# How to compute adjusted R-squared (in R)

## Task

If we have fit a multivariate linear model, how can we compute the Adjusted $R^2$ for that model, to measure its goodness of fit?

Related tasks:

## Solution

We assume you have already fit a multivariate linear model to the data, as in the code below. (If you’re unfamiliar with how to do so, see how to fit a multivariate linear model.) The data shown below is fake, and we assume you will replace it with your own real data if you use this code.

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x1 <- c(2, 7, 4, 3, 11, 18, 6, 15, 9, 12)
x2 <- c(4, 6, 10, 1, 18, 11, 8, 20, 16, 13)
x3 <- c(11, 16, 20, 6, 14, 8, 5, 23, 13, 10)
y <- c(24, 60, 32, 29, 90, 45, 130, 76, 100, 120)
model <- lm(y ~ x1 + x2 + x3)

You can get a lot of information about your model from its summary.

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summary(model)

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Call:
lm(formula = y ~ x1 + x2 + x3)
Residuals:
Min 1Q Median 3Q Max
-25.031 -20.218 -8.373 22.937 35.640
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 77.244 27.366 2.823 0.0302 *
x1 -2.701 2.855 -0.946 0.3806
x2 7.299 2.875 2.539 0.0441 *
x3 -4.861 2.187 -2.223 0.0679 .
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 30.13 on 6 degrees of freedom
Multiple R-squared: 0.5936, Adjusted R-squared: 0.3904
F-statistic: 2.921 on 3 and 6 DF, p-value: 0.1222

In particular, that printout contains the Adjusted $R^2$ value; it is the second value in the right-hand column, near the top.

You can also obtain it directly, as follows:

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summary(model)$adj.r.squared

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[1] 0.3903924

In this case, the Adjusted $R^2$ is $0.3904$.

Content last modified on 24 July 2023.

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Contributed by Elizabeth Czarniak (CZARNIA_ELIZ@bentley.edu)