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# ACONFIDENCE procedure

Calculates simultaneous confidence intervals for `ANOVA` means (D.M. Smith).

### Options

`PRINT` = string token Controls printed output (`intervals`); default `inte` Type of interval (`individual`, `smm`, `product`, `Bonferroni`, `Scheffe`); default `smm` Limit on the number of factors in each term; default 3 The required significance level; default 0.05 Save structure to provide the tables of means and associated information; default uses the save structure from the most recent `ANOVA`

### Parameters

`TERMS` = formula Treatment terms whose means are to be required Saves the means Saves the lower limits Saves the upper limits

### Description

`ACONFIDENCE` calculates sets 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 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 `individual` setting calculates the intervals as if they were independent, each with the input probability. The `smm` setting 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 pivoted F statistics; see Hsu (1996, Section 1.3.7). The default setting is `smm` because it produces exact simultaneous confidence intervals.

The `TERMS` parameter specifies a model formula to define the treatment terms whose means and confidence intervals are required. The means (and the necessary associated information) are usually taken from the most recent analysis of variance (performed by `ANOVA`), but you can set the `SAVE` option to a save structure from another `ANOVA` if you want to examine means from an earlier analysis. As in `ANOVA`, the `FACTORIAL` option sets a limit on the number of factors in each term (default 3). Note: intervals cannot be formed for means whose effects are estimated in different strata.

The `MEANS` parameter can save the means. If the `TERMS` parameter specifies a single term, `MEANS` should be set to a table. If `TERMS` specifies several terms, you must supply a pointer which will then be set up to contain as many tables as there are terms. Similarly the `LOWER` parameter can save the lower bounds of the confidence intervals, and the `UPPER` parameter can save the upper bounds.

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

Options: `PRINT`, `METHOD`, `FACTORIAL`, `PROBABILITY`, `SAVE`.

Parameters: `TERMS`, `MEANS`, `LOWER`, `UPPER`.

### Method

The methodology implemented is based on that described and reviewed in Hsu (1996), and Bechhofer, Santner & Goldsman (1995). For specific details of the tests these books should be referred to.

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.

Directive: `ANOVA`.

Procedures: `AMCOMPARISON`, `CONFIDENCE`.

Commands for: Analysis of variance.

### Example

```CAPTION     'ACONFIDENCE example',!t('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
ACONFIDENCE [METHOD=smm] Age
ACONFIDENCE [METHOD=individual] Age
ACONFIDENCE [METHOD=product] Age
ACONFIDENCE [METHOD=bonferroni] Age
ACONFIDENCE [METHOD=scheffe] Age
```