Supplementary Material to : Ranking Procedures for Repeated Measures Designs with Missing Data-Estimation , Testing and Asymptotic Theory

We develop purely nonparametric methods for the analysis of repeated measures designs with missing values. Hypotheses are formulated in terms of purely nonparametric treatment effects. In particular, data can have different shapes even under the null hypothesis and therefore, a solution to the nonparametric Behrens-Fisher problem in repeated measures designs will be presented. Moreover, global testing and multiple contrast test procedures as well as simultaneous confidence intervals for the treatment effects of interest will be developed. All methods can be applied for the analysis of metric, discrete, ordinal, and even binary data in a unified way. Extensive simulation studies indicate a satisfactory control of the nominal type-I error rate, even for small sample sizes and a high amount of missing data (up to 30%). We apply the newly developed methodology to a real data set, demonstrating its application and interpretation.


the ATS
in (17) with the proposed F -approximation, 3. the ATS (2) in (18) with the proposed F -approximation, 4. the MCTP T 0 in (19) with a t n−1 (0, R n ) approximation, and compared them with 5. the WTS and ATS statistics for testing H F 0 as proposed by Domhof et al. 1 in different homo-and heteroscedastic repeated measures designs with different rates of missing values. Even though Domhof et al. 1 reported a liberal behavior of the WTS (for testing H F 0 ), we added the method as a competing procedure for completeness. We thus also investigated their robustness to variance heteroscedasticity. Since all of the methods above use all-available data, we additionally compared them with two MCTPbased approaches: a complete case analysis and a naive imputation approach, in which we either 6. deleted the whole observation vector X k of subject k if any X ik was missing (λ ik = 0), or 7. if X ik was missing (λ ik = 0), we calculated median(λ i1 X i1 , . . . , λ in X in ), and assigned it to X ik and set λ ik = 1.
Data have been generated using discretized, by rounding to integers, normal and lognormal distributions with varying numbers of time points d ∈ {3, 4}, sample sizes n ∈ {20, 30, 50}, amount of missing values r ∈ {0%, 10%, 30%} and six different types of covariance matrices The covariance matrices were chosen to model a broad selection of dependency patterns, including homoscedastic (Σ 1 and Σ 2 ) as well as heteroscedastic marginals. Note that H F 0 holds only under Σ 1 and Σ 2 . We furthermore investigate the methods sensitivity to both MCAR and MAR data to cover realistic scenarios. In order to generate the former, we multiplied the observations with randomly chosen indicators λ ik ∼ B(1 − r), with a zero entry being interpreted as a missing observation, whereas we followed Santos et. al 2 for the latter. Hereby we defined pairs of observations {X obs , X miss }, where X obs determines the probability that X miss was actually observed. For instance, in case of d = 4 we defined the pairs {X 1k , X 2k } and {X 3k , X 4k }. Following the idea of Amro et al. 3 , we investigated two different types of MAR scenarios, MAR (1) and MAR (2). First, for the MAR (1) scenario, we divided X i,obs into three groups: (1) i is the variance of X i,obs . Then, we assigned a missing rate of 10% to the first and third group and a missing rate of 30% to the second group. Second, in the MAR (2) scenario, data was divided into two groups using the median, following the idea of Zhu et al 4 . Specifically, we defined (1) {X ik = X i,obs ∈ (−∞, median(X i,obs ), k = 1, ..., n} and (2) {X ik = X i,obs ∈ (median(X i,obs ), ∞), k = 1, ..., n}. Here, we assigned a missing rate of 0% to the first group and a missing rate of 10% to the second group.
In order to investigate the power of the procedures, a simulation study was conducted using four-dimensional normal and log-normal distributions with µ = (µ 1 , µ 2 , µ 3 , µ 4 ) and covariance matrices Σ 2 , Σ 4 , Σ 5 , Σ 6 . In particular, three different types of shiftalternatives were considered with ranging δ = (0.2, 0.4, 0.6, 0.8, 1, 1.5) and different amount of missing values. As the WTS turned out to be inappropriate for small sample sizes, it was not included into the power analysis. Moreover, since the second version of the ATS for testing H p 0 showed a more accurate behaviour than the first version, we only present results for the second version.
For each design, 10, 000 simulation runs were performed using the R software package of statistical computing, version R 3.6.4 5 . The complete simulation code is available on https://github.com/KerstinRubarth/RM_Miss.

Type-I error
The type-I error rates under the MCAR mechanism for different sample sizes n, covariances matrices, missing rates r and discretized distributions can be found in Tables 1 -3. A graphical comparisons of rather small and large sample sizes and no, medium and high missing rates can be found in Figure 1.

Power
The power simulation results for n = 30 and Σ 4 are given in Tables 4 and 5 for discretized normal and log-normal distributions respectively. An exploration of the power under different MAR scenarios, covariance matrices, alternatives and discretized distributions can be found in Figures 2 -5.

Proofs
In this section we will provide the proofs of the theoretical results achieved in the paper. The strong consistency of the point and variance estimators follows from the following generalization of the Glivenko-Cantelli Theorem: denote the leftcontinuous and right-continuous versions of the distribution function of X i1 and let denote their empirical counterparts, where denote the normalized versions of the distribution functions as used in the manuscript. Then, is the mean of the left-and right continuous version, it follows The proof of the almost sure convergence of the last remaining terms is given by 6 , page 111. Furthermore, by triangle inequality, the convergence G − G ∞ a.s.
→ 0 follows, which completes the proof.

Proof of Proposition 1
1. The estimator p = ( p 1 , ..., p d ) is asymptotically unbiased, because 2. The strong consistency of p follows from

Proof of Theorem 1
It is sufficient to prove the theorem only for the i-th component. By adding and subtracting Gd F i and GdF i , it holds for all i = 1, . . . , d that If it can be shown, that the proof will be complete. It is technically easier to proof the stronger result E( √ nZ i ) 2 → 0. Therefore, consider and thus It follows with the same arguments as used by Dobler et al. 7 (see equation (8)), Gao et al. 8 (Lemma 2.1) or Munzel 9 that for any fixed pair of (s, t) under (A1) and (A2). It remains to show that which completes the proof.

Proof of Theorem 2
Note that √ nB is a mean (multiplied by √ n) of independent random variables Furthermore, note that the random variables Ψ k are unformly bounded by Assumption (A2) in (??). Thus, the multivariate Lindeberg Feller Theorem implies that √ nB is asymptotically normal with covariance matrix V n and the result follows from Theorem 1 and Slutzky.  − − → 0, n → ∞, ∀i, First, we compute the bound of With the same arguments, it follows | Ψ ik | ≤ N 0 , |β ik | ≤ N 0 and | β ik | ≤ N 0 .
As n → ∞, it follows that