G04AGF : NAG Library, Mark 24

Note: before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

G04AGF performs an analysis of variance for a two-way hierarchical classification with subgroups of possibly unequal size, and also computes the treatment group and subgroup means. A fixed effects model is assumed.

In a two-way hierarchical classification, there are

k

(

\geq 2

) treatment groups, the

i

th of which is subdivided into

l_{i}

treatment subgroups. The

j

th subgroup of group

i

contains

n_{i j}

observations, which may be denoted by

y_{1 i j}, y_{2 i j}, \dots, y_{n_{i j} i j} .

The general observation is denoted by

y_{m i j}

, being the

m

th observation in subgroup

j

of group

i

, for

1 \leq i \leq k

1 \leq j \leq l_{i}

1 \leq m \leq n_{i j}

The following quantities are computed

(i)

The subgroup means

{\bar{y}}_{. i j} = \frac{\sum_{m = 1}^{n_{i j}} y_{m i j}}{n_{i j}}

(ii)

The group means

{\bar{y}}_{. i .} = \frac{\sum_{j = 1}^{l_{i}} \sum_{m = 1}^{n_{i j}} y_{m i j}}{\sum_{j = 1}^{l_{i}} n_{i j}}

(iii)

The grand mean

{\bar{y}}_{\dots} = \frac{\sum_{i = 1}^{k} \sum_{j = 1}^{l_{i}} \sum_{m = 1}^{n_{i j}} y_{m i j}}{\sum_{i = 1}^{k} \sum_{j = 1}^{l_{i}} n_{i j}}

(iv)

The number of observations in each group

n_{i .} = \sum_{j = 1}^{l_{i}} n_{i j}

(v)

Sums of squares

\begin{array}{l} Between groups & = {SS}_{g} = \sum_{i = 1}^{k} n_{i .} {({\bar{y}}_{. i .} - {\bar{y}}_{\dots})}^{2} \\ Between subgroups within groups & = {SS}_{s g} = \sum_{i = 1}^{k} \sum_{j = 1}^{l_{i}} n_{i j} {(y_{. i j} - {\bar{y}}_{. i .})}^{2} \\ Residual (within subgroups) & = {SS}_{res} = \sum_{i = 1}^{k} \sum_{j = 1}^{l_{i}} \sum_{m = 1}^{n_{i j}} {(y_{m i j} - {\bar{y}}_{. i j})}^{2} = {SS}_{tot} - {SS}_{g} - {SS}_{s g} \\ Corrected total & = {SS}_{tot} = \sum_{i = 1}^{k} \sum_{j = 1}^{l_{i}} \sum_{m = 1}^{n_{i j}} {(y_{m i j} - {\bar{y}}_{\dots})}^{2} \end{array}

(vi)

Degrees of freedom of variance components

Between groups:	$k - 1$
Subgroups within groups:	$l - k$
Residual:	$n - l$
Total:	$n - 1$

where

$l = \sum_{i = 1}^{k} l_{i}$ ,
$n = \sum_{i = 1}^{k} n_{i .}$

(vii)

F

ratios. These are the ratios of the group and subgroup mean squares to the residual mean square.

Groups	$F_{1} = \frac{Between groups sum of squares / (k - 1)}{Residual sum of squares / (n - l)} = \frac{{SS}_{g} / (k - 1)}{{SS}_{res} / (n - l)}$
Subgroups	$F_{2} = \frac{Between subgroups (within group) sum of squares / (l - k)}{Residual sum of squares / (n - l)} = \frac{{SS}_{s g} / (l - k)}{{SS}_{res} / (n - l)}$

If either

F

ratio exceeds

9999.0

, the value

9999.0

is assigned instead.

(viii)

F significances. The probability of obtaining a value from the appropriate

F

-distribution which exceeds the computed mean square ratio.

Groups	$p_{1} = Prob (F_{(k - 1), (n - l)} > F_{1})$
Subgroups	$p_{2} = Prob (F_{(l - k), (n - l)} > F_{2})$

where

F_{ν_{1}, ν_{2}}

denotes the central

F

-distribution with degrees of freedom

ν_{1}

and

ν_{2}

If any

F_{i} = 9999.0

, then

p_{i}

is set to zero,

i = 1, 2

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

Moore P G, Shirley E A and Edwards D E (1972) Standard Statistical Calculations Pitman

On entry: the elements of Y must contain the observations

y_{mij}

in the following order:

y_{111}, y_{211}, \dots, y_{n_{11} 11}, y_{112}, y_{212}, \dots, y_{n_{12} 12}, \dots, y_{11 l_{1}}, \dots,

y_{n_{1 l_{1}} 1 l_{1}}, \dots, y_{1 i j}, \dots, y_{n_{i j} i j}, \dots, y_{1 k l_{k}}, \dots, y_{n_{k l_{k}} k l_{k}} .

In words, the ordering is by group, and within each group is by subgroup, the members of each subgroup being in consecutive locations in Y.

On entry:

n

, the total number of observations.

On entry:

k

, the number of groups.

Constraint:

K \geq 2

On entry: the number of subgroups within group

i

l_{i}

, for

i = 1, 2, \dots, k

Constraint:

LSUB (i) > 0

, for

i = 1, 2, \dots, k

On entry: the numbers of observations in each subgroup,

n_{i j}

, in the following order:

n_{11}, n_{12}, \dots, n_{1 l_{1}}, n_{21}, \dots, n_{2 l_{2}}, \dots, n_{k 1}, \dots, n_{k l_{k}}

Constraint:

n = \sum_{i = 1}^{k} \sum_{j = 1}^{l_{i}} n_{i j}

, that is

N = \sum_{i = 1}^{l} NOBS (i)

and

NOBS (i) > 0

, for

i = 1, 2, \dots, l

On entry:

l

, the total number of subgroups.

Constraint:

L = \sum_{i = 1}^{k} LSUB (i)

On exit: the total number of observations in group

i

n_{i .}

, for

i = 1, 2, \dots, k

On exit: the mean for group

i

{\bar{y}}_{. i .}

, for

i = 1, 2, \dots, k

On exit: the subgroup means,

{\bar{y}}_{. i j}

, in the following order:

{\bar{y}}_{.11}, {\bar{y}}_{.12}, \dots, {\bar{y}}_{{.1 l}_{1}}, {\bar{y}}_{.21}, {\bar{y}}_{.22}, \dots, {\bar{y}}_{{.2 l}_{2}}, \dots, {\bar{y}}_{. k 1}, {\bar{y}}_{. k 2}, \dots, {\bar{y}}_{{. k l}_{k}} .

On exit: the grand mean,

{\bar{y}}_{\dots}

On exit: contains the sums of squares for the analysis of variance, as follows;

$SS (1) =$ Between group sum of squares, ${SS}_{g}$ ,
$SS (2) =$ Between subgroup within groups sum of squares, ${SS}_{s g}$ ,
$SS (3) =$ Residual sum of squares, ${SS}_{res}$ ,
$SS (4) =$ Corrected total sum of squares, ${SS}_{tot}$ .

On exit: contains the degrees of freedom attributable to each sum of squares in the analysis of variance, as follows:

$IDF (1) =$ Degrees of freedom for between group sum of squares,
$IDF (2) =$ Degrees of freedom for between subgroup within groups sum of squares,
$IDF (3) =$ Degrees of freedom for residual sum of squares,
$IDF (4) =$ Degrees of freedom for corrected total sum of squares.

On exit: contains the mean square ratios,

F_{1}

and

F_{2}

, for the between groups variation, and the between subgroups within groups variation, with respect to the residual, respectively.

On exit: contains the significances of the mean square ratios,

p_{1}

and

p_{2}

respectively.

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is

0

. When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

The total corrected sum of squares is zero, indicating that all the data values are equal. The means returned are therefore all equal, and the sums of squares are zero. No assignments are made to IDF, F, and FP.

The residual sum of squares is zero. This arises when either each subgroup contains exactly one observation, or the observations within each subgroup are equal. The means, sums of squares, and degrees of freedom are computed, but no assignments are made to F and FP.

The computations are believed to be stable.

The time taken by G04AGF increases approximately linearly with the total number of observations,

n

This example has two groups, the first of which consists of five subgroups, and the second of three subgroups. The numbers of observations in each subgroup are not equal. The data represent the percentage stretch in the length of samples of sack kraft drawn from consignments (subgroups) received over two years (groups). For details see Moore et al. (1972).

NAG Library Routine Document

G04AGF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine DocumentG04AGF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

G04AGF