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IJASETR Research Paper ISSN: 1839-7239
June – 2012 Volume – 1, Issue – 3 Article #04
www.setscholars.org Knowledge is Power
A Modified F test for testing Homogeneity of Means against One-sided alternatives 1S.
M. Sayem, 2Rumana Rois*, 3Tapati Basak, 4Mhod. Muzibur Rahman and 5Ajit Kumar Majumder 1 Lecturer, Department of Statistics, Bangladesh Agricultural Unuversity, Bangladesh 2,3Assistant Professor, Department of Statistics, Jahangirnagar University, Bangladesh 4, 5 Professor, Department of Statistics, Jahangirnagar University, Bangladesh *Corresponding
author’s email:
[email protected],
[email protected]
Abstract Usual F and 2 test is suitable for testing the homogeneity of means against two-sided alternatives of ANOVA or MANOVA model. This test is sensitive for restricted alternatives. But, for the slippage types ordered alternatives, max t is suitable. We propose weighted mixture of F type test for restricted alternatives. Our Monte Carlo study suggests that, proposed test perform better than existing two-sided F test in terms of power properties. Keywords: Balanced model, Fixed- effect model, Global test, max t test, Ordered restricted inference, Slippage. Citation: Sayem SM. et al. (2012), A Modified F test for testing Homogeneity of Means against One-sided alternatives. IJASETR 1(3): p. 36 - 41. Received: 01-06-2012
Accepted: 12-06-2012
Copyright: @ 2012 Sayem SM. et al. This is an open access article distributed under the terms of the Creative Common Attribution 3.0 License.
1. Introduction In applied research, numbers of factors or independent variables influence the response variable in the model. Based on the signs of the estimated parameters of the model, one of the variables associated with an incorrectly signed parameter is deleted from the equation and the model is re-estimated. This procedure is repeated until all of the variables left in the equation about which the researcher has a priori beliefs have correctly signed estimated coefficient. Test for restricted alternative allows the researchers to asses in a hypothesis testing framework, whether or not the data is consistent with true values of the parameters, satisfying the sign restrictions imposed on the estimated coefficient (see Walok, 1987).
The Likelihood ratio principal is one of the most widely used methods of deriving statistical tests. To test the restricted parameter, the likelihood ratio principal is a way, whose yields a test criterion provided that the likelihood of the sample under the alternative hypothesis can be maximized. Consider the one-way analysis of variance model yij i ij , j 1, 2,..., ni , i 1, 2,..., k
(1.1) 2
where ij ’s are identically independently distributed N (0, i ) . In the analysis of the normal theory linear model, the conventional F test is frequently used. For testing the hypothesis of means against restricted alternatives, 2 and E 2 can be used (see for example, Barlow et al., 1961). For ordered or monotone alternative max t can be used (see IJASETR (ISSN: 1839-7239) | June 2012 | Vol. 1 | Issue 3
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Hirotsu and Srivastava, 2000). In this paper, we propose weighted mixture of F -type test for testing homogeneity of treatment means against restricted alternatives for experimental design model. This paper is organized into the following sections. Section 2 introduces Likelihood-based weighted mixture of F for ordered and restricted alternatives. In Section 3 briefly discuss the procedure of computing weights. Section 4 represents the simulation studies. Finally, section 5 contains some concluding remarks.
2. Proposed weighted mixture of F -type test for restricted alternatives In order to test the homogeneity of means for experimental design model, as defined in (1.1) with disturbance term ij , the hypotheses are as follows:
H 01 : i 0 versus H a1 : i 0 ,
(2.1)
H02 : 1 2 ... k versus H a 2 : 1 2 ... k
(2.2)
For (2.1), to get the maximum likelihood estimate for the parameter of one-way ANOVA model, we need to optimize objective function
1 k 1 ni min 2 ( yij i ) 2 , i 2 i 1 i j 1 subject to the restriction
i 0 . For (2.2), to get the maximum likelihood estimate for the parameter of one-way ANOVA model, we need to optimize objective function
1 k 1 ni min 2 ( yij i ) 2 , i 2 i 1 i j 1 subject to the constraint
i 1 i i 1 . Now, the test statistic isk
F
2 ni ( i ) ( k 1)
i 1
k 2 k 2 ( yij ) ni ( i ) ( ni k ) i i 1 1 j 1 i 1 k ni
,
where i the optimum value of the treatment means with weight wi which optimize the likelihood function. The simplification of the proposed test statistic is
F
ni ( i )2 i 1 ˆ 2 k
,
which is distributed as weighted mixture of F distribution, namely IJASETR (ISSN: 1839-7239) | June 2012 | Vol. 1 | Issue 3
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p
P( F c) wi ( p, p k , 2 I ) P( Fk , N k C ) . k 1
where, Fk , N k is the F distribution with
(k , N k )
degrees of freedom and wi are non negative weights such
that w1 w2 ... wq 1 and C is a polyhedron.
3. Computing the weights Computation of the weights has been a subject of extensive research for many years. Closed-form solutions for the weights are available for less than 7 parameters. Shapiro (1985) provided alternative closed form equation for this weights for the case in which P 4 . Siskind (1976) computed a Taylor expansion of null distribution of the test statistic for Bartholomew’s (1959a, b) hypothesis test for cases in which P 4 , and therefore avoided the numerical methods necessary to compute the weights and critical values for testing procedure. Unfortunately, his technique is not straight forward to apply to general problems and it only provides to the cases in which P 7 . A final methodology for computing these weights for the cases P 8 is to use Monte Carlo technique. Wolak (1987), Shapiro (1988) discussed the procedure for computing weights. However there has been hardly any research on practical approaches to computing the weights for more general case. In this section, we describe a computational procedure for such cases. Suppose that C is a polyhedral. Let Y be distributed as N (0, 2 I ) , and (Y ) be the projection of Y onto C .
Now, Shapiro (1985) discuss about wi ( p, p k , 2 I ) where wi ( p, p k , 2 I ) is the probability that ( Z ) lies on a
i dimensional face of C i.e., wi ( p, p k , 2 I ) P[ (Y ) lies on a i dimensional face C] . Based on this we can develop the following simulation algorithm: (1) Generate an observation Y from N (0, 2 I ) and compute its projection (Y ) on C . (2) Compute dim( ) , the dimension of the linear space generated by where denotes the face of
C on which (Y ) lies. Computation of dimension depends on the nature of the constraint defining C . (3) Repeat the forgoing two steps a large number of times and (4) An estimate of wi ( p, p k , 2 I ) is the proportions of times dim( ) turned out to be i .
4. Simulation Studies In this study, Monte Carlo simulations are carried out to compare the power of the usual F test and the newly proposed modified F test for testing the homogeneity of means for restricted alternatives. We consider one-way fixed effect analysis of variance model
yij i ij , j 1(1)n , i 1(1)k IJASETR (ISSN: 1839-7239) | June 2012 | Vol. 1 | Issue 3
(4.1)
38
where ij
N (0, 2 ) are independent random variables and i are the mean of the ith treatment where i 1(1)k , k=2
or 3 and 2 is nuisance parameter. In order to carry out Monte Carlo simulation, we generate random error term ij
N (0, 2 ) and consequently we
generate the model (4.1). We perform at least 5000 replications to calculate simulated powers of the new and usual tests. The estimated simulated powers of the two tests are presented in Table 1 to 2.
F and weighted F - type test for hypothesis H 01 versus H a1 when k=2,
Table 1 Simulated power comparison of usual mixture of
F and weighted F - type test for hypothesis H 01 versus H a1 when k=3,
Table 2 Simulated power comparison of usual mixture of
n 20 and 2 20 .
n 20 and 2 16 .
1
0
0.5
0.7
2
3
F
F
0
0.05
0.05
1.5
0.18
0.25
2.5
0.42
0.51
3.5
0.73
0.84
4.5
0.93
0.96
5.5
0.97
0.99
0.05
0.5
0.05
0.05
0.08
0.1
2.5
0.25
0.33
2
0.27
0.31
3.5
0.5
0.64
3
0.52
0.55
4.5
0.82
0.88
4
0.76
0.8
5.5
0.96
0.97
5
0.93
0.94
6
0.96
0.98
2.5
0.11
0.2
7
1
1
3.5
0.32
0.35
0.7
0.05
0.05
4.5
0.62
0.68
2
0.22
0.25
5.5
0.85
0.9
3
0.46
0.51
4
0.73
0.76
5
0.91
0.93
2
F
F
0
0.05
0.05
1
0.16
0.21
2
0.38
0.48
3
0.61
0.77
4
0.85
0.94
5
0.95
0.97
6
0.99
1
0.5
0.05
1
1
0
0.5
1
0
0.5
2
We observe that the power of modified F test is higher than F test in all cases. For example, the power of the F test and F test are 0.51 and 0.46 respectively, for 1 0.7 , 2 3 , k 2 and n1 n2 n 20 .
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1
Power
Power
1.2
1.2 1 0.8 0.6 0.4 0.2 0
0.8 0.6 0.4 0.2
0
2
4
6
0
8
0
2
Treatm ent m ean F
F
16 .
2
1.5
6
1
5 4
0.5 0 0
2
mod F*
fixed 1 0 and 2 0 and 20 .
Power
Power
fixed 1 0 and
6
Treatm ent m ean
mod F* 2
4
4
6
Treatment mean
3 2 1 0 0
1
2
3
4
Treatm ent m ean
F
F* F
fixed 1 0 and 2 1 and
Figure 1: Power curves of mod
2
20 .
mod F*
fixed 1 1 and 2 3 and
2
20 .
F and F tests for hypothesis H 01 versus H a1 , for n=20, k=2 and 3.
If there are three treatment means in one-way ANOVA model, we observe from the figures and tables that the simulated power of the F test is higher than the F test in all cases. For example, the simulated powers of F and F tests are 0.84 and 0.73 respectively, for 1 0, 2 0, 3 3.5 , k 3 and n1 n2 n3 n 20 .
5. Conclusions In this paper, we develop a procedure for weighted mixture of F - type test for restricted alternatives. Testing the homogeneity of means for one-way ANOVA model, we observe that our proposed test gives higher power than usual
F test for strictly one-sided alternatives.
References [1] [2] [3] [4] [5]
Barlow, R.E., Bartholomew, D.J., Bremner, J.N., and Brunk, K.D. (1972). Statistical inference under order restriction, John Wiley, New York. Bartholomew, D. J. (1959a). A test of homogeneity for ordered alternatives, Biometrika, 46, 36-48. Bartholomew, D. J. (1959b). A test of homogeneity for ordered alternatives II, Biometrika, 46, 328-35. Bartholomew, D.J. (1961). A Test of Homogeneity of Means under Restricted Alternatives, Journal of the Royal Statistical Society, B, 23, 239-281. Hirotsu,C. and Srivastava, M.S.(2000). Simultaneous Confidence Intervals Based on One-Sided max t Test, Statistics and Probability Letters, 49, 25-37.
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[6] [7] [8] [9]
Shapiro, A. (1985). Asymptotic distribution of test statistics in the analysis of moment structures under inequality constraints, Biometrika, 72,133-144. Shapiro, A. (1988). Towards a unified theory of inequality constrained testing in multivariate analysis, International Statistical Review, 56, 49-62. Silvapulle, M.J., and Silvapulle, P. (1995). A score test against one-sided alternatives, Journal of The American Statistical Association, 90, 342-349. Wolak, F.A.(1987). An Exact Test for Multiple Inequality and Equality constraints in the Linear Regression Model,Journal of the American Statistical Association,74, 365-367.
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