NAG Library Function Document

nag_anova_confid_interval (g04dbc)

+− 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

1 Purpose

nag_anova_confid_interval (g04dbc) computes simultaneous confidence intervals for the differences between means. It is intended for use after nag_anova_random (g04bbc) or nag_anova_row_col (g04bcc).

2 Specification

#include <nag.h>

#include <nagg04.h>

void	nag_anova_confid_interval (Nag_IntervalType type, Integer nt, const double tmean[], double rdf, const double c[], Integer tdc, double clevel, double cil[], double ciu[], Integer isig[], NagError *fail)

3 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 five percent 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. 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_deviates_students_t (g01fbc).

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)}^{*}

4 References

Kotz S and Johnson N L (ed.) (1985a) Multiple range and associated test procedures Encyclopedia of Statistical Sciences 5 Wiley, New York

Kotz S and Johnson N L (ed.) (1985b) Multiple comparison Encyclopedia of Statistical Sciences 5 Wiley, New York

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

5 Arguments

1: type – Nag_IntervalTypeInput

On entry: indicates which method is to be used.

$type = Nag_TukeyInterval$: The Tukey–Kramer method is used.
$type = Nag_BonferroniInterval$: The Bonferroni method is used.
$type = Nag_DunnInterval$: The Dunn–Sidak method is used.
$type = Nag_FisherInterval$: The Fisher LSD method is used.
$type = Nag_ScheffeInterval$: The Scheffe's method is used.

Constraint:

type = Nag_TukeyInterval

Nag_BonferroniInterval

Nag_DunnInterval

Nag_FisherInterval

Nag_ScheffeInterval

2: nt – IntegerInput

On entry: the number of treatment means,

t

Constraint:

nt \geq 2

3: tmean[nt] – const doubleInput

On entry:

tmean [i - 1]

contains the treatment means,

{\hat{τ}}_{i}

i = 1, 2, \dots, t

4: rdf – doubleInput

On entry: the residual degrees of freedom,

ν

Constraint:

rdf \geq 1.0

5: c[ $nt \times tdc$ ] – const doubleInput

On entry: the strictly lower triangular part of c must contain the standard errors of the differences between the means as returned by nag_anova_random (g04bbc) and nag_anova_row_col (g04bcc). That is

c [(i - 1) \times tdc + j - 1]

i > j

, contains the standard error of the difference between the

i

th and

j

th mean in tmean.

Constraint:

c [(i - 1) \times tdc + j - 1] > 0.0

, for

i = 2, 3, \dots, t

and

j = 1, 2, \dots, i - 1

6: tdc – IntegerInput

On entry: the stride separating matrix column elements in the array c.

Constraint:

tdc \geq nt

7: clevel – doubleInput

On entry: the required confidence level for the computed intervals,

(1 - α)

Constraint:

0.0 < clevel < 1.0

8: cil[ $nt \times (nt - 1) / 2$ ] – doubleOutput

On exit:

cil [(i - 1) (i - 2) / 2 + j - 1]

contains the lower limit to the confidence interval for the difference between

i

th and

j

th means in tmean, for

i = 2, 3, \dots, t

and

j = 1, 2, \dots, i - 1

9: ciu[ $nt \times (nt - 1) / 2$ ] – doubleOutput

On exit:

ciu [(i - 1) (i - 2) / 2 + j - 1]

contains the upper limit to the confidence interval for the difference between

i

th and

j

th means in tmean, for

i = 2, 3, \dots, t

and

j = 1, 2, \dots, i - 1

10: isig[ $nt \times (nt - 1) / 2$ ] – IntegerOutput

On exit:

isig [(i - 1) (i - 2) / 2 + j - 1]

indicates if the difference between

i

th and

j

th means in tmean is significant, for

i = 2, 3, \dots, t

and

j = 1, 2, \dots, i - 1

. If the difference is significant then the returned value is 1; otherwise the returned value is 0.

11: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_2_INT_ARG_LT: On entry, $tdc = ⟨value⟩$ while $nt = ⟨value⟩$ . These arguments must satisfy $tdc \geq nt$ .
NE_2D_REAL_ARRAY_CONS: On entry, $c [(⟨value⟩) \times tdc + ⟨value⟩] = ⟨value⟩$ .
Constraint: $c [(i) \times tdc + j] > 0.0$ , for $i = 1, 2, \dots, nt - 1$ and $j = 0, 1, \dots, i - 1$ .
NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BAD_PARAM: On entry, argument type had an illegal value.
NE_INT_ARG_LT: On entry, $nt = ⟨value⟩$ .
Constraint: $nt \geq 2$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_REAL: On entry, $clevel = ⟨value⟩$ .
Constraint: $0.0 < clevel < 1.0$ .
NE_REAL_ARG_LT: On entry, rdf must not be less than 1.0: $rdf = ⟨value⟩$ .
NE_STUDENTIZED_STAT: There has been a failure in the computation of the studentized range statistic. Try using a smaller value of clevel.

7 Accuracy

For the accuracy of the percentage point statistics see nag_deviates_students_t (g01fbc).

8 Parallelism and Performance

Not applicable.

9 Further Comments

An alternative approach to one used in nag_anova_confid_interval (g04dbc) 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.

10 Example

In the example taken from Winer (1970) a completely randomized design with unequal treatment replication is analysed using nag_anova_random (g04bbc) and then confidence intervals are computed by nag_anova_confid_interval (g04dbc) using the Tukey–Kramer method.

NAG Library Function Documentnag_anova_confid_interval (g04dbc)

+− 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 Library Function Document

nag_anova_confid_interval (g04dbc)