proprate2.neyman {surv2sample}R Documentation

Neyman's Smooth Test of Proportional Rates in Two Samples of Censored Data

Description

Checks the assumption of proportional rates (proportional hazards, proportional odds) in two samples of right-censored data using (possibly data-driven) Neyman's smooth test.

Usage

proprate2.neyman(x, group, model = 0, data.driven = TRUE,
                 d = ifelse(data.driven, 5, 3),
                 basis = "legendre", time.transf = "F",
                 beta.init = 0, maxiter = 20, eps = 1e-09,
                 choltol = 1e-07)

## S3 method for class 'proprate2.neyman':
summary(object, ...)

Arguments

x a "Surv" object, as returned by the Surv function.
group a vector indicating to which group each observation belongs. May contain values 1 and 2 only.
model the type of model. Possible values are 0 for proportional hazards, 1 for proportional odds.
data.driven Should the test be data-driven?
d the number of basis functions for the test with fixed dimension, the maximum number of basis functions for the data-driven test.
basis the basis of functions. Possible values are "legendre" for Legendre polynomials and "cos" for cosines.
time.transf the time transformation for basis functions. Possible values are "F" for the distribution function (F(t)/F(tau)) estimated from the pooled sample (recommended), "A" for the cumulative hazard (A(t)/A(tau)) and "I" for no transformation (the linear transformation t/tau).
beta.init the initial parameter value for iteration in the simplified partial likelihood estimation.
maxiter the maximum number of iterations.
eps the convergence tolerance parameter. The convergence criterion is |(l-l_old)/l|<eps.
choltol a tolerance parameter for the Cholesky decomposition.
object an object of class "proprate2.neyman", as returned by proprate2.neyman.
... further parameters for printing.

Details

This function tests the hypothesis that transformation rates (currently hazard rates or odds functions) are proportional (their ratio is constant in time) in two samples of censored survival data.

The proportional rate model is estimated by a two-sample simplification of the partial likelihood. Then Neyman's smooth test of fit is performed. In general, Neyman's smooth tests are based on embedding the null hypothesis in a d-dimensional alternative. Here it consists of expressing the logarithm of the possibly time-varying ratio of rates as a linear combination of d basis functions (Legendre polynomials or cosines) in transformed time. Their significance is tested by a score test. The score is derived from the simplified partial likelihood. See Kraus (2007a) for details. The quadratic test statistic is asymptotically chi-square distributed with d degrees of freedom.

A data-driven choice of the number of basis functions is possible. The selection is based on a Schwarz-type criterion which is the maximiser of penalised score statistics for dimensions 1,...,d. For the p-value of the data-driven test a two-term approximation is used (see Kraus (2007b), eq. (12)), as the asymptotic chi-square with 1 d.f. is inaccurate.

If the test is data-driven, the summary method prints details on the selection procedure (statistics and penalised statistics for each dimension). This is equivalent to print(x, detail=TRUE, ...).

Value

proprate2.neyman returns a list of class "proprate2.neyman" and "neyman.test". Its main components are:

stat the test statistic.
pval the p-value (based on the chi-square distribution for the fixed-dimension test and on the two-term approximation for the data-driven test).
stats, stats.penal statistics and penalised statistics for dimensions 1,...,d (only for data-driven tests).
S.dim the selected dimension (only for data-driven tests).

Most input parameters and some further components are included.

Author(s)

David Kraus (http://www.davidkraus.net/)

References

Bagdonavicius, V. and Nikulin, M. (2000) On goodness-of-fit for the linear transformation and frailty models. Statist. Probab. Lett. 47, 177–188.

Kraus, D. (2007a) Checking proportional rates in the two-sample transformation model. Research Report 2203, Institute of Information Theory and Automation, Prague. Available at http://www.davidkraus.net/surv2sample/.

Kraus, D. (2007b) Data-driven smooth tests of the proportional hazards assumption. Lifetime Data Anal. 13, 1–16.

See Also

proprate2.ks, proprate2.gs for other tests of the proportional rate assumption

proprate2 for estimation

Examples

## chronic active hepatitis data
data(hepatitis)

## Neyman's test of proportional odds
## test with fixed dimension
proprate2.neyman(Surv(hepatitis$time, hepatitis$status),
    hepatitis$treatment, model = 1, data.driven = FALSE)
## data-driven test
print(a <- proprate2.neyman(Surv(hepatitis$time, hepatitis$status),
    hepatitis$treatment, model = 1, data.driven = TRUE))
## details of the selection procedure
summary(a)

[Package surv2sample version 0.1-2 Index]