GLTOBITPOISSON procedure

Uses the Tobit method to fit a generalized linear mixed model with censored Poisson data (R.W. Payne).

Options

Parameters

Description

When an experiment generates a mixture of small and very large counts, it may be convenient to count only the observations less than a specified boundary value, and enter that value for the larger observations. The data then come from a right-censored Poisson distribution. In the similar (but less common) left-censored situation, the emphasis is on the larger observations. It may then not be worth recording the small observations in detail, only that they are no larger than the boundary value. Censored Poisson data can be analysed by the Tobit method (Terza 1985), which is implemented in this procedure.

In the Tobit model, the probabilities for the uncensored observations are standard Poisson probabilities. The probabilities for right-censored observations are cumulative upper Poisson probabilities for values greater than or equal to the boundary value. Probabilities for left-censored observations cumulative lower Poisson probabilities for values less than or equal to the boundary value. The Tobit method uses an E-M (expectation-maximization) algorithm to estimate values for the censored observations. It starts with initial estimates for the censored observations, which can be specified by the INITIAL parameter in either a variate or a scalar. For right-censored data the default is to use the boundary value plus one. For left-censored data the default is the boundry value minus one. In each iteration, the method first fits a Poisson-log generalized linear mixed model, saving the resulting fitted values to provide estimated means for the Poisson distributions of the censored observations. The new estimates for the censored observations are then given by the expected values for the upper parts of those Poisson distributions. The process continues either until the updates to the estimates are less than or equal to the value specified by the TOLERANCE option (default 0.001), or until the number of iterations equals the number specified by the MAXCYCLE option (default 100). The EXIT parameter can be set to a scalar that will be set to zero for a successful fit, one for failure in the E-M algorithm, two if the generalized linear mixed model has failed to fit, or a missing value for an earlier fault.

The fixed and random models are specified by the FIXED and RANDOM options, respectively. By default the variance components are constrained to be positive, but you can set option VCONSTRAINTS to none to allow them to become negative. The model can also contain an offset, specified by the OFFSET parameter. The CONSTANT option indicates whether the constant is to be estimated or omitted, and the FACTORIAL option sets a limit on the number of variates and/or factors in the model terms, in the usual way. As in the REML directive, the MVINCLUDE option specifies whether units with missing values in explanatory factors and variates and/or y-variates are to be included. The DISPERSION option specifies the dispersion parameter (default 1). A missing value indicates that the dispersion parameter is to be estimated.

The response variate is specified by the Y parameter, and the NEWY parameter can save a variate where the censored observations are replaced by their estimates. The BOUND option specifies the boundary value for the censoring (and the value that has been entered to indicate the censored observations in the Y variate). The DIRECTION option specifies whether the data are left or right censored. The default is that they are right censored.

The GLMAXCYCLE and GLTOLERANCE options specify the maximum number of iterations (default 20) and tolerance (default 0.0001) for the fit of the generalized linear model by the GLMM procedure. The FMETHODGLMM option specifies the method used to form the fitted values in the GLMM analysis, and thus determines the fitting method to be used. The default setting all specifies that both fixed and random terms should be used to form fitted values which gives the method of Schall (1991); setting fixed indicates that only fixed terms are used to form fitted values which gives the marginal method of Breslow & Clayton (1993).

The VMAXCYCLE option sets the limit on the number of iterations in the REML algorithm (default 30). The VMETHOD option specifies the algorithm to use in the REML steps of the GLMM algorithm: either Fisher or AI(default). By default any covariates are centred for the REML fitting by subtracting their means, weighted according to the iterative weights of the generalized linear model. Alternatively you can set option CADJUST=none to request that the uncentred covariates are used instead. The WORKSPACE option specifies the number of blocks of internal memory to be set up for use by the REML algorithm; default 1

The PRINT option controls the printed output. The settings are as in the GLMM procedure, except that the monitoring setting prints monitoring information for the E-M algorithm, and that there is an additional setting censored to print the estimates of the censored observations. The options PTERMS and PSE all operate like those of GLMM and REML.

The GLSAVE parameter can save a pointer, with information about the GLMM analysis, for use by procedures like GLDISPLAY or GLKEEP. Alternatively the SAVE parameter can save a REML save structure to provide a more direct route to the REML part of the analysis. This can be used by the directives VDISPLAY , VKEEP etc. Options: PRINT, DISPERSION, RANDOM, FIXED, CONSTANT, FACTORIAL, PTERMS, PSE, MVINCLUDE, MAXCYCLE, TOLERANCE, DIRECTION, GLMAXCYCLE, GLTOLERANCE, FMETHODGLMM, CADJUST, WORKSPACE, VCONSTRAINTS, VMETHOD, VMAXCYCLE. Parameters: Y, BOUND, INITIAL, NEWY, OFFSET, EXIT, GLSAVE, SAVE.

Method

The generalized linear mixed model is fitted by the GLMM procedure. The expected values for the upper parts of the Poisson distributions are calculated by the EUPOISSON procedure, and those for the lower parts of the distributions are calculated by the ELPOISSON procedure.

Action with RESTRICT

If the Y-variate is restricted, only the units not excluded by the restriction will be analysed.

References

Breslow, N.E. & Clayton, D.G. (1993). Approximate inference in generalized linear mixed models. Journal of the American Statistical Association, 88, 421, 9-25.

Schall, R. (1991) Estimation in generalized linear models with random effects. Biometrika, 78, 719-727.

Terza, J.V. (1985). A Tobit-type estimator for the censored Poisson regression model. Economics Letters, 18, 361-365.

See also

Directive: REML.
Procedures: CENSOR, ELPOISSON, EUPOISSON, GLMM, GLDISPLAY, GLKEEP, GLPERMTEST, GLPLOT, GLPREDICT, GLRTEST, HGTOBITPOISSON, RTOBITPOISSON, TOBIT.
Commands for: Regression analysis.

Example

CAPTION        'GLTOBITPOISSON example',\
               !t('Nematode data from Cochran & Cox (1957) p.46,',\
               'analysed in Section 4.3 of the Statistics Guide,',\
               'with the unfumigated plots removed to simplify the analysis.',\
               'Suppose that counting stopped at 400.',\
               'Units 18-20, 24 & 27 are then censored.');  STYLE=meta,plain
SPLOAD         '%data%/Nematode.gsh'
SUBSET         [Fumigant.IN.'Fumigated'; SETLEVELS=yes]\
               Blocks,Amount,Type,Count,Priorcount
CALCULATE      Logpriorcount = LOG(Priorcount)
GLTOBITPOISSON [PRINT=model,components,fittedvalues,means,backmeans,\
               waldtests,censored; RANDOM=Blocks; FIXED=Amount*Type;\
               DISPERSION=*] Count; BOUND=400; OFFSET=Logpriorcount