# Rule of Thumb for multivariable linear regression analysis

Rule-of-thumb suggested by Green(1991)(https://doi.org/10.1207/s15327906mbr2603_7) : "Some support was obtained for a rule-of-thumb that $$N ≥ 50 + 8 m$$ for the multiple correlation and $$N ≥104 + m$$ for the partial correlation."

I would like to determine the upper limit of the number of covariates $$(m)$$ according to the rule-of-thumb suggested by Green in an analysis using a general linear model with a fixed sample size $$N$$. The purpose is to verify the statistical significance of the regression coefficient of a predictor after controlling for the effects of several covariates.

In this case, which formula is appropriate for the model? $$N ≥ 50 + 8 m$$ or $$N ≥104 + m$$?

The "multiple correlation" is the positive square root of the multiple regression model's $$R^2$$. The "partial correlation" is referring to a specific coefficient within that model. Since you want to verify a pre-specified coefficient, you want the latter (i.e., $$N ≥104 + m$$).
A better approach would be to conduct a power analysis. Specifically, you want to conduct a sensitivity type or a post-hoc type power analysis. That is, given your sample size, what is the smallest correlation you would have your preferred level of power (often 80%) to detect (s), or what would be your power to detect your preferred correlation (ph). First, subtract $$1$$ from your $$N$$ for every degree of freedom your covariates will consume, set alpha at, oh, I don't know, say, $$.05$$, and solve for the correlation by stipulating a level of power, or solve for the power by stipulating a correlation. It is possible your analysis would not be worth pursuing, even if your $$N$$ exceeds the rule of thumb, or that you are likely to be OK, even if your $$N$$ does not exceed the rule of thumb.