A New Test for Simple Tree Alternative in a 2 x k Table

Parthasarathi Chakrabarti; Uttam Bandyopadhyay

doi:10.2991/jsta.2018.17.2.7

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Volume 17, Issue 2, June 2018, Pages 271 - 282

A New Test for Simple Tree Alternative in a 2 x k Table

Authors

Parthasarathi Chakrabartipartho.stat@gmail.com

Department of Statistics, R.K.M. Residential College, Narendrapur, Kolkata 700 103, India

Uttam Bandyopadhyay^*^,ubandyopadhyay08@gmail.com

Department of Statistics, University of Calcutta, 35 Ballygunge Circular Road, Kolkata 700 019, India

^*corresponding author.

Corresponding Author

Uttam Bandyopadhyayubandyopadhyay08@gmail.com

Received 20 October 2016, Accepted 31 May 2017, Available Online 30 June 2018.

DOI: 10.2991/jsta.2018.17.2.7 How to use a DOI?
Keywords: order restriction; simple tree; empirical size; empirical power; bootstrap
Abstract: This paper considers simple tree order restriction in 2×k cohort study and provides a consistent test in which the usual multiple comparison test statistics are modified by using the characteristic roots of a consistent estimator of the associated correlation matrix. The relevant performance measures of the proposed test are obtained and are compared numerically with existing competitors via simulation. It is shown that the proposed test is comparable to or better than the competitors in terms of type I error rate and power. Finally, data study illustrates the use of such a test.
Copyright: Copyright © 2018, the Authors. Published by Atlantis Press.
Open Access: This is an open access article under the CC BY-NC license (http://creativecommons.org/licences/by-nc/4.0/).

1. Introduction

Testing the equality of multiple mortality rates from different exposure categories against an ordered alternative occurs frequently in epidemiological studies. For example, consider the cohort study by Gupta and Mehta (2000) in which the age adjusted mortality rates among women in Mumbai, India using mishri (roasted, powdered form of tobacco used to clean teeth) and betel nut are, respectively, 12.3 and 12.6 per 1000 per annum, whereas such rate for control group is 9.9. Hence, it would be reasonable to assume the simple tree restriction π₁ ≤ π₂,π₃, where π₁, π₂ and π₃ represent, respectively, the risks of dying among women for the control group, for those who use mishri and for those who chew betel nuts. In general, if H : π₁ = π₂ = ⋯ = π_k represents no restriction on mortality rates for k exposure categories, H can be tested against the patterned alternative H_st − H, where H_st : π₁ ≤ π₂,π₃,…π_k.

Several tests are available in the literature for testing H against H_st − H. These are, for example, based on restricted maximum likelihood estimator (RMLE), multiple comparison procedures and non parametric kernels (see, for example, Fligner and Wolfe, 1982; Magel, 1988; Desu et al., 1996). While detecting order restrictions on binomial probabilities based on a 2 × k cohort study, multi-nomial allocation probabilities corresponding to the exposure levels play an important role. The existing tests to detect simple tree order restriction in a 2 × k table, where allocation probabilities are unbalanced, occasionally fail to attain the nominal level for small values of π₁. Our aim is to propose a multiple comparison consistent test using the characteristic roots of a consistent estimator of the associated correlation matrix based on the multinomial allocation probabilities, in which this short fall has been overcome.

Among the RMLE based approaches, the work on confidence interval estimation subject to order restriction (Hwang and Peddada, 1994) is based on modified generalised isotonic regression estimator (MGIRE). A number of testing procedures are obtained following MGIRE (see, for example, Peddada et al., 2001; Peddada and Haseman, 2006; Teoh et al., 2008). In this paper we choose an MGIRE based test as competitor and is referred to as the MGIRE test. Other RMLE based procedures to detect simple tree alternative are, for example, due to Wright and Tran (1985), Conaway et al. (1991), Singh et al. (1993), Futschik and Pflug (1998), Tsai (2004). Multiple comparison procedure (Bretz et al., 2001, 2003; Genz, 2004; Schaarschmidt et al., 2008; Hothorn et al., 2009), based on normal and binary responses, is proposed as a method in which the cut off points of the related tests are obtained from the distribution functions of multivariate normal and multivariate t distributions and are provided numerically through the R-packages mnormt and mvtnorm. In our setting we also choose one of such tests under binary response as another competitor and call the corresponding test as the GBH (Genz-Bretz-Hothorn) test. Besides these multiple comparison tests some single contrast tests are available to detect order restriction among binomial probabilities (see, for example, Leuraud and Benichou, 2001, 2004; Bretz and Hothorn, 2003; Bandyopadhyay and Chakrabarti, 2013 and the references there in). Our numerical computation shows that for small sample size the MGIRE and GBH tests often fail to attain the nominal level under unbalanced allocation as compared to that under balanced allocation. The proposed test overcomes such shortfall and increases its power locally.

The outline of the paper is as follows. Section 2 provides the data layout and notations. Section 3 contains some asymptotics and formulation of the proposed test. Section 4 describes competitors of the proposed test. Simulation results on size and power of the tests are given in Section 5. Section 6 contains data study. The paper concludes with some discussions in Section 7, followed by some technical details in Appendices A and B.

2. Data layout and notations

Consider a cohort study on n individuals, where the dichotomous response variable Y, indicating survival status, is recorded for the exposure X consisting of the levels x₁,x₂,…,x_k, measured in a nominal scale, satisfying x₁ ≲ x₂,x₃,…,x_k. Let p_j = P(X = x_j) > 0, the chance of occurrence of the exposure level x_j, j = 1,2,…,k with ∑j=1kpj=1, and π_j = P(Y = 1|X = x_j) = 1−P(Y = 0|X = x_j), the mortality rate at x_j, j = 1,2,…,k. Define n_j = #(X = x_j) as the number of individuals observed at x_j and s_j = #(Y = 1|X = x_j) as the disease count at x_j, j = 1,2,…,k, where n=∑j=1knj.

Let us write n^T = (n₁,n₂,…,n_k), p^T = (p₁, p₂,…,p_k) and π^T = (π₁,π₂,…,π_k). Evidently, the distribution of n is multinomial on k categories with index n and parameter p. Further (s₁,s₂,…,s_k), conditioning on n, constitutes k–independent binomial random variables, where s_j follows binomial distribution with index n_j and parameter π_j, j = 1,2,…,k. In order to understand the simple tree order of the mortality rates at different exposure levels, H is tested against H_st − H.

In the subsequent discussions, pˆj and πˆj are used to denote, respectively, the observed proportions of individuals and successes at x_j, where pˆj=nj/n and πˆj=sj/nj. Then the overall proportion of success is obtained by πˆ=1n∑j=1knjπˆj=pˆTπˆ, where pˆT=(pˆ1,pˆ2,…,pˆk) and πˆT=(πˆ1,πˆ2,…,πˆk). If n_j vanishes for some j = 1,2,…,k, dirichlet prior is used to choose pˆj=nj+1/kn+1, j = 1,2,…,k. Similarly, if πˆ is found to be 0 or 1 for a specific sample, we choose πˆ=∑j=1knjπˆj+1/2n+1 by use of beta prior.

3. Proposed test and related asymptotic results

A naive test, analogous to Dunnett’s procedure (1955), can be constructed through Bonferroni’s correction in which H is rejected at level α against H_st − H if and only if

T=max {tj=n(πˆj−πˆ1)πˆ(1−πˆ) (1pˆ1+1pˆj),j=2,3,…,k}

exceeds τ_α/(k−1), where τ_α is the (1 − α)^th quantile of standard normal distribution, 0 < α < 1. Such a test is referred to as the T-test. In this paper a modification of the T-test is proposed by standardizing t = (t₂,t₃,…,t_k)^T through the estimators of the characteristic roots of the correlation matrix of t. Towards such modification, H is expressed in terms of multiple contrasts of π by

H:Cπ=0,

where C^(k−1)×k = (−1_k−1 e₁ e₂ … e_k−1) with e_j, j = 1,2,…,k − 1 as (k − 1) component independent unit vectors and 1k−1=∑j=1k−1ej. Then, H is rejected against H_st −H if and only if H_j is rejected against Hja for at least one j, where H_j : π₁ = π_j and Hja:π1<πj, j = 2,3,…,k. Furthermore, an upper tail test based on t_j is appropriate for the testing problem (Hj,Hja), j = 2,3,…,k. Hence, combining all such component tests, the resulting test becomes the T-test. Now, we consider the proposed modification.

Modifying T:

It is not difficult to see that, for 0 < p_j < 1, j = 1,2,…,k, as n → ∞,

{n(πˆj−πj)πˆj(1−πˆj)pˆj, j=1,2,…,k}

converges in distribution to N_k(0,I), the k-variate normal distribution with mean vector 0 and unit dispersion matrix I. That means, for large n,

n(πˆ−π)~aNk(0,Σ(pˆ)),

where

Σ(pˆ)=Diag (π1(1−π1)pˆ1,π2(1−π2)pˆ2,…,πk(1−πk)pˆk) ,

which under H reduces to π(1−π)Diag(pˆ1−1,pˆ2−1,…,pˆk−1), and the notation “un~avn” is used to mean that asymptotic distributions of the random variables u_n and v_n are same. Therefore, the asymptotic distribution of t, shown in Appendix A, is N_k−1 (0,R(p)), under H, where R(p) is the correlation matrix with elements

rij(p)=1if 1≤i=j≤k−1,=pi+1pj+1(p1+pi+1)(p1+pj+1)if 1≤i≠j≤k−1.

Let λ_j = λ_j(p) > 0, j = 1,2,…,k − 1 be the characteristic roots of R(p) and w_j be the unit norm characteristic vector corresponding to λ_j, j = 1,2,…,k − 1. Then, setting W = (w₁ w₂ … w_k−1), it follows that

WTR(p)W=Λ(p)=Diag(λ1,λ2,…,λk−1).

Hence, there exists a positive definite matrix R^1/2(p) for which, as n → ∞,

R−1/2(p)t→Nk−1(0,I)

in distribution under H, where

R−1/2(p)=WΛ−1/2(p)WT, Λ−1/2(p)=Diag (1λ1,1λ2,…,1λk−1).

Since, R−1/2(pˆ)→R−1/2(p) in probability, we get, as n → ∞,

(1)tm=(t2m,t3m,…,tkm)T=R−1/2(pˆ)t→Nk−1(0,I)

in distribution under H, and hence T can be modified by

(2)Tm=max {t2m,t3m,…,tkm}.

As usual, an upper tail test based on T_m would be appropriate. Such a test can be described by the critical region

(3)w : Tm>Tmα,

where, for given α : 0 < α < 1, T_mα is obtained approximately from the relation

limn→∞PH{Tm≤Tmα}=1−α,

which, by use of (1) and 2), yields the approximate relation

(4){Φ(Tmα)}k−1=1−α

with Φ(.) as the distribution function of standard normal variable. Thus the test (referred to as the T_m-test), given by (3) and (4), is asymptotically level α test for the testing problem (H,H_st − H) and is a modification of the T-test. It is shown (see Appendix B) that the test is consistent.

4. Competitors

MGIRE test

Here components of π are estimated ( subject to a general order restriction) by

π˜1=min {∑j=1lpˆjπˆj∑j=1lpˆj,l=1,2,…,k}

and

π˜j=max {π˜1,πˆj}, j=2,3,…,k.

Then, incorporating Bonferroni’s corrections, the test, described by the critical region

TMGIRE=max {n(π˜j−π˜1)πˆ(1−πˆ)(1pˆj+1pˆ1), j=2,3,…,k}>τα/(k−1),

is asymptotically level α test for the testing problem (H,H_st − H) and is used as a competitor of the proposed tests. From Appendix B, it is not difficult to see that under any π,

1nTMGIRE→max {θj,j=1,2,…,k−1}

in probability, which is ‘zero’ or positive according as π ∈ H or π ∈ H_st − H. This implies that the MGIRE test is consistent for testing H against π ∈ H_st − H.

GBH test

Here H is rejected at level α against H_st − H if and only if

TGBH=max {tj=n(πˆj−πˆ1)πˆ(1−πˆ)(1pˆ1+1pˆj), j=2,3,…,k}

exceeds the 100(1 − α) equi-percentage point, ck−1,R(pˆ),1−α, of Nk−1(0,R(pˆ)), the approximated null distribution of t. The consistency of this test for testing H against π ∈ H_st − H can be established by the same technique as in the previous test.

5. Simulation study

We perform a simulation study with hundred thousand replications taking k = 3 and, for the purpose of illustration, the nominal level (α) is chosen at 0.05. The proposed test and the competitors are compared with respect to both empirical type I error rate and empirical power. Empirical type I error rate (power) of a test is computed by that proportion of hundred thousand replications of the experiment under H (H − H_st), in which the test statistic exceeds the 0.95^th quantile of its asymptotic null distribution.

For a 2 × k cohort data, setting p=pˆ and the common success probability under H at π=πˆ, 100,000 tables, similar to the data, are generated. If the bootstrap percentile points of the simulated null distributions of the statistics agree with the percentile points of the asymptotic null distributions of the respective statistics, P-values of the tests are obtained using the approximate null distributions, otherwise P-values are determined by bootstrapping (See, for example, Efron and Tibshirani (1993), Noreen (1989) and Romano (1988, 1989) for details) in which the proportion of cases the test statistics, evaluated from all such 100,000 tables, exceed the respective observed values obtained from the data set.

Similarly, if the empirical type I error rates do not agree with the nominal level, the powers of the corresponding test are evaluated using empirical cut-off point (0.95^th quantile of the simulated null distribution of the test statistic) instead of the approximate cut-off point.

The simulation study is performed for different choices of n and p. For illustration, we choose n = 100,200,300,400 and 500 for both balanced (p₁ = p₂ = p₃) and unbalanced situations. As most of the cohort studies indicate highly unbalanced situations , we take p = (0.9,0.05,0.05) (more allocation towards control) and p = (0.1,0.45,0.45) (less allocation towards control) for the present computation. For balanced allocation ρ = r₁₂(p) is equal to 0.5 and for p = (0.9,0.05,0.05), (0.1,0.45,0.45) ρ is, respectively, equal to 0.053 and 0.818. π is chosen from {0.1,0.3,0.5} in order to ensure the conformity of the type I error rates to the nominal level. The empirical powers of the tests are obtained under the following cases of the parametric configurations:

Case A: π lying in the boundary of the alternative region, such as: π₁ = π₃ < π₂.
Case B: π is well within the alternative region, such as: (B1) π₁ < π₂ = π₃, (B2) π₁ < π₃ < π₂.

For revealing the behaviour of the tests under Case A, we choose π = (0.1,0.2,0.1), and that under Case B, we choose π = (0.1,0.2,0.2) and (0.1,0.3,0.2) for (B1) and (B2), respectively.

Simplification: k = 3.

Setting ρˆ=pˆ2pˆ3(pˆ1+pˆ2)(pˆ1+pˆ3), we get R(pˆ)=(1ρˆρˆ1), which givesΛ(pˆ)=(1−ρˆ001+ρˆ) and W=(1/21/2−1/21/2) ,

and hence

R−1/2(pˆ)=12(11+ρˆ+11−ρˆ11+ρˆ−11−ρˆ11+ρˆ−11−ρˆ11+ρˆ+11−ρˆ).

Consequently, T_m becomes

Tm=max {a(ρˆ)t2+b(ρˆ)t3,b(ρˆ)t2+a(ρˆ)t3},

where

a(ρˆ)=12(11+ρˆ+11−ρˆ) and b(ρˆ)=12(11+ρˆ−11−ρˆ).

Result:

Computation of Type I error rate

In Table 1, the entries, showing maximum departure of the type I error rates from the nominal level (more than 10 %departure from the nominal level) for different choices of π and n, are marked in bold faces. The table shows that under balanced allocation and unbalanced allocation probabilities (0.9,0.05,0.05) the T_m test and its competitors, except one exception, have similar behaviour. Again, unlike the T_m test, type I error rates of the MGIRE test do not agree with the nominal level under the allocation probabilities (0.1,0.45,0.45). However, in this situation, the GBH test maintains the nominal level except for small values of π. The more the increase in ρ, more is the deviation of the type I error rates for the MGIRE and GBH tests from the nominal level.

π	n	T_m	MGIRE	GBH	T_m	MGIRE	GBH	T_m	MGIRE	GBH
π	n	p = (1/3,1/3,1/3)			p = (0.09,0.05,0.05)			p = (0.1,0.45,0.45)
0.1	100	0.058	0.052	0.053	0.099	0.099	0.098	0.039	0.004	0.009
	200	0.055	0.050	0.052	0.083	0.083	0.083	0.404	0.014	0.026
	300	0.055	0.052	0.053	0.082	0.082	0.081	0.045	0.023	0.036
	400	0.053	0.050	0.050	0.077	0.077	0.076	0.045	0.029	0.040
	500	0.054	0.051	0.051	0.073	0.074	0.073	0.048	0.028	0.041

0.3	100	0.053	0.049	0.051	0.070	0.069	0.070	0.050	0.029	0.043
	200	0.050	0.049	0.051	0.063	0.064	0.063	0.047	0.033	0.043
	300	0.052	0.052	0.053	0.060	0.061	0.060	0.047	0.033	0.044
	400	0.049	0.049	0.050	0.061	0.061	0.061	0.050	0.034	0.046
	500	0.049	0.047	0.048	0.059	0.060	0.060	0.048	0.033	0.046

0.5	100	0.052	0.049	0.050	0.042	0.042	0.043	0.053	0.040	0.053
	200	0.049	0.048	0.049	0.048	0.048	0.048	0.050	0.039	0.051
	300	0.050	0.050	0.051	0.048	0.049	0.048	0.049	0.038	0.050
	400	0.049	0.046	0.047	0.052	0.052	0.052	0.052	0.040	0.051
	500	0.051	0.048	0.050	0.048	0.049	0.048	0.051	0.037	0.049

Table 1.

Empirical type I error rate: T_m, MGIRE and GBH tests (α= 0.05).

Computation of empirical power

Table 2 and Table 3 show, respectively, the empirical powers of the tests under Case A and Case B. For each of the given choices of p, π and n maximum powers are marked in bold faces.

Case A:

Table 2 shows the empirical powers of all the tests for the given choices of π lying in a boundary of parametric space under H_st − H. For the given choices of n, the T_m test is found to be more powerful than the MGIRE and GBH tests under both balanced and unbalanced allocation probabilities. Based on this empirical power comparison, an approximate ordering of the tests is T_m, GBH, MGIRE, in which the T_m-test is the best in terms of having maximum empirical power.
Case B: Table 3 shows numerical computations of empirical power under both Case B1 and Case B2. Here, under Case B1, the GBH test is found to be be more powerful than the MGIRE and T_m tests under both balanced and unbalanced allocation probabilities. Based on this empirical power comparison, like Case A, an approximate ordering of the tests is GBH, MGIRE, T_m. Under Case B2 for balanced allocation probabilities and allocation probabilities (0.9,0.05,0.05) ordering of the tests with respect to empirical powers remains unaltered with an insignificant variation among the empirical powers. Under allocation probabilities (0.1,0.45,0.45) corresponding to Case B2 , for n ≥ 300, T_m test is found to be more powerful, whereas ordering of the tests remains same as in Case B1 for n = 100 and 200.

π	n	T_m	MGIRE	GBH	T_m	MGIRE	GBH	T_m	MGIRE	GBH
π	n	p = (1/3,1/3,1/3)			p = (0.09,0.05,0.05)			p = (0.1,0.45,0.45)
(0.1,0.2,0.1)	100	0.261	0.230	0.240	0.116	0.115	0.116	0.280	0.240	0.243
	200	0.461	0.397	0.407	0.166	0.162	0.164	0.463	0.289	0.290
	300	0.623	0.549	0.561	0.292	0.290	0.292	0.620	0.384	0.384
	400	0.745	0.668	0.679	0.267	0.264	0.267	0.735	0.463	0.463
	500	0.832	0.764	0.772	0.307	0.306	0.307	0.813	0.535	0.537

Table 2.

Empirical power: T_m,MGIRE and GBH tests (α= 0.05, Case A).

π	n	T_m	MGIRE	GBH	T_m	MGIRE	GBH	T_m	MGIRE	GBH

		p = (1/3,1/3,1/3)			p = (0.09,0.05,0.05)			p = (0.1,0.45,0.45)
(0.1,0.2,0.2)	100	0.245	0.302	0.312	0.175	0.178	0.178	0.137	0.259	0.260
	200	0.422	0.507	0.521	0.265	0.268	0.270	0.206	0.311	0.312
	300	0.572	0.667	0.681	0.346	0.351	0.352	0.282	0.397	0.398
	400	0.683	0.775	0.788	0.404	0.409	0.411	0.347	0.492	0.493
	500	0.792	0.866	0.874	0.486	0.490	0.494	0.392	0.563	0.564

(0.1,0.3,0.2)	100	0.519	0.566	0.581	0.285	0.285	0.287	0.374	0.468	0.469
	200	0.810	0.847	0.854	0.444	0.448	0.450	0.629	0.634	0.635
	300	0.939	0.957	0.960	0.577	0.580	0.582	0.803	0.777	0.779
	400	0.980	0.989	0.990	0.683	0.688	0.690	0.903	0.879	0.880
	500	0.995	0.997	0.998	0.759	0.763	0.765	0.951	0.905	0.926

Table 3.

Empirical power: T_m,MGIRE and GBH tests (α = 0.05, Case B).

6. Data study

Example 1:

The data, given in Table 4, are extracted from the cohort study (Gupta and Mehta, 2000) on the risk of mortality among tobacco users in Mumbai, India,

category	frequency	pˆ	mortality risk ( πˆ)
Control	64414	0.5225	0.0099
Mishri	56515	0.4585	0.0123
Betel nut	2343	0.0190	0.0126
total	123272	1	-

Table 4.

Mortality risk by use of mishri and betel nut among women.

where users are classified gender-wise into smoking groups (smoking cigarette and bidi (tobacco hand rolled in temburni leaf and flaked)) and consuming smokeless tobacco (mishri, betel quid, betel nut, etc). Table 4 shows the risk of mortality among women in Mumbai from the use of mishri and betel nut. Here, the P-values of all the tests, proposed and competitors, are obtained by bootstrapping. In addition the bootstrap 0.95^th percentile points of the simulated null distributions of such test statistics are obtained at various sample sizes. Furthermore, setting p=pˆ and π=πˆ, another 100,000 tables are generated for those sample sizes. From each such tables the test statistics are computed. Finally, the powers of the tests are obtained as the proportions of cases in which such test statistics exceed the respective bootstrap percentile points. Estimated P-values and powers of the tests corresponding to Example 1 are given in Table 5.

P-value/Power	T_m	MGIRE	GBH
P-value	0.00020	0.00025	0.00025
Power
n = 123272	0.979	0.980	0.981
n = 50000	0.722	0.728	0.735
n = 25000	0.434	0.442	0.451
n = 10000	0.208	0.209	0.216
n = 5000	0.139	0.134	0.139
n = 1000	0.075	0.073	0.074

Table 5.

P-values and powers of the tests obtained by bootstrapping .

It is observed (Table 5) that all the tests, proposed and competitors, strongly reject the null hypothesis of no difference among the risks of mortality, where the T_m test has the least P-value. For different choices of n in Table 5 we see that empirical powers of the T_m, GBH and MGIRE tests are approximately equal. For n > 5,000, an approximate ordering of the tests with respect to empirical power is GBH, MGIRE, T_m, in which the GBH-test is the best in terms of having maximum power. However, for n ≤ 5,000, the ordering becomes T_m, GBH, MGIRE.

Example 2:

All the tests are applied to another data set (Graubard and Korn, 1987) relating to the effect of maternal alcoholism on congenital sex organ malformation among infants. The information on alcohol consumption is collected from would-be mothers after the first trimester and the malformations among infants are recorded following childbirth. Alcohol consumption categories are classified as average number of drinks per day. The data set is summarized in Table 6.

average number of drinks/day	frequency of mothers	pˆ	risk of malformation
< 1	31616	0.9706	0.0027
1 − 2	793	0.0243	0.0063
> 2	165	0.0051	0.0121
Total	32574	1	–

Table 6.

Risk of infant’s sex organ malformation for maternal alcoholism.

Adopting the similar technique, as used in Example 1, P-values and powers of the tests are determined and exhibited in Table 7.

P-value/Power	T_m	MGIRE	GBH
P-value	0.062	0.062	0.062
Power
n = 32574	0.629	0.631	0.633
n = 10000	0.323	0.323	0.323
n = 5000	0.264	0.228	0.228
n = 1000	0.130	0.130	0.130

Table 7.

P-values and powers of the tests obtained by resampling.

Table 6 shows that pˆ1 is almost unity and pˆ2 is significantly larger than pˆ3. Thus, the sample corresponds to an extremely unbalanced situation. Here, as the P-values suggest, all the T_m, GBH, MGIRE tests strongly reject the null hypothesis. The T_m test is found to be more powerful than its competitors for n ≤ 10,000.

7. Discussion

The failure of the type I error rate to attain the nominal level occurs more frequently in the MGIRE and GBH tests than in the T_m test under unbalanced allocation probabilities. On the boundary of the parameter space under H_st − H, that is, under Case A, the T_m test is found to be locally more powerful than its competitors. Power of the T_m test in this case becomes significantly more as compared to that of its competitors with the increase in the value of ρ. Thus, for unbalanced allocation probabilities yielding high values of r_ij(p)′s, the T_m test can be preferred for its agreement of type I error rate with the nominal level.

Appendix A

Asymptotic Distribution under H

Setting

M(pˆ)=Diag (pˆ1pˆ2pˆ1+pˆ2,pˆ1pˆ3pˆ1+pˆ3,…,pˆ1pˆkpˆ1+pˆk ) ,

it follows that, under H, for large n,

nM(pˆ)Cπˆ~aNk−1(0,ΣH(pˆ)),

where

ΣH(pˆ)=M(pˆ)CΣ(pˆ)CT MT(pˆ)=π(1−π)R(pˆ)

with R(pˆ) given by Section 3 when p is estimated by pˆ. Hence the statistic t is identified as

t=nπˆ(1−πˆ)M(pˆ)Cπˆ.

Now, using the fact that pˆ→p in probability, it follows that, under H,

t→Nk−1(0,R(p))

in distribution as n → ∞.

Appendix B

Consistency

First, we prove the following result.

Result B.1: Let A = (a₁a₂ ⋯ a_d) be a positive definite symmetric matrix, and α = (α₁,α₂,…,α_d)^T be a vector of non-negative elements with α ≠ 0. Then

max {ajTα,j=1,2,…,d}>0.

Proof:

Assume that the assertion is false. Then, by the given conditions, we have α^T Aα ≤ 0. But this is a contradiction as A is positive definite. Hence the result follows.

Next, writing θ = (θ₁,θ₂,…,θ_k−1)^T with

θj=(πj+1−πj)π(1−π)p1pj+1p1+pj+1,j=1, 2,…,k−1,

we can find a vector valued function g = (g₁,g₂,…,g_k−1)^T of θ such that as n → ∞,

1ntm=1nR−12(pˆ)t=1πˆ(1−πˆ)R−12(pˆ)M(pˆ)Cπˆ

converges to

g(θ)=R−12(p)θ

in probability for any π, where M(pˆ) is defined in Appendix A. It is obvious that θ = 0 when π ∈ H and θ_j ≥ 0, j = 1,2,…,k − 1 with θ ≠ 0 when π ∈ H_st − H. Furthermore, as n → ∞,

1nTm→μ(θ)=max {gj(θ), j=1,2,…,k−1}

in probability, where gj(θ)=ajTθ with a_j as the j^th column of the symmetric positive definite matrix R−12(p). Hence, by use of Result B.1, we get

μ(θ)=0 if π∈H>0 if π∈Hst−H.

This implies that the proposed test, described by (3), is consistent for testing H against any π under H_st − H.

References

[1]A Futschik and GC Pflug, The likelihood ratio test for simple tree order: A useful asymptotic expansion, J. Stat. Plan. Inference, Vol. 70, 1998, pp. 57-68.

[2]A Genz, Numerical computation of rectangular bivariate and trivariate normal and t-probabilities, Stat. Comput, Vol. 14, 2004, pp. 251-260.

[3]B Efron and R Tibshirani, An Introduction to the Bootstrap, Chapman & Hall, New York, 1993.

[4]BI Graubard and EL Korn, Choice of column scores for testing independence in ordered 2 x K contingency tables, Biometrics, Vol. 43, 1987, pp. 471-476.

[5]B Singh, MJ Schell, and FT Wright, The power functions of the likelihood ratio tests for a simple tree ordering in normal means: Unequal weights, Comm. Statist. Theory Methods A, Vol. 22, 1993, pp. 425-449.

[6]CW Dunnett, A multiple comparison procedure for comparing several treatments with a control, Canad. J. Statist, Vol. 50, 1955, pp. 1096-1121.

[7]E Noreen, Computer Intensive Methods for Testing Hypotheses: An Introduction, Wiley, New York, 1989.

[8]E Teoh, A Nyska, U Wormser, and SD Peddada, Statistical inference under order restrictions on both rows and columns of a matrix, with an application in toxicology, IMS Collections, Vol. 1, 2008, pp. 62-77. Beyond Parametrics in Interdisciplinary Research: Festschrift in Honor of Professor Pranab K. Sen

[9]F Bretz, A Genz, and LA Hothorn, On the numerical availability of multiple comparison procedures, Biom. J, Vol. 43, No. 5, 2001, pp. 645-656.

[10]F Bretz and LA Hothorn, Statistical analysis of monotone or non-monotone dose–response data from in vitro toxicological assays, ATLA-ALTERN LAB ANIM, Vol. 31, No. 1, 2003, pp. 81-96.

[11]F Schaarschmidt, M Sill, and LA Hothorn, Approximate simultaneous confidence intervals for multiple contrasts of binomial proportions, Biom, J, Vol. 50, No. 5, 2008, pp. 782-792.

[12]FT Wright and T Tran, Approximating the level probabilities in order restricted inference:The simple tree ordering, Biometrika, Vol. 71, No. 2, 1985, pp. 429-439.

[13]J Hwang and S Peddada, Confidence interval estimation subject to order restrictions, Ann. Stat, Vol. 22, 1994, pp. 67-93.

[14]J Romano, A bootstrap revival of some nonparametric distance tests, J. Amer. Statist. Assoc, Vol. 83, 1988, pp. 698-708.

[15]J Romano, Bootstrap and randomization tests of some nonparametric hypotheses, Ann. Stat, Vol. 17, 1989, pp. 141-159.

[16]K Leuraud and J Benichou, A comparison of several methods to test for the existence of a monotonic dose-response relationship in clinical and epidemiological studies, Stat. Med, Vol. 20, 2001, pp. 3335-3351.

[17]K Leuraud and J Benichou, Tests for monotonic trend from case-control data: Cochran-Armitage-Mantel trend test, isotonic regression and single and multiple contrast tests, Biom. J, Vol. 46, 2004, pp. 731-749.

[18]LA Hothorn, M Vaeth, and T Hothorn, Trend tests for the evaluation of exposure-response relationships in epidemiological exposure studies, Epidemiol Perspect Innov, Vol. 6, 2009, pp. 1.

[19]MA Fligner and DA Wolfe, Distribution-free tests for comparing several treatments with a control, Stat. Neerl, Vol. 36, 1982, pp. 119-127.

[20]M Conaway, C Pillers, T Robertson, and J Sconing, A circular-cone test for testing homogeneity against a simple tree order, Canad. J. Statist, Vol. 19, 1991, pp. 283-296.

[21]MM Desu, S Park, and S Chakraborti, Linear rank statistics for the simple tree alternatives, Biom. J, Vol. 38, No. 3, 1996, pp. 359-373.

[22]MT Tsai, Maximum likelihood estimation of covariance matrices under simple tree ordering, J. Multivar. Anal, Vol. 89, 2004, pp. 292-303.

[23]PC Gupta and HC Mehta, Cohort study of all-cause mortality among tobacco users in Mumbai, India, Bull. World Health Organ, Vol. 78, No. 7, 2000, pp. 877-883.

[24]R Magel, A test for the equality of k medians against the simple tree alternative under right censorship, Comm. Statist. Simul. Comput. B, Vol. 17, 1988, pp. 917-925.

[25]SD Peddada and JK Haseman, Tests for a simple tree order restriction with application to dose-response studies, Appl. Stat, Vol. 55, No. 4, 2006, pp. 493-506.

[26]S Peddada, K Prescott, and M Conaway, Tests for order restrictions in binary response, Biometrics, Vol. 57, 2001, pp. 1219-1227.

[27]U Bandyopadhyay and P Chakrabarti, Single contrast tests for detecting trends in binomial proportions, J. Korean Statist. Soc, Vol. 42, 2013, pp. 235-246.

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Journal: Journal of Statistical Theory and Applications
Volume-Issue: 17 - 2
Pages: 271 - 282
Publication Date: 2018/06/30
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TY  - JOUR
AU  - Parthasarathi Chakrabarti
AU  - Uttam Bandyopadhyay
PY  - 2018
DA  - 2018/06/30
TI  - A New Test for Simple Tree Alternative in a 2 x k Table
JO  - Journal of Statistical Theory and Applications
SP  - 271
EP  - 282
VL  - 17
IS  - 2
SN  - 2214-1766
UR  - https://doi.org/10.2991/jsta.2018.17.2.7
DO  - 10.2991/jsta.2018.17.2.7
ID  - Chakrabarti2018
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