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    Non-parametric statistics and small sample sizes

    Friedman's ANOVA Kruskal-Wallis Mann-Whitney U Non-parametric Statistics Psychometric Tests Sample Size Wilcoxon

    Non-parametric statistics are robust to small sample sizes

    The right way to conduct statistics

    Mark Twain said it best, "There are lies, damn lies, and statistics." Statistics can be misleading from both the standpoint of the person conducting the statistics and the person that is interpreting the analyses. Many between-subjects studies have small sample sizes (n < 20) and statistical assumptions for parametric statistics cannot be met.

    For basic researchers that operate day in and day out with small sample sizes, the answer is to use non-parametric statistics. Non-parametric statistical tests such as the Mann-Whitney U, Kruskal-Wallis, Wilcoxon, and Friedman's ANOVA are robust to violations of statistical assumptions and skewed distributions. These tests can yield interpretable medians, interquartile ranges, and p-values.

    Non-parametric statistics are also useful in the social sciences due to the inherent measurement error associated with assessing human behaviors, thoughts, feelings, intelligence, and emotional states. The underlying algebra associated with psychometrics relies on intercorrelations amongst constructs or items.  Correlations can easily be skewed by outlying observations and measurement error.  Therefore, in concordance with mathematical and empirical reasoning, non-parametric statistics should be used often for between-subjects comparisons of surveys, instruments, and psychological measures.
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    Ordinal measures becoming continuous with normality

    ANOVA Test Chi-square Classical Test Theory Continuous Friedman's ANOVA Independent Samples T-test Item Response Theory Kruskal-Wallis Kurtosis Logistic Regression Mann-Whitney U Non-parametric Statistics Normality Ordinal Parametric Statistics Regression Repeated-measures ANOVA Skewness Wilcoxon

    Ordinal measures and normality

    Ordinal level measurement can become interval level with assumed normality

    Here is an interesting trick I picked up along the way when it comes to ordinal outcomes and some unvalidated measures. If you run skewness and kurtosis statistics on the ordinal variable and its distribution meets the assumption of normality (skewness and kurtosis statistics are less than an absolute value of 2.0), then you can "upgrade" the variable to a continuous level of measurement and analyze it using more powerful parametric statistics.  

    This type of thinking is the reason that the SAT, ACT, GRE, MCAT, LSAT, and validated psychological instruments are perceived at a continuous level. The scores yielded from these instruments, by definition, are not continuous because a "true zero" does not exist. Scores from these tests are often norm- or criterion-referenced to the population so that they can be interpreted in the correct context. Therefore, with the subjectivity and measurement error associated with classical test theory and item response theory, the scores are actually ordinal.

    With that being said, if the survey instrument or ordinal outcome is used in the empirical literature often and it meets the assumption of normality as per skewness and kurtosis statistics, treat the ordinal variable as a continuous variable and run analyses using parametric statistics (t-tests, ANOVA, regression) versus non-parametric statistics (Chi-square, Mann-Whitney U, Kruskal-Wallis, McNemar's, Wicoxon, Friedman's ANOVA, logistic regression). 
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    Statistical Designs

    ANOVA Test Between-subjects Chi-square Chi-square Goodness-of-fit Friedman's ANOVA Independent Samples T-test Kruskal-Wallis Mann-Whitney U Odds Ratio With 95% CI Relative Risk Repeated-measures ANOVA Repeated-measures T-test Wilcoxon Within-subjects

    Research questions lead to choice of statistical design

    Differences between-subjects and within-subjects designs

    There are terms in statistics that many people do not understand from a practical standpoint. I'm a biostatistical scientist and it took me YEARS to wrap my head around some fundamental aspects of statistical reasoning, much less the lexicon. I would hypothesize that 90% of the statistics reported in the empirical literature as a whole fall between two different categories of statistics, between-subjects and within-subjects. Here is a basic breakdown of the differences in these types of statistical tests:

    1. Between-subjects - When you are comparing independent groups on a categorical, ordinal, or continuous outcome variable, you are conducting between-subjects analyses. The "between-" denotes the differences between mutually exclusive groups or levels of a categorical predictor variable. Chi-square, Mann-Whitney U, independent-samples t-tests, odds ratio, Kruskal-Wallis, and one-way ANOVA are all considered between-subjects analyses because of the comparison of independent groups.  

    2. Within-subjects - When you are comparing THE SAME GROUP on a categorical, ordinal, or continuous outcome ACROSS TIME OR WITHIN THE SAME OBJECT OF MEASUREMENT MULTIPLE TIMES, then you are conducting within-subjects analyses. The "within-" relates to the differences within the same object of measurement across multiple observations, time, or literally, "within-subjects." Chi-square Goodness-of-fit, Wilcoxon, repeated-measures t-tests, relative risk, Friedman's ANOVA, and repeated-measures ANOVA are within-subjects analyses because the same group or cohort of individuals is measured at several different time-points or observations.
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    Non-parametric statistics as post hoc tests

    Homogeneity Of Variance Kruskal-Wallis Mann-Whitney U Non-parametric Statistics Normality Post Hoc Statistical Assumptions Wilcoxon

    Mann-Whitney U and Wilcoxon as post hoc tests

    Explain significant main effects from Kruskal-Wallis tests and Friedman's ANOVA

    Non-parametric statistics are used when analyzing categorical and ordinal outcomes.  These statistics are also used with smaller sample sizes (n < 20) and when the assumptions of certain statistical tests are violated.

    The Mann-Whitney U test is employed when comparing two independent groups on an ordinal outcome.  It is also used when the assumptions of an independent samples or unpaired t-test are violated (normality, homogeneity of variance).

    The Wilcoxon test is used when comparing ordinal outcomes at two different points in time or within-subjects.  It is further used when the assumptions of a repeated measures t-test are violated (independence of observations, normality of difference scores).

    A lesser known use for these two non-parametric tests is when significant main effects are found for non-parametric Kruskal-Wallis and Friedman's ANOVA tests.  Much like with a parametric one-way ANOVA or repeated-measures ANOVA, if a significant main effect is found using non-parametric statistics, then a post hoc analysis must be undertaken to explain the significant main effect. Non-parametric statistics do not have Tukey, Scheffe, and Dunnett tests like parametric statistics!

    When a significant main effect is found using a Kruskal-Wallis test, subsequent Mann-Whitney U tests must be employed in a post hoc fashion to explain where amongst the independent groups the actual differences exist.

    The same holds true for Friedman's ANOVA.  If a significant main effect is found, then Wilcoxon tests must be used in a post hoc fashion to explain where the significant changes occur amongst the observations or within-subjects.