# RE: [R] evaluation of discriminant functions+multivariate homosce dasticity

From: Liaw, Andy (andy_liaw@merck.com)
Date: Wed 21 Jan 2004 - 08:55:06 EST

```Message-id: <3A822319EB35174CA3714066D590DCD504AF7669@usrymx25.merck.com>

```

While I don't know anything about Box's M test, I googled around and found a
Matlab M-file that computes it. Below is my straight-forward translation of
the code, without knowing Matlab or the formula (and done in a few minutes).
I hope this demonstrates one of Prof. Ripley's point: If you really want to
shoot yourself in the foot, you can probably program R to do that for you.

[BTW: I left the original comments largely intact. The output I get from R
for the example is the same as what is shown in the comments.]

[BTW #2: I'd imagine the original Matlab code probably could be improved a
bit...]

Andy

=======================================================================
BoxMTest <- function(X, cl, alpha=0.05) {
## Multivariate Statistical Testing for the Homogeneity of Covariance
## Matrices by the Box's M.
##
## Syntax: function [MBox] = BoxMTest(X,alpha)
##
## Inputs:
## X - data matrix (Size of matrix must be n-by-(1+p); sample=column
1,
## variables=column 2:p).
## alpha - significance level (default = 0.05).
## Output:
## MBox - the Box's M statistic.
## Chi-sqr. or F - the approximation statistic test.
## df's - degrees' of freedom of the approximation statistic test.
## P - observed significance level.
##
## If the groups sample-size is at least 20 (sufficiently large),
## Box's M test takes a Chi-square approximation; otherwise it takes
## an F approximation.
##
## Example: For a two groups (g = 2) with three independent variables
## (p = 3), we are interested in testing the homogeneity of covariances
## matrices with a significance level = 0.05. The two groups have the
## same sample-size n1 = n2 = 5.
## Group
## ---------------------------------------
## 1 2
## ---------------------------------------
## x1 x2 x3 x1 x2 x3
## ---------------------------------------
## 23 45 15 277 230 63
## 40 85 18 153 80 29
## 215 307 60 306 440 105
## 110 110 50 252 350 175
## 65 105 24 143 205 42
## ---------------------------------------
##
## Total data matrix must be:
## X=[1 23 45 15;1 40 85 18;1 215 307 60;1 110 110 50;1 65 105
24;
## 2 277 230 63;2 153 80 29;2 306 440 105;2 252 350 175;2 143 205
42];
##
## Calling on Matlab the function:
## MBoxtest(X,0.05)
##
##
## ------------------------------------------------------------
## MBox F df1 df2 P
## ------------------------------------------------------------
## 27.1622 2.6293 6 463 0.0162
## ------------------------------------------------------------
## Covariance matrices are significantly different.
##

## Created by A. Trujillo-Ortiz and R. Hernandez-Walls
## Universidad Autonoma de Baja California
## Apdo. Postal 453
## Mexico.
## atrujo@uabc.mx
## And the special collaboration of the post-graduate students of the
2002:2
## Multivariate Statistics Course: Karel Castro-Morales,
## Alejandro Espinoza-Tenorio, Andrea Guia-Ramirez, Raquel Muniz-Salazar,
## Jose Luis Sanchez-Osorio and Roberto Carmona-Pina.
## November 2002.
##
## To cite this file, this would be an appropriate format:
## Trujillo-Ortiz, A., R. Hernandez-Walls, K. Castro-Morales,
## A. Espinoza-Tenorio, A. Guia-Ramirez and R. Carmona-Pina. (2002).
## MBoxtest: Multivariate Statistical Testing for the Homogeneity of
## Covariance Matrices by the Box's M. A MATLAB file. [WWW document].
## URL
3&objectType=FILE
##
## References:
##
## Stevens, J. (1992), Applied Multivariate Statistics for Social
Sciences.
## 2nd. ed., New-Jersey:Lawrance Erlbaum Associates Publishers. pp.
260-269.

if (alpha <= 0 || alpha >= 1)
stop('significance level must be between 0 and 1')

g = nlevels(cl) ## Number of groups.
n = table(cl) ## Vector of groups-size.

N = nrow(X)
p = ncol(X)

bandera = 2
if (any(n >= 20)) bandera = 1

## Partition of the group covariance matrices.
covList <- tapply(X, rep(cl, ncol(X)), function(x, nc) cov(matrix(x,
nc=nc)),
ncol(X))

deno = sum(n) - g
suma = array(0, dim=dim(covList[[1]]))

for (k in 1:g)
suma = suma + (n[k] - 1) * covList[[k]]

Sp = suma / deno ## Pooled covariance matrix.
Falta=0

for (k in 1:g)
Falta = Falta + ((n[k] - 1) * log(det(covList[[k]])))

MB = (sum(n) - g) * log(det(Sp)) - Falta ## Box's M statistic.
suma1 = sum(1 / (n[1:g] - 1))
suma2 = sum(1 / ((n[1:g] - 1)^2))
C = (((2 * p^2) + (3 * p) - 1) / (6 * (p + 1) * (g - 1))) *
(suma1 - (1 / deno)) ## Computing of correction factor.
if (bandera == 1) {
X2 = MB * (1 - C) ## Chi-square approximation.
v = as.integer((p * (p + 1) * (g - 1)) / 2) ## Degrees of freedom.
## Significance value associated to the observed Chi-square statistic.
P = pchisq(X2, v, lower=TRUE)

cat('------------------------------------------------\n');
cat(' MBox Chi-sqr. df P\n')
cat('------------------------------------------------\n')
cat(sprintf("%10.4f%11.4f%12.i%13.4f\n", MB, X2, v, P))
cat('------------------------------------------------\n')
if (P >= alpha) {
cat('Covariance matrices are not significantly different.\n')
} else {
cat('Covariance matrices are significantly different.\n')
}
return(list(MBox=MB, ChiSq=X2, df=v, pValue=P))
} else {
## To obtain the F approximation we first define Co, which combined to
## the before C value are used to estimate the denominator degrees of
## freedom (v2); resulting two possible cases.
Co = (((p-1) * (p+2)) / (6 * (g-1))) * (suma2 - (1 / (deno^2)))
if (Co - (C^2) >= 0) {
v1 = as.integer((p * (p + 1) * (g - 1)) / 2) ## Numerator DF.
v21 = as.integer(trunc((v1 + 2) / (Co - (C^2)))) ## Denominator DF.
F1 = MB * ((1 - C - (v1 / v21)) / v1) ## F approximation.
## Significance value associated to the observed F statistic.
P1 = pf(F1, v1, v21, lower=FALSE)

cat('\n------------------------------------------------------------\n')
cat(' MBox F df1 df2 P\n')
cat('------------------------------------------------------------\n')
cat(sprintf("%10.4f%11.4f%11.i%14.i%13.4f\n", MB, F1, v1, v21, P1))
cat('------------------------------------------------------------\n')
if (P1 >= alpha) {
cat('Covariance matrices are not significantly different.\n')
} else {
cat('Covariance matrices are significantly different.\n')
}
return(list(MBox=MB, F=F1, df1=v1, df2=v21, pValue=P1))
} else {
v1 = as.integer((p * (p + 1) * (g - 1)) / 2) ## Numerator df.
v22 = as.integer(trunc((v1 + 2) / ((C^2) - Co))) ## Denominator df.
b = v22 / (1 - C - (2 / v22))
F2 = (v22 * MB) / (v1 * (b - MB)) ## F approximation.
## Significance value associated to the observed F statistic.
P2 = pf(F2, v1, v22, lower=FALSE)

cat('\n------------------------------------------------------------\n')
cat(' MBox F df1 df2 P\n')
cat('------------------------------------------------------------\n')
cat(sprintf('%10.4f%11.4f%11.i%14.i%13.4f\n', MB, F2, v1, v22, P2))
cat('------------------------------------------------------------\n')

if (P2 >= alpha) {
cat('Covariance matrices are not significantly different.\n')
} else {
cat('Covariance matrices are significantly different.\n')
}
return(list(MBox=MB, F=F2, df1=v1, df2=v22, pValue=P2))
}
}
}

> From: Janke ten Holt
>
> Hello,
>
> I am switching from SPSS-Windows to R-Linux. My university is very
> SPSS-oriented so maybe that's the cause of my problems. I am
> a beginner
> in R and my assignments are SPSS-oriented, so I hope I don't annoy
> anyone with my questions...
>
> Right now I've got 2 problems:
> -I have to evaluate discriminant functions I have calculated with
> lda(MASS). I can't find a measure that evaluates their significance
> (Wilk's lambda in my textbook (Stevens,(2002),"Applied multivariate
> statistics for the social sciences")and in SPSS). Is there a Wilk's
> lambda for discriminant functions in R? or can I use an alternative
> measure? or am I thinking in the wrong direction? I have searched the
> help-archive to find similar questions to mine but no answer to them.
>
> -My second problem: to check the assumption of multivariate
> homoscedasticity I have to test if the variance-covariance
> matrices for
> my variables are homogene. My textbook suggests Box's M test. I can't
> find this statistic in R. Again I have found similar questions in the
> help-archives, but no answers. Is there a way to calculate
> Box's M in R?
> Or is there an alternative way to check for multivariate
> homoscedasticity?
>
> Any suggestion would be greatly appreciated!
>
> Cheers,
> Janke ten Holt

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