While many different types of distributions exist (e.g., normal, binomial, Poisson), working with SEM generally only needs to distinguish normal from non-normal distributions.

PLS-SEM is a nonparametric statistical method. Different from maximum likelihood (ML)–based CB-SEM, it does not require the data to be normally distributed.

However, it is nevertheless worthwhile to consider the distribution when working with PLS-SEM.

"It is important to verify that the data are not too far from normal as extremely non-normal data prove problematic in the assessment of the parameters’ significance. Specifically, extremely non-normal data inflate standard errors obtained from bootstrapping and thus decrease the likelihood that some relationships will be assessed as significant"

(Hair et al., 2016)

What is Normality?

• Refers to the shape of data distribution for individual variable.

• If variation from normal distribution is large, all resulting statistical tests are invalid, because F & t-statistics assume normality (Hair et al., 2010).

• Normality can have serious effects in small samples (<50), but the impact effectively diminishes when sample sizes > 200.

How to Test for Normality?

• Histogram: Compare the observed data values with a distribution approximating normal distribution.

• Normal Probability Plot: Compare the cumulative distribution of actual data values with the cumulative distribution of a normal distribution.

• Skewness and Kurtosis Statistics.

• Shapiro-Wilks (sample < 2000).

• Kolmogorov-Smirnov (sample > 2000).

Skewness

The degree of symmetry in the variable distribution.

Threshold: -2 ≤ skewness ≤ 2 (Curran et al., 1996; West et al., 1995; Gliselli et al., 1981).

Kurtosis

The degree peakedness/flatness in the variable distribution.

Threshold: -7 ≤ Kurtosis ≤ 7 (Curran et al., 1996; West et al., 1995).

Data is extremely not normal, how to remedy?

• Check and remove outlier cases.

• Remove non-normal item from the model.

• Bootstrapping (i.e., re-sampling process in the existing data-set with replacement).

Multivariate Outlier

To test for multivariate outliers, Hair et al. (2010) and Byrne (2010) suggested to identify the extreme score on two or more constructs by using Mahalanobis distance (Mahalanobis D²). It evaluates the position of a particular case from the centroid of the remaining cases. Centroid is defined as the point created by the means of all the variables (Tabachnick & Fidell, 2007).

Based on a rule of thumb, the maximum Mahalanobis distance should not exceed the critical chi-square value, given the number of predictors as degree of freedom. Otherwise, the data may contain multivariate outliers (Hair, Tatham, Anderson, & Black, 1998)