Hi,

If you want to change your data in normal form, you can first use basic functions as log, exp, square root, cubic root, inverse function....It depends on the distribution of your data...

Jonathan

The best answer I can provide is that the premise of your question is not correct. 

You should not apply statistical methods for normally distributed data to data that is not normally distributed. But this doesn't suggest that you should try to make all of your data look normally distributed.

If you are working with data that is best described by a Poisson, exponential, binomial or another parametric model, then you will usually be better served by those parametric models. 

If you can provide a more specific question and/or example, then perhaps I can provide a more useful answer.

Hello Manvendra,

In SEM, you have a couple of options in dealing with non-normality data. The above discussion on transformation is one option. I’ll discuss the other. The main issue with non-normality within SEM (using ML; maximum likelihood) is that the standard errors are inaccurate (too low) and the chi-squares are biases (too high). However, there are current methods which help to overcome these issues which are quite prevalent nowadays. Examples include the asymptotically distribution-free (ADF) approach (Muthen & Kaplan, 1992; Muthén & Kaplan, 1985). However, best known are methods which use the Satorra-Bentler statistic (Satorra & Bentler, 1994) which adjusts downward the value of chi-square from standard ML by an amount that reflects the degree of kurtosis (Kline, 2011, pp. 176-177).

Some good background reading on aspects of nonnormality in SEM can be found in Chapter 5 of Kaplan’s book (Kaplan, 2009). It’s a good read.

If you are using Mplus, using MLR (robust maximum likelihood) as your estimator is a very good (and easy) solution – it uses the Satorra-Bentler correction and is regularly used in social science empirical works, because almost ‘everything’ is non-normally distributed in that space (ok, a bit tongue in cheek) . In general:

-          ADF is not always an easy remedy, due in part to its heavy reliance on very large sample size

-          Bootstrapping can be an effective remedy, although it rests heavily on the representativeness of the parent sample

-          Satorra-Bentler corrections are strongly recommended due to their empirical support … and increasing availability in software.

References:

Kaplan, D. (2009). Structural equation modeling foundations and extensions (2nd ed. ed.). Los Angeles, [Calif.]

Kline, R. B. (2011). Principles and Practice of Structural Equation Modeling (3rd ed.). London: The Guilford Press.

Muthen, B., & Kaplan, D. (1992). A comparison of some methodologies for the factor analysis of non‐normal Likert variables: A note on the size of the model. British Journal of Mathematical and Statistical Psychology, 45(1), 19-30. doi:10.1111/j.2044-8317.1992.tb00975.x

Muthén, B., & Kaplan, D. (1985). A comparison of some methodologies for the factor analysis of non‐normal Likert variables. British Journal of Mathematical and Statistical Psychology, 38(2), 171-189. doi:10.1111/j.2044-8317.1985.tb00832.x

Satorra, A., & Bentler, P. M. (1994). Corrections to test statistics and standard errors in covariance structure analysis. In A. von Eye & C. C. Clogg (Eds.), Latent variables analysis: Applications for developmental research (pp. 399--419). Thousand Oaks, CA: Sage.

Wishing you success in your research,

Tim 

@Tim Edward Powers,Chalamalla Srinivas, Shane McGee McMahon

Thanks for ur valuable suggestions and literature. Though, in connection to  the above question, my additional question is: The above discussion was for time series data or  likert scale data.

I am working on behavioral science and usually get primary data. Kindly suggest the above method can also be applied for likert scale data too.

Hello Manvendra,

All of the above applies to primary data and your likert scale measurement. Likert scale data on behavioral science and your prior comments would suggest that you expect (or would like) to use normal distribution modeling. However, it is worth noting Shane's comment above about the type of measures you are using (e.g., if you are using a count, then a Poisson distribution model would be appropriate). Without knowing more, it is hard to say. But I would be guessing that if you are using Likert scales you are in the normal distributed population realm. 

Your approach in handling non-normally distributed data remains the same in a SEM framework whether it is cross sectional (one time) or over multiple timepoints. There are some further concerns if you are using factors (i.e., invariance of items on a factor over time). But that may not be an issue for you if you are simply using a path analysis (one measure for each construct). 

Kindly,

Tim