# How to get confidence interval on population r-square change

For sake of a simple example assume that there are two linear regression models

• Model 1 has three predictors, x1a, x2b, and x2c
• Model 2 has three predictors from model 1 and two additional predictors x2a and x2b

There is a population regression equation where the population variance explained is $\rho^2_{(1)}$ for Model 1 and $\rho^2_{(2)}$ for Model 2. The incremental variance explained by Model 2 in the population is $\Delta\rho^2 = \rho^2_{(2)} - \rho^2_{(1)}$

I am interested in obtaining standard errors and confidence intervals for an estimator of $\Delta\rho^2$. While the example involves 3 and 2 predictors respectively, my research interest concerns a wide range of different numbers of predictors (e.g., 5 and 30). My first thought was to use $\Delta r^2_{adj} = r^2_{adj(2)} - r^2_{adj(1)}$ as an estimator and bootstrap it, but I wasn't sure whether this would be appropriate.

### Questions

• Is $\Delta r^2_{adj}$ a reasonable estimator of $\Delta \rho^2$?
• How can a confidence interval be obtained for the population r-square change (i.e., $\Delta\rho^2$)?
• Would bootstrapping $\Delta\rho^2$ be appropriate for confidence interval calculation?

Any references to simulations or the published literature would also be most welcome.

### Example code

If it helps, I created a little simulation dataset in R which could be used to demonstrate an answer:

n <- 100
x <- data.frame(matrix(rnorm(n *5), ncol=5))
names(x) <- c('x1a', 'x1b', 'x1c', 'x2a', 'x2b')
beta <- c(1,2,3,1,2)
model2_rho_square <- .7
error_rho_square <- 1 - model2_rho_square
error_sd <- sqrt(error_rho_square / model2_rho_square* sum(beta^2))
model1_rho_square <- sum(beta[1:3]^2) / (sum(beta^2) + error_sd^2)
delta_rho_square <- model2_rho_square - model1_rho_square

x$y <- rnorm(n, beta * x$x1a + beta * x$x1b + beta * x$x1c +
beta * x$x2a + beta * x$x2b, error_sd)

c(delta_rho_square, model1_rho_square, model2_rho_square)
summary(lm(y~., data=x))$adj.r.square - summary(lm(y~x1a + x1b + x1c, data=x))$adj.r.square


### Reason for concern with bootstrap

I ran a bootstrap on some data with around 300 cases, and 5 predictors in the simple model and 30 predictors in the full model. While the sample estimate using adjusted r-square difference was 0.116, the boostrapped confidence interval were mostly larger CI95%(0.095 to 0.214) and the mean of the bootstraps was nowhere near the sample estimate. Rather the mean of the boostrapped samples appeared to be centred on the sample estimate of the difference between r-squares in the sample. This is despite the fact that I was using the sample adjusted r-squares to estimate the difference.

Interestingly, I tried an alternative way of computing $\Delta\rho^2$ as

1. calculate sample r-square change
2. adjust the sample r-square change using the standard adjusted r-square formula

When applied to the sample data this reduced the estimate of $\Delta \rho^2$ to .082 but the confidence intervals seemed appropriate for the the method I mentioned first, CI95% (.062, .179) with mean of .118.

Broadly, I'm concerned that bootstrapping assumes that the sample is the population, and therefore estimates that reduce for overfitting may not perform appropriately.

• "However, I am concerned that bootstrapping such a population adjusted value might be problematic." -- why? – January Jun 28 '13 at 4:22
• @January I've edited the question and tried to articulate my concern about bootstrapping with adjusted r-square. – Jeromy Anglim Jun 28 '13 at 5:54
• What is the population R squared ? I've took a look at the definition given here but for me the variance $\sigma^2_y$ makes no sense because the $y_i$ are not identically distributed. – Stéphane Laurent Jun 28 '13 at 7:00
• @StéphaneLaurent it is the percentage of variance explained in the population by the population regression equation. Or you could define it asymptotically as the proportion of variance explained in your sample as your sample size approaches infinity. See also this answer regarding unbiased estimates of population r-square. It is particularly relevant in psychology where we are often more interested in the true relationship rather than actually applying our estimated prediction equation. – Jeromy Anglim Jun 28 '13 at 8:16
• An F-test can be thought of as test of the hypothesis $\Delta\rho^2 = 0$. Can that be used to derive the standard error and confidence interval you are looking for? – Maarten Buis Jun 28 '13 at 9:09

## Population $R^2$

I'm firstly trying to understand the definition of the population R-squared.

Or you could define it asymptotically as the proportion of variance explained in your sample as your sample size approaches infinity.

I think you mean this is the limit of the sample $R^2$ when one replicates the model infinitely many times (with the same predictors at each replicate).

So what is the formula for the asymptotic value of the sample $R^²$ ? Write your linear model $\boxed{Y=\mu+\sigma G}$ as in https://stats.stackexchange.com/a/58133/8402, and use the same notations as this link.
Then one can check that the sample $R^2$ goes to $\boxed{popR^2:=\dfrac{\lambda}{n+\lambda}}$ when one replicates the model $Y=\mu+\sigma G$ infinitely many times.

As example:

> ## design of the simple regression model lm(y~x0)
> n0 <- 10
> sigma <- 1
> x0 <- rnorm(n0, 1:n0, sigma)
> a <- 1; b <- 2 # intercept and slope
> params <- c(a,b)
> X <- model.matrix(~x0)
> Mu <- (X%*%params)[,1]
>
> ## replicate this experiment k times
> k <- 200
> y <- rep(Mu,k) + rnorm(k*n0)
> # the R-squared is:

### References

• Algina, J., Keselman, H. J., & Penfield, R. D. Confidence Intervals for the Squared Multiple Semipartial Correlation Coefficient. PDF
• Smithson, M. (2001). Correct confidence intervals for various regression effect sizes and parameters: The importance of noncentral distributions in computing intervals. Educational and Psychological Measurement, 61(4), 605-632.
• It seems that nobody here (including you) knows the definition of your population R-squared. Hence IMHO this is a seriously problematic approach. – Stéphane Laurent Jul 1 '13 at 5:35
• @StéphaneLaurent Thanks for that. I confess that up to this point I have not seen population r-square as a property of contention. For example, I could propose a data generating process and there would be an r-square that is approached as my simulation sample size approaches infinity. And likewise I assume that there is a data generating process for my data, and therefore if it were possible to get an infinite sample, I could calculate the true population r-square. – Jeromy Anglim Jul 1 '13 at 7:20
• Yes but I'm under the impression you also assume a generating process for the predictors. I cannot figure how this could make sense for a general linear model. – Stéphane Laurent Jul 1 '13 at 8:01