This paper addresses two related topics in applied statistics and social psychology research. The first section outlines how a multivariate analysis of variance (MANOVA) can be applied to examine the influence of different course delivery methods — online, face-to-face, and hybrid — on students' statistical learning, stress perceptions, and tutoring behavior. Research questions, null and alternative hypotheses, and variable structures are developed. The second section reviews a meta-analysis by Pieterse et al. (2012) examining the relationship between perceived racism and mental health among Black Americans across 66 studies, covering effect sizes, moderator variables, and the study's conclusions regarding psychological distress outcomes.
A multivariate analysis of variance (MANOVA) is a statistical procedure for comparing the population means of several groups when two or more dependent variables exist (StatSoft, 2011). One of the strengths of a MANOVA is that individual p-values are produced for each dependent variable. This enables a comparison of the levels of significance of the variables and can show interactions among the dependent variables (StatSoft, 2011). It is also possible to determine whether changes to the independent variables will create significant effects on the dependent variables (StatSoft, 2011).
If we consider the question about the influence of course delivery methods on the academic performance of students enrolled in statistics classes, we may find a MANOVA approach helpful in our analysis. The three course delivery methods under consideration are: online, face-to-face, and hybrid. The course delivery methods serve as the independent variable, and the statistical learning of the students is the dependent variable. Generally, when the topic of a research study is student learning — not simply performance on a single assessment — we are interested in both pre-test and post-test scores. Adding a repeated measures component to the research design can clarify the extent of learning that students achieved.
Another independent variable added to this analysis is college major, along with the number of mathematics courses previously taken. A second dependent variable is tutoring requests made during the course, and a third dependent variable is students' feelings about the course — specifically, distress or confidence — rated on a Likert scale administered alongside each test. Through these additions, the MANOVA can assist in analyzing the three types of course delivery with regard to effectiveness for students at various levels of competency. With these changes — adopting a MANOVA approach and adding the independent variable of entry-level statistics knowledge — we are ready to articulate the research questions and hypotheses.
As before, the dependent variable reflects learning of statistical skills and may be measured as a quantitative, interval variable through a structural skills test (SST), which is a 100-item test scoring students on a scale of 1 to 100. In this version of the research design, the SST will be administered to students on the first and last days of the courses, regardless of course delivery type.
RQ1: How are the various course delivery methods related to students' pre- and post-statistics skills?
The corresponding null and alternative hypotheses are:
H0: There are no discernible differences in pre- and post-statistics scores across course delivery methods.
H1: There are significant differences in pre- and post-statistics scores across course delivery methods.
RQ2: How are the various course delivery methods related to students' perceptions of stress or confidence?
The corresponding null and alternative hypotheses are:
H0: There are no discernible differences in course delivery methods and students' perceptions of stress or confidence.
H1: There are significant differences in the relationship of SST scores to perceptions of stress or confidence.
RQ3: How are students' SST scores related to their college majors?
The corresponding null and alternative hypotheses are:
H0: There are no discernible differences in SST scores across college majors.
H1: There are significant differences in SST scores across college majors.
"Variable correlation concerns and MANCOVA extension"
A MANOVA carries the same difficulties with regard to multiple post hoc comparisons as an ANOVA. Essentially, MANCOVA is an extension of ANCOVA, but the dependent variables are adjusted for differences in covariates, thereby reducing error "noise" in the analysis.
Racism is pervasive in many societies and resists efforts to eradicate it from the cultural practices and mindsets of many people. Among the many negative outcomes of racism, one of the most important is the impact that perceived racism has on the very people against whom it is directed — or the groups to which they belong. While it is natural to assume that the psychological and physiological outcomes of perceived prejudice have adverse effects on the perceivers, relatively little research exists to demonstrate these relationships conclusively. Pieterse et al. (2012) conducted a meta-analysis review of 66 studies (representing a total sample size of 18,140 participants) conducted between January 1996 and April 2011, in an effort to better understand the relationship between racism and mental health in Black Americans.
The research question and hypotheses for the meta-analysis are as follows:
RQ: What is the magnitude of the relationship between perceived racism and mental health among Black Americans?
H0: No relationship exists between perceived racism and mental health among Black Americans.
H1: A relationship does exist between perceived racism and mental health among Black Americans.
The initial search generated 138 empirical studies, which were further scrutinized for inclusion criteria and sufficient statistical information to productively determine effect sizes. Inclusion criteria required that studies specifically analyze indicators of mental health and perceived racism, focus on Black American subjects living in the United States, and have been published in peer-reviewed journals or completed as dissertations between 1996 and 2011. A coding protocol and coder training manual were developed; university students were trained to code the articles and dissertations, and inter-rater reliability of 90% was achieved.
Five moderator variables were identified. Racism scale type measured the frequency of perceived prejudicial events and the associated level of stress. Measurement precision was the reported reliability estimate on the racism scale. Sample type indicated whether subjects were drawn from the community, from colleges, or from a combined population. Publication type indicated whether the study originated from a dissertation or a peer-reviewed journal article. Outcome type indicated the effect size for mental health clusters, including: anxiety or depression; psychiatric symptoms; life satisfaction or self-esteem; and general distress.
Effect sizes were measured using the Pearson product-moment correlation to indicate positive correlations between perceived racism and mental distress. Using a random effects model, the effects from individual studies were aggregated to calculate an overall effect size with weighted contributions from each study. The researchers checked for publication bias using a funnel plot, which accounts for the tendency of published works to contain larger samples and statistically significant effect sizes.
"Effect sizes, moderators, and study recommendations"
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