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I am working with logistic regression in R by means of glm. I have fitted a logistic (0-1) regression model with seven predictor variables. I obtain a model where the variables have high p-values (>0.1) (not significative) but the r^2 of Mcfadden is high (0.6).

McFadden is equivalent to Pearson's r^2 in linear regression (I am not totally sure). Therefore, how it is possible to obtain a high correlation with no statistically significant variables?

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    $\begingroup$ What are the correlations like? How high are they? $\endgroup$ – user2974951 Jan 14 '19 at 8:44
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    $\begingroup$ What is the sample size? $\endgroup$ – AdamO Jan 14 '19 at 17:20
  • $\begingroup$ sample size is n=40. the correlations between the explicative variables are low or near to zero. $\endgroup$ – Genebrando Martinez Merino Jan 14 '19 at 23:02
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There are several possible reasons which could be responsible for the scenario you describe.

1. Collinearity among some or all of your predictor variables

Did you check if some of your predictor variables are engaged in collinearity? That might be one possible explanation. You can use the vif() function in the car package to check for collinearity. If you find any predictor variables which have high VIF (variance inflation factor) values - say larger than 5 - you may need to exclude some of those from your model and see if that resolves your collinearity.

2. Too many predictor variables in your model relative to the number of events

How many observations do you have in your model and how many events? (An event corresponds to Y = 1, where Y is your response variable.) There are rules of thumb for how many events you should have per predictor variable included in your model (e.g., 10 events per variable), which you can use to determine if your model includes too many predictor variables relative to the available number of events. If it does, you will need to include fewer predictor variables in your model.

3. Too few observations in your model

Perhaps the predictor variables included in your model have significant effects on the log odds of success (i.e., achieving a value of 1 for Y), but you just don't have enough observations in your model to detect these effects. To see if this might be the case, look at the confidence intervals for each predictor in relation to zero and see (i) how wide the confidence intervals are and (ii) how far the centers of the confidence intervals are from 0. (Use the confint() function to extract the 95% confidence intervals from your model and the coef() function to extract the centers of these intervals.) If the centers of the intervals are not too close to zero and the intervals are wide, that would suggest you need more data in your study to be able to detect the effects of interest. Recall that R works on the log odds scale by default.

There may be other reasons as well - perhaps others on this forum can highlight them.

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    $\begingroup$ To add to this just a bit, very little intuition can be gained from relating OLS to logistic models. Indeed in most cases a high Pearson correlation could indicate a strong logistic coefficient. However, even with orthogonal design (no correlation between predictors), while OLS gradually loses power due to degrees of freedom, logistic regression coefficients tend to perfect separation, and log ORs tend to a bias of a factor of 2 and the power of the model goes to 0 (SEs explode and all $p$-values > $\alpha$). $\endgroup$ – AdamO Jan 14 '19 at 17:30
  • $\begingroup$ McFadden's pseudo R squared in a logistic regression model is not equivalent to the multiple coefficient of determination R squared in a linear regression model, though it is similar in nature. $\endgroup$ – Isabella Ghement Jan 14 '19 at 18:02
  • $\begingroup$ certainly, the sample size is small n=40. Obviously, the number of 1 per variable is low $\endgroup$ – Genebrando Martinez Merino Jan 14 '19 at 23:03

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