1. Home
2. CONFIDENCE procedure

# CONFIDENCE procedure

Calculates simultaneous confidence intervals (D.M. Smith).

### Options

`PRINT` = string token Controls printed output (`intervals`); default `inte` Type of interval (`individual`, `smm`, `product`, `Bonferroni`, `Scheffe`); default `smm` Value for population mean checked as to whether in the confidence interval; default `*` i.e. no checking The required significance level; default 0.05

### Parameters

`MEANS` = tables or variates Mean values Number(s) of observations per mean Estimate of variance Degrees of freedom Matrix of coefficients of orthogonal contrasts Identifiers of mean values Lower values of confidence intervals Upper values of confidence intervals

### Description

`CONFIDENCE` calculates a set of simultaneous confidence intervals i.e. intervals whose formation takes account of the number of intervals formed and the fact that the intervals are (slightly) correlated because of the use of a common variance (see Hsu 1996 and Bechhofer, Santner & Goldsman 1995). The methodology implemented in the procedure closely follows that described in Section 1.3 of Hsu (1996).

The means are input using the `MEANS` parameter, either in a table saved e.g. from `AKEEP`, or in a variate. The replication (or number of observations in each mean) is supplied by the `REPLICATIONS` parameter, either in a scalar (if all the replications are equal) or in a structure of the same type as the means. The estimate of the variance (usually a pooled estimate as given by the residual mean square in `ANOVA`, and accessible using the `VARIANCE` parameter of `AKEEP`) and its corresponding degrees of freedom are input as scalars using the `VARIANCE` and `DF` parameters respectively. Confidence limits can be formed for contrasts amongst the means by supplying the matrix defining the contrasts using the `XCONTRASTS` parameter. Each row of the matrix contains a contrast similarly to the specification in the `REG` function in `ANOVA` but, unlike `REG`, the contrasts must all be orthogonal. The `LABELS` parameter can be used to supply labels for the means or for the contrasts, while the `LOWER` and `UPPER` parameters allow the limits of the confidence intervals to be saved.

The type of interval to be formed is specified by the `METHOD` option, with settings `individual`, `smm` (studentized maximum modulus), `product` (inequality), `Bonferroni` and `Scheffe`. The setting `individual` calculates the intervals as if they were independent, each with the input probability. The setting `smm` calculates the intervals as correlated, each with a probability adjusted for the multiplicity of intervals. The two settings `product` and `Bonferroni` calculate the intervals as independent, but with a probability adjusted for the multiplicity of intervals. These two settings produce very similar intervals although the Bonferroni intervals are always slightly larger. The final setting `Scheffe` calculates the intervals using privoted F statistics. Hsu (1996, Section 1.3.7) should be referred to for details of this last setting. The default setting is `smm` because it produces exact simultaneous confidence intervals.

The `MU` option allows you to supply a (population) mean to be tested for inclusion in each interval, and the `PROBABILITY` option allows the experiment-wise significance level for the intervals to be changed from the default of 0.05 (i.e. 5%). The interval-wise significance level is calculated according to the setting of `METHOD`.

You can set option `PRINT=*` to suppress printing of the intervals; by default `PRINT=intervals`.

Options: `PRINT`, `METHOD`, `MU`, `PROBABILITY`.

Parameters: `MEANS`, `REPLICATIONS`, `VARIANCE`, `DF`, `XCONTRASTS`, `LABELS`, `LOWER`, `UPPER`.

### Method

The methodology implemented is based on that described and reviewed in Hsu (1996), and Bechhofer, Santner & Goldsman (1995).

### References

Bechhofer, R.E., Santner, T.J. & Goldsman, D.M. (1995). Design and Analysis of Experiments for Statistical Selection, Screening, and Multiple Comparisons. Wiley, New York.

Hsu, J.C. (1996). Multiple Comparisons Theory and Methods. Chapman & Hall, London.

Procedures: `AMCOMPARISON`, `AUMCOMPARISON`, `AMDUNNETT`, `VMCOMPARISON`.

Commands for: Analysis of variance.

### Example

```CAPTION  'CONFIDENCE example',!t('1) Hsu (1996), Multiple Comparisons,',\
'Theory and Methods, Table 1.1'); STYLE=meta,plain
FACTOR   [LABELS=!t('20-29','30-39','40-49','50-59','60-69');\
VALUES=6(1...5)] Age
VARIATE  Standard,New; VALUES=\
!(57,53,28,60,40,48,70,85,50,61,83,51,55,36,31,28,41,32,\
18,39,53,44,63,80,76,67,75,78,67,80) ,\
!(72,27,26,71,60,45,52,26,46,52,53,58,83,65,40,66,50,44,\
40,55,70,55,61,60,60,37,45,58,54,69)
TREATMENT Age
ANOVA Standard-New
AKEEP Age; MEAN=Mean; REP=Rep; VARIANCE=Var; RTERM=Units
AKEEP #Units; DF=Resdf
CONFIDENCE [METHOD=smm]        Mean; REP=Rep; VARIANCE=Var; DF=Resdf
CONFIDENCE [METHOD=individual] Mean; REP=Rep; VARIANCE=Var; DF=Resdf
CONFIDENCE [METHOD=product]    Mean; REP=Rep; VARIANCE=Var; DF=Resdf
CONFIDENCE [METHOD=bonferroni] Mean; REP=Rep; VARIANCE=Var; DF=Resdf
CONFIDENCE [METHOD=scheffe]    Mean; REP=Rep; VARIANCE=Var; DF=Resdf

CAPTION '2) Bechhofer, Santner & Goldsman (1995), Example 4.2.5.'
FACTOR  [LEVELS=!(0,4,8,12)] Labels
TABLE   [CLASSIFICATION=Labels; VALUES=34.8,41.1,42.6,41.8] Means
MATRIX  [ROWS=CLabels; COLUMNS=4; VALUES=-3,-1,+1,+3,+1,-1,-1,+1,-1,+3,-3,+1]\
Contrasts
CONFIDENCE [METHOD=BONFERRONI] Means; REPLICATIONS=8; VARIANCE=11.9; DF=28;\
XCONTRASTS=Contrasts
CONFIDENCE [METHOD=SMM] Means; REPLICATIONS=8; VARIANCE=11.9; DF=28;\
XCONTRASTS=Contrasts
CONFIDENCE [METHOD=PRODUCT] Means; REPLICATIONS=8; VARIANCE=11.9; DF=28;\
XCONTRASTS=Contrasts
CONFIDENCE [METHOD=SCHEFFE] Means; REPLICATIONS=8; VARIANCE=11.9; DF=28;\
XCONTRASTS=Contrasts
CONFIDENCE [METHOD=INDIVIDUAL] Means; REPLICATIONS=8; VARIANCE=11.9; DF=28;\
XCONTRASTS=Contrasts
```
Updated on June 20, 2019