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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_anova_factorial (g04ca)

## Purpose

nag_anova_factorial (g04ca) computes an analysis of variance table and treatment means for a complete factorial design.

## Syntax

[table, itotal, tmean, e, imean, semean, bmean, r, ifail] = g04ca(y, lfac, nblock, inter, irdf, mterm, maxt, 'n', n, 'nfac', nfac)
[table, itotal, tmean, e, imean, semean, bmean, r, ifail] = nag_anova_factorial(y, lfac, nblock, inter, irdf, mterm, maxt, 'n', n, 'nfac', nfac)

## Description

An experiment consists of a collection of units, or plots, to which a number of treatments are applied. In a factorial experiment the effects of several different sets of conditions are compared, e.g., three different temperatures, T1${T}_{1}$, T2${T}_{2}$ and T3${T}_{3}$, and two different pressures, P1${P}_{1}$ and P2${P}_{2}$. The conditions are known as factors and the different values the conditions take are known as levels. In a factorial experiment the experimental treatments are the combinations of all the different levels of all factors, e.g.,
 T1P1, T2P1, T3P1 T1P2, T2P2, T3P2
$T1P1, T2P1, T3P1 T1P2, T2P2, T3P2$
The effect of a factor averaged over all other factors is known as a main effect, and the effect of a combination of some of the factors averaged over all other factors is known as an interaction. This can be represented by a linear model. In the above example if the response was yijk${y}_{ijk}$ for the k$k$th replicate of the i$i$th level of T$T$ and the j$j$th level of P$P$ the linear model would be
 yijk = μ + ti + pj + γij + eijk $yijk = μ+ ti+ pj+ γij+ eijk$
where μ$\mu$ is the overall mean, ti${t}_{i}$ is the main effect of T$T$, pj${p}_{j}$ is the main effect of P$P$, γij${\gamma }_{ij}$ is the T × P$T×P$ interaction and eijk${e}_{ijk}$ is the random error term. In order to find unique estimates constraints are placed on the parameters estimates. For the example here these are:
 3 ∑ t̂i = 0, i = 1
 2 ∑ p̂j = 0, j = 1
 3 ∑ γ̂ij = 0, i = 1
for ​j = 1,2​ and
 2 ∑ γ̂ij = 0, j = 1
for ​ i = 1,2,3 ,
$∑i=13t^i=0, ∑j=12p^j=0, ∑ i=1 3 γ^ij = 0 , for ​j=1,2​ and ∑ j=1 2 γ^ ij = 0 , for ​ i=1,2,3 ,$
where ​ ​^$\stackrel{^}{\text{​ ​}}$ denotes the estimate.
If there is variation in the experimental conditions (e.g., in an experiment on the production of a material different batches of raw material may be used, or the experiment may be carried out on different days), then plots that are similar are grouped together into blocks. For a balanced complete factorial experiment all the treatment combinations occur the same number of times in each block.
nag_anova_factorial (g04ca) computes the analysis of variance (ANOVA) table by sequentially computing the totals and means for an effect from the residuals computed when previous effects have been removed. The effect sum of squares is the sum of squared totals divided by the number of observations per total. The means are then subtracted from the residuals to compute a new set of residuals. At the same time the means for the original data are computed. When all effects are removed the residual sum of squares is computed from the residuals. Given the sums of squares an ANOVA table is then computed along with standard errors for the difference in treatment means.
The data for nag_anova_factorial (g04ca) has to be in standard order given by the order of the factors. Let there be k$k$ factors, f1,f2,,fk${f}_{1},{f}_{2},\dots ,{f}_{k}$ in that order with levels l1,l2,,lk${l}_{1},{l}_{2},\dots ,{l}_{k}$ respectively. Standard order requires the levels of factor f1${f}_{1}$ are in order 1,2,,l1$1,2,\dots ,{l}_{1}$ and within each level of f1${f}_{1}$ the levels of f2${f}_{2}$ are in order 1,2,,l2$1,2,\dots ,{l}_{2}$ and so on.
For an experiment with blocks the data is for block 1$1$ then for block 2$2$, etc. Within each block the data must be arranged so that the levels of factor f1${f}_{1}$ are in order 1,2,,l1$1,2,\dots ,{l}_{1}$ and within each level of f1${f}_{1}$ the levels of f2${f}_{2}$ are in order 1,2,,l2$1,2,\dots ,{l}_{2}$ and so on. Any within block replication of treatment combinations must occur within the levels of fk${f}_{k}$.
The ANOVA table is given in the following order. For a complete factorial experiment the first row is for blocks, if present, then the main effects of the factors in their order, e.g., f1${f}_{1}$ followed by f2${f}_{2}$, etc. These are then followed by all the two factor interactions then all the three factor interactions, etc., the last two rows being for the residual and total sums of squares. The interactions are arranged in lexical order for the given factor order. For example, for the three factor interactions for a five factor experiment the 10$10$ interactions would be in the following order:
 f1f2f3 f1f2f4 f1f2f5 f1f3f4 f1f3f5 f1f4f5 f2f3f4 f2f3f5 f2f4f5 f3f4f5
$f1f2f3 f1f2f4 f1f2f5 f1f3f4 f1f3f5 f1f4f5 f2f3f4 f2f3f5 f2f4f5 f3f4f5$

## References

Cochran W G and Cox G M (1957) Experimental Designs Wiley
Davis O L (1978) The Design and Analysis of Industrial Experiments Longman
John J A and Quenouille M H (1977) Experiments: Design and Analysis Griffin

## Parameters

### Compulsory Input Parameters

1:     y(n) – double array
n, the dimension of the array, must satisfy the constraint
• n4${\mathbf{n}}\ge 4$
• if nblock > 1${\mathbf{nblock}}>1$, n must be a multiple of nblock
• n must be a multiple of the number of treatment combinations, that is a multiple of i = 1klfac(i)$\prod _{i=1}^{k}{\mathbf{lfac}}\left(i\right)$
• .
The observations in standard order, see Section [Description].
2:     lfac(nfac) – int64int32nag_int array
nfac, the dimension of the array, must satisfy the constraint nfac1${\mathbf{nfac}}\ge 1$.
lfac(i)${\mathbf{lfac}}\left(\mathit{i}\right)$ must contain the number of levels for the i$\mathit{i}$th factor, for i = 1,2,,k$\mathit{i}=1,2,\dots ,k$.
Constraint: lfac(i)2${\mathbf{lfac}}\left(\mathit{i}\right)\ge 2$, for i = 1,2,,k$\mathit{i}=1,2,\dots ,k$.
3:     nblock – int64int32nag_int scalar
The number of blocks. If there are no blocks, set nblock = 0${\mathbf{nblock}}=0$ or 1$1$.
Constraints:
• nblock0${\mathbf{nblock}}\ge 0$;
• if nblock2${\mathbf{nblock}}\ge 2$, ${\mathbf{n}}/{\mathbf{nblock}}$ must be a multiple of the number of treatment combinations, that is a multiple of i = 1klfac(i)$\prod _{i=1}^{k}{\mathbf{lfac}}\left(i\right)$.
4:     inter – int64int32nag_int scalar
The maximum number of factors in an interaction term. If no interaction terms are to be computed, set inter = 0${\mathbf{inter}}=0$ or 1$1$.
Constraint: 0internfac$0\le {\mathbf{inter}}\le {\mathbf{nfac}}$.
5:     irdf – int64int32nag_int scalar
The adjustment to the residual and total degrees of freedom. The total degrees of freedom are set to ${\mathbf{n}}-{\mathbf{irdf}}$ and the residual degrees of freedom adjusted accordingly. For examples of the use of irdf see Section [Further Comments].
Constraint: irdf0${\mathbf{irdf}}\ge 0$.
6:     mterm – int64int32nag_int scalar
The maximum number of terms in the analysis of variance table, see Section [Further Comments].
Constraint: mterm${\mathbf{mterm}}$ must be large enough to contain the terms specified by nfac, inter and nblock. If the function exits with ${\mathbf{ifail}}\ge {\mathbf{2}}$, the required minimum value of mterm is returned in itotal.
7:     maxt – int64int32nag_int scalar
The maximum number of treatment means to be computed, see Section [Further Comments]. If the value of maxt is too small for the required analysis then the minimum number is returned in imean(1)${\mathbf{imean}}\left(1\right)$.
Constraint: maxt${\mathbf{maxt}}$ must be large enough for the number of means specified by lfac and inter; if ${\mathbf{inter}}={\mathbf{nfac}}$ then maxti = 1k(lfac(i) + 1)1${\mathbf{maxt}}\ge \prod _{i=1}^{k}\left({\mathbf{lfac}}\left(i\right)+1\right)-1$.

### Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The dimension of the array y.
The number of observations.
Constraints:
• n4${\mathbf{n}}\ge 4$;
• if nblock > 1${\mathbf{nblock}}>1$, n must be a multiple of nblock;
• n must be a multiple of the number of treatment combinations, that is a multiple of i = 1klfac(i)$\prod _{i=1}^{k}{\mathbf{lfac}}\left(i\right)$.
2:     nfac – int64int32nag_int scalar
Default: The dimension of the array lfac.
k$k$, the number of factors.
Constraint: nfac1${\mathbf{nfac}}\ge 1$.

iwk

### Output Parameters

1:     table(mterm,5$5$) – double array
The first itotal rows of table contain the analysis of variance table. The first column contains the degrees of freedom, the second column contains the sum of squares, the third column (except for the row corresponding to the total sum of squares) contains the mean squares, i.e., the sums of squares divided by the degrees of freedom, and the fourth and fifth columns contain the F$F$ ratio and significance level, respectively (except for rows corresponding to the total sum of squares, and the residual sum of squares). All other cells of the table are set to zero.
The first row corresponds to the blocks and is set to zero if there are no blocks. The itotalth row corresponds to the total sum of squares for y and the (itotal1)$\left({\mathbf{itotal}}-1\right)$th row corresponds to the residual sum of squares. The central rows of the table correspond to the main effects followed by the interaction if specified by inter. The main effects are in the order specified by lfac and the interactions are in lexical order, see Section [Description].
2:     itotal – int64int32nag_int scalar
The row in table corresponding to the total sum of squares. The number of treatment effects is itotal3${\mathbf{itotal}}-3$.
3:     tmean(maxt) – double array
The treatment means. The position of the means for an effect is given by the index in imean. For a given effect the means are in standard order, see Section [Description].
4:     e(maxt) – double array
The estimated effects in the same order as for the means in tmean.
5:     imean(mterm) – int64int32nag_int array
Indicates the position of the effect means in tmean. The effect means corresponding to the first treatment effect in the ANOVA table are stored in tmean(1)${\mathbf{tmean}}\left(1\right)$ up to tmean(imean(1))${\mathbf{tmean}}\left({\mathbf{imean}}\left(1\right)\right)$. Other effect means corresponding to the i$i$th treatment effect, i = 1,2,,itotal3$i=1,2,\dots ,{\mathbf{itotal}}-3$, are stored in tmean(imean(i1) + 1)${\mathbf{tmean}}\left({\mathbf{imean}}\left(i-1\right)+1\right)$ up to tmean(imean(i))${\mathbf{tmean}}\left({\mathbf{imean}}\left(i\right)\right)$.
6:     semean(mterm) – double array
The standard error of the difference between means corresponding to the i$i$th treatment effect in the ANOVA table.
7:     bmean(nblock + 1${\mathbf{nblock}}+1$) – double array
bmean(1)${\mathbf{bmean}}\left(1\right)$ contains the grand mean, if nblock > 1${\mathbf{nblock}}>1$, bmean(2)${\mathbf{bmean}}\left(2\right)$ up to bmean(nblock + 1)${\mathbf{bmean}}\left({\mathbf{nblock}}+1\right)$ contain the block means.
8:     r(n) – double array
The residuals.
9:     ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

## Error Indicators and Warnings

Errors or warnings detected by the function:
ifail = 1${\mathbf{ifail}}=1$
 On entry, n < 4${\mathbf{n}}<4$, or nfac < 1${\mathbf{nfac}}<1$, or nblock < 0${\mathbf{nblock}}<0$, or inter < 0${\mathbf{inter}}<0$, or ${\mathbf{inter}}>{\mathbf{nfac}}$, or irdf < 0${\mathbf{irdf}}<0$.
ifail = 2${\mathbf{ifail}}=2$
 On entry, lfac(i) ≤ 1${\mathbf{lfac}}\left(i\right)\le 1$, for some i = 1,2, … ,nfac$i=1,2,\dots ,{\mathbf{nfac}}$, or the value of maxt is too small, or the value of mterm is too small, or nblock > 1${\mathbf{nblock}}>1$ and n is not a multiple of nblock, or the number of plots per block is not a multiple of the number of treatment combinations.
ifail = 3${\mathbf{ifail}}=3$
 On entry, the values of y are constant.
ifail = 4${\mathbf{ifail}}=4$
There are no degrees of freedom for the residual or the residual sum of squares is zero. In either case the standard errors and F$F$-statistics cannot be computed.

## Accuracy

The block and treatment sums of squares are computed from the block and treatment residual totals. The residuals are updated as each effect is computed and the residual sum of squares computed directly from the residuals. This avoids any loss of accuracy in subtracting sums of squares.

The number of rows in the ANOVA table and the number of treatment means are given by the following formulae.
Let there be k$k$ factors with levels li${l}_{i}$ for i = 1,2,,k$i=1,2,\dots ,k$, and let t$t$ be the maximum number of terms in an interaction then the number of rows in the ANOVA tables is:
t
 ( k ) i
+ 3.
i = 1
$∑i=1t k i +3.$
The number of treatment means is:
 t ∑ ∏ lj, i = 1 j ∈ Si
$∑ i =1 t ∏ j ∈ S i l j ,$
where Si${S}_{i}$ is the set of all combinations of the k$k$ factors i$i$ at a time.
To estimate missing values the Healy and Westmacott procedure or its derivatives may be used, see John and Quenouille (1977). This is an iterative procedure in which estimates of the missing values are adjusted by subtracting the corresponding values of the residuals. The new estimates are then used in the analysis of variance. This process is repeated until convergence. A suitable initial value may be the grand mean. When using this procedure irdf should be set to the number of missing values plus one to obtain the correct degrees of freedom for the residual sum of squares.
For analysis of covariance the residuals are obtained from an analysis of variance of both the response variable and the covariates. The residuals from the response variable are then regressed on the residuals from the covariates using, say, nag_correg_linregs_noconst (g02cb) or nag_correg_linregm_fit (g02da). The coefficients obtained from the regression can be examined for significance and used to produce an adjusted dependent variable using the original response variable and covariate. An approximate adjusted analysis of variance table can then be produced by using the adjusted dependent variable. In this case irdf should be set to one plus the number of fitted covariates.
For designs such as Latin squares one more of the blocking factors has to be removed in a preliminary analysis before the final analysis. This preliminary analysis can be performed using nag_anova_random (g04bb) or a prior call to nag_anova_factorial (g04ca) if the data is reordered between calls. The residuals from the preliminary analysis are then input to nag_anova_factorial (g04ca). In these cases irdf should be set to the difference between n and the residual degrees of freedom from preliminary analysis. Care should be taken when using this approach as there is no check on the orthogonality of the two analyses.

## Example

```function nag_anova_factorial_example
y = [274;
361;
253;
325;
317;
339;
326;
402;
336;
379;
345;
361;
352;
334;
318;
339;
393;
358;
350;
340;
203;
397;
356;
298;
382;
376;
355;
418;
387;
379;
432;
339;
293;
322;
417;
342;
82;
297;
133;
306;
352;
361;
220;
333;
270;
388;
379;
274;
336;
307;
266;
389;
333;
353];
lfac = [int64(6);3];
nblock = int64(3);
inter = int64(2);
irdf = int64(0);
mterm = int64(6);
maxt = int64(27);
[table, itotal, tmean, e, imean, semean, bmean, r, ifail] = ...
nag_anova_factorial(y, lfac, nblock, inter, irdf, mterm, maxt)
```
```

table =

1.0e+05 *

0.0000    0.3012    0.1506    0.0001    0.0000
0.0001    0.7301    0.1460    0.0001    0.0000
0.0000    0.2160    0.1080    0.0001    0.0000
0.0001    0.3119    0.0312    0.0000    0.0000
0.0003    0.6663    0.0196         0         0
0.0005    2.2254         0         0         0

itotal =

6

tmean =

254.7778
339.0000
333.3333
367.7778
330.7778
360.6667
334.2778
353.7778
305.1111
235.3333
332.6667
196.3333
342.6667
341.6667
332.6667
309.3333
370.3333
320.3333
395.0000
370.3333
338.0000
373.3333
326.6667
292.3333
350.0000
381.0000
351.0000

e =

-76.2778
7.9444
2.2778
36.7222
-0.2778
29.6111
3.2222
22.7222
-25.9444
-22.6667
55.1667
-32.5000
0.4444
-20.0556
19.6111
-27.2222
14.2778
12.9444
24.0000
-20.1667
-3.8333
39.3333
-26.8333
-12.5000
-13.8889
-2.3889
16.2778

imean =

6
9
27
0
0
0

semean =

20.8681
14.7560
36.1446
0
0
0

bmean =

331.0556
339.5556
354.7778
298.8333

r =

30.1667
19.8333
48.1667
-26.1667
-33.1667
-2.1667
8.1667
23.1667
7.1667
-24.5000
-33.8333
14.5000
-29.8333
-1.1667
17.1667
-19.5000
3.5000
-1.5000
90.9444
-16.3889
-17.0556
30.6111
-9.3889
-58.3889
48.9444
-18.0556
10.9444
-0.7222
-7.0556
17.2778
34.9444
-11.3889
-23.0556
-51.7222
12.2778
-32.7222
-121.1111
-3.4444
-31.1111
-4.4444
42.5556
60.5556
-57.1111
-5.1111
-18.1111
25.2222
40.8889
-31.7778
-5.1111
12.5556
5.8889
71.2222
-15.7778
34.2222

ifail =

0

```
```function g04ca_example
y = [274;
361;
253;
325;
317;
339;
326;
402;
336;
379;
345;
361;
352;
334;
318;
339;
393;
358;
350;
340;
203;
397;
356;
298;
382;
376;
355;
418;
387;
379;
432;
339;
293;
322;
417;
342;
82;
297;
133;
306;
352;
361;
220;
333;
270;
388;
379;
274;
336;
307;
266;
389;
333;
353];
lfac = [int64(6);3];
nblock = int64(3);
inter = int64(2);
irdf = int64(0);
mterm = int64(6);
maxt = int64(27);
[table, itotal, tmean, e, imean, semean, bmean, r, ifail] = ...
g04ca(y, lfac, nblock, inter, irdf, mterm, maxt)
```
```

table =

1.0e+05 *

0.0000    0.3012    0.1506    0.0001    0.0000
0.0001    0.7301    0.1460    0.0001    0.0000
0.0000    0.2160    0.1080    0.0001    0.0000
0.0001    0.3119    0.0312    0.0000    0.0000
0.0003    0.6663    0.0196         0         0
0.0005    2.2254         0         0         0

itotal =

6

tmean =

254.7778
339.0000
333.3333
367.7778
330.7778
360.6667
334.2778
353.7778
305.1111
235.3333
332.6667
196.3333
342.6667
341.6667
332.6667
309.3333
370.3333
320.3333
395.0000
370.3333
338.0000
373.3333
326.6667
292.3333
350.0000
381.0000
351.0000

e =

-76.2778
7.9444
2.2778
36.7222
-0.2778
29.6111
3.2222
22.7222
-25.9444
-22.6667
55.1667
-32.5000
0.4444
-20.0556
19.6111
-27.2222
14.2778
12.9444
24.0000
-20.1667
-3.8333
39.3333
-26.8333
-12.5000
-13.8889
-2.3889
16.2778

imean =

6
9
27
0
0
0

semean =

20.8681
14.7560
36.1446
0
0
0

bmean =

331.0556
339.5556
354.7778
298.8333

r =

30.1667
19.8333
48.1667
-26.1667
-33.1667
-2.1667
8.1667
23.1667
7.1667
-24.5000
-33.8333
14.5000
-29.8333
-1.1667
17.1667
-19.5000
3.5000
-1.5000
90.9444
-16.3889
-17.0556
30.6111
-9.3889
-58.3889
48.9444
-18.0556
10.9444
-0.7222
-7.0556
17.2778
34.9444
-11.3889
-23.0556
-51.7222
12.2778
-32.7222
-121.1111
-3.4444
-31.1111
-4.4444
42.5556
60.5556
-57.1111
-5.1111
-18.1111
25.2222
40.8889
-31.7778
-5.1111
12.5556
5.8889
71.2222
-15.7778
34.2222

ifail =

0

```