Jaume Arnau

Using the linear mixed model to analyze nonnormal data distributions in longitudinal designs

Using a Monte Carlo simulation and the Kenward–Roger (KR) correction for degrees of freedom, in this article we analyzed the application of the linear mixed model (LMM) to a mixed repeated measures design. The LMM was first used to select the covariance structure with three types of data distribution: normal, exponential, and log-normal. This showed that, with homogeneous between-groups covariance and when the distribution was normal, the covariance structure with the best fit was the unstructured population matrix. However, with heterogeneous between-groups covariance and when the pairing between covariance matrices and group sizes was null, the best fit was shown by the between-subjects heterogeneous unstructured population matrix, which was the case for all of the distributions analyzed. By contrast, with positive or negative pairings, the within-subjects and between-subjects heterogeneous first-order autoregressive structure produced the best fit. In the second stage of the study, the robustness of the LMM was tested. This showed that the KR method provided adequate control of Type I error rates for the time effect with normally distributed data. However, as skewness increased—as occurs, for example, in the log-normal distribution—the robustness of KR was null, especially when the assumption of sphericity was violated. As regards the influence of kurtosis, the analysis showed that the degree of robustness increased in line with the amount of kurtosis.

General Linear Mixed Model for Analysing Longitudinal Data in Developmental Research

Many areas of psychological, social, and health research are characterised by hierarchically structured data. Growth curves are usually represented by means of a two-level hierarchical structure in which observations are the first-level units nested within subjects, the second-level units. With data such as these, the best option for analysis is the general linear mixed model, which can be used even with longitudinal data series in which intervals are not constant or for which over the passage of time there is loss of data. In this paper an overview is given of the general linear mixed model approach to the analysis of longitudinal data in developmental research. The advantages of this model in comparison with the traditional approaches for analysing longitudinal data are shown, emphasising the usefulness of modelling the covariance structure properly to achieve a precise estimation of the parameters of the model.

The effect of skewness and kurtosis on the robustness of linear mixed models.

This study analyzes the robustness of the linear mixed model (LMM) with the Kenward–Roger (KR) procedure to violations of normality and sphericity when used in split-plot designs with small sample sizes. Specifically, it explores the independent effect of skewness and kurtosis on KR robustness for the values of skewness and kurtosis coefficients that are most frequently found in psychological and educational research data. To this end, a Monte Carlo simulation study was designed, considering a split-plot design with three levels of the between-subjects grouping factor and four levels of the within-subjects factor. Robustness is assessed in terms of the probability of type I error. The results showed that (1) the robustness of the KR procedure does not differ as a function of the violation or satisfaction of the sphericity assumption when small samples are used; (2) the LMM with KR can be a good option for analyzing total sample sizes of 45 or larger when their distributions are normal, slightly or moderately skewed, and with different degrees of kurtosis violation; (3) the effect of skewness on the robustness of the LMM with KR is greater than the corresponding effect of kurtosis for common values; and (4) when data are not normal and the total sample size is 30, the procedure is not robust. Alternative analyses should be performed when the total sample size is 30.

Short Time Series Analysis: C Statistic vs Edgington Model

Youngs C statistic (1941) makes it possible to compare the randomization of a set of sequentially organized data and constitutes an alternative of appropriate analysis in short time series designs. On the other hand, models based on the randomization of stimuli are also very important within the behavioral content applied. For this reason, a comparison is established between the C statistic and the Edgington model. The data analyzed in the comparative study have been obtained from graphs in studies published in behavioral journals. According to the results obtained, it is concluded that the Edgington model in experimental designs AB involves many measurements while the C statistic requires fewer observations to reach the conventional significance level.

Should we rely on the Kenward–Roger approximation when using linear mixed models if the groups have different distributions?

The study explores the robustness to violations of normality and sphericity of linear mixed models when they are used with the Kenward-Roger procedure (KR) in split-plot designs in which the groups have different distributions and sample sizes are small. The focus is on examining the effect of skewness and kurtosis. To this end, a Monte Carlo simulation study was carried out, involving a split-plot design with three levels of the between-subjects grouping factor and four levels of the within-subjects factor. The results show that: (1) the violation of the sphericity assumption did not affect KR robustness when the assumption of normality was not fulfilled; (2) the robustness of the KR procedure decreased as skewness in the distributions increased, there being no strong effect of kurtosis; and (3) the type of pairing between kurtosis and group size was shown to be a relevant variable to consider when using this procedure, especially when pairing is positive (i.e., when the largest group is associated with the largest value of the kurtosis coefficient and the smallest group with its smallest value). The KR procedure can be a good option for analysing repeated-measures data when the groups have different distributions, provided the total sample sizes are 45 or larger and the data are not highly or extremely skewed.

Asymptotic Distribution Free Interval Estimation for an Intraclass Correlation Coefficient with Applications to Longitudinal Data

Abstract. Confidence intervals for the intraclass correlation coefficient (ICC) have been proposed under the assumption of multivariate normality. We propose confidence intervals which do not require distributional assumptions. We performed a simulation study to assess the coverage rates of normal theory (NT) and asymptotically distribution free (ADF) intervals. We found that the ADF intervals performed better than the NT intervals when kurtosis was greater than 4. When violations of distributional assumptions were not too severe, both the intervals performed about the same. The point estimate of the ICC was robust to distributional violations. We provide R code for computing the ADF confidence intervals for the ICC.

Evaluating Effects in Short Time Series: Alternative Models of Analysis

In the applied context, short time-series designs are suitable to evaluate a treatment effect. These designs present serious problems given autocorrelation among data and the small number of observations involved. This paper describes analytic procedures that have been applied to data from short time series, and an alternative which is a new version of the generalized least squares method to simplify estimation of the error covariance matrix. Using the results of a simulation study and assuming a stationary first-order autoregressive model, it is proposed that the original observations and the design matrix be transformed by means of the square root or Cholesky factor of the inverse of the covariance matrix. This provides a solution to the problem of estimating the parameters of the error covariance matrix. Finally, the results of the simulation study obtained using the proposed generalized least squares method are compared with those obtained by the ordinary least squares approach. The probability of Type I error associated with the proposed method is close to the nominal value for all values of ρ1 and n investigated, especially for positive values of ρ1. The proposed generalized least squares method corrects the effect of autocorrelation on the tests power.

AUTOCORRELATION PROBLEMS IN SHORT TIME SERIES

The work of Huitema (1985) on autocorrelation in behavioral data suggests that the use of conventional statistical methods is justified. The present study restates the problem of autocorrelation by analyzing 100 baselines of small samples designs published in the Journal of Applied Behavior Analysis during 1992. The results show a negative bias in the autocorrelations, especially with very small samples. The autocorrelation values are normally distributed, and the method of Davies, Trigg, and Newbold (1977) is the most accurate in calculating the standard deviation.

Effect of variance ratio on ANOVA robustness: Might 1.5 be the limit?

Inconsistencies in the research findings on F-test robustness to variance heterogeneity could be related to the lack of a standard criterion to assess robustness or to the different measures used to quantify heterogeneity. In the present paper we use Monte Carlo simulation to systematically examine the Type I error rate of F-test under heterogeneity. One-way, balanced, and unbalanced designs with monotonic patterns of variance were considered. Variance ratio (VR) was used as a measure of heterogeneity (1.5, 1.6, 1.7, 1.8, 2, 3, 5, and 9), the coefficient of sample size variation as a measure of inequality between group sizes (0.16, 0.33, and 0.50), and the correlation between variance and group size as an indicator of the pairing between them (1, .50, 0, −.50, and −1). Overall, the results suggest that in terms of Type I error a VR above 1.5 may be established as a rule of thumb for considering a potential threat to F-test robustness under heterogeneity with unequal sample sizes.

Non-normal Distributions Commonly Used in Health, Education, and Social Sciences: A Systematic Review

Statistical analysis is crucial for research and the choice of analytical technique should take into account the specific distribution of data. Although the data obtained from health, educational, and social sciences research are often not normally distributed, there are very few studies detailing which distributions are most likely to represent data in these disciplines. The aim of this systematic review was to determine the frequency of appearance of the most common non-normal distributions in the health, educational, and social sciences. The search was carried out in the Web of Science database, from which we retrieved the abstracts of papers published between 2010 and 2015. The selection was made on the basis of the title and the abstract, and was performed independently by two reviewers. The inter-rater reliability for article selection was high (Cohen’s kappa = 0.84), and agreement regarding the type of distribution reached 96.5%. A total of 262 abstracts were included in the final review. The distribution of the response variable was reported in 231 of these abstracts, while in the remaining 31 it was merely stated that the distribution was non-normal. In terms of their frequency of appearance, the most-common non-normal distributions can be ranked in descending order as follows: gamma, negative binomial, multinomial, binomial, lognormal, and exponential. In addition to identifying the distributions most commonly used in empirical studies these results will help researchers to decide which distributions should be included in simulation studies examining statistical procedures.