Uses the Tobit method to perform analysis of variance with censored data (R.W. Payne & V.M. Cave).
Options
PRINT = string tokens |
Controls printed output from the analysis (aovtable, information, covariates, effects, residuals, contrasts, means, %cv, missingvalues, monitoring, censored); default aovt, mean |
FACTORAL = scalar |
Limit on number of factors in a treatment term; default 3 |
CONTRAST = scalar |
Limit on the order of a contrast of a treatment term; default 4 |
DEVIATIONS = scalar |
Limit on the number of factors in a treatment term for the deviations from its fitted contrasts to be retained in the model; default 9 |
PFACTORIAL = scalar |
Limit on number of factors in printed tables of means or effects; default 9 |
PCONTRASTS = scalar |
Limit on order of printed contrasts; default 9 |
PDEVIATIONS = scalar |
Limit on number of factors in a treatment term whose deviations from the fitted contrasts are to be printed; default 9 |
FPROBABILITY = string token |
Printing of probabilities for variance ratios (yes, no); default no |
PSE = string token |
Standard errors to be printed with tables of means, PSE=* requests s.e.’s to be omitted (differences, lsd, means); default diff |
TWOLEVEL = string tokens |
Representation of effects in 2n experiments (responses, Yates, effects); default resp |
WEIGHTS = variate |
Weights for each unit; default * i.e. all units with weight one |
ORTHOGONAL = string token |
Whether or not design to be assumed orthogonal (notassumed, assumed, compulsory); default nota |
SEED = scalar |
Seed for random numbers to generate dummy variate for determining the design; default 12345 |
MAXCYCLE = scalar |
Sets a limit on the number of iterations performed by the E-M algorithm; default 100 |
TOLERANCE = scalar |
Sets tolerance limits for convergence of the E-M algorithm on the estimates of the censored observations; default 0.001 |
RMETHOD = string token |
Which random terms to use (all, final); default fina |
DIRECTION = string tokens |
Whether the data are left or right censored (left, right); default left |
AMAXCYCLE = scalar |
Maximum number of iterations for estimating missing values; default 20 |
ATOLERANCE = variate |
Allows you to redefine the tolerances for zero used by various parts of the algorithm |
NOMESSAGE = string tokens |
Which warning messages to suppress (nonorthogonal, residual); default * |
LSDLEVEL = scalar |
Significance level (%) to use in the calculation of least significant differences; default 5 |
Parameters
Y = variates |
Each of these contains the data values for an analysis |
RESIDUALS = variates |
Saves the residuals from each analysis |
FITTEDVALUES = variate |
Saves 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; 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 from each analysis (0 for success, 1 for failure to converge, 2 for failure in the ANOVA analysis) |
SAVE= identifiers |
ANOVA save structures from the analyses of the Y variates with censored observations replaced by their estimates |
Description
The ATOBIT procedure performs an analysis of variance 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 by the BLOCKSTRUCTURE, TREATMENTSTRUCTURE and COVARIATE directives, as with the ANOVA directive (which is used by ATOBIT).
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 does an analysis of variance, 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, two for a failure in the analysis of variance, or a missing value for an earlier fault.
The RMETHOD specifies how strata other than the bottom stratum in the analysis of variance are used when estimating values for the censored observations during the E-step of the E-M algorithm. With the default, RMETHOD=final, the censored observations are estimated from the treatment and the block residuals from the higher strata, and the variance of their Normal distributions is given by variance component of the bottom stratum (i.e. its residual mean square). Alternatively, when RMETHOD=all, the censored observations are estimated from the treatment effects only, and variance of the Normal distributions is the sum of the variance components of all the strata.
The FACTORIAL, CONSTRASTS, DEVIATIONS, PFACTORIAL, PCONSTRASTS, PDEVIATIONS, FPROBABILITY, PSE, TWOLEVEL, WEIGHTS, ORTHOGONAL, SEED, NOMESSAGE, and LSDLEVEL options operate as in the ANOVA directive to control the operation and output of the analysis of variance. The AMAXCYCLE and ATOLERANCES options supply the settings for the MAXCYCLE and TOLERANCES options of ANOVA, as ATOBIT has its own MAXCYCLE and TOLERANCE options (described above). The PRINT option contains the same settings are as the PRINT option of ANOVA 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.
The RESIDUALS and FITTEDVALUES parameters can save the residuals and fitted values, respectively.
Options: PRINT, FACTORIAL,CONSTRASTS, DEVIATIONS, PFACTORIAL , PCONSTRASTS, PDEVIATIONS, FPROBABILITY, PSE, TWOLEVEL, WEIGHTS, ORTHOGONAL, SEED, MAXCYCLE , TOLERANCE , RMETHOD, DIRECTION, AMAXCYCLE, ATOLERANCES, NOMESSAGE, LSDLEVEL
Parameters: Y, RESIDUALS, FITTEDVALUES, BOUND, INITIAL,NEWY,EXIT, SAVE
Action with RESTRICT
If the Y variate is restricted, only the units not excluded by the restriction will be analysed.
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.
See also
Directives: ANOVA
Procedures: AUTOBIT, CENCOR, GLTOBITPOISSON, HGTOBITPOISSONRGTOBIT, RNTOBIT RNBTOBIT, RTOBITPOISSON, TOBIT
GenStat Reference Manual 1 Summary section on: Analysis of variance
Example
CAPTION 'ATOBIT example',\
!t('Split plot design, see Guide to Anova and Design',\
' in Genstat, Section 5.1.'); STYLE=meta,plain
SPLOAD [PRINT=summary] '%Data%/Oats.gsh'
CAPTION 'Yield left-censored data at 70.',\
'Results match the TOBIT analyses (by REML) when RMETHOD=all.';\
STYLE=meta,plain
BLOCKS blocks/wplots/subplots
TREATMENTS nitrogen*variety
ATOBIT [PRINT=aovtable,means,censored; RMETHOD=all; NOMESSAGE=residual;\
FPROBABILITY=yes] yield; BOUND=70
CAPTION !t('However, results do not match the TOBIT analyses when',\
'RMETHOD=final. (The ANOVA residuals are least-squares estimates,',\
'whereas the REML residuals are BLUPs.) See the TOBIT example.')
ATOBIT [PRINT=aovtable,means,censored; RMETHOD=final; NOMESSAGE=residual;\
FPROBABILITY=yes] yield; BOUND=70