Structural equation modeling is based on covariances, and these are less stable when estimations are based on small samples. Personally I usually consider as minimum sample size around 10 observations for every free parameter estimated (target frequently recommended for SEM analyses). About this subject, I recommend the reading of the following paper:

Westland, J.C. (2010). Lower bounds on sample size in structural equation modeling, Electronic Commerce Research and Applications, 9(6), 476-487.

available in the following link:

http://indigo.uic.edu/bitstream/handle/10027/7655/Lower%20bounds%20on%20sample%20size%20in%20structural%20equation%20modeling.pdf

I don't think there is a simple answer to this question. As a rule of thumb thought I would never do an SEM with less than 500. In general though in the thousands is what you want. Researchers have done tons of simulations to attempt to answer this question but it seems that there are too many variables to get a clear answer. You can also use MPLUS to run a power analysis and make some inferences.

I agree with Nadra Lisha. There is no unique answer. Any way, if you use a non-parametric estimation method, as Partial Least Squares, you can use bootstrap to evaluate the significance of your estimated parameters. In this case, a sample size equal to 100 is enough.

I remember there was a guideline from Bentler for 5 observations per parameter estimated. But don't remember the year. It may have changed, it's a developing methodology.

Dear Gerry,

I have been working with SEM and CFA in MPlus in the last two year, I am not sure I understood what you mean by "distribution-free parameters". If the variables are continuous, then multivariate normality should be checked, irrespective of sample size. If variables are ordinal, up to 7 or 10 point, polychoric correlations are a better option and no distribution is assumed.

Concerning the sample size, as I have read, it does not depend on the number of parameters estimated. If you have too many parameters, the model is underidentified, irrespectively of the sample size. However, the sample size is important for other reasons, there is a nice reference: Costello & Osborne (2005) Pract Ass Res & Evaluation. They authors show that smaller samples are more likely to have the wrong number of factors, Heywood cases, misclassified items, etc.

Normally you either go for samples >500 as Nadra pointed (better is >1000) or 20 individuals per item. No one knows exactly the best way to calculate this, as it has been said here.

Interesting note on the number of parameters dependence. I also read that with large samples (over 1000) the chi squared becomes really sensitive and you get significance where you are not supposed to. Of course the alternative is to use other fit indexes. But it's crazy how the number varies from 5 to 20 people per item. And then of course there was this whole debate on whether it should be per item or per parameter.

If you are talking about asymptotic distribution free (ADF) estimation, conventional guidelines suggest that more cases are needed than for maximum likelihood estimation. The Amos program won't even let you run a model with ADF unless you have at least 20 cases per observed variable.

As Roger, I am confused about what you mean by "distribution-free parameters". Rules of thumb are very very tricky. Take a look at the following papers:

MacCallum, R. C., Browne, M. W., & Sugawara, H. M. (1996). Power analysis

and determination of sample size for covariance structure modeling.

Psychological Methods, 1, 130-149.

Muthen, L. K. & Muthen, B. O. 2002. How to use a Monte Carlo study to

decide on sample size and determine power. Structural Equation Modeling,

9(4): 599-620.

Preacher, K. J., & Coffman, D. L. (2006, May). Computing power and minimum

sample size for RMSEA [Computer software]. Available

fromhttp://www.quantpsy.org/rmsea/rmsea.htm

More than 200 of sample size

i have used SEM in my PhD thesis and i think that there is no agreement about sample size but the minimum can be 200.

Structural equation modeling is based on covariances, and these are less stable when estimations are based on small samples. Personally I usually consider as minimum sample size around 10 observations for every free parameter estimated (target frequently recommended for SEM analyses). About this subject, I recommend the reading of the following paper:

Westland, J.C. (2010). Lower bounds on sample size in structural equation modeling, Electronic Commerce Research and Applications, 9(6), 476-487.

available in the following link:

http://indigo.uic.edu/bitstream/handle/10027/7655/Lower%20bounds%20on%20sample%20size%20in%20structural%20equation%20modeling.pdf

Regarding the parameter estimation with ADF, "Boomsma and Hoogland (2001, p. 148) found that for a sample size of N ≤ 200, "ADF is a disaster with more than one-third of the solutions being improper" (Schermelleh-Engel & Moosbrugger, 2003, p. 49).

Available in: http://www.cob.unt.edu/slides/paswan/busi6280/Y-Muller_Erfurt_2003.pdf

With respect to categories of sample size (small, medium or large), personally I think that we need to consider the rule of thumb of N = 200 as minimum (Flora & Curran, 2004), and any sample size below that, it could be categorized as small, a "Common N (300, 400 and 500)" (Myers, Ahn & Jin, 2011, p.416) could be categorized as medium sample size, and every sample size bigger than that, could be categorized as large sample size.

I suggest to read Myers, Ahn & Jin (2011), available in the following link:

http://www.statmodel.com/download/Myers,%20Ahn,%20&%20Jin,%202011.pdf

And Flora & Curran (2004), available in:http://www.statmodel.com/download/floracurran.pdf

I am using more than 700 as my sample size for SEM

Hi Gerry,

In my opinion, the best option is to consider the model complexity (i.e., every free parameter estimated ) and the following rules of thumb: minimum ratio 5:1 (Bentler & Chou, 1987; Worthington & Whittaker, 2006); recommended ratio 10:1 (Bentler & Chou, 1987; 2001; Kline, 2005; Worthington & Whittaker, 2006); recommended ratio for data with no normal distribution 15:1 (Hair, et al., 2006).

References:

Bentler, P., & Chou, C. (1987). Practical issues in structural modeling. Sociological Methods and Research, 16(1), 78-117.

Hair, J., Black, W., Babin, B., Anderson, R., & Tatham, R. (2006). Multivariate Data Analysis (6th ed.). New Jersey: Pearson Educational, Inc.

Kline, R. (2005). Principles and Practice of Structural Equation Modeling (2nd ed.). New York: The Guilford Press.

Worthington, R., & Whittaker, T. (2006). Scale Development Research. A Content Analysis and Recommendations for Best Practices. The Counseling Psychologist, 34(6), 806-838.

Thank you for your comments, advice and recommendations. I was not aware of the Hair et al 2006 article and this is helpful. The paper I have found useful in this discussion is by Ory & Mokhtarian (2010) The impact of non-normality, sample size and estimation technique on goodness-of-fit measures in structural equation modeling. Qual Quant 44:427-445. It suggests using multiple estimation techniques to 'double check model specifications and learn more about the data'

The minimum recommended is usually 10 cases per variable but the bigger the dataset the better - except that you need to be careful about which fit indices you then select. The chi-square statistic is of course sample size dependent so the larger the sample the less likely it is that you will achieve non-significance. As this is now an old and very outdated fit statistic, this is not really a problem (and certainly not a reason to use a smaller rather than a larger dataset). However, I find that with well-fitting models testing strong theoretical propositions even very large datasets will achieve a non-significant chi-squared statistic when fitted.

You can check also R Package sem and others similar packages.

For example:

http://cran.r-project.org/web/packages/sem/index.html

I am considering of following advice from Hair (2010) as follows :

1. 5 or less latent constructs. Each latent construct has > 3 items (Min 100 sample)

2. 7 or less latent constructs. Each latent construct has > 3 items (Min 150 sample)

3. 7 or less latent constructs. Each latent constructs have less than 3 items (just identified model) (Min 300 sample)

4. More than 7 latent constructs. Some constructs have less than 3 items (just identified model) (Min 500 sample)

I agree with Tze Leong Chan suggestions, 200 is minimum sufficient but for more latent

constructs better the more sample size to give accurate results for your model. Regarding insignificant chisquare value, it is sensitive to sample size and yiu may not be able to see insignificant p value. It happened with my model too. Rather consider relative chisquare value in comparisuon with other fit indices.

Hi Gerry. I use only non-parametric, non linear approaches. My vision is that media is not a parameter but an inner property of the sample when you assume Laplace criterion ( each X is the media of a sector of equal frequence 1/n, etc.) But standard deviation is not a useful parameter in my view and I do not need it when modelling. All of us agree that the bigger the data size the better the results, but small data sizes may produce acceptable results. I have made SEM for N=6 that sometimes result better than those made for 10 values, but this is not a rule. N=20 is good, N=100 is excelent, more may be unnecesary. As an example I wrote a 4 pages mini-paper in RG showing a case with graphs titled "How to obtain Lorenz and CDF models from data samples" . I hope it is not too heterodox for your vision. Maybe it helps to be more clear. Thanks, emilio

Dear Emilio

Thank you for your very interesting contribution. It looks radical and controversial and this is exactly what these pages are for. I would like to view your mini-paper and I wanted to check that you have posted this on your RG personal pages? If I have difficulty in finding it I may request a copy via email please. It may help in this discussion under what circumstances you believe that small data sets may be appropriate for SEM.

With thanks

Gerry

Gerry: Here I insert the minipaper. emilio

Hi Gerry and others. Sorry for not reading carefully your question: "under what circumstances you believe that small data sets may be appropriate for SEM?". The chance that a 10 points (Xacumulated, Kaverage) match exactly is indeed remote; it requires that the data set be "representative" and this term is vague because it requires knowing a priori the answer we want to obtain. In general, I "believe for the moment" that if a number close to 45% of the N=20 acumulated points are in the left side with respect to 0.5, and 55% in the right side, then the chance of getting an acceptable SEM is good. But this may only be checked with simulations that require: 1)We define a real model with precision; 2) We generate N random numbers between 0 and 1; 3) We apply the model to the random numbers to obtain Kaverage for each 1/n frequence and 4) compare theoretical results with trial results. The smaller N, the smaller the chance of a nice representative data set, and some "good luck" is needed. This is my first guess and I welcome your ideas and critics to improve this key point of data analysis and its interpretation. Thanks, emilio