Could someone explain to the statistically naive what the difference between Multiple R-squared and Adjusted R-squared is? I am doing a single-variate regression analysis as follows:
v.lm <- lm(epm ~ n_days, data=v)
print(summary(v.lm))
Results:
Call:
lm(formula = epm ~ n_days, data = v)
Residuals:
Min 1Q Median 3Q Max
-693.59 -325.79 53.34 302.46 964.95
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 2550.39 92.15 27.677 <2e-16 ***
n_days -13.12 5.39 -2.433 0.0216 *
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 410.1 on 28 degrees of freedom
Multiple R-squared: 0.1746, Adjusted R-squared: 0.1451
F-statistic: 5.921 on 1 and 28 DF, p-value: 0.0216
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Note that, in addition to number of predictive variables, the Adjusted R-squared formula above also adjusts for sample size. A small sample will give a deceptively large R-squared.
Ping Yin & Xitao Fan, J. of Experimental Education 69(2): 203-224, "Estimating R-squared shrinkage in multiple regression", compares different methods for adjusting r-squared and concludes that the commonly-used ones quoted above are not good. They recommend the Olkin & Pratt formula.
However, I've seen some indication that population size has a much larger effect than any of these formulas indicate. I am not convinced that any of these formulas are good enough to allow you to compare regressions done with very different sample sizes (e.g., 2,000 vs. 200,000 samples; the standard formulas would make almost no sample-size-based adjustment). I would do some cross-validation to check the r-squared on each sample.
The R-squared is not dependent on the number of variables in the model. The adjusted R-squared is.
The adjusted R-squared adds a penalty for adding variables to the model that are uncorrelated with the variable your trying to explain. You can use it to test if a variable is relevant to the thing your trying to explain.
Adjusted R-squared is R-squared with some divisions added to make it dependent on the number of variables in the model.
The Adjusted R-squared is close to, but different from, the value of R2. Instead of being based on the explained sum of squares SSR and the total sum of squares SSY, it is based on the overall variance (a quantity we do not typically calculate), s2T = SSY/(n - 1) and the error variance MSE (from the ANOVA table) and is worked out like this: adjusted R-squared = (s2T - MSE) / s2T.
This approach provides a better basis for judging the improvement in a fit due to adding an explanatory variable, but it does not have the simple summarizing interpretation that R2 has.
If I haven't made a mistake, you should verify the values of adjusted R-squared and R-squared as follows:
s2T <- sum(anova(v.lm)[[2]]) / sum(anova(v.lm)[[1]])
MSE <- anova(v.lm)[[3]][2]
adj.R2 <- (s2T - MSE) / s2T
On the other side, R2 is: SSR/SSY, where SSR = SSY - SSE
attach(v)
SSE <- deviance(v.lm) # or SSE <- sum((epm - predict(v.lm,list(n_days)))^2)
SSY <- deviance(lm(epm ~ 1)) # or SSY <- sum((epm-mean(epm))^2)
SSR <- (SSY - SSE) # or SSR <- sum((predict(v.lm,list(n_days)) - mean(epm))^2)
R2 <- SSR / SSY
Source: Stackoverflow.com