Summary Method for Multivariate Analysis of Variance

Description

A summary method for class "manova".

Usage

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## S3 method for class 'manova'
summary(object,
        test = c("Pillai", "Wilks", "Hotelling-Lawley", "Roy"),
        intercept = FALSE, tol = 1e-7, ...)

Arguments

object

An object of class "manova" or an aov object with multiple responses.

test

The name of the test statistic to be used. Partial matching is used so the name can be abbreviated.

intercept

logical. If TRUE, the intercept term is included in the table.

tol

tolerance to be used in deciding if the residuals are rank-deficient: see qr.

...

further arguments passed to or from other methods.

Details

The summary.manova method uses a multivariate test statistic for the summary table. Wilks' statistic is most popular in the literature, but the default Pillai–Bartlett statistic is recommended by Hand and Taylor (1987).

The table gives a transformation of the test statistic which has approximately an F distribution. The approximations used follow S-PLUS and SAS (the latter apart from some cases of the Hotelling–Lawley statistic), but many other distributional approximations exist: see Anderson (1984) and Krzanowski and Marriott (1994) for further references. All four approximate F statistics are the same when the term being tested has one degree of freedom, but in other cases that for the Roy statistic is an upper bound.

The tolerance tol is applied to the QR decomposition of the residual correlation matrix (unless some response has essentially zero residuals, when it is unscaled). Thus the default value guards against very highly correlated responses: it can be reduced but doing so will allow rather inaccurate results and it will normally be better to transform the responses to remove the high correlation.

Value

An object of class "summary.manova". If there is a positive residual degrees of freedom, this is a list with components

row.names

The names of the terms, the row names of the stats table if present.

SS

A named list of sums of squares and product matrices.

Eigenvalues

A matrix of eigenvalues.

stats

A matrix of the statistics, approximate F value, degrees of freedom and P value.

otherwise components row.names, SS and Df (degrees of freedom) for the terms (and not the residuals).

References

Anderson, T. W. (1994) An Introduction to Multivariate Statistical Analysis. Wiley.

Hand, D. J. and Taylor, C. C. (1987) Multivariate Analysis of Variance and Repeated Measures. Chapman and Hall.

Krzanowski, W. J. (1988) Principles of Multivariate Analysis. A User's Perspective. Oxford.

Krzanowski, W. J. and Marriott, F. H. C. (1994) Multivariate Analysis. Part I: Distributions, Ordination and Inference. Edward Arnold.

See Also

manova, aov

Examples

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## Example on producing plastic film from Krzanowski (1998, p. 381)
tear <- c(6.5, 6.2, 5.8, 6.5, 6.5, 6.9, 7.2, 6.9, 6.1, 6.3,
          6.7, 6.6, 7.2, 7.1, 6.8, 7.1, 7.0, 7.2, 7.5, 7.6)
gloss <- c(9.5, 9.9, 9.6, 9.6, 9.2, 9.1, 10.0, 9.9, 9.5, 9.4,
           9.1, 9.3, 8.3, 8.4, 8.5, 9.2, 8.8, 9.7, 10.1, 9.2)
opacity <- c(4.4, 6.4, 3.0, 4.1, 0.8, 5.7, 2.0, 3.9, 1.9, 5.7,
             2.8, 4.1, 3.8, 1.6, 3.4, 8.4, 5.2, 6.9, 2.7, 1.9)
Y <- cbind(tear, gloss, opacity)
rate     <- gl(2,10, labels = c("Low", "High"))
additive <- gl(2, 5, length = 20, labels = c("Low", "High"))

fit <- manova(Y ~ rate * additive)
summary.aov(fit)             # univariate ANOVA tables
summary(fit, test = "Wilks") # ANOVA table of Wilks' lambda
summary(fit)                # same F statistics as single-df terms

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