Solved – Violation of linearity assumption in Logistic Regression

goodness of fitlogisticmodelingregressionsplines

I built a logistic model with 7 continuous independent variable and 8 categorical (dummy) variables in R. I tested for the linearity assumption using the Box Tidwell test and all my variables show high significance.

I am wondering now, if the assumption is violated what it would mean for the coefficient estimates and if I still can run the model with caution.

I read about the approach of using regression splines but I am confused how I would determine those and interpret the model then.

Best Answer

Well, if you have indication (you say Box-Tidwell indicated non-linearity), you must be careful with interpretations. The best solution is probably to model in such a way that the model assumptions is pretty probable to hold, upfront.

With continuous predictors in logistic regression, linearity is a pretty strong assumption. My preferred choice would be to model your seven continuous predictors using regression splines. In R that would look like:

    library(splines)
    mod <- glm(response ~ ns(x1, df=4)+ns(x2,df=4)+ ..., <etc>)

Related Solutions

Solved – Testing nonlinearity in logistic regression (or other forms of regression)

I would suggest to use restricted cubic splines (rcs in R, see the Hmisc and Design packages for examples of use), instead of adding power of $X$ in your model. This approach is the one that is recommended by Frank Harrell, for instance, and you will find a nice illustration in his handouts (§2.5 and chap. 9) on Regression Modeling Strategies (see the companion website).

You can compare the results with your Box-Tidwell test by using the boxTidwell() in the car package.

Transforming continuous predictors into categorical ones is generally not a good idea, see e.g. Problems Caused by Categorizing Continuous Variables.

Solved – Investigating robustness of logistic regression against violation of linearity of logit

The linearity assumption is so commonly violated in regression that it should be called a surprise rather than an assumption. Like other regression models, the logistic model is not robust to nonlinearity when you falsely assume linearity. Rather than detect nonlinearity using residuals or omnibus goodness of fit tests, it is better to use direct tests. For example, expand continuous predictors using regression splines and do a composite test of all the nonlinear terms. Better still don't test the terms and just expect nonlinearity. This approach is much better than trying different single-slope choices of transformations such as square root, log, etc., because statistical inference arise after such analyses will be incorrect because it does not have large enough numerator degrees of freedom.

Here's an example in R.

require(rms)
f <- lrm(y ~ rcs(age,4) + rcs(blood.pressure,5) + sex + rcs(height,4))
# Fits restricted cubic splines in 3 variables with default knots
# 4, 5, 4 knots = 2, 3, 2 nonlinear terms
Function(f)   # display algebraic form of fit
anova(f)      # obtain individual + combined linearity tests

Best Answer

Related Solutions

Solved – Testing nonlinearity in logistic regression (or other forms of regression)

Solved – Investigating robustness of logistic regression against violation of linearity of logit

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