Nonparametric procedures for partially paired data in two groups

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Abstract

A fully nonparametric method is developed for comparing samples with partially paired data. Partially-paired (correlated) data naturally arise, for example, as a result of missing values, in incomplete block designs or meta analysis. In the nonparametric setup, treatment effects are characterized in terms of functionals of distribution functions and the only assumption needed is that the marginal distributions are non-degenerate. The setup accommodates binary, ordered categorical, discrete and continuous data in a seamless fashion. The use of nonparametric effects addresses the Behrens–Fisher problem from the nonparametric point of view and allows construction of confidence intervals. Although the nonparametric methods are mainly asymptotic, methods for small sample approximations are also proposed. Size and power simulation results show numerical evidence of favorable performance of the nonparametric methods. The new nonparametric method has overwhelming power advantage when treatment effects are in the shape of the distributions and they perform comparably well with parametric methods for location-type alternatives. Data from randomized trials in public health are used to illustrate the application of the method.

Introduction

In typical two-arm randomized controlled trials or observational studies, the efficacy of the active treatment is validated by comparing it with a control treatment. Such trials are typically conducted in a pre-post setting. The primary question of interest is whether the treatment is effective. The two assessments (pre and post) of the outcome can be viewed as levels of a within-subject factor and the two groups as levels of a between-subject factor. Therefore, in the terminology of repeated measures analysis, the primary question of interest becomes investigation of time-by-treatment interaction (Davis, 2002). In practice, data may not be available at either of the assessment points. Such instances may arise in randomized incomplete block designs or meta analysis. For an example of a randomized incomplete block design (RIBD), consider the special case of two units per block. These units could be two organs of the same person or a pair of twins. Suppose a unit is randomly selected from the two units and a treatment is applied to it whereas the other unit does not receive the treatment. For some of the blocks, it may happen that one of the units may not be available due to reasons other than treatment assignment or aspects unrelated to the outcome of interest, for example, due to transportation problems or work schedules. This missing mechanism is completely at random in the sense that data are missing for reasons unrelated to the study variables. This design leads to data for which the proposed methods can be applied. For another example, suppose the data to be analyzed are obtained from multiple sources where for some of the sources both versions of a treatment have been applied and for the others only one version of the treatment is applied. In the classical linear model paradigm, the corresponding models would include a random cluster (block) effect. In the nonparametric setting, the random effects are typically accounted for by allowing a general dependence (covariance) model.

Under the assumption of normality, data analysis for the above inferential problem can be carried out by a t test with equal or unequal variances as appropriate. In the absence of normality, Generalized Estimating Equations (GEE) or Generalized Linear Mixed Model (GLMM) can be applied if the data fit certain parametric models (Diggle et al., 2013). When data are not measured using a metric scale (such as ordered categorical data), or when data have heavy tails or are skewed for any parametric model to be appropriate, nonparametric methods are preferred. Often, nonparametric methods have as much power as parametric methods when the assumptions of parametric methods are met, and have even more power when the data come from non-normal populations (DeWayne et al., 2010).

Modern nonparametric methods forego parametric or semi-parametric assumptions by formulating hypotheses in terms of distribution functions or some suitable functionals of the distribution functions. These functionals, referred to as nonparametric relative effects, are also used to quantify the magnitude of the effects of interest (Brunner et al., 2002). The use of empirical distribution functions to estimate these functionals leads to rank-based methods in a natural way. Although most of the rank-based procedures seem to sacrifice too much of the basic information in the samples, theoretical efficiency investigations have shown that this is not the case (Hodges and Lehmann, 1956, Hallin and Tribel, 2000).

Nonparametric methods for dependent data have been developed in a body of literature spanning over four decades. One of the earlier attempts was the use of rank-based methods for repeated measures by Brunner and Neumann (1982), which was later generalized by Thompson, 1990, Thompson, 1991 under continuity assumption. The idea to formulate nonparametric hypotheses for the marginal models in terms of the distribution functions, where marginal distributions are used to describe treatment effects and to formulate hypotheses, was introduced by Akritas and Arnold (1994), and further developed in the mixed effects model context by Akritas and Brunner (1997). These ideas were elaborated for repeated measures design by Brunner et al. (1999). Motivated by the difficulty in interpreting alternatives when hypotheses are formulated in terms of distribution functions, Konietschke et al. (2012) developed methods for repeated measures in one group for hypotheses formulated in terms of nonparametric relative effects. These methods allow construction of confidence intervals and they also address the nonparametric Behrens–Fisher problem (Brunner and Munzel, 2000) in the sense that the distribution of the data in the various groups could still be different under the null hypothesis. The results of Konietschke et al. (2012) were later generalized to a factorial design setup by Brunner et al. (2017).

The seemingly elementary data analysis problem discussed above could quickly turn into a challenge if the data are subject to missing values and the assumption of normality is grossly violated. In today’s data collection methods, unavailability of data from study units due to, for example, failure of devices or progression of disease, happen quite commonly. Data could be missing completely at random (MCAR ) for a reason unrelated to the study variables or missing at random (MAR) for a reason unrelated to the actual missing value, but may be related to other study variables. In this paper we focus on the former type. Albeit simple, sometimes this missing type can be motivated from design and survey perspectives, for example, see the discussion at the end of Section 2. See also Fong et al. (2017), Fuchs et al. (2017), Samawi and Vogel, 2014, Samawi and Vogel, 2015 and Xu and Harrar (2012) for examples in medicine and public health.

Under the assumptions of parametric models, Expectation–Maximization (EM) or Multiple Imputation (MI) algorithms (Little and Rubin, 2002) can be applied in conjunction with the assumed models for the complete data. Recent semi-parametric approaches that use all available information, but at least require the existence of the first few moments, include Xu and Harrar, 2012, Samawi and Vogel, 2014, Samawi and Vogel, 2015, Amro and Pauly (2017), and the references therein. The methods of Samawi and Vogel, 2014, Samawi and Vogel, 2015 are in the context of partially paired data and work by combining paired and independent sample tests for the paired and unpaired portion of the data, respectively. Apart from using permutations to determine the null distribution, Amro and Pauly (2017) also combine t statistics separately calculated from the complete and incomplete data.

Nonparametric methods for incomplete paired data received the attention of researchers fairly recently (Akritas et al., 2002, Akritas et al., 2006, Konietschke et al., 2012, Samawi and Vogel, 2014, Samawi and Vogel, 2015, Fong et al., 2017). The papers by Akritas et al., 2002, Akritas et al., 2006 introduced nonparametric tests for one and multiple-group paired data under the so-called random missingness. However, the hypotheses of interest are formulated in terms of marginal distributions and the asymptotic variances are estimated under the null hypotheses. Therefore, the tests cannot be used for construction of confidence intervals. The methods of Fong et al. (2017) and Samawi and Vogel, 2014, Samawi and Vogel, 2015, which are constructed by combining the Wilcoxon-Signed-Rank and Wilcoxon–Mann–Whitney tests, also suffer from this very problem. Further, these methods are intended for partially paired data in one group. Konietschke et al. (2012) considered a nonparametric method for partially paired data. The strengths of their method are that hypotheses are formulated in terms of the nonparametric relative effect and the asymptotic variance of the test statistic is derived under general conditions. Therefore, the asymptotic results can be used to construct confidence intervals. However, as the method is designed for a single group situation and it cannot be applied to compare treatments in two-arms and two-assessments studies. Hence, this paper attempts to address the shortcoming outlined above. More specifically, this paper aims to develop a nonparametric test for two-arm trials where the outcome, assessed at two time points, is subject to missing values. The nonparametric method accommodates binary, ordered categorical, discrete and continuous data in a unified manner. The approach can also be used to construct a confidence interval for the primary effect of interest, which is the interaction effect. Unlike some of the other methods (e.g. Samawi and Vogel, 2014, Samawi and Vogel, 2015, Xu and Harrar, 2012, Amro and Pauly, 2017), the new method compares the complete and incomplete pieces of the data to construct a test and a confidence interval. For further clarity we display the schematic layout of the dataset in the paper as given in Table 1.

The remainder of the paper is organized as follows. The statistical model and the nonparametric effect size are detailed in Section 2. An estimator for the nonparametric effect is proposed and a rank-based expression for the estimator is derived. The asymptotic theory for the nonparametric method, which constitutes the main contribution of the paper, is presented in Section 3. The application of the theory for carrying out a significance test and constructing a confidence interval calls for derivation of a consistent estimator for the asymptotic variance. This is done in Section 4. The applications of the asymptotic theory for tests and confidence intervals for the nonparametric effect are outlined in Section 5. More importantly, Section 5 proposes a small-sample test and a confidence interval. Simulation studies are carried out in Section 6 to evaluate the numerical accuracy of the asymptotic results and finite sample approximations. Another aim of Section 6 is to numerically compare the performance of the nonparametric test developed and to compare its finite sample approximation with competing parametric method in terms of size and power of tests. We close Section 6 by analyzing real datasets from public health to illustrate the application of the methods. Further discussion of the methods, conclusions and recommendations are provided in Section 7. All proofs and technical  details are placed in the Appendix.

Section snippets

Model and interaction-effect size measure

Suppose there are ng subjects from group g (g=1,2) that can be partitioned into ncg complete cases with measurements at both time points, ng1 incomplete cases that are observed only at time point t=1, and ng2 incomplete cases that are observed only at time point t=2. For instance, n11 denotes the number of incomplete subjects in group 1 at time point 1. Denote the paired observations from the complete cases by (Xg1k(c),Xg2k(c)) for g=1,2, and k=1,2,,ncg. Further, denote the observations from

Asymptotic theory

Under the asymptotic framework of Assumption 1, the estimator qˆI,θ is asymptotically unbiased and strongly consistent. These facts are stated and proved in Proposition 3.

Proposition 3

Under Assumption 1,

  • (i)

    qˆI,θ is an asymptotically unbiased estimator of qI.

  • (ii)

    qˆI,θ is a strongly consistent estimator of qI, i.e. |qˆI,θqI|a.s0.

Next, we investigate the asymptotic distribution of qˆI,θ. Our method of derivation involves obtaining an asymptotic equivalent version of n(qˆI,θqI) which can be expressed as the sum of

Estimation of the asymptotic variance

In order to be able to use the asymptotic distribution result for statistical inference, we need a consistent estimator of σθ2. From the Weak Law of Large Numbers, we know that σ̃cg2pσcg2andσ̃gt2pσgt2,for g,t=1,2 where σ̃cg21ncg1k=1ncg(Zgk(c)Z̄g.(c))2andσ̃gt21ngt1k=1ngt(Ygtk(i)Ȳgt.(i))2.However, σ̃cg2 and σ̃gt2 cannot directly be used to construct a consistent estimator of σθ2 because they are defined in terms of unobservable random variables Zgk(c) and Ygtk(i). In the following, we

Test procedures and confidence intervals

Recall that in Eq. (2), the quantity pI(p12p11)(p22p21) is defined as an interaction effect. So, pI=0 means that there is no interaction effect, which in turn is equivalent to saying qI=1. Therefore, testing the hypothesis of no interaction effect can be formulated as H0:qI=1 versus H1:qI1.The statistic TM,θ and its limiting distribution, under Assumption 1, are TM,θ=n(qˆI,θqI)σˆθDN(0,1),which can be used to test the hypothesis of no interaction effect. The asymptotic (1α)100%

Simulation study

In this subsection we investigate the numerical accuracy of the asymptotic results of Sections 3 Asymptotic theory, 4 Estimation of the asymptotic variance via a simulation study. We also use simulations to perform numerical comparisons of the nonparametric test proposed in this paper with parametric tests for the same problem developed elsewhere. We carry out comparisons in terms of the size and power of the tests. In all the simulations, the run size is 10,000. The methods compared are:

    TM,θ:

Discussion and conclusion

We have developed a nonparametric method for partially paired (correlated) data. The method is particularly suited to the situation where two treatments are compared by assessing outcomes at two occasions, and it is designed for the situation where data are available at either one of the occasions. The procedures proposed in the manuscript are geared towards testing significance and constructing confidence interval for the difference in the change induced by the two treatments. The method can

Acknowledgments

The authors are grateful to the associate editor and the three anonymous referees for critically reading the original version of the manuscript and making valuable suggestions that led to substantial improvements. The authors are also thankful to the editor for the orderly handling of the manuscript. The authors would like to express their gratitude to Prof. Derek S. Young for his willingness and time to read through the final version of manuscript and make editorial suggestions. Part of the

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