statistics - What is the difference between Multiple R-squared and Adjusted R-squared in a single-variate least squares regression? - Stack Overflow
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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))
lm(formula = epm ~ n_days, data = v)
Residuals:
-693.59 -325.79
Coefficients:
Estimate Std. Error t value Pr(&|t|)
(Intercept)
&2e-16 ***
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
27.9k27126252
25.2k167193
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:
6,42812336
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
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
17.8k864104
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.
3,48922032
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 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.
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Stack Overflow works best with JavaScript enabledR-squared - Minitab
In This TopicWhat is R-squared?
R2 is the percentage of response variable variation that is explained by its relationship with one or more predictor variables. Usually, the higher the R2, the better the model fits your data. R2 is always between 0 and 100%. R-squared is also known as the coefficient of determination or multiple determination (in multiple linear regression).Graphical illustration of R-squared
You can plot observed values by fitted values to graphically illustrate R2 values for regression models.
Plots of Observed Responses Versus Fitted ResponsesThe first regression model explains 85.5% of the variance while the second one explains 22.6%. The more variance that is explained by the regression model the closer the data points will fall to the fitted regression line. Theoretically, if a model could explain 100% of the variance, the fitted values would always equal the observed values and, therefore, all the data points would fall on the fitted regression line. What is R-squared adjusted?
R2 adjusted is the percentage of response variable variation that is explained by its relationship with one or more predictor variables, adjusted for the number of predictors in the model. This adjustment is important because the R2 for any model will always increase when a new term is added. A model with more terms can seem to have a better fit because it has more terms.
Use R2 adjusted to determine how well the model fits your data when you want to adjust for the number of predictors in the model. The adjusted R2 value incorporates the number of predictors in the model to help you choose the correct model.
Example of R-squared adjusted
For example, you work for a potato chip company that examines the factors which affect the percentage of crumbled potato chips per container. You include the percentage of potato relative to other ingredients, cooling rate, and cooking temperature as predictors in the regression model. You receive the following results as you add the predictors in a forward stepwise approach:
Cooling rate
Cooking temp
R2 adjusted
Regression p-value
The first step yields a statistically significant regression model. Adding the second term, you see that the adjusted R2 has increased indicating that "cooling rate" has improved the model. You add the third term, cooking temperature, and while the R2 increases, the adjusted R2 does not. Because cooking temperature has not improved the model, you might consider removing it from the model. What is predicted R-squared?
Use predicted R2 to determine how well the model predicts responses for new observations. Larger values of predicted R2 indicate models of greater predictive ability.
Predicted R2 is calculated by systematically removing each observation from the data set, estimating the regression equation, and determining how well the model predicts the removed observation. Predicted R2 ranges between 0 and 100% and is calculated from the PRESS statistic.
Predicted R2 can prevent over-fitting the model and can be more useful than adjusted R2 for comparing models because it is calculated using observations not included in model estimation. Over-fitting refers to models that seem to explain the relationship between the predictor and response variables for the data set used for model calculation but fail to provide valid predictions for new observations.
Example of predicted R-squared
For example, you work for a financial consulting company and are developing a model to predict future market conditions. The model looks promising because it has an R2 of 87%. However, when you calculate the predicted R2 you see that it drops to 52%. This might indicate an over-fitted model and indicates that your model will not predict new observations as accurately as it fits your existing data. Englishfran?aisDeutschportuguêsespa?ol日本語???中文（简体）By using this site you agree to the use of cookies for analytics and personalized content.&&当前位置：>>新闻资讯>>正文
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