CONFIDENCE procedure

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 lets you 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
Updated on January 10, 2018

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