Exercise 25. Localised melanoma : generating and analysing a nested case-control (NCC) study


##  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.

##                         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
##                          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.

##        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
##                   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