Pre-processes censored data before analysis by `ANOVA`

(P.W. Lane).

### Options

`PRINT` = string token |
Whether to monitor convergence (`monitor` ); default `*` implies no monitoring |
---|---|

`TERM` = formula |
Formula for lowest stratum residual term; no default – this option must be set |

`DESIGN` = pointer |
Identifier specifying design information for `ANOVA` , or to save design information; default `*` |

`MAXCYCLE` = scalar |
Maximum number of iterations; default 20 |

### Parameters

`Y` = variates |
Observed variate with censored values represented by values greater than or equal to the bound; no default – this parameter must be set |
---|---|

`BOUND` = scalars or variates |
Upper bound for censoring for each unit; no default – this parameter must be set |

`DF` = scalars |
Estimated residual d.f. for lowest stratum, adjusting for censoring; default `*` |

`NEWY` = variates |
Saves a variate with the censored values replaced by their estimates; if unset, the censored values are replaced in the original `Y` variate |

`SAVE` = identifiers |
Save details of each analysis for use in subsequent `ADISPLAY` or `AKEEP` statements |

### Description

An observation is said to be censored if it is known only that it is less than (or greater than) a particular value. Such observations can occur in designed experiments when the observed variable is the time until some event takes place for each experimental unit. For example, if the observed measurement is the lifetime of electric light-bulbs, it may happen that some bulbs are still alight when the experiment has to be concluded.

The response variate should be specified using the `Y`

parameter, representing the censored values as values that are greater than or equal to the censoring bound. The bound is specified using the `BOUND`

parameter, either as a scalar – if the bound is constant over the experiment – or as a variate of the same length as the response variate. Missing values in `Y`

will be treated as usual, not as censored values. The procedure deals with the case of censoring with an upper bound. For a problem involving a lower bound, the structures in `Y`

and `BOUND`

should be multiplied by -1 before using the procedure (and the analysis interpreted accordingly).

The results for any experiment analysable by the `ANOVA`

directive may be processed by the procedure. You must give `BLOCKSTRUCTURE`

, `TREATMENTSTRUCTURE`

and `COVARIATE`

statements, as relevant, before using the procedure. If the analysis of the experiment requires a setting of the `WEIGHTS`

, `FACTORIAL`

, `CONTRASTS`

or `DEVIATIONS`

options of the `ANOVA`

directive, you should give an `ANOVA`

statement with these settings before using the procedure, setting the `DESIGN`

option and then using the same identifier in the `DESIGN`

option of `CENSOR`

. The lowest stratum of the experiment must be identified explicitly in the `BLOCKSTRUCTURE`

statement, rather than being implicitly taken as the `*units*`

stratum by `ANOVA`

; the model term representing this stratum must be specified using the `TERM`

option of `CENSOR`

. For example, a split-plot experiment with blocks might be specified by

`BLOCKS block/plot/subplot`

`CENSOR [TERM=block.plot.subplot; ...`

If you set the option `PRINT=monitor`

, the procedure will print the values of the standard error of the lowest stratum at each cycle of the iterative estimation process. The maximum number of iterations is specified by the `MAXCYCLE`

option, with a default of 20. The `NEWY`

parameter allows you to specify a copy of the `Y`

variate with the censored values replaced by their estimates. If `NEWY`

is unset, the censored values are replaced in the original `Y`

variate. The analysis of this variate can be displayed with `ADISPLAY`

, or results saved with `AKEEP`

. The save structure for the corresponding analysis of variance can be saved using the `SAVE`

parameter.

The analysis will not be exact, due to the estimation of the censored values. However, the residual degrees of freedom in the lowest stratum may be corrected to the value output by the `DF`

parameter (this is the absolute correction referred to by Taylor 1973; see the Method section).

Options: `PRINT`

, `TERM`

, `DESIGN`

, `MAXCYCLE`

. Parameters: `Y`

, `BOUND`

, `DF`

, `NEWY`

, `SAVE`

.

### Method

The censored units in the observed variate are replaced by estimated values, using the method outlined by Taylor (1973). This method estimates the expected value of each censored observation iteratively conditional on the fact that the value must be greater than the fixed bound, and using the relevant information from the other observations in the experiment. The iterative process is deemed to have converged when the relative change in each fitted value, and in the stratum standard error, is less than 0.1%. If convergence is not reached within the number of iteractions specified by the `MAXCYCLE`

option, a message will be printed and the process will terminate. This should occur only if there is a high proportion of censored values, or if the design affords no information about one or more of the censored values.

### Action with `RESTRICT`

The `Y`

variate can be restricted, but the `BOUNDS`

variate must not be.

### Reference

Taylor, J. (1973). The analysis of designed experiments with censored observations. *Biometrics*, **29**, 35-43.

### See also

Directive: `ANOVA`

.

Procedures: `RTOBITPOISSON`

, `TOBIT`

.

Commands for: Analysis of variance.

### Example

CAPTION 'CENSOR example',!t(\ 'Analysis of an experiment testing control of bolting in',\ 'sugar beet. Two chemicals were tested, Gibberellin and Ancymidol.',\ 'The number of days to bolting was recorded, but some plants did',\ 'not bolt before harvest (78 days after chemical treatment):',\ 'these were recorded as 99 days.'); STYLE=meta,plain VARIATE [NVALUES=90] days FACTOR [NVALUES=90; LEVELS=3] gibb,ancy & [LEVELS=10] block READ days 10 11 3 15 14 12 99 11 16 99 18 12 14 23 11 12 10 14 10 26 14 24 12 99 23 16 99 99 20 99 11 11 11 10 7 7 10 11 1 10 9 11 12 8 7 9 11 12 11 10 1 3 12 11 11 14 12 11 11 15 7 4 7 10 7 3 10 11 11 10 3 7 5 2 7 7 10 11 11 12 10 24 11 10 10 7 12 11 17 7 : GENERATE gibb,ancy,block CALCULATE logdays = LOG10(50+days) & bound = LOG10(78) CAPTION !T('The experiment was a 3*3 factorial with 10 blocks:',\ 'ensure that the lower stratum is defined explicitly in the BLOCKS',\ 'statement so it can be referred to in the TERM option of CENSOR.')\ BLOCKS block/(gibb.ancy) TREATMENTS gibb*ancy CENSOR [TERM=block.gibb.ancy] Y=logdays; BOUND=bound; DF=resdf CAPTION !T('Display the analysis, based on the values estimated for the',\ 'censored plants. The analysis should be adjusted using the',\ 'effective degrees of freedom calculated by the procedure.') ADISPLAY [PRINT=aov,means] AKEEP [FITTED=fitted] PRINT [SERIAL=yes] resdf RESTRICT fitted; logdays>=bound CALCULATE fitted,newdays = 10**fitted,logdays-50 CAPTION 'Display the values fitted to the censored observations.' PRINT days,newdays,fitted