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
data(melanoma)
mel <- subset(melanoma, stage=="Localised") # restrict the cohort to stage==1
mel <- transform(mel,
dc = (mel$status=="Dead: cancer" & surv_mm<120)+0,
surv_10y = pmin(120, surv_mm))
table(mel$dc , mel$status) # check
## Factor w/ 2 levels "Male","Female": 2 2 2 2 2 2 2 1 2 2 ...
## Factor w/ 2 levels "Diagnosed 75-84",..: 1 1 1 1 1 1 1 1 1 1 ...
## 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)
##
## 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 2097 87.5 0 Male Diagnosed 75-84 45-59 0 2937
## 1915 115 4275 98.5 1 Male Diagnosed 85-94 60-74 1 6168
## 1916 115 2694 98.5 0 Female Diagnosed 85-94 0-44 0 3816
## 1917 116 5287 2.5 1 Male Diagnosed 85-94 45-59 1 7713
## 1918 116 3013 2.5 0 Male Diagnosed 85-94 0-44 0 4279
## 1919 117 5314 119.5 1 Male Diagnosed 85-94 60-74 1 7753
## 1920 117 609 119.5 0 Male Diagnosed 75-84 45-59 0 868
(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 -0.51826 0.59556 0.09452 -5.483 4.18e-08 ***
## year8594Diagnosed 85-94 -0.40636 0.66607 0.09611 -4.228 2.36e-05 ***
## agegrp45-59 0.16839 1.18340 0.12815 1.314 0.189
## agegrp60-74 0.63602 1.88895 0.12902 4.930 8.24e-07 ***
## agegrp75+ 1.11502 3.04962 0.16348 6.821 9.06e-12 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## exp(coef) exp(-coef) lower .95 upper .95
## sexFemale 0.5956 1.6791 0.4948 0.7168
## year8594Diagnosed 85-94 0.6661 1.5013 0.5517 0.8041
## agegrp45-59 1.1834 0.8450 0.9206 1.5213
## agegrp60-74 1.8890 0.5294 1.4669 2.4324
## agegrp75+ 3.0496 0.3279 2.2136 4.2014
##
## Concordance= 0.64 (se = 0.015 )
## Likelihood ratio test= 96.12 on 5 df, p=<2e-16
## Wald test = 89.64 on 5 df, p=<2e-16
## Score (logrank) test = 93.9 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.
comp <- exp(data.frame(coef(out_coh),coef(out_ncc)))
colnames(comp) <- c("cohort HR", "NCC HR")
comp # print the HR estimates from the cohort and the NCC
## cohort HR NCC HR
## sexFemale 0.5882470 0.5955570
## year8594Diagnosed 85-94 0.7164882 0.6660677
## agegrp45-59 1.3268817 1.1834004
## agegrp60-74 1.8590433 1.8889544
## agegrp75+ 3.3804475 3.0496221
var <- data.frame(diag(vcov(out_coh)), diag(vcov(out_ncc)), diag(vcov(out_coh))/diag(vcov(out_ncc)))
colnames(var) <- c("cohort var", "NCC var", "ratio coh/ncc" )
var # print the variances of estimates from the cohort and the NCC and their ratio
## cohort var NCC var ratio coh/ncc
## sexFemale 0.004283749 0.008933339 0.4795238
## year8594Diagnosed 85-94 0.004380163 0.009238029 0.4741447
## agegrp45-59 0.008868630 0.016422877 0.5400168
## agegrp60-74 0.008258394 0.016646103 0.4961158
## agegrp75+ 0.010906329 0.026724461 0.4081029
(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.
M <- 20 # Number of loops: change the M value to change the number of loops
param <- matrix(0,M,5) # Define the matrice of the coefficients
for (i in 1:M) { # Start of the loop, create NCC data and analyse it
nccdata <-ccwc( entry=0, exit=surv_10y , fail=dc, origin=0, controls=1, include=list(sex,year8594,agegrp), data=mel )
out_ncc <- clogit(Fail ~ sex + year8594 + agegrp + strata(Set), data=nccdata)
param [ i , 1:5] <- coef(out_ncc) # store the 5 coefficients M times
} # End of the loop
param <- as.data.frame(param) # data frame with the coefficients
param <- exp(param) # data frame with the HR
param <- rbind (summary(out_coh)$coef[c(1:5),2], param)
colnames(param) <- names(coef(out_coh))
rownames(param) <- c("cohort", c(1:M))
param
## sexFemale year8594Diagnosed 85-94 agegrp45-59 agegrp60-74 agegrp75+
## cohort 0.5882470 0.7164882 1.326882 1.859043 3.380448
## 1 0.5486474 0.6381542 1.497273 1.772094 4.122505
## 2 0.5328482 0.7181448 1.313150 2.073535 3.868298
## 3 0.5464003 0.7443230 1.413770 1.953145 4.337926
## 4 0.5839390 0.7573494 1.409873 1.907685 2.761936
## 5 0.5598362 0.7143165 1.419559 1.928466 3.554149
## 6 0.5603335 0.7141989 1.268408 1.814712 3.287115
## 7 0.5875319 0.7264022 1.512866 1.993000 3.649585
## 8 0.5606054 0.7206263 1.449527 1.828567 3.958375
## 9 0.5680247 0.6848129 1.261659 1.883825 3.504086
## 10 0.5381000 0.6972992 1.282885 2.059576 3.631084
## 11 0.5505851 0.6923363 1.535384 2.189954 4.108926
## 12 0.5476165 0.6629938 1.430209 2.027819 4.652872
## 13 0.5452970 0.6431432 1.163497 1.879994 3.914656
## 14 0.5616197 0.5961417 1.357514 1.807640 4.055455
## 15 0.5346850 0.6160208 1.415101 2.057579 4.236180
## 16 0.5021241 0.6733884 1.241578 1.550081 3.709983
## 17 0.5703349 0.6609033 1.447120 2.014933 4.560851
## 18 0.5522950 0.7516153 1.389732 1.957561 3.632394
## 19 0.5539184 0.7861786 1.190621 1.737093 2.939583
## 20 0.5134505 0.6675301 1.261959 1.890939 3.498083
mean_param <- apply(param[c(2:M),], 2, mean) # compute the mean of the log HR for the M loops
sd_param <- apply(param[c(2:M),], 2, sd) # compute the sd of the log HR for the M loops
par_sum <- rbind (summary(out_coh)$coef[c(1:5),1], mean_param, sd_param)
rownames(par_sum) <- c("cohort log HR","mean log HR NCC","sd log HR NCC")
colnames(par_sum) <- names(coef(out_coh))
par_sum
## sexFemale year8594Diagnosed 85-94 agegrp45-59 agegrp60-74 agegrp75+
## cohort log HR -0.53060828 -0.33339352 0.2828316 0.6200620 1.2180081
## mean log HR NCC 0.55288118 0.69464993 1.3684067 1.9177506 3.8150506
## sd log HR NCC 0.01917491 0.05025401 0.1086644 0.1473112 0.4947221
par(mfrow=c(2,3) ) # allow 2*3 graphs on the same page
for (i in 1:5) {
hist(param[,i], main=colnames(par_sum)[i], xlab="log HR value") # histogram of HR for variable sex (female vs male)
abline(v=summary(out_coh)$coef[i,2], col="green")
abline(v=mean_param[i], col="red")
}
plot.new()
legend(0, 0.5, c("Cox","Mean NCC"),lty=1,col=c("green","red"),bty="n")