Single-case experiments have become increasingly popular in psychological and educational research. However, the analysis of single-case data is often complicated by the frequent occurrence of missing or incomplete data. If missingness or incompleteness cannot be avoided, it becomes important to know which strategies are optimal, because the presence of missing data or inadequate data handling strategies may lead to experiments no longer \"meeting standards\" set by, for example, the What Works Clearinghouse. For the examination and comparison of strategies to handle missing data, we simulated complete datasets for ABAB phase designs, randomized block designs, and multiple-baseline designs. We introduced different levels of missingness in the simulated datasets by randomly deleting 10%, 30%, and 50% of the data. We evaluated the type I error rate and statistical power of a randomization test for the null hypothesis that there was no treatment effect under these different levels of missingness, using different strategies for handling missing data: (1) randomizing a missing-data marker and calculating all reference statistics only for the available data points, (2) estimating the missing data points by single imputation using the state space representation of a time series model, and (3) multiple imputation based on regressing the available data points on preceding and succeeding data points. The results are conclusive for the conditions simulated: The randomized-marker method outperforms the other two methods in terms of statistical power in a randomization test, while keeping the type I error rate under control.

Multilevel models (MLMs) have been proposed in single-case research, to synthesize data from a group of cases in a multiple-baseline design (MBD). A limitation of this approach is that MLMs require several statistical assumptions that are often violated in single-case research. In this article we propose a solution to this limitation by presenting a randomization test (RT) wrapper for MLMs that offers a nonparametric way to evaluate treatment effects, without making distributional assumptions or an assumption of random sampling. We present the rationale underlying the proposed technique and validate its performance (with respect to Type I error rate and power) as compared to parametric statistical inference in MLMs, in the context of evaluating the average treatment effect across cases in an MBD. We performed a simulation study that manipulated the numbers of cases and of observations per case in a dataset, the data variability between cases, the distributional characteristics of the data, the level of autocorrelation, and the size of the treatment effect in the data. The results showed that the power of the RT wrapper is superior to the power of parametric tests based on F distributions for MBDs with fewer than five cases, and that the Type I error rate of the RT wrapper is controlled for bimodal data, whereas this is not the case for traditional MLMs.

The conditional power (CP) of the randomization test (RT) was investigated in a simulation study in which three different single-case effect size (ES) measures were used as the test statistics: the mean difference (MD), the percentage of nonoverlapping data (PND), and the nonoverlap of all pairs (NAP). Furthermore, we studied the effect of the experimental design on the RT\'s CP for three different single-case designs with rapid treatment alternation: the completely randomized design (CRD), the randomized block design (RBD), and the restricted randomized alternation design (RRAD). As a third goal, we evaluated the CP of the RT for three types of simulated data: data generated from a standard normal distribution, data generated from a uniform distribution, and data generated from a first-order autoregressive Gaussian process. The results showed that the MD and NAP perform very similarly in terms of CP, whereas the PND performs substantially worse. Furthermore, the RRAD yielded marginally higher power in the RT, followed by the CRD and then the RBD. Finally, the power of the RT was almost unaffected by the type of the simulated data. On the basis of the results of the simulation study, we recommend at least 20 measurement occasions for single-case designs with a randomized treatment order that are to be evaluated with an RT using a 5% significance level. Furthermore, we do not recommend use of the PND, because of its low power in the RT.