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.