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    The assumption of independence of observations

    Generalized Estimating Equations (GEE) Homogeneity Of Variance Independence Of Observations Assumption Kurtosis Levene's Test Non-parametric Statistics Normality Outliers P-value Skewness Statistical Assumptions

    Independence of observations

    Each participant in a sample can only be counted as one observation

    As a biostatistician, I spend a lot of time testing for normality and homogeneity of variance.

    Skewness and kurtosis statistics are used to assess the normality of a continuous variable's distribution.  A skewness or kurtosis statistic above an absolute value of 2.0 is considered to be non-normal.  Distributions are often non-normal due to outliers in the distribution.  Any observation that falls more than 3.29 standard deviations away from the mean is considered an outlier.

    Levene's Test of Equality of Variances is used to measure for meeting the assumption of homogeneity of variance. Any Levene's Test with a p-value below .05 means that the assumption has been violated.  In the event that the assumption is violated, non-parametric tests can be employed.

    There is one more important statistical assumption that exists coincident with the aforementioned two, the assumption of independence of observations.  Simply stated, this assumption stipulates that study participants are independent of each other in the analysis. They are only counted once.

    In between-subjects designs, each study participant is a mutually exclusive observation that is completely independent from all other participants in all other groups.

    For within-subjects designs, each participant is independent of other participants.  There are just multiple observations of the outcome, per participant.

    With this being said, it is prevalent for researchers to take multiple measurements of an outcome and compare these multiple measurements in an independent fashion (oftentimes with differing numbers of observations across participants) or within-subjects (ALWAYS with differing numbers of observations of the outcome).  By default, these are not independent measures and violate the assumption of independence of observations.  What is one to do?

    The answer is generalized estimating equations (GEE).  This family of statistical tests are robust to multiple observations (or correlated observations) of an outcome and can be used for between-subjects, within-subjects, factorial, and multivariate analyses.
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    Logarithmic transformations for skewed variables

    Interquartile Range Logarithmic Transformations Median Normality P-value

    Logarithmic transformations adjust skewed distributions

    Analyze skewed data using more powerful parametric statistics

    Logarithmic transformations are powerful statistical tools when employed and interpreted in the correct fashion. Transforming the distribution of a continuous variable due to violating normality allows researchers to account for outlying observations and use more powerful parametric statistics to assess any significant associations. 

    Also, some continuous variables are naturally skewed.  One particular outcome that is prevalent in medicine is LOS or length of stay in the hospital.  Most patients will be in the hospital between one and three days, VERY FEW will be in the hospital for weeks and months at a time.  In order to include these outlying patients in analyses, transformations must be performed.  Naturally skewed variables can be analyzed with parametric statistics with transformations! 

    An important thing to remember when conducting logarithmic transformations is that only the p-value associated with inferential statistics can be interpreted, NOT the means and standard deviations of the transformed observations. Instead, researchers should report the median and interquartile range for the distribution.
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    Parametric vs. non-parametric statistics

    ANCOVA Test ANOVA Test Chi-square Chi-square Goodness-of-fit Cochran-Mantel-Haenszel Cochran's Q Test Cox Regression Factorial ANOVA Friedman's ANOVA Homogeneity Of Variance Independent Samples T-test Kruskal-Wallis Log-Rank Test Logistic Regression MANCOVA Mann-Whitney U MANOVA McNemar's Test Multinomial Logistic Regression Multiple Regression Non-parametric Statistics Normality Normality Of Difference Scores Odds Ratio With 95% CI One-sample Median Tests One-sample T-test Parametric Statistics Proportional Odds Regression Relative Risk Repeated-measures ANOVA Statistical Assumptions Wilcoxon

    Parametric statistics are more powerful statistics

    Non-parametric statistics are used with categorical and ordinal outcomes

    As we continue our journey to break through the barriers associated with statistical lexicons, here is another dichotomy of popular statistical terms that are spoken commonly but not always understood by everyone.  

    Parametric statistics are used to assess differences and effects for continuous outcomes. These statistical tests include one-sample t-tests, independent samples t-tests, one-way ANOVA, repeated-measures ANOVA, ANCOVA, factorial ANOVA, multiple regression, MANOVA, and MANCOVA. 

    Non-parametric statistics are used to assess differences and effects for:

    1. Ordinal outcomes - One-sample median tests, Mann-Whitney U, Wilcoxon, Kruskal-Wallis, Friedman's ANOVA, Proportional odds regression

    2. Categorical outcomes - Chi-square, Chi-square Goodness-of-fit, odds ratio, relative risk, McNemar's, Cochran's Q, Kaplan-Meier, log-rank test, Cochran-Mantel-Haenszel, Cox regression, logistic regression, multinomial logistic regression

    3. Small sample sizes (n < 30) - Smaller sample sizes make it harder to meet the statistical assumptions associated with parametric statistics.  Non-parametric statistics can generate valid statistical inferences in these situations.

    4. Violations of statistical assumptions for parametric tests - Normality, Homogeneity of variance, Normality of difference scores
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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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    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.
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    Meeting statistical assumptions

    Homogeneity Of Variance Kurtosis Levene's Test Missing Data Normality Skewness Statistical Assumptions

    Meeting statistical assumptions is IMPORTANT

    Statistics is a flawed mathematical science and assumptions MUST be met

    I've read in the literature that somewhere between 30-90% of all statistics reported in the medical literature are incorrectly conducted. First of all, that's a WIDE range and either extreme should be pretty frightening to consumers of healthcare and other related services. If your practitioner is using evidence-based practices, then one would hope that your treatment regimen does NOT fall within that range!

    Many times, statistics are incorrect because researchers do not check for the statistical assumptions associated with using their statistical tests. There are three fundamental statistical assumptions that all researchers should check before running any type of statistic:

    1. Normality - If you are using ANY continuous variables, then use skewness and kurtosis statistics to assess their normality. Any variables that have a skewness or kurtosis statistics above an absolute value of 2.0 are assumed to be non-normal.

    2. Homogeneity of variance - If you are using between-subjects analyses to compare independent groups on a continuous outcome, then use Levene's test to check for meeting the assumption of homogeneity of variance between your independent groups. This assumption assesses if the independent groups have similar variances associated with the outcome. If the p-value for Levene's test is LESS THAN .05, then the assumption has been violated.  

    3. "Missingness" - Missing data is a constant battle when conducting research. There are a litany of different reasons that lead to missing data but regardless, missing data can skew the results of a study by under-representation of the population of interest. If ANY of your variables have MORE THAN 20% of their observations missing, then that variable should be discarded.