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

### Options

`PRINT` = string token |
Controls printed output (`intervals` ); default `inte` |
---|---|

`METHOD` = string token |
Type of interval (`individual` , `smm` , `product` , `Bonferroni` , `Scheffe` ); default `smm` |

`MU` = scalar |
Value for population mean checked as to whether in the confidence interval; default `*` i.e. no checking |

`PROBABILITY` = scalar |
The required significance level; default 0.05 |

### Parameters

`MEANS` = tables or variates |
Mean values |
---|---|

`REPLICATIONS` = scalars or tables or variates |
Number(s) of observations per mean |

`VARIANCE` = scalars |
Estimate of variance |

`DF` = scalars |
Degrees of freedom |

`XCONTRASTS` = matrices |
Matrix of coefficients of orthogonal contrasts |

`LABELS` = texts |
Identifiers of mean values |

`LOWER` = tables or variates |
Lower values of confidence intervals |

`UPPER` = tables or variates |
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.

### See also

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 TEXT [NVALUES=3] CLabels; VALUES=!T('Linear','Quadratic','Cubic') 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