Logistic regression and ordinal independent variables

Question

I have found this post:

Yes. The coefficient reflects the change in log odds for each increment of change in the ordinal predictor. This (very common) model specification assumes the the predictor has a linear impact across its increments. To test the assumption, you can compare a model in which you use the ordinal variable as a single predictor to one in which you discretize the responses and treat them as multiple predictors (as you would if the variable were nominal); if the latter model doesn't result in a significantly better fit, then treating each increment as having a linear effect is reasonable.

– @dmk38 Dec 12 '10 at 5:21

Could you please tell me where can find something published that supports this claim? I am working with data and I would like to use ordinal independent variables in logistic regression.

Answer 1

As @Scortchi notes, you can also use orthogonal polynomials. Here is a quick demonstration in R:

set.seed(3406)
N      = 50
real.x = runif(N, 0, 10)
ord.x  = cut(real.x, breaks=c(0,2,4,6,8,10), labels=FALSE)
ord.x  = factor(ord.x, levels=1:5, ordered=TRUE)
lo.lin = -3 + .5*real.x
p.lin  = exp(lo.lin)/(1 + exp(lo.lin))
y.lin  = rbinom(N, 1, prob=p.lin)

mod.lin = glm(y.lin ~ ord.x, family=binomial)
summary(mod.lin)
# ...
# Coefficients:
#             Estimate Std. Error z value Pr(>|z|)   
# (Intercept)  0.05754    0.36635   0.157  0.87520   
# ord.x.L      2.94083    0.90304   3.257  0.00113 **
# ord.x.Q      0.94049    0.85724   1.097  0.27260   
# ord.x.C     -0.67049    0.77171  -0.869  0.38494   
# ord.x^4     -0.09155    0.73376  -0.125  0.90071   
# ...

Answer 2

Any good book on logistic regression will have this, although perhaps not in exactly those words. Try Agresti's Categorical Data Analysis for a very authoritative source.

It also follows from the definition of logistic regression (or other regressions). There are few methods explicitly for ordinal independent variables. The usual options are treating it as categorical (which loses the order) or as continuous (which makes the assumption stated in what you quoted). If you treat it as continuous then the program doing the analysis doesn't know it's ordinal. E.g. suppose your IV is "How much do you like President Obama?" and your answer choices are a Likert scale from 1. "Very much" to 5. "Not at all". If you treat this as continuous then (from the program's point of view) a "5" answer is 5 times a "1" answer. This may or may not be unreasonable.