# Solved – Rules of thumb for minimum sample size for multiple regression

regressionrule-of-thumbsample-sizestatistical-power

Within the context of a research proposal in the social sciences, I was asked the following question:

I have always gone by 100 + m (where m
is the number of predictors) when
determining minimum sample size for
multiple regression. Is this
appropriate?

I get similar questions a lot, often with different rules of thumb.
I've also read such rules of thumb quite a lot in various textbooks.
I sometimes wonder whether popularity of a rule in terms of citations is based on how low the standard is set.
However, I'm also aware of the value of good heuristics in simplifying decision making.

### Questions:

• What is the utility of simple rules of thumb for minimum sample sizes within the context of applied researchers designing research studies?
• Would you suggest an alternative rule of thumb for minimum sample size for multiple regression?
• Alternatively, what alternative strategies would you suggest for determining minimum sample size for multiple regression? In particular, it would be good if value is assigned to the degree to which any strategy can readily be applied by a non-statistician.

I'm not a fan of simple formulas for generating minimum sample sizes. At the very least, any formula should consider effect size and the questions of interest. And the difference between either side of a cut-off is minimal.

### Sample size as optimisation problem

• Bigger samples are better.
• Sample size is often determined by pragmatic considerations.
• Sample size should be seen as one consideration in an optimisation problem where the cost in time, money, effort, and so on of obtaining additional participants is weighed against the benefits of having additional participants.

### A Rough Rule of Thumb

In terms of very rough rules of thumb within the typical context of observational psychological studies involving things like ability tests, attitude scales, personality measures, and so forth, I sometimes think of:

• n=200 as good
• n=400+ as great

These rules of thumb are grounded in the 95% confidence intervals associated with correlations at these respective levels and the degree of precision that I'd like to theoretically understand the relations of interest. However, it is only a heuristic.

### Multiple Regression tests multiple hypotheses

• Any power analysis question requires consideration of effect sizes.
• Power analysis for multiple regression is made more complicated by the fact that there are multiple effects including the overall r-squared and one for each individual coefficient. Furthermore, most studies include more than one multiple regression. For me, this is further reason to rely more on general heuristics, and thinking about the minimal effect size that you want to detect.

• In relation to multiple regression, I'll often think more in terms of the degree of precision in estimating the underlying correlation matrix.

### Accuracy in Parameter Estimation

I also like Ken Kelley and colleagues' discussion of Accuracy in Parameter Estimation.

• See Ken Kelley's website for publications
• As mentioned by @Dmitrij, Kelley and Maxwell (2003) FREE PDF have a useful article.
• Ken Kelley developed the MBESS package in R to perform analyses relating sample size to precision in parameter estimation.