ISSUE: How to determine minimum sample size for multiple regression model?

MODEL: The model as described in the question may be written in the form:

(1)   Y = B0 + B1X1 + B2X2 + ... + B5X5 + e

... where Y = response (dependent variable) ; B0 = intercept; Bi = parameters to be determined; and Xi = explanatory factors (independent variables). Recall that the test statistic used for polynomial model of this form is the F-test given by:

(2)   F = MSR / MSE

... where the following definitions applied:

(3)   MSE = SSE / n - p -1       

(4)   SSE = Σ(Yi - Yhat)2               with degree of freedom = n - p -1

(5)   MSR = SSR / p

(6)   SSR = Σ(Yihat - Y*)2            

... where Y* = mean. Therefore, in a long form F may be written as:

(7)   F = (Σ(Yihat - Y*)2) / p) / Σ(Yi - Yhat)2 / n - p - 1

This long-form is helpful when we put the terms in the spread sheet for calculation.  See attached Excel. Now solve for n as our expected minimum sample size. After re-arranging term, n is given as:

(8)   n = [Fp(Σ(Yi - Yhat)2) / Σ(Yihat - Y*)2 ] p + 1

Under this approach, using the F-test for multiple regression model, it is necessary to take a test sample. Attached is an excel of a test sample of 10 counts. The sample run produces a minimum sample of 188. Re-rum many tests, i.e. Monte Carlo simulation to obtain minimum sample size that could be related to the F-table. Why F-table? because you are using multiple regression, equation (1) supra.

A common suggestion for minimum sample size is n > 30.

http://www.stat.ufl.edu/~aa/

http://www.itl.nist.gov/div898/handbook/prc/section2/prc241.htm

http://www.nss.gov.au/nss/home.nsf/pages/Sample+size+calculator

You can use the software G*Power to calculate appropriate sample sizes, see

http://www.gpower.hhu.de/

In order to be able to estimate the appropriate sample size, you have to decide on your alpha and beta values and estimate the size of effect that you expect from past experiments or from what seems reasonable from your perspective (i.e., to have a sample size that is large enough that you have a fair chance of actually finding the effect in your experiment, given that it exists). For basic reading, see for example:

Cohen, J. (1992). A power primer. Psychological bulletin, 112(1), 155.

Here are a few books/articles that might help:

Ryan, Thomas P. Sample size determination and power. John Wiley & Sons, 2013.

Desu, M. M. Sample size methodology. Elsevier, 2012.

Lenth, Russell V. "Some practical guidelines for effective sample size determination." The American Statistician 55.3 (2001): 187-193.

MacCallum, Robert C., Michael W. Browne, and Hazuki M. Sugawara. "Power analysis and determination of sample size for covariance structure modeling." Psychological methods 1.2 (1996): 130.

Thank you everyone, I am absolutely sure the material and the theory regarding sample size calculation are worthy and will go over the same. Once again, thank you. 

The following article mya be helpful in determinig the sample size in different situations:

Eng J. Sample size estimation: How many individuals should be studied? Radiology 2003; 227:309 –313.

Available at: https://biotime.uchicago.edu/eng_samplesize.pdf