Biostatistics III in R

In biostat3: Utility Functions, Datasets and Extended Examples for Survival Analysis

Exercise 10. Examining the proportional hazards hypothesis (localised melanoma)

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Load the diet data using time-on-study as the timescale with a maximum of 10 years follow-up.

library('knitr')
read_chunk('../q10.R')
opts_chunk$set(cache=FALSE)

You may have to install the required packages the first time you use them. You can install a package by install.packages("package_of_interest") for each package you require.

Load melanoma data and explore it.

(a) ##

If we look at the hazard curves, at their peak the ratio is approximately $0.038/0.048\approx0.79$. The ratio is similar at other follow-up times.

(b) ##

There is no strong evidence against an assumption of proportional hazards since we see (close to) parallel curves when plotting the instantaneous cause-specific hazard on the log scale.

(c) ##

If the proportional hazards assumption is appropriate then we should see parallel lines. This looks okay; we shouldn’t put too much weight on the fact that the curves cross early in the follow-up since there are so few deaths there. The difference between the two log-cumulative hazard curves is similar during the part of the follow-up where we have the most information (most deaths). Note that these curves are not based on the estimated Cox model (i.e., they are unadjusted).

(d) ##

The estimated hazard ratio from the Cox model is $0.78$ which is similar (as it should be) to the estimate made by looking at the hazard function plot.

(e) ##

The plot of the scaled Schoenfeld residuals for the effect of period. Under proportional hazards, the smoother will be a horizontal line. The line does vary (with possibly an increasing log hazard ratio at around 4 years), but not in a linear pattern, which is consistent with the p-value of 0.10 for linear trend.

(f) ##

Based on the tests for linear trends, we expect a trend for the log hazard ratio comparing ages less than 45 years with those aged 75 years and over. All three log hazard ratios tend to decrease by time of diagnosis, while the trend is more marked for the oldest age group (75+). There is evidence for non-proportionality, particularly for those patients aged 75 years and over.

(g) ##

It seems that there is evidence of non-proportional hazards by age (particularly for the comparison of the oldest to youngest) but not for calendar period. The plot of Schoenfeld residuals suggested non-proportionality for period but this was not statistically significant.

(h) ##

The hazard ratios for age in the top panel are for the first two years subsequent to diagnosis. To obtain the hazard ratios for the period two years or more following diagnosis we multiply the hazard ratios in the top and bottom panel. That is, during the first two years following diagnosis patients aged 75 years or more at diagnosis have 5.4 times higher cancer-specific mortality than patients aged 0–44 at diagnosis. During the period two years or more following diagnosis the corresponding hazard ratio is $5.4 \times 0.49=2.66$.\ Using survSplit to split on time will give you the same results as above. We see that the age:follow up interaction is statistically significant.

The regression equation for the cox2p8Split1 model is \begin{align} h(t|\text{year8594},\text{sex},\text{agegrp},\text{fu}) &= h_0(t) \exp(\beta_1 I(\text{sex}=\text{"Female"})+\beta_2 I(\text{year8594}=\text{"Diagnosed 85-94"})+\ &\qquad\beta_3 I(\text{agegrp}=\text{"45-59"})+\beta_4 I(\text{agegrp}=\text{"60-74"})+\beta_5 I(\text{agegrp}=\text{"75+"}) + \ &\qquad \beta_6 I(\text{agegrp}=\text{"45-59"} \&\ \text{fu}=2)+\beta_7 I(\text{agegrp}=\text{"60-74"} \&\ \text{fu}=2)+\beta_8 I(\text{agegrp}=\text{"75+"} \&\ \text{fu}=2)) \end{align} where $h(t|\text{year8594},\text{sex},\text{agegrp},\text{fu})$ is the hazard at time $t$ given covariates $\text{year8594}$, $\text{sex}$, $\text{agegrp}$ and $\text{fu}$, with baseline hazard $h_0(t)$ and regression coefficients representing log hazard ratios for $\beta_1$ for females, $\beta_2$ for the calendar period 1985--1994, $\beta_3$ for those aged 45--59 years at diagnosis, $\beta_4$ for those aged 60--74 years and $\beta_5$ for those aged 75 years and over, with interaction terms for the change in log hazard ratio for the second follow-up period being $\beta_6$ for those aged 45--59 years at diagnosis, $\beta_7$ for those aged 60--74 years and $\beta_8$ for those aged 75 years and over.

(i) ##

          0–2 years   2+ years

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Agegrp0-44 1.00 1.00 Agegrp45-59 1.70 1.23 Agegrp60-74 2.46 1.70 Agegrp75+ 5.42 2.67

The regression equation for the cox2p8Split2 model is \begin{align} h(t|\text{year8594},\text{sex},\text{agegrp},\text{fu}) &= h_0(t) \exp(\beta_1 I(\text{sex}=\text{"Female"})+\beta_2 I(\text{year8594}=\text{"Diagnosed 85-94"})+\ &\qquad\beta_3 I(\text{agegrp}=\text{"45-59"} \&\ \text{fu}=1)+\beta_4 I(\text{agegrp}=\text{"45-59"} \&\ \text{fu}=2)+\beta_5 I(\text{agegrp}=\text{"60-74"} \&\ \text{fu}=1) + \ &\qquad \beta_6 I(\text{agegrp}=\text{"60-74"} \&\ \text{fu}=2)+\beta_7 I(\text{agegrp}=\text{"75+"} \&\ \text{fu}=1)+\beta_8 I(\text{agegrp}=\text{"75+"} \&\ \text{fu}=2)) \end{align} where $h(t|\text{year8594},\text{sex},\text{agegrp},\text{agegrp},\text{fu})$ is the hazard at time $t$ given covariates $\text{year8594}$, $\text{sex}$ and $\text{agegrp}$, $\text{agegrp}$ and $\text{fu}$, with baseline hazard $h_0(t)$ and regression coefficients representing log hazard ratios for $\beta_1$ for the calendar period 1985--1994, $\beta_2$ for females, with log hazard ratios for the first and second follow-up period being $\beta_3$ and $\beta_4$ for those aged 45--59 years at diagnosis, $\beta_5$ and $\beta_6$ for those aged 60--74 years and $\beta_7$ and $\beta_8$ for those aged 75 years and over.

We can also use the tt argument in coxph for modelling for time-varying effects:

The regression equation for the cox2p8tvct model is \begin{align} h(t|\text{year8594},\text{sex},\text{agegrp}) &= h_0(t) \exp(\beta_1 I(\text{sex}=\text{"Female"})+\beta_2 I(\text{year8594}=\text{"Diagnosed 85-94"})+\ &\qquad\beta_3 I(\text{agegrp}=\text{"45-59"})+\beta_4 I(\text{agegrp}=\text{"60-64"}) + \beta_5 I(\text{agegrp}=\text{"75+"}) + \ &\qquad \beta_6 I(\text{agegrp}=\text{"45-59"}) t +\beta_7 I(\text{agegrp}=\text{"60-74"})t + \beta_8 I(\text{agegrp}=\text{"75+"}) t) \end{align} where $h(t|\text{year8594},\text{sex},\text{agegrp},\text{agegrp})$ is the hazard at time $t$ given covariates $\text{year8594}$, $\text{sex}$ and $\text{agegrp}$ and $\text{agegrp}$, with baseline hazard $h_0(t)$ and regression coefficients representing log hazard ratios for $\beta_1$ for the calendar period 1985--1994, $\beta_2$ for females, with log hazard ratios at time 0 for those aged 45--59 years, 60--74 years and 75 years and over are $\beta_3$, $\beta_4$ and $\beta_5$, respectively, while the change in log hazard ratios per year for those aged those aged 45--59 years, 60--74 years and 75 years and over are $\beta_3$, $\beta_4$ and $\beta_5$, respectively.

The hazard ratio for model cox2p8tvct for the those aged 75 years and over compared with those aged less than 45 years is \begin{align} \frac{h(t|\text{year8594},\text{sex},\text{agegrp}=\text{"75+"})}{h(t|\text{year8594},\text{sex},\text{agegrp}=\text{"0-44"})} &= \frac{h_0(t) \exp(\beta_1 I(\text{sex}=\text{"Female"})+\beta_2 I(\text{year8594}=\text{"Diagnosed 85-94"})+ \beta_5 + \beta_8 t)}{h_0(t)\exp(\beta_1 I(\text{sex}=\text{"Female"})+\beta_2 I(\text{year8594}=\text{"Diagnosed 85-94"}))} \ &= \exp(\beta_5 + \beta_8 t) \end{align}

We have shown several ways to use the tt functionality for a factor variable, including using different tt arguments for each factor level (as per model cox2p8tvc2) and using a tt term that returns a set of columns (as per model cox2p8tvct). We have used the lincom function to estimate the hazard ratio for agegrp75+. We will later describe a more flexible approach to modelling time-dependent effects using stpm2.

(j) ##

We use the predictnl function from rstpm2 to calculate the hazard ratios and confidence intervals based on the delta method. We see clear evidence for a rate ratio for those aged 75 years and over compared with those aged less than 45 years declining by time since diagnosis. Fitting for an interaction between age 75+ and time from diagnosis, we get a rate ratio for those aged 75 years at 0.5 years from diagnosis of 5.19 (95% confidence interval (CI): 3.56, 7.29), while at 2 years the rate ratio is 3.82 (95% CI: 3.00, 4.86).

biostat3 documentation built on Oct. 29, 2024, 5:07 p.m.