Confidence intervals construction for difference of two means with incomplete correlated data

Background Incomplete data often arise in various clinical trials such as crossover trials, equivalence trials, and pre and post-test comparative studies. Various methods have been developed to construct confidence interval (CI) of risk difference or risk ratio for incomplete paired binary data. But, there is little works done on incomplete continuous correlated data. To this end, this manuscript aims to develop several approaches to construct CI of the difference of two means for incomplete continuous correlated data. Methods Large sample method, hybrid method, simple Bootstrap-resampling method based on the maximum likelihood estimates (B1) and Ekbohm’s unbiased estimator (B2), and percentile Bootstrap-resampling method based on the maximum likelihood estimates (B3) and Ekbohm’s unbiased estimator (B4) are presented to construct CI of the difference of two means for incomplete continuous correlated data. Simulation studies are conducted to evaluate the performance of the proposed CIs in terms of empirical coverage probability, expected interval width, and mesial and distal non-coverage probabilities. Results Empirical results show that the Bootstrap-resampling-based CIs B1, B2, B4 behave satisfactorily for small to moderate sample sizes in the sense that their coverage probabilities could be well controlled around the pre-specified nominal confidence level and the ratio of their mesial non-coverage probabilities to the non-coverage probabilities could be well controlled in the interval [0.4, 0.6]. Conclusions If one would like a CI with the shortest interval width, the Bootstrap-resampling-based CIs B1 is the optimal choice.


Background
Incomplete data often arise in various research fields such as crossover trials, equivalence trials, and pre and post-test comparative studies. For instance, ( [1] pp. 212) designed a crossover clinical trial to measure the onset of action of two doses of formoterol solution aerosol: 12 ug and 24 ug. In this study, twenty-four patients were randomly allocated in equal numbers to one of the six possible sequences of two treatments at a time. Each patient was received two aerosols at each of visits 2 and 4. After four weeks, researchers measured the forced expiratory Kunming, China Full list of author information is available at the end of the article volume of a second (FEV 1 ) indicators for twenty-four patients. Due to the fact that researches did not consider all possible combinations of three treatments (e.g., placebo, 12 ug and 24 ug aerosols), which indicates that the missing data mechanism is missing completely at random (MCAR) thus FEV 1 was only observed for 7 patients under both treatments (e.g., 12 ug and 24 ug aerosols), 9 patients only for 12 ug aerosol, and 8 patients only for 24 ug aerosol. The resultant data are shown in Table 1, which consist of two parts: the complete observations and the incomplete observations.
For the above crossover clinical trial, our main interest is to test the equivalence between 12 ug and 24 ug formoterol solution aerosols with respect to the FEV 1 value. To this end, we can construct a (1 − α)100 % confidence interval for the difference of two FEV 1 values. If the resultant confidence interval (CI) lies entirely in the interval (−δ 0 , δ 0 ) with δ 0 (> 0) being some pre-specified clinical acceptable threshold, we thus could conclude the equivalence between two doses of formoterol solution aerosol at the α significance level. As a result, reliable CIs for the difference in the presence of incomplete data are necessary. The problem of testing the equality and constructing CI for the difference of two correlated proportions in the presence of incomplete paired binary data has received considerable attention in past years. For example, ones can refer to [2][3][4][5][6] for the large sample method, and [7] for the corrected profile likelihood method. When sample size is small, [8] proposed the exact unconditional test procedure for testing equality of two correlated proportions with incomplete correlated data. Tang, Ling and Tian [9] developed the exact unconditional and approximate unconditional CIs for proportion difference in the presence of incomplete paired binary data. Lin et al. [10] presented a Bayesian method to test equality of two correlated proportions with incomplete correlated data. Li et al. [11] discussed the confidence interval con-struction for rate ratio in matched-pair studies with incomplete data. However, all the aforementioned methods were developed for incomplete paired binary data.
Statistical inference on the difference of two means with incomplete correlated data has received a limited attention. For example, [12] discussed the problem of testing the equality of two means with missing data on one response and recommended [13] statistic when the variances were not too different. Lin and Stivers [14] also gave a similar comparison. Lin and Stivers [15] and [12] suggested some test statistics for testing the equality of two means with incomplete data on both response. However, to our knowledge, little work has been done on CI construction for the difference of two means with incomplete correlated data under the MCAR assumption.
Inspired by [16][17][18][19], we develop several CIs for the difference of two means with incomplete correlated data under the MCAR assumption based on the large sample method, hybrid method and Bootstrap-resampling method. The presented Bootstrap-resampling CIs have not been considered in the literature related to missing observations.
The rest of this article is organized as follows. Several methods are presented to construct CIs for the difference of the two means with incomplete correlated data in Section "Methods". Simulation studies and an example are conducted to evaluate the finite performance of the proposed CIs in terms of coverage probability, expected interval width, and mesial and distal non-coverage probabilities in Section "Results". A brief discussion is given in Section "Discussion". Some concluding remarks are given in Section "Conclusion".

Confidence interval based on the large sample method
To make a comparison with the following proposed methods, we assume that x follows a bivariate normal distribution in this subsection. In this case, if only variable x 1 or x 2 is subject to missingness (i.e., n 1 = 0 or n 2 = 0), one can obtain the closed forms of the maximum likelihood estimates (MLEs) of μ and [22]. However, there are no closed forms of the MLEs for μ and when variables x 1 and x 2 are simultaneously subject to missingness (i.e., n 1 = 0 and n 2 = 0), though one can find the MLEs of μ and using an iterative algorithm [23]. To get the closed forms of MLEs for μ and , [15] proposed the modified MLEs using a non-iterative procedure and provided several test statistics based on the obtained estimators of μ and . (
Following [12], when σ 1 = σ 2 , another statistic for testing H 0 can be expressed as , which is asymptotically distributed as t distribution with degrees ν σ of freedom under Thus, an approximate 100(1 − α) % CI of δ for T 5 is denoted as T 5 -CI.

Confidence interval based on the generalized estimating equations(GEEs)
To relax the bivariate normality assumption of x, the method of the generalized estimating equations (GEEs) with exchangeable working correlation structure (e.g., [25]) can be adopted to make statistical inference on δ in the incomplete correlated data because the GEE approach have become one of the most widely used methods in dealing with correlated response data [26,27]. Following [28], the GEEs with exchangeable working correlation structure can be used to estimate parameter vector μ; the so-called sandwich variance estimator can be used to consistently estimate the covariance matrix of μ; and the ML method under a bivariate normal assumption via available paired observations is used to estimate the correlation parameter. Thus, an approximate 100(1 − α) % CI of δ based on GEE method is denoted as T g -CI.

Bootstrap-resampling-based confidence intervals
When the distribution of x is known, one can obtain the approximate CIs of δ based on the asymptotic distributions of the constructed test statistics under the null hypotheses H 0 : δ = δ 0 . However, when the distribution of x is unknown, the asymptotic distributions of the constructed test statistics may not be reliable, especially with small sample size. On the other hand, estimators of some nuisance parameters have not the closed-form solutions even if the approximate distribution is reliable, and they must be obtained by using some iterative algorithms, which are computationally intensive. In this case, the Bootstrap method is often adopted to construct CIs of parameter of interest. The Bootstrap CIs can be constructed via the following steps.
Step 3. Repeating the above steps 1 and 2 for a total of G times yields G Bootstrap estimates δ * Step 4. Based on the bootstrap estimates δ * g , g = 1, 2, . . . , G , Bootstrap-resampling-based CIs for δ can be constructed as follows.
Generally, the standard error se (δ) ofδ can be estimated by the sample standard deviation of the G replications, G is approximately normally distributed, an approximate 100(1 − α) % Bootstrap CI for δ is given by δ − z α/2ŝ e(δ),δ + z α/2ŝ e(δ) , where z α/2 is the upper α/2-percentile of the standard normal distribution, which is referred as the simple Bootstrap confidence interval. Whenδ = ax (n) Alternatively, if δ * g : g = 1, · · · , G is not normally distributed, it follows from ( [16] p.132) that the approximate represents the integer part of a, which is referred as the percentile Bootstrap CI. Whenδ = ax , the corresponding percentile Bootstrap CI is denoted as B 4 .

Simulation studies
In this subsection, we investigate the finite performance of various CIs in terms of empirical coverage probability (ECP), empirical confidence widths (ECW), and distal and mesial non-coverage probabilities (DNP and MNP) in various parameter settings via Monte Carlo simulation studies. A summary of abbreviation for various confidence intervals is presented in Table 2.
In the second Monte Carlo simulation study, we assume that the random samples of bivariate variables x 1 and x 2 are generated from a bivariate t-distribution with five degrees of freedom, and mean μ and scale parameter specified in the first simulation study. The corresponding results with (n, n 1 , n 2 ) = (5, 5, 5) are given in Tables 6, 7 and 8. Similarly, we calculate the corresponding results for T 3 , T 4 , T 5 , hybrid CIs, Bootstrap-resampling-based CIs when σ 2 = 4 and (n, n 1 , n 2 ) = (5, 5, 2), which are given in Tables 9, 10 and 11.
To investigate powers for the proposed CIs, we calculated the power in both the first and second simulation study. The results are shown in Tables 12 and 13. There is very little power in both the first and second simulation study to exclude a difference of zero.

Results of simulation studies
From Tables 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 and 13, we have the following findings. First, when is unknown, the CIs based on the the Bootstrap-resampling-based methods except for B 3 behave satisfactorily in the sense that their ECPs are close to the pre-specified confidence level 95 % (e.g., see Tables 3 and 6); the CI based on the Bootstrapresampling-based method B 1 generally yielded shorter Table 3 ECPs of various confidence intervals under bivariate normal distribution with different ρ and δ, μ 1 , μ 2 σ 2 1 and (n, n 1 , n 2 ) = (5, 2, 2) and σ 2 2 = 4       Table 9 ECPs of various confidence intervals with different ρ and δ, μ 1 , μ 2 , (n, n 1 , n 2 ) = (5, 5, 2), when σ 2 1 = σ 2 2 = 4 Bivariate normal distribution    ECWs than others (e.g., see Tables 4 and 7); the CIs corresponding to bivariate t-distribution are generally wider than those corresponding to bivariate normal distribution; the ECWs decrease as the correlation coefficient ρ increases. Second, the RNCPs of all the considered CIs lie in the interval [0.4,0.6] (e.g., see Tables 5 and 8), which show that our derived CIs generally demonstrate symmetry. Third, when σ 2 1 = σ 2 2 , the CIs based on statistics T 3 , T 4 and T 5 behave unsatisfactory (e.g., see Tables 9 and 10) because their corresponding ECPs are almost less than the pre-specified confidence level 95 %. Fourth, powers corresponding to W a and B 1 are larger than others (e.g., see Tables 12 and 13). From the above findings, we would recommend the usage of the Bootstrap-resampling-based CI (i.e., B 1 ) because its coverage probability is generally close to the pre-chosen confidence level, it consistently yields the shortest interval width even when sample size is small, it usually guarantees its ratios of the MNCPs to the noncoverage probabilities lying in [0.4, 0.6], and its power is usually larger than others.

An worked example
In this subsection, the data introduced in Section for the action of two doses of formoterol solution aerosol are used to illustrate the proposed methodologies. In this example, we are interested in CI construction of the difference of two FEV 1 values for two doses of formoterol solution aerosol. Under the previously given notation, we have n = 7, n 1 = 9, n 2 = 8,δ = ax  Table 14. Examination of Table 14 shows that the actions of two

Discussion
Although testing equivalence of two correlated means with incomplete data has been studied, there is little work done on their interval estimators. To address the issue, this paper proposes various interval estimators of the difference of two correlated means for known and unknown cases based on the large sample method, hybrid method and Bootstrap-resampling method. Extensive simulation studies are conducted to evaluate the finite performance of the proposed CIs in terms of the empirical coverage probability, empirical interval width and ratio of the mesial non-coverage probability to the non-coverage probability (RNCP). Empirical results evidence that the Bootstrap-resampling-based CIs B 1 , B 2 , B 4 behave satisfactorily for small to moderate sample sizes in the sense that their coverage probabilities could be well controlled around the pre-specified nominal confidence level and their RNCPs almost lie in the interval [0.4, 0.6]. However, confidence intervals based on the large sample method and hybrid method behave unsatisfactory for small sample sizes because the distributions of statistics T 1 , · · · , T 5 are asymptotical, and these asymptotical distributions are proper only when N i → ∞. When is unknown, using GEE method to estimate variance is less efficient. It is interesting to investigate confidence interval construction of the difference of two means with incomplete correlated data under missing at random and non-ignorable missing data mechanism assumptions of bivariate variables. We are working on the topics.