testing hypotheses about the common mean of normal distributions

Journal of Statistical Planning and Inference 9 (1984) 207-227

North-Holland

207

TESTING HYPOTHESES ABOUT THE COMMON MEAN OF NORMAL DISTRIBUTIONS *

Arthur COHEN and H.B. SACKROWITZ

Department of Statisrics, Rutgers University, New Brunswick, NJ 08903, USA

Received 7 May 1982; revised manuscript received 13 April 1983

Recommended by S. Zacks

Abstract: An overview of hypothesis testing for the common mean of independent normal

distributions is given. The case of two populations is studied in detail. A number of different

types of tests are studied. Among them are a test based on the maximum of the two available

t-tests, Fisher’s combined test, a test based on Graybill-Deal’s estimator, an approximation to

the likelihood ratio test, and some tests derived using some Bayesian considerations for improper

priors along with intuitive considerations. Based on some theoretical findings and mostly based

on a Monte Carlo study the conclusions are that for the most part the Bayes-intuitive type tests

are superior and can be recommended. When the variances of the populations are close the ap-

proximate likelihood ratio test does best.

Key words: Common mean; Hypothesis testing; t-test; Admissibility; Asymptotically Bahadur

optimal; Convex acceptance sections.

1. Introduction

An extensive number of studies have been made on statistical inference pro-

cedures for the common mean of normal distributions with different unknown

variances. There is good reason for such interest in the problem since the model is

appropriate for many real applied situations. For example it is appropriate in some

Balanced Incomplete Blocks Designs; for situations where different instruments,

different methods, or different laboratories are used to measure like substances or

products; and it can be appropriate in situations where one tests the hypothesis that

means are equal and decides to accept that hypothesis.

Most of the early literature on the subject is concerned with point estimation. An

important paper was written by Graybill and Deal (1959) who considered the case

of two independent populations. They studied the intuitive estimator

p = (X& + P&/[& + $1 (1.1)

where X and P are sample means and sx, sp are sample standard errors of these

* Research supported by NSF Grant #MC.%81 18564.

0378-3758/84/$3.00 @ 1984, Elsevier Science Publishers B.V. (North-Holland)

208 A. Cohen, H.B. Sackrowitz / Testing the common mean

means. If the variance of the populations were known and used instead of sample variances, this estimator would be the best unbiased estimator. Many studies follow- ed Graybill and Deal (1959). We mention Brown and Cohen (1974), Khatri and Shah (1974), a recent thesis by Bhattacharya (1981), where most of the literature is cited.

The latest paper on confidence sets for the common mean of two independent populations is by Marie and Graybill (1979). The recommendation there is that if both sample sizes are at least 13 then use a method first suggested by Meier (1935). This method is to use for equal sample sizes n the interval

where Za,2 is such that 1 - @(Z,,,) = a/2, @ is the c.d.f. of the standard normal distribution, and P=s~s$/&+s$). For smaller sample sizes Marie and Graybill (1979) have another recommendation.

There are only a few papers concerned with testing hypotheses about the common mean. Among these are Cohen and Sackrowitz (1977) and Christiansen (1976). Both papers consider 2 independent populations. Cohen and Sackrowitz (1977) prove that each t-test (t, or t,,) based on a single sample is an admissible test for the hypothesis of zero mean vs nonzero mean or zero mean vs positive mean. Such a result is of theoretical interest as opposed to practical interest since one would not wish to ig- nore the data provided by the other sample. In fact Cohen and Sackrowitz (1977) went on to show that if the variance of, say, the X-population is bounded away from 0, then the test based on tx is inadmissible. In an effort to suggest a practical test procedure, Christiansen studied power properties of the test which rejects if max (ItI, jt,,l) exceeds a constant.

The main purpose of the present paper is to provide an overview of the hypothesis testing problem and to make recommendations for practical use. Tests which com- bine tx and t,,, regardless of s,$ and sb, have many desirable properties. For example, they are similar, invariant under scale transformations, are easy to evaluate, and easy to carry out since critical values are readily attainable. The tests based on tx alone, t,, alone, max(lt,l, I$[) are admissible. Fisher’s test, based on a simple function of the two significance levels (see Section 2), also depends only on t, and t,,. It can be shown to be asymptotically Bahadur optimal for the one sided testing problem. Despite the above optimality properties of tests based on tx and t,, only, each such test has serious deficiencies from the point of view of power and some have other obvious undesirable features. These will be discussed in Section 3.

Another test statistic that is logical to study is the normalized Graybill-Deal estimator. Such a test corresponds to Meier’s confidence interval given in (1.2). Thus the normalized Graybill-Deal statistic for equal sample sizes is (fi/P”‘). Under the null hypothesis of zero mean this statistic is asymptotically normal with mean zero and variance one. The test based on this statistic is locally asymptotically Bahadur optimal for the one sided case (see Section 3) and fares reasonably well in the Monte Carlo simulation study of size and power in Section 4 for large or moderate sample sizes. There are cases however, for small sample sizes, and for the

A. Cohen, H.B. Sackrowitz / Testing the common mean 209

customary levels of significance, where this test can be proven inadmissible. More importantly, in these cases, the reason for inadmissibility is that for some fixed values of the sufficient statistic, fixed except for 8 (or Y), the sections of the acceptance region for the test are not convex (not intervals). This is a most objectionable and counter intuitive property.

Another test worthy of study is an approximation to the likelihood ratio test (LRT). Since the maximum likelihood estimator of the common mean requires the solution of a cubic equation, only an iterative solution is computationally feasible. This leads to an approximate LRT statistic and an approximate LRT. This latter test performs well in terms of power whenever the population variances are close to one another.

An approach, somewhat but not entirely Bayesian, leads us to tests which have some of the desirable properties of the test based on the Graybill-Deal estimator and the approximate LRT statistic. We prove that these new tests oftentimes have convex acceptance sections. They fare very well in the Monte Carlo study of size and power when compared to all other tests studied. Furthermore the size functions of these tests do not vary very much. Hence if one is not overly concerned about the property of having constant size it appears that these latter tests can be recommended.

The model, to be given in the next section, will be for two independent populations and the sample sizes will be assumed equal. Essentially all the mathematical results hold true for k populations and also for varying sample sizes. Remark 2.1 is concerned with varying sample sizes. Asymptotic results would require all sample sizes going to infinity at the same rate. In the next section we list the various tests to be studied. In Section 3 we state the properties and theorems concerned with the tests. Section 4 is devoted to the Monte Carlo study and discussion.

2. Model and test statistics

Let X=(X1,X2,..., X,,) be a random sample from a normal population with unknown mean B and unknown variance a,. Let 8= Cy=, Xi/n, S,‘= C (Xi-X)*, T,= C:=, Xf, s,‘=Sz/(n-I), si=sjf/n. Let Y=(Yt, Y,,..., Y,,) be a random sample from a normal population with unknown mean 0 and unknown variance a;. The Y sample is independent of the X sample. Define Y, S$ Ty, si, sg in analogy with their counterparts in the X sample. Assume n?2. The problem is to test H,,: f?=O vs H,: 0~0 or H,,: 8=0 vs W;: 8>0. Let t,=fiX/s, and tr=fi Y/s,,. Clearly t, and t,, are independent, each with Student’s t distribution with (n - 1) degrees of freedom. The statistic Z’ =(X, T,, y, Ty) is a sufficient statistic whose joint probability density is multivariate exponential family with respect to p, a measure, absolutely continuous with respect to Lebesgue measure. The vector of corresponding natural parameters is <’ = (rt, t2, &, &) = (ne/a~, - 0,212, ne/o$ -@2). See Lehmann (1959, p. 168). Test functions are denoted by p(z) and


&,,(0, oz, cr$ denotes for cp the probability of rejecting He. The function &JO, oz, 0:) is the size function and (r(, = sup,;, 01 &,(O, oz, cry) denotes the size of the test cp. Define ,Q =o,,/oX. All tests studied will depend only on the sufficient statistic, will be scale invariant, and with the exception of the two trivial tests based on tx alone or t,, alone, will be permutation invariant in X, Y. For any test to be considered seriously an initial requirement is that if $4 0, s: f, 0, X+0, or s: + 0, .sz % 0, P#O, the test must reject.

We proceed to list some of the tests to be studied and make some remarks. The tests will be given for I& vs HI although it is obvious how to make them appropriate for He vs I&‘. Consider

v,(z) = 1 if IL > 4-d2(n - 11, = 0 otherwise. (2.1)

This test is the one based on tx alone. Although it is admissible it does not satisfy the initial requirement above when s: -+ 0. Hence it is not recommended and will not be discussed further. (Similarly for the test based on ty alone.) There are many combined tests based on tx and t,, only. Many such tests are studied by Marden (1980) for the problem of testing whether two noncentrality parameters, corresponding to two independent t-distributions are zero against the alternative that they are positive. All such tests can be appropriate for the one sided problem here. Tests based on tx and tu only have the advantage of being similar tests and critical values are easily obtained. We will study two of them in particular. That is,

Ip2(z)= 1 if maW,l, It,l)> t,*,2(n - 11, = 0 otherwise,

(2.2)

where CI = 1 - (1 - a*)2. Also Fisher’s test is

ps(z) = 1 if -logL(t,)-logL(t,,)>&4)/2, = 0 otherwise,

(2.3)

where x:(4) is the critical value of a chi-square distribution with 4 degrees of freedom and L(t,)= 1 -F(ltxl), L(t,)= 1 -F(lt,,l), with F the c.d.f. of It,1 when e=o.

The test based on the normalized Graybill-Deal statistic is

q4(z)=1 if ji*/P>Ki(a;n), = 0 otherwise. (2.4)

where K,(a;n) -+ Z,,, as n -+ 0, but is determined for each individual n so that the approximate size of the test is a. Finding K,(a;n) involves considering various limiting values of Q and also using the Monte Carlo simulation.


The LRT is

ps(z)= 1 if

n{log[(l +@(n- l))/(l +noz(X-jj)z/s;)]

+log[(l + $/(n - l))/(l + n(l - G)~(X-j+/S;)]} >&(a, n), (2.5)

ps(z) = 0 otherwise.

where u^ = c$/(c$ + r$), ~9: and 6; are maximum likelihood estimates. The quantity I,? is determined iteratively after an initial estimate by S,‘/(S, + S:) and so the test is an approximation to the LRT. &(a, n) -+x;(l) as n + 00, but KS needs to be determined for each individual moderate or small n so that the approximate size of the test is a. Thus the determination of Ks requires simulation.

Two other tests that are somewhat Bayesian are

p6, C,(z) = 1 if n&$(X, Y, S,, S,)/[(I/T,)+(I/T,)]‘>K,(a, n), 0 otherwise. (2.6)

v77, c,(z) = 1

:, if nH$,(X, r’, S,, S,)/[(l/T,)+ l/(T,+ Ty)]2>K7(~, n), otherwise.

(2.7)

where

Hcj(X, F, S,, Sy) = /(sgnX/S$ + (sgn Y/S:)1

+Ci/(Tx+ T,-+n(X+ F’)‘), i=6,7. (2.8)

Remark 2.1. Suppose the sample sizes are m for the X population and n for the Y population. Tests (p, , p2, and (p3 are defined in the same way as they are now defined. For p4 the test statistic is unchanged and the critical value’s determination would require simulation. The LRT is as in (2.5) with the following changes: I? = nc$/(mnc$ + t&); n in front of the brackets is deleted and m multiplies the first term in the bracket and m replaces n wherever it appears in the first term. Also n multiplies the second term in the bracketed expression. The analogous test statistic for (2.7) is

where

[(m(m-1)sgn ~/S,2+.(n_l)sgn~/~~l+C6/~~+p]

/[(m(m - 1)/T,) + (n(n - 1)/T,) + mn(m + n - 2)/(m + n)(T,+ T,)],

s~+p=((m+n)/mn(m+n-2))i~, (Xj-(mR+n~/(m+n))2

+ ig, (Yi-(m8+nF)/(m+n))2.


3. Properties of tests

We start out by giving a reason why we would expect tests based on tx and t,, only, to have poor power properties. The Monte Carlo study of Section 4 bears out this reasoning. We also indicate another undesirable intuitive property of both the max test and Fisher’s test. Following these criticisms we prove that the max test is admissible and indicate an asymptotic optimality property for Fisher’s test. Never- theless we cannot recommend either test unless one felt that they must have a similar test. Next we show that the test based on Graybill-Deal’s statistic can sometimes be proven inadmissible. In particular it can be shown to be inadmissible for small sample sizes when significance levels are 0.05 or 0.01. The proof involves showing that this test has the highly undesirable property of non-convex acceptance sections in R and/or Y. We also prove as a corollary that the confidence interval based on Graybill-Deal’s statistic is also inadmissible in cases where the test is inadmissible. We indicate that the test given in (2.4) has a large sample optimality property in spite of the fact that it sometimes can be inadmissible by virtue of non convex acceptance sections. Finally we indicate the rationale for tests (2.6) and (2.7) and prove that they often have convex acceptance sections in R and P.

The power of tests based on tx and t,, will depend on noncentrality parameters &=13/a, and $,=fVaY. If BfO, &>a,, provided ~~<a,,. Hence if s,’ is con- siderably less than sy using tx more often than t,, should yield a test with greater power since this will enable detection of 8#0 more readily than a test that uses tx and tu in a symmetric way. Another criticism of tests (2.2) and (2.3) is as follows: suppose that X= -P and sf is close to sz with 1 txj and It,, moderately large. These tests would reject and they should not. The LRT and approximate LRT have some favorable properties. These tests reject when X+ 0 and sz + 0, sj % 0. These tests accept if R= - P and s: =sy . The LRT (which is not used - the approximation is used) can be shown to have convex (X, 0 sections for fixed TX, Ty.

We now prove

Theorem 3.1. Test (2.2) is admissible.

Proof. The test (2.1) and its counterpart based on tr yield admissible acceptance regions for any critical value. This is proved in Theorem 2.1 of Cohen and Sackrowitz (1977). Since the test (2.1) is an intersection of admissible acceptance regions it follows from Schwartz (1967), Corollary 1 that it too is admissible. 0

Remark 3.2. In a personal communication K. Singh has demonstrated that Fisher’s test is asymptotically Bahadur optimal (ABO) for the one sided testing problem. For the definition of ABO we refer to Bahadur (1967). Furthermore the test (2.4) has a local asymptotic optimality property. Namely, it is Pitman efficient for fixed

o,, oy*

Next we prove


Theorem 3.3. The test v)~ as defined by (2.4) has convex acceptance sections in x

(F) for ali fixed Y, TX, TY (X, T, T,) if and only if Ki(a; n) I 8(n - 1).

Proof. We will study only the acceptance sections in X for fixed F, TX, TY as the analagous study can be done for P. First we recall that the sample space is restricted by -m’X’p. The acceptance region for the test q4 is the set of points for which (for simplicity we let K* = K$(a, n)/(n - l)),

.Z\/;i(xS;+ yS,2)-KSXSYjw<0 (3.2)

where Z= 1 if XS$+ YS:>O, - 1 if XSj+ P$z<O. (3.3)

For convenience let us call the left side of (3.2) R(X; Y, TX, T,). We note that R-+flS;>O as X+ +m and P, TX, TY are kept fixed. Then it follows that the acceptance region (in X for fixed Y, TX, TY) is not convex if and only if, as a function of X, R(X; Y, TX, T,) has at least 4 distinct roots between - m and m. Furthermore R(X; 7, TX, TY) cannot have 4 distinct roots unless it makes at least one upward crossing of the x-axis for R_c 0 or at least one downward crossing of the z-axis for Xl0 (or both). To get this needed information on R we take the partial derivative with respect to x. The derivative is (remembering that $ = TX - nK*)

s = Z(Sj - 2nx P)\l;? + nK.%,,(2Sz + $)/(S,~~). (3.4)

Since R(X; -Y, TX, TY) = R(-8; y, TX, TY) to determine the existence of convex acceptance sections we need only consider sections for which P>O. Of course any crossing of zero entails R = 0.

Case I: Rz 0 (downward crossing). It follows from (3.2) that for Xr 0, Z= 1 and so R = 0 if and only if

Substituting this value for P into (3.4) it is seen that aR/aX<O if and only if

But the left side of (3.6) is greater than 2Jzsf for all 2, S,, S,,. Thus if K*_( 8 !? there can be no downward crossing for positive __.

Case II: x< 0 (upward crossing). If R+S: + PSz<O, so that Z = - 1, then it is seen from (3.4) that aR/az<O so that there can be no upward crossings for such Xvalues. If XSj+ YS:>O, so that Z= + 1, then it follows from (3.4) that (as now XC 0) aR/az’> 0 if and only if (3.6) holds. Thus if K’s 8 there can be no upward crossing for negative R.

(3.5)

(3.6)


We have now established that all acceptance sections must be convex if K2 I 8. Conversely if K2> 8 consider the section for P=Kn-“2(4im - 16)/ (K2 - S), TX = 3K2, T, = n P2 + 32K2/(K2 - 8). For this section R = 0 and aR/aX< 0 at X0 = @%. Therefore there exist O<E < 1 such that R > 0 at X=&,-E and R < 0 at 8=X0 + E. Also R < 0 at X= -X0. Thus the acceptance region for this section is not convex.

Corollary 3.1. Suppose Ki(a;n)> 8(n - l), Then the test in (2.4) is inadmissible.

Proof. We study acceptance sections in X for fixed Y, TX, Ty. The conditional distribution of X given Y, TX, Ty is exponential family with parameter <r and the conditional density is positive for all X such that --m<R<m. It is clear from the proof of Theorem 3.3 that there exists for Ki(a, n) < 8(n - l), a set of positive probability in the space of (Y, TX, Ty) such that for each fixed point in the set, the acceptance section in X will consist of disjoint intervals. Since the conditional distribution of X given Y is exponential family and hence monotone likelihood ratio it follows that conditionally the test is inadmissible. (See, for example, Karlin (1956).) Furthermore since the test statistic in (2.4) is a continuous function of X, Y, TX, Ty it follows that a better test can be constructed so that it is measurable by constructing better conditional tests which we know exist. Continuity of the test statistic will also imply that if we find an interior point in the space of Y, TX, T,, for which the acceptance section is not convex then there is a set of positive probability for which this will be true. 0

Remark 3.4. One implication of Theorem 3.3 is that for every n there exists some size such that the test (2.4) is inadmissible for that size and all smaller sizes. From a practical point of view the Monte Carlo study of Section 4 indicates that the test (2.4) is inadmissible for (r = 0.01, n = 2,3,4 and (Y = 0.05, n = 2,3. (For these cases K&;n)>S(n- 1))

Remark 3.5. We want to stress that from a practical point of view inadmissibility of a test may not be a serious shortcoming in this problem. However, lack of convex acceptance sections for a test is a serious shortcoming.

Next we discuss a corollary concerned with confidence sets. Define a confidence set resulting from a non-randomized test p as S, = {p: cp(X- p, Y-p) = 01, where x-/l=(x,-/l,xz-/l,..., X,-P) and Y-p=(Y,-p, Y2-p,..., Y,,-P). The fact that F is an equivariant estimator of 0 (under translation) and P is invariant implies that the confidence set resulting from the test in (2.4) is

(3.7)

Suppose now we wish to evaluate a confidence set by probability of covering the true value and by the Lebesgue measure of the set. Then we prove


Corollary 3.6. If the test (2.4) is inadmissible then the confidence interval in (3.7)

is inadmissible.

Proof. The result follows from Cohen and Strawderman (1973). See that reference for the definition of inadmissibility of a confidence set.

The remainder of this section is concerned with the tests (2.6) and (2.7). Both tests were derived by first using Bayesian considerations. A Bayes test statistic results from the ratio

where 0, is the null space, Q, is the alternative space, to, and <i are prior distributions. The prior (generalized prior) r, was chosen to be a mixture of three distributions. The first distribution was put on 6i =8/a,, ot = l/a,, a,,= y. Note S,= 8/cr,,=6icrX/y. The first distribution was a product of distributions. The uniform distribution on the real line for 6i, (of) raised to a suitable negative power, and au had a point mass at y. Subsequently the limit as y-+ 03 was obtained. The second distribution was the same as the first except that the roles of x and y were interchanged. The third distribution treated a,= ou and then required a uniform distribution on 6, times of raised to a suitable power. This third distribution had to be multiplied by a function of y, chosen so as to make it of the same order as the other distributions since y was allowed to tend to 03. The prior to for (2.7) was chosen as was t, except now 0 = 0. The prior to for (2.6) was just a mixture of the first two distributions above with 8=0. In both cases (2.6) and (2.7) we let y-+ 00 and so obtained the limit of sequences of Bayes tests. The tests were modified by replacing ((sgn x)/S:) for (l/S:) and ((sgn n/S$ for (l/S:). This was done so that if S,’ and S; were close and K= - F then we would accept Ho. The constants C, and C, appearing in the numerators of (2.6) and (2.7) are multipliers of the third distribution in ri and were selected so that the tests would have appropriate size functions. The choices of C were initially based on what was learned from Monte Carlo simulation.

We conclude this section with a sufficient condition for the tests 96,C,, p7,c, to have convex acceptance sections.

Theorem 3.7. The tests qi,c, in (2.6) and (2.7) have convex acceptance sections in R(P) ifOSCiS2, i=6,7.

Proof. It suffices to show (to establish convex sections in x) that for each fixed r, TX, Ty the set (2: H&X, F, TX, T,)rk} is convex for all k>O where Hc is as defined in (2.8). We will accomplish this by showing that Hc as a function of 8 is first monotone decreasing and then monotone increasing on [-(TX/n)“2, (TX/n)“2].


Let Si = TX+ Ty - +n(X+ n2 and for fixed y, TX, Ty define

if TxzS:,

x0=x& TX, Ty)=

if Tx<SF.

Note that Sx<S$ if 81x0.

Case I, P>O. For Y>O

SL2 - ST2 + CSg2 if X5 min(O, x0), H&P, Y, TX, Ty) = S;‘- Si2 + CSG2 if min(0, x0) r8< 0,

S;2+Si2+CS*2 if O<X.

It is easy to see that

$H,(X Y, TX, 7J = -~KKS;~ + 02(X+ P)K4 if min(0, x0)5X< 0,

~~z_K_S;~ + Cn(X+ P)SG4 otherwise.

(a) Say x05 -Y then

8% -Z-IO if ZIiImin(O,xo)=xo, ax

We also note that for x0 <X1- P we have -2X> -C(X+ P) for C12 and

Sz4= T,+T,-;(X+P)’ [

-2

1 [ = s;+s;+; (X- Y)2 1 -2 5 s,-4.

Thus

aHc - 10 aX

if min(0, x0)5X.

(b) Say -Plx,. Here it is clear that

a& -50 if 25-Y, ax

a& Y 20 if x05X. ax

Also a2H, - =2n(T,+ 3nX2)$?+ Cn(T,+ Ty+ tn(X+ P)2)S;6z0 ax2

if - P~X~x, so that anjr extreme point must be a minimum which completes the proof when P>O.


Case 11, P<O. This follows immediately from Case I as Hc(x - Y, TX, T,) = H&-x Y, TX, Ty). To show convex acceptance sections in Y we need only reverse the roles of X and Y and TX and Ty in the above.

Remark 3.8. The assumption that O~Ci~2 used in Theorem 3.7 is a sufficient condition for convex acceptance sections. We do not know whether or not the tests ps,og and v,,c, have convex acceptance sections for Ci> 2 and reasonable levels of significance. However it should be noted that since Ho is an increasing function in

- - C for all X, Y, TX, Ty it follows that the power function of plc, is an increasing function of Ci, i = 6,7, for all 0, Q. Thus if C1>2 and the size of (ai,c; is the same as the size of pi,* then pi,c; would be preferred as it would have uniformly better power on the alternative space. In Section 4 we will see that this is often the case. It should also be noted that the greatest effects of altering Ci are felt when .Q is near 1. As e-0 or 00 the power function becomes independent of the choice of C;.

Remark 3.9. Clearly there are many other tests that are intuitively appealing for this problem. A natural approach is to weight tx and t,, in relation to the value of s,/s,,. We studied many such tests including one based on a preliminary test of Q = o/ax = 1. All such weighted tests turned out to have some nonconvex acceptance sections in X for fixed (Y, TX, Ty).

4. Monte Carlo results

This section contains evaluations (accomplished through a Monte Carlo study) of the tests (p2 through (p, for tests with size 0.01 for n = 4,7,10, and for tests with size 0.05 for n =3,4,7,10,15. The first step in the evaluation is to determine the appropriate critical values. As indicated in Section 2 the critical values for o2 and ps are easily obtained from the Student’s t and chi-square table respectively. For p4, the Graybill-Deal test, the critical value was determined through simulation as the normal approximation is not good when n is small. For p5 the critical value was determined by simulation since for small n the x2 approximation is not good. For the test p6 and (p7 the constants (C6 and C,) and critical values must be determined together. It is easy to see that the test statistics are increasing in C6 and C,. Thus there are an infinite number of choices’of critical value and constant for every level of significance. Our approach is as follows. Note that as Q= o,,/crx approaches either 0 or 03 both tests p6 and p7 become (in probability) equivalent to a one sample f-test. Thus a value can easily be obtained from the Student’s t table which will guarantee any particular size, say a, for Q near 0 or Q very large. We then use this value as critical value for the test statistic and choose the largest C6 and C, values (through the use of simulation) which will have size a for all Q. This seems only to require choosing C6 and C, so that the size is a when Q= 1. This method seems to give a test which is nearly similar. Although we have only been able to show


that (Do and q7 have convex sections when 0% Ci I 2, i = 6,7 (see Section 3) it is clear that if the size of the test can be maintained by taking Ci>2 the power of the test will be increased. In our tables we include the test for Ci = 2 whenever the ‘best’ (determined through simulation) Ci is greater than 2. All simulations are based on 10 000 replications of the experiment. Some estimate of the accuracy of the tables can be obtained by considering the size for q2,(p3 and p6,a)7 for large Q as the critical values should be fairly precise at these points. For ps ten iterations were done in determining 0 unless successive iterations were within 0.01 of each other in which case iteration stopped.

Remark 4.1. Since all the tests tabled are symmetric in X and Y and the power is a function of a, and au only through Q it suffices to consider only er 1 (as the power will be the same at l/e).

Tables 1 through 8 contain the power computations for (Y = 0.01; n = 4,7,10; and a = 0.05; n = 3,4,7,10,15. The last row of each table gives the critical values for each test. For p4 we list Ki(a; n) as in (2.4). The tables indicate that when Q is near 1 the approximate LRT is strongly preferred to all others. However for other values of Q and in most other instances for n small and even n moderate the tests 96 and/or

(p7, especially 697, are/is preferred. These latter tests certainly have more of a minimax flavor than the LRT and are clearly superior to all others. When n = 10 the advantage of (pg or q7 over the others is less marked and when n = 15 the approximate LRT is doing best.

Table I

A. Cohen, H.B. Sackrowitz / Testing fhe common mean 219

Power calculations, a = 0.01, n = 4

Graybill-

Fisher maximum Deal LRT

0 e @3 @2 @4 063.4 tb.9.2 @s @a @7,2

0.00 1.00 0.0105 0.0094 0.0100 0.0099 0.0099 0.0097

0.00 5.00 0.0105 0.0094 0.0075 0.0103 0.0102 0.0069

0.00 9.00 0.0105 0.0094 0.0069 0.0102 0.0101 0.0066

0.00 13.00 0.0105 0.0094 0.0069 0.0103 0.0102 0.0065

0.00 17.00 0.0105 0.0094 0.0067 0.0103 0.0102 0.0065

0.00 21.00 0.0105 0.0094 0.0065 0.0103 0.0103 0.0067

0.50 1.00 0.027 1 0.0374 0.0279 0.0253 0.0361 0.0442

0.50 5.00 0.0180 0.0189 0.0156 0.0225 0.0225 0.0172

0.50 9.00 0.0179 0.0179 0.0145 0.0237 0.023 1 0.0151

0.50 13.00 0.0175 0.0181 0.0143 0.0245 0.0239 0.0148

0.50 17.00 0.0175 0.0182 0.0141 0.0246 0.0244 0.0149

0.50 21.00 0.0175 0.0182 0.0141 0.0247 0.0246 0.0144

1.00 1.00 0.0807 0.1717 0.1000 0.1103 0.1626 0.2109

1.00 5.00 0.0471 0.0539 0.0482 0.0711 0.0701 0.0573

1.00 9.00 0.0464 0.0503 0.0463 0.0732 0.0728 0.0520

1.00 13.00 0.0461 0.0487 0.0451 0.0742 0.0736 0.0503 1.00 17.00 0.0461 0.0482 0.0450 0.075 1 0.0744 0.0497

1.00 21.00 0.0460 0.0478 0.0450 0.0754 0.0750 0.0491

1.50 1.00 0.1897 0.4605 0.2525 0.3135 0.4460 0.5239

1.50 5.00 0.1061 0.1227 0.1128 0.1557 0.1551 0.1479

1.50 9.00 0.1034 0.1097 0.1071 0.1648 0.1619 0.1272

1.50 13.00 0.1033 0.1056 0.1057 0.1695 0.1678 0.1210

1.50 17.00 0.1032 0.1049 0.1048 0.1716 0.1702 0.1191

1.50 21.00 0.1030 0.1046 0.1044 0.1720 0.1711 0.1181

2.00 1.00 0.3345 0.7515 0.4697 0.5839 0.7461 0.8137

2.00 5.00 0.1909 0.2228 0.2052 0.2716 0.2737 0.2810

2.00 9.00 0.1876 0.1908 0.1969 0.2887 0.2818 0.2332

2.00 13.00 0.1868 0.1809 0.1944 0.2995 0.2935 0.2226

2.00 17.00 0.1869 0.1778 0.1934 0.3038 0.2997 0.2186

2.00 21.00 0.1868 0.1776 0.1922 0.3058 0.3034 0.2182

2.50 1.00 0.5103 0.9252 0.6934 0.8184 0.9279 0.9561

2.50 5.00 0.3070 0.3485 0.3343 0.4118 0.4227 0.4473

2.50 9.00 0.3028 0.2884 0.3190 0.4342 0.4256 0.3771 2.50 13.00 0.3012 0.2746 0.3144 0.4475 0.4403 0.3565 2.50 17.00 0.3010 0.2679 0.3125 0.4549 0.4480 0.3482 2.50 21.00 0.3010 0.265 1 0.3115 0.4584 0.4544 0.3451

3.00 1.00 0.6687 0.9856 0.8561 0.9423 0.9866 0.993 1 3.00 5.00 0.4312 0.4859 0.4706 0.5541 0.5733 0.6140 3.00 9.00 0.4259 0.3970 0.4489 0.5774 0.5682 0.5249 3.00 13.00 0.4241 0.3762 0.4426 0.5919 0.5829 0.4996 3.00 17.00 0.4236 0.3675 0.4397 0.6009 0.5937 0.4900 3.00 21.00 0.4236 0.3630 0.4381 0.6075 0.6008 0.4839

Critical value 55.4551 6.6399 13.3291 12.3721 12.3721 2.8500

0.0092 0.0067

0.0103 0.0096

0.0102 0.0101

0.0102 0.0102

0.0102 0.0102

0.0103 0.0102

0.0221 0.0163

0.0220 0.0205

0.0236 0.0225

0.0241 0.0232

0.0246 0.0240

0.0247 0.0243

0.0903 0.0633

0.0690 0.0595

0.0728 0.0694

0.0739 0.0720

0.0750 0.0735

0.0753 0.0741

0.2504 0.1825

0.1501 0.1328

0.1626 0.1539

0.1683 0.1619

0.1706 0.1663

0.1717 0.1695

0.4905 0.3820

0.2598 0.2206

0.2836 0.2637

0.2970 0.2815

0.3022 0.2919

0.3047 0.2979

0.7292 0.6097

0.3895 0.3319

0.4246 0.3888

0.4433 0.4203

0.4520 0.4362

0.4569 0.4450

0.8905 0.8024

0.5248 0.4441

0.5647 0.5170

0.5857 0.5540

0.5972 0.5749 0.6054 0.5874

12.3721 12.3721

220

Table 2

A. Cohen, H.B. Sackrowitz / Testing the common mean


Graybill-

Fisher maximum Deal LRT e e @2 @3 @4 @6,2.5 h.4.7 @5 @a2 h.2

0.00 1.00 0.0093 0.0090 0.0099 0.0099

0.00 5.00 0.0093 0.0090 0.0078 0.0087

0.00 9.00 0.0093 0.0090 0.0073 0.0088

0.00 13.00 0.0093 0.0090 0.0071 0.0088

0.00 17.00 0.0093 0.0090 0.0072 0.0088

0.00 21.00 0.0093 0.0090 0.0072 0.0088

0.30 1.00 0.0261 0.0316 0.0392 0.0411

0.30 5.00 0.0177 0.0186 0.0207 0.0237

0.30 9.00 0.0175 0.0179 0.0203 0.0237

0.30 13.00 0.0177 0.0180 0.0198 0.0239

0.30 17.00 0.0177 0.0181 0.0199 0.0237

0.30 21 .OO 0.0177 0.0179 0.0200 0.0237

0.60 1.00 0.0963 0.1597 0.1740 0.1923

0.60 5.00 0.0558 0.0586 0.0777 0.0848

0.60 9.00 0.0552 0.0553 0.0733 0.0857

0.60 13.00 0.055 1 0.0542 0.0731 0.0857

0.60 17.00 0.0550 0.0543 0.0726 0.0863

0.60 21.00 0.0552 0.0543 0.0723 0.0864

0.90 1.00 0.2473 0.4594 0.4635 0.4903

0.90 5.00 0.1431 0.1439 0.2009 0.2174

0.90 9.00 0.1405 0.1327 0.1938 0.2153

0.90 13.00 0.1402 0.1298 0.1899 0.2153

0.90 17.00 0.1403 0.1290 0.1891 0.2160

0.90 21.00 0.1400 0.1290 0.1896 0.2165

1.20 1.00 0.4780 0.7734 0.7669 0.7964

1.20 5.00 0.2874 0.2861 0.3938 0.4155

1.20 9.00 0.2837 0.2632 0.3768 0.4101

1.20 13.00 0.2834 0.2552 0.3717 0.4103

1.20 17.00 0.2831 0.2521 0.3682 0.4112

1.20 21.00 0.2831 0.2507 0.3681 0.4116

1.50 1.00 0.7203 0.9478 0.9419 0.9522

1.50 5.00 0.4766 0.4694 0.6071 0.6262

1.50 9.00 0.4733 0.4286 0.5876 0.6233

1.50 13.00 0.4723 0.4145 0.5820 0.6228

1.50 17.00 0.4722 0.4083 0.5804 0.6239

1.50 21.00 0.4718 0.4062 0.5787 0.6251

1.80 1.00 0.8885 0.9942 0.9926 0.9947

1.80 5.00 0.6711 0.6473 0.7966 0.8114

1.80 9.00 0.6677 0.5966 0.7733 0.8014

1.80 13.00 0.6670 0.5818 0.7672 0.7993

1.80 17.00 0.6667 0.5773 0.7648 0.7996

1.80 21.00 0.6666 0.575 1 0.7637 0.8006

Critical value 18.6152 6.6399 2.1607 3.2908

0.0099 0.0099 0.0077 0.0031 0.0085 0.0082 0.0085 0.0077

0.0089 0.0082 0.0088 0.0086

0.0088 0.0085 0.0088 0.0088 0.0088 0.0085 0.0088 0.0088 0.0088 0.0086 0.0088 0.0088

0.0482 0.0545 0.0323 0.0140 0.0230 0.0243 0.0229 0.0202 0.0228 0.0242 0.0231 0.0221

0.0233 0.0237 0.0237 0.0224 0.0236 0.0237 0.0237 0.0229

0.0236 0.0234 0.0237 0.023 1

0.2213 0.2450 0.1528 0.0756

0.0824 0.0916 0.0817 0.0691

0.0846 0.0868 0.0850 0.0785

0.0856 0.0855 0.0857 0.0829

0.0857 0.0849 0.0859 0.0842

0.0861 0.0844 0.0863 0.0847

0.5493 0.5832 0.4250 0.2563

0.2123 0.2346 0.2100 0.1768

0.2123 0.2201 0.2129 0.2017

0.2139 0.2165 0.2144 0.2082

0.2148 0.2143 0.2152 0.2115

0.2154 0.2148 0.2156 0.2132

0.8408 0.8619 0.7396 0.5539

0.4073 0.4397 0.4029 0.3400

0.4054 0.4202 0.4056 0.3796 0.4070 0.4127 0.4079 0.3952

0.4091 0.4095 0.4100 0.4011

0.4104 0.4085 0.4109 0.4042

0.9684 0.9742 0.9282 0.8187 0.6184 0.6560 0.6104 0.5349 0.6157 0.6267 0.6174 0.5842

0.6196 0.6223 0.6203 0.6028

0.6216 0.6207 0.6223 0.6122

0.6229 0.6209 0.6234 0.6167

0.9970 0.9975 0.9887 0.9547

0.8039 0.8293 0.7964 0.7106

0.7942 0.8067 0.7959 0.7618

0.7959 0.8043 0.7967 0.7830

0.7975 0.8002 0.7892 0.7892

0.7980 0.8000 0.7985 0.7933

3.2908 1.2000 3.2908 3.2908

Table 3



Graybill-


0 e $2 @3 @4 &.I.3 @7,3.0 @5 @6,2 e7.2

0.00

0.00

0.00

0.00

0.00

0.00

0.30

0.30

0.30

0.30

0.30

0.30

0.60

0.60

0.60

0.60

0.60

0.60

0.90

0.90

0.90

0.90

0.90

0.90

1.20

1.20

1.20

1.20

1.20

1.20

1 so

1.50

1 so

1.50

1 so

1.50

1.80

1.80

1.80

1.80

1.80

1.80

1.00 0.0094

5.00 0.0094

9.00 0.0094

13.00 0.0094

17.00 0.0094

21.00 0.0094

1.00 0.0414

5.00 0.0250

9.00 0.0249

13.00 0.0248

17.00 0.0248

21.00 0.0248

1.00 0.1867

5.00 0.1007

9.00 0.0995

13.00 0.0991

17.00 0.0992

21.00 0.0992

1.00 0.4926

5.00 0.2922

9.00 0.2897

13.00 0.2892

17.00 0.2890

21 .OO 0.2890

1.00 0.8119

5.00 0.5707

9.00 0.5671

13.00 0.5667

17.00 0.5663

21.00 0.5662

1.00 0.9628

5.00 0.8102

9.00 0.8072

13.00 0.8068

17.00 0.8067

21.00 0.8064

1.00 0.9955

5.00 0.9415

9.00 0.9405

13.00 0.9402

17.00 0.9400

21 .OO 0.9400

0.0094

0.0094

0.0094

0.0094

0.0094

0.0094

0.0533

0.0278

0.0278

0.0277

0.0275

0.0272

0.2991

0.1067

0.1004

0.1002

0.0998

0.0999

0.7353

0.2845

0.2641

0.2596

0.2569

0.2560

0.9623

0.5449

0.5066

0.4995

0.4979

0.4968

0.9980

0.7850

0.7504

0.7413

0.7361

0.7345

1.0000

0.9263

0.9033

0.8977

0.8940

0.8929

Critical value 13.6017 6.6399 1.1642 2.1735 2.1735 0.8200

0.0098 0.0095

0.0082 0.0102

0.0083 0.0105

0.0081 0.0107

0.0082 0.0107

0.0082 0.0108

0.0625 0.0587

0.0305 0.0342

0.0302 0.0349

0.0300 0.0360

0.0296 0.0361

0.0295 0.0364

0.3240 0.3005

0.1360 0.1445

0.1297 0.1465

0.1276 0.1481

0.1271 0.1483

0.1270 0.1486

0.7552 0.7263

0.3745 0.3867

0.3589 0.3927

0.3564 0.3940

0.3553 0.3954

0.3551 0.3960

0.9646 0.9522

0.6766 0.6861

0.6525 0.6857

0.6468 0.6891

0.6448 0.6904

0.6433 0.6908

0.9983 0.9971

0.8837 0.8855

0.8687 0.8870

0.8646 0.8885

0.8638 0.8895

0.8630 0.8901

1.0000 0.9999

0.9732 0.9729

0.9665 0.9744

0.965 1 0.9748

0.9646 0.9753

0.9647 0.9754

0.0099 0.0101

0.0100 0.0087

0.0104 0.0089

0.0107 0.0088

0.0107 0.0085

0.0108 0.0084

0.0713 0.0743

0.0336 0.0325

0.0347 0.0317

0.0357 0.0314

0.0362 0.0313

0.0363 0.0312

0.3532 0.3764

0.1418 0.143 1

0.1456 0.1349

0.1469 0.1329

0.1474 0.1315

0.1482 0.1311

0.7789 0.7974

0.3832 0.3903

0.3898 0.3710

0.3925 0.3645

0.3943 0.3634

0.3953 0.3624

0.9683 0.9740

0.6807 0.6898

0.6805 0.6647

0.6868 0.6576

0.6890 0.6548

0.6902 0.6541

0.9986 0.9989

0.8816 0.8860

0.8850 0.8717

0.8869 0.8707

0.8888 0.8699

0.8893 0.8695

1.0000 1.0000

0.9711 0.9689

0.973 1 0.9666

0.9742 0.9664 0.9748 0.9663

0.9752 0.9662

0.0197 0.0043

0.0108 0.0088

0.0109 0.0101

0.0107 0.0106

0.0108 0.0107

0.0108 0.0107

0.1048 0.0313

0.0368 0.0311

0.0362 0.0337

0.0364 0.0347

0.0365 0.0358

0.0367 0.0360

0.4521 0.2070

0.1577 0.1254

0.1502 0.1411

0.1493 0.1444

0.1497 0.1464

0.1497 0.1473

0.8433 0.6020

0.4155 0.3510

0.4007 0.3771

0.3976 0.3880

0.3975 0.3916

0.3976 0.3932

0.9832 0.9085

0.7153 0.6338

0.6977 0.6673

0.6947 0.6781

0.6932 0.6840

0.6927 0.6865

0.9993 0.9926

0.9053 0.8512

0.8947 0.8739

0.8920 0.8812

0.8917 0.8849

0.8915 0.8869

1.0000 0.9999

0.9800 0.9558

0.9772 0.9672

0.9760 0.9716

0.9759 0.9733

0.9758 0.9742

2.1735 2.1735

222

Table 4



Graybill-


e e @2 @3 @4 h.O.5 e7.4.a @s $62 @7,2

0.00

0.00

0.00

0.00

0.00

1.00

5.00

9.00

13.00

17.00

0.0496 0.0503 0.0531 0.0501 0.0500 0.0491 0.0536 0.0457

0.0496 0.0503 0.0439 0.0478 0.0477 0.0346 0.0484 0.0465

0.0496 0.0503 0.0381 0.0482 0.0478 0.0321 0.0482 0.0469

0.0496 0.0503 0.0358 0.0480 0.0481 0.0323 0.0480 0.0476

0.0496 0.0503 0.0343 0.0478 0.0479 0.0319 0.0480 0.0478

0.30

0.30

0.30

0.30

0.30

1.00

5.00

9.00

13.00

17.00

0.0659 0.0680 0.0718 0.0658 0.0677 0.0762 0.0711 0.0593

0.0599 0.0581 0.0553 0.0622 0.0611 0.0477 0.0632 0.0601

0.0594 0.0584 0.0507 0.0620 0.0615 0.0461 0.0622 0.0608

0.0595 0.0582 0.0474 0.0624 0.0618 0.0456 0.0625 0.0616

0.0595 0.0583 0.0463 0.0625 0.0623 0.0452 0.0625 0.0623

0.60

0.60

0.60

0.60

0.60

1.00

5.00

9.00

13.00

17.00

0.1027 0.1257 0.1202 0.1012 0.1144 0.1497

0.0785 0.0865 0.0794 0.0930 0.0916 0.0764

0.0781 0.0841 0.0745 0.0952 0.0940 0.0710

0.0776 0.0834 0.0707 0.0959 0.0954 0.0696

0.0774 0.0830 0.0688 0.0960 0.0957 0.0698

0.90

0.90

0.90

0.90

0.90

1.00

5.00

9.00

13.00

17.00

0.1588 0.2280 0.1977 0.1703 0.2043 0.2848

0.1086 0.1226 0.1156 0.1432 0.1415 0.1238

0.1074 0.1195 0.1089 0.1484 0.1467 0.1126

0.1071 0.1173 0.1050 0.1503 0.1489 0.1110

0.1068 0.1179 0.1022 0.1524 0.1507 0.1097

1.20

1.20

1.20

1.20

1.20

1.00

5.00

9.00

13.00

17.00

0.2338 0.3780 0.3068 0.2688 0.3289 0.4522

0.1483 0.1728 0.1661 0.2037 0.2046 0.1867

0.1460 0.1647 0.1565 0.2154 0.2131 0.1716

0.1461 0.1627 0.1532 0.2198 0.2182 0.1667

0.1460 0.1617 0.1501 0.2215 0.2205 0.1645

1.50 1.00 0.3241 0.5453 0.4242 0.3850 0.4800 0.6323

1.50 5.00 0.2000 0.2272 0.2296 0.2813 0.2792 0.2659

1.50 9.00 0.1962 0.2180 0.2166 0.2975 0.2926 0.2368

1.50 13.00 0.1951 0.2145 0.2114 0.3050 0.3015 0.2291

1.50 17.00 0.1949 0.2135 0.2095 0.3081 0.3061 0.227 I

1.80

1.80

1.80

1.80

1.80

1.00

5.00

9.00

13.00

17.00

0.4132 0.7016 0.5584 0.5155 0.6339 0.7828

0.2582 0.2941 0.3006 0.3573 0.3549 0.3549

0.2523 0.2722 0.2868 0.3850 0.3776 0.3200

0.2512 0.2662 0.2812 0.3946 0.3913 0.3105

0.2508 0.2&7 0.2788 0.4010 0.3972 0.3074

2.10

2.10

2.10

2.10

2.10

1.00

5.00

9.00

13.00

17.00

0.5082 0.8188 0.6804 0.6397 0.763 1 0.8847

0.3251 0.3615 0.3781 0.4379 0.4401 0.4480

0.3178 0.3277 0.3591 0.4694 0.4638 0.403 1

0.3170 0.3194 0.3540 0.4849 0.4790 0.3903

0.3166 0.3173 0.3508 0.4907 0.4869 0.385 1

Critical I ralue 38.0000 4.7458 10.6643 10.2564 10.2564 2.7000

0.1136 0.0937

0.0946 0.0880

0.0959 0.0931

0.0964 0.0950

0.0965 0.0954

0.2014 0.1588

0.1467 0.1354

0.1502 0.1442

0.1510 0.1474

0.1529 0.1497

0.3121 0.2579

0.2120 0.1943

0.2181 0.2092

0.2214 0.2159

0.2228 0.2186

0.4525 0.3762

0.2902 0.2626

0.3026 0.2856

0.3077 0.2971

0.3099 0.3035

0.5970 0.5096

0.3109 0.3340

0.3912 0.3680

0.3988 0.3847

0.4029 0.3923

0.7258 0.6377

0.4563 0.4090

0.477 1 0.4496

4.8840 0.4716

0.4933 0.4820

10.2564 10.2564

Table 5



Graybill-


e e @, @1 04 &G I 7 Qb 4 a @5 @h 7 h 7

0.00 1 .oo 0.0516 0.0501 0.0512 0.0508 0.0513 0.0500 0.0529 0.0380

0.00 5.00 0.0516 0.0501 0.0417 0.0522 0.0515 0.0419 0.0524 0.0493

0.00 9.00 0.0516 0.0501 0.0375 0.0519 0.0517 0.0410 0.0520 0.0512

0.00 13.00 0.0516 0.0501 0.0356 0.0520 0.0521 0.0408 0.0520 0.0516

0.00 17.00 0.0516 0.0501 0.0348 0.0519 0.0519 0.0411 0.0519 0.0519

0.00 21.00 0.0516 0.0501 0.0345 0.0519 0.0519 0.0413 0.0520 0.0519

0.40 1.00 0.0926 0.1044 0.1065 0.1014 0.1135 0.1265

0.40 5.00 0.0718 0.0751 0.0671 0.0824 0.0818 0.0738

0.40 9.00 0.0716 0.0734 0.0615 0.0845 0.0837 0.0696

0.40 13.00 0.0704 0.0732 0.0592 0.0848 0.0846 0.0686

0.40 17.00 0.0700 0.0728 0.0587 0.0853 0.0848 0.0679

0.40 21.00 0.0703 0.0730 0.0582 0.0856 0.0853 0.0672

0.80 1.00 0.2248 0.3108 0.2976 0.2919 0.3380 0.3862

0.80 5.00 0.1427 0.1535 0.1581 0.1995 0.1985 0.1802

0.80 9.00 0.1402 0.1504 0.1488 0.2066 0.2037 0.1701

0.80 13.00 0.1394 0.1489 0.1446 0.2080 0.2073 0.1697

0.80 17.00 0.1395 0.1469 0.1434 0.2091 0.2082 0.1699

0.80 21.00 0.1388 0.1476 0.1420 0.2089 0.2088 0.1699

1.20 1.00 0.4236 0.6166 0.5692 0.5711 0.6384 0.6945

1.20 5.00 0.2641 0.2775 0.3022 0.3637 0.3630 0.3492

1.20 9.00 0.2609 0.2634 0.2871 0.3719 0.3693 0.3267

1.20 13.00 0.2602 0.2575 0.2810 0.3756 0.3733 0.3220

1.20 17.00 0.2598 0.2574 0.2793 0.3776 0.3758 0.3221

1.20 21.00 0.2595 0.2569 0.2789 0.3784 0.3776 0.3212

0.1081 0.0761

0.0837 0.0772

0.0847 0.0811

0.0848 0.0830

0.0853 0.0844

0.0857 0.0851

0.3084 0.2274

0.2024 0.1822

0.2073 0.1967

0.2083 0.2037

0.2091 0.2063

0.2092 0.2078

0.5937 0.4806

0.3689 0.3298

0.3729 0.3580

0.3759 0.3673

0.3780 0.3719

0.3786 0.3753

1.60 1.00 0.6284 0.8612 0.8102 0.8210 0.8754 0.9069

1.60 5.00 0.4111 0.4314 0.4757 0.5524 0.5516 0.5428

1.60 9.00 0.4052 0.4009 0.4546 0.5662 0.5662 0.5124

1.60 13.00 0.4041 0.3932 0.4480 0.5724 0.5691 0.5059

1.60 17.00 0.4036 0.3898 0.4459 0.5751 0.5725 0.5043

1.60 21.00 0.4030 0.3866 0.4434 0.5766 0.5748 0.5027

2.00 1.00 0.8045 0.9683 0.9419 0.9510 0.973 1 0.9827

2.00 5.00 0.5779 0.5878 0.6553 0.7219 0.7259 0.7182

2.00 9.00 0.5704 0.5414 0.6279 0.7346 0.7321 0.6888 2.00 13.00 0.5700 0.5284 0.6219 0.7430 0.7394 0.6810

2.00 17.00 0.5694 0.5239 0.6187 0.7473 0.7445 0.6792 2.00 21.00 0.5685 0.5203 0.6169 0.7497 0.747 1 0.6779

2.40 1.00 0.9153 0.9956 0.9879 0.9908 0.9955 0.9982

2.40 5.00 0.7228 0.7205 0.7976 0.8431 0.8436 0.8380 2.40 9.00 0.7151 0.6715 0.7737 0.8613 0.8561 0.8192

2.40 13.00 0.7146 0.6566 0.7662 0.8661 0.8633 0.8163

2.40 17.00 0.7142 0.6507 0.7628 0.8696 0.8679 0.8163 2.40 21.00 0.7140 0.6494 0.7623 0.8714 0.8696 0.8175

Critical value 17.2761 4.7458 3.7320 4.3760 4.3760 1.6500

0.8405 0.7415

0.5600 0.5046

0.5696 0.5415

0.5746 0.5588

0.5758 0.5663

0.5768 0.5708

0.9595 0.9099

0.7302 0.6651

0.7390 0.7062

0.7445 0.7259

0.7487 0.7351

0.7507 0.7420

0.9932 0.9788

0.8499 0.7921

0.8638 0.8297

0.8681 0.8494

0.8706 0.8588 0.8717 0.8648

4.3760 4.3760


Table 6

Power calculations, (x = 0.05, n = 7

Graybill-


e e @2 @3 @4 ti6.1.3 @7,2.8 @5 @6,2 @7,2

0.00

0.00

0.00

0.00

0.00

0.00

0.30

0.30

0.30

0.30

0.30

0.30

0.60

0.60

0.60

0.60

0.60

0.60

0.90

0.90

0.90

0.90

0.90

0.90

1.20

1.20

1.20

1.20

1.20

1.20

1.50

1.50

1.50

1.50

1.50

1.50

1.80

1.80

1.80

1.80

1.80

1.80

Critical

1.00

5.00 9.00

13.00 17.00 21.00

1.00

5.00

9.00

13.00

17.00

21.00

1.00

5.00

9.00

13.00

17.00

21.00

1.00

5.00

9.00

13.00

17.00

21.00

1.00

5.00

9.00

13.00

17.00

21.00

1.00

5.00

9.00

13.00

17.00

21.00

1.00

5.00

9.00

13.00

17.00

21.00

0.0451

0.0451

0.0451

0.0451

0.045 1

0.0451

0.1161 0.1241 0.1557

0.0846 0.0866 0.0937

0.0836 0.085 1 0.0896

0.0830 0.0846 0.0879

0.0830 0.0846 0.0882

0.0828 0.0845 0.0879

0.3130 0.4023 0.4704

0.1995 0.2013 0.2579

0.1954 0.1948 0.2473

0.1939 0.1919 0.2448

0.1934 0.1909 0.2439

0.1933 0.1910 0.2433

0.6030 0.7474 0.8053

0.3949 0.3960 0.4972

0.3910 0.3768 0.4815

0.3888 0.3723 0.4786

0.3874 0.3696 0.4773

0.3869 0.3691 0.4764

0.8482 0.9492 0.9650

0.6294 0.6134 0.7378

0.6244 0.5882 0.7202

0.6236 0.5821 0.7147

0.6221 0.5778 0.7124

0.6215 0.5754 0.7121

0.965 1 0.9957 0.9974

0.8195 0.7996 0.8983

0.8154 0.7714 0.8851

0.8144 0.7648 0.8839

0.8141 0.7619 0.8824

0.8132 0.7597 0.8813

0.9945 0.9997 0.9998

0.9327 0.9161 0.9697

0.9299 0.8979 0.9640

0.9294 0.8927 0.9629

0.9291 0.8910 0.9621

0.9290 0.8896 0.9619

8.7548 4.7458 0.9668

0.0459 0.0494

0.0459 0.0395

0.0459 0.0384

0.0459 0.0379

0.0459 0.0381

0.0459 0.0382

0.0493 0.0501

0.0444 0.0430

0.0462 0.0455

0.0463 0.0466

0.0468 0.0466

0.0470 0.0469

0.1569 0.1621

0.1016 0.0993

0.1011 0.1006

0.1034 0.1028

0.1032 0.1030

0.1033 0.1032

0.4672 0.4859

0.2718 0.2683

0.2745 0.2719

0.2753 0.2740

0.2759 0.2753

0.2766 0.2762

0.8033 0.8209

0.5127 0.5065

0.5170 0.5130

0.5195 0.5181

0.5198 0.5193

0.5209 0.5200

0.9605 0.9678

0.7541 0.7470

0.7520 0.7472

0.7529 0.7512

0.7538 0.7521

0.7548 0.7538

0.9965 0.9977

0.9050 0.9007

0.9058 0.9033

0.9071 0.9052

0.9069 0.9061 0.9074 0.9069

0.9998 0.9999

0.9707 0.9676

0.9714 0.9696

0.9715 0.9710

0.9716 0.9715

0.9720 0.9715

1.9979 1.9979

0.0483

0.0423

0.0436

0.0437

0.0437

0.0439

0.1687

0.1019

0.0996

0.0999

0.0997

0.0991

0.4970

0.2771

0.2698

0.2679

0.2676

0.2679

0.8290

0.5242

0.5105

0.5085

0.5086

0.5084

0.9696

0.7607

0.7472

0.7457

0.7441

0.7420

0.9972

0.9068

0.9007

0.9000

0.8999

0.9004

1.0000

0.9707

0.9696

0.9688

0.9690

0.9701

0.7100

0.0833 0.0305

0.0483 0.0401

0.0475 0.0445

0.0471 0.0459

0.0473 0.0463

0.0473 0.0465

0.2363 0.1063

0.1072 0.0920

0.1043 0.0984

0.1037 0.1008

0.1037 0.1023

0.1037 0.1025

0.5951 0.3717

0.2898 0.2516

0.2808 0.2661

0.2783 0.2712

0.2779 0.2737

0.2774 0.2749

0.8833 0.7175

0.5400 0.4796

0.5261 0.5049

0.5233 0.5136

0.5223 0.5164

0.5220 0.5182

0.9821 0.9315

0.7786 0.7174

0.7610 0.7382

0.7572 0.7453

0.7568 0.7496

0.7575 0.7510

0.9990 0.9918

0.9189 0.8792

0.9119 0.8957

0.9091 0.9020

0.9093 0.9046

0.9091 0.9049

1.0000 0.9997

0.9761 0.9583

0.9740 0.9664

0.9734 0.9690

9.7290 0.9702

0.9724 0.9705

1.9979 1.9979

Table 7


Power calculations, cf= 0.05, n = 10

Graybill-


I9 e @2 @3 @4 kO.il6 G7.2.14 @5 @6.2 b.2

0.00

0.00

0.00

0.00

0.00

0.00

0.25

0.25

0.25

0.25

0.25

0.25

0.50

0.50

0.50

0.50

0.50

0.50

0.75

0.75

0.75

0.75

0.75

0.75

1.00

1.00

1.00

1.00

1.00

1.00

1.25

1.25

1.25

1.25

1.25

1.25

1.50

1.50

1.50

1.50

1.50

1.50

Critical

1.00

5.00

9.00

13.00

17.00

21.00

1.00

5.00

9.00

13.00

17.00

21.00

1.00

5.00

9.00

13.00

17.00

21.00

1.00

5.00

9.00

13.00

17.00

21.00

1.00

5.00

9.00

13.00

17.00

21.00

1.00

5.00

9.00

13.00

17.00

21.00

1.00

5.00

9.00

13.00

17.00

21.00

value

0.0483

0.0483

0.0483

0.0483

0.0483

0.0483

0.1209

0.0872

0.0870

0.0873

0.0873

0.0870

0.3519

0.2237

0.2197

0.2186

0.2193

0.2194

0.6739

0.4521

0.4472

0.4453

0.4444

0.4440

0.9048

0.7053

0.6983

0.6971

0.6964

0.6962

0.9853

0.8844

0.8805

0.8793

0.8792

0.8788

0.9990

0.9704

0.9692

0.9686

0.9682

0.9682

7.1676

0.0484

0.0484

0.0484

0.0484

0.0484

0.0484

0.1356

0.0909

0.0886

0.0883

0.0884

0.088 1

0.4399

0.2244

0.2178

0.2158

0.2149

0.2151

0.7937

0.4460

0.4300

0.4268

0.4255

0.4252

0.9663

0.685 1

0.6644

0.6603

0.6566

0.6572

0.9980

0.8646

0.8467

0.8423

0.8402

0.8366

1.0000

0.9585

0.9500

0.9459

0.9444

0.9442

4.7458

0.0493

0.0420

0.0425

0.0420

0.0420

0.0418

0.1729

0.1042

0.1015

0.1024

0.1029

0.1029

0.5242

0.2921

0.2812

0.2792

0.2773

0.2762

0.8567

0.5629

0.5486

0.5432

0.5414

0.5404

0.9808

0.8052

0.7906

0.7874

0.7871

0.7870

0.9994

0.9421

0.9346

0.9324

0.9307

0.9308

1.0000

0.9889

0.9868

0.9864

0.9864

0.9864

0.5477

0.0497 0.0495

0.0444 0.0432

0.0458 0.0454

0.0464 0.0460

0.0468 0.0465

0.0470 0.0467

0.1743 0.1780

0.1062 0.1033

0.1119 0.1111

0.1134 0.1132

0.1141 0.1138

0.1145 0.1143

0.5219 0.5330

0.2876 0.2827

0.2931 0.2906

0.2958 0.2943

0.2971 0.2965

0.2978 0.2972

0.8503 0.8594

0.5594 0.5530

0.5599 0.5562

0.5640 0.5621

0.5655 0.5651

0.5657 0.5655

0.9779 0.9809

0.7986 0.7917

0.7989 0.7962

0.8003 0.7989

0.8013 0.8004

0.8022 0.8017

0.9989 0.9991

0.9362 0.9322

0.9378 0.9363

0.9384 0.9378

0.9392 0.9387

0.9395 0.9393

1.0000 1.0000

0.9866 0.9851

0.9882 0.9877

0.9892 0.9885

0.9890 0.9889

0.9892 0.9892

1.5686 1.5686

0.0476

0.0451

0.0461

0.0456

0.0451

0.0448

0.1773

0.1098

0.1076

0.1086

0.1090

0.1097

0.5333

0.3024

0.2939

0.2914

0.2899

0.2898

0.8591

0.5723

0.5628

0.5591

0.5585

0.5581

0.9810

0.8116

0.8019

0.7984

0.7981

0.7967

0.9988

0.9422

0.9400

0.9388

0.9385

0.9379

1.0000

0.9889 0.9878

0.9873

0.9873

0.9875

0.4600

0.2673 0.0404

0.0533 0.0421

0.0490 0.045 1

0.0478 0.0458

0.0477 0.0463

0.0477 0.0465

0.4839 0.1575

0.1291 0.1018

0.1179 0.1104

0.1158 0.1129

0.1152 0.1137

0.1153 0.1143

0.8184 0.4989

0.3344 0.2774

0.3083 0.2897

0.3035 0.2934

0.3015 0.2960

0.3001 0.2970

0.9712 0.8382

0.6137 0.5465

0.5806 0.5545

0.5733 0.5611

0.5715 0.5641

0.5708 0.5649

0.9971 0.9749

0.8412 0.7855

0.8151 0.7939

0.8093 0.7982

0.8068 0.7997

0.8064 0.8011

1.0000 0.9988 0.9573 0.9279

0.9475 0.9350

0.9434 0.9376

0.9417 0.9382

0.9408 0.9390

1.0000 1.0000 0.9933 0.9835

0.9906 0.9874 0.9903 0.9881 0.9898 0.9887 0.9898 0.9892

1.5686 1.5686

226

Table 8


Power calculations, a=0.05, n = 15

Graybill-


8 e @2 @3 @4 @6,0.51 @7,1.67 @5 %2 @7,2

0.00 1.00 0.0480 0.0492 0.0497

0.00 5.00 0.0480 0.0492 0.0492

0.00 9.00 0.0480 0.0492 0.0474

0.00 13.00 0.0480 0.0492 0.0471

0.00 17.00 0.0480 0.0492 0.0470

0.00 21.00 0.0480 0.0492 0.0471

0.25 1.00 0.1725 0.1937 0.2462

0.25 5.00 0.1165 0.1139 0.1447

0.25 9.00 0.1150 0.1124 0.1415

0.25 13.00 0.1148 0.1126 0.1420

0.25 17.00 0.1144 0.1125 0.1416

0.25 21.00 0.1148 0.1129 0.1418

0.50 1.00 0.5361 0.6401 0.7227

0.50 5.00 0.3398 0.3385 0.4415

0.50 9.00 0.3349 0.3269 0.4314

0.50 13.00 0.3346 0.3257 0.4284

0.50 17.00 0.3342 0.3248 0.4279

0.50 21.00 0.3344 0.3253 0.4277

0.75 1.00 0.8852 0.9473 0.9711

0.75 5.00 0.6716 0.6611 0.7733

0.75 9.00 0.6659 0.6433 0.7620

0.75 13.00 0.6656 0.6384 0.7579

0.75 17.00 0.6656 0.6347 0.7574

0.75 21.00 0.6655 0.6331 0.7565

1.00 1.00 0.9915 0.9986 0.9995

1.00 5.00 0.903 1 0.8905 0.9493

1.00 9.00 0.8996 0.8783 0.9444

1.00 13.00 0.8989 0.8741 0.9427

1.00 17.00 0.8990 0.8745 0.9426

1.00 21.00 0.8991 0.8737 0.9424

Critical values 6.2649 4.7458 0.3207

0.0503 0.0490

0.0453 0.0435

0.0491 0.0484

0.0504 0.0502

0.0511 0.0509

0.0515 0.0514

0.2420 0.2446

0.1337 0.1314

0.1432 0.1424

0.1465 0.1458

0.1480 0.1474

0.1491 0.1486

0.7102 0.7191

0.4144 0.4099

0.4304 0.4285

0.4370 0.4361

0.4390 0.4385

0.4400 0.4393

0.9667 0.9695

0.7474 0.7418

0.7599 0.7580

0.7655 0.7645

0.7674 0.7667

0.7693 0.7685

0.9993 0.9993

0.9392 0.9374

0.9432 0.9423

0.9447 0.9443

0.9453 0.9452

0.9461 0.9457

1.3286 1.3286

0.0480

0.0513

0.0505

0.0501

0.0504

0.0501

0.2471

0.1497

0.1465

0.1462

0.1466

0.1468

0.7241

0.4489

0.4410

0.4382

0.4375

0.4378

0.9709

0.7777

0.7681

0.7668

0.7666

0.7665

0.9991

0.9522

0.9470

0.9461

0.9465

0.9461

0.2880

0.4900 0.1016

0.0701 0.0482

0.0566 0.0504

0.0548 0.0508

0.0534 0.0514

0.0529 0.0518

0.7215 0.3753

0.1873 0.1402

0.1580 0.1448

0.1542 0.1475

0.1526 0.1482

0.1519 0.1494

0.9477 0.8330

0.5121 0.4295

0.4608 0.4350

0.4509 0.4393

0.4472 0.4403

0.4451 0.4409

0.9976 0.9879

0.8258 0.7596

0.7862 0.7644

0.7791 0.7668

0.7762 0.7691

0.7743 0.7699

1.0000 0.9998

0.9682 0.9450

0.9542 0.9450

0.9504 0.9460

0.9491 0.9459

0.9487 0.9466

1.3286 1.3286

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