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Most psychological research employs tables to report descriptive and inferential statistics. Unfortunately, those tables often misrepresent critical information on the shape and variability of the datas distribution. In addition, certain information such as the modality and score probability density is hard to report succinctly in tables and, indeed, not reported typically in published research. This paper discusses the importance of using graphical techniques not only to explore data but also to report it effectively. In so doing, the role of exploratory data analysis in detecting Type I and Type II errors is considered. A small data set resembling a Type II error is simulated to demonstrate this procedure, using a conventional parametric test. A potential analysis routine to explore data is also presented. The paper proposes that essential summary statistics and information about the shape and variability of data should be reported via graphical techniques.

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This piece is meant to help you understand and master two-level linear modeling in an accessible, swift, and fun way (while being based on rigorous and up-to-date research). It is divided into four parts: PART 1 presents the three key principles of two-level linear modeling. PART 2 presents a three-step procedure for conducting two-level linear modeling using SPSS, Stata, R, or Mplus (from centering variables to interpreting the cross-level interactions). PART 3 presents the results from a series of simulations comparing the performances of SPSS, Stata, R, and Mplus. PART 4 gives a Q&A addressing multilevel modeling issues pertaining to statistical power, effect sizes, complex design, and nonlinear two-level regression. The empirical example used in this tutorial is based on genuine data pertaining to ʼ90s and post-ʼ00s boy band member hotness and Instagram popularity. In reading this paper, you will have the opportunity to win a signed picture of Justin Timberlake.

Effect sizes and confidence intervals are important statistics to assess the magnitude and the precision of an effect. The various standardized effect sizes can be grouped in three categories depending on the experimental design: measures of the difference between two means (the $d$ family), measures of strength of association (e.$g$., $r$, $R^2$, $\eta ^2$, $\omega ^2$), and risk estimates (e.g., odds ratio, relative risk, phi; Kirk, 1996). Part I of this study reviews the $d$ family, with a special focus on Cohen's $d$ and Hedges' $g$ for two-independent groups and two-repeated measures (or paired samples) designs. The present paper answers questions concerning the $d$ family via Monte Carlo simulations. First, four different denominators are often proposed to standardize the mean difference in a repeated measures design. Which one should be used? Second, the literature proposes several approximations to estimate the standard error. Which one most closely estimates the true standard deviation of the distribution? Lastly, central and noncentral methods have been proposed to construct a confidence interval around $d$. Which method leads to more precise coverage, and how to calculate it? Results suggest that the best way to standardize the effect in both designs is by using the pooled standard deviation in conjunction with a correction factor to unbias $d$. Likewise, the best standard error approximation is given by substituting the gamma function from the true formula by its approximation. Lastly, results from the confidence interval simulations show that, under the normality assumption, the noncentral method is always superior, especially with small sample sizes. However, the central method is equivalent to the noncentral method when $n$ is greater than 20 in each group for a between-group design and when $n$ is greater than 24 pairs of observations for a repeated measures design. A practical guide to apply the findings of this study can be found after the general discussion.

A common problem in displaying within-subject data is that of how to show confidence intervals that accurately reflect the pattern of significant differences between conditions. The Cousineau-Morey method \parencite {c05,m08} is a widely used solution to this issue; however, this method only applies to experimental designs with only one repeated-measures factor (e.g., subjects). Many experimental designs in fields such as psycholinguistics and social psychology use crossed random effect designs where, e.g., there are repeated measures both for subjects and stimuli. For such designs, extant methods for showing within-subject intervals would require first aggregating over stimuli, and thus such intervals might be a less accurate reflection of the statistical significance patterns if the data are actually analyzed using a method that takes both random effects into account (e.g., linear mixed-effects models). The present paper proposes an extension of the method described above to address this problem; the proposal is to scale the data using a mixed-effects model, rather than using the means from each subject, and then calculate confidence intervals from the data scaled thusly. Analysis of a sample of crossed random effect datasets reveals that intervals calculated using this method give a slightly more accurate reflection of the pattern of statistical significance in the between-condition differences.

The results of the method for estimating confidence intervals according to Steiger and Fouladi (1997) were not displayed correctly in our published paper (Figure 7 in Appendix C). We correct the figure here. The new figure changes our interpretation of the method to a more favorable appreciation.