Regression does not require data normality, only normality of residuals.  What analysis are you doing?

Yasser -

I am not clear as to what is your application here.  What you are doing might require normality, but most uses of statistics do not have such a requirement. For example, I spent about a quarter of a century working on estimation for establishment sample surveys, and those data are highly skewed.  The methodology I developed depended on that.

It is true that distributions of errors may be 'normal,' and similarly, in regression, it may be desirable that the random factors of (WLS) estimated residuals be approximately normal or t-distributed.  Also, the Central Limit Theorem leads us to normal distributions, but that may often be misconstrued as saying that we should want data to all be normally distributed, and that something might be wrong otherwise.  Unnecessary transformations can create confusion when trying to interpret results. This is included in Jochen's responses and mine in answer to the question found through the following link:

 https://www.researchgate.net/post/A_generic_transformation_to_normality

Your application may have certain requirements, or it might be that normality is not actually a requirement, if you look into this.

Cheers - Jim 

Generally, multivariate analyses including regression require that data are normally distributed. If not, one has to use some other solution which accept not normally distributed data, but must be clear of type of distribution and relationships because a function of relationships has to be specified and initial estimates of parameters are given. However, in other case like Cobb–Douglas production function, logarithmic transformation is used to solve the function using normal regression analysis. Similarly, in many cases multivariate analysis is carried out fulfilling normality criterion by transforming variables taking log, square root, cube root or conversely taking anti log, squaring and raising variable to power 3. Distribution of values variables against their ranks in a variable, gives some idea as how a variable is transformed  to become normal. Latter parameters may be transformed  to conform original distributions. However, one cannot be sure what function has actually underlay the model, if transformation of parameters is carried out because it is possible different variable are differently distributed.

 prof. Oleksiy (Alex) Chadyuk 

 prof.  James R Knaub

prof . Mohammad Firoz Khan 

thank you very much for your answers

I am using SEM AMOS Software. but the data not normal.

(Mardia’s coefficient = 228.527, CR = 39.88).

Is there a reason that the data should be normal?  

James's question is key here - it is not as important that the data are not normally distributed as it is that you have a handle on why you have those results.  If, for example, the data are continuous but skewed, SEM is fairly robust in its estimation (although, you may want to consider Bayesian estimations).  But if your data are not continuous, drawing inferences from SEM will lead to strange results unless you have carefully articulated your measurement parameters.  If you can share your area of study, I can come up with some articles (for example, this is a topic in both marketing research and biostatistics, but the literature doesn't cross over particularly well).

In response to “Is there a reason that the data should be normal?”

There are three reasons why this might be so:

1. Mathematical Simplicity. It turns out that this distribution is relatively easy to work with, so it is easy to obtain multivariate methods based on this particular distribution.

2. Multivariate version of the Central Limit Theorem. As is well-known that in the univariate cases that we have a central limit theorem for the sample mean for large samples of random variables. A similar result is available in multivariate statistics that says if we have a collection of random vectors X1, X2,...,Xn that are independent and identically distributed, then the sample mean vector, x¯, is going to be approximately multivariate normally distributed for large samples.

3. Many natural phenomena may also be modelled using this distribution, just as in the univariate case.

For details, pleas see:

http://support.sas.com/publishing/pubcat/chaps/56903.pdf

http://www.idosi.org/mejsr/mejsr13%282%2913/21.pdf

http://jtemplin.coe.uga.edu/files/multivariate/mv11ersh8350/mv11ersh8350_R3.3.pdf

Generally,  data should follow normal distribution, but many statistical techniques are used for approximately normally distributed data. 

Prof James R Knaub

prof Julia B. Smith

prof Hassan Fikury 

thank you very much for your answers

 The problem is that, I continued the analysis, and ignored the non normality based on recommended from some friend. But now I need literature to support that.

I need literature to support that " In case of multivariate normality if the data not normal the researcher can ignore the normality and continue analysis .

prof  Mohammad Firoz Khan

prof Abhay Tiwari

Thank you very much for your answer

As Alex noted in the first response and others have similarly noted, What analysis are you trying to do?  Few analyses require normality, but no one can give you a reference on any topic until that topic is described. What analysis is it that you want to continue? 

The study was to develop integrated model for the successful adoption of information systems. The questionnaire used to collect primary data, and analyzes the data by using AMOS version 21. Confirmatory factor analysis used in the study and hypothesized model were tested with SEM techniques.

Prof. James R Knaub .

Thank you very much

Ah.  Factor analysis.  I haven't used that, at least not in a very long time, if ever, but a quick look at the internet seems to indicate normality is desirable (I don't know why), but maybe not strictly required. I don't know, but now that it has been stated that you are doing factor analysis, someone expert in that can reply.

I did see some similar questions on ResearchGate that you may want to examine. Just search on that topic.  - Cheers! 

PS -  I don't know if the fact that you are looking at confirmatory factor analysis makes this different re assumptions from factor analysis, but some people reading your question may know. 

Dear James and Yaser, 

I think that for factorial analysis i would firts explore the data to see if there is symmetry. If so, it implies that Mode = Mean. As FA analyse means that you know that the interpretation of Means are correct , and can procede with your model. After that you have several measures to test the quality of your FA.

Usually we have no problem if  we obtain at least symetry and the distributions are  not platykurtic .

PS:  you have to attend also to the dimension of your sample. In terms of dimension, James as a very interesting paper about the way you should interpret p.

Have a nice time

Helena

prof. Helena Pestana thank you for your answer

Dear Yaser, 

It is a pleasure. If you need further help, please let me know.

Helena

A better question would be "how do departures from the assumptions of the model influence model performance?" You could also ask "does my data depart from normality to such an extent that I am unwilling to accept the outcome?" It is nearly given, that with a large enough sample size all data will depart significantly from normal.

In factor analysis (Proc Factor in SAS) we have the following choices:

1) There are nine methods for factor analysis.

2) There are six methods for selecting priors (not related as far as I can tell to the priors used in Bayesian methods)

3) There are eight rotation methods.

In Proc Calis (the SAS procedure for confirmatory factor analysis: http://support.sas.com/documentation/cdl/en/statug/65328/HTML/default/viewer.htm#statug_calis_examples43.htm)

I think it has the same options, but there are then additional features that enable more specific tests of hypotheses.

So you will need to figure out which models are appropriate, and the strengths/weaknesses of the different approaches. From the reasonable approaches, are there differences in the results when you try different models?

This is not a game of try to find the model that gives you the answer that you want. Rather think of this as a game where you see how many solutions there are. If only one is present, then use the simplest model and explain that other models were tried. If there are multiple outcomes then you will need to show at least two of them. If necessary, you can include "supplemental material" that shows an alternate interpretation of the data.

prof. Timothy A Ebert  thank you very much for your answer

This 1987 piece is the one I see referenced most often concerning assumptions for factor analysis. If you trace who references the Browne study, you should find any subsequent discussion about your issue.

prof. Julia B. Smith thank you very much for your answer