Purpose
Computes delta beta values for specific parameters in a model.
Programs
GMBO, PEANUTS, AMFIT
Syntax
DFBETA #pnlist {STANDARD | UNSTANDARD} @
Arguments and Subcommands
#pnlist
A list of the parameter numbers for which delta beta values are to be computed. The variables are called %DB_nn where nn is the parameter number (with a leading 0 if the number is less than 10).
STANDARD
Standardize the delta beta values by dividing the residuals by the asymptotic standard error of the associated parameter. This has no effect in GMBO or AMFIT.
UNSTANDARD
Do not standardize the delta beta values. This is the default each time this command is used. This has no effect in GMBO or AMFIT.
Remarks
The delta beta value for parameter j in the ith record is an estimate of the difference between the parameter estimate based on all of the data and the estimate that would be obtained if the model were fit to a sample that does not include the ith record. In linear regression Belsley et al. (1980) refer to unstandardized delta beta values as DFBETA, and standardized values as DFBETAS.
For Poisson and binomial regression models these values are simple functions of the Pearson residual and the generalized hat diagonal for parameter j. In particular, where is the Pearson residual (see EPICURE User’s Guide) and is the generalized hat matrix diagonal for parameter j. Generalized hat matrix diagonals for logistic regression were developed by Pregibon (1981) and are described by Hosmer and Lemeshow (1989, p. 151). The extension to Poisson or other generalized linear models is discussed in McCullagh and Nelder (1989 p. 397). Further extension to the general class of non-linear models available in EPICURE is straightforward, involving the replacement of covariate values by derivatives of the rate function.
For survival data fit using partial-likelihood methods, the delta beta vector for an observation is a defined as where is the information matrix evaluated for the current parameter estimates and is the score residual vector for observation i. (Score residuals for these models are discussed in the Remarks section of the PEANUTS RESIDUAL command description). The delta beta value for a given parameter is standardized by dividing it by the estimated standard error of the parameter. Unstandardized delta beta values can be used to compute robust variance and covariance estimates. The robust variance estimate for a parameter is equal to the sum of the squares of the unstandardized delta betas for that parameter. The sum of products of delta beta values for two parameters provides a robust estimate of the covariance between the two parameters. These estimates are not computed directly, but can easily be computed from unstandardized delta beta estimates using simple transformations as illustrated in the examples shown below.
The DFBETA command is used in the V2FEAT.PNT, MLUNG.PNT examples.
Examples
a) Fit a model with sex and treatment effects and then compute delta beta values for these effects. For this example, we assume that we are fitting a model using the default subterm in AMFIT or GMBO. If there are ties, this value will be recoded for the first in a set of records with tied failure times. For other records with failures at this time, the value is missing.
FIT sex trmnt @
DFBETA 2 3 @
b) Compute standardized dfbeta estimates for sex and treatment parameters in a proportional hazards model fit using PEANUTS. There is no constant term in this model so the parameters are numbered 1 and 2. Plot the delta beta value for sex against its rank (sort order).
FIT sex trmnt @
DFBETA 1 2 STANDARD @
PLOT #DB_01 @
c) Compute unstandardized delta beta estimates for sex and treatment parameters in a proportional hazards model fit using PEANUTS. Then use the resulting variables to compute robust variance estimates for these parameters.
FIT sex trmnt @
DFBETA 1 2 @
CONS #rvsex = 0; #rvtrt = 0 @
TRAN #rvsex = #rvsex + %DB_01*%DB_01 ;
#rvtrt = #rvtrt + %DB_02*%DB_02 ;
@