What is the difference between Multiple R-squared and Adjusted R-squared in a single-variate least squares regression?

Question

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  

Answer 1

The "adjustment" in adjusted R-squared is related to the number of variables and the number of observations.

If you keep adding variables (predictors) to your model, R-squared will improve - that is, the predictors will appear to explain the variance - but some of that improvement may be due to chance alone. So adjusted R-squared tries to correct for this, by taking into account the ratio (N-1)/(N-k-1) where N = number of observations and k = number of variables (predictors).

It's probably not a concern in your case, since you have a single variate.

Some references:

1. How high, R-squared?
2. Goodness of fit statistics
3. Multiple regression
4. Re: What is "Adjusted R^2" in Multiple Regression