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I am using SPSS AMOS to do a factor analysis and it produces statistics related to both univariate and multivariate normality.

p35 of the AMOS Users Guide states that "[T]he observed variables must meet some distributional requirements. If the observed variables have a multivariate normal distribution, that will suffice". I have also been told in person by a friend that univariate normality should also be reported. In addition to that, this paper states on p19 that "[A]ssessment of univariate and multivariate normality should be provided."

Is it the case that if I have a problem with univariate normality then that will also be reflected in the multivariate normality statistic? If so, would that be a good argument for me just reporting multivariate normality?

user1205901 - Reinstate Monica
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    If data are multivariate normal they have marginal univariate normality automatically. – ttnphns Jun 03 '14 at 07:13
  • I googled "marginal univariate normality". Does it just mean that each variable will be univariate normal? – user1205901 - Reinstate Monica Jun 03 '14 at 12:03
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    Yes. If the X,Y cloud is bivariate normal, X and Y (besides other possible projections) are normal too. The opposite is not necessarily true: normal X and Y doesn't imply normal cloud X,Y. – ttnphns Jun 03 '14 at 12:16

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As @ttnphns commented, multivariate normality implies the variables involved are approximately normal as well. Exceptions will be very rare, if even possible (IDK, TBH). If there's any harm in reporting both univariate and multivariate normality assessments in your context, you can probably get by with just the multivariate normality assessment (assuming it's favorable)...but I have trouble imagining why it would be harmful to just report both and make everyone happy. Nobody's going to object too strenuously to the redundancy of including both. Just make sure you're performing the assessment conscientiously. Consider the advice in, "Is normality testing 'essentially useless'?"

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Nick Stauner
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