Exercise 10. Examining the proportional hazards hypothesis (localised melanoma)
Load the diet data using time-on-study as the timescale with a maximum of 10 years follow-up.
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.
library(biostat3)
library(dplyr) # for data manipulation
library(ggplot2)
library(car) # car::linearHypothesis -> biostat3::lincomLoad melanoma data and explore it.
localised <- dplyr::filter(biostat3::melanoma, stage == "Localised") %>%
dplyr::mutate(death_cancer = ifelse( status == "Dead: cancer" & surv_mm <= 120, 1, 0), #censoring for > 120 months
trunc_yy = pmin(surv_mm/12,10)) #scale to years and truncate to 10 years(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.
# Using muhaz2 to smooth the Kaplan-Meier hazards by strata
hazDiaDate <- muhaz2(Surv(trunc_yy,death_cancer)~year8594, data=localised)
hazDiaDateDf <- as.data.frame(hazDiaDate)
## Comparing hazards
plot(hazDiaDate, haz.scale=1000,
xlab="Time since diagnosis (years)",
ylab="Hazard per 1000 person-years")
# or using ggplot2
ggplot(hazDiaDateDf, aes(x=x, y=y*1000, colour= strata)) + geom_line() +
xlab("Time since diagnosis (years)") +
ylab("Hazard per 1000 person-years")
(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).
## Calculating -log cumulative hazards per strata
survfit1 <- survfit(Surv(trunc_yy,death_cancer)~year8594, data=localised)
plot(survfit1, col=1:2, fun=function(S) -log(-log(S)), log="x",
xlab="log(time)", ylab="-log(H)")
legend("topright",legend=levels(localised$year8594),col=1:2,lty=1)
(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.
# Cox regression with time-since-entry as the timescale
# Note that R uses the Efron method for approximating the likelihood in the presence of ties
# whereas Stata (and some other software) use the Breslow method
cox1 <- coxph(Surv(trunc_yy, death_cancer==1) ~ year8594, data=localised)
summary(cox1)## Call:
## coxph(formula = Surv(trunc_yy, death_cancer == 1) ~ year8594,
## data = localised)
##
## n= 5318, number of events= 960
##
## coef exp(coef) se(coef) z Pr(>|z|)
## year8594Diagnosed 85-94 -0.25297 0.77649 0.06579 -3.845 0.000121 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## exp(coef) exp(-coef) lower .95 upper .95
## year8594Diagnosed 85-94 0.7765 1.288 0.6825 0.8834
##
## Concordance= 0.533 (se = 0.008 )
## Likelihood ratio test= 14.83 on 1 df, p=1e-04
## Wald test = 14.78 on 1 df, p=1e-04
## Score (logrank) test = 14.86 on 1 df, p=1e-04(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 is not, however, perfectly horizontal; it appears that the effect of period is greater earlier in the follow-up.
cox2 <- coxph(Surv(trunc_yy, death_cancer==1) ~ sex + year8594 + agegrp, data=localised)
summary(cox2)## Call:
## coxph(formula = Surv(trunc_yy, death_cancer == 1) ~ sex + year8594 +
## agegrp, data = localised)
##
## n= 5318, number of events= 960
##
## coef exp(coef) se(coef) z Pr(>|z|)
## sexFemale -0.53061 0.58825 0.06545 -8.107 5.19e-16 ***
## year8594Diagnosed 85-94 -0.33339 0.71649 0.06618 -5.037 4.72e-07 ***
## agegrp45-59 0.28283 1.32688 0.09417 3.003 0.00267 **
## agegrp60-74 0.62006 1.85904 0.09088 6.823 8.90e-12 ***
## agegrp75+ 1.21801 3.38045 0.10443 11.663 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## exp(coef) exp(-coef) lower .95 upper .95
## sexFemale 0.5882 1.7000 0.5174 0.6688
## year8594Diagnosed 85-94 0.7165 1.3957 0.6293 0.8157
## agegrp45-59 1.3269 0.7536 1.1032 1.5959
## agegrp60-74 1.8590 0.5379 1.5557 2.2215
## agegrp75+ 3.3804 0.2958 2.7547 4.1483
##
## Concordance= 0.646 (se = 0.009 )
## Likelihood ratio test= 212.7 on 5 df, p=<2e-16
## Wald test = 217.9 on 5 df, p=<2e-16
## Score (logrank) test = 226.8 on 5 df, p=<2e-16## Plot of the scaled Schoenfeld residuals for calendar period 1985–94.
## The smooth line shows the estimated log hazard ratio as a function of time.
cox2.phtest <- cox.zph(cox2, transform="identity") #Stata appears to be using 'identity'
plot(cox2.phtest[2],resid=TRUE, se=TRUE, main="Schoenfeld residuals", ylim=c(-4,4))
(f)
No solution written for this part.
(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.
## chisq df p
## sex 1.17 1 0.2784
## year8594 1.57 1 0.2096
## agegrp 15.93 3 0.0012
## GLOBAL 20.45 5 0.0010(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.
melanoma2p8Split <- survSplit(localised, cut=c(2), end="trunc_yy", start="start",
event="death_cancer", episode="fu") %>%
mutate(fu = as.factor(fu))
##Tabulate ageband including risk_time
melanoma2p8Split %>% select(id, start, trunc_yy) %>% filter(id<=3) %>% arrange(id, trunc_yy)## id start trunc_yy
## 1 1 0 2.000000
## 2 1 2 2.208333
## 3 2 0 2.000000
## 4 2 2 4.625000
## 5 3 0 2.000000
## 6 3 2 10.000000## sex age stage mmdx yydx surv_mm surv_yy status subsite
## 1 Female 81 Localised 2 1981 26.5 2.5 Dead: other Head and Neck
## 2 Female 81 Localised 2 1981 26.5 2.5 Dead: other Head and Neck
## 3 Female 75 Localised 9 1975 55.5 4.5 Dead: other Head and Neck
## 4 Female 75 Localised 9 1975 55.5 4.5 Dead: other Head and Neck
## 5 Female 78 Localised 2 1978 177.5 14.5 Dead: other Limbs
## 6 Female 78 Localised 2 1978 177.5 14.5 Dead: other Limbs
## year8594 dx exit agegrp id ydx yexit start trunc_yy
## 1 Diagnosed 75-84 1981-02-02 1983-04-20 75+ 1 1981.088 1983.298 0 2.000000
## 2 Diagnosed 75-84 1981-02-02 1983-04-20 75+ 1 1981.088 1983.298 2 2.208333
## 3 Diagnosed 75-84 1975-09-21 1980-05-07 75+ 2 1975.720 1980.348 0 2.000000
## 4 Diagnosed 75-84 1975-09-21 1980-05-07 75+ 2 1975.720 1980.348 2 4.625000
## 5 Diagnosed 75-84 1978-02-21 1992-12-07 75+ 3 1978.140 1992.934 0 2.000000
## 6 Diagnosed 75-84 1978-02-21 1992-12-07 75+ 3 1978.140 1992.934 2 10.000000
## death_cancer fu
## 1 0 1
## 2 0 2
## 3 0 1
## 4 0 2
## 5 0 1
## 6 0 2cox2p8Split1 <- coxph(Surv(start, trunc_yy, death_cancer) ~ sex + year8594 + agegrp*fu,
data=melanoma2p8Split)
summary(cox2p8Split1)## Call:
## coxph(formula = Surv(start, trunc_yy, death_cancer) ~ sex + year8594 +
## agegrp * fu, data = melanoma2p8Split)
##
## n= 9856, number of events= 960
##
## coef exp(coef) se(coef) z Pr(>|z|)
## sexFemale -0.52742 0.59012 0.06543 -8.061 7.58e-16 ***
## year8594Diagnosed 85-94 -0.33548 0.71499 0.06623 -5.065 4.08e-07 ***
## agegrp45-59 0.53058 1.69992 0.19634 2.702 0.00689 **
## agegrp60-74 0.90046 2.46074 0.18741 4.805 1.55e-06 ***
## agegrp75+ 1.68918 5.41503 0.19175 8.809 < 2e-16 ***
## fu2 NA NA 0.00000 NA NA
## agegrp45-59:fu2 -0.32093 0.72547 0.22382 -1.434 0.15161
## agegrp60-74:fu2 -0.36715 0.69270 0.21467 -1.710 0.08720 .
## agegrp75+:fu2 -0.70783 0.49271 0.23207 -3.050 0.00229 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## exp(coef) exp(-coef) lower .95 upper .95
## sexFemale 0.5901 1.6946 0.5191 0.6709
## year8594Diagnosed 85-94 0.7150 1.3986 0.6280 0.8141
## agegrp45-59 1.6999 0.5883 1.1569 2.4978
## agegrp60-74 2.4607 0.4064 1.7043 3.5529
## agegrp75+ 5.4150 0.1847 3.7186 7.8853
## fu2 NA NA NA NA
## agegrp45-59:fu2 0.7255 1.3784 0.4678 1.1250
## agegrp60-74:fu2 0.6927 1.4436 0.4548 1.0550
## agegrp75+:fu2 0.4927 2.0296 0.3126 0.7765
##
## Concordance= 0.645 (se = 0.009 )
## Likelihood ratio test= 222.5 on 8 df, p=<2e-16
## Wald test = 224.5 on 8 df, p=<2e-16
## Score (logrank) test = 238 on 8 df, p=<2e-16cox2p8Split1b <- coxph(Surv(start, trunc_yy, death_cancer) ~ sex + year8594 + agegrp +
I(agegrp=="45-59" & fu=="2") + I(agegrp=="60-74" & fu=="2") +
I(agegrp=="75+" & fu=="2"), data=melanoma2p8Split)
summary(cox2p8Split1b)## Call:
## coxph(formula = Surv(start, trunc_yy, death_cancer) ~ sex + year8594 +
## agegrp + I(agegrp == "45-59" & fu == "2") + I(agegrp == "60-74" &
## fu == "2") + I(agegrp == "75+" & fu == "2"), data = melanoma2p8Split)
##
## n= 9856, number of events= 960
##
## coef exp(coef) se(coef) z Pr(>|z|)
## sexFemale -0.52742 0.59012 0.06543 -8.061 7.58e-16 ***
## year8594Diagnosed 85-94 -0.33548 0.71499 0.06623 -5.065 4.08e-07 ***
## agegrp45-59 0.53058 1.69992 0.19634 2.702 0.00689 **
## agegrp60-74 0.90046 2.46074 0.18741 4.805 1.55e-06 ***
## agegrp75+ 1.68918 5.41503 0.19175 8.809 < 2e-16 ***
## I(agegrp == "45-59" & fu == "2")TRUE -0.32093 0.72547 0.22382 -1.434 0.15161
## I(agegrp == "60-74" & fu == "2")TRUE -0.36715 0.69270 0.21467 -1.710 0.08720 .
## I(agegrp == "75+" & fu == "2")TRUE -0.70783 0.49271 0.23207 -3.050 0.00229 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## exp(coef) exp(-coef) lower .95 upper .95
## sexFemale 0.5901 1.6946 0.5191 0.6709
## year8594Diagnosed 85-94 0.7150 1.3986 0.6280 0.8141
## agegrp45-59 1.6999 0.5883 1.1569 2.4978
## agegrp60-74 2.4607 0.4064 1.7043 3.5529
## agegrp75+ 5.4150 0.1847 3.7186 7.8853
## I(agegrp == "45-59" & fu == "2")TRUE 0.7255 1.3784 0.4678 1.1250
## I(agegrp == "60-74" & fu == "2")TRUE 0.6927 1.4436 0.4548 1.0550
## I(agegrp == "75+" & fu == "2")TRUE 0.4927 2.0296 0.3126 0.7765
##
## Concordance= 0.645 (se = 0.009 )
## Likelihood ratio test= 222.5 on 8 df, p=<2e-16
## Wald test = 224.5 on 8 df, p=<2e-16
## Score (logrank) test = 238 on 8 df, p=<2e-16The 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 | |
|---|---|---|
| Agegrp0-44 | 1.00 | 1.00 |
| Agegrp45-59 | 1.70 | 1.23 |
| Agegrp60-74 | 2.46 | 1.70 |
| Agegrp75+ | 5.42 | 2.67 |
cox2p8Split2 <- coxph(Surv(start, trunc_yy, death_cancer) ~ sex + year8594 + fu + fu:agegrp, data=melanoma2p8Split)
summary(cox2p8Split2)## Call:
## coxph(formula = Surv(start, trunc_yy, death_cancer) ~ sex + year8594 +
## fu + fu:agegrp, data = melanoma2p8Split)
##
## n= 9856, number of events= 960
##
## coef exp(coef) se(coef) z Pr(>|z|)
## sexFemale -0.52742 0.59012 0.06543 -8.061 7.58e-16 ***
## year8594Diagnosed 85-94 -0.33548 0.71499 0.06623 -5.065 4.08e-07 ***
## fu2 NA NA 0.00000 NA NA
## fu1:agegrp45-59 0.53058 1.69992 0.19634 2.702 0.00689 **
## fu2:agegrp45-59 0.20965 1.23325 0.10774 1.946 0.05167 .
## fu1:agegrp60-74 0.90046 2.46074 0.18741 4.805 1.55e-06 ***
## fu2:agegrp60-74 0.53331 1.70456 0.10479 5.089 3.59e-07 ***
## fu1:agegrp75+ 1.68918 5.41503 0.19175 8.809 < 2e-16 ***
## fu2:agegrp75+ 0.98135 2.66806 0.13157 7.458 8.75e-14 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## exp(coef) exp(-coef) lower .95 upper .95
## sexFemale 0.5901 1.6946 0.5191 0.6709
## year8594Diagnosed 85-94 0.7150 1.3986 0.6280 0.8141
## fu2 NA NA NA NA
## fu1:agegrp45-59 1.6999 0.5883 1.1569 2.4978
## fu2:agegrp45-59 1.2332 0.8109 0.9985 1.5232
## fu1:agegrp60-74 2.4607 0.4064 1.7043 3.5529
## fu2:agegrp60-74 1.7046 0.5867 1.3881 2.0932
## fu1:agegrp75+ 5.4150 0.1847 3.7186 7.8853
## fu2:agegrp75+ 2.6681 0.3748 2.0616 3.4530
##
## Concordance= 0.645 (se = 0.009 )
## Likelihood ratio test= 222.5 on 8 df, p=<2e-16
## Wald test = 224.5 on 8 df, p=<2e-16
## Score (logrank) test = 238 on 8 df, p=<2e-16The 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:
## Alternative approach using tt():
## http://cran.r-project.org/web/packages/survival/vignettes/timedep.pdf
cox2p8tvc2 <- coxph(Surv(trunc_yy, death_cancer) ~ sex + year8594 + agegrp +
tt(agegrp=="45-59") + tt(agegrp=="60-74") + tt(agegrp=="75+"),
data=localised,
tt = function(x, t, ...) x*(t>=2))
summary(cox2p8tvc2)## Call:
## coxph(formula = Surv(trunc_yy, death_cancer) ~ sex + year8594 +
## agegrp + tt(agegrp == "45-59") + tt(agegrp == "60-74") +
## tt(agegrp == "75+"), data = localised, tt = function(x, t,
## ...) x * (t >= 2))
##
## n= 5318, number of events= 960
##
## coef exp(coef) se(coef) z Pr(>|z|)
## sexFemale -0.52742 0.59012 0.06543 -8.061 7.58e-16 ***
## year8594Diagnosed 85-94 -0.33548 0.71499 0.06623 -5.065 4.08e-07 ***
## agegrp45-59 0.53058 1.69992 0.19634 2.702 0.00689 **
## agegrp60-74 0.90046 2.46074 0.18741 4.805 1.55e-06 ***
## agegrp75+ 1.68918 5.41503 0.19175 8.809 < 2e-16 ***
## tt(agegrp == "45-59") -0.32093 0.72547 0.22382 -1.434 0.15161
## tt(agegrp == "60-74") -0.36715 0.69270 0.21467 -1.710 0.08720 .
## tt(agegrp == "75+") -0.70783 0.49271 0.23207 -3.050 0.00229 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## exp(coef) exp(-coef) lower .95 upper .95
## sexFemale 0.5901 1.6946 0.5191 0.6709
## year8594Diagnosed 85-94 0.7150 1.3986 0.6280 0.8141
## agegrp45-59 1.6999 0.5883 1.1569 2.4978
## agegrp60-74 2.4607 0.4064 1.7043 3.5529
## agegrp75+ 5.4150 0.1847 3.7186 7.8853
## tt(agegrp == "45-59") 0.7255 1.3784 0.4678 1.1250
## tt(agegrp == "60-74") 0.6927 1.4436 0.4548 1.0550
## tt(agegrp == "75+") 0.4927 2.0296 0.3126 0.7765
##
## Concordance= 0.645 (se = 0.009 )
## Likelihood ratio test= 222.5 on 8 df, p=<2e-16
## Wald test = 224.5 on 8 df, p=<2e-16
## Score (logrank) test = 238 on 8 df, p=<2e-16## The tt labels do not play nicely with lincom:(
cox2p8tvct <- coxph(Surv(trunc_yy, death_cancer) ~ sex + year8594 + agegrp + tt(agegrp),
data=localised,
tt = function(x, t, ...) cbind(`45-59`=(x=="45-59")*t,
`60-74`=(x=="60-74")*t,
`75+`=(x=="75+")*t))
summary(cox2p8tvct)## Call:
## coxph(formula = Surv(trunc_yy, death_cancer) ~ sex + year8594 +
## agegrp + tt(agegrp), data = localised, tt = function(x, t,
## ...) cbind(`45-59` = (x == "45-59") * t, `60-74` = (x ==
## "60-74") * t, `75+` = (x == "75+") * t))
##
## n= 5318, number of events= 960
##
## coef exp(coef) se(coef) z Pr(>|z|)
## sexFemale -0.52429 0.59198 0.06541 -8.016 1.10e-15 ***
## year8594Diagnosed 85-94 -0.33768 0.71342 0.06627 -5.095 3.48e-07 ***
## agegrp45-59 0.49614 1.64237 0.18573 2.671 0.007555 **
## agegrp60-74 0.99490 2.70445 0.17856 5.572 2.52e-08 ***
## agegrp75+ 1.89383 6.64478 0.20455 9.259 < 2e-16 ***
## tt(agegrp)45-59 -0.05256 0.94880 0.04054 -1.297 0.194801
## tt(agegrp)60-74 -0.09839 0.90630 0.04040 -2.435 0.014875 *
## tt(agegrp)75+ -0.21623 0.80555 0.05739 -3.768 0.000165 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## exp(coef) exp(-coef) lower .95 upper .95
## sexFemale 0.5920 1.6893 0.5208 0.6729
## year8594Diagnosed 85-94 0.7134 1.4017 0.6265 0.8124
## agegrp45-59 1.6424 0.6089 1.1412 2.3635
## agegrp60-74 2.7044 0.3698 1.9058 3.8377
## agegrp75+ 6.6448 0.1505 4.4501 9.9218
## tt(agegrp)45-59 0.9488 1.0540 0.8763 1.0273
## tt(agegrp)60-74 0.9063 1.1034 0.8373 0.9810
## tt(agegrp)75+ 0.8055 1.2414 0.7198 0.9015
##
## Concordance= 0.645 (se = 0.009 )
## Likelihood ratio test= 229.7 on 8 df, p=<2e-16
## Wald test = 231.3 on 8 df, p=<2e-16
## Score (logrank) test = 245.9 on 8 df, p=<2e-16## Estimate 2.5 % 97.5 % Chisq Pr(>Chisq)
## agegrp75+ 6.644778 4.450099 9.921819 85.72268 2.070247e-20## Estimate 2.5 % 97.5 % Chisq Pr(>Chisq)
## agegrp75+ + tt(agegrp)75+ 5.35269 3.915088 7.318171 110.5215 7.532419e-26## Estimate 2.5 % 97.5 % Chisq Pr(>Chisq)
## agegrp75+ + 2*tt(agegrp)75+ 4.31185 3.373556 5.511113 136.2282 1.778638e-31cox2p8tvclogt <- coxph(Surv(trunc_yy, death_cancer) ~ sex + year8594 + agegrp +
tt(agegrp),
data=localised,
tt = function(x, t, ...) cbind(`45-59`=(x=="45-59")*log(t),
`60-74`=(x=="60-74")*log(t),
`75+`=(x=="75+")*log(t)))
summary(cox2p8tvclogt)## Call:
## coxph(formula = Surv(trunc_yy, death_cancer) ~ sex + year8594 +
## agegrp + tt(agegrp), data = localised, tt = function(x, t,
## ...) cbind(`45-59` = (x == "45-59") * log(t), `60-74` = (x ==
## "60-74") * log(t), `75+` = (x == "75+") * log(t)))
##
## n= 5318, number of events= 960
##
## coef exp(coef) se(coef) z Pr(>|z|)
## sexFemale -0.52385 0.59224 0.06540 -8.010 1.15e-15 ***
## year8594Diagnosed 85-94 -0.33810 0.71313 0.06628 -5.101 3.38e-07 ***
## agegrp45-59 0.56930 1.76703 0.19477 2.923 0.00347 **
## agegrp60-74 1.02025 2.77389 0.18503 5.514 3.51e-08 ***
## agegrp75+ 1.89910 6.67989 0.18956 10.018 < 2e-16 ***
## tt(agegrp)45-59 -0.23758 0.78853 0.14336 -1.657 0.09747 .
## tt(agegrp)60-74 -0.34509 0.70816 0.13838 -2.494 0.01264 *
## tt(agegrp)75+ -0.73311 0.48041 0.15837 -4.629 3.67e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## exp(coef) exp(-coef) lower .95 upper .95
## sexFemale 0.5922 1.6885 0.5210 0.6732
## year8594Diagnosed 85-94 0.7131 1.4023 0.6263 0.8121
## agegrp45-59 1.7670 0.5659 1.2063 2.5884
## agegrp60-74 2.7739 0.3605 1.9302 3.9864
## agegrp75+ 6.6799 0.1497 4.6070 9.6855
## tt(agegrp)45-59 0.7885 1.2682 0.5954 1.0444
## tt(agegrp)60-74 0.7082 1.4121 0.5399 0.9288
## tt(agegrp)75+ 0.4804 2.0816 0.3522 0.6553
##
## Concordance= 0.643 (se = 0.009 )
## Likelihood ratio test= 237.2 on 8 df, p=<2e-16
## Wald test = 235.8 on 8 df, p=<2e-16
## Score (logrank) test = 257.7 on 8 df, p=<2e-16lincom(cox2p8tvclogt, "agegrp75+ - 0.6931472*tt(agegrp)75+",eform=TRUE) # t=0.5 => log(t)=-0.6931472## Estimate 2.5 % 97.5 % Chisq Pr(>Chisq)
## agegrp75+ - 0.6931472*tt(agegrp)75+ 11.10348 6.347921 19.42166 71.20567 3.218622e-17## Estimate 2.5 % 97.5 % Chisq Pr(>Chisq)
## agegrp75+ 6.679893 4.606969 9.685537 100.3693 1.264746e-23## Estimate 2.5 % 97.5 % Chisq Pr(>Chisq)
## agegrp75+ + 0.6931472*tt(agegrp)75+ 4.018649 3.17166 5.091825 132.6644 1.070656e-30The 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)
library(splines)
time.cuts <- seq(0,10,length=100)
delta <- diff(time.cuts)[1]
## split and collapse
melanoma2p8Split2 <- survSplit(Surv(trunc_yy,death_cancer)~., data=localised,
cut=time.cuts, end="tstop", start="tstart",
event="death_cancer") %>%
mutate(fu=cut(tstop,time.cuts),
mid=time.cuts[unclass(fu)]+delta/2) %>%
group_by(mid,sex,year8594,agegrp) %>%
summarise(pt=sum(tstop-tstart), death_cancer=sum(death_cancer)) %>%
mutate(age75 = (agegrp=="75+")+0)## `summarise()` has grouped output by 'mid', 'sex', 'year8594'. You can override using the `.groups`
## argument.poisson2p8tvc <- glm(death_cancer ~ sex + year8594 + agegrp + ns(mid,df=3) +
age75:ns(mid,df=3) + offset(log(pt)),
data=melanoma2p8Split2, family=poisson)
## utility function to draw a confidence interval
polygon.ci <- function(time, interval, col="lightgrey")
polygon(c(time,rev(time)), c(interval[,1],rev(interval[,2])), col=col, border=col)
## define exposures
newdata <- data.frame(mid=seq(0,max(time.cuts),length=100), year8594="Diagnosed 85-94",
sex="Male", agegrp="75+", age75=1, pt=1)
library(rstpm2)
logirr <- rstpm2::predictnl(poisson2p8tvc,
fun=function(fit,newdata) predict(fit, newdata) -
predict(fit, transform(newdata, agegrp='0-44', age75=0)),
newdata=newdata)
pred <- exp(logirr$fit)
ci <- exp(confint(logirr))
## plot
matplot(newdata$mid, ci, type="n", xlab="Time since diagnosis (months)",
ylab="Rate ratio", main="Ages 75+ compared with ages 0-44 years")
polygon.ci(newdata$mid, ci)
lines(newdata$mid, pred)