Abstract
In clinical studies, paired binary metadata are often encountered during collecting information from paired organs or parts. The risk difference, relative risk ratio, and odds ratios of response rates are widely used to measure cured effects for such data. Under Dallal’s model, the paper extends these measures to the general case in groups. We propose eight test statistics for various functions when the dependency measures are equal or not. Simulation results show that the Score and Rosner-type tests can produce robust empirical type I error rates, while Wald-type and likelihood ratio tests have better power in testing the risk difference, relative risk ratio, and odds ratios. Finally, two real examples illustrate our proposed method’s practicability.
1. Introduction
Researchers often obtain bilateral data from patients’ paired organs in clinical studies. For example, in ophthalmology, patients are randomly divided into treatment groups. All clinical outcomes of patients’ two eyes could be summarized as unilateral, bilateral, and no response(s). Since ignoring the correlation of paired data may bring about a series of misleading results, it is essential to consider this issue. On this account, many corresponding statistical models have been developed by statisticians. Rosner [1] proposed a classical intercorrelation model assuming that the dependence between two eyes is a constant . That is to say, with one eye responding, the probability of another eye responding is times that of an unconditional one. Until now, there have been valuable results on the homogeneity test under Rosner’s model. Tang et al. [2] developed eight statistics to test the equality of response rates in two groups. Ulteriorly, Ma et al. [3] proposed asymptotic test statistics to analyze the abovementioned problems in the case of groups , which can be seen as the generalization of results obtained in [2]. Furthermore, the rate difference and ratio tests were investigated between two proportions of the bilateral data in [4, 5].
However, Dallal [6] designated that Rosner’s model would give ill-fitting results if the bilateral response occurred almost certainly under the population-specific prevalence rate. Given this, Dallal proposed a substitutable model assuming that the probability of one organ responding is independent of the probability of another organ responding. In other words, , where is the th eye response of the th patient in the group, where , , and . Further research has shown that Dallal’s model is more suitable for the correlated data in [7]. Therefore, the correlational research on Dallal’s model has received considerable attention, and some interesting results have been reported in [8–10]. In the case of , MIan et al. [8] proposed three objective Bayesian methods to investigate the risk difference, relative risk ratio, and odds ratio of paired data. Chen et al. [9] put forward eight statistics to test whether the response rates of each treatment group were equal when the parameter is not equal. Moreover, Sun et al. [10] studied the homogeneity test of risk differences between response rates of different treatment groups.
To date, under Rosner’s model, numerous results compare the effectiveness of different treatments (i.e., risk difference and relative risk ratio). Significantly, a general function relationship usually represents the response rate between multiple bilateral data. However, hypothesis testing is less considered under Dallal’s model, let alone the homogeneity testing with general function relationship under Dallal’s model. The above discussion arouses our research interest. For this, this paper proposes a novel hypothesis to test the homogeneity of response rate functions in bilateral data.
The structure of this article is as follows: Section 2 sorts out the data structure, obtains the likelihood function, and establishes the function hypothesis test under Dallal’s model according to two cases: (i) the parameter is not equal; (ii) . In Section 3, the iterative algorithm calculates the maximum likelihood estimates (MLEs) of unknown parameters in these two cases. In Section 4, we derive the general expressions of eight test statistics in the function hypothesis and present three specific forms in combination with particular problems. In Section 5, simulation experiments are carried out in different cases to compare the performance of the statistics of Section 4 in terms of the empirical type I error rates (TIEs) and power. Section 6 uses two real examples to illuminate our proposed method. The summary conclusion and prospects of the article are given in Section 7.
2. Preliminaries
Assume that the patients are randomly divided into groups with different treatments. The number of patients in the th group is assumed to be ; thus, . Let be the number of patients with exact ( = 0, 1, 2) responses in the th group, then . Moreover, denotes the total number of patients with responses, then obviously, . Table 1 summarizes the data structure for the patient frequencies.
Let a binary variable = 1 when the th site (the th body site) of the th subject is disease-free in the th treatment group, otherwise = 0 ( = 0, 1, = 1, , and = 1, 2). We take the ophthalmologic diseases under Dallal’s model for example. We suppose that and for . Let the probability of no response, one response, and both responses be , , and , then for any fixed . Let , according to Table 1, the probability density function of the th group can be written as
After derivation, we can get , , and .
The observed data are represented by vector , and its maximum likelihood function is as follows:where and . Then, the log-likelihood function is expressed bywhere is a constant and .
Unlike previous research, we propose a function hypothesis test, including the risk difference, relative risk ratio, and odds ratio. Let be continuously differentiable and let it have an inverse mapping . To study the prevalence of ophthalmologic diseases among patients with different treatments, some relevant hypotheses are given as follows: Case (i): for some , the hypotheses are given by : versus : , . Case (ii): , the hypotheses are given by : versus : , .
3. Parameter Estimation
3.1. MLEs under and
Taking different function forms of can study various problems in bilateral correlated data. To this end, the functional forms of are set as follows:(i)When , it can be used to investigate the risk difference (RD) of bilateral correlated data. In this case, the null hypothesis is .(ii)When , it can be used to investigate the relative risk ratio (RR) of bilateral correlated data. In this case, the null hypothesis is .(iii)When , it can be used to investigate the odds ratio (OR) of bilateral correlated data. In this case, the null hypothesis is .
Under the null hypothesis , we have . Since has an inverse mapping , let . Then, the log-likelihood function iswhere is a constant. To determine the MLEs of parameters, we set the partial differentiations equal to zero, i.e.,where
Since there is no explicit solution for the abovementioned equation, we use the Newton–Raphson algorithm to solve the equations. The solution to the abovementioned equations is obtained by using the two-step algorithm:(1)We take the initial value and , where can be reduced to a second-order polynomial:(2)The update of is given by the Newton–Raphson algorithm as follows:where
For , we have , as well as . There are only two unknown parameters and . In this case, the corresponding log-likelihood function can be expressed aswhere is a constant. To obtain the MLEs of parameters, we take the first partial derivative of each parameter equal to zero, i.e.,where
When , we have
When the function is complex, the system of equations may have no explicit solution. In this case, the Fisher-score algorithm is used to solve the equations because of its strong convergence. The update of is given by the Fisher-score algorithm as follows:where is the inverse matrix of the Fisher information matrix satisfyingwhere
3.2. MLEs under and
Under the alternative hypothesis , let the global MLEs of and be and , respectively. Then,
For the alternative hypothesis , we suppose that and are the global MLEs of and , respectively. Then,
4. The Proposed Methods
4.1. Likelihood Ratio Tests
To test the hypothesis , the likelihood ratio test statistic is given bywhere , , , and . It can be simplified as
For convenience, we next provide three special expressions of the likelihood ratio test under the risk difference, relative risk ratio, and odds ratio.(i)RD: . From (20), it is easy to get(ii)RR: . It follows that(iii)OR: . Through (20), we have
For testing the hypothesis , the likelihood ratio test statistic is given bywhere and . It can be simplified as
Similar to , can be given different forms to obtain corresponding to three shapes of the risk difference, risk ratio, and odds ratio. Under and , and are, respectively, asymptotically distributed as the chi-squared distribution with degrees of freedom [11].
4.2. Wald-Type Tests
The null hypothesis : is equivalent to : . For , that is, . Then, the null hypothesis can be written as in the matrix form, where and
Let . Following Agresti [14], the Wald-type test statistic is given bywhere is the information matrix in the Appendix A. By calculation, we havewhere
We denote . Then,where
Note that is a symmetric tridiagonal matrix. The elements of can be derived as follows:in which
By the abovementioned calculation, the Wald-type statistic can be simplified to
To investigate specific problems, we only need to give the corresponding function forms of or .(i)RD: . In this case, the Wald-type statistic becomes where and .(ii)RR: . From (34), we have where(iii)OR: . Hence,whereand .
The hypothesis can be written as in the matrix form, where and
Let . Then, another Wald-type test statistic is given bywhere is the information matrix in the Appendix B. We denote . Then, can be simplified to
See the Appendix C for the derivation process of .
Identically, can be obtained by giving the forms of corresponding to the risk difference, relative risk ratio, and odds ratio. Under the hypotheses and , and asymptotically obey the chi-square distribution with the degree of freedom .
4.3. Score Tests
For Case (i), we note that for some . Let . Under the null hypothesis : , according to [12, 13], the score test statistic can be defined as follows:where is the Fisher information matrix. Hence, can be simplified aswhere
For Case (ii) , let . Under the null hypothesis : , the score test statistic can be given bywhere is the Fisher information matrix. can be simplified aswhere
Under and , and asymptotically follow the chi-square distribution with degree of freedom [11], respectively.
4.4. Rosner-Type Tests
Let and be the MLEs under and , respectively. To test the null hypothesis , the Rosner-type test statistic is given bywhere
Appendix D shows the specific calculation process. Then, one has
To test , the test statistic is given by the following expression:where is given in Appendix E. Then, one has
Here, and are the MLEs under and , respectively. Under and , and asymptotically submit to the chi-square distribution with degree of freedom.
5. Comparison with Test Methods
This section will evaluate the eight test statistics with different forms of Section 4 from the empirical TIEs and power through simulation experiments. We randomly generate 10000 replications from null hypotheses and . The empirical TIEs are calculated as (the number of rejections)/10000 at the significance level . According to [15], a test is defined as liberal (or conservative) if its TIE is greater than 0.06 (or less than 0.04). Otherwise, it is robust.
Recently, there have been some exciting research results under Dallal’s model. Our proposed method extends the existing studies to a more general situation. For this, we make take the different forms, and its three conditions are as follows: (i) risk difference: , (ii) relative risk ratio: , and (iii) odds ratio: .
The specific value of is shown in Table 2, where . Table 3 shows the parameter settings of empirical power.
Tables 4 and 5 present only the empirical TIEs of the partial cases under and , and the rest of the cases are shown in the Appendix F (i.e., Tables 6 and 7). It can be easily seen that the TIEs of and are the most inflated, followed by , , , , , and . Moreover, the TIEs of , , , and are close to 0.05. Thus, , , , and are robust, while , , , and are liberal.
Further, we consider the general result that in cases DR, RR, and OR are unequal. Firstly, 1000 parameters and are randomly generated under and , respectively. In each parameter setting, = 0, = 1, and , , , and are randomly generated results, which are drawn in Figures 1 and 2.
Whether to study the risk difference, relative risk ratio, or odds ratio, the empirical TIEs of the eight test statistics under and will approach 0.05 as the sample size increases. However, when the sample size increases in , the change of TIEs of is not significant. The empirical TIEs of increase; thereinto, are the most significant. Since , , , and change around 0.05, they are all robust, while and always have the unstable TIEs, followed by and .
Next, we compare the power of the eight test statistics of and set the parameters under and , respectively. Without loss of generality, we only list the parameter settings for cases EQ and RR in Table 3 and the rest of the cases are similar.
Figure 3 shows that the power of each test statistic increases as increases, and when is large, their power is close to each other.
As can be seen from the box plots (Figures 1 and 2), in the same number of groups and sample size, the type I error rate of and will be obviously different under different response rate function forms; however, the other seven test statistics have no obvious changes.
6. Two Real Examples
In this section, two real examples are given to investigate the performance of the proposed methods at the nominal level .
The first example is a retinitis pigmentosa (RP) clinical trial in [1]. In this trial, 216 patients aged 20–39 with PR from different families are divided into four genetic groups, that is, autosomal dominant RP (DOM), autosomal recessive RP (AR), sex-linked RP (SL), and isolate RP (ISO). Table 8 shows the response conduction of patients’ eyes.
Li et al. [16] derived by using the likelihood ratio, Score, and Wald-type statistics. Under Dallal’s model, it is interesting to test whether the response rate of four groups is a functional relationship, i.e., . The test statistics and their values are listed in Table 9. The value of is less than 0.05 in cases of EQ, DR, and OR, but the value of is more significant than 0.05 in cases RR. The function setting of the risk ratio is more suitable for our data in cases RR.
Another example is the two-arm multi-center phase II double placebo control clinical trial given by Pei et al. [17]. One hundred seventy patients with diffuse scleroderma were randomly assigned to receive 500 g/day of oral natural collagen or a similar placebo. The total duration of the treatment phase was 12 months, and the safety follow-up was conducted in the 15th month (3 months after drug withdrawal). The MRSS measured disease improvement within 170 patients is shown in Table 10.
In the hypothesis test, we take the function . Table 11 describes the eight test statistics, , , and its values. It is easy to see that the values of eight statistics are all greater than 0.05. Among them, the value of case DR is more significant than that of other cases, which shows that the function setting of the risk difference (case DR) is most suitable for our data.
7. Conclusions
In this paper, a novel general hypothesis test was proposed to test the homogeneity of response rate function values of each group. Our proposed method can effectively test the consistency of paired data with a general function form, which is a generalization of the existing parametric hypothesis test method. Furthermore, it is of great significance to make the function of null hypotheses take the corresponding form for different research problems. The eight test statistics and were proposed to test null hypotheses.
Simulation studies are given to explore test statistics’ performance in power and TIEs under general hypothesis tests. When parameter is unknown or not all equal in the risk difference, relative risk ratio, and odds ratio analysis, and have satisfactory power. However, they produce inflated TIEs, especially for . It is worth noting that , , , and always have robust TIEs and higher power. Therefore, , , , and are recommended under the abovementioned circumstances.
We notice little research to investigate the general hypothesis test for small samples. Moreover, when there is a random zero structure in the contingency table, this may lead to the failure of test statistics. It is meaningful to study the above problems, which will be considered in our future work.
Appendix
A. Derivation of the Fisher Information Matrix
If for some , by taking the second partial derivative of with respect to and , we get
It follows that the information matrix is given bywhere
B. Derivation of the Information Matrix
If , by taking the second partial derivative of with respect to and , we getand . So, the information matrix is given bywhere
Furthermore, we havewhere
C. Derivation of the Inverse Matrix
For , it is easy to get . If , then
When , we havewhere . Thus, and is the adjoint matrix of .
D. Derivation of for Rosner-Type Statistic
Due to , , and , the expectation and variance of , are, respectively, , , , , and . Based on the abovementioned conditions, the covariance of and is calculated as
Then, we have
Furthermore, the expression of is
E. Derivation of the Rosner-Type Statistic
For case (ii), since and , we can get
The expression of is
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this article.
Acknowledgments
This research was supported by the National Undergraduate Training Program for Innovation and Entrepreneurship (grant no.: 202210755079), the Research Innovation Program for Postgraduates of Xinjiang Uygur Autonomous Region (grant nos.: XJ2023G016, XJ2023G017, and XJ2022G020), the Innovation Project of Excellent Doctoral Students of Xinjiang University (grant nos.: XJU2023BS017 and XJU2022BS027), the National Natural Science Foundation of China (grant no.: 12061070), and the Science and Technology Department of Xinjiang Uygur Autonomous Region (grant no.: 2021D01E13).