chapter  14
glrt – New R Package for Analyzing Interval-Censored Sur- vival Data
Pages 21

Department of Mathematics, Texas State University-San Marcos, San Marcos,

Texas, USA

14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378

14.2 Generalized Logrank Tests for Comparing Survival Functions . . 379

14.2.1 Basic Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379

14.2.2 Generalized Logrank Test I (gLRT1) . . . . . . . . . . . . . . . . . . . . 380

14.2.3 Generalized Logrank Test II (gLRT2) . . . . . . . . . . . . . . . . . . . 381

14.2.4 Generalized Logrank Test III (gLRT3) . . . . . . . . . . . . . . . . . . 382

14.2.5 Generalized Logrank Test IV (gLRT4 or Score Test) . . . 384

14.3 Software: glrt Package . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385

14.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387

14.4.1 Conducting Generalized Logrank Tests . . . . . . . . . . . . . . . . . 387

14.4.2 Estimating the Survival Function . . . . . . . . . . . . . . . . . . . . . . . 393

14.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394

378 Interval-Censored Time-to-Event Data: Methods and Applications

In survival studies, one of the main goals is to compare the survival of indi-

viduals in different treatment groups. For the problem, when right-censored

failure time data are available, the well-known logrank test is a powerful and

widely used method and is available in major statistical software packages

such as SAS and S-Plus. For interval-censored survival data, which arise nat-

urally from studies in which there is a periodic follow-up, several authors

have discussed this problem. For example, Peto and Peto (1972) considered

the two-sample comparison problem under Lehmann-type alternatives. In this

case, the comparison problem reduces to a score test, which they referred to as

the logrank test for interval-censored data. Finkelstein (1986) later took a re-

gression approach and developed a score test under the proportional hazards

model when the covariates are treatment indicators. This score test allows

k-sample treatment comparisons and is a generalization of the logrank test.

Following Finkelstein (1986), Sun (1996) studied the same problem without

assuming the proportional hazards model and developed a nonparametric test

using the idea behind the logrank test for right-censored data. However, it

does not reduce to the logrank test in the case of right-censored data. Also,

it may not have the right size and good power if the proportion of strictly

interval-censored observations is small. Zhao and Sun (2004) improved this

test by making adjustments to the observed failure and risk numbers so that

the resulting test has a higher power and reduces to the logrank test when

right-censored data are available. Other existing test procedures for interval-

censored data can be found in Sun (1998). Given that most existing test proce-

dures for interval-censored data are ad hoc methods with unknown properties

and/or the variance estimation of the test statistic is complicated, Sun et al.

(2005) proposed a new class of generalized logrank tests for interval-censored

data without exact observations and established their asymptotic properties.