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The PHREG Procedure |
MODEL Statement |
The MODEL statement identifies the variables to be used as the failure time variables, the optional censoring variable, and the explanatory effects, including covariates, main effects, interactions, nested effects; see the section Specification of Effects of Chapter 39, The GLM Procedure, for more information. A note of caution: specifying the effect T*A in the MODEL statement, where T is the time variable and A is a CLASS variable, does not make the effect time-dependent. See the section Clarification of the Time and CLASS Variables Usage for more information.
Two forms of MODEL syntax can be specified; the first form allows one response variable, while the second form allows two variables for the counting process style of input (see the section Counting Process Style of Input for more information).
In the first MODEL statement, preceding the equal sign, is the name of the failure time variable. This can optionally be followed by an asterisk, the name of the censoring variable, and a list of censoring values (separated by blanks or commas if there is more than one) enclosed in parentheses. If the censoring variable takes on one of these values, the corresponding failure time is considered to be censored. The variables following the equal sign are the explanatory variables (sometimes called independent variables or covariates) for the model.
Instead of a single failure time variable, the second MODEL statement identifies a pair of failure time variables. Their names are enclosed in parentheses, and they signify the endpoints of a semi-closed interval during which the subject is at risk. If the censoring variable takes on one of the censoring values, the time
is considered to be censored.
The censoring variable and the explanatory variables must be numeric. The failure time variables must contain nonnegative values. Any observation with a negative failure time is excluded from the analysis, as is any observation with a missing value for any of the variables listed in the MODEL statement.
Table 64.3 summarizes the options available in the MODEL statement, which can be specified after a slash (/). Four convergence criteria are allowed for the maximum likelihood optimization: ABSFCONV=, FCONV=, GCONV=, and XCONV=. If you specify more than one convergence criterion, the optimization is terminated as soon as one of the criteria is satisfied. If none of the criteria is specified, the default is GCONV=1E8.
Option |
Description |
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Model Specification Options |
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suppresses model fitting |
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specifies offset variable |
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specifies effect selection method |
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Effect Selection Options |
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controls the number of models displayed for SCORE selection |
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requests detailed results at each step |
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specifies whether and how hierarchy is maintained and whether a single effect or multiple effects are allowed to enter or leave the model per step |
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specifies number of effects included in every model |
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specifies maximum number of steps for STEPWISE selection |
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adds or deletes effects in sequential order |
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specifies significance level for entering effects |
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specifies significance level for removing effects |
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specifies number of variables in first model |
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specifies number of variables in final model |
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adds or deletes variables by residual chi-square criterion |
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Maximum Likelihood Optimization Options |
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specifies absolute function convergence criterion |
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specifies relative function convergence criterion |
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specifies Firth’s penalized likelihood method |
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specifies relative gradient convergence criterion |
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specifies relative parameter convergence criterion |
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specifies maximum number of iterations |
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specifies the initial ridging value |
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specifies the technique to improve the log likelihood function when its value is worse than that of the previous step |
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specifies tolerance for testing singularity |
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Confidence Interval Options |
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specifies |
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specifies profile-likelihood convergence criterion |
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computes confidence intervals for hazard ratios |
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Display Options |
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displays correlation matrix |
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displays covariance matrix |
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displays iteration history |
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suppresses "Class Level Information" table |
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Miscellaneous Options |
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specifies the delayed entry time variable |
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specifies the method of handling ties in failure times |
sets the significance level used for the confidence limits for the hazard ratios. The value must be between 0 and 1. The default is the value of the ALPHA= option in the PROC PHREG statement, or 0.05 if that option is not specified. This option has no effect unless the RISKLIMITS option is specified.
specifies the absolute function convergence criterion. Termination requires a small change in the objective function (log partial likelihood function) in subsequent iterations,
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where is the value of the objective function at iteration
.
is used exclusively with the SCORE model selection method. The BEST=n option specifies that n models with the highest-score chi-square statistics are to be displayed for each model size. If the option is omitted and there are no more than 10 explanatory variables, then all possible models are listed for each model size. If the option is omitted and there are more than 10 explanatory variables, then the number of models selected for each model size is, at most, equal to the number of explanatory variables listed in the MODEL statement.
See Example 64.2 for an illustration of the SCORE selection method and the BEST= option.
displays the estimated correlation matrix of the parameter estimates.
displays the estimated covariance matrix of the parameter estimates.
produces a detailed display at each step of the model-building process. It produces an "Analysis of Variables Not in the Model" table before displaying the variable selected for entry for FORWARD or STEPWISE selection. For each model fitted, it produces the "Analysis of Maximum Likelihood Estimates" table.
See Example 64.1 for a discussion of these tables.
specifies the name of the variable that represents the left truncation time. This option has no effect when the counting process style of input is specified. See the section Left Truncation of Failure Times for more information.
specifies the relative function convergence criterion. Termination requires a small relative change in the objective function (log partial likelihood function) in subsequent iterations,
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where is the value of the objective function at iteration
.
performs Firth’s penalized maximum likelihood estimation to reduce bias in the parameter estimates (Heinze and Schemper; 2001; Firth; 1993). This method is useful when the likelihood is monotone—that is, the likelihood converges to finite value while at least one estimate diverges to infinity.
specifies the relative gradient convergence criterion. Termination requires that the normalized prediction function reduction is small,
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where is the log partial likelihood,
is the gradient vector (first partial derivatives of the log partial likelihood), and
is the negative Hessian matrix (second partial derivatives of the log partial likelihood), all at iteration
.
specifies whether and how the model hierarchy requirement is applied and whether a single effect or multiple effects are allowed to enter or leave the model in one step. You can specify that only CLASS variable effects, or both CLASS and continuous variable effects, be subject to the hierarchy requirement. The HIERARCHY= option is ignored unless you also specify the FORWARD, BACKWARD, or STEPWISE selection method.
Model hierarchy refers to the requirement that, for any term to be in the model, all effects contained in the term must be present in the model. For example, in order for the interaction A*B to enter the model, the main effects A and B must be in the model. Likewise, neither effect A nor B can leave the model while the interaction A*B is in the model.
indicates that the model hierarchy is not maintained. Any single effect can enter or leave the model at any given step of the selection process.
indicates that only one effect can enter or leave the model at one time, subject to the model hierarchy requirement. For example, suppose that you specify the main effects A and B and the interaction of A*B in the model. In the first step of the selection process, either A or B can enter the model. In the second step, the other main effect can enter the model. The interaction effect can enter the model only when both main effects have already been entered. Also, before A or B can be removed from the model, the A*B interaction must first be removed. All effects (CLASS and continuous variables) are subject to the hierarchy requirement.
is the same as HIERARCHY=SINGLE except that only CLASS effects are subject to the hierarchy requirement.
indicates that more than one effect can enter or leave the model at one time, subject to the model hierarchy requirement. In a forward selection step, a single main effect can enter the model, or an interaction can enter the model together with all the effects that are contained in the interaction. In a backward elimination step, an interaction itself, or the interaction together with all the effects that the interaction contains, can be removed. All effects (CLASS and continuous variable) are subject to the hierarchy requirement.
is the same as HIERARCHY=MULTIPLE except that only CLASS effects are subject to the hierarchy requirement.
The default value is HIERARCHY=SINGLE, which means that model hierarchy is to be maintained for all effects (that is, both CLASS and continuous variable effects) and that only a single effect can enter or leave the model at each step.
includes the first effects in the MODEL statement in every model. By default, INCLUDE=0. The INCLUDE= option has no effect when SELECTION=NONE.
displays the iteration history, including the last evaluation of the gradient vector.
specifies the maximum number of iterations allowed. The default value for n is 25. If convergence is not attained in n iterations, the displayed output and all data sets created by PROC PHREG contain results that are based on the last maximum likelihood iteration.
specifies the maximum number of times the explanatory variables can move in and out of the model before the STEPWISE model-building process ends. The default value for n is twice the number of explanatory variables in the MODEL statement. The option has no effect for other model selection methods.
suppresses the "Class Level Information" table, which shows how the design matrix columns for the CLASS variables are coded.
performs the global score test, which tests the joint significance of all the explanatory variables in the MODEL statement. No parameters are estimated. If the NOFIT option is specified along with other MODEL statement options, NOFIT takes precedence, and all other options are ignored except the TIES= option.
specifies the name of an offset variable, which is an explanatory variable with a regression coefficient fixed as one. This option can be used to incorporate risk weights for the likelihood function.
controls the convergence criterion for confidence intervals based on the profile-likelihood function. The quantity value must be a positive number, with a default value of 1E4. The PLCONV= option has no effect if profile-likelihood based confidence intervals are not requested.
specifies that the diagonal elements of the negative (expected) Hessian be inflated by adding the ridge value.
specifies that the diagonal elements be inflated by the factor equal to 1 plus the ridge value.
specifies the crude line-search method of taking half a step be used instead of ridging.
specifies the initial ridge value. The maximum ridge value is 2000 times the maximum of 1 and the initial ridge value. The initial ridge value is raised to 1E–4 if it is less than 1E–4. By default, RIDGEINIT=1E–4. This option has no effect for RIDGING=ABSOLUTE.
requests profile-likelihood confidence limits.
requests confidence limits based on the Wald tests.
request both profile-likelihood and Wald confidence limits.
fits the complete model specified in the MODEL statement. This is the default value.
requests best subset selection. It identifies a specified number of models with the highest-score chi-square statistic for all possible model sizes ranging from one explanatory variable to the total number of explanatory variables listed in the MODEL statement. This option is not allowed if an explanatory effect in the MODEL statement contains a CLASS variable.
For more information, see the section Effect Selection Methods.
forces variables to be added to the model in the order specified in the MODEL statement or to be eliminated from the model in the reverse order of that specified in the MODEL statement.
specifies the singularity criterion for determining linear dependencies in the set of explanatory variables. The default value is .
specifies the significance level (a value between 0 and 1) for entering an explanatory variable into the model in the FORWARD or STEPWISE method. For all variables not in the model, the one with the smallest p-value is entered if the p-value is less than or equal to the specified significance level. The default value is 0.05.
specifies the significance level (a value between 0 and 1) for removing an explanatory variable from the model in the BACKWARD or STEPWISE method. For all variables in the model, the one with the largest p-value is removed if the p-value exceeds the specified significance level. The default value is 0.05.
begins the FORWARD, BACKWARD, or STEPWISE selection process with the first effects listed in the MODEL statement. The value of
ranges from 0 to
, where
is the total number of effects in the MODEL statement. The default value of
is
for the BACKWARD method and 0 for the FORWARD and STEPWISE methods. Note that START=
specifies only that the first
effects appear in the first model, while INCLUDE=
requires that the first
effects be included in every model. For the SCORE method, START=
specifies that the smallest models contain
effects, where
ranges from 1 to
; the default value is 1. The START= option has no effect when SELECTION=NONE.
specifies the maximum (FORWARD method) or minimum (BACKWARD method) number of effects to be included in the final model. The effect selection process is stopped when effects are found. The value of
ranges from 0 to
, where
is the total number of effects in the MODEL statement. The default value of
is
for the FORWARD method and 0 for the BACKWARD method. For the SCORE method, STOP=
specifies that the smallest models contain
effects, where
ranges from 1 to
; the default value of
is
. The STOP= option has no effect when SELECTION=NONE or STEPWISE.
specifies that the addition and deletion of variables be based on the result of the likelihood score test for testing the joint significance of variables not in the model. This score chi-square statistic is referred to as the residual chi-square. In the FORWARD method, the STOPRES option enters the explanatory variables into the model one at a time until the residual chi-square becomes insignificant (that is, until the p-value of the residual chi-square exceeds the SLENTRY= value). In the BACKWARD method, the STOPRES option removes variables from the model one at a time until the residual chi-square becomes significant (that is, until the p-value of the residual chi-square becomes less than the SLSTAY= value). The STOPRES option has no effect for the STEPWISE method.
uses the approximate likelihood of Breslow (1974). This is the default value.
replaces the proportional hazards model by the discrete logistic model
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uses the approximate likelihood of Efron (1977).
computes the exact conditional probability under the proportional hazards assumption that all tied event times occur before censored times of the same value or before larger values. This is equivalent to summing all terms of the marginal likelihood for that are consistent with the observed data (Kalbfleisch and Prentice; 1980; DeLong, Guirguis, and So; 1994).
specifies the relative parameter convergence criterion. Termination requires a small relative parameter change in subsequent iterations,
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where
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where is the estimate of the
th parameter at iteration
.
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