Integer, Intent (In)	::	nt, irep(nt), nc, ldct, ldtabl
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (In)	::	tmean(nt), rms, rdf, ct(ldct,nc), tol, tx(nt)
Real (Kind=nag_wp), Intent (Inout)	::	tabl(ldtabl,*)
Real (Kind=nag_wp), Intent (Out)	::	est(nc)
Logical, Intent (In)	::	usetx

C Header Interface

#include <nag.h>

void	g04daf_ (const Integer nt, const double tmean[], const Integer irep[], const double rms, const double rdf, const Integer nc, const double ct[], const Integer ldct, double est[], double tabl[], const Integer ldtabl, const double tol, const logical usetx, const double tx[], Integer *ifail)

The routine may be called by the names g04daf or nagf_anova_contrasts.

3 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).

4 References

Cochran W G and Cox G M (1957) Experimental Designs Wiley

John J A (1987) Cyclic Designs Chapman and Hall

Winer B J (1970) Statistical Principles in Experimental Design McGraw–Hill

5 Arguments

1: $nt$ – Integer Input: On entry: $t$ , the number of treatment means.

Constraint: $nt \geq 2$ .
2: $tmean (nt)$ – Real (Kind=nag_wp) array Input: On entry: the treatment means, ${\hat{τ}}_{i}$ , for $i = 1, 2, \dots, t$ .
3: $irep (nt)$ – Integer array Input: On entry: the replication for each treatment mean, $n_{i}$ , for $i = 1, 2, \dots, t$ .
4: $rms$ – Real (Kind=nag_wp) Input: On entry: the residual mean square, ${\hat{σ}}^{2}$ .

Constraint: $rms > 0.0$ .
5: $rdf$ – Real (Kind=nag_wp) Input: On entry: the residual degrees of freedom.

Constraint: $rdf \geq 1.0$ .
6: $nc$ – Integer Input: On entry: the number of contrasts.

Constraint: $nc \geq 1$ .
7: $ct (ldct, nc)$ – Real (Kind=nag_wp) array Input: On entry: the columns of ct must contain the nc contrasts, that is $ct (i, j)$ must contain $λ_{i j}$ , for $i = 1, 2, \dots, t$ and $j = 1, 2, \dots, nc$ .
8: $ldct$ – Integer Input: On entry: the first dimension of the array ct as declared in the (sub)program from which g04daf is called.

Constraint: $ldct \geq nt$ .
9: $est (nc)$ – Real (Kind=nag_wp) array Output: On exit: the estimates of the contrast, $Λ_{j}$ , for $j = 1, 2, \dots, nc$ .
10: $tabl (ldtabl *)$ – Real (Kind=nag_wp) array Input/Output: Note: the second dimension of the array tabl must be at least $5$ .

On entry: the elements of tabl that are not referenced as described below remain unchanged.

On exit: the rows of the analysis of variance table for the contrasts. For each row column 1 contains the degrees of freedom, column 2 contains the sum of squares, column 3 contains the mean square, column 4 the $F$ -statistic and column 5 the significance level for the contrast. Note that the degrees of freedom are always one and so the mean square equals the sum of squares.
11: $ldtabl$ – Integer Input: On entry: the first dimension of the array tabl as declared in the (sub)program from which g04daf is called.

Constraint: $ldtabl \geq nc$ .
12: $tol$ – Real (Kind=nag_wp) Input: On entry: the tolerance, $ε$ used to check if the contrasts are orthogonal and if they are orthogonal to the mean. If $tol \leq 0.0$ the value machine precision is used.
13: $usetx$ – Logical Input: On entry: if $usetx = .TRUE.$ the means $τ_{i}^{*}$ are provided in tx and the formula (2) is used instead of formula (1).
If $usetx = .FALSE.$ formula (1) is used and tx is not referenced.
14: $tx (nt)$ – Real (Kind=nag_wp) array Input: On entry: if $usetx = .TRUE.$ tx must contain the means $τ_{i}^{*}$ , for $i = 1, 2, \dots, t$ .
15: $ifail$ – Integer Input/Output: On entry: ifail must be set to $0$ , $- 1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $- 1$ means that an error message is printed while a value of $1$ means that it is not.

If halting is not appropriate, the value $- 1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $- 1$ is recommended since useful values can be provided in some output arguments even when $ifail \neq 0$ on exit. 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).

6 Error Indicators and Warnings

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:

Note: in some cases g04daf may return useful information.

$ifail = 1$: On entry, $ldct = 〈value〉$ and $nt = 〈value〉$ .
Constraint: $ldct \geq nt$ .

On entry, $ldtabl = 〈value〉$ and $nc = 〈value〉$ .
Constraint: $ldtabl \geq nc$ .

On entry, $nc = 〈value〉$ .
Constraint: $nc \geq 1$ .

On entry, $nt = 〈value〉$ .
Constraint: $nt \geq 2$ .

On entry, $rdf = 〈value〉$ .
Constraint: $rdf \geq 1.0$ .

On entry, $rms = 〈value〉$ .
Constraint: $rms > 0.0$ .

$ifail = 2$: The $〈value〉$ and $〈value〉$ contrasts are not orthogonal. Full results are returned but they should be interpreted with care.

The $〈value〉$ contrast is not orthogonal to the mean. Full results are returned but they should be interpreted with care.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

The computations are stable.

8 Parallelism and Performance

g04daf is not threaded in any implementation.

9 Further Comments

If the treatments have a factorial structure g04caf should be used and if the treatments have no structure the means can be compared using g04dbf.

10 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 g02bbf. Two contrasts are analysed, one comparing the control with use of sulphur, the other comparing spring with autumn application.

g04da: FL CL CPP AD

NAG FL Interfaceg04daf (contrasts)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG FL Interface
g04daf (contrasts)