Statistical tests under Dallal's model: Asymptotic and exact methods.

This paper proposes asymptotic and exact methods for testing the equality of correlations for multiple bilateral data under Dallal's model. Three asymptotic test statistics are derived for large samples. Since they are not applicable to small data, several conditional and unconditional exact methods are proposed based on these three statistics. Numerical studies are conducted to compare all these methods with regard to type I error rates (TIEs) and powers. The results show that the asymptotic score test is the most robust, and two exact tests have satisfactory TIEs and powers. Some real examples are provided to illustrate the effectiveness of these tests.

Introduction

In clinical medicine, we often encounter bilateral data taken from paired organs of patients such as eyes and ears. For the same patient, the intraclass correlation between responses of paired parts should be considered to avoid misleading results. There have been in the past various models to analyze such data. For example, Rosner [1] introduced a positive constant R as a measure of the dependency by assuming that the probability of a response at one side of the paired body given a response at the other side is R times to the response rate. Donner [2] provided an alternative approach and considered the common correlation coefficient in each of two groups. Under these two models, asymptotic and exact methods have been studied for many years and achieved significant progress.

Under Rosner’s model, Tang et al. [3] developed exact and approximate procedures when sample size is small or the data structure is sparse. Qiu et al. [4] derived sample formulas for testing difference between two proportions. Shan and Ma [5], and Ma et al. [6] presented several asymptotic and exact methods to investigate the equality of proportions. Peng et al. [7] constructed asymptotic confidence intervals (CIs) of proportion ratio for correlated paired data. Under Donner’s model, Pei et al. [8, 9] used asymptotic methods to analyze test statistics and CIs in two treated groups. Liu et al. [10, 11] provided exact methods to test the homogeneity of prevalence from multiple groups. Generally, asymptotic methods can produce empirical type I error rates (TIEs) close to the pre-specified nominal level for large samples. However, they may yield inflation TIEs for small samples. Thus, exact tests become alternative to deal with the problem.

Dallal [12] indicated that Rosner’s model may give a poor fit if the characteristic was almost certain to occur bilaterally with widely varying group-specific prevalence. Suppose the probability of response at one organ given response at the other organ was independent of its probability. He introduced likelihood ratio test for large samples. However, the approach performs poorly with unsatisfactory TIE control in small samples. Up to now, statistical inferences on Dallal’s model have been less considered, including asymptotic and exact methods. This paper aims to propose asymptotic and exact methods for testing homogeneity of correlations among multiple bilateral data under Dallal’s model.

The remainder of the work is organized as follows. In Section 2, we review bilateral data structure and introduce Dallal’s model. The maximum likelihood estimations (MLEs) are derived for different hypotheses. Three asymptotic statistics and six exact procedures are proposed in Section 3. In Section 4, some numerical studies are conducted to compare these methods in terms of TIEs and power. Two examples are provided to illustrate these proposed approaches in Section 5. Conclusions are given in Section 6.

Dallal’s model and estimators

Suppose that N patients is randomly allocated into g groups. There are m patients in the ith (i = 1, …, g) group. Let m be the number of patients who have l(l = 0, 1, 2) organ(s) with improvement response(s) in the ith (i = 1, …, g) group, and S be the total number of patients with l(l = 0, 1, 2) response(s). Obviously, and . The data structure is shown in Table 1. Let p be the probability that a patient has l(l = 0, 1, 2) response(s) in the ith (i = 1, …, g) group. The vector m ≜ (m0, m1, m2) follows a multinomial distribution M(m;p0, p1, p2). The probability density satisfies

Table 1

Bilateral data structure with g groups.

Response (l)	Group						Total
	1	2	…	i	…	g
0	m01	m02	…	m0i	…	m0g	S0
1	m11	m12	…	m1i	…	m1g	S1
2	m21	m22	…	m2i	…	m2g	S2
Total	m1	m2	…	mi	…	mg	N

Let Z = 1 if the kth organ of the jth patient has improvement response in the ith group for k = 1, 2, i = 1, 2, …, g, and j = 1, 2, …, m, and 0 otherwise. Under Dallal’s model, we assume where 0 ≤ π, γ ≤ 1. Especially, γ = π means that two organ responses of each patient are completely independent, and γ = 1 represents that they are completely dependent in ith group. By using the Eq (1), the probabilities of no, one or both response(s) are where 0 ≤ p ≤ 1, p01 + p1 + p2 = 1, and . In the work, we are interested to test whether the correlations of g groups are identical. Thus, the hypotheses are given by

Denote m = (m1, …, m), = (π1, …, π) and = (γ1, …, γ). Given the observation m, the log-likelihood function where . Let and be the unconstrained MLEs of π and γ under H1. Differentiate (2) with respect to π and γ, and set them to 0. The MLEs and are the solutions of the following equations

Then,

Let and be the constrained MLEs of π and γ under H0. Similarly, they are the solution of the equations

For the first equation, we have . The second equation can be simplified as γ(S1 + 2S2) − 2S2 = 0. Then, the constrained MLEs are obtained

Test methods

An information matrix

Denote = (γ1, …, γ, π1, …, π). According to the Eq (3), the second-order derivatives of l with respect to π and γ are for i = 1, …, g, and for i ≠ j. Thus, the information matrix I with respect to is where

Otherwise, I = 0. By calculation, its inverse matrix is where for i = 1, …, g.

Asymptotic test statistics

In this section, we propose three asymptotic tests for large samples based on the unconstrained and constrained MLEs.

Likelihood ratio test. Let , be the unconstrained MLEs, and , be the constrained MLEs under H0. Denote , and . Given observation m, likelihood ratio statistic is given by where l(, |m) is defined in (2) and

From (4) and (5), likelihood ratio test can be represented as

Score test. Denote Under H0, score test statistic can be defined as

A direct calculation shows that the simplified form of T is

Wald-type test. Let . The null hypothesis H0: γ1 = … = γ is equivalent to C = 0, where 0 is a zero vector, and

Hence, Wald-type test statistic can be written as where is defined in (6). Let

Then,

For convenience, denote . Obviously, A is a symmetric tridiagonal matrix of order g − 1. Let , for j = 2, …, g − 1, and , . Following [13], A−1 is also a symmetric matrix denoted by where

Since , we obtain the simplified form

Next we provide the expressions of T for g = 2, 3, 4. If g = 2, it follows that

If g = 3, we have

Denote . If g = 4, then where a is defined in (7).

Under H0, test statistic T(= T, T or T) has asymptotic chi-square distribution with g − 1 degrees of freedom. Let be the (1 − α)th quantile of the chi-square distribution with g − 1 degree of freedom. Given the nominal level α, the null hypothesis H0 will be rejected if the value of T is larger than .

Exact methods

Given the observed data m = (m1, …, m), let T(m) be the value of the aforementioned statistic T(l = L, SC, W). The asymptotic (A) p-values of these statistics are defined by where m* is an observed data of m. For convenience, we call , and as “A approach” based on statistics T, T and T. Asymptotic tests work well when the sample size is large. However, they have some limitations if the sample size is relatively small. Several exact conditional and unconditional methods are proposed for small samples based on these statistics.

An exact conditional method is introduced under the assumption that all of m(i = 1, …, g) and S(l = 0, 1, 2) are fixed in Table 1. Thus, the cell values of the table follow a hypergeometric distribution. Define the tail area of statistics T, T and T as where . According to the tail area Ψ(m*), the exact conditional (C) p-values can be calculated by

Here, and are described as “C approach” based on statistics T, T and T.

Another exact p-value is from Basu’s maximization approach [14]. It can be obtained by maximizing the tail probability over all nuisance parameters instead of the constrained MLEs under H0. In this case, we define the tail area of statistic T(l = L, SC, W) as for a given table m*. Denote Θ = {: π ∈ [0, 1], i = 1, …, g} and where = (π1, …, π) and = (γ1, …, γ). Hence, under maximization (M) method, three exact unconditional p-value of are given by where L(, |m) = exp(l(, |m)) and l(, |m) is defined in (2). Corresponding to “A approach” and “C approach”, , and are called “M approach” based on T, T and T.

Numerical studies

In this section, we investigate the performance of the proposed asymptotic and exact tests in terms of TIEs and powers under different parameter settings.

We first compare asymptotic methods T, T and T with empirical TIEs. Let g = 2, 3, 4, π = 0.3: 0.02: 0.5, γ = 0.3: 0.02: 0.8 and m = m1 = ⋯ = m = 15, 50, 100. Here, a: b: c means increasing from a to c by b. For each parameter setting, 10,000 samples are randomly generated from the null hypothesis H0. Given the nominal level α = 0.05, empirical TIE is calculated by the proportion of rejecting H0, i.e., the number of rejections/10,000. Figs 1, 2 and 3 show the distribution surfaces of empirical TIEs for all the tests under π = π and γ = γ(i = 1, 2, …, g;g = 2, 3, 4). According to Tang et al. [3], a test is liberal if its empirical TIE is greater than 0.06, conservative if it is less than 0.04, otherwise it is robust. We observe that score test is more robust than other tests since its TIEs are closer to the pre-determined level α = 0.05. All the tests work well for larger sample size. However, likelihood ratio and Wald-type tests have inflated TIEs and are especially liberal when sample size is small. Some of their TIEs may be less than 0.04 or greater than 0.06.

Fig 1

Empirical TIE surfaces of asymptotic tests for g = 2, π = π and γ = γ.

Fig 2

Empirical TIE surfaces of asymptotic tests for g = 3, π = π and γ = γ.

Fig 3

Empirical TIE surfaces of asymptotic tests for g = 4, π = π and γ = γ.

Next we calculate the empirical powers of these tests according to the parameter settings for m = 15, 50, 100: (i) g = 2, = (0.2, 0.3), γ1 = 0.2: 0.05: 0.95, γ2 = 0.1, (ii) g = 3, = (0.2, 0.3, 0.3), γ1 = 0.2: 0.05: 0.95, γ2 = γ3 = 0.1, and (iii) g = 4, = (0.2, 0.3, 0.3, 0.3), γ1 = 0.2: 0.05: 0.95, γ2 = γ3 = γ4 = 0.1. For each parameter setting, we randomly choose 10,000 samples from the alternative hypothesis H1. The empirical power is computed by the proportion of rejecting H0 for all samples. Fig 4 reflects the empirical powers of three proposed tests for g = 2, 3, 4. The powers will increase when sample size is larger or the group number increases. Especially, the powers of all the tests are very close when m = 50, 100. However, there exists some differences between these tests for smaller samples. Wald-type test has higher power and likelihood ratio test has lower power.

Fig 4

Empirical power curves of asymptotic tests for g = 2, 3, 4.

Considering the limitations of asymptotic methods, we analyse A, C and M approaches for small samples. Unlike 10,000 random samples of asymptotic tests, we need to generate all possible tables with random cell values. For m = 10 and g = 2, 3, there are totally 4,356 and 287,492 tables. The TIEs and powers are obtained for m = m1 = ⋯ = m = 10 and g = 2, 3 according to the cases: π = 0: 0.04: 1, γ = 0: 0.04: 1, satisfying 0 ≤ p ≤ 1(l = 0, 1, 2, i = 2, 3). At the given nominal level α = 0.05, the probabilities are calculated by the log-likelihood (2) of all possible tables. We will reject the null hypothesis H0 if the probability is less than 0.05. Figs 5 and 6 show TIE surfaces of all the exact methods for π = π and γ = γ (i = 1, …, g; g = 2, 3). We observe that A approach is closer to the pre-specified nominal level α = 0.05 for m = 10 and g = 2, 3. However, and have the inflated TIEs. For C approach, is better than and since they have the inflated TIEs. The M approaches and can produce satisfactory TIEs.

Fig 5

TIE surfaces of exact approaches for m = 10, g = 2, π = π and γ = γ.

Fig 6

TIE surfaces of exact approaches for m = 10, g = 3, π = π and γ = γ.

Fig 7 provides the powers of exact methods according to parameter settings for m = 10: (i) g = 2, = (0.2, 0.3), γ1 = 0.2: 0.05: 0.9 and γ2 = 0.1, and (ii) g = 3, = (0.2, 0.3, 0.3), γ1 = 0.2: 0.05: 0.9 and γ2 = γ3 = 0.1. We observe that the powers will increase when m or γ1 increases under other fixed parameters. The powers of A, C and M approaches are relatively close based on statistics T(l = L, SC).

Fig 7

Power curves of exact approaches for m = 10 and g = 2, 3.

Note that all parameter settings of asymptotic and exact methods are studied under balanced designs, that is, m = m1 = ⋯ = m. For unbalanced case, we can handle it through some examples.

Real examples

In this section, two real examples with unbalanced designs are provided to illustrate our proposed methods at the nominal level α = 0.05. We first show an example with large samples based on asymptotic test statistics.

Example 1 [15] There were 216 patients aged 20-39 with retinitis pigmentosa (RP) at the Massachusetts Eye and Ear infirmary. They were divided into four genetic groups (Table 2): autosomal dominant RP (DOM), autosomal recessive RP (AR), sex-linked RP (SL) and isolate RP (ISO).

Table 2

The number of patients for genetic types.

Response	DOM	AR	SL	ISO	Total
0	15	7	3	67	92
1	6	5	2	24	37
2	7	9	14	57	87
Total	28	21	19	148	216

Let m be the number of patients with l(l = 0, 1, 2) affected eyes in the ith (i = 1, 2, 3, 4) group. Under Dallal’s model, we are interested to test if the correlations of these four groups are equal, i.e., . Table 3 provides the results of statistics, p-values and constrained MLEs. Moreover, the unconstrained MLEs and . Given the nominal level α = 0.05, T, T, T = 7.81 and p-values are greater than 0.05. Thus, there is no evidence to reject H0. That is to say, the correlations of four groups are equal: γ1 = γ2 = γ3 = γ4 = 0.8246.

Table 3

Test statistics, p-values and constrained MLEs under H0.

Value	Test statistics			π˜=(π˜_1,π˜_2,π˜_3,π˜_4)	γ˜
	TL	TSC	TW
Statistic value	4.4569	4.1831	5.7594	(0.3950, 0.5675, 0.7165, 0.4656)	0.8246
p-value	0.2162	0.2424	0.1239

For small sample case, we provide another example to compare the effectiveness of asymptotic and exact methods.

Example 2 [16] A double-blind clinical trial was conducted to study amoxicillin treatment of acute otitis media with effusion (OME) in twenty-four children at 14 days. Each child underwent no, unilateral or bilateral OME and was assigned into three groups according to ages: <2, 2-5 and ≤6 years (Table 4). Denote m* = (2, 2, 11, 5, 1, 3, 6, 0, 7). Next we apply asymptotic and exact methods to test H0: γ1 = γ2 = γ3 = γ.

Table 4

14-day OME status.

Response	< 2 years	2-5 years	≥6 years	Total
0	2	5	6	13
1	2	1	0	3
2	11	3	1	15
Total	15	9	7	31

Through calculating (4) and (5), the unconstrained MLEs , , and the constrained MLEs , under H0. Then, T(m*) = 0.2445, T(m*) = 0.2525 and T(m*) = 0.2285. Table 5 provides the comparison of asymptotic and exact methods. The result shows that there is no significant difference among the correlations of two groups regardless of any approaches.

Table 5

Comparison of asymptotic and exact p-values.

Method	A approach			C approach			M approach
	pLA	pSCA	pWA	pLC	pSCC	pWC	pLM	pSCM	pWM
p-value	0.8849	0.8814	0.8920	0.8643	0.9022	0.7686	0.8963	0.9899	0.8042

Conclusions

In this paper, we propose asymptotic statistics and exact procedures to test if the correlations of multiple bilateral data are equal under Dallal’s model. Three asymptotic test statistics are likelihood ratio T, score T and Wald-type T for large sample. The explicit expressions of these tests are obtained, and their asymptotic p-values are denoted by A approach. For small sample, six exact methods are derived based on statistics T, T and T, including three conditional exact C procedures and three unconditional exact M approaches .

Numerical studies are conducted to investigate the performance of asymptotic and exact methods in terms of TIEs and powers. When the samples is larger, empirical TIEs and powers of T, T and T are close to each other. In general, score test T is more robust than other two tests. However, these tests may produce unacceptable TIEs such as Wald-type test when the samples is smaller. The results are similar to those of Rosner’s and Donner’s models, see Ma et al. [6] and Liu et al. [10]. For small sample, we obtain TIE surfaces and power curves of exact C and M approaches with two and three groups, comparing with A approach. As for TIEs, the A approaches and are liberal, and is close to the nominal level 0.05 under different parameter configurations. The C approaches and tend to be more inflated than . The M approach is better than and . On the other hand, the powers of exact methods are very close based on likelihood ratio T and score T. For C approach, has higher power, while has lower power. Moreover, has higher power, but has lower power in M approach.

The ideas of asymptotic and exact methods can be extended other data structures with larger or small samples such as crash data. For example, Zeng et al. [17-19] proposed some models for the analysis of crash rates by injury severity. Dong et al. [20] introduced mixed logit model to investigate the difference between single- and multi-vehicle accident probability. Chen et al. [21-23] analyzed unbalance panel models by using real-time environmental and traffic big data. For these problems, we will leave these for future research.

28 Sep 2020

PONE-D-20-26283

Statistical tests under Dallal's model: Asymptotic and exact methods

PLOS ONE

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Reviewer #1: This study proposes some asymptotic and exact methods for testing the equality of correlations for multiple bilateral data under Dallal's model. Their performance is compared via two numerical studies. The paper is generally well organized and written. A minor suggestion is that more references on the MLE should be acknowledged, such as:

A multivariate random parameters Tobit model for analyzing highway crash rate by injury severity. Accident Analysis and Prevention, 2017, 99: 184-191.

Jointly modeling area-level crash rates by severity: A Bayesian multivariate random-parameters spatio-temporal Tobit regression. Transportmetrica A: Transport Science, 2019, 15(2): 1867-1884.

Spatial joint analysis for zonal daytime and nighttime crash frequencies using a Bayesian bivariate conditional autoregressive model. Journal of Transportation Safety and Security, 2020, 12(4): 566-585.

Besides, some more directions for future research are suggested to draw in the Conclusion Section.

Reviewer #2: The topic of this paper is interesting. The methods sound. The results are meaningful and useful. There is one suggestion to improve this paper.

Some related references about likelihood ratio test or maximum likelihood estimations could be added.

[1] Investigating the Differences of Single- and Multi-vehicle Accident Probability Using Mixed Logit Model, Journal of Advanced Transportation, 2018, UNSP 2702360.

[2] Analysis of hourly crash likelihood using unbalanced panel data mixed logit model and real-time driving environmental big data. 2018, JOURNAL OF SAFETY RESEARCH. 65: 153-159.

[3] Investigation on the Injury Severity of Drivers in Rear-End Collisions Between Cars Using a Random Parameters Bivariate Ordered Probit Model, International Journal of Environmental Research and Public Health, 2019, 16(14) , 2632.

[4] Crash Frequency Modeling Using Real-Time Environmental and Traffic Data and Unbalanced Panel Data Models, International Journal of Environmental Research and Public Health, 2016, 13(6), 609.

**********

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2 Nov 2020

Reviewer 1:

This study proposes some asymptotic and exact methods for testing the equality of correlations for multiple bilateral data under Dallal's model. Their performance is compared via two numerical studies. The paper is generally well organized and written. A minor suggestion is that more references on the MLE should be acknowledged, such as:

[1] Zeng Qiang, Wen Huiying, Huang Helai, Pei Xin, Wong S.C. A multivariate random parameters Tobit model for analyzing highway crash rate by injury severity. Accident Analysis and Prevention, 2017, 99: 184-191. https://doi.org/10.1016/j.aap.2016.11.018.

[2] Qiang Zeng, Qiang Guo, S. C. Wong, Huiying Wen, Heilai Huang \\& Xin Pei. Jointly modeling area-level crash rates by severity: A Bayesian multivariate random-parameters spatio-temporal Tobit regression. Transportmetrica A: Transport Science, 2019, 15(2): 1867-1884. https://doi.org/10.1080/23249935.2019.1652867.

[3] Zeng, Qiang, Wen Huiying, Wong S.C., Huang Helai, Guo Qiang, Pei Xin. Spatial joint analysis for zonal daytime and nighttime crash frequencies using a Bayesian bivariate conditional autoregressive model. Journal of Transportation Safety and Security, 2020, 12(4): 566-585. 10.1080/19439962.2018.1516259.

Besides, some more directions for future research are suggested to draw in the Conclusion Section.

Response. Thank your suggestion. We have added the above references in the revised version. The ideas of asymptotic and exact methods can be extended other data structures such as crash data. For the problem, it is worthy of researching and exploring (see the Conclusion Section).

Reviewer 2:

The topic of this paper is interesting. The methods sound. The results are meaningful and useful. There is one suggestion to improve this paper.

Some related references about likelihood ratio test or maximum likelihood estimations could be added.

[1] Dong Bowen, Ma Xiaoxiang, Chen Feng, Chen Suren. Investigating the Differences of Single- and Multi-vehicle Accident Probability Using Mixed Logit Model. Journal of Advanced Transportation, 2018, UNSP 2702360. DOI：10.1155/2018/2702360.

[2] Chen Feng, Chen Suren, Ma Xiaoxiang. Analysis of hourly crash likelihood using unbalanced panel data mixed logit model and real-time driving environmental big data. 2018, Journal of Safety Research. 65: 153-159. DOI：10.1016/j.jsr.2018.02.010.

[3] Feng Chen, Mingtao Song and Xiaoxiang Ma. Investigation on the Injury Severity of Drivers in Rear-End Collisions Between Cars Using a Random Parameters Bivariate Ordered Probit Model. International Journal of Environmental Research and Public Health, 2019, 16(14) , 2632. https://doi.org/10.3390/ijerph16142632.

[4] Chen Feng, Chen Suren, Ma Xiaoxiang. Crash Frequency Modeling Using Real-Time Environmental and Traffic Data and Unbalanced Panel Data Models. International Journal of Environmental Research and Public Health, 2016, 13(6), 609. DOI：10.3390/ijerph13060609.

Response. Thank your suggestion. Some related references have been added in our revision.

Yours Sincerely,

Zhiming Li

9 Nov 2020

Statistical tests under Dallal's model: Asymptotic and exact methods

PONE-D-20-26283R1

Dear Dr. Li,

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Reviewer #1: No

16 Nov 2020

PONE-D-20-26283R1

Statistical tests under Dallal’s model: Asymptotic and exact methods

Dear Dr. Li:

I'm pleased to inform you that your manuscript has been deemed suitable for publication in PLOS ONE. Congratulations! Your manuscript is now with our production department.

If your institution or institutions have a press office, please let them know about your upcoming paper now to help maximize its impact. If they'll be preparing press materials, please inform our press team within the next 48 hours. Your manuscript will remain under strict press embargo until 2 pm Eastern Time on the date of publication. For more information please contact onepress@plos.org.

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on behalf of

Dr. Feng Chen

Academic Editor

PLOS ONE