Exercise 25. Localised melanoma : generating and analysing a nested case-control (NCC) study
library(biostat3) # cox and conditional logistic analyses
library(Epi) # sample a nested case-control study from a cohort## Get the data for exercise 25 and have a look at it
mel <- subset(biostat3::melanoma, stage=="Localised") |>
transform(dc = (status=="Dead: cancer" & surv_mm<120)+0,
surv_10y = pmin(120, surv_mm))
with(mel,table(dc, status))## 'data.frame': 5318 obs. of 3 variables:
## $ sex : Factor w/ 2 levels "Male","Female": 2 2 2 2 2 2 2 1 2 2 ...
## $ year8594: Factor w/ 2 levels "Diagnosed 75-84",..: 1 1 1 1 1 1 1 1 1 1 ...
## $ agegrp : Factor w/ 4 levels "0-44","45-59",..: 4 4 4 4 4 4 4 4 4 4 ...## Call:
## coxph(formula = Surv(surv_10y, dc) ~ sex + year8594 + agegrp,
## data = mel)
##
## 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(a)
There are 5318 individuals in the study that we would need to collect data for if we were to use the complete cohort of patients.
## [1] 5318(b)
960 cancer patients die from melanoma during the first 10 years of follow-up.
##
## 0 1 <NA>
## 4358 960 0## 1
## 960(c1)
set.seed(12345)
nccdata <- ccwc(entry=0, exit=surv_10y , fail=dc, origin=0, controls=1,
include=list(sex,year8594,agegrp,dc,id), data=mel)Sampling risk sets: ………………………………………………………………………………………………………
| Set | Map | Time | Fail | sex | year8594 | agegrp | dc | id | |
|---|---|---|---|---|---|---|---|---|---|
| 1913 | 114 | 2674 | 87.5 | 1 | Female | Diagnosed 85-94 | 0-44 | 1 | 3784 |
| 1914 | 114 | 1566 | 87.5 | 0 | Female | Diagnosed 75-84 | 0-44 | 0 | 2202 |
| 1915 | 115 | 4275 | 98.5 | 1 | Male | Diagnosed 85-94 | 60-74 | 1 | 6168 |
| 1916 | 115 | 242 | 98.5 | 0 | Male | Diagnosed 75-84 | 60-74 | 0 | 348 |
| 1917 | 116 | 5287 | 2.5 | 1 | Male | Diagnosed 85-94 | 45-59 | 1 | 7713 |
| 1918 | 116 | 625 | 2.5 | 0 | Male | Diagnosed 75-84 | 45-59 | 0 | 894 |
| 1919 | 117 | 5314 | 119.5 | 1 | Male | Diagnosed 85-94 | 60-74 | 1 | 7753 |
| 1920 | 117 | 1364 | 119.5 | 0 | Male | Diagnosed 75-84 | 0-44 | 0 | 1922 |
(c2)
## Call:
## coxph(formula = Surv(rep(1, 1920L), Fail) ~ sex + year8594 +
## agegrp + strata(Set), data = nccdata, method = "exact")
##
## n= 1920, number of events= 960
##
## coef exp(coef) se(coef) z Pr(>|z|)
## sexFemale -0.58162 0.55899 0.09588 -6.066 1.31e-09 ***
## year8594Diagnosed 85-94 -0.23264 0.79244 0.09646 -2.412 0.015877 *
## agegrp45-59 0.41896 1.52038 0.12593 3.327 0.000878 ***
## agegrp60-74 0.70976 2.03351 0.12527 5.666 1.46e-08 ***
## agegrp75+ 1.35147 3.86310 0.16460 8.210 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## exp(coef) exp(-coef) lower .95 upper .95
## sexFemale 0.5590 1.7889 0.4632 0.6746
## year8594Diagnosed 85-94 0.7924 1.2619 0.6559 0.9574
## agegrp45-59 1.5204 0.6577 1.1878 1.9460
## agegrp60-74 2.0335 0.4918 1.5908 2.5994
## agegrp75+ 3.8631 0.2589 2.7979 5.3339
##
## Concordance= 0.653 (se = 0.015 )
## Likelihood ratio test= 109.1 on 5 df, p=<2e-16
## Wald test = 100.9 on 5 df, p=<2e-16
## Score (logrank) test = 106.3 on 5 df, p=<2e-16(e)
Note that, since every nested case-control study is different, the parameter estimates you obtain will not be identical to those above. However, the hazard ratios from the two models should be very similar. The standard errors are slightly larger for the nested case-control study since the estimates are based on a sample from the full cohort. Loss of precision is the trade-off we have to make when designing a nested case-control study. The precision can be improved by adding more controls to each case.
## Full cohort:## # A tibble: 5 × 7
## term estimate std.error statistic p.value conf.low conf.high
## <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 sexFemale 0.58825 0.065450 -8.1070 5.1870e-16 0.51743 0.66876
## 2 year8594Diagnosed 85-94 0.71649 0.066183 -5.0375 4.7174e- 7 0.62932 0.81573
## 3 agegrp45-59 1.3269 0.094173 3.0033 2.6706e- 3 1.1032 1.5959
## 4 agegrp60-74 1.8590 0.090876 6.8232 8.9042e-12 1.5557 2.2215
## 5 agegrp75+ 3.3804 0.10443 11.663 1.9694e-31 2.7547 4.1483## Nested case-control study:## # A tibble: 5 × 7
## term estimate std.error statistic p.value conf.low conf.high
## <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 sexFemale 0.55899 0.095884 -6.0659 1.3125e- 9 0.46322 0.67456
## 2 year8594Diagnosed 85-94 0.79244 0.096459 -2.4117 1.5877e- 2 0.65594 0.95736
## 3 agegrp45-59 1.5204 0.12593 3.3269 8.7804e- 4 1.1878 1.9460
## 4 agegrp60-74 2.0335 0.12527 5.6660 1.4615e- 8 1.5908 2.5994
## 5 agegrp75+ 3.8631 0.16460 8.2104 2.2041e-16 2.7979 5.3339## # A tibble: 5 × 2
## term variance.ratio
## <chr> <dbl>
## 1 sexFemale 2.1462
## 2 year8594Diagnosed 85-94 2.1242
## 3 agegrp45-59 1.7881
## 4 agegrp60-74 1.9001
## 5 agegrp75+ 2.4843tidy_coh <- tidy(out_coh)
tidy_ncc <- tidy(out_ncc)
f <- function(x,mu,sigma) {
mu=as.numeric(mu)
sigma=as.numeric(sigma)
xp=exp(mu+x*sigma)
cbind(x=xp,y=dlnorm(xp,mu,sigma))
}
x <- seq(-4,4,length=301)
par(mfrow=c(2,3))
for (i in 1:5) {
f_coh = f(x,tidy_coh[i,"estimate"], tidy_coh[i,"std.error"])
f_ncc = f(x,tidy_ncc[i,"estimate"], tidy_ncc[i,"std.error"])
plot(rbind(f_coh,f_ncc), type="n", xlab="Hazard ratio", ylab="Density",
main=as.character(tidy_coh$term[i]))
polygon(f_coh,
col=alpha("green",0.2), border=alpha("green",0.2))
polygon(f_ncc,
col=alpha("blue",0.2), border=alpha("blue",0.2))
}
legend("topright", c("Cohort","NCC"), col=c(alpha("green",0.2),alpha("blue",0.2)), lwd=5,
bty="n")
(f)
We see that there is sampling variation in the parameter estimates from the five nested case-control studies but they are centered on the full cohort estimate. We see that the standard errors of the estimates from the nested case-control studies are larger than for the full cohort but there is some sampling variation.
set.seed(54321)
M <- 20
tidys_ncc <- lapply(1:M, function(i)
ccwc(entry=0, exit=surv_10y , fail=dc, origin=0, controls=1,
include=list(sex,year8594,agegrp), data=mel, silent=TRUE) |>
clogit(formula = Fail ~ sex + year8594 + agegrp + strata(Set)) |>
suppressWarnings() |> tidy())
tidy_coh <- tidy(out_coh)
f <- function(x,mu,sigma) {
mu=as.numeric(mu)
sigma=as.numeric(sigma)
xp=exp(mu+x*sigma)
cbind(x=xp,y=dlnorm(xp,mu,sigma))
}
x <- seq(-4,4,length=301)
par(mfrow=c(2,3))
for (i in 1:5) {
f_coh = f(x,tidy_coh[i,"estimate"], tidy_coh[i,"std.error"])
f_ncc = lapply(tidys_ncc, function(object)
f(x,object[i,"estimate"], object[i,"std.error"]))
plot(do.call(what=rbind,c(list(f_coh),f_ncc)), type="n", xlab="Hazard ratio", ylab="Density",
main=as.character(tidy_coh$term[i]))
for (j in 1:length(tidys_ncc))
polygon(f_ncc[[j]],
col=alpha("blue",0.02), border=alpha("blue",0.02))
polygon(f_coh,
col=alpha("green",0.2), border=alpha("green",0.2))
}
plot(0:1,0:1,type="n",axes=FALSE, xlab="",ylab="")
legend("center", c("Cohort","NCC"), col=c(alpha("green",0.2),alpha("blue",0.2)), lwd=5,
bty="n")