g04db:: Analysis of Variance (NAG Toolbox)

Description

In the computation of analysis of a designed experiment 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}

, the residual degrees of freedom,

ν

, and the (variance ratio)

F

-statistic for the

t

treatments. The second stage of the analysis is to compare the treatment means. If the treatments have no structure, for example the treatments are different varieties, rather than being structured, for example a set of different temperatures, then a multiple comparison procedure can be used.

A multiple comparison procedure looks at all possible pairs of means and either computes confidence intervals for the difference in means or performs a suitable test on the difference. If there are

t

treatments then there are

t (t - 1) / 2

comparisons to be considered. In tests the type

1

error or significance level is the probability that the result is considered to be significant when there is no difference in the means. If the usual

t

-test is used with, say, a

6 %

significance level then the type

1

error for all

k = t (t - 1) / 2

tests will be much higher. If the tests were independent then if each test is carried out at the

100 α

percent level then the overall type

1

error would be

α^{*} = 1 - {(1 - α)}^{k} ≃ k α

. In order to provide an overall protection the individual tests, or confidence intervals, would have to be carried out at a value of

α

such that

α^{*}

is the required significance level, e.g., five percent.

The

100 (1 - α)

percent confidence interval for the difference in two treatment means,

{\hat{τ}}_{i}

and

{\hat{τ}}_{j}

is given by

({\hat{τ}}_{i} - {\hat{τ}}_{j}) \pm T_{(α, ν, t)}^{*} s e ({\hat{τ}}_{i} - {\hat{τ}}_{j}),

where

s e ()

denotes the standard error of the difference in means and

T_{(α, ν, t)}^{*}

is an appropriate percentage point from a distribution. There are several possible choices for

T_{(α, ν, t)}^{*}

. These are:

(a)	$\frac{1}{2} q_{(1 - α, ν, t)}$ , the studentized range statistic, see nag_stat_inv_cdf_studentized_range (g01fm). It is the appropriate statistic to compare the largest mean with the smallest mean. This is known as Tukey–Kramer method.
(b)	$t_{(α / k, ν)}$ , this is the Bonferroni method.
(c)	$t_{(α_{0}, ν)}$ , where $α_{0} = 1 - {(1 - α)}^{1 / k}$ , this is known as the Dunn–Sidak method.
(d)	$t_{(α, ν)}$ , this is known as Fisher's LSD (least significant difference) method. It should only be used if the overall $F$ -test is significant, the number of treatment comparisons is small and were planned before the analysis.
(e)	$\sqrt{(k - 1) F_{1 - α, k - 1, ν}}$ where $F_{1 - α, k - 1, ν}$ is the deviate corresponding to a lower tail probability of $1 - α$ from an $F$ -distribution with $k - 1$ and $ν$ degrees of freedom. This is Scheffe's method.

In cases (b), (c) and (d),

t_{(α, ν)}

denotes the

α

two tail significance level for the Student's

t

-distribution with

ν

degrees of freedom, see nag_stat_inv_cdf_students_t (g01fb).

The Scheffe method is the most conservative, followed closely by the Dunn–Sidak and Tukey–Kramer methods.

To compute a test for the difference between two means the statistic,

\frac{{\hat{τ}}_{i} - {\hat{τ}}_{j}}{s e ({\hat{τ}}_{i} - {\hat{τ}}_{j})}

is compared with the appropriate value of

T_{(α, ν, t)}^{*}

References

Parameters

Compulsory Input Parameters

Optional Input Parameters

Output Parameters

Error Indicators and Warnings

Accuracy

Further Comments

An alternative approach to one used in nag_anova_confidence (g04db) is the sequential testing of the Student–Newman–Keuls procedure. This, in effect, uses the Tukey–Kramer method but first ordering the treatment means and examining only subsets of the treatment means in which the largest and smallest are significantly different. At each stage the third argument of the Studentized range statistic is the number of means in the subset rather than the total number of means.

Example

function g04db_example


fprintf('g04db example results\n\n');

n1 = int64(1);
iblock = n1;
nt = 4*n1;
y  = [ 3  2  4  3  1  5  7  8  4 10 ...
       6  3  2  1  2  4  2  3  1 10 ...
      12  8  5 12 10  9];
it = [n1  1  1  1  1  1  2  2  2  2 ...
       2  3  3  3  3  3  3  3  3  4 ...
       4  4  4  4  4  4];


% 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 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';
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));
fprintf('\nTreatment Means\n\n');
for j = 1:8:nt
  fprintf('%8.3f', tmean(j:min(j+7,nt)));
  fprintf('\n');
end
fprintf('\n');

% Extract the residual degrees of freedom
rdf = table(3,1);

% Calculate simultaneous CIs
typ = 'T';
clevel = 0.95;
[cil, ciu, isig, ifail] = g04db( ...
                                 typ, tmean, rdf, c, clevel);

% Display CI results
fprintf('\nSimultaneous Confidence Intervals\n\n');
star(2) = '*';
star(1) = ' ';
ij = 0;
for i = 1:nt
  for j = 1:i-1
    ij = ij + 1;
    fprintf(' %2d%2d%15.3f%15.3f%5s\n', i, j, cil(ij), ciu(ij), ...
            star(isig(ij)+1));
  end
end

g04db example results

ANOVA table

 Source        df         SS          MS          F        Prob

Treatments      3       239.9        80.0      24.029     0.0000
Residual       22        73.2         3.3
Total          25       313.1

Treatment Means

   3.000   7.000   2.250   9.429


Simultaneous Confidence Intervals

  2 1          0.933          7.067    *
  3 1         -3.486          1.986     
  3 2         -7.638         -1.862    *
  4 1          3.610          9.247    *
  4 2         -0.538          5.395     
  4 3          4.557          9.800    *

On entry,	$nt < 2$ ,
or	$ldc < nt$ ,
or	$rdf < 1.0$ ,
or	$clevel \leq 0.0$ ,
or	$clevel \geq 1.0$ ,
or	$typ \neq'T'$ , $'B'$ , $'D'$ , $'L'$ or $'S'$ .

NAG Toolbox: nag_anova_confidence (g04db)

▸▿ Contents

Purpose

Syntax