I have 4 predictors, and 1 binary response. I fitted a logistic regression model. A strange thing is that all the coefficient of the model are negative. Is that possible? Probably I did something wrong. My interpretation is that the odds ratio of either variable is less than 1. That is, neither variable actually do any good to the response. Even the intercept is negative. Please share your intelligence.
Yes, it is possible. Couple of things here. The direction of your predictors are critical to the interpretation; if they were scaled in the opposite direction, they would be positive. Second, it would seem on the face that your predictors lower the log odds given a unit change, which in itself might be good if say the outcome is death. Also, you might consider centering some of your predictors if the interecpt does not seem interpretable.
I didn't really get logistic regression until I thought about it this way:
- Picture the S curve (the logistic) going from -3 to 3
- Look at the coefficient estimate on your constant term. Mark it on the x axis of the S curve.
- Each coefficient moves you $\beta$ units along the X axis of the S curve. If you want to know what probability that corresponds to, go up from the X axis to the S curve and then over to the Y axis.
So, say that your intercept is, like, -.5. This is something like 40% probability (or so). Say your first beta is -.2 or something. This means that you follow the X axis over to -.7, which has a lower probability. Say you have a coefficient that is -5. That'll take you way out left, where chances are basically zero.
Its really pretty simple when you break it down simply.
From your description I see nothing out of the ordinary.
That the intercept is negative corresponds to that the estimated probability of the response is less than 50% when all model covariates equal zero.
If the coefficients of the model covariates are negative, then yes, the corresponding odds ratios are smaller than 1. If this is unexpected given your data, you may need to check how your covariates are coded.