# Skewed Distributions for Logistic Regression

I have been developing a logistic regression model based on retrospective data from a national trauma database of head injury in the UK. The key outcome is 30 day mortality (denoted as Outcome30 measure). Other measures across the whole database with published evidence of significant effect on outcome in previous studies include:

Yeardecimal - Date of procedure = 1994.0-2013.99
inctoCran - Time from head injury to craniotomy in minutes = 0-2880 (After 2880 minutes is defined as a separate diagnosis)
ISS - Injury Severity Score = 1-75
Age - Age of patient = 16.0-101.5
GCS - Glasgow Coma Scale = 3-15
Sex - Gender of patient = Male or Female
rcteyemi - Pupil reactivity (1 = neither, 2 = one, 3 = both)
neuroFirst2 - Location of admission (Neurosurgical unit or not)
Other - other traums (0 - No, 1 - Yes)
othopYN - Other operation required
LOS - Length of stay in days
LOSCC - Length of stay in critical care in days


When I conduct univariate analysis of the variables, I have conducted a logistic regression for each continuous variable. I am unable to model Yeardecimal however, with the following result:

> rcs.ASDH<-lrm(formula = Survive ~ Yeardecimal, data = ASDH_Paper1.1)
singular information matrix in lrm.fit (rank= 1 ).  Offending variable(s):
Yeardecimal
Error in lrm(formula = Survive ~ Yeardecimal, data = ASDH_Paper1.1) :
Unable to fit model using “lrm.fit”


However, the restricted cubic spline works:

> rcs.ASDH<-lrm(formula = Survive ~ rcs(Yeardecimal), data = ASDH_Paper1.1)
>
> rcs.ASDH

Logistic Regression Model

lrm(formula = Survive ~ rcs(Yeardecimal), data = ASDH_Paper1.1)

Model Likelihood     Discrimination    Rank Discrim.
Ratio Test            Indexes          Indexes
Obs          5998    LR chi2     106.61    R2       0.027    C       0.578
0           1281    d.f.             4    g        0.319    Dxy     0.155
1           4717    Pr(> chi2) <0.0001    gr       1.376    gamma   0.160
max |deriv| 2e-08                          gp       0.057    tau-a   0.052
Brier    0.165

Coef     S.E.    Wald Z Pr(>|Z|)
Intercept      -68.3035 45.8473 -1.49  0.1363
Yeardecimal      0.0345  0.0229  1.51  0.1321
Yeardecimal'     0.1071  0.0482  2.22  0.0262
Yeardecimal''   -2.0008  0.6340 -3.16  0.0016
Yeardecimal'''  11.3582  4.0002  2.84  0.0045


Could anyone explain why this is? I am nervous about using a mode complicated model if I am unable to model with a simpler approach.

I am currently using restricted cubic splines to model Age, ISS and Yeardecimal. Would anyone recommend any alternative approach?

• Required no, but it might be sensible to transform some of the more skewed predictors before creating the spline basis, to avoid having some data point with very high leverage. – Scortchi Dec 19 '14 at 14:01
• To see if @StasK's answer is correct, consider trying this with the vanilla glm function, as per r.789695.n4.nabble.com/… – shadowtalker Dec 20 '14 at 5:10

## 2 Answers

The date as a predictor may be failing because it is highly collinear with the constant. If you enter it as a year, it's variability is about 10/2000 = 0.005 (in fact less because most of your data are in the more recent years), and when squared it becomes 4e-6. When inverting a matrix with eigenvalues 1 and 4e-6, the package that you use may decide it is a zero in finite precision arithmetics, and throw this error message. The solution is simple -- center your data, at least approximately, by subtracting 2000 from the year.

• This resolved the problem, thank you for the explanation. – Dan Fountain Dec 19 '14 at 16:24
• I don't see that. That is a pretty nice range of dates. – Frank Harrell Dec 19 '14 at 21:05

Restricted cubic splines would be expected to work well here. You are worried slightly too much about marginal distributions of predictors.

Length of stay is in the wrong part of the causal pathway to use it as a predictor of death. And watch out for other operation required.

I don't see much value in univariable analyses.

• Thank you. I agree but traditionally univariate results are asked for in the literature. Any thoughts why the logistic regression is running an error for Yeardecimal when a linear term? – Dan Fountain Dec 19 '14 at 15:06
• Frank Harrell IS the literature. – StasK Dec 19 '14 at 15:26
• Apologies, I was referring to studies of outcome in sub-dural haematoma. – Dan Fountain Dec 19 '14 at 15:52