R: *g*oodness *o*f *f*it test for a 'coxph' object

gof {survMisc} | R Documentation

## goodness of fit test for a `coxph` object

### Description

goodness of fit test for a `coxph` object

### Usage

    gof(x, ...)

    ## S3 method for class 'coxph'
    gof(x, ..., G = NULL)

### Arguments

`x` | An object of class `coxph`
`...` | Additional arguments (not implemented)
`G` | Number of groups into which to divide risk score. If `G=NULL` (the default), uses closest integer to G = max(2, min(10, ne/40)) where ne is the number of events overall.

### Details

In order to verify the overall goodness of fit, the risk score r[i] for each observation i is given by

r[i] = B.X[i]

where B is the vector of fitted coefficients and X[i] is the vector of predictor variables for observation i.
This risk score is then sorted and 'lumped' into a grouping variable with G groups, (containing approximately equal numbers of observations).
The number of observed (e) and expected (exp) events in each group are used to generate a Z statistic for each group, which is assumed to follow a normal distribution with Z \sim N(0,1).
The indicator variable `indicG` is added to the original model and the two models are compared to determine the improvement in fit via the likelihood ratio test.

### Value

A `list` with elements:

`groups` | A `data.table` with one row per group G. The columns are n Number of observations e Number of events exp Number of events expected. This is exp = ∑ e_i - M_i where e_i are the events and M_i are the martingale residuals for each observation i z Z score, calculated as Z = (e - exp) / exp^0.5 p p-value for Z, which is p = 2 * pnorm(-|z|) where `pnorm` is the normal distribution function with mean 0 and standard deviation 1 and |z| is the absolute value.
`lrTest` | Likelihood-ratio test. Tests the improvement in log-likelihood with addition of an indicator variable with G-1 groups. This is done with `survival:::anova.coxph`. The test is distributed as chi-square with G-1 degrees of freedom

### Note

The choice of G is somewhat arbitrary but rarely should be > 10.
As illustrated in the example, a larger value for G makes the Z test for each group more likely to be significant. This does not affect the significance of adding the indicator variable `indicG` to the original model.
The Z score is chosen for simplicity, as for large sample sizes the Poisson distribution approaches the normal. Strictly speaking, the Poisson would be more appropriate for e and exp as per Counting Theory.
The Z score may be somewhat conservative as the expected events are calculated using the martingale residuals from the overall model, rather than by group. This is likely to bring the expected events closer to the observed events.
This test is similar to the Hosmer-Lemeshow test for logistic regression.

### Source

Method and example are from:
May S, Hosmer DW 1998. A simplified method of calculating an overall goodness-of-fit test for the Cox proportional hazards model. Lifetime Data Analysis 4(2):109–20. Springer (paywall)

### References

Default value for G as per:
May S, Hosmer DW 2004. A cautionary note on the use of the Gronnesby and Borgan goodness-of-fit test for the Cox proportional hazards model. Lifetime Data Analysis 10(3):283–91. Springer (paywall)

Changes to the `pbc` dataset in the example are as detailed in:
Fleming T, Harrington D 2005. Counting Processes and Survival Analysis. New Jersey: Wiley and Sons. Chapter 4, section 4.6, pp 188. Wiley (paywall)

### Examples

    data("pbc", package="survival")
    pbc <- pbc[!is.na(pbc$trt), ]
    ## make corrections as per Fleming
    pbc[pbc$id==253, "age"] <-  54.4
    pbc[pbc$id==107, "protime"] <-  10.7
    ### misspecified; should be log(bili) and log(protime) instead
    c1 <- coxph(Surv(time, status==2) ~
                age + log(albumin) + bili + edema + protime,
                data=pbc)
    gof(c1, G=10)
    gof(c1)

[Package survMisc version 0.5.4 Index]