When we check our data set for univariate (e.g., SPSS) and Multivariate Skewness and Kurtosis as measures of non-normality as defined by Mardia (via Mardia' s web of Power) in order to establish if our data set is normally distributed or data is non-normal and on the basis of this we decide to proceed with parametric or non-parametric test such as SmartPLS SEM. My question is that our data is a multivariate type of data, so do still we need to test and report a Univariate normality test performed in SPPS, or we should only report Marida' Multivariate Skewness and Kurtosis results. Particularly, when our aim is to use later on SmartPLS, SEM or AMOS for further data analysis.
Reply to this question from researchers and scholars will be highly appreciated.
Thank you
Reporting Skewness and Kurtosis is enough while applying SEM. Moreover, in Smart PLS, assumption of normal data is not required.
Hello Shoaib Muhammad,
"Multivariate" can mean several things. Do you have (1) multiple explanatory variables, (2) multiple response variables, (3) lots of variables with no clear distinction between response (dependent) and explanatory variables?
For (1) normality does not matter because it is the response variable that is assumed to be normal. For (2) it matters. However, the assumptions are checked with the residuals from that analysis. For (3) we don't do hypothesis testing, we do things like cluster analysis or principle component analysis.
For checking assumptions you will need a stat package that will produce the residuals and fitted values. If the package won't do that, it is inadequate for statistical analysis.
I think it is important that we back up our advice online with sound references to the statistical literature. So here are the references.
The assumptions for the general linear model (including regression, ANOVA, and ANCOVA) are that the errors (residuals) are normal and homogeneous (Eisenhart 1947, Seber 1966, Neter et al 1983 pp 31& 49, Quinn and Keogh 2002 pp 110 & 280). Evaluation of assumptions in advanced texts, where it occurs, often entails a residual versus fit plot (for homogeneity) and a normal score plot or Quantile-Quantile plot (for normality of the residuals).
The statistical literature warns against statistical tests to evaluate assumptions and advocates graphical tools (Montgomery & Peck 1992; Draper & Smith 1998, Quinn & Keough 2002). Läärä (2009) gives several reasons for not applying preliminary tests for normality, including: most statistical techniques based on normal errors are robust against violation; for larger data sets the central limit theory implies approximate normality; for small samples the power of the tests is low; and for larger data sets the tests are sensitive to small deviations (contradicting the central limit theory).
That preliminary evaluation is not correct can come as a surprise, as can advice against statistical tests to evaluate assumptions. But I think best practice, as in the literature, needs to be said.
To check assumptions for residuals see:
https://www.datanovia.com/en/lessons/anova-in-r/#check-assumptions-1
Good luck with your research,
~David S
Chatfield, C. 1995. Problem Solving: A statistician's guide, Second edition (Chapman & Hall/CRC Texts in Statistical Science) 2nd Edition,
Draper, NR and H. Smith 1998 Applied Regression Analysis, 3rd Edition. Wiley
Eisenhart, C. 1947. Biometrics 3:1-21
Gelman, A C Pasarica & R. Dodhia (2002) Let's Practice What We Preach, The American Statistician, 56:2, 121-130, DOI: 10.1198/000313002317572790
Läärä, E. 2009. Statistics: reasoning on uncertainty, and the insignificance of testing null. — Ann. Zool. Fennici 46: 138–157.
Montgomery, DC EA Peck, GG Vining 2012 Introduction to Linear Regression Analysis, 5th Edition. Wiley
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