To what extent does sample size impact upon Cronbach's Alpha and Multiple Regression?
Any recommendations in terms of literature?
Cronbach alpha is not dependent on sample size, however, it is influenced by the number of items used to measure a latent variable. Sample size affects the "accuracy" of Cronbach alpha though. It is a measure of internal consistency that is based on inter-item correlations. Therefore, the more items you use for a latent construct, the alpha will tend to be higher. About desired sample size, a study indicated that the minimum sample size may be 50 (please see attachment 1). However, this is open to interpretation by other scholars as well. This is just a reference, and other studies might differ.
For regression, there are also various ways like power analysis or "rules of thumbs" to determine minimum sample size requirement (please see the attachment 2). Many authors have put different views about minimum sample size based on number of independent variables in the model. Usually, the higher the number of independent variables, the larger should be the sample size. You can still run regression with smaller sample sizes, but statistical power would be compromised.
Hello again,
And one more article on the issue you are interested in.
All the best
Conduct a power analysis for your multiple regression, especially if you're doing a frequentist analysis. For Cronbach's alpha, it depends on the number of items in the scale, not the sample size. It measures internal consistency and tends to go up with the number of items. You don't want lots of duplication in your items, but you don't want too few items, either. A factor analysis of a new scale may help you to reduce to the essential items first. If it's an established scale, then just calculate and report the Cronbach's alpha for your sample.
As a general rule the ratio of participants(observations) to independent variables should not fall below 5 to 1.However, the desired level is between 15 to 20 observations for each independent variable.
The 15 to 20 observations per independent variable rule is a very old one, before we were able to do more sophisticated power calculations. It is not a good rule. There is no substitute for doing more precise power calculations--software is readily available and there are free cites on the Internet for doing them.
Sample size impacts on alpha in terms of the precision with which you can measure it. So while a small sample might have relatively unbiased estimation (there is usually some small sample bias) you mainly suffer because your estimate is very imprecise. That is why it is a good idea to calculate an interval estimate for alpha (See linked paper.)
In terms of multiple regression the key issue is statistical power. All the common rules of thumb are pretty unhelpful here because the impact of sample size will depend on what you are trying to do with the regression model (e.g., hypothesis testing, model comparison, interval estimation), what your focus is (the overall model or individual effects) and very importantly collinearity multicollinearity.
Thus you could have a huge sample but almost no power to detect effects of individual predictors if collinearity was high. Or a small sample with considerable power because collinearity is low and the effect of interest very large.
The same model could have low power to detect one effect and high power to detect another.
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