Sample size estimation for regression coefficient

Question

I am looking into the method to calculate the sample size if we would want the specific coefficient significantly different from zero. I have 4 independent variables and 1 dependent variable. And I have 50 observations so far. I would like to know how many more observations do I need to get one of the coefficients significant because it's trending.

I have read this paper from Ken Kelley and Scott E. Maxwell, "Sample Size for Multiple Regression: Obtaining Regression Coefficients That Are Accurate, Not Simply Significant". And they have this formula to estimate sample size.

$N = \Big(\frac{z_{(1-\alpha/2)}}{w}\Big)^2\Big(\frac{1-R^2}{1-R^2_{XX_j}}\Big) + p + 1$

where $R^2$ represents the population multiple correlation coefficient predicting the criterion (dependent) variable $Y$ from the $p$ predictor variables and $R^2_{XX_j}$ represents the population multiple correlation coefficient predicting the $j$th predictor from the remaining $p − 1$ predictors. The calculated $N$ should be rounded to the next larger integer for sample size. The $w$ in the above equation is the desired half-width of the confidence interval.

$R^2$ is easy to calculate. But I don't know how to calculate $R^2_{XX_j}$ for this. Do you guys have any idea?

Answer 1

Strictly speaking, you run the same regression as if the $j$-th variable was your dependent variable (and you exclude that from your independent variables). The corresponding $R^2$ will be your $R^2_{XX_j}$. For instance, imagine you ran the following model:

# estimate main model:
mod <- lm(mpg ~ disp + drat + wt*qsec, data=mtcars)

and now you want $R^2_2$, which corresponds to disp. You can run

mod2 <- lm(disp ~ drat + wt*qsec, data=mtcars)
Rsq_disp <- summary(mod2)$r.squared
Rsq_disp
>  0.8933448

Note that $1/(1-R^2_{XX_j})$ is also called the variance inflation factor of variable $j$. See ??vif if you are using R. For instance, let's use the car package in R:

library(car)
# get variance inflation factors:
vif(mod)
      disp       drat         wt       qsec    wt:qsec 
  9.376007   2.499659 455.406742  41.596850 428.288269 

These numbers correspond to

$$ \frac{1}{1-R^2_j} $$

To see this, note that

1/(1-Rsq_disp)
> 9.376007

You see that the number is identical to the first one obtained earlier with vif.