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.

Analysis using AMOS is a parametric statistical method. Different from PLS-SEM, the maximum likelihood (ML)-based CB-SEM requires the data to be normally distributed.

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).

-1 ≤ skewness ≤ 1 (Byrne, 2016; Awang, 2015)

Critical Ratio (CR) of skewness should be > 3 (Kline, 2015, Latan et al., 2020).

Kurtosis

The degree peakedness/flatness in the variable distribution.

Threshold:

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

-3 ≤ Kurtosis ≤ 3 (Awang, 2015)

CR of kurtosis should be <10 (Kline, 2015, Latan et al., 2020).

Insights from Awang (2015)

The normality can be assessed by assessing the measure of skewness for every item. The absolute value of skewness 1.0 or lower indicates the data is normally distributed. However, SEM using the Maximum Likelihood Estimator (MLE) like AMOS is fairly robust to skewness greater than 1.0 in absolute value if the sample size is large. Meaning, the researcher could proceed into further analysis (SEM) since the estimator used is MLE. Normally, the sample size greater than 200 is considered large enough in MLE even though the data distribution is slightly non-normal.

Another method for normality assessment is by looking at the multivariate kurtosis statistic. However, SEM using MLE is also robust to kurtotic violations of multivariate normality as long the sample size is large.

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).

In the case when the normality assumption is not fulfilled, the researchers still have many options to take. One of them is to remove the non-normal items from the measurement model (based on the measure of skewness) and continue with the analysis. Another option is to remove the farthest observation from the center (outlier) of distribution.

However, the most popular method lately is to continue with the analysis with MLE (without deleting any item and also without removing any observation) and reconfirm the result of analysis through Bootstrapping.

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)