AUTOBIT procedure

Uses the Tobit method for analysis of variance of an unbalanced design with censored data (R.W. Payne & V.M. Cave).

Options

`PRINT` = string tokens	Controls printed output (aovtable, effects, means, residuals, %cv, censored, monitoring); default aovt, mean, cens
`FACTORIAL` = scalar	Limit on number of factors in a treatment term; default 3
`PFACTORIAL` = scalar	Limit on number of factors in printed tables of predicted means; default 3
`FPROBABILITY` = string token	Printing of probabilities for variance ratios in the analysis-of-variance table (yes, no); default no
`TPROBABILITY` = string token	Printing of probabilities for t-tests of effects (yes, no); default no
`COMBINATIONS` = string token	Factor combinations for which to form predicted means (present, estimable); default esti
`ADJUSTMENTS` = string token	Type of adjustment to be made when predicting means (marginal, equal, observed); default marg
`PSE` = string token	Types of standard errors to be printed with the predicted means (differences, alldifferences, lsd, alllsd, means, ese); default diff
`MAXCYCLE` = scalar	Sets a limit on the number of iterations performed by the E-M algorithm; default 100
`TOLERANCE` = variate	Sets tolerance limits for convergence of the E-M algorithm on the estimates of the censored observations; default 0.001
`DIRECTION` = string tokens	Whether the data are left or right censored (left, right); default left
`NOMESSAGE` = string tokens	Which warning messages to suppress (dispersion, leverage, residual, aliasing, marginality, vertical, df, inflation); default * i.e. none
`LSDLEVEL` = scalar	Significance level (%) for least significant differences; default 5

Parameters

`Y` = variates	Response variates to be analysed; must be set
`RESIDUALS` = variates	Variate to save the residuals from each analysis
`FITTEDVALUES` = variate	Variate to save the fitted values from each analysis
`BOUND` = scalars, variates or pointers	Censoring thresholds; must be set
`INITIAL` = scalar or variates	Scalar or a variate providing starting values for the censored observations in the E-M algorithm for each analysis; default BOUND+1 for right-censored data and BOUND−1 for left-censored data
`NEWY` = variates	Saves a copy of each response variate with the censored observations replaced by their estimates
`EXIT`= scalars	Exit status (0 for success, 1 for failure to converge)
`SAVE`= regression save structures	Save structures from the analyses of the data with censored observations replaced by their estimates

Description

The AUTOBIT procedure performs an analysis of variance for an unbalanced design with censored data. For example, with the default, left-censoring, some observations may be below the reliable detection limit of a measuring device. Alternatively, with right-censoring (specified by setting option DIRECTION = right), some observations may be so large that it is impracticable to measure them exactly. You can also set DIRECTION = left,right to have censoring in both directions.

The values at which the measurements are censored must be specified by the BOUND parameter. For censoring in a single direction, this can be a scalar if all observations are censored at the same point, or a variate if they are censored at different points. If there is both left and right censoring, BOUND supplies a pointer containing, first, a scalar or variate to define the left-hand bounds, and then a scalar or variate to define the right-hand bounds.

Censored observations in the data, supplied by the Y parameter, are represented as values at or outside the boundary. The NEWY parameter can save a copy of the y-variate with the censored observations replaced by their estimates.

The model to be fitted in the analysis of variance must be specified beforehand in the same way as for the AUNBALANCED procedure. The treatment terms are specified by the TREATMENTSTRUCTURE directive. Similarly, any covariates are defined by the COVARIATE directive. AUTOBIT also takes account of any blocking structure specified by the BLOCKSTRUCTURE directive. However, it cannot produce stratified analyses like those generated by ANOVA, and is able to estimate treatments and covariates only in the “bottom stratum”. So, for example, the full analysis can be produced for a randomized block design, where the treatments are all estimated on the plots within blocks, but it cannot produce the whole-plot analysis in a split plot design. Instead you can analyse these by REML using the TOBIT procedure.

In the Tobit model (Tobin 1958), the probabilities for the uncensored observations are standard Normal probabilities. The probabilities for right-censored observations are cumulative upper Normal probabilities for values greater than or equal to the boundary value. Probabilities for left-censored observations are cumulative lower Normal 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. (See Dempster, Laird, N.M. & Rubin 1977.) 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 boundary value minus one. In each iteration, the method uses the FIT directive to fit the model, saving the resulting fitted values to provide estimated means for the distributions of the censored observations. The new estimates for the censored observations are then given by the expected values for the lower or upper parts of the Normal distributions, according to whether the observations are left- or right-censored. 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 which will be set to zero for a successful fit, one for failure in the E-M algorithm, or a missing value for an earlier fault.

The FACTORIAL, FPACTORIAL, FPROBABILITY, TPROBABILITY, COMBINATIONS, ADJUSTMENT, PSE, NOMESSAGE, and LSDLEVEL options operate as in the AUNBALANCED procedure to control the operation and output of the analysis of variance. The PRINT option contains the same settings are as the PRINT option of AUNBALANCED directive, as well as a monitoring setting to print monitoring information for the E-M algorithm, and a censored setting to print the estimates of the censored observations.

Following the analysis, you can display further output, or save information, using the procedures AUDISPLAY, AUGRAPH, AUPREDICT, AUMCOMPARISON, and AUKEEP, as with the AUNBALANCED procedure. You can also use relevant regression procedures, such as RCHECK for plots of residuals. The SAVE parameter can save the regression save structure used by these procedures, so that you can display output from this analysis even if there have been other regression analyses in the intervening period.

The RESIDUALS and FITTEDVALUES parameters can save the residuals and fitted values, respectively

Options: PRINT, FACTORIAL, FPACTORIAL, FPROBABILITY, TPROBABILITY, COMBINATIONS, ADJUSTMENT, PSE, MAXCYCLE, TOLERANCE,DIRECTION, NOMESSAGE, LSDLEVEL

Parameters: Y, RESIDUALS, FITTEDVALUES, BOUND, INITIAL,NEWY,EXIT, SAVE

Action with `RESTRICT`

As in FIT, the y-variate or any of the model variates or factors can be restricted to analyse a subset of the data.

References

Dempster, A.P., Laird, N.M. & Rubin, D.B. (1977). Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society, Series B, 39, 1-38.
Tobin, J. (1958). Estimation of relationships for limited dependent variables. Econometrica, 26, 24-36.

Example

CAPTION  'AUTOBIT example',\
        !t('Experiment on foster feeding of rats from Scheffe (1959)',\
        'The Analysis of Variance; also see McConway, Jones & Taylor (1999)',\
        'Statistical Modelling using GENSTAT, Example 7.6.',\
        'To illustrate AUTOBIT, suppose rats larger than 65',\
        'could not be weighed accurately and are treated as censored.');\
        STYLE=meta,plain

FACTOR    [NVALUES=61; LABELS=!t('A','B','I','J')] litter

READ      litter; FREPRESENTATION=labels
A A A A A A A A A A A A A A A A A B B B B B B B B B B B B B B B I I I I I I
I I I I I I I I J J J J J J J J J J J J J J J :

FACTOR     [NVALUES=61; LABELS=!t('A','B','I','J')] mother

READ      mother; FREPRESENTATION=labels
A A A A A B B B I I I I J J J J J A A A A B B B B B I I I I J J A A A B B B
I I I I I J J J A A A A B B B I I I J J J J J :

VARIATE    [NVALUES=61] littwt

READ       littwt
61.5 68.2 64 65 59.7 55 42 60.2 52.5 61.8 49.5 52.7 42 54 61 48.2 39.6 60.3
51.7 49.3 48 50.8 64.7 61.7 64 62 56.5 59 47.2 53 51.3 40.5 37 36.3 68 56.3
69.8 67 39.7 46 61.3 55.3 55.7 50 43.8 54.5 59 57.4 54 47 59.5 52.8 56 45.2
57 61.4 44.8 51.5 53 42 54 :

TREATMENTSTRUCTURE  litter * mother

AUTOBIT    [PRINT=aovtable,means,censored; FPROBABILITY=yes; DIRECTION=right]\
        littwt; BOUND=65

Updated on April 15, 2024

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