Checking proportional hazards

The proportional-hazards test and the Schoenfeld residual plot for the model of the Cox example, the log-log plot, and what to do when the assumption fails, including a stratified model.

The Cox model of the previous example reports one hazard ratio per variable, valid at every time of follow-up. That is an assumption, and it has to be checked (see proportional hazards).

Step 1: the test

In the same Cox analysis, tick Proportional hazards test (under Model) and Schoenfeld residuals (under Plots):

jamovi syntax 
r <- prognosis::cox(data = lung, elapsed = time, event = event, eventLevel = "dead",
                    factors = vars(sex, ecog), covs = age, ph = TRUE, phPlot = TRUE,
                    timeUnit = "months", showExplanations = TRUE)

Proportional hazards test

χ² df p
sex 2.440 1 .118
ecog 2.871 2 .238
age 0.163 1 .686
Global 4.992 4 .288
Note. Test based on Schoenfeld residuals; a small p suggests that the hazard ratio changes over time (non-proportional hazards).

There is one row per variable (ECOG, with two coefficients, is tested on 2 df) and a global test for the whole model. A small p-value would suggest that the HR of that variable changes over time. Here none is small: p = 0.12 for sex, 0.24 for ECOG and 0.69 for age, and 0.29 globally. There is no evidence against proportional hazards, but “no evidence” is not proof: with 163 deaths the test can miss a moderate change. Look at the plot too.

Step 2: the residual plot

Figure 1: Scaled Schoenfeld residuals, one panel per coefficient, against time. Each dot is one death. The dashed red line is the coefficient β of the Cox model (constant by assumption); the blue curve, with its 95% band, is a smooth estimate of how the coefficient changes over time, β(t). The p-value in each title tests that coefficient alone.

How to read it:

  • The residuals of a factor fall in bands (one per level), so look at the smooth curve, not at the dots.
  • If the assumption holds, the curve is flat around the dashed line. A curve that rises or falls over time means the effect grows or fades; for a factor, a curve that crosses 0 means the HR crosses 1.
  • The band is wide where there are few deaths, at the start and especially at the right end. Bends there mean little.

Here the plot shows something the table does not. In roughly the first six months, the curves for both ECOG coefficients rise above the dashed line, and their bands briefly exclude it: the effect of ECOG looks stronger early. This fits the log-log plot of the Kaplan-Meier example, where the ECOG 1 curve starts below ECOG 0 (the very first deaths happened to be in ECOG 0) and overtakes it within a few months. Age shows a similar early wiggle. After that, the curves settle around the dashed lines. For sex the curve drifts upward (towards less protection) late in follow-up, where there are few deaths and the band is wide.

Why do the test and the plot seem to disagree? The test looks for a steady trend of β(t) over the whole follow-up, and a bump in the first months followed by a flat stretch is not a trend. The band is pointwise: over many time points, it will exclude the line somewhere by chance fairly often. The fair reading is: there is no strong evidence against proportional hazards, the single HR for ECOG is a reasonable summary, and there are hints that its effect is largest in the first half-year. A paper about ECOG itself could report that, for example with survival at 6 and 12 months or separate HRs for early and late follow-up (step 3).

The log-log plot of the Kaplan-Meier analysis is the graphical check for one factor at a time without adjustment; the residuals check each coefficient of the adjusted model, including continuous covariates such as age, which the log-log plot cannot handle.

Step 3: if the assumption fails

When a variable clearly violates proportional hazards, there are honest options:

  1. Stratify on it, if you do not need its HR: each level gets its own baseline hazard, and the other HRs are estimated within levels. See step 4.
  2. Report survival at fixed times (Kaplan-Meier or adjusted curves) instead of, or as well as, an HR. These need no proportional-hazards assumption. The restricted mean survival time is another summary that does not (Royston and Parmar 2013).
  3. Split follow-up: estimate one HR for the early period and another for the late period. The cut-off should be chosen on clinical grounds and reported. This, and models with time-varying coefficients, need software beyond this module (for example R’s survival package, which this module uses).
  4. Report the average HR with a clear statement that it averages an effect that changes over time, and show the residual plot. Acceptable when the change is mild.

What not to do: drop the variable because its test was significant, or switch to another test because its p-value is smaller.

Step 4: a stratified model

Suppose sex had failed the test. Move sex from Factors to Strata:

jamovi syntax 
rs <- prognosis::cox(data = lung, elapsed = time, event = event, eventLevel = "dead",
                     factors = ecog, covs = age, strata = sex, ph = TRUE)

Model summary

95% CI
N Events C-index Lower Upper
226 163 0.604 0.552 0.656
Note. Stratified by sex

Hazard ratios

95% CI
HR Lower Upper p
ecog ECOG 0 1.00
ECOG 1 1.44 0.972 2.14 .069
ECOG 2 2.46 1.570 3.84 < .001
age per unit 1.01 0.992 1.03 .277

Men and women now have separate baseline hazards of any shape, and sex has no HR any more. The ECOG hazard ratios (1.44 and 2.46) are close to those of the unstratified model (1.51 and 2.47), as expected when sex’s effect was in fact roughly proportional. The C-index is lower (0.604) because it now only compares patients of the same sex: the part of the prediction that came from sex is gone.

A stratified model cannot give adjusted survival curves: with a separate baseline per stratum, there is no single curve to adjust.

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References

Royston, Patrick, and Mahesh K. B. Parmar. 2013. “Restricted Mean Survival Time: An Alternative to the Hazard Ratio for the Design and Analysis of Randomized Trials with a Time-to-Event Outcome.” BMC Medical Research Methodology 13: 152. https://doi.org/10.1186/1471-2288-13-152.