R: Rank Correlation for Censored Data

rcorr.cens {Hmisc} | R Documentation

## Rank Correlation for Censored Data

### Description

Computes the c index and the corresponding generalization of Somers' Dxy rank correlation for a censored response variable. Also works for uncensored and binary responses, although its use of all possible pairings makes it slow for this purpose. Dxy and c are related by \var{Dxy} = 2*(\var{c} - 0.5).

`rcorr.cens` handles one predictor variable. `rcorrcens` computes rank correlation measures separately by a series of predictors. In addition, `rcorrcens` has a rough way of handling categorical predictors. If a categorical (factor) predictor has two levels, it is coverted to a numeric having values 1 and 2. If it has more than 2 levels, an indicator variable is formed for the most frequently level vs. all others, and another indicator for the second most frequent level and all others. The correlation is taken as the maximum of the two (in absolute value).

### Usage

    rcorr.cens(x, S, outx=FALSE)

    ## S3 method for class 'formula'
    rcorrcens(formula, data=NULL, subset=NULL,
              na.action=na.retain, exclude.imputed=TRUE, outx=FALSE,
              ...)

### Arguments

`x` | a numeric predictor variable
`S` | an `Surv` object or a vector. If a vector, assumes that every observation is uncensored.
`outx` | set to `TRUE` to not count pairs of observations tied on `x` as a relevant pair. This results in a Goodman–Kruskal gamma type rank correlation.
`formula` | a formula with a `Surv` object or a numeric vector on the left-hand side
`data, subset, na.action` | the usual options for models. Default for `na.action` is to retain all values, NA or not, so that NAs can be deleted in only a pairwise fashion.
`exclude.imputed` | set to `FALSE` to include imputed values (created by `impute`) in the calculations.
`...` | extra arguments passed to `biVar`.

### Value

`rcorr.cens` returns a vector with the following named elements: `C Index`, `Dxy`, `S.D.`, `n`, `missing`, `uncensored`, `Relevant Pairs`, `Concordant`, and `Uncertain`

`n` | number of observations not missing on any input variables
`missing` | number of observations missing on `x` or `S`
`relevant` | number of pairs of non-missing observations for which `S` could be ordered
`concordant` | number of relevant pairs for which `x` and `S` are concordant.
`uncertain` | number of pairs of non-missing observations for which censoring prevents classification of concordance of `x` and `S`.

`rcorrcens.formula` returns an object of class `biVar` which is documented with the `biVar` function.

### Author(s)

Frank Harrell
Department of Biostatistics
Vanderbilt University
f.harrell@vanderbilt.edu

### References

Newson R: Confidence intervals for rank statistics: Somers' D and extensions. Stata Journal 6:309-334; 2006.

### See Also

`somers2`, `biVar`, `rcorrp.cens`

### Examples

    set.seed(1)
    x <- round(rnorm(200))
    y <- rnorm(200)
    rcorr.cens(x, y, outx=TRUE)   # can correlate non-censored variables
    library(survival)
    age <- rnorm(400, 50, 10)
    bp  <- rnorm(400,120, 15)
    bp[1]  <- NA
    d.time <- rexp(400)
    cens   <- runif(400,.5,2)
    death  <- d.time <= cens
    d.time <- pmin(d.time, cens)
    rcorr.cens(age, Surv(d.time, death))
    r <- rcorrcens(Surv(d.time, death) ~ age + bp)
    r
    plot(r)

    # Show typical 0.95 confidence limits for ROC areas for a sample size
    # with 24 events and 62 non-events, for varying population ROC areas
    # Repeat for 138 events and 102 non-events
    set.seed(8)
    par(mfrow=c(2,1))
    for(i in 1:2) {
     n1 <- c(24,138)[i]
     n0 <- c(62,102)[i]
     y <- c(rep(0,n0), rep(1,n1))
     deltas <- seq(-3, 3, by=.25)
     C <- se <- deltas
     j <- 0
     for(d in deltas) {
      j <- j + 1
      x <- c(rnorm(n0, 0), rnorm(n1, d))
      w <- rcorr.cens(x, y)
      C[j]  <- w['C Index']
      se[j] <- w['S.D.']/2
     }
     low <- C-1.96*se; hi <- C+1.96*se
     print(cbind(C, low, hi))
     errbar(deltas, C, C+1.96*se, C-1.96*se,
            xlab='True Difference in Mean X',
            ylab='ROC Area and Approx. 0.95 CI')
     title(paste('n1=',n1,'  n0=',n0,sep=''))
     abline(h=.5, v=0, col='gray')
     true <- 1 - pnorm(0, deltas, sqrt(2))
     lines(deltas, true, col='blue')
    }
    par(mfrow=c(1,1))

[Package Hmisc version 4.3-0 Index]