g04da:: Analysis of Variance (NAG Toolbox)

Description

In the analysis of designed experiments the first stage is to compute the basic analysis of variance table, the estimate of the error variance (the residual or error mean square),

{\hat{σ}}^{2}

, and the (variance ratio)

F

-statistic for the

t

treatments. If this

F

-test is significant then the second stage of the analysis is to explore which treatments are significantly different.

If there is a structure to the treatments then this may lead to hypotheses that can be defined before the analysis and tested using linear contrasts. For example, if the treatments were three different fixed temperatures, say

18

20

and

22

, and an uncontrolled temperature (denoted by

N

) then the following contrasts might be of interest.

\begin{array}{r} 18 & 20 & 22 & N \\ (a) & \frac{1}{3} & \frac{1}{3} & \frac{1}{3} & - 1 \\ (b) & - 1 & 0 & 1 & 0 \end{array}

The first represents the average difference between the controlled temperatures and the uncontrolled temperature. The second represents the linear effect of an increasing fixed temperature.

For a randomized complete block design or a completely randomized design, let the treatment means be

{\hat{τ}}_{i}

i = 1, 2, \dots, t

, and let the

j

th contrast be defined by

λ_{i j}

i = 1, 2, \dots, t

, then the estimate of the contrast is simply:

Λ_{j} = \sum_{i = 1}^{t} {\hat{τ}}_{i} λ_{i j}

and the sum of squares for the contrast is:

{SS}_{j} = \frac{Λ_{j}^{2}}{\sum_{i = 1}^{t} λ_{i j}^{2} / n_{i}}

(1)

where

n_{i}

is the number of observations for the

i

th treatment. Such a contrast has one degree of freedom so that the appropriate

F

-statistic is

{SS}_{j} / {\hat{σ}}^{2}

The two contrasts

λ_{i j}

and

λ_{i j^{'}}

are orthogonal if

\sum_{i = 1}^{t} λ_{i j} λ_{i j^{'}} = 0

and the contrast

λ_{i j}

is orthogonal to the overall mean if

\sum_{i = 1}^{t} λ_{i j} = 0

. In practice these sums will be tested against a small quantity,

ε

. If each of a set of contrasts is orthogonal to the mean and they are all mutually orthogonal then the contrasts provide a partition of the treatment sum of squares into independent components. Hence the resulting

F

-tests are independent.

If the treatments come from a design in which treatments are not orthogonal to blocks then the sum of squares for a contrast is given by:

{SS}_{j} = \frac{Λ_{j} Λ_{j}^{*}}{\sum_{i = 1}^{t} λ_{i j}^{2} / n_{i}}

(2)

where

Λ_{j}^{*} = \sum_{i = 1}^{t} τ_{i}^{*} λ_{i j}

with

τ_{i}^{*}

, for

i = 1, 2, \dots, t

, being adjusted treatment means computed by first eliminating blocks then computing the treatment means from the block adjusted observations without taking into account the non-orthogonality between treatments and blocks. For further details see John (1987).

References

Parameters

Compulsory Input Parameters

Optional Input Parameters

Output Parameters

Error Indicators and Warnings

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

Accuracy

Further Comments

Example

The data is from a completely randomized experiment on potato scab with seven treatments representing amounts of sulphur applied, whether the application was in spring or autumn and a control treatment. The one-way anova is computed using nag_correg_coeffs_pearson_miss_case (g02bb). Two contrasts are analysed, one comparing the control with use of sulphur, the other comparing spring with autumn application.

function g04da_example


fprintf('g04da example results\n\n');

n1 = int64(1);
iblock = n1;

y  = [12 10 24 29 30 18 32 26  9  9 ...
      16  4 30  7 21  9 16 10 18 18 ...
      18 24 12 19 10  4  4  5 17  7 ...
      16 17];
it = [n1  1  1  1  1  1  1  1  2  2 ...
       2  2  3  3  3  3  4  4  4  4 ...
       5  5  5  5  6  6  6  6  7  7 ...
       7  7];

% Contrasts
nt = 7*n1;
nc = 2*n1;
ct = [6,  0;
     -1,  1;
     -1, -1;
     -1,  1;
     -1, -1;
     -1,  1;
     -1, -1];
names = {'Cntl v S    '; 'Spring v A  '};

% Calculate ANOVA table
tol  = 0;
irdf = int64(0);
[gmean, bmean, tmean, table, c, irep, r, ef, ifail] = ...
  g04bb( ...
         y, iblock, nt, it, tol, irdf);

% Display ANOVA table results
fprintf('ANOVA table\n\n');
fprintf(' Source        df         SS          MS          F        Prob\n\n');
fmt5 = '%s%5.0f%12.1f%12.1f%12.3f%11.4f\n';
fmt3 = '%s%5.0f%12.1f%12.1f\n';
fmt2 = '%s%5.0f%12.1f\n\n';
if iblock > 1
  fprintf(fmt5, 'Blocks      ', table(1,1:5));
end
fprintf(fmt5, 'Treatments  ', table(2,1:5));
fprintf(fmt3, 'Residual    ', table(3,1:3));
fprintf(fmt2, 'Total       ', table(4,1:2));

% Extract the residual mean square and degrees of freedom from table
rms = table(3,3);
rdf = table(3,1);

% Compute sums of squares for contrast
usetx = false;
tx    = zeros(nt, 1);
table = zeros(nc,5);
[est, table, ifail] = g04da( ...
                             tmean, irep, rms, rdf, ct, table, tol, usetx, tx);

% Display results
fprintf('Orthogonal Contrasts\n\n');
for i = 1:nc
  fprintf(fmt5, names{i}, table(i,1:5));
end

g04da example results

ANOVA table

 Source        df         SS          MS          F        Prob

Treatments      6       972.3       162.1       3.608     0.0103
Residual       25      1122.9        44.9
Total          31      2095.2

Orthogonal Contrasts

Cntl v S        1       518.0       518.0      11.533     0.0023
Spring v A      1       228.2       228.2       5.080     0.0332

On entry,	$nc < 1$ ,
or	$nt < 2$ ,
or	$ldct < nt$ ,
or	$ldtabl < nc$ ,
or	$rms \leq 0.0$ ,
or	$rdf < 1.0$ .

On entry,	a contrast is not orthogonal to the mean,
or	at least two contrasts are not orthogonal.

NAG Toolbox: nag_anova_contrasts (g04da)

▸▿ Contents

Purpose

Syntax