An ANOVA test for the equivalency of means under unequal variances

www.elsevier.com/locate/csda

An ANOVA test for the equivalency

of means under unequal variances

Shun-Yi Chen

a; ∗

_{, Hubert J. Chen}

b

_{, Cherng G. Ding}

c a_{Department of Mathematics, Tamkang University, Tamsui 251, Taiwan} b_{Department of Statistics, The University of Georgia, Athens, GA 30602, USA} c_{Institute of Business and Management, National Chiao Tung University, Taipei, 100, Taiwan}

Received 1 December 1998; received in revised form 1 June 1999

Abstract

In this paper, we present a two-stage test procedure for testing the hypothesis that the normal means are falling into a practically indierent zone. Both the level and the power associated with the proposed test are controllable and are completely independent of the unknown variances. Relation to a single-stage procedure is discussed when the two-stage sampling procedure cannot be completely carried through. An example and tables needed for implementation are given. c 2000 Elsevier Science B.V. All rights

reserved.

Keywords: Analysis of variance; Two stage; Single stage; Power

1. Introduction

It is well known that, for a large sample size (100 observations will suce in many

applications), the null hypothesis of equal means 

1

= · · · = 

k

(k ≥ 2) can almost

surely be rejected if the underlying distribution is continuous. In applications,

practi-tioners often wish to know whether the means of interest fall into some meaningful

preference region under a hypothesis. This idea leads to the interval null hypothesis

H

0

:

Pki=1

(

i

− )

2

=k ≤ 

2

against the alternative H

a

:

Pki=1

(

i

− )

2

=k ¿ 

2

; where

 is the average of 

1

; : : : ; 

k

and  (≥ 0) is a zone of indierence which must be

speciÿed in advance by an expert in his experiment. The null hypothesis H

0

can be

interpreted as saying that there is little deviation among the means and the constant

 can be interpreted as the amount of variation among means about which we are

∗_{Corresponding author.}

0167-9473/00/$ - see front matter c 2000 Elsevier Science B.V. All rights reserved.

indierent. Stating the null hypothesis as a region rather than as a point necessitates

the use of an unbiased test (see, e.g., Lehmann, 1986) which requires the evaluation

of percentage points of some noncentral distributions and, especially, the calculation

of a power function for a speciÿed alternative (see, e.g., Ding, 1999).

In a one-way layout ÿxed-eects model, let there be k treatment populations



1

; : : : ; 

k

such that observations obtained from population 

i

are independent and

normally distributed with unknown mean 

i

and unknown variance 

2i

(i = 1; : : : ; k):

Even if the variances are equal but unknown, the power of the noncentral F test for

testing the hypothesis H

0

:

P

(

i

− )

2

=

2

≤ 

2

depends upon the unknown common

variance, which renders it dicult to plan an experiment reasonably. Moreover, when

the variances are unknown and unequal, there does not exist an exact one-stage

sta-tistical procedure to solve the analysis of variance test problem (see, e.g., Bishop and

Dudewicz, 1978) if one wishes to have both the level and the power of the test

con-trollable at some ÿxed values. In this paper, we will employ the two-stage sampling

procedure (see, e.g., Bishop and Dudewicz, 1978; Stein, 1945; Rasch et al., 1997) and

propose an ˜F (analog of F) test for testing the null hypothesis H

0

:

P

(

i

− )

2

=k ≤ 

2

against the alternative H

a

:

P

(

i

− )

2

=k ¿ 

2

: The distribution of the test ˜F by the

two-stage procedure drops the assumptions of unknown variances, hence the test has

the level and the power independent of the variances. We ÿrst introduce the two-stage

sampling procedure for the one-way layout in Section 2 and then calculate the

criti-cal values and the power of the test in Section 3. Statisticriti-cal tables to implement the

procedure are provided. A numerical example for one-way layout to illustrate the

use of the ˜F test is given in Section 4. When the required sample sizes cannot be

reached in an experiment, the single-stage procedure (see Chen and Chen, 1998) can

provide a useful solution which is discussed in Section 5. The two-stage procedure is

a design-oriented procedure while the single-stage procedure is a data-analysis

pro-cedure with data being already available on hand. Their relation is also discussed.

In Section 6, an extension to two-way ÿxed-eects model is investigated. Finally, in

Section 7, a relation to the single-stage procedure in the two-way layout is outlined.

2. The ˜F test

In a one-way layout, the ÿxed-eects model is given by

X

ij

= 

i

+ e

ij

; i = 1; : : : ; k; j = 1; : : : ; n

i

;

where the e

ij

’s are independent and normally distributed with mean zero and variance



2

i

; denoted by e

ij

∼ N(0; 

i2

): We may denote 

i

by 

i

=  +

i

; where  =

Pki=1



i

=k;

and 

i

is the treatment eect. The variances 

2i

’s are unknown and possibly unequal.

Our goal is to test the null hypothesis that the normal means fall into a zone of

practical indierence of size 

2

_{(≥ 0); i.e., to test the hypothesis}

H

0

:

k X i=1

(

i

− )

2

=k ≤ 

2

vs: H

a

:

k X i=1

(

i

− )

2

=k ≥ 

∗2

(

∗2

¿ 

2

)

(1)

or equivalently to test the hypothesis with 

i

= 

i

− 

H

0

:

k X i=1



2 i

=k ≤ 

2

vs: H

a

:

k X i=1



2 i

=k ≥ 

∗2

(

∗2

¿ 

2

)

in such a way that both the level and the power of the test are controllable and are

not dependent upon the unknown variances. A two-stage sampling procedure (P

1

)

for this problem is given as follows:

P

1

: Choose a number z ¿ 0 (z is determined by the required power of the test to

be discussed later), and take an initial random sample of size n

0

(at least 2, but 10 or

more will give better results) from each of the k populations. For the ith population

let S

2

i

be the usual unbiased estimate of 

2i

based on the initial n

0

observations, and

deÿne

N

i

= max

(

n

0

+ 1;

"

S

2 i

z

#

+ 1

)

;

(2)

where [x] denotes the greatest integer less than or equal to x: Then, take N

i

− n

0

additional random observations (assuming no time trend) from the ith population so

we have a total of N

i

observations denoted by X

i1

; : : : ; X

in0

; : : : ; X

iNi

: For each i; set

the coecients a

i1

; : : : ; a

in0

; : : : ; a

iNi

; so that

a

i1

= · · · = a

in0

=

1 − (N

i

− n

0

)b

i

n

0

= a

i

;

a

i;n0+1

= · · · = a

iNi

=

1

N

i  

_{1 +}

s

n

0

(N

i

z − S

i2

)

(N

i

− n

0

)S

i2  

_{= b}

_i

_;

and then compute the weighted mean

˜X

i:

= a

i n0 X j=1

X

ij

+ b

i Ni X j=n0+1

X

ij

(3)

which is a linear combination of the ÿrst-stage sample data (X

i1

; : : : ; X

in0

) and the

second-stage sample data (X

in0+1

; : : : ; X

iNi

): Note that the coecients a

ij

’s are so

de-termined to satisfy the equations

(a)

XNi j=1

a

ij

= 1; (b) a

i1

= · · · = a

in0

; and (c) S

i2 Ni X j=1

a

2 ij

= z

that the random variables t

i

=( ˜X

i:

−

i

)=

√

z; i =1; : : : ; k; have independent and

identi-cally distributed (i.i.d.) t distributions each with n

0

− 1 d.f. (see, e.g., Dudewicz and

Dalal, 1975; Chen and Chen, 1998). The condition (a) is to ensure the unbiasedness

of ˜X

i:

for 

i

; (b) guarantees that the sample mean X

i

and the sample variance S

i2

based on the ÿrst-stage observations are independent, and (c) is the variance estimate

of ˜X

i:

controlled at a power-speciÿed value z which makes the choices of a

ij

possible

and guarantees that the {t

i

} have independent t distributions.

Finally, compute the test statistic

˜F =

Xk

i=1

( ˜X

i:

− ˜X

::

)

2

z

(4)

where ˜X

::

is the arithmetic mean of ˜X

1:

; : : : ; ˜X

k:

; and we reject H

0

at level  if and

only if

˜F ¿ ˜F



;

(5)

where the level  critical value ˜F



= ˜F



(; z; k; n

0

) and the P

∗

-power-related z value

are determined such that the following simultaneous equations are satisÿed:

P

˜F ¿ ˜F



| H

0

:

k X i=1

(

i

− )

2

=k ≤ 

2 !

≤ 

(6)

and

P

˜F ¿ ˜F



| H

a

:

k X i=1

(

i

− )

2

=k ≥ 

∗2

¿ 

2 !

≥ P

∗

_;

₍₇₎

where  ∈ (0; 1) and P

∗

_{∈ (0; 1) are predetermined values.}

We can rewrite the test statistic ˜F in (4) as

˜F =

Xk i=1

˜X

i:

_√

− 

i

z

−

˜X

::

_√

− 

z

+



i

_√

− 

z

!2

=

Xk i=1 

t

i

− t +

√



i

_z

2

;

(8)

where t =

Pk i=1

t

i

=k:

It should be noted that if ˜X

::

in (4) were taken to be a weighted average of ˜X

i:

’s

of the form

P

N

i

˜X

i:

=

P

N

i

, then the test statistic in (4) would be

˜F =

Xk i=1 

t

i

−

P

N

i

t

i P

N

i

+



i

−

P

_√

N

i



i

=

P

N

i

z

2

which is a function of the unknown but random sample sizes N

i

’s. This contradicts

the two-stage procedure and it fails to determine the sample sizes in (2) with a

prespeciÿed power when the population variances are unknown and unequal.

Fur-thermore, if ˜X

::

were taken to be the weighted average of ˜X

i:

’s, the noncentrality

parameter of ˜F would be a function of the unknown N

i

’s and it would not consist

of the form of the parameters to be tested in (1). Finally, if all N

i

are taken to be

equal and are larger than or equal to n

0

+ 1, either the required power P

∗

cannot be

met or the procedure is not economical for the design. Hence, the ˜X

::

in (4) is taken

to be the arithmetic mean in order to reach a satisfactory solution to the problem

under investigation.

It is clear that the distribution of the test statistic ˜F in (8) is independent of

the unknown variances 

2



i

−  6= 0 for some i it is intuitively clear (for analytical proof, see Bishop and

Dudewicz, 1978) that

lim

z→0

P( ˜F ¿ ˜F



) = 1:

It is easy to see that the limiting distribution of ˜F is a noncentral chi-square with

k −1 degrees of freedom and noncentrality parameter
=

Pk

i=1

(

i

− )

2

=z=

Pki=1



2i

=z:

As discussed in Lehmann (1986), the noncentral 

2

_{has monotone likelihood ratio}

property in
; thus, as n

0

is large, ˜F is an asymptotically UMP test for H

0

vs. H

a

:

For the test procedure ˜F to be of practical usage for small n

0

the critical values ˜F



and its power-related design parameter z must be determined, which will be discussed

in the following section.

3. The critical values and the power of ˜F

The critical values ˜F



and the power of ˜F were obtained by Monte Carlo

simu-lation when n

0

is small (n

0

= 5; 10; 15): In our calculation we consider the

asymp-totically least-favorable conÿguration of the means for the power of ˜F; subject to

P

(

i

− )

2

= c (see, e.g., Bishop and Dudewicz, 1978).

It can be seen from David et al. (1972) that, for ÿxed ; the minimum range

of the 

i

(or 

i

) under H

0

:

Pki=1

(

i

− )

2

=k ≤ 

2

occurs at the asymptotically

least-favorable conÿguration, for even k



0

_{= (−; : : : ; −; ; : : : ; )}

₍₉₎

with half of the ’s being − and half being , and for odd k,



0

₌

 

_−

s

k − 1

k + 1

; : : : ; −

s

k − 1

k + 1

; 

s

k + 1

k − 1

; : : : ; 

s

k + 1

k − 1

 

₍₁₀₎

with (k + 1)=2 of the ’s being −

p

(k − 1)=(k + 1) and the rest being



p

(k + 1)=(k − 1). Similarly, the maximum range of the 

i

under H

a

:

Pki=1

(

i

− )

2

=k ≥ 

∗2

¿ 

2

; for ÿxed 

∗

; occurs at the asymptotically least-favorable

conÿguration



∗

_{= (−}

∗q

_{k=2; 0; : : : ; 0; }

∗q

_k=2):

₍₁₁₎

For each k (k = 2(1)6; 8) and each n

0

(n

0

= 5; 10; 15); k independent t random

variates, t

1

; : : : ; t

k

were generated by the formula t =Y=

p

u=r; where Y is the standard

normal random variate generated from RANNOR (SAS Institute, Inc., 1990) and u

is the independent chi-square random variate with r = n

0

− 1 degrees of freedom

generated from the gamma random number generator RANGAM. The quantity  in

(9) and (10) is replaced by (=

∗

_)(

∗

_{). For selected ; }

∗

_{and z; we formulate the}

ratios =

∗

_{and }

∗

₌

√

_{z in the calculation of (8) according to }

0

in (9) or (10) and

according to 

∗

_{in (11). The reason to use the ratio =}

∗

_{instead of  and the ratio}

 and 

∗

_{. Similarly when }

∗

_{in (11) is substituted for ’s in (8) we have seen the}

ratio 

∗

₌

√

_{z in ˜F: For example, when k = 2; we use}

˜F = [t

1

− t − (=

∗

)(

∗

=

√

z)]

2

+ [t

2

− t + (=

∗

)(

∗

=

√

z)]

2

for calculating the critical values of ˜F under (9), and

˜F = (t

1

− t − 

∗

=

√

z)

2

+ (t

2

− t + 

∗

=

√

z)

2

for calculating the power of ˜F under (11); when k = 3; we use

˜F = [t

1

− t − (1=

√

2)(=

∗

_)(

∗

₌

√

_z)]

2

+[t

2

− t − (1=

√

2)(=

∗

_)(

∗

₌

√

_z)]

2

+[t

3

− t +

√

2(=

∗

_)(

∗

₌

√

_z)]

2

for calculating the critical values of ˜F under (10), and

˜F = (t

1

− t −

√

1:5

∗

₌

√

_z)

2

_{+ (t}

2

− t)

2

+ (t

3

− t +

√

1:5

∗

₌

√

_z)

2

for calculating the power of ˜F under (11). In each simulation run, for a speciÿed

pair of values of =

∗

_{and }

∗

₌

√

_{z; k i.i.d. t random variates were generated and ˜F}

in (8) under (9) or (10) for H

0

was calculated. After 20,000 simulation runs, all

˜F values were ranked in ascending order. Then the 99th, 95th and 90th percentiles

were used to estimate the level 1%, 5% and 10% critical values ˜F

0:01

; ˜F

0:05

and

˜F

0:10

; respectively. Similarly, for given ˜F



and 

∗

=

√

z; ˜F in (8) under (11) for H

a

was calculated. This process was repeated 20,000 times and the power of (7) was

estimated by

P

∗

∼

₌

No: times( ˜F ¿ ˜F



)

20; 000

:

(12)

The estimated critical values and the estimated power are given in Table 3 in the

appendix for  = 0:01; 0:05; 0:10; k = 2(1)6; 8; =

∗

_{= 0; 0:2; 0:4; 0:5; 0:6; }

∗

₌

√

_{z =}

1(:5)10; and n

0

= 5; 10; 15: To reduce the table size without losing practical

useful-ness, we delete the cases of P

∗

_{greater than 0.99 (using  = 0:05 as the guideline).}

The critical values are reported to the ÿrst decimal place, and the power are accurate

to the second decimal place. An example of how to use Table 3 is illustrated as

fol-lows: If one has k = 4 treatments in his experiments, and the initial sample available

is n

0

= 10 observations, at the price of  = 5% risk, he will feel comfortably

indier-ent among these treatmindier-ents if they are within a one-half unit of variation ( = 0:5)

among the means; on the other hand, if these treatments have variation larger than

one unit (

∗

_{= 1:0) among means, he would like to detect such a dierence with}

a required power, say, P

∗

_{= 0:82: From Table 3, he can ÿnd the ratio }

∗

₌

√

_{z = 3:0}

corresponding to the ratio =

∗

_{= 0:5 and the required power P}

∗

_{= 0:82: Then, the}

design constant is found to be z = (

∗

_=3:0)

2

_{or z = 0:1111 which will be employed}

in (2) to determine the required total sample size N

i

in the experiment. Simulation

study shows that linear interpolation in 

∗

₌

√

_{z would give satisfactory results for}

Table 1

Bacterial killing ability example (ÿrst 15 observations) and intermediate statistics

Solvent 1 Solvent 2 Solvent 3 Solvent 4 96.44 93.63 93.58 97.18 96.87 93.99 93.02 97.42 97.24 94.61 93.86 97.65 95.41 91.69 92.90 95.90 95.29 93.00 91.43 96.35 95.61 94.17 92.68 97.13 95.28 92.62 91.57 96.06 94.63 93.41 92.87 96.33 95.58 94.67 92.65 96.71 98.20 95.28 95.31 98.11 98.29 95.13 95.33 98.38 98.30 95.68 95.17 98.35 98.65 97.52 98.59 98.05 98.43 97.52 98.00 98.25 98.41 97.37 98.79 98.12 Intermediate statistics S2 i 2.10995 3.17085 5.88428 0.77969 ai 0.05200 0.03024 0.01803 0.04424 bi 0.05501 0.03902 0.01920 0.33637 Ni 19 29 53 16 ˜ Xi: 97.192 95.381 95.391 97.547 z = 0:1111 ˜F = 35:981

For moderate or large n

0

; the critical values and the power of the ˜F test can be

obtained by using the noncentral chi-square approximation with k − 1 degrees of

freedom and noncentrality parameter
= k

2

_{=z = k[(=}

∗

_)(

∗

₌

√

_z)]

2

_{under H}

0

and

=k(

∗

₌

√

_z)

2

_{under H}

a

: The critical values and the power of the chi-square test can

be computed by using the CINV and PROBCHI functions (SAS Institute, Inc., 1990).

They are given in Table 4 in the appendix for =0:01; 0:05; 0:10; k =2(1)6; 8; =

∗

₌

0; 0:2; 0:4; 0:5; 0:6; and 

∗

₌

√

_{z = 1(:5)7:}

4. A numerical example

The data in Table 1 is from an experiment reported in Bishop and Dudewicz

(1978) for studying the bacterial killing ability of four solvents. The percentage

of fungus destroyed was recorded. Let 

i

denote the mean percentage of fungus

destroyed by solvent i. If the experimenter regards a dierence of  = 0:5 unit of

variation among the means to be irrelevant, and he wishes to detect a dierence of

at least 

∗

_{= 1:0 unit of variation among the means, then he can translate it into the}

null hypothesis

H

0

:

4 X i=1

(

i

− )

2

=4 ≤ (0:5)

2

Table 2

Bacterial killing ability (second-stage observations)

Solvent 1 Solvent 2 Solvent 3 Solvent 4 98.59 96.97 96.36 93.43 98.15 97.97 98.20 97.21 96.69 92.72 96.73 98.37 97.44 96.89 93.56 97.55 98.57 96.86 96.13 94.13 94.44 97.26 97.65 93.57 93.61 98.27 97.81 96.27 93.61 97.57 97.71 98.05 94.20 97.81 97.48 97.67 94.20 98.20 97.96 98.93 93.92 94.30 97.23 93.86 93.29 95.95 92.57 94.21 97.79 93.32 92.90 97.41 92.15 93.02 96.94 93.43 97.08

against the alternative hypothesis

H

a

:

4 X

i=1

(

i

− )

2

=4 ≥ (1:0)

2

:

In the ÿrst stage of experiment n

0

= 15 observations (a random sample of size 15)

were run with each solvent. Wen and Chen (1994) discovered that these data are not

normally distributed. So, we have conducted a robust Levene’s test for homogeneity

of variances (see, e.g., Conover et al., 1981) and found a signiÿcant dierence among

the variances (p value¡ 0:001). Further, Dudewicz and van der Meulen (1983)

have also shown robustness results which applies to the two-stage procedure for

general non-normal distributions. If the experimenter decides the level of the test

to be 5% and a power of at least 0.85, he can use the two-stage test procedure

by taking the initial sample of size n

0

= 15 observations from each population. The

critical value ˜F



= 25:9 and 

∗

=

√

z = 3:0 at =

∗

= 0:5 are found using Table 3, so

z = (1:0=3:0)

2

_{= 0:1111: The initial sample variances based on the ÿrst-stage samples,}

the coecients for calculating the weighted sample means, and the ÿnal weighted

sample means, S

2

i

; a

i

; b

i

; N

i

and ˜X

i:

deÿned in (2) and (3) are given at the bottom of

Table 1. The remaining N

i

− 15 observations taken at the second stage are given in

Table 2. Using formula (4) we found the test statistic ˜F = 35:981; which exceeds

the critical value of 25.9, so H

0

is rejected.

5. Relation to the single-stage procedure

The two-stage procedure discussed in Section 2 is a design-oriented method which

determines the necessary sample sizes N

i

in order to meet a prespeciÿed power

requirement. In situations where the two-stage experiment is terminated earlier due

to budget shortage or some other uncontrollable factors, the required total sample

size N

i

in (2) cannot be reached, one may have to use the available n

i

(n

i

≥ (n

0

+1))

observations on hand and recalculate the coecients a

ij

’s according to the so-called

single-stage sampling procedure (see Chen and Chen, 1998) such that the statistical

inference theory can still work. The general single-stage procedure (P

2

) is described

below.

P

2

: Given a random sample of size n

i

from normal population (or treatment) 

i

with unknown mean 

i

and unknown variance 

2i

(1 ≤ i ≤ k): Employ the ÿrst (or

randomly chosen) n

0

(2 ≤ n

0

¡ n

i

) observations and calculate the usual unbiased

sample mean and unbiased sample variance, respectively, by

X

i

=

n0 X j=1

X

ij

=n

0

and

S

2 i

=

n0 X j=1

(X

ij

− X

i

)

2

=(n

0

− 1):

Then, calculate the coecients

U

i

=

_n

1

i

+

1

n

i s

n

i

− n

0

n

0

(n

i

z

∗

_=S

2 i

− 1) ;

V

i

=

_n

1

i

−

1

n

i r

_n

0

n

i

− n

0

(n

i

z

∗

_=S

2 i

− 1) ;

where z

∗

_{is the maximum of {S}

2

j

=n

j

; j = 1; : : : ; k}: Let the ÿnal weighted sample

mean be deÿned by

˜X

i:

=

ni X j=1

W

ij

X

ij

;

(13)

where

W

ij

=



U

i

for 1 ≤ j ≤ n

0

;

V

i

for (n

0

+ 1) ≤ j ≤ n

i

;

and W

ij

satisfy the following conditions:

ni X j=1

W

ij

= 1; W

i1

= · · · = W

in0

; S

i2 ni X j=1

W

2 ij

= z

∗

:

It is well known (see Chen and Chen, 1998) that given the sample variances

S

2

i

; i = 1; : : : ; k; the weighted sample mean ˜X

i

has a conditional normal distribution

with mean 

i

and variance

Pj

W

ij2



2i

: Furthermore, the transformations

t

i

=

q

˜X

i:

− 

i

S

2 i P_ni j=1

W

ij2

=

˜X

i:

√

− 

i

z

∗

; i = 1; : : : ; k

have i.i.d. t distributions each with n

0

− 1 degrees of freedom. Note that in the

single-stage procedure, the data-dependent z

∗

_{is used to replace the design constant}

z for the two-stage procedure. Thus, the power of the single-stage procedure is not

controllable.

The statistic

˜F

1

=

Xk i=1

( ˜X

i:

− ˜X

::

)

2

z

∗

;

(14)

where ˜X

::

is the arithmetic mean of the ˜X

i:

’s, is used as a test statistic for testing

the hypothesis H

0

vs. H

a

in (1). Further, ˜F

1

can be written as

˜F

1

=

Xk i=1

˜X

_√

i:

− 

i

z

∗

−

˜X

_√

::

− 

z

∗

+



_√

i

− 

z

∗ !2

=

Xk i=1 

t

i

− t +



√

i

− 

_z

_∗ 2

:

Note that if ˜X

::

in (14) were taken to be the weighted average of ˜X

i

’s, it would lead

to testing dierent hypotheses (could be meaningless ones) rather than (1).

The critical values of ˜F

1

for testing H

0

and its power against H

a

can be obtained

at the asymptotically least-favorable conÿgurations given in (9)–(11) by using the

tables in which 

∗

₌

√

_{z is replaced by }

∗

₌

√

_z

∗

_{: For example, if k = 4; n}

₀

_{= 10;  =}

0:05;  = 1; 

∗

_{= 2 and z}

∗

_{= max(S}

2

j

=n

j

) = 0:64. Thus, =

∗

= 0:5 and 

∗

=

√

z

∗

=

2=

√

0:64 = 2:5: From Table 3, we ÿnd the critical value ˜F

0:05

= 23:0 and the power

P

∗

_{= 0:67:}

The actual power of the test using the single-stage procedure is data-dependent.

Its power could be larger than, equal to, or smaller than the required one using the

two-stage procedure whose sample sizes are determined by the prespeciÿed power.

This point is elaborated as follows: If the sample size n

i

¿ n

0

+ 1; i = 1; : : : ; k, were

given by the single-stage procedure and the following cases.

Case 1. If S

2

i

=n

i

= S

j2

=n

j

for all i 6= j, then the two- and single-stage procedures

have the same power because S

2

i

=n

i

= z, except for a rounding error in sample size

by deÿnition (2) and S

2

i

=n

i

= max{S

j2

=n

j

; j = 1; : : : ; k} = z

∗

deÿned by single-stage

procedure. Thus, z = z

∗

_{gives the same power.}

Case 2. If z

∗

_{= max}

1≤j≤k

(S

j2

=n

j

) ¡ z, then the single-stage procedure has a power

larger than that of the two-stage one. A smaller z

∗

_{-value means a larger sample size}

n

i

than the required one by two-stage procedure and hence, it carries a larger power.

Case 3. If min

1≤j≤k

(S

j2

=n

j

) ¿ z, then the power of the single-stage test is smaller

than that of the two-stage test.

Case 4. All other situations, the single-stage procedure could have power larger

than, equal to, or smaller than that of the two-stage test depending on the actual

sample data and the true population variances.

6. The two-way layout

The two-way ÿxed-eects model in the analysis of variance is usually deÿned by

X

ijk

=  + 

i

+ ÿ

j

+ (ÿ)

ij

+ e

ijk

(i = 1; : : : ; I; j = 1; : : : ; J; k = 1; : : : ; n

ij

)

where the random errors e

ijk

’s are independently and normally distributed with mean

zero and unknown (and possibly unequal) variances 

2

ij

, and by convention,

I X i=1



i

=

J X j=1

ÿ

j

= 0;

I X i=1

(ÿ)

ij

= 0 for every j

(15)

and

J X j=1

(ÿ)

ij

= 0 for every i:

The null and alternative hypotheses under consideration are

H

1 0

:

I X i=1



2 i

=I ≤ 

21

vs: H

1a

:

I X i=1



2 i

=I ≥ 

∗21

¿ 

21

;

H

2 0

:

J X j=1

ÿ

2 j

=J ≤ 

22

vs: H

2a

:

J X j=1

ÿ

2 j

=J ≥ 

∗22

¿ 

22

;

(16)

and

H

3 0

:

I X i=1 J X j=1

(ÿ)

2 ij

=IJ ≤ 

23

vs: H

3a

:

I X i=1 J X j=1

(ÿ)

2 ij

=IJ ≥ 

∗23

¿ 

23

:

The purpose is to seek tests of these hypotheses based on statistics whose

distri-butions are independent of the unknown variances and the unknown means. In the

two-way layout, there are I ∗ J possible treatment combinations. We refer cell (i; j)

to the treatment combination of level i of the ÿrst factor and level j of the second

factor. In each cell (i; j), the two-stage sampling procedure (P

3

) is given below.

P

3

: Choose a number z ¿ 0 (to be determined by the power), and in each cell (i; j)

take an initial sample of size n

0

; X

ij1

; : : : ; X

ijn0

: Compute the usual unbiased variance

estimate S

2

ij

of 

ij2

based on the ÿrst n

0

random observations, and deÿne

N

ij

= max

(

n

0

+ 1;

"

S

2 ij

z

#

+ 1

)

:

(17)

Then, take N

ij

− n

0

additional random observations from cell (i; j) so we have a

total of N

ij

observations denoted by X

ij1

; : : : ; X

ijn0

; : : : ; X

ijNij

: For each cell (i; j), set

the coecients a

ij1

; : : : ; a

ijn0

; : : : ; a

ijNij

, so that

a

ij1

= · · · = a

ijn0

=

1 − (N

ij

− n

0

)b

ij

n

0

= a

ij

;

a

ij;n0+1

= · · · = a

ijNij

= b

ij

;

where

b

ij

=

_N

1

ij "

1 +

s

n

0

(N

ij

z − S

ij2

)

(N

ij

− n

0

)S

ij2 #

;

and then compute the weighted sample mean

˜X

ij:

=

Nij X k=1

a

ij

X

ijk

= a

ij n0 X k=1

X

ijk

+ b

ij Nij X k=n0+1

X

ijk

:

As in Section 2, it can be shown that the random variables

t

ij

=

˜X

ij:

− ( + 

√

i

+ ÿ

_z

j

+ (ÿ)

ij

)

(18)

have independent t distribution with n

0

− 1 degrees of freedom, denoted by t

n0−1

.

Finally, compute

˜X

i::

=

1

_J

J X j=1

˜X

ij:

;

˜X

:j:

=

1

_I

I X i=1

˜X

ij:

;

˜X

:::

=

_IJ

1

I X i=1 J X j=1

˜X

ij:

(19)

These means in (19) are taken to be the unweighted ones carrying the same

argument raised immediately after expression (8).

Similar to the usual two-way layout argued by Bishop and Dudewicz (1978, p.

422), our test statistic for H

1

0

vs. H

a1

is

˜F

1

= J

I X i=1

( ˜X

i::

− ˜X

:::

)

2

z

:

(20)

At level  the hypothesis H

1 0

:

P_I

i=1



2i

=I ≤ 

21

is rejected if and only if

˜F

1

¿ ˜F

1

;

where the level  critical value ˜F

₁

= ˜F

₁

(

1

; z; I; J; n

0

) and the P

∗

-power-related z value

are determined such that

P

˜F

1

¿ ˜F

1

| H

10

:

I X i=1



2 i

=I ≤ 

21 !

≤ 

and

P

˜F

1

¿ ˜F

1

| H

1a

:

I X i=1



2 i

=I ≥ 

∗21

¿ 

21 !

≥ P

∗

_:

₍₂₁₎

We can rewrite the test statistic ˜F

1

in (20) by applying conditions (15) and the

deÿnition (19) as

˜F

1

= J

I X i=1

(t

i:

− t

::

+ 

i

=

√

z)

2

;

(22)

where t

i:

=

PJj=1

t

ij

=J; t

::

=

PIi=1 P_J j=1

t

ij

=IJ.

Similarly, the hypothesis H

2

0

:

P_J

j=1

ÿ

2j

=J ≤ 

22

is tested using the statistic

˜F

2

= I

J X j=1

( ˜X

:j:

− ˜X

:::

)

2

z

:

(23)

The null hypothesis H

2

0

is rejected at level  if and only if

˜F

2

¿ ˜F

2

;

where the level  critical value ˜F

₂

= ˜F

₂

(

2

; z; I; J; n

0

) and the P

∗

-power-related z value

are determined by the simultaneous equations

P

 

_˜F

₂

_{¿ ˜F}

₂

_{| H}

2 0

:

J X j=1

ÿ

2 j

=J ≤ 

22  

_≤

and

P

 

_˜F

₂

_{¿ ˜F}

₂

_{| H}

2 a

:

J X j=1

ÿ

2 j

=J ≥ 

∗22

¿ 

22  

_{≥ P}

∗

_:

₍₂₄₎

The test statistic ˜F

2

in (23) using conditions (15) can be rewritten as

˜F

2

= I

J X

j=1

(t

:j

− t

::

+ ÿ

j

=

√

z)

2

;

(25)

where t

:j

=

PIi=1

t

ij

=I:

Finally, H

3 0

is tested using

˜F

3

=

I X i=1 J X j=1

( ˜X

ij:

− ˜X

i::

− ˜X

:j:

+ ˜X

:::

)

2

z

:

(26)

The hypothesis H

3 0

:

P_I i=1 P_J

j=1

(ÿ)

2ij

=IJ ≤ 

23

is rejected if and only if

˜F

3

¿ ˜F

3

where ˜F

₃

and z value are determined by the simultaneous equations

P

 

_˜F

₃

_{¿ ˜F}

₃

_{| H}

3 0

:

I X i=1 J X j=1

(ÿ)

2 ij

=IJ ≤ 

23  

_≤

and

P

 

_˜F

₃

_{¿ ˜F}

₃

_{| H}

3 a

:

I X i=1 J X j=1

(ÿ)

2 ij

=IJ ≥ 

∗23

¿ 

23  

_{≥ P}

∗

_:

₍₂₇₎

As in the previous case the statistic ˜F

3

in (26) can be rewritten as

˜F

3

=

I X i=1 J X j=1

(t

ij

− t

i:

− t

:j

+ t

::

+ (ÿ)

ij

=

√

z)

2

:

(28)

The statistic ˜F

1

in (22), ˜F

2

in (25) and ˜F

3

in (28), respectively, are used to simulate

the critical values ˜F

₁

; ˜F

₂

; ˜F

₃

and their powers at their asymptotically least-favorable

conÿgurations of 

i

’s, ÿ’s and (ÿ)

ij

’s similar to (9)–(11).

It is easy to see that the limiting distributions of ˜F

1

; ˜F

2

and ˜F

3

are

noncen-tral chi-square with degrees of freedom I − 1; J − 1 and (I − 1)(J − 1), and with

noncentrality parameters

1

=

PIi=1



2i

=z;

2

=

PJj=1

ÿ

2j

=z and

3

=

PIi=1

P_J

respectively. The tables of the critical values and the power can be produced by

us-ing the noncentral chi-square distribution for moderate or large n

0

: The r-way model

of analysis of variance and its associated hypotheses can be similarly extended by

the analogue of the two-way model.

7. The single-stage procedure for two-way ANOVA

When the required sample sizes N

ij

(17) in the two-way layout cannot be reached

by the two-stage sampling procedure, one may employ the feasible single-stage

pro-cedure for a reasonable solution. The single-stage sampling propro-cedure (P

4

) for testing

the hypotheses of (16) proceeds as follows.

P

4

: Initially, we employ the ÿrst (or randomly chosen) n

0

observations within each

cell and compute the usual sample mean and unbiased sample variance, respectively,

X

ij

=

n0 X k=1

X

ijk

=n

0

and

S

2 ij

=

n0 X k=1

(X

ijk

− X

ij

)

2

=(n

0

− 1):

Then the weights of the observations in cell (i; j) are

U

ij

=

_n

1

ij

+

1

n

ij s

n

ij

− n

0

n

0

(n

ij

z

∗

_=S

2 ij

− 1);

V

ij

=

_n

1

ij

−

1

n

ij s

n

0

n

ij

− n

0

(n

ij

z

∗

_=S

2 ij

− 1);

(29)

where z

∗

_{is the maximum value of {S}

2

ij

=n

ij

; i = 1; : : : ; I; j = 1; : : : ; J}. Let the ÿnal

weighted sample mean for cell (i; j) be deÿned by

˜X

ij:

=

nij X k=1

W

ijk

X

ijk

;

(30)

where

W

ijk

=



U

ij

for 1 ≤ k ≤ n

0

;

V

ij

for (n

0

+ 1) ≤ k ≤ n

ij

:

Therefore, we compute

˜X

i::

=

1

_J

J X j=1

˜X

ij:

;

˜X

:j:

=

1

_I

I X i=1

˜X

ij:

;

˜X

:::

=

_IJ

1

I X i=1 J X j=1

˜X

ij:

The test statistic we consider to use for H

1

_is

˜F

1 1

=

I X i=1 J X j=1

˜X

i::

_√

− ˜X

:::

z

∗ !2

= J

XI i=1 

t

i:

− t

::

+

√



_z

i_∗ 2

;

(31)

for H

2

_:

˜F

1 2

=

I X i=1 J X j=1

˜X

:j:

_√

− ˜X

:::

z

∗ !₂

= I

XJ j=1 

t

:j

− t

::

+

√

ÿ

_z

j_∗ 2

;

(32)

and for H

3

_:

˜F

1 3

=

I X i=1 J X j=1

˜X

ij:

− ˜X

i::

_√

− ˜X

:j:

+ ˜X

:::

z

∗ !₂

=

XI i=1 J X j=1 

t

ij

− t

i:

− t

:j

+ t

::

+

(ÿ)

√

_z

_∗ij 2

;

(33)

where

t

i:

=

_J

1

J X j=1

t

ij

; t

:j

=

1

_I

I X i=1

t

ij

; t

::

=

_IJ

1

I X i=1 J X j=1

t

ij

:

It can be shown (see Chen and Chen, 1998) that

t

ij

=

˜X

ij:

− ( + 

√

i

+ ÿ

j

+ (ÿ)

ij

)

z

∗

;

(34)

for i=1; : : : ; I; j =1; : : : ; J; are distributed as independent Student’s t each with n

0

−1

degrees of freedom. This result is due to the fact that given the sample variances

S

2

ij

’s, the weighted sample mean ˜X

ij:

has a conditional normal distribution with mean

 + 

i

+ ÿ

j

+ (ÿ)

ij

and variance

Pk

W

ijk2



2ij

, as described in Section 5.

Similar to the case of one-way layout, the critical values and the power for the

single-stage procedure can be obtained by using the tables prepared for the two-stage

procedure.The relationship between the single- and two-stage procedure is similar to

the argument in Section 5.

In the situation where all n

ij

’s are equal to n; i.e., a balanced design, and the



1

; 

2

and 

3

are equal to zero, the critical values of ˜F

11

; ˜F

1 2

; and ˜F

1

3

for small n

0

and

selected numbers of I and J were calculated by Chen and Chen (1998).

Acknowledgements

The authors wish to express their sincere appreciation to the editor and the referees

for their constructive and valuable suggestions and comments which improved the

original manuscript.This research was partially supported by Tamkang University

Research Fund and by the National Science Council, Taiwan.

Appendix

The critical values and the power of the ˜F test and chi-test are given in Tables 3

and 4, respectively.

Table 3

Critical values and power of the ˜F test

 ∗  ∗ √ z ( ˜F0:01; P∗) ( ˜F0:05; P∗) ( ˜F0:10; P∗) ∗  ∗ √ z ( ˜F0:01; P∗) ( ˜F0:05; P∗) ( ˜F0:10; P∗) k = 2; n0= 5 0.6 3.0 37.3 0.08 22.2 0.34 17.3 0.53 3.5 43.6 0.10 27.3 0.40 21.4 0.61 0.0 1.0 19.0 0.02 7.7 0.14 4.8 0.26 4.0 49.6 0.13 31.1 0.53 25.1 0.71 1.5 19.0 0.05 7.7 0.29 4.8 0.48 4.5 55.8 0.18 36.0 0.62 29.4 0.78 2.0 19.0 0.11 7.7 0.52 4.8 0.70 5.0 63.0 0.24 41.8 0.70 34.4 0.84 2.5 19.0 0.25 7.7 0.74 4.8 0.86 5.5 67.7 0.35 47.2 0.77 39.7 0.88 3.0 19.0 0.46 7.7 0.88 4.8 0.94 6.0 77.8 0.38 53.2 0.83 45.1 0.92 3.5 19.0 0.70 7.7 0.95 4.8 0.97 6.5 84.4 0.50 59.2 0.89 50.9 0.94 4.0 19.0 0.85 7.7 0.98 4.8 0.99 7.0 91.7 0.61 66.1 0.92 57.1 0.96 4.5 19.0 0.93 7.7 0.99 4.8 0.99 7.5 97.5 0.73 72.5 0.94 63.4 0.97 0.2 1.0 19.2 0.02 7.9 0.14 5.0 0.26 8.0 109.1 0.76 80.0 0.96 70.2 0.98 1.5 19.5 0.04 8.4 0.26 5.3 0.44 8.5 115.4 0.85 87.7 0.97 77.8 0.99 2.0 20.1 0.09 9.0 0.44 5.7 0.65 9.0 126.1 0.88 96.6 0.98 85.6 0.99 2.5 20.4 0.22 9.4 0.65 6.2 0.81 9.5 140.5 0.90 106.0 0.99 94.8 0.99 3.0 21.3 0.34 9.9 0.82 6.7 0.91 3.5 22.3 0.58 10.5 0.91 7.2 0.96 k = 2; n0= 10 4.0 23.5 0.75 11.5 0.95 7.9 0.98 4.5 26.1 0.85 12.5 0.98 8.7 0.99 0.0 1.0 9.7 0.07 5.1 0.22 3.4 0.35 5.0 26.6 0.93 13.4 0.99 9.3 0.99 1.5 9.7 0.18 5.1 0.46 3.4 0.60 0.4 1.0 20.4 0.02 8.9 0.11 5.7 0.22 2.0 9.7 0.39 5.1 0.70 3.4 0.82 1.5 21.3 0.04 10.0 0.20 6.7 0.35 2.5 9.7 0.65 5.1 0.88 3.4 0.93 2.0 24.7 0.06 11.6 0.31 8.0 0.50 3.0 9.7 0.85 5.1 0.96 3.4 0.98 2.5 25.5 0.11 13.4 0.46 9.4 0.65 3.5 9.7 0.95 5.1 0.99 3.4 0.99 3.0 29.4 0.17 15.7 0.60 11.3 0.77 0.2 1.0 9.9 0.06 5.5 0.20 3.7 0.32 3.5 33.0 0.25 17.9 0.73 13.1 0.86 1.5 10.6 0.15 5.7 0.40 4.0 0.54 4.0 34.6 0.42 19.8 0.84 15.1 0.92 2.0 11.5 0.31 6.3 0.63 4.2 0.76 4.5 38.1 0.57 22.3 0.90 17.1 0.95 2.5 12.2 0.51 6.8 0.80 4.8 0.89 5.0 38.4 0.77 24.6 0.95 19.5 0.97 3.0 12.8 0.74 7.5 0.91 5.3 0.96 5.5 46.9 0.79 28.9 0.96 22.8 0.98 3.5 14.4 0.86 8.3 0.97 5.9 0.99 6.0 49.0 0.88 31.2 0.98 25.0 0.99 4.0 14.8 0.95 8.9 0.99 6.6 0.99 6.5 52.7 0.93 34.6 0.99 28.0 0.99 0.4 1.0 11.6 0.04 6.3 0.16 4.3 0.27 0.5 1.0 21.5 0.02 9.3 0.10 6.2 0.19 1.5 12.7 0.10 7.4 0.29 5.3 0.44 1.5 23.9 0.03 11.1 0.17 7.4 0.31 2.0 15.2 0.17 8.9 0.44 6.5 0.60 2.0 26.3 0.05 13.5 0.24 9.4 0.42 2.5 17.6 0.27 10.9 0.59 8.1 0.74 2.5 29.1 0.07 15.6 0.36 11.5 0.55 3.0 19.8 0.42 12.4 0.75 9.7 0.85 3.0 32.2 0.12 18.7 0.47 14.1 0.66 3.5 22.0 0.59 14.5 0.85 11.4 0.92 3.5 37.1 0.18 22.1 0.59 16.8 0.76 4.0 24.7 0.74 16.9 0.92 13.6 0.96 4.0 41.7 0.25 25.1 0.71 19.9 0.84 4.5 28.9 0.82 19.5 0.96 15.7 0.98 4.5 46.3 0.36 29.2 0.78 23.3 0.89 5.0 30.6 0.92 21.7 0.98 18.1 0.99 5.0 52.0 0.45 33.0 0.86 26.4 0.93 0.5 1.0 12.0 0.04 6.7 0.14 4.6 0.25 5.5 57.7 0.56 37.4 0.91 30.5 0.95 1.5 14.9 0.06 8.6 0.23 6.1 0.37 6.0 63.3 0.68 41.7 0.94 34.4 0.97 2.0 17.8 0.11 10.7 0.35 8.0 0.50 6.5 68.0 0.78 46.7 0.96 38.7 0.98 2.5 19.9 0.20 13.1 0.47 10.1 0.63 7.0 73.3 0.86 50.9 0.98 42.8 0.99 3.0 23.8 0.27 15.8 0.60 12.6 0.74 7.5 78.6 0.91 56.6 0.98 48.1 0.99 3.5 26.9 0.41 18.5 0.72 15.1 0.83 8.0 84.4 0.94 61.8 0.99 53.0 0.99 4.0 31.3 0.53 22.0 0.81 18.2 0.89 0.6 1.0 23.6 0.01 10.3 0.09 6.8 0.17 4.5 34.6 0.67 25.4 0.88 21.1 0.94 1.5 23.7 0.02 12.3 0.13 8.6 0.25 5.0 39.5 0.77 28.9 0.94 24.4 0.97 2.0 31.1 0.03 15.8 0.18 11.4 0.33 5.5 43.8 0.85 32.9 0.97 28.0 0.98 2.5 32.9 0.05 18.2 0.27 13.9 0.44 6.0 48.8 0.91 37.1 0.98 31.9 0.99

Table 3 (Continued.)  ∗  ∗ √_z ( ˜F_0:01; P∗) ( ˜F_0:05; P∗) ( ˜F_0:10; P∗) __∗ √∗_z ( ˜F_0:01; P∗) ( ˜F_0:05; P∗) ( ˜F_0:10; P∗) 0.6 1.0 13.1 0.03 7.5 0.12 5.4 0.21 0.6 1.0 12.2 0.03 7.0 0.12 5.0 0.22 1.5 16.1 0.04 9.7 0.18 7.2 0.29 1.5 14.7 0.06 9.1 0.20 6.9 0.32 2.0 19.7 0.07 12.5 0.26 9.8 0.39 2.0 18.0 0.10 11.9 0.27 9.3 0.41 2.5 24.3 0.10 15.7 0.35 12.4 0.51 2.5 22.2 0.14 15.1 0.38 12.2 0.52 3.0 27.4 0.18 19.2 0.45 15.6 0.60 3.0 26.0 0.21 18.7 0.47 15.3 0.62 3.5 32.0 0.26 22.8 0.57 19.0 0.71 3.5 31.5 0.26 22.8 0.57 19.1 0.71 4.0 36.8 0.35 27.3 0.66 23.1 0.79 4.0 35.9 0.38 26.9 0.68 23.0 0.80 4.5 42.6 0.44 32.0 0.74 27.5 0.85 4.5 40.4 0.51 31.0 0.78 26.7 0.87 5.0 48.8 0.53 37.1 0.81 31.9 0.90 5.0 46.7 0.59 36.1 0.84 31.4 0.92 5.5 54.5 0.65 42.5 0.88 36.9 0.94 5.5 53.5 0.67 41.5 0.90 36.5 0.95 6.0 61.7 0.72 48.4 0.91 42.5 0.96 6.0 57.8 0.80 46.8 0.94 41.3 0.97 6.5 67.9 0.81 54.0 0.95 47.8 0.98 6.5 65.3 0.85 53.3 0.96 47.2 0.98 7.0 75.0 0.87 60.5 0.97 54.0 0.99 7.0 72.7 0.90 59.6 0.97 53.5 0.99 7.5 84.4 0.90 67.9 0.98 60.7 0.99 7.5 81.1 0.94 66.7 0.99 60.0 0.99 8.0 89.7 0.95 73.9 0.99 67.1 1.00 k = 3; n0= 5 k = 2; n0= 15 0.0 1.0 30.5 0.02 13.4 0.12 8.9 0.25 0.0 1.0 8.3 0.08 4.5 0.25 3.1 0.37 1.5 30.5 0.03 13.4 0.26 8.9 0.47 1.5 8.3 0.23 4.5 0.50 3.1 0.64 2.0 30.5 0.08 13.4 0.51 8.9 0.74 2.0 8.3 0.47 4.5 0.74 3.1 0.84 2.5 30.5 0.20 13.4 0.77 8.9 0.90 2.5 8.3 0.73 4.5 0.91 3.1 0.95 3.0 30.5 0.43 13.4 0.91 8.9 0.96 3.0 8.3 0.90 4.5 0.97 3.1 0.99 3.5 30.5 0.71 13.4 0.97 8.9 0.99 0.2 1.0 9.1 0.07 5.0 0.22 3.4 0.34 0.2 1.0 30.7 0.02 13.5 0.11 9.0 0.23 1.5 9.3 0.19 5.1 0.44 3.6 0.59 1.5 31.0 0.04 13.8 0.24 9.3 0.45 2.0 9.7 0.40 5.6 0.67 3.9 0.80 2.0 31.7 0.08 14.3 0.47 9.8 0.69 2.5 11.2 0.57 6.2 0.84 4.4 0.91 2.5 32.3 0.18 15.2 0.70 10.4 0.86 3.0 11.6 0.79 6.8 0.94 5.0 0.97 3.0 33.6 0.38 16.1 0.87 11.1 0.95 3.5 12.5 0.91 7.6 0.98 5.6 0.99 3.5 35.3 0.59 16.6 0.95 11.7 0.98 0.4 1.0 9.9 0.05 5.7 0.18 4.0 0.29 4.0 36.1 0.82 17.8 0.98 12.8 0.99 1.5 11.9 0.11 7.0 0.30 5.0 0.45 0.4 1.0 31.1 0.02 14.4 0.10 9.9 0.21 2.0 13.7 0.21 8.5 0.46 6.2 0.62 1.5 32.0 0.03 15.4 0.20 10.9 0.36 2.5 16.0 0.33 10.3 0.62 7.8 0.76 2.0 35.7 0.05 17.5 0.34 12.5 0.56 3.0 17.8 0.51 12.0 0.78 9.3 0.88 2.5 39.9 0.08 19.9 0.51 14.2 0.74 3.5 20.6 0.65 14.0 0.87 11.2 0.93 3.0 41.2 0.19 22.3 0.70 17.0 0.85 4.0 23.8 0.77 16.0 0.94 13.1 0.97 3.5 42.1 0.40 25.2 0.83 19.4 0.93 4.5 26.0 0.89 18.4 0.97 15.1 0.99 4.0 50.2 0.48 29.2 0.91 22.3 0.96 5.0 29.6 0.94 21.2 0.99 17.6 0.99 4.5 52.8 0.70 31.9 0.96 24.9 0.98 0.5 1.0 11.0 0.04 6.3 0.15 4.4 0.26 5.0 57.9 0.83 35.6 0.98 28.8 0.99 1.5 13.3 0.08 8.1 0.25 6.0 0.38 5.5 63.3 0.90 39.4 0.99 32.1 0.99 2.0 15.8 0.14 10.2 0.36 7.8 0.51 0.5 1.0 32.1 0.02 15.2 0.09 10.4 0.19 2.5 19.1 0.21 12.5 0.50 9.8 0.66 1.5 35.0 0.02 17.1 0.16 12.2 0.31 3.0 21.8 0.35 15.1 0.63 12.2 0.77 2.0 37.3 0.04 19.8 0.26 14.5 0.46 3.5 25.2 0.47 18.0 0.75 14.8 0.85 2.5 42.5 0.07 23.1 0.40 17.6 0.60 4.0 29.1 0.60 21.3 0.84 17.6 0.91 3.0 48.1 0.11 27.1 0.55 21.0 0.74 4.5 33.6 0.70 24.7 0.91 20.7 0.95 3.5 51.4 0.21 31.0 0.70 24.5 0.85 5.0 37.8 0.81 28.3 0.95 24.0 0.98 4.0 57.3 0.33 35.5 0.82 28.5 0.91 5.5 41.9 0.89 32.1 0.98 27.5 0.99 4.5 65.2 0.44 40.2 0.90 33.4 0.95 6.0 47.1 0.94 36.0 0.99 31.3 1.00 5.0 70.0 0.63 46.3 0.93 38.3 0.97

Table 3 (Continued.)  ∗  ∗ √_z ( ˜F_0:01; P∗) ( ˜F_0:05; P∗) ( ˜F_0:10; P∗) __∗ √∗_z ( ˜F_0:01; P∗) ( ˜F_0:05; P∗) ( ˜F_0:10; P∗) 0.5 5.5 74.4 0.79 51.4 0.96 43.3 0.98 0.6 2.0 26.8 0.08 17.6 0.30 14.0 0.46 6.0 84.1 0.86 57.8 0.98 48.8 0.99 2.5 30.2 0.17 21.5 0.44 17.6 0.60 6.5 94.9 0.90 64.7 0.99 55.5 0.99 3.0 37.0 0.24 26.5 0.56 22.2 0.71 0.6 1.0 33.5 0.01 15.9 0.08 11.2 0.16 3.5 42.7 0.36 31.6 0.69 26.9 0.82 1.5 37.6 0.02 18.6 0.13 13.4 0.26 4.0 49.9 0.48 37.7 0.79 32.5 0.89 2.0 43.9 0.03 22.7 0.19 16.9 0.36 4.5 56.3 0.63 44.0 0.87 38.4 0.94 2.5 46.8 0.05 27.0 0.28 20.7 0.49 5.0 66.0 0.71 51.4 0.92 45.3 0.96 3.0 55.1 0.06 32.4 0.38 25.4 0.60 5.5 73.0 0.84 58.4 0.96 52.1 0.98 3.5 62.2 0.09 37.9 0.51 30.9 0.70 6.0 83.6 0.88 67.0 0.98 60.2 0.99 4.0 66.6 0.18 43.5 0.64 35.7 0.81 6.5 93.0 0.93 75.9 0.99 68.7 0.99 4.5 75.9 0.25 50.5 0.75 42.0 0.87 5.0 83.7 0.37 58.6 0.82 49.2 0.92 k = 3; n0= 15 5.5 93.3 0.49 66.4 0.88 56.5 0.95 6.0 102.5 0.62 74.0 0.93 63.6 0.97 0.0 1.0 12.0 0.08 7.3 0.26 5.4 0.38 6.5 111.2 0.75 83.1 0.95 72.5 0.98 1.5 12.0 0.26 7.3 0.54 5.4 0.68 7.0 124.4 0.80 93.6 0.97 82.3 0.98 2.0 12.0 0.56 7.3 0.82 5.4 0.89 7.5 135.1 0.88 102.4 0.98 91.0 0.99 2.5 12.0 0.83 7.3 0.95 5.4 0.98 8.0 147.6 0.92 113.6 0.99 101.4 0.99 0.2 1.0 12.7 0.07 7.6 0.24 5.8 0.36 1.5 12.9 0.21 8.0 0.49 6.0 0.64 k = 3; n0= 10 2.0 13.9 0.47 8.5 0.76 6.4 0.85 2.5 14.5 0.74 9.2 0.91 7.1 0.96 0.0 1.0 14.3 0.06 8.2 0.22 6.0 0.35 3.0 15.6 0.90 10.1 0.98 7.7 0.99 1.5 14.3 0.18 8.2 0.48 6.0 0.64 0.4 1.0 13.8 0.05 8.5 0.20 6.5 0.31 2.0 14.3 0.44 8.2 0.77 6.0 0.87 1.5 15.8 0.13 10.1 0.36 7.7 0.52 2.5 14.3 0.74 8.2 0.93 6.0 0.97 2.0 18.1 0.27 12.0 0.56 9.4 0.71 3.0 14.3 0.92 8.2 0.99 6.0 0.99 2.5 20.7 0.47 14.0 0.76 11.2 0.86 0.2 1.0 14.8 0.05 8.5 0.21 6.3 0.33 3.0 23.7 0.66 16.8 0.87 13.6 0.94 1.5 15.4 0.15 8.9 0.44 6.6 0.60 3.5 26.3 0.83 19.4 0.95 15.8 0.98 2.0 15.8 0.37 9.5 0.71 7.0 0.83 4.0 31.4 0.90 22.1 0.98 18.5 0.99 2.5 16.3 0.66 10.1 0.89 7.7 0.94 0.5 1.0 14.4 0.05 9.3 0.16 7.0 0.27 3.0 17.6 0.85 10.9 0.97 8.3 0.99 1.5 17.9 0.09 11.6 0.28 9.0 0.43 0.4 1.0 15.8 0.04 9.5 0.17 7.1 0.28 2.0 20.3 0.20 14.2 0.45 11.3 0.60 1.5 17.9 0.10 11.0 0.32 8.3 0.48 2.5 24.7 0.32 17.3 0.62 14.0 0.75 2.0 19.9 0.22 12.8 0.52 9.8 0.69 3.0 28.5 0.49 20.7 0.77 17.2 0.87 2.5 22.8 0.39 14.9 0.72 11.7 0.84 3.5 33.5 0.63 24.9 0.87 21.0 0.93 3.0 25.9 0.59 17.3 0.86 14.0 0.93 4.0 37.7 0.80 28.9 0.94 24.7 0.97 3.5 29.2 0.76 20.4 0.93 16.6 0.97 4.5 43.3 0.89 33.4 0.97 29.0 0.99 4.0 33.2 0.88 23.2 0.98 19.3 0.99 5.0 48.5 0.95 38.8 0.99 33.8 1.00 0.5 1.0 16.4 0.04 10.1 0.15 7.6 0.25 0.6 1.0 15.4 0.04 10.0 0.13 7.7 0.22 1.5 19.3 0.08 12.4 0.25 9.5 0.40 1.5 19.2 0.07 12.8 0.23 10.2 0.36 2.0 23.4 0.13 15.0 0.41 11.7 0.58 2.0 24.5 0.10 16.6 0.33 13.4 0.48 2.5 26.5 0.27 17.9 0.59 14.5 0.74 2.5 29.0 0.19 20.7 0.46 17.2 0.62 3.0 31.3 0.40 21.8 0.73 17.6 0.85 3.0 34.3 0.30 25.4 0.60 21.6 0.73 3.5 35.7 0.57 25.6 0.85 21.3 0.92 3.5 40.7 0.42 30.8 0.72 26.2 0.84 4.0 41.7 0.70 30.2 0.92 25.5 0.96 4.0 47.2 0.56 36.7 0.82 31.9 0.90 4.5 46.1 0.84 34.8 0.96 29.7 0.98 4.5 53.3 0.70 42.9 0.89 37.8 0.94 5.0 52.3 0.91 39.8 0.98 34.5 0.99 5.0 61.9 0.80 50.0 0.94 44.4 0.97 0.6 1.0 18.0 0.03 11.0 0.12 8.4 0.21 5.5 69.3 0.88 57.9 0.97 51.8 0.99 1.5 21.9 0.05 14.0 0.19 10.9 0.33 6.0 78.9 0.93 65.8 0.98 59.4 0.99

Table 3 (Continued.)  ∗  ∗ √ z ( ˜F0:01; P∗) ( ˜F0:05; P∗) ( ˜F0:10; P∗) ∗  ∗ √ z ( ˜F0:01; P∗) ( ˜F0:05; P∗) ( ˜F0:10; P∗) k = 4; n0= 5 0.0 2.0 17.8 0.53 10.9 0.83 8.3 0.91 2.5 17.8 0.83 10.9 0.97 8.3 0.99 0.0 1.0 41.0 0.02 18.6 0.12 12.7 0.24 0.2 1.0 18.3 0.05 11.4 0.21 8.7 0.35 1.5 41.0 0.03 18.6 0.25 12.7 0.48 1.5 19.0 0.18 11.8 0.49 8.9 0.65 2.0 41.0 0.07 18.6 0.52 12.7 0.77 2.0 19.6 0.45 12.3 0.77 9.5 0.88 2.5 41.0 0.19 18.6 0.80 12.7 0.93 2.5 20.9 0.74 13.4 0.93 10.3 0.97 3.0 41.0 0.46 18.6 0.94 12.7 0.98 3.0 21.8 0.92 14.0 0.99 11.0 0.99 3.5 41.0 0.76 18.6 0.99 12.7 1.00 0.4 1.0 19.6 0.04 12.6 0.16 9.6 0.29 0.2 1.0 41.4 0.01 18.8 0.11 12.8 0.23 1.5 21.8 0.11 14.0 0.36 10.9 0.53 1.5 42.3 0.03 19.2 0.24 13.4 0.44 2.0 24.4 0.27 16.0 0.61 12.9 0.75 2.0 43.1 0.07 20.1 0.47 14.0 0.72 2.5 27.9 0.48 18.8 0.80 15.3 0.90 2.5 43.7 0.15 20.8 0.74 14.8 0.89 3.0 31.9 0.70 22.0 0.92 18.0 0.96 3.0 44.4 0.37 21.5 0.91 15.4 0.97 3.5 37.0 0.85 25.6 0.97 21.4 0.99 3.5 45.9 0.65 22.4 0.97 16.0 0.99 0.5 1.0 20.9 0.03 13.2 0.15 10.3 0.26 0.4 1.0 41.5 0.01 19.5 0.10 13.6 0.21 1.5 24.5 0.08 15.7 0.29 12.5 0.44 1.5 42.7 0.03 20.8 0.20 15.0 0.37 2.0 27.6 0.19 18.8 0.49 15.3 0.64 2.0 46.3 0.05 23.7 0.35 17.2 0.58 2.5 32.6 0.33 23.0 0.67 18.8 0.81 2.5 48.8 0.10 25.9 0.58 19.3 0.79 3.0 37.7 0.52 27.3 0.82 22.7 0.91 3.0 52.1 0.22 29.5 0.76 22.2 0.90 3.5 43.4 0.71 32.4 0.91 27.4 0.96 3.5 56.4 0.41 33.1 0.89 25.8 0.95 4.0 49.6 0.85 37.9 0.97 32.4 0.99 4.0 62.2 0.60 36.8 0.95 28.9 0.98 4.5 57.4 0.92 44.3 0.98 38.4 0.99 4.5 66.1 0.81 41.1 0.98 33.0 0.99 0.6 1.0 22.8 0.02 14.1 0.12 11.0 0.23 0.5 1.0 41.6 0.01 20.1 0.09 14.2 0.19 1.5 26.0 0.06 17.7 0.22 14.2 0.36 1.5 44.3 0.02 23.1 0.14 17.2 0.30 2.0 31.6 0.11 21.9 0.36 18.0 0.53 2.0 49.0 0.03 26.5 0.27 19.6 0.49 2.5 37.8 0.20 27.3 0.50 22.9 0.66 2.5 53.9 0.07 29.7 0.45 23.0 0.67 3.0 45.1 0.31 33.2 0.66 28.5 0.79 3.0 58.1 0.14 35.0 0.62 27.7 0.80 3.5 52.9 0.46 40.4 0.78 35.0 0.88 3.5 65.8 0.24 40.7 0.76 32.0 0.90 4.0 61.7 0.60 47.5 0.88 41.6 0.94 4.0 69.4 0.46 45.6 0.88 37.3 0.95 4.5 69.8 0.77 56.0 0.94 49.5 0.97 4.5 81.6 0.56 52.5 0.93 43.6 0.97 5.0 80.7 0.85 65.4 0.97 58.3 0.99 5.0 84.8 0.79 58.4 0.97 49.2 0.99 5.5 91.8 0.92 75.5 0.98 67.8 0.99 5.5 98.5 0.85 66.8 0.98 56.3 0.99 0.6 1.0 43.3 0.01 21.1 0.09 15.1 0.17 k = 4; n0= 15 1.5 48.6 0.02 25.3 0.12 18.6 0.25 2.0 52.8 0.03 30.1 0.19 22.7 0.38 0.0 1.0 15.0 0.09 9.6 0.28 7.5 0.41 2.5 59.7 0.05 34.6 0.32 27.2 0.54 1.5 15.0 0.31 9.6 0.60 7.5 0.74 3.0 68.5 0.07 41.3 0.44 33.3 0.66 2.0 15.0 0.66 9.6 0.88 7.5 0.94 3.5 74.9 0.14 47.8 0.61 39.3 0.79 2.5 15.0 0.90 9.6 0.98 7.5 0.99 4.0 83.5 0.23 55.9 0.73 46.8 0.87 0.2 1.0 15.5 0.08 9.9 0.26 7.7 0.40 4.5 93.2 0.34 64.4 0.83 54.9 0.92 1.5 16.3 0.25 10.2 0.57 8.2 0.70 5.0 104.3 0.49 73.6 0.90 63.3 0.96 2.0 17.0 0.57 11.2 0.82 8.8 0.90 5.5 116.1 0.63 83.9 0.94 73.0 0.97 2.5 18.3 0.82 12.1 0.95 9.4 0.98 6.0 130.4 0.73 95.7 0.96 83.7 0.98 0.4 1.0 17.3 0.06 11.0 0.21 8.7 0.33 6.5 140.2 0.85 107.1 0.98 94.3 0.99 1.5 19.0 0.16 12.8 0.41 10.4 0.55 7.0 155.5 0.91 119.2 0.99 106.5 0.99 2.0 22.3 0.33 15.2 0.64 12.2 0.77 2.5 25.5 0.57 17.9 0.83 14.7 0.91 k = 4; n0= 10 3.0 28.4 0.80 20.9 0.94 17.2 0.97 3.5 32.8 0.91 24.1 0.98 20.4 0.99 0.0 1.0 17.8 0.06 10.9 0.23 8.3 0.37 0.5 1.0 17.8 0.05 11.7 0.18 9.3 0.29 1.5 17.8 0.22 10.9 0.54 8.3 0.69 1.5 21.1 0.12 14.3 0.34 11.6 0.48

Table 3 (Continued.)  ∗  ∗ √ z ( ˜F0:01; P∗) ( ˜F0:05; P∗) ( ˜F0:10; P∗) ∗  ∗ √ z ( ˜F0:01; P∗) ( ˜F0:05; P∗) ( ˜F0:10; P∗) 0.5 2.0 25.4 0.23 17.8 0.53 14.5 0.68 0.6 3.5 91.3 0.14 57.8 0.67 48.2 0.84 2.5 29.9 0.40 21.6 0.71 17.9 0.83 4.0 100.2 0.26 67.9 0.80 57.1 0.91 3.0 35.3 0.60 25.9 0.85 22.0 0.92 4.5 112.2 0.40 78.1 0.88 66.7 0.95 3.5 40.3 0.78 30.9 0.94 26.6 0.97 5.0 123.6 0.60 88.3 0.94 77.1 0.97 4.0 46.2 0.90 36.1 0.98 31.6 0.99 5.5 141.0 0.70 101.0 0.96 89.1 0.98 0.6 1.0 19.5 0.03 13.0 0.14 10.4 0.24 6.0 152.7 0.85 114.8 0.98 101.5 0.99 1.5 23.5 0.08 16.3 0.25 13.2 0.39 2.0 28.8 0.14 20.8 0.38 17.4 0.53 k = 5; n0= 10 2.5 35.0 0.25 25.8 0.55 21.8 0.70 3.0 41.5 0.40 32.1 0.69 27.6 0.82 0.0 1.0 21.1 0.07 13.3 0.25 10.4 0.40 3.5 49.6 0.55 38.3 0.83 33.5 0.91 1.5 21.1 0.25 13.3 0.59 10.4 0.75 4.0 57.8 0.70 46.2 0.90 40.6 0.95 2.0 21.1 0.60 13.3 0.88 10.4 0.94 4.5 66.5 0.82 54.1 0.95 48.3 0.98 2.5 21.1 0.89 13.3 0.98 10.4 0.99 5.0 76.4 0.90 62.9 0.98 56.8 0.99 0.2 1.0 21.3 0.06 13.8 0.23 10.9 0.37 1.5 22.3 0.21 14.2 0.55 11.2 0.70 k = 5; n0= 5 2.0 23.0 0.52 14.9 0.83 12.0 0.91 2.5 24.0 0.83 15.8 0.96 12.5 0.99 0.0 1.0 49.4 0.02 23.2 0.11 16.2 0.24 0.4 1.0 23.3 0.05 15.0 0.19 11.8 0.31 1.5 49.4 0.03 23.2 0.26 16.2 0.50 1.5 26.1 0.12 16.9 0.41 13.5 0.58 2.0 49.4 0.07 23.2 0.56 16.2 0.80 2.0 29.5 0.30 19.7 0.66 15.9 0.80 2.5 49.4 0.21 23.2 0.85 16.2 0.95 2.5 32.5 0.57 22.5 0.86 18.6 0.93 3.0 49.4 0.51 23.2 0.96 16.2 0.99 3.0 36.8 0.80 26.2 0.95 22.0 0.98 3.5 49.4 0.82 23.2 0.99 16.2 1.00 3.5 41.8 0.93 30.5 0.99 25.8 0.99 0.2 1.0 49.6 0.02 23.4 0.11 16.7 0.23 0.5 1.0 23.7 0.04 15.8 0.17 12.5 0.28 1.5 49.9 0.04 24.0 0.24 17.1 0.46 1.5 28.7 0.08 19.0 0.32 15.4 0.48 2.0 50.6 0.07 24.5 0.52 17.6 0.76 2.0 32.2 0.23 22.8 0.54 18.8 0.70 2.5 51.2 0.17 25.5 0.80 18.4 0.93 2.5 37.2 0.43 27.0 0.75 22.8 0.86 3.0 52.1 0.48 26.9 0.94 19.2 0.98 3.0 43.7 0.64 32.6 0.88 27.7 0.95 3.5 54.4 0.75 27.4 0.99 20.2 1.00 3.5 51.0 0.80 38.8 0.95 33.5 0.98 0.4 1.0 51.9 0.02 24.5 0.10 17.4 0.21 4.0 57.0 0.93 44.8 0.99 39.4 0.99 1.5 52.5 0.03 26.2 0.19 19.2 0.38 0.6 1.0 24.9 0.04 16.9 0.14 13.6 0.24 2.0 56.2 0.05 29.4 0.36 21.7 0.62 1.5 29.8 0.07 21.2 0.24 17.2 0.39 2.5 57.9 0.11 32.4 0.61 24.3 0.83 2.0 37.1 0.13 26.7 0.39 22.0 0.57 3.0 61.9 0.26 36.3 0.81 27.9 0.93 2.5 44.0 0.25 32.8 0.57 27.8 0.73 3.5 67.7 0.47 40.0 0.93 31.6 0.97 3.0 52.5 0.41 40.2 0.73 34.6 0.85 4.0 70.9 0.75 44.5 0.97 35.9 0.99 3.5 61.5 0.58 47.9 0.85 42.1 0.92 0.5 1.0 53.2 0.01 26.1 0.08 18.6 0.18 4.0 72.4 0.73 57.6 0.92 50.6 0.97 1.5 54.8 0.02 27.9 0.16 20.8 0.33 4.5 82.3 0.86 67.1 0.97 59.9 0.99 2.0 58.1 0.04 32.3 0.29 24.8 0.50 5.0 95.2 0.92 78.0 0.99 70.6 0.99 2.5 63.6 0.08 36.7 0.49 28.4 0.72 3.0 68.6 0.17 41.6 0.70 33.4 0.86 k = 5; n0= 15 3.5 75.0 0.33 47.9 0.84 38.9 0.94 4.0 85.2 0.50 55.9 0.92 45.9 0.97 0.0 1.0 17.5 0.10 11.7 0.31 9.3 0.45 4.5 95.6 0.68 63.5 0.97 53.0 0.99 1.5 17.5 0.36 11.7 0.67 9.3 0.79 5.0 100.7 0.87 70.5 0.98 60.2 0.99 2.0 17.5 0.75 11.7 0.92 9.3 0.96 0.6 1.0 52.5 0.01 26.5 0.08 19.4 0.16 2.5 17.5 0.95 11.7 0.99 9.3 0.99 1.5 59.4 0.02 30.8 0.13 23.2 0.26 0.2 1.0 17.6 0.10 12.1 0.28 9.6 0.42 2.0 64.2 0.03 36.0 0.21 27.6 0.41 1.5 18.7 0.32 12.5 0.62 10.1 0.75 2.5 70.3 0.05 41.4 0.36 33.1 0.59 2.0 19.6 0.65 13.4 0.88 10.8 0.94 3.0 79.6 0.08 49.4 0.51 40.4 0.73 2.5 21.3 0.89 14.2 0.98 11.6 0.99

Table 3 (Continued.)  ∗  ∗ √ z ( ˜F0:01; P∗) ( ˜F0:05; P∗) ( ˜F0:10; P∗) ∗  ∗ √ z ( ˜F0:01; P∗) ( ˜F0:05; P∗) ( ˜F0:10; P∗) 0.4 1.0 19.9 0.07 13.4 0.22 10.7 0.36 0.6 2.0 71.5 0.03 41.7 0.24 32.8 0.44 1.5 22.1 0.20 15.6 0.45 12.6 0.61 2.5 80.0 0.06 48.8 0.39 39.1 0.63 2.0 25.0 0.44 17.9 0.73 14.7 0.84 3.0 91.7 0.09 58.4 0.55 47.9 0.77 2.5 28.8 0.69 21.2 0.89 17.7 0.95 3.5 99.6 0.21 66.8 0.74 56.4 0.88 3.0 33.6 0.87 24.9 0.97 21.0 0.99 4.0 113.3 0.34 79.0 0.85 67.4 0.94 0.5 1.0 20.6 0.06 14.4 0.19 11.6 0.31 4.5 126.9 0.53 91.4 0.92 79.5 0.97 1.5 24.6 0.14 17.4 0.37 14.2 0.53 5.0 141.8 0.70 105.1 0.96 91.7 0.98 2.0 28.9 0.30 21.2 0.60 17.6 0.74 5.5 158.1 0.82 119.3 0.98 104.9 0.99 2.5 34.5 0.50 25.5 0.79 21.6 0.89 6.0 177.8 0.90 134.2 0.99 120.5 0.99 3.0 41.4 0.70 31.3 0.91 26.9 0.96 3.5 47.6 0.86 36.9 0.97 32.1 0.99 k = 6; n0= 10 0.6 1.0 22.8 0.04 15.6 0.15 12.6 0.26 1.5 27.6 0.09 19.6 0.28 16.2 0.42 0.0 1.0 24.2 0.07 15.6 0.26 12.4 0.42 2.0 33.7 0.18 24.5 0.46 20.7 0.61 1.5 24.2 0.28 15.6 0.63 12.4 0.78 2.5 40.8 0.32 31.1 0.62 26.8 0.76 2.0 24.2 0.67 15.6 0.92 12.4 0.96 3.0 48.6 0.49 38.2 0.77 33.2 0.87 2.5 24.2 0.93 15.6 0.99 12.4 0.99 3.5 56.6 0.70 45.7 0.89 41.0 0.94 0.2 1.0 25.1 0.06 16.1 0.25 12.8 0.40 4.0 68.9 0.79 55.6 0.95 49.7 0.97 1.5 25.6 0.23 16.8 0.58 13.5 0.73 4.5 78.4 0.90 65.1 0.98 58.9 0.99 2.0 26.6 0.59 17.6 0.87 14.1 0.94 2.5 28.5 0.86 18.4 0.98 15.0 0.99 k = 6; n0= 5 0.4 1.0 25.9 0.05 17.4 0.20 13.9 0.34 1.5 29.3 0.15 19.8 0.44 16.0 0.61 0.0 1.0 57.7 0.01 27.6 0.11 19.7 0.24 2.0 32.5 0.38 22.6 0.73 18.7 0.85 1.5 57.7 0.03 27.6 0.27 19.7 0.51 2.5 37.2 0.65 26.5 0.90 22.0 0.96 2.0 57.7 0.08 27.6 0.60 19.7 0.83 3.0 42.0 0.87 30.2 0.98 25.6 0.99 2.5 57.7 0.23 27.6 0.88 19.7 0.96 0.5 1.0 28.1 0.04 18.6 0.17 15.1 0.29 3.0 57.7 0.56 27.6 0.98 19.7 0.99 1.5 31.4 0.11 21.9 0.35 17.9 0.53 0.2 1.0 57.9 0.02 28.0 0.11 19.9 0.24 2.0 37.1 0.26 26.7 0.58 21.9 0.75 1.5 58.6 0.03 28.9 0.26 20.3 0.49 2.5 41.8 0.52 31.6 0.80 26.8 0.90 2.0 60.1 0.06 29.8 0.52 21.1 0.78 3.0 50.2 0.71 38.2 0.92 32.6 0.96 2.5 60.8 0.23 30.3 0.84 21.7 0.95 3.5 58.3 0.88 44.9 0.97 39.0 0.99 3.0 61.1 0.50 31.4 0.96 23.4 0.99 0.6 1.0 29.5 0.03 19.8 0.14 16.1 0.24 3.5 63.4 0.80 33.2 0.99 24.5 1.00 1.5 34.8 0.07 24.5 0.27 20.1 0.43 0.4 1.0 58.2 0.02 29.4 0.10 21.2 0.21 2.0 41.0 0.17 30.4 0.46 25.7 0.63 1.5 61.0 0.03 31.1 0.20 23.1 0.39 2.5 50.3 0.30 37.8 0.64 32.3 0.79 2.0 62.1 0.05 33.7 0.41 25.5 0.66 3.0 59.6 0.48 46.1 0.80 40.5 0.89 2.5 67.6 0.11 38.0 0.66 29.5 0.85 3.5 69.5 0.68 56.0 0.90 49.3 0.95 3.0 69.5 0.33 41.6 0.87 32.6 0.95 4.0 82.3 0.82 66.8 0.95 59.4 0.98 3.5 76.2 0.58 46.9 0.95 37.3 0.98 4.5 95.4 0.91 78.2 0.98 70.9 0.99 4.0 83.5 0.80 51.0 0.99 41.9 0.99 0.5 1.0 60.0 0.01 30.3 0.09 22.3 0.19 k = 6; n0= 15 1.5 61.7 0.02 32.8 0.17 24.8 0.34 2.0 68.5 0.04 37.5 0.31 28.8 0.56 0.0 1.0 20.0 0.11 13.7 0.32 11.1 0.47 2.5 72.8 0.09 42.7 0.53 33.9 0.76 1.5 20.0 0.40 13.7 0.70 11.1 0.82 3.0 78.6 0.20 48.6 0.76 39.4 0.89 2.0 20.0 0.81 13.7 0.95 11.1 0.98 3.5 88.1 0.36 56.5 0.88 46.4 0.95 0.2 1.0 20.6 0.10 14.2 0.30 11.6 0.44 4.0 95.6 0.62 63.9 0.95 53.5 0.98 1.5 21.8 0.34 15.0 0.65 12.3 0.78 4.5 106.5 0.80 73.3 0.98 62.0 0.99 2.0 22.3 0.73 15.5 0.92 12.7 0.96 0.6 1.0 61.0 0.01 31.0 0.08 23.0 0.17 2.5 23.2 0.95 16.7 0.99 13.7 0.99 1.5 65.4 0.02 36.1 0.13 27.4 0.28 0.4 1.0 22.2 0.08 15.5 0.25 12.6 0.38

Table 3 (Continued.)  ∗  ∗ √ z ( ˜F0:01; P∗) ( ˜F0:05; P∗) ( ˜F0:10; P∗) ∗  ∗ √ z ( ˜F0:01; P∗) ( ˜F0:05; P∗) ( ˜F0:10; P∗) 0.4 1.5 24.7 0.24 17.9 0.51 14.9 0.65 0.6 4.5 160.3 0.66 116.8 0.96 102.4 0.99 2.0 28.8 0.49 20.9 0.78 17.4 0.88 5.0 175.6 0.85 134.3 0.98 118.0 0.99 2.5 33.3 0.76 24.7 0.93 20.7 0.97 3.0 38.4 0.92 28.9 0.99 24.4 0.99 k = 8, n0= 10 0.5 1.0 23.4 0.06 16.7 0.20 13.6 0.33 1.5 27.7 0.16 20.1 0.41 16.7 0.57 0.0 1.0 29.4 0.08 20.0 0.30 16.3 0.46 2.0 32.8 0.36 24.5 0.66 20.9 0.78 1.5 29.4 0.36 20.0 0.72 16.3 0.84 2.5 39.1 0.59 29.8 0.84 25.6 0.91 2.0 29.4 0.80 20.0 0.96 16.3 0.98 3.0 46.5 0.79 35.9 0.94 31.2 0.98 0.2 1.0 29.9 0.08 20.8 0.27 17.0 0.43 3.5 54.6 0.92 43.0 0.99 37.8 0.99 1.5 31.1 0.29 21.2 0.66 17.5 0.80 0.6 1.0 24.8 0.05 17.9 0.17 14.8 0.28 2.0 31.3 0.75 22.1 0.94 18.4 0.97 1.5 30.0 0.12 22.4 0.31 19.0 0.46 0.4 1.0 32.0 0.06 22.0 0.23 18.2 0.37 2.0 38.4 0.21 28.6 0.50 24.6 0.65 1.5 35.3 0.21 24.9 0.53 20.9 0.69 2.5 46.3 0.38 35.6 0.70 30.9 0.82 2.0 39.3 0.51 28.7 0.81 24.1 0.91 3.0 55.4 0.58 44.3 0.83 39.3 0.91 2.5 44.8 0.80 33.3 0.95 28.4 0.98 3.5 66.3 0.75 53.9 0.93 48.2 0.96 0.5 1.0 32.8 0.05 23.6 0.19 19.5 0.32 4.0 76.1 0.89 63.9 0.97 57.8 0.99 1.5 38.3 0.15 27.5 0.43 23.0 0.60 4.5 89.5 0.95 76.2 0.99 68.6 1.00 2.0 44.3 0.36 33.2 0.69 28.4 0.82 2.5 53.0 0.61 39.7 0.88 34.3 0.95 k = 8, n0= 5 3.0 60.6 0.86 47.5 0.97 41.6 0.99 0.6 1.0 35.1 0.04 25.2 0.15 20.6 0.28 0.0 1.0 71.2 0.02 36.0 0.12 26.4 0.25 1.5 42.1 0.09 30.9 0.31 26.0 0.48 1.5 71.2 0.03 36.0 0.30 26.4 0.55 2.0 51.1 0.21 38.4 0.53 33.1 0.70 2.0 71.2 0.09 36.0 0.67 26.4 0.88 2.5 60.2 0.43 47.3 0.75 41.5 0.86 2.5 71.2 0.30 36.0 0.93 26.4 0.98 3.0 74.9 0.60 58.7 0.88 51.9 0.94 3.0 71.2 0.70 36.0 0.99 26.4 0.99 3.5 86.9 0.82 71.0 0.95 63.7 0.98 0.2 1.0 71.6 0.01 36.8 0.11 27.0 0.24 4.0 101.3 0.93 84.5 0.98 76.6 0.99 1.5 72.1 0.03 37.2 0.27 27.6 0.53 2.0 73.2 0.07 37.9 0.61 28.5 0.83 k = 8, n0= 15 2.5 73.7 0.27 38.6 0.90 29.2 0.97 3.0 76.5 0.60 40.3 0.98 30.6 0.99 0.0 1.0 24.4 0.13 17.4 0.37 14.5 0.52 0.4 1.0 72.3 0.02 37.9 0.10 28.3 0.21 1.5 24.4 0.52 17.4 0.80 14.5 0.89 1.5 76.6 0.03 40.0 0.22 30.8 0.42 2.0 24.4 0.91 17.4 0.98 14.5 0.99 2.0 78.9 0.06 43.8 0.46 33.8 0.72 0.2 1.0 24.8 0.13 18.0 0.35 15.0 0.50 2.5 80.6 0.18 47.6 0.77 37.2 0.92 1.5 26.3 0.44 18.8 0.75 15.7 0.85 3.0 88.9 0.39 54.8 0.92 43.4 0.98 2.0 27.0 0.85 19.8 0.96 16.6 0.98 3.5 95.0 0.70 58.1 0.98 48.6 0.99 0.4 1.0 26.9 0.09 19.8 0.28 16.6 0.42 0.5 1.0 74.3 0.02 39.9 0.08 29.2 0.19 1.5 30.6 0.29 22.8 0.58 19.2 0.73 1.5 80.5 0.02 43.3 0.18 33.0 0.36 2.0 35.0 0.63 26.2 0.87 22.3 0.94 2.0 90.4 0.03 48.4 0.36 37.5 0.62 2.5 40.0 0.88 31.0 0.97 26.6 0.99 2.5 92.9 0.09 54.9 0.61 43.9 0.83 0.5 1.0 29.0 0.07 21.2 0.23 17.9 0.36 3.0 100.3 0.23 62.5 0.84 51.7 0.94 1.5 33.2 0.22 25.3 0.49 21.6 0.65 3.5 108.9 0.49 70.9 0.94 59.8 0.98 2.0 40.2 0.46 30.7 0.75 26.5 0.86 4.0 120.1 0.75 82.8 0.98 69.8 0.99 2.5 46.9 0.75 37.0 0.92 32.5 0.96 0.6 1.0 73.9 0.01 40.6 0.08 30.5 0.18 3.0 56.8 0.91 45.1 0.98 39.9 0.99 1.5 82.6 0.02 45.8 0.15 35.7 0.29 0.6 1.0 31.0 0.05 22.6 0.18 19.2 0.30 2.0 90.0 0.03 54.0 0.26 42.9 0.48 1.5 37.3 0.14 28.7 0.36 24.5 0.52 2.5 103.2 0.06 62.6 0.46 51.4 0.70 2.0 46.5 0.28 35.8 0.59 31.3 0.73 3.0 113.4 0.12 74.0 0.65 62.2 0.83 2.5 57.0 0.50 44.5 0.80 39.9 0.89 3.5 124.1 0.28 86.2 0.82 73.6 0.93 3.0 68.3 0.72 55.7 0.92 50.1 0.96 4.0 139.2 0.49 100.1 0.92 86.4 0.97 3.5 80.9 0.89 67.8 0.97 61.5 0.99