g03dac computes a test statistic for the equality of within-group covariance matrices and also computes matrices for use in discriminant analysis.

2 Specification

#include <nag.h>

void	g03dac (Integer n, Integer m, const double x[], Integer tdx, const Integer isx[], Integer nvar, const Integer ing[], Integer ng, const double wt[], Integer nig[], double gmean[], Integer tdg, double det[], double gc[], double stat, double df, double sig, NagError fail)

The function may be called by the names: g03dac or nag_mv_discrim.

3 Description

Let a sample of

n

observations on

p

variables come from

n_{g}

groups with

n_{j}

observations in the

j

th group and

\sum n_{j} = n

. If the data is assumed to follow a multivariate Normal distribution with the variance-covariance matrix of the

j

th group

Σ_{j}

, then to test for equality of the variance-covariance matrices between groups, that is,

Σ_{1} = Σ_{2} = \dots = Σ_{n_{g}} = Σ

, the following likelihood-ratio test statistic,

G

, can be used;

G = C \{({n - n}_{g}) \log |S| - \sum_{j = 1}^{n_{g}} (n_{j} - 1) \log |S_{j}|\},

where

C = 1 - \frac{2 p^{2} + 3 p - 1}{6 (p + 1) (n_{g} - 1)} (\sum_{j = 1}^{n_{g}} \frac{1}{(n_{j} - 1)} - \frac{1}{({n - n}_{g})}),

and

S_{j}

are the within-group variance-covariance matrices and

S

is the pooled variance-covariance matrix given by

S = \frac{\sum_{j = 1}^{n_{g}} (n_{j} - 1) S_{j}}{({n - n}_{g})} .

For large

n

G

is approximately distributed as a

χ^{2}

variable with

\frac{1}{2} p (p + 1) (n_{g} - 1)

degrees of freedom, see Morrison (1967) for further comments. If weights are used, then

S

and

S_{j}

are the weighted pooled and within-group variance-covariance matrices and

n

is the effective number of observations, that is, the sum of the weights.

Instead of calculating the within-group variance-covariance matrices and then computing their determinants in order to calculate the test statistic, g03dac uses a

Q R

decomposition. The group means are subtracted from the data and then for each group, a

Q R

decomposition is computed to give an upper triangular matrix

R_{j}^{*}

. This matrix can be scaled to give a matrix

R_{j}

such that

S_{j} = R_{j}^{T} R_{j}

. The pooled

R

matrix is then computed from the

R_{j}

matrices. The values of

|S|

and the

|S_{j}|

can then be calculated from the diagonal elements of

R

and the

R_{j}

This approach means that the Mahalanobis squared distances for a vector observation

x

can be computed as

z^{T} z

, where

R_{j} z = (x - {\bar{x}}_{j})

{\bar{x}}_{j}

being the vector of means of the

j

th group. These distances can be calculated by g03dbc. The distances are used in discriminant analysis and g03dcc uses the results of g03dac to perform several different types of discriminant analysis. The differences between the discriminant methods are, in part, due to whether or not the within-group variance-covariance matrices are equal.

4 References

Aitchison J and Dunsmore I R (1975) Statistical Prediction Analysis Cambridge

Kendall M G and Stuart A (1976) The Advanced Theory of Statistics (Volume 3) (3rd Edition) Griffin

Krzanowski W J (1990) Principles of Multivariate Analysis Oxford University Press

Morrison D F (1967) Multivariate Statistical Methods McGraw–Hill

5 Arguments

1: $n$ – Integer Input

On entry: the number of observations,

n

Constraint:

n \geq 1

2: $m$ – Integer Input

On entry: the number of variables in the data array x.

Constraint:

m \geq nvar

3: $x [n \times tdx]$ – const double Input

On entry:

x [(k - 1) \times tdx + l - 1]

must contain the

k

th observation for the

l

th variable, for

k = 1, 2, \dots, n

and

l = 1, 2, \dots, m

4: $tdx$ – Integer Input

On entry: the stride separating matrix column elements in the array x.

Constraint:

tdx \geq m

5: $isx [m]$ – const Integer Input

On entry:

isx [l - 1]

indicates whether or not the

l

th variable in x is to be included in the variance-covariance matrices.

isx [l - 1] > 0

the

l

th variable is included, for

l = 1, 2, \dots, m

; otherwise it is not referenced.

Constraint:

isx [l - 1] > 0

for nvar values of

l

6: $nvar$ – Integer Input

On entry: the number of variables in the variance-covariance matrices,

p

Constraint:

nvar \geq 1

7: $ing [n]$ – const Integer Input

On entry:

ing [k - 1]

indicates to which group the

k

th observation belongs, for

k = 1, 2, \dots, n

Constraint:

1 \leq ing [k - 1] \leq ng

, for

k = 1, 2, \dots, n

The values of ing must be such that each group has at least nvar members

8: $ng$ – Integer Input

On entry: the number of groups,

n_{g}

Constraint:

ng \geq 2

9: $wt [n]$ – const double Input

On entry: the elements of wt must contain the weights to be used in the analysis and the effective number of observations for a group is the sum of the weights of the observations in that group. If

wt [k - 1] = 0.0

then the

k

th observation is excluded from the calculations.

If weights are not provided then wt must be set to NULL and the effective number of observations for a group is the number of observations in that group.

Constraints:

if wt is not NULL, $wt [k - 1] \geq 0.0$ , for $k = 1, 2, \dots, n$ ;
the effective number of observations for each group must be greater than 1.

10: $nig [ng]$ – Integer Output

On exit:

nig [j - 1]

contains the number of observations in the

j

th group, for

j = 1, 2, \dots, n_{g}

11: $gmean [ng \times tdg]$ – double Output

Note: the

(i, j)

th element of the matrix is stored in

gmean [(i - 1) \times tdg + j - 1]

On exit: the

j

th row of gmean contains the means of the

p

selected variables for the

j

th group, for

j = 1, 2, \dots, n_{g}

12: $tdg$ – Integer Input

On entry: the stride separating matrix column elements in the array gmean.

Constraint:

tdg \geq nvar

13: $\det [ng]$ – double Output

On exit: the logarithm of the determinants of the within-group variance-covariance matrices.

14: $gc [\dim]$ – double Output

Note: the dimension, dim, of the array gc must be at least

(ng + 1) \times nvar \times (nvar + 1) / 2

On exit: the first

p (p + 1) / 2

elements of gc contain

R

and the remaining

n_{g}

blocks of

p (p + 1) / 2

elements contain the

R_{j}

matrices. All are stored in packed form by columns.

15: $stat$ – double * Output

On exit: the likelihood-ratio test static,

G

16: $df$ – double * Output

On exit: the degrees of freedom for the distribution of

G

17: $sig$ – double * Output

On exit: the significance level for

G

18: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_2_INT_ARG_LT: On entry, $m = 〈value〉$ while $nvar = 〈value〉$ . These arguments must satisfy $m \geq nvar$ .

On entry, $tdg = 〈value〉$ while $nvar = 〈value〉$ . These arguments must satisfy $tdg \geq nvar$ .

On entry, $tdx = 〈value〉$ while $m = 〈value〉$ . These arguments must satisfy $tdx \geq m$ .
NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_GROUP_OBSERV: On entry, group $〈value〉$ has $〈value〉$ effective observations.
Constraint: in each group the effective number of observations must be $\geq 1$ .
NE_GROUP_VAR: On entry, group $〈value〉$ has $〈value〉$ members, while $nvar = 〈value〉$ .
Constraint: number of members in each group $\geq nvar$ .
NE_GROUP_VAR_RANK: The variables in group $〈value〉$ are not of full rank.
NE_INT_ARG_LT: On entry, $n = 〈value〉$ .
Constraint: $n \geq 1$ .

On entry, $ng = 〈value〉$ .
Constraint: $ng \geq 2$ .

On entry, $nvar = 〈value〉$ .
Constraint: $nvar \geq 1$ .
NE_INTARR_INT: On entry, $ing [〈value〉] = 〈value〉$ , $ng = 〈value〉$ .
Constraint: $1 \leq ing [i - 1] \leq ng$ , for $i = 1, 2, \dots, n$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_NEG_WEIGHT_ELEMENT: On entry, $wt [〈value〉] = 〈value〉$ .
Constraint: when referenced, all elements of wt must be non-negative.
NE_VAR_INCL_INDICATED: The number of variables, nvar in the analysis $= 〈value〉$ , while number of variables included in the analysis via array $isx = 〈value〉$ . Constraint: these two numbers must be the same.
NE_VAR_RANK: The variables are not of full rank.

7 Accuracy

The accuracy is dependent on the accuracy of the computation of the

Q R

decomposition.

8 Parallelism and Performance

g03dac is not threaded in any implementation.

9 Further Comments

The time will be approximately proportional to

n p^{2}

10 Example

The data, taken from Aitchison and Dunsmore (1975), is concerned with the diagnosis of three ‘types’ of Cushing's syndrome. The variables are the logarithms of the urinary excretion rates (mg/24hr) of two steroid metabolites. Observations for a total of 21 patients are input and the statistics computed by g03dac. The printed results show that there is evidence that the within-group variance-covariance matrices are not equal.

Interfaces: FL CL AD

NAG CL Interface Introduction

G03 (Mv) Chapter Contents

G03 (Mv) Chapter Introduction

g03da: FL CL AD

NAG CL Interfaceg03dac (discrim)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG CL Interface
g03dac (discrim)