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Planned missing data (PMD) designs allow researchers to collect additional data under time constraints and to reduce participant burden, both of which can occur in social, behavioral, and education...

Although the bias-corrected (BC) bootstrap is an often-recommended method for testing mediation due to its higher statistical power relative to other tests, it has also been found to have elevated Type I error rates with small sample sizes. Under limitations for participant recruitment, obtaining a larger sample size is not always feasible. Thus, this study examines whether using alternative corrections for bias in the BC bootstrap test of mediation for small sample sizes can achieve equal levels of statistical power without the associated increase in Type I error. A simulation study was conducted to compare Efron and Tibshirani’s original correction for bias, z 0, to six alternative corrections for bias: (a) mean, (b–e) Winsorized mean with 10%, 20%, 30%, and 40% trimming in each tail, and (f) medcouple (robust skewness measure). Most variation in Type I error (given a medium effect size of one regression slope and zero for the other slope) and power (small effect size in both regression slopes) was found with small sample sizes. Recommendations for applied researchers are made based on the results. An empirical example using data from the ATLAS drug prevention intervention study is presented to illustrate these results. Limitations and future directions are discussed.

Introductory statistics courses often focus on equations and null hypothesis significance testing. While this approach may work if the students already have an understanding of research methods, the reality is that for many students, an introductory statistics course will be the only methodology course they ever take. For those students, it is more important to provide a context and rationale for the use of statistics in research than to have them memorize equations. This chapter presents a case study in which a series of additional readings was used to stimulate engaged learning and provide a broader introduction to the field of statistics.

The present study examined relations between nightly bedtime routines and sleep outcome measures in a sample of 185 toddlers aged 30 months. Parents reported on their toddler's sleep duration and the length and activities included in the bedtime routine each night for approximately 2 weeks. Toddlers wore actigraphs to track their sleep during the same time period. Correlation, mean difference, and regression analyses indicated that toddlers experienced different bedtime routines and exhibited differences in parent reported sleep duration between weeknights and weekends. Multi-level models revealed that variability in the bedtime routine on an individual night most consistently affected parent reported sleep duration on that night. Differences in the bedtime routines between weeknights and weekends also affected actigraph recorded sleep duration and sleep efficiency. Results suggest that keeping consistent bedtime routines between weeknights and weekends is important for optimal sleep outcomes.

The use of anabolic-androgenic steroids (AAS) is problematic for youth because of negative effects such as reduced fertility, increased aggression and exposure to toxic chemicals. An effective programme for addressing this problem is Adolescents Training and Learning to Avoid Steroids (ATLAS). This secondary analysis expands prior research by identifying prominent mechanisms of change and highlighting key longitudinal processes that contributed to the success of ATLAS. The current sample consists of high-school football players (N = 1.068; Mage = 15.25) who began ATLAS in grades nine through eleven and participated in booster sessions for two years post-baseline. Knowledge of AAS effects, belief in media ads, reasons not to use AAS, perceived severity of and susceptibility to AAS effects and ability to resist drug offers were critical mediators of the relations between ATLAS and outcomes. Modern applications of the ATLAS programme are also discussed.

Mediation analysis requires a number of strong assumptions be met in order to make valid causal inferences. Failing to account for violations of these assumptions, such as not modeling measurement error or omitting a common cause of the effects in the model, can bias the parameter estimates of the mediated effect. When the independent variable is perfectly reliable, for example when participants are randomly assigned to levels of treatment, measurement error in the mediator tends to underestimate the mediated effect, while the omission of a confounding variable of the mediator to outcome relation tends to overestimate the mediated effect. Violations of these two assumptions often co-occur, however, in which case the mediated effect could be overestimated, underestimated, or even, in very rare circumstances, unbiased. In order to explore the combined effect of measurement error and omitted confounders in the same model, the impact of each violation on the single-mediator model is first examined individually. Then the combined effect of having measurement error and omitted confounders in the same model is discussed. Throughout, an empirical example is provided to illustrate the effect of violating these assumptions on the mediated effect.

Several years ago, the American Psychological Association began requiring that effect size estimates be reported to provide a better indication of the associative strength between factors and dependent measures in empirical studies (Publication manual of the American Psychological Association, 2010, Author, Washington, DC). Accordingly, developmental journals require/strongly recommend effect size estimates be included in published work. Potentially, this trend has important benefits for infancy research given some of the inherent difficulties in establishing conceptually strong findings when often facing highly variable performance in typically small samples. This study examined recent infant research from select journals for accuracy and interpretative value of effect size estimates. Demographics, sample size, design, and statistical data were coded from 158 published (2007–2012) articles presenting 878 effect size estimates from experimental findings with infants using behavioral methods. Descriptive and distribution statistics were calculated for the following variables: (1) statistical tests, (2) effect size parameters, and (3) effect size interpretations. Although partial eta squared () and eta squared (η2) were most common (49 and 42%, respectively), “η confusion” was apparent, and interpretation of effect size estimates was virtually nonexistent. Thus, effect size estimates are not impacting infant development research in spite of criticisms of sole dependence on null hypothesis (e.g. American Psychologist, 49, 1994 and 997). Suggestions for increasing accuracy of effect size estimate selection and interpretative effect size estimate cutoffs are offered to improve empirical clarity.

As noted by Paul Jose in Doing Statistical Mediation & Moderation, although there are many articles on mediation and moderation, most do not focus on the actual procedures for estimating these mode...

This study (a) examined the role of anger rumination as a mediator of the relation between social anxiety and the experience of anger, hostility, and aggression, in the general population, and (b) evaluated the degree to which the presence of autism spectrum disorder characteristics moderates the indirect influence of anger rumination. We then explored whether social cognition and perseveration characteristic of autism spectrum disorder uniquely accounted for the predicted moderation. In this survey study of young adults ( n = 948), anger rumination mediated the relation between social anxiety and hostility, as well as verbal and physical aggression, as predicted. Greater autism spectrum disorder characteristics augmented the effect of social anxiety on hostility and physical aggression by increasing the effect of anger rumination, but not by increasing the effect of social anxiety on anger rumination. Implications for developing treatment approaches that target hostility and aggression among young adults who may not be formally diagnosed but have characteristics of autism spectrum disorder are discussed.

Inadequate statistical power to detect treatment effects in health research is a problem that is compounded when testing for mediation. In general, the preferred strategy for increasing power is to increase the sample size, but there are many situations where additional participants cannot be recruited, necessitating the use of other methods to increase statistical power. Many of these other strategies, commonly applied to analysis of variance and multiple regression models, can be applied to mediation models with similar results. Additional predictors or blocking variables will increase or decrease statistical power, however, depending on whether these variables are related to the mediator, the outcome, or both. The effect of these two methods on the power for tests of mediation is illustrated through the use of simulations. Implications for health researchers using these methods are discussed.

Moderation occurs when the magnitude and/or direction of the relation between two variables depend on the value of a third variable called a moderator variable. Moderator variables are distinct from mediator variables, which are intermediate variables in a causal chain between two other variables, and confounder variables, which can cause two otherwise unrelated variables to be related. Determining whether a variable is a moderator of the relation between two other variables requires statistically testing an interaction term. When the interaction term contains two categorical variables, analysis of variance (ANOVA) or multiple regression may be used, though ANOVA is usually preferred. When the interaction term contains one or more continuous variables, multiple regression is used. Multiple moderators may be operating simultaneously, in which case higher-order interaction terms can be added to the model, though these higher-order terms may be challenging to probe and interpret. In addition, interaction effects are often small in size, meaning most studies may have inadequate statistical power to detect these effects. When multilevel models are used to account for the nesting of individuals within clusters, moderation can be examined at the individual level, the cluster level, or across levels in what is termed a cross-level interaction. Within the structural equation modeling (SEM) framework, multiple group analyses are often used to test for moderation. Moderation in the context of mediation can be examined using a conditional process model, while moderation of the measurement of a latent variable can be examined by testing for factorial invariance. Challenges faced when testing for moderation include the need to test for treatment by demographic or context interactions, the need to account for excessive multicollinearity, and the need for care when testing models with multiple higher-order interactions terms.

Mediator variables are variables that lie between the cause and effect in a causal chain. In other words, mediator variables are the mechanisms through which change in one variable causes change in a subsequent variable. The single-mediator model is deceptively simple because it has only three variables: an antecedent, a mediator, and a consequent. Determining that a variable functions as a mediator is a difficult process, however, because causation can be inferred only when many strict assumptions are met, including, but not limited to, perfectly reliable measures, correct temporal design, and no omitted confounders. Since many of these assumptions are difficult to assess and rarely met in practice, the significance of a statistical test of mediation alone usually provides only weak evidence of mediation. New methodological approaches are constantly being developed to circumvent these limitations. Specifically, new methods are being created for the following purposes: (1) to assess the impact of violating assumptions (e.g., sensitivity analyses) and (2) to make fewer assumptions and provide more flexible analysis techniques (e.g., Bayesian analysis or bootstrapping) that may be more robust to assumption violations. Despite these advances, the importance of the design of a study cannot be overstated. A statistical analysis, no matter how sophisticated, cannot redeem a study that measured the wrong variables or used an incorrect temporal design.

Developmental psychopathology research often involves hypotheses about multiple variables interacting and affecting each other over time to produce adaptive and maladaptive behavior and thus analytic methods for examining moderation and mediation in longitudinal data are particularly germane in this area. This chapter describes methods for testing moderation and mediation in cross sectional data, outlines two frameworks frequently used for the analysis of longitudinal data, and describes a number of models for testing longitudinal moderation and mediation, including ax lag as moderator model, multilevel models with time-varying covariate interactions, a model of mediation change over time, autoregressive panel models, latent growth curve models, latent change score models, and exponential decay models. Methods for combining moderation and mediation and causal inference in longitudinal data are also discussed. Illustrative analyses are conducted using example data on kindergarten through fifth-graders' attention-deficit/hyperactivity disorder (ADHD) symptoms, interpersonal skills, academic performance, and depression variables from the ECLS-K data set. Keywords: mediation; moderation; longitudinal data; interactions; change over time

Mediation models are widely used, and there are many tests of the mediated effect. One of the most common questions that researchers have when planning mediation studies is, “How many subjects do I need to achieve adequate power when testing for mediation?” This article presents the necessary sample sizes for six of the most common and the most recommended tests of mediation for various combinations of parameters, to provide a guide for researchers when designing studies or applying for grants.

On mental timing tasks, erroneous knowledge of results (KR) leads to incorrect performance accompanied by the subjective judgment of accurate performance. Using the start-stop technique (an analogue of the peak interval procedure) with both reproduction and production timing tasks, the authors analyze what processes erroneous KR alters. KR provides guidance (performance error information) that lowers decision thresholds. Erroneous KR also provides targeting information that alters response durations proportionately to the magnitude of the feedback error. On the production task, this shift results from changes in the reference memory, whereas on the reproduction task this shift results from changes in the decision threshold for responding. The idea that erroneous KR can alter different cognitive processes on related tasks is supported by the authors' demonstration that the learned strategies can transfer from the reproduction task to the production task but not visa versa. Thus effects of KR are both task and context dependent.

This paper investigates baseline by treatment interactions (BTI) of a randomized anabolic steroid prevention program delivered to high school football players. Baseline by treatment interactions occur when a participant's score on an outcome variable is associated with both their pretreatment standing on the outcome variable and the treatment itself. The program was delivered to 31 high school football teams (Control=16, Treatment=15) in Oregon and Washington over the course of 3 years (Total N=3207). Although most interactions were nonsignificant, consistent baseline by treatment interactions were obtained for knowledge of the effects of steroid use and intentions to use steroids. Both of these interactions were beneficial in that they increased the effectiveness of the program for participants lower in knowledge and higher in intentions at baseline.

Mediation analysis is often used to investigate mechanisms of change in prevention research. Results finding mediation are strengthened when longitudinal data are used because of the need for temporal precedence. Current longitudinal mediation models have focused mainly on linear change, but many variables in prevention change nonlinearly across time. The most common solution to nonlinearity is to add a quadratic term to the linear model, but this can lead to the use of the quadratic function to explain all nonlinearity, regardless of theory and the characteristics of the variables in the model. The current study describes the problems that arise when quadratic functions are used to describe all nonlinearity and how the use of nonlinear functions, such as exponential decay, addresses many of these problems. In addition, nonlinear models provide several advantages over polynomial models including usefulness of parameters, parsimony, and generalizability. The effects of using nonlinear functions for mediation analysis are then discussed and a nonlinear growth curve model for mediation is presented. An empirical example using data from a randomized intervention study is then provided to illustrate the estimation and interpretation of the model. Implications, limitations, and future directions are also discussed.

Previous studies of different methods of testing mediation models have consistently found two anomalous results. The first result is elevated Type I error rates for the bias-corrected and accelerated bias-corrected bootstrap tests not found in nonresampling tests or in resampling tests that did not include a bias correction. This is of special concern as the bias-corrected bootstrap is often recommended and used due to its higher statistical power compared with other tests. The second result is statistical power reaching an asymptote far below 1.0 and in some conditions even declining slightly as the size of the relationship between X and M, a, increased. Two computer simulations were conducted to examine these findings in greater detail. Results from the first simulation found that the increased Type I error rates for the bias-corrected and accelerated bias-corrected bootstrap are a function of an interaction between the size of the individual paths making up the mediated effect and the sample size, such that elevated Type I error rates occur when the sample size is small and the effect size of the nonzero path is medium or larger. Results from the second simulation found that stagnation and decreases in statistical power as a function of the effect size of the a path occurred primarily when the path between M and Y, b, was small. Two empirical mediation examples are provided using data from a steroid prevention and health promotion program aimed at high school football players (Athletes Training and Learning to Avoid Steroids; Goldberg et al., 1996), one to illustrate a possible Type I error for the bias-corrected bootstrap test and a second to illustrate a loss in power related to the size of a. Implications of these findings are discussed.

Mediation analysis is widely used in the social sciences. Despite the popularity of mediation models, few researchers have used graphical methods, other than structural path diagrams, to represent their models. Plots of the mediated effect can help a researcher better understand the results of the analysis and convey these results to others. This article presents a method for creating and interpreting plots of the mediated effect for a variety of mediation models, including models with (1) a dichotomous independent variable, (2) a continuous independent variable, and (3) an interaction between an independent variable and the mediating variable. An empirical example is then presented to illustrate these plots. Sample code for creating plots of the mediated effect in R and SAS is also included, and may be downloaded from www.psychonomic.org/archive.

This article describes a program, PRODCLIN (distribution of the PRODuct Confidence Limits for INdirect effects), written for SAS, SPSS, and R, that computes confidence limits for the product of two normal random variables. The program is important because it can be used to obtain more accurate confidence limits for the indirect effect, as demonstrated in several recent articles (MacKinnon, Lockwood, & Williams, 2004; Pituch, Whittaker, & Stapleton, 2005). Tests of the significance of and confidence limits for indirect effects based on the distribution of the product method have more accurate Type I error rates and more power than other, more commonly used tests. Values for the two paths involved in the indirect effect and their standard errors are entered in the PRODCLIN program, and distribution of the product confidence limits are computed. Several examples are used to illustrate the PRODCLIN program. The PRODCLIN programs in rich text format may be downloaded from www.psychonomic.org/archive.