
I noticed that one theoretically important coefficient, call it $x_0$, is highly significant in both models but in OLS it is estimated as a strong negative effect and in logistic regression it is a small but positive effect.
How is this possible? The variable is binary and I am comparing the standardized coefficients from both models. My assumption is that $x_0$ either increases the rate or not and it should be of the same sign in both models.
UPDATE
When my only regressor is $x_0$ this doesn't happen, it's only when I add the other covariates (of which there is about $200$). I have pasted the output of the full models below, excluding the estimates of the other covariates.
OLS
Call:
lm(formula = fx, data = fit_data, weights = weights)
Weighted Residuals:
Min 1Q Median 3Q Max
-5.044177447 -0.616615480 0.070124542 0.692928438 4.539252302
Coefficients
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.33734283e+04 4.75932218e+03 -2.80994 0.00495879 **
x0 -1.06636833e-01 2.63420263e-02 -4.04816 5.1773e-05 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.01868241 on 25500 degrees of freedom
Multiple R-squared: 0.59443022, Adjusted R-squared: 0.591090233
F-statistic: 177.973843 on 210 and 25500 DF, p-value: < 2.220446e-16
GLM
Call:
glm(formula = fx, family = "binomial", data = fit_data, weights = weights)
Deviance Residuals:
Min 1Q Median 3Q Max
-103.51742894 -3.06309844 -0.24575206 2.27364818 165.10350046
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 2.83201590e+03 1.31990079e+03 2.14563 0.03190268 *
x0 1.77428670e-02 1.90814767e-03 9.29848 < 2.22e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for binomial family taken to be 1)
Null deviance: 9542439.111 on 25710 degrees of freedom
Residual deviance: 2224838.019 on 25500 degrees of freedom
AIC: 2340850.46
Number of Fisher Scoring iterations: 8
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