A New Test for Simple Tree Alternative in a 2 x k Table
 DOI
 10.2991/jsta.2018.17.2.7How 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 BYNC license (http://creativecommons.org/licences/bync/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 π_{1} ≤ π_{2},π_{3}, where π_{1}, π_{2} and π_{3} 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 : π_{1} = π_{2} = ⋯ = 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} : π_{1} ≤ π_{2},π_{3},…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, multinomial 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 π_{1}. 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 Rpackages 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 (GenzBretzHothorn) 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 x_{1},x_{2},…,x_{k}, measured in a nominal scale, satisfying x_{1} ≲ x_{2},x_{3},…,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
Let us write n^{T} = (n_{1},n_{2},…,n_{k}), p^{T} = (p_{1}, p_{2},…,p_{k}) and π^{T} = (π_{1},π_{2},…,π_{k}). Evidently, the distribution of n is multinomial on k categories with index n and parameter p. Further (s_{1},s_{2},…,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,
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
Modifying T:
It is not difficult to see that, for 0 < p_{j} < 1, j = 1,2,…,k, as n → ∞,
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 = (w_{1} w_{2} … w_{k−1}), it follows that
Hence, there exists a positive definite matrix R^{1/2}(p) for which, as n → ∞,
Since,
As usual, an upper tail test based on T_{m} would be appropriate. Such a test can be described by the critical region
4. Competitors
MGIRE test
Here components of π are estimated ( subject to a general order restriction) by
Then, incorporating Bonferroni’s corrections, the test, described by the critical region
GBH test
Here H is rejected at level α against H_{st} − H if and only if
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
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 cutoff point (0.95^{th} quantile of the simulated null distribution of the test statistic) instead of the approximate cutoff 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 (p_{1} = p_{2} = p_{3}) 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 = r_{12}(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: π_{1} = π_{3} < π_{2}.
Case B: π is well within the alternative region, such as: (B1) π_{1} < π_{2} = π_{3}, (B2) π_{1} < π_{3} < π_{2}.
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.
Consequently, T_{m} becomes
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 

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 
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 

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 
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 
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 

mortality risk (


Control  64414  0.5225  0.0099 
Mishri  56515  0.4585  0.0123 
Betel nut  2343  0.0190  0.0126 
total  123272  1   
Mortality risk by use of mishri and betel nut among women.
Pvalue/Power  T_{m}  MGIRE  GBH 

Pvalue  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 
Pvalues 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 Pvalue. 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 GBHtest 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 wouldbe 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 

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  – 
Risk of infant’s sex organ malformation for maternal alcoholism.
Adopting the similar technique, as used in Example 1, Pvalues and powers of the tests are determined and exhibited in Table 7.
Pvalue/Power  T_{m}  MGIRE  GBH 

Pvalue  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 
Pvalues and powers of the tests obtained by resampling.
Table 6 shows that
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
Now, using the fact that
Appendix B
Consistency
First, we prove the following result.
Result B.1: Let A = (a_{1}a_{2} ⋯ a_{d}) be a positive definite symmetric matrix, and α = (α_{1},α_{2},…,α_{d})^{T} be a vector of nonnegative elements with α ≠ 0. Then
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 = (θ_{1},θ_{2},…,θ_{k−1})^{T} with
This implies that the proposed test, described by (3), is consistent for testing H against any π under H_{st} − H.
References
Cite this article
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  22141766 UR  https://doi.org/10.2991/jsta.2018.17.2.7 DO  10.2991/jsta.2018.17.2.7 ID  Chakrabarti2018 ER 