My research includes a binomial variable as DV and numerous continuous variables as IV. From descriptive analysis I can tell that distribution for most IVs is not normal and couldn't find any transformation that can solve the problem. I applied log, 1/x, sqr, sqrt, … . Here is the histogram of one of the main variables. What type of analysis suits these data (my choice was logistic regression) and how can I solve the normality issue?
Solved – Normality distribution and logistic regression
data transformationlogisticnormality-assumption
Related Solutions
There are several things in your description that are a bit confusing, for example you state that taking the log transform reverses the direction of the coding, but the log by itself does not reverse coding.
Your main question seems to be that when you look at individual pairwise correlations the sign of the correlation is as expected, but some of the signs of the slopes in a multiple regression are opposite what you expect. This is not uncommon since the interpretation of slopes is much more complex in multiple regression models.
Consider this example (I read this recently, I don't get the credit for thinking of it): Collect data on the change in various peoples pockets, the variables to collect are the total value of the change (y), the total number of coins (x1), and the total number of coins that are not quarters, or if using non-US coins then number of coins not the highest common coin carried (x2). Generally x1 and x2 will both be positivly correlated with y, but if you do a multiple regression using both x1 and x2 then the slope on x2 will be negative because to increase the number of non-quarters without changing the total number of coins we need to trade quarters for other coins of lesser value which decreases y. You could have something similar happening with your data, does it really make sense to increase the religeous variable without the others changing? What is often more meaningful is to compare predicted outcomes for what would be considered common combinations of your predictor variables.
When referring to Box-Cox transformations there are really 2 concepts that look like they are being mixed up. The first is what the original paper was about, the methodology of finding a transformation within a family of transformations that gives the "best" transformation assuming the truth results in normal residuals with equal variance and a linear relationship. This is what you already did with the response (dependent) variable.
In the paper Box and Cox presented a family of transformations (actually the paper has more than 1, but variations on the main one are what people mainly refer to), this family of transformations (or variations on it) is also often called the Box-Cox transformations. If you want to apply one of this family of transformations to a predictor variable, then just plug the variable into the formula.
To determine what types of transformations to apply to predictor variables, the most important thing to use is knowledge about your data and the science behind it. How do you expect weight to change with age, height, or income? (I would be surprised if a Box-Cox transform is the best for any of these).
There are tools like spline fits, ACE, AVAS, and others that can suggest transformations, but you need to use knowledge and common sense to convert these into meaningful transformations.
Best Answer
No assumptions are made or needed about the marginal distribution of the independent variables in logistic regression. You can safely not worry about this.