Explicit formulae for

Denote

Then

Likelihood Ratio Test

Score Test

Wald’s Test

Robust Score Test

where is the score residual of the th subject at =; that is, =, where the score process is defined in the section Residuals.

Robust Wald’s Test

where is the sandwich variance estimate (see the section Robust Sandwich Variance Estimate for details).

Likelihood Ratio Statistic

Score Statistic

Wald’s Statistic

where is the estimated covariance matrix, which can be the model-based covariance matrix or the sandwich covariance matrix (see the section Robust Sandwich Variance Estimate for details).

Point Estimate

The hazard ratio is estimated by , where is the maximum likelihood estimate of the .

Wald’s Confidence Limits

The confidence limits for the hazard ratio are calculated as

where is estimated covariance matrix, and is the percentile point of the standard normal distribution.

Profile-Likelihood Confidence Limits

The construction of the profile-likelihood-based confidence interval is derived from the asymptotic distribution of the generalized likelihood ratio test of Venzon and Moolgavkar (1988). Suppose that the parameter vector is and you want to compute a confidence interval for . The profile-likelihood function for is defined as

where is the set of all with the th element fixed at , and is the log-likelihood function for . If is the log likelihood evaluated at the maximum likelihood estimate , then has a limiting chi-square distribution with one degree of freedom if is the true parameter value. Let , where is the percentile of the chi-square distribution with one degree of freedom. A % confidence interval for is

The endpoints of the confidence interval are found by solving numerically for values of that satisfy equality in the preceding relation. To obtain an iterative algorithm for computing the confidence limits, the log-likelihood function in a neighborhood of is approximated by the quadratic function

where is the gradient vector and is the Hessian matrix. The increment for the next iteration is obtained by solving the likelihood equations

where is the Lagrange multiplier, is the th unit vector, and is an unknown constant. The solution is

By substituting this into the equation , you can estimate as

The upper confidence limit for is computed by starting at the maximum likelihood estimate of and iterating with positive values of until convergence is attained. The process is repeated for the lower confidence limit, using negative values of .

Convergence is controlled by value specified with the PLCONV= option in the MODEL statement (the default value of is 1E4). Convergence is declared on the current iteration if the following two conditions are satisfied:

and

The profile-likelihood confidence limits for the hazard ratio are obtained by exponentiating these confidence limits.

Optimal Weights for the AVERAGE option in the TEST Statement

Let , where is a subset of regression coefficients. For any vector of length ,

To find such that has the minimum variance, it is necessary to minimize subject to . Let be a vector of 1’s of length . The expression to be minimized is

where is the Lagrange multipler. Differentiating with respect to and , respectively, yields

Solving these equations gives

This provides a one degree-of-freedom test for testing the null hypothesis with normal test statistic

This test is more sensitive than the multivariate test specified by the TEST statement

Multivariate: test X1, ..., Xs;

where X, ..., X are the variables with regression coefficients , respectively.

Marginal Cox Models for Multiple Events Data

Suppose there are subjects and each subject can experience up to potential events. Let be the covariate process associated with the th event for the th subject. The marginal Cox models are given by

where is the (event-specific) baseline hazard function for the th event and is the (event-specific) column vector of regression coefficients for the th event. WLW estimates by the maximum partial likelihood estimates , respectively, and uses a robust sandwich covariance matrix estimate for to account for the dependence of the multiple failure times.

By using a properly prepared input data set, you can estimate the regression parameters for all the marginal Cox models and compute the robust sandwich covariance estimates in one PROC PHREG invocation. For convenience of discussion, suppose each subject can potentially experience =3 events and there are two explanatory variables Z1 and Z2. The event-specific parameters to be estimated are for the first marginal model, for the second marginal model, and for the third marginal model. Inference of these parameters is based on the robust sandwich covariance matrix estimate of the parameter estimators. It is necessary that each row of the input data set represent the data for a potential event of a subject. The input data set should contain the following:

• an ID variable for identifying the subject so that all observations of the same subject have the same ID value

• an Enum variable to index the multiple events. For example, Enum=1 for the first event, Enum=2 for the second event, and so on.

• a Time variable to represent the observed time from some time origin for the event. For recurrence events data, it is the time from the study entry to each recurrence.

• a Status variable to indicate whether the Time value is a censored or uncensored time. For example, Status=1 indicates an uncensored time and Status=0 indicates a censored time.

• independent variables (Z1 and Z2)

The WLW analysis can be carried out by specifying the following:

proc phreg covs(aggregate);
   model Time*Status(0)=Z11 Z12 Z13 Z21 Z22 Z23;
   strata Enum;
   id ID;
   Z11= Z1 * (Enum=1);
   Z12= Z1 * (Enum=2);
   Z13= Z1 * (Enum=3);
   Z21= Z2 * (Enum=1);
   Z22= Z2 * (Enum=2);
   Z23= Z2 * (Enum=3);
   run;

The variable Enum is specified in the STRATA statement so that there is one marginal Cox model for each distinct value of Enum. The variables Z11, Z12, Z13, Z21, Z22, and Z23 in the MODEL statement are event-specific variables derived from the independent variables Z1 and Z2 by the given programming statements. In particular, the variables Z11, Z12, and Z13 are event-specific variables for the explanatory variable Z1; the variables Z21, Z22, and Z23 are event-specific variables for the explanatory variable Z2. For , and , variable Zjk contains the same values as the explanatory variable Zj for the rows that correspond to th marginal model and the value 0 for all other rows; as such, is the regression coefficient for Zjk. You can avoid using the programming statements in PROC PHREG if you create these event-specific variables in the input data set by using the same programming statements in a DATA step.

The option COVS(AGGREGATE) is specified in the PROC statement to obtain the robust sandwich estimate of the covariance matrix, and the score residuals used in computing the middle part of the sandwich estimate are aggregated over identical ID values. You can also include TEST statements in the PROC PHREG code to test various linear hypotheses of the regression parameters based on the robust sandwich covariance matrix estimate.

Consider the AIDS study data in Wei, Lin, and Weissfeld (1989) from a randomized clinical trial to assess the antiretroviral capacity of ribavirin over time in AIDS patients. Blood samples were collected at weeks 4, 8, and 12 from each patient in three treatment groups (placebo, low dose of ribavirin, and high dose). For each serum sample, the failure time is the number of days before virus positivity was detected. If the sample was contaminated or it took a longer period of time than was achievable in the laboratory, the sample was censored. For example:

• Patient #1 in the placebo group has uncensored times 9, 6, and 7 days (that is, it took 9 days to detect viral positivity in the first blood sample, 6 days for the second blood sample, and 7 days for the third blood sample).

• Patient #14 in the low-dose group of ribavirin has uncensored times of 16 and 17 days for the first and second sample, respectively, and a censored time of 21 days for the third blood sample.

• Patient #28 in the high-dose group has an uncensored time of 21 days for the first sample, no measurement for the second blood sample, and a censored time of 25 days for the third sample.

For a full-rank parameterization, two design variables are sufficient to represent three treatment groups. Based on the reference coding with placebo as the reference, the values of the two dummy explanatory variables Z1 and Z2 representing the treatments are as follows:

The bulk of the task in using PROC PHREG to perform the WLW analysis lies in the preparation of the input data set. As discussed earlier, the input data set should contain the ID, Enum, Time, and Status variables, and event-specific independent variables Z11, Z12, Z13, Z21, Z22, and Z23. Data for the three patients described earlier are arrayed as follows:

The first three rows are data for Patient #1 with event times at 9, 6, and 7 days, one row for each event. The next three rows are data for Patient #14, who has an uncensored time of 16 days for the first serum sample, an uncensored time of 17 days for the second sample, and a censored time of 21 days for the third sample. The last two rows are data for Patient #28 of the high-dose group (Z1=0 and Z2=1). Since the patient did not have a second serum sample, there are only two rows of data.

To perform the WLW analysis, you specify the following statements:

proc phreg covs(aggregate);
   model Time*Status(0)=Z11 Z12 Z13 Z21 Z22 Z23;
   strata Enum;
   id ID;
   Z11= Z1 * (Enum=1);
   Z12= Z1 * (Enum=2);
   Z13= Z1 * (Enum=3);
   Z21= Z2 * (Enum=1);
   Z22= Z2 * (Enum=2);
   Z23= Z2 * (Enum=3);
   EqualLowDose: test Z11=Z12, Z12=Z23;
   AverageLow: test Z11,Z12,Z13 / e average;
run;

Two linear hypotheses are tested using the TEST statements. The specification

   EqualLowDose: test Z11=Z12, Z12=Z13;

tests the null hypothesis of identical low-dose effects across three marginal models. The specification

 AverageLow: test Z11,Z12,Z13 / e average;

tests the null hypothesis of no low-dose effects (that is, ). The AVERAGE option computes the optimal weights for estimating the average low-dose effect and performs a 1 DF test for testing the null hypothesis that . The E option displays the coefficients for the linear hypotheses, including the optimal weights.

Marginal Cox Models for Clustered Data

Suppose there are clusters with members in the th cluster, . Let be the covariate process associated with the th member of the th cluster. The marginal Cox model is given by

where is an arbitrary baseline hazard function and is the vector of regression coefficients. Lee, Wei, and Amato (1992) estimate by the maximum partial likelihood estimate under the independent working assumption, and use a robust sandwich covariance estimate to account for the intracluster dependence.

To use PROC PHREG to analyze the clustered data, each member of a cluster is represented by an observation in the input data set. The input data set to PROC PHREG should contain the following:

• an ID variable to identify the cluster so that members of the same cluster have the same ID value

• a Time variable to represent the observed survival time of a member of a cluster

• a Status variable to indicate whether the Time value is an uncensored or censored time. For example, Status=1 indicates an uncensored time and Status=0 indicates a censored time.

• the explanatory variables thought to be related to the failure time

Consider a tumor study in which one of three female rats of the same litter was randomly subjected to a drug treatment. The failure time is the time from randomization to the detection of tumor. If a rat died before the tumor was detected, the failure time was censored. For example:

• In litter #1, the drug-treated rat has an uncensored time of 101 days, one untreated rat has a censored time of 49 days, and the other untreated rat has a failure time of 104 days.

• In litter #3, the drug-treated rat has a censored time of 104 days, one untreated rat has a censored time of 102 days, and the other untreated rat has a censored time of 104 days.

In this example, a litter is a cluster and the rats of the same litter are members of the cluster. Let Trt be a 0-1 variable representing the treatment a rat received, with value 1 for drug treatment and 0 otherwise. Data for the two litters of rats described earlier contribute six observations to the input data set:

The analysis of Lee, Wei, and Amato (1992) can be performed by PROC PHREG as follows:

proc phreg covs(aggregate);
   model Time*Status(0)=Treatment;
   id Litter;
run;

Intensity and Rate/Mean Models for Recurrent Events Data

Suppose each subject experiences recurrences of the same phenomenon. Let be the number of events a subject experiences over the interval [0,] and let be the covariate process of the subject.

The intensity model (Andersen and Gill; 1982) is given by

where represents all the information of the processes and up to time , is an arbitrary baseline intensity function, and is the vector of regression coefficients. This model consists of two components: (1) all the influence of the prior events on future recurrences, if there is any, is mediated through the time-dependent covariates, and (2) the covariates have multiplicative effects on the instantaneous rate of the counting process. If the covariates are time invariant, the risk of recurrences is unaffected by the past events.

The proportional rates and means models (Pepe and Cai 1993; Lawless and Nadeau 1995; Lin et al. 2000) assume that the covariates have multiplicative effects on the mean and rate functions of the counting process. The rate function is given by

where is an unknown continuous function and is the vector of regression parameters. If is time invariant, the mean function is given by

For both the intensity and the proportional rates/means models, estimates of the regression coefficients are obtained by solving the partial likelihood score function. However, the covariance matrix estimate for the intensity model is computed as the inverse of the observed information matrix, while that for the proportional rates/means model is given by a sandwich estimate. For a given pattern of fixed covariates, the Nelson estimate for the cumulative intensity function is the same for the cumulative mean function, but their standard errors are not the same.

To fit the intensity or rate/mean model by using PROC PHREG, the counting process style of input is needed. A subject with events contributes +1 observations to the input data set. The th observation of the subject identifies the time interval from the th event or time 0 (if ) to the th event, . The th observation represents the time interval from the th event to time of censorship. The input data set should contain the following variables:

• a TStart variable to represent the th recurrence time or the value 0 if

• a TStop variable to represent the th recurrence time or the follow-up time if

• a Status variable indicating whether the TStop time is a recurrence time or a censored time; for example, Status=1 for a recurrence time and Status=0 for censored time

• explanatory variables thought to be related to the recurrence times

If the rate/mean model is used, the input data should also contain an ID variable for identifying the subjects.

Consider the chronic granulomatous disease (CGD) data listed in Fleming and Harrington (1991). The disease is a rare disorder characterized by recurrent pyrogenic infections. The study is a placebo-controlled randomized clinical trial conducted by the International CGD Cooperative Study to assess the effect of gamma interferon to reduce the rate of infection. For each study patient the times of recurrent infections along with a number of prognostic factors were collected. For example:

• Patient #17404, age 38, in the gamma interferon group had a follow-up time of 293 without any infection.

• Patient #204001, age 12, in the placebo group had an infection at 219 days, a recurrent infection at 373 days, and was followed up to 414 days.

Let Trt be the variable representing the treatment status with value 1 for gamma interferon and value 2 for placebo. Let Age be a covariate representing the age of the CGD patient. Data for the two CGD patients described earlier are given in the following table.

Since Patient #174054 had no infection through the end of the follow-up period (293 days), there is only one observation representing the period from time 0 to the end of the follow-up. Data for Patient #204001 are broken into three observations, since there are two infections. The first observation represents the period from time 0 to the first infection, the second observation represents the period from the first infection to the second infection, and the third time period represents the period from the second infection to the end of the follow-up.

The following specification fits the intensity model:

proc phreg;
   model (TStart,TStop)*Status(0)=Trt Age;
   run;

You can predict the cumulative intensity function for a given pattern of fixed covariates by specifying the CUMHAZ= option in the BASELINE statement. Suppose you are interested in two fixed patterns, one for patients of age 30 in the gamma interferon group and the other for patients of age 1 in the placebo group. You first create the SAS data set as follows:

data Pattern;
   Trt=1; Age=30;
   output;
   Trt=2; Age=1;
   output;
   run;

You then include the following BASELINE statement in the PROC PHREG specification. The CUMHAZ=_all_ option produces the cumulative hazard function estimates, the standard error estimates, and the lower and upper pointwise confidence limits.

baseline covariates=Pattern out=out1 cumhaz=_all_;

The following specification of PROC PHREG fits the mean model and predicts the cumulative mean function for the two patterns of covariates in the Pattern data set:

proc phreg covs(aggregate);
   model (Tstart,Tstop)*Status(0)=Trt Age;
   baseline covariates=Pattern out=out2 cmf=_all_;
   id ID;

The COV(AGGREGATE) option, along with the ID statement, computes the robust sandwich covariance matrix estimate. The CMF=_ALL_ option adds the cumulative mean function estimates, the standard error estimates, and the lower and upper pointwise confidence limits to the OUT=Out2 data set.

PWP Models for Recurrent Events Data

Let be the number of events a subject experiences by time . Let be the covariate vectors of the subject at time . For a subject who has events before censorship takes place, let , let be the th recurrence time, , and let be the censored time. Prentice, Williams, and Peterson (1981) consider two time scales, a total time from the beginning of the study and a gap time from immediately preceding failure. The PWP models are stratified Cox-type models that allow the shape of the hazard function to depend on the number of preceding events and possibly on other characteristics of { and }. The total time and gap time models are given, respectively, as follows:

where is an arbitrary baseline intensity functions, and is a vector of stratum-specific regression coefficients. Here, a subject moves to the th stratum immediately after his th recurrence time and remains there until the th recurrence occurs or until censorship takes place. For instance, a subject who experiences only one event moves from the first stratum to the second stratum after the event occurs and remains in the second stratum until the end of the follow-up.

You can use PROC PHREG to carry out the analyses of the PWP models, but you have to prepare the input data set to provide the correct risk sets. The input data set for analyzing the total time is the same as the AG model with an additional variable to represent the stratum that the subject is in. A subject with events contributes +1 observations to the input data set, one for each stratum that the subject moves to. The input data should contain the following variables:

• a TStart variable to represent the th recurrence time or the value 0 if

• a TStop variable to represent the th recurrence time or the time of censorship if

• a Status variable with value 1 if the Time value is a recurrence time and value 0 if the Time value is a censored time

• an Enum variable representing the index of the stratum that the subject is in. For a subject who has only one event at and is followed to time , Enum=1 for the first observation (where Time= and Status=1) and Enum=2 for the second observation (where Time= and Status=0).

• explanatory variables thought to be related to the recurrence times

To analyze gap times, the input data set should also include a GapTime variable that is equal to (TStop TStart).

Consider the data of two subjects in CGD data described in the previous section:

• Patients #174054, age 38, in the gamma interferon group had a follow-up time of 293 without any infection.

• Patient #204001, age 12, in the placebo group had an infection at 219 days, a recurrent infection at 373 days, and a follow-up time of 414 days.

To illustrate, suppose all subjects have at most two observed events. The data for the two subjects in the input data set are as follows:

Subject #174054 contributes only one observation to the input data, since there is no observed event. Subject #204001 contributes three observations, since there are two observed events.

To fit the total time model of PWP with stratum-specific slopes, either you can create the stratum-specific explanatory variables (Trt1, Trt2, and Trt3 for Trt, and Age1, Age2, and Age3 for Age) in a DATA step, or you can specify them in PROC PHREG by using programming statements as follows:

proc phreg;
   model (TStart,TStop)*Status(0)=Trt1 Trt2 Trt3 Age1 Age2 Age3;
   strata Enum;
   Trt1= Trt * (Enum=1);
   Trt2= Trt * (Enum=2);
   Trt3= Trt * (Enum=3);
   Age1= Age * (Enum=1);
   Age2= Age * (Enum=2);
   Age3= Age * (Enum=3);
   run;

To fit the total time model of PWP with the common regression coefficients, you specify the following:

proc phreg;
   model (TStart,TStop)*Status(0)=Trt Age;
   strata Enum;
   run;

To fit the gap time model of PWP with stratum-specific regression coefficients, you specify the following:

proc phreg;
   model Gaptime*Status(0)=Trt1 Trt2 Trt3 Age1 Age2 Age3;
   strata Enum;
   Trt1= Trt * (Enum=1);
   Trt2= Trt * (Enum=2);
   Trt3= Trt * (Enum=3);
   Age1= Age * (Enum=1);
   Age2= Age * (Enum=2);
   Age3= Age * (Enum=3);
run;

To fit the gap time model of PWP with common regression coefficients, you specify the following:

proc phreg;
   model Gaptime*Status(0)=Trt Age;
   strata Enum;
   run;

Product-Limit Estimates

Let denote the set of individuals censored in the half-open interval , where and . Let denote the censoring times in ; l ranges over .

The likelihood function for all individuals is given by

where is empty. The likelihood is maximized by taking for and allowing the probability mass to fall only on the observed event times , , . By considering a discrete model with hazard contribution at , you take , where . Substitution into the likelihood function produces

If you replace with estimated from the partial likelihood function and then maximize with respect to , , , the maximum likelihood estimate of becomes a solution of

When only a single failure occurs at , can be found explicitly. Otherwise, an iterative solution is obtained by the Newton method.

The estimated baseline cumulative hazard function is

where is the estimated baseline survivor function given by

For details, refer to Kalbfleisch and Prentice (1980). For a given realization of the explanatory variables , the product-limit estimate of the survival function at is

Empirical Cumulative Hazards Function Estimates

Let be a given realization of the explanatory variables. The empirical cumulative hazard function estimate at is

The variance estimator of is given by the following (Tsiatis; 1981):

where is the estimated covariance matrix of and

For the marginal model, the variance estimator computation follows Spiekerman and Lin (1998).

The empirical cumulative hazard function (CH) estimate of the survivor function for is

Confidence Intervals for the Survivor Function

Let and correspond to the product-limit (PL) and empirical cumulative hazard function (CH) estimates of the survivor function for , respectively. Both the standard error of log() and the standard error of log() are approximated by , which is the square root of the variance estimate of ; refer to Kalbfleisch and Prentice (1980, p. 116). By the delta method, the standard errors of and are given by

respectively. The standard errors of log[–log()] and log[–log()] are given by

respectively.

Let be the upper percentile point of the standard normal distribution. A confidence interval for the survivor function is given in the following table.

Checking the Functional Form of a Covariate

To check the functional form of the th covariate, consider the partial-sum process of :

Under that null hypothesis that the model holds, can be approximated by the zero-mean Gaussian process

where are independent standard normal variables that are independent of , .

You can assess the functional form of the th covariate by plotting a small number of realizations (the default is 20) of on the same graph as the observed and visually comparing them to see how typical the observed pattern of is of the null distribution samples. You can supplement the graphical inspection method with a Kolmogorov-type supremum test. Let be the observed value of and let . The -value is approximated by , which in turn is approximated by generating a large number of realizations (1000 is the default) of .

Checking the Proportional Hazards Assumption

Consider the standardized empirical score process for the th component of

Under the null hypothesis that the model holds, can be approximated by

where is the th component of , and are independent standard normal variables that are independent of , .

You can assess the proportional hazards assumption for the th covariate by plotting a few realizations of on the same graph as the observed and visually comparing them to see how typical the observed pattern of is of the null distribution samples. Again you can supplement the graphical inspection method with a Kolmogorov-type supremum test. Let be the observed value of and let . The -value is approximated by , which in turn is approximated by generating a large number of realizations (1000 is the default) of .