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Sample size for Regression Analysis. What is Sample Size? Since we cannot work with population data (due to constraints of time and money), we always prefer to work with sample data. Therefore, it becomes important to know how many data points (or sample size) are required in the sample. Usually the sample size determination is dependent on the following parameters 1. Significance Level or alpha 2. Power of the test or (1-Beta) 3. Effect or the difference to be detected Smaller the alpha, Higher the Power of test, smaller the effect that needs to be detected --> Higher is the sample size required. Sample size for Regression Analysis depends on the following (in addition to the parameters already listed for sample size selection above and hence starting with number 4 below) 4. Type of Regression being done (Linear, Multiple, Ordinal etc.) 5. Purpose of Regression - a. Determine the effectiveness of the model (looking at R-square value) b. Determine the statistically important predictors (or determining the Beta values for each predictor) 6. Level of correlation between the predictors For point 4, generally, simpler the regression lesser the sample size required. Hence, a lower sample size if I'm carrying out a linear regression vs a multiple regression. For point 5, if the purpose if only to check the fit of the model a smaller sample size would suffice as compared to determining the significant factors from all the potential ones For point 6, higher the correlation higher the sample size (applicable only if there are multiple predictors) Now that we know the factors affecting sample size for regression, how should be check if we have the required number of samples for doing regression. The best way is to follow the theory behind sampling - higher the size, better it is But this arises another question, what sample is sufficiently high? There are a few empirical formulae that can be of help here. I am listing a few of them below 1. One common rule of thumb and the most famous one is that sample size should be 10 times the number of predictors. So if you have 4 predictors, you should have a minimum of 40 samples for running regression 2. As suggested by Green (1991) a. Sample size = 50+8*k, k --> number of predictors; applicable if we are doing regression for point 5a b. Sample size = 104 + k, k--> number of predictors; applicable if we are doing regression for point 5b There are some more depending on the kind of regression (ordinal, log etc.) that you plan to run. Sometimes, it is difficult to have answers to the 6 parameters before one decides the sample size. A more practical approach is to work backwards i.e. since we know the number of samples or we know how many we could collect, we could always do the Power Analysis (given the other factors are kept constant or pre-decided).

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