Test for equality of within-group covariance matrices

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NAG[g03dac] NAG[nag_mv_discrim] - Test for equality of within-group covariance matrices

Calling Sequence

g03dac(x, isx, ing, nig, gmean, det, gcov, stat, df, sig, 'n'=n, 'm'=m, 'tdx'=tdx, 'nvar'=nvar, 'ng'=ng, 'wt'=wt, 'tdg'=tdg, 'fail'=fail)

nag_mv_discrim(. . .)

Parameters

x - Matrix(1..n, 1..tdx, datatype=float[8], order=C_order);

On entry: must contain the th observation for the th variable, for ; , m.

isx - Vector(1..m, datatype=integer[kernelopts('wordsize')/8]);

On entry: indicates whether or not the th variable in x is to be included in the variance-covariance matrices.

If the th variable is included, for ; otherwise it is not referenced.

Constraint: for nvar values of .

ing - Vector(1..n, datatype=integer[kernelopts('wordsize')/8]);

On entry: indicates to which group the th observation belongs, for .

Constraint: , for .

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

nig - Vector(1..ng, datatype=integer[kernelopts('wordsize')/8]);

On exit: contains the number of observations in the th group, for .

gmean - Matrix(1..ng, 1..tdg, datatype=float[8], order=C_order);

On exit: the th row of gmean contains the means of the selected variables for the th group, for .

det - Vector(1..ng, datatype=float[8]);

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

gcov - Vector(1.., datatype=float[8]);

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

On exit: the first elements of gcov contain and the remaining blocks of elements contain the matrices. All are stored in packed form by columns.

stat - assignable;

Note: On exit the variable stat will have a value of type float.

On exit: the likelihood-ratio test static, .

df - assignable;

Note: On exit the variable df will have a value of type float.

On exit: the degrees of freedom for the distribution of .

sig - assignable;

Note: On exit the variable sig will have a value of type float.

On exit: the significance level for .

'n'=n - integer; (optional)

Default value: the first dimension of the arrays x, ing, wt.

On entry: the number of observations, .

Constraint: . .

'm'=m - integer; (optional)

Default value: the first dimension of the array isx and the second dimension of the array isxthe array x.

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

Constraint: . .

'tdx'=tdx - integer; (optional)

On entry: the second dimension of the array x as declared in the function from which nag_mv_discrim (g03dac) is called.

Constraint: . .

'nvar'=nvar - integer; (optional)

Default value: the second dimension of the array gmean.

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

Constraint: . .

'ng'=ng - integer; (optional)

Default value: the first dimension of the arrays nig, gmean, det.

On entry: the number of groups, .

Constraint: . .

'wt'=wt - Vector(1..n, datatype=float[8]); (optional)

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 then the th observation is excluded from the calculations.

Constraints:

, for ;

The effective number of observations for each group must be greater than 1.

Note: If wt is set to the null pointer NULL, i.e., (double *)0, then wt is not referenced and the effective number of observations for a group is the number of observations in that group.

'tdg'=tdg - integer; (optional)

On entry: the second dimension of the array gmean as declared in the function from which nag_mv_discrim (g03dac) is called.

Constraint: . .

'fail'=fail - table; (optional)

The NAG error argument, see the documentation for NagError.

Description

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

Description

Let a sample of observations on variables come from groups with observations in the th group and . If the data is assumed to follow a multivariate Normal distribution with the variance-covariance matrix of the th group , then to test for equality of the variance-covariance matrices between groups, that is, , the following likelihood-ratio test statistic, , can be used;

G=C{((n-n)[g]),log|S|,-,(∑),(n[j],-,1),log|S[j]|}

where

C=1-(2p^2+3p-1)/(6(p,+,1)(n[g],-,1))((∑),1/(n[j],-,1),-,1/((n-n)[g]))

and are the within-group variance-covariance matrices and is the pooled variance-covariance matrix given by

S=((∑)(n[j],-,1)S[j])/((n-n)[g])

For large , is approximately distributed as a variable with degrees of freedom, see Morrison (1967) for further comments. If weights are used, then and are the weighted pooled and within-group variance-covariance matrices and 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, nag_mv_discrim (g03dac) uses a decomposition. The group means are subtracted from the data and then for each group, a decomposition is computed to give an upper triangular matrix . This matrix can be scaled to give a matrix such that . The pooled matrix is then computed from the matrices. The values of and the can then be calculated from the diagonal elements of and the .

This approach means that the Mahalanobis squared distances for a vector observation can be computed as , where , being the vector of means of the th group. These distances can be calculated by g03dbc (nag_mv_discrim_mahaldist). The distances are used in discriminant analysis and g03dcc (nag_mv_discrim_group) uses the results of nag_mv_discrim (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.

Error Indicators and Warnings

"NE_2_INT_ARG_LT"

On entry, while . These arguments must satisfy .

"NE_ALLOC_FAIL"

Dynamic memory allocation failed.

"NE_GROUP_OBSERV"

On entry, group has effective observations. Constraint: in each group the effective number of observations must be .

"NE_GROUP_VAR"

On entry, group has members, while . Constraint: number of members in each group

"NE_GROUP_VAR_RANK"

The variables in group are not of full rank.

"NE_INT_ARG_LT"

On entry, n must not be less than 1: .

"NE_INTARR_INT"

On entry, , . Constraint: , .

"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 consult NAG for assistance.

"NE_NEG_WEIGHT_ELEMENT"

On entry, . Constraint: when referenced, all elements of wt must be non-negative.

"NE_VAR_INCL_INDICATED"

The number of variables, nvar in the analysis , while number of variables included in the analysis via array . Constraint: these two numbers must be the same.

"NE_VAR_RANK"

The variables are not of full rank.

	Accuracy
	The accuracy is dependent on the accuracy of the computation of the decomposition. See f01qcc (nag_real_qr) for further details.

	Further Comments
	The time will be approximately proportional to .

Examples

n := 21:
m := 2:
tdx := 2:
nvar := 2:
ng := 3:
tdg := 2:
x := Matrix([[1.1314, 2.4596], [1.0986, 0.2624], [0.6419, -2.3026], [1.335, -3.2189], [1.411, 0.0953], [0.6419, -0.9163], [2.1163, 0], [1.335, -1.6094], [1.361, -0.5108], [2.0541, 0.1823], [2.2083, -0.5108], [2.7344, 1.2809], [2.0412, 0.47], [1.8718, -0.9163], [1.7405, -0.9163], [2.6101, 0.47], [2.3224, 1.8563], [2.2192, 2.0669], [2.2618, 1.1314], [3.9853, 0.9163], [2.76, 2.0281]], datatype=float[8], order='C_order'):
isx := Vector([1, 1], datatype=integer[kernelopts('wordsize')/8]):
ing := Vector([1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3], datatype=integer[kernelopts('wordsize')/8]):
wt := Vector([], datatype=float[8]):
nig := Vector(3, datatype=integer[kernelopts('wordsize')/8]):
gmean := Matrix(3, 2, datatype=float[8], order='C_order'):
det := Vector(3, datatype=float[8]):
gcov := Vector(12, datatype=float[8]):
NAG:-g03dac(x, isx, ing, nig, gmean, det, gcov, stat, df, sig, 'n' = n, 'm' = m, 'tdx' = tdx, 'nvar' = nvar, 'ng' = ng, 'wt' = wt, 'tdg' = tdg):

nig;

gmean;

det;

gcov;

stat;

df;

sig;

	See Also
	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 g03 Chapter Introduction. NAG Toolbox Overview. NAG Web Site.