# Interpreting mortality trends shown as relative hazard

I am analysing trends in survival. I was recommended to use the following package and parameterization.

I have a model as follows (done with rms package):

survobj12 = with(data, Surv(time_12months,status_12months))

model2 = cph(survobj12 ~ rcs(time_to_diagnosis_in_years, 4), data=data,x=TRUE ,

y=TRUE)



Result

Cox Proportional Hazards Model

cph(formula = survobj12 ~ rcs(time_to_diagnosis_in_years, 4), data = data,
x = TRUE, y = TRUE)

Model Tests       Discrimination
Indexes
Obs     11491    LR chi2      0.56    R2       0.000
Events   3534    d.f.            3    Dxy      0.009
Center 0.0343    Pr(> chi2) 0.9048    g        0.013
Score chi2   0.56    gr       1.013
Pr(> chi2) 0.9053

Coef    S.E.   Wald Z Pr(>|Z|)
time_to_diagnosis_in_years    0.0145 0.0330  0.44  0.6597
time_to_diagnosis_in_years'  -0.0271 0.0946 -0.29  0.7741
time_to_diagnosis_ine_years''  0.0741 0.2853  0.26  0.7952


This is the plotted result: #plot ggplot(Predict(model2), vnames = "names")+ xlab("Time from diagnosis in years")+

This seems really nice; however, how should I interpret the plot? Is relative hazard equal to hazard ratios? The interpretation of hazard ratios is quite simple: e.g. HR = 2 means that females have two times higher risk for dying than males during a certain time period. However, my plot does not have a reference category at all.

So, are these two estimates (Relative Hazard and Hazard Ratio) equal to each other and what is the reference on my plot/analysis?

• Please add to your question the specific command that you used to produce the plot that you show, as there might be some built-in default settings specific to the software. It would help if you could add to your question the result of the cph model and of anova() applied to the model. Finally, what specifically do you mean by "these two estimates"?
– EdM
Feb 6, 2021 at 16:04
• Good points! Edited the OP. Feb 6, 2021 at 16:19

The most important point in this example is that your model isn't different from a null model. The p-values for both the likelihood-ratio (LR) and score tests for the model are very high, over 0.9. With a 4-knot restricted cubic spline you are estimating coefficients for the equivalent of 3 predictors, with no values significantly different from zero here.

For the plot itself, it's important to learn about what your software tools are doing; see the manual page for the rms Predict and datadist functions. An example on the manual page for Predict shows that the reference value for a continuous predictor is taken from the second value in the limits vector for that predictor in the datadist summary of the data. Resetting that value allows you to choose any reference value that you want; in that manual page example:

ddist$$limits$$age[2] <- 30


resets the reference value for a variable age to 30. If you don't specify your own reference, the manual page for datadist shows that the median value is chosen for a continuous predictor:

Adjustment values are 0 for binary variables, the most frequent category (or optionally the first category level) for categorical (factor) variables, the middle level for ordered factor variables, and medians for continuous variables.

That adjustment/reference value is shown in this example as the time value at which the "log Relative Hazard" crosses 0. The plot then shows changes in "log Hazard" relative to that reference time value as you go to lower or higher times. If your model had more predictors, the 0 reference for the "log Relative Hazard" would also be based on the reference values for all the other predictors, so depending on the data and the model this type of plot might never cross 0.

• Thank you! I use this plot for showing/illustrating non-significant temporal trend, using plotting of the Cox model's results. Adjusting is not necessary. Is this a problem that my model is not significant from a null model? IMHO it is not a problem. Mar 3, 2021 at 8:26
• @st4co4 Finding this model not to be significantly different from a null model means there is no evidence of a significant association of hazard with time, as including that predictor is no different than including no predictor at all. That seems to have been exactly what you wished to examine. The result would be more compelling if your model also incorporated other covariates associated with survival, however. Also, be careful that the way you defined time_to_diagnosis_in_years isn't leading to survivorship bias.
– EdM
Mar 3, 2021 at 15:55