Questions tagged [r-squared]

The coefficient of determination, usually symbolized by $R^2$, is the proportion of the total response variance explained by a regression model. Can also be used for various pseudo R-squared proposed, for instance for logistic regression (and other models.)

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When is it ok to remove the intercept in a linear regression model?

I am running linear regression models and wondering what the conditions are for removing the intercept term. In comparing results from two different regressions where one has the intercept and the ...
analyticsPierce's user avatar
287 votes
6 answers
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Is $R^2$ useful or dangerous?

I was skimming through some lecture notes by Cosma Shalizi (in particular, section 2.1.1 of the second lecture), and was reminded that you can get very low $R^2$ even when you have a completely linear ...
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Is my model any good, based on the diagnostic metric ($R^2$/ AUC/ accuracy/ RMSE etc.) value?

I've fitted my model and am trying to understand whether it's any good. I've calculated the recommended metrics to assess it ($R^2$/ AUC / accuracy / prediction error / etc) but do not know how to ...
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How to motivate the definition of $R^2$ in `sklearn.metrics.r2_score`?

TLDR: What motivates the definition of $R^2$ in the Python function sklearn.metrics.r2_score? DETAILS The Python machine learning package ...
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Removal of statistically significant intercept term increases $R^2$ in linear model

In a simple linear model with a single explanatory variable, $\alpha_i = \beta_0 + \beta_1 \delta_i + \epsilon_i$ I find that removing the intercept term improves the fit greatly (value of $R^2$ ...
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9 answers
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Which pseudo-$R^2$ measure is the one to report for logistic regression (Cox & Snell or Nagelkerke)?

I have SPSS output for a logistic regression model. The output reports two measures for the model fit, Cox & Snell and ...
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Interpreting nonlinear regression $R^2$

In ordinary least squares linear regression, $R^2=1-\frac{SSRes}{SSTotal}$ is described as the “proportion of variance explained”. Does this apply to nonlinear regression, too?
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Logistic regression with poor goodness of fit (hosmer lemeshow)?

I built a model with 9 categorical predictor variables. Using SPSS, my omnibus test was significant ($\chi^2$=220.01), my -2loglikelihood was 1335.2 (Nagelkerke $R^2$ 0.231), but my Hosmer and ...
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4 answers
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When is R squared negative? [duplicate]

My understanding is that $R^2$ cannot be negative as it is the square of R. However I ran a simple linear regression in SPSS with a single independent variable and a dependent variable. My SPSS output ...
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$R^2$ on out-sample data set

The conventional definition of $R^2$ is: $R^2 = 1-SSE/SST$, where SSE denotes sum of squared errors and SST is total sum of squares ($n\times variance$, n being number of sample points in train set). ...
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Is the proportion classified correctly a reasonable analogue of $R^2$ for a classification model?

Let's do some classification and evaluate the prediction quality. The easiest metric to understand is the prediction accuracy, which can be reported as the proportion classified correctly to put the ...
Dave's user avatar
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60 votes
6 answers
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Relationship between $R^2$ and correlation coefficient

Let's say I have two 1-dimensional arrays, $a_1$ and $a_2$. Each contains 100 data points. $a_1$ is the actual data, and $a_2$ is the model prediction. In this case, the $R^2$ value would be: $$ R^2 = ...
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Geometric interpretation of multiple correlation coefficient $R$ and coefficient of determination $R^2$

I am interested in the geometric meaning of the multiple correlation $R$ and coefficient of determination $R^2$ in the regression $y_i = \beta_1 + \beta_2 x_{2,i} + \dots + \beta_k x_{k,i} + \...
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What does negative R-squared mean?

Let's say I have some data, and then I fit the data with a model (a non-linear regression). Then I calculate the R-squared ($R^2$). When R-squared is negative, what does that mean? Does that mean my ...
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What is the difference between "coefficient of determination" and "mean squared error"?

For regression problem, I have seen people use "coefficient of determination" (a.k.a R squared) to perform model selection, e.g., finding the appropriate penalty coefficient for regularization. ...
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What is the distribution of $R^2$ in linear regression under the null hypothesis? Why is its mode not at zero when $k>3$?

What is the distribution of the coefficient of determination, or R squared, $R^2$, in linear univariate multiple regression under the null hypothesis $H_0:\beta=0$? How does it depend on the number ...
amoeba's user avatar
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Is R-squared truly an invalid metric for non-linear models?

I have read that R-squared is invalid for non-linear models, because the relationship that SSR + SSE = SSTotal no longer holds. Can somebody explain why this is true? SSR and SSE are just the ...
Greg's user avatar
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4 answers
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Importance of predictors in multiple regression: Partial $R^2$ vs. standardized coefficients

I am wondering what the exact relationship between partial $R^2$ and coefficients in a linear model is and whether I should use only one or both to illustrate the importance and influence of factors. ...
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Is there an elegant/insightful way to understand this linear regression identity for multiple $R^2$?

In linear regression I have come across a delightful result that if we fit the model $$E[Y] = \beta_1 X_1 + \beta_2 X_2 + c,$$ then, if we standardize and centre the $Y$, $X_1$ and $X_2$ data, $$R^...
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How to interpret the UCLA "adjusted count" logistic regression pseudo $R^2?$

Here, UCLA gives a number of pseudo $R^2$ values for evaluating logistic regression models. Despite the issues with doing this, the last two deal with hard classifications rather than the ...
Dave's user avatar
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44 votes
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Manually calculated $R^2$ doesn't match up with randomForest() $R^2$ for testing new data

I know this is a fairly specific R question, but I may be thinking about proportion variance explained, $R^2$, incorrectly. Here goes. I'm trying to use the ...
Stephen Turner's user avatar
22 votes
3 answers
32k views

How to calculate out of sample R squared?

I know this probably has been discussed somewhere else, but I have not been able to find an explicit answer. I am trying to use the formula $R^2 = 1 - SSR/SST$ to calculate out-of-sample $R^2$ of a ...
crazydriver's user avatar
15 votes
2 answers
4k views

What is an unbiased estimate of population R-square?

I am interested in getting an unbiased estimate of $R^2$ in a multiple linear regression. On reflection, I can think of two different values that an unbiased estimate of $R^2$ might be trying to ...
Jeromy Anglim's user avatar
30 votes
4 answers
75k views

Pseudo R squared formula for GLMs

I found a formula for pseudo $R^2$ in the book Extending the Linear Model with R, Julian J. Faraway (p. 59). $$1-\frac{\text{ResidualDeviance}}{\text{NullDeviance}}$$. Is this a common formula for ...
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24 votes
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How to split r-squared between predictor variables in multiple regression?

I have just read a paper in which the authors carried out a multiple regression with two predictors. The overall r-squared value was 0.65. They provided a table which split the r-squared between the ...
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21 votes
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Calculating $R^2$ in mixed models using Nakagawa & Schielzeth's (2013) R2glmm method

I have been reading about calculating $R^2$ values in mixed models and after reading the R-sig FAQ, other posts on this forum (I would link a few but I don't have enough reputation) and several other ...
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9 answers
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Measuring accuracy of a logistic regression-based model

I have a trained logistic regression model that I am applying to a testing data set. The dependent variable is binary (boolean). For each sample in the testing data set, I apply the logistic ...
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2 answers
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Why not use the R squared to measure forecast accuracy?

Why in literature usually the common accuracy measures like MAD, MSE, RMSE, MAPE ... are used. Why not use the $R^2$ (coefficient of determination)? I was thinking about the difference: By using the ...
NMe's user avatar
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11 votes
1 answer
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MAPE vs R-squared in regression models

Usually regression models are evaluated using $R^2$. I understand this metric can be misleading too at times but as far as I understand the first parameter we look at is $R^2$. There is another ...
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2 answers
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Can you calculate $R^2$ from correlation coefficents in multiple linear regression?

In simple linear regression, $R^2$ is equivalent to the squared correlation of a dependent and an independent variable. Is this also true for multiple linear regression? For example, I measured trait ...
E_H's user avatar
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23 votes
4 answers
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Linear regression what does the F statistic, R squared and residual standard error tell us?

I'm really confused about the difference in meaning regarding the context of linear regression of the following terms: F statistic R squared Residual standard error I found this webstie which gave ...
KingBoomie's user avatar
19 votes
2 answers
7k views

Why squaring $R$ gives explained variance?

This may be a basic question, but I was wondering why an $R$ value in a regression model can simply be squared to give a figure of explained variance? I understand that $R$ coefficient can give the ...
David's user avatar
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13 votes
2 answers
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Ridge regression in R with p values and goodness of fit [closed]

Doing ridge regression in R I have discovered linearRidge in the ridge package - which fits a model, reports coefficients and p ...
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4 votes
2 answers
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Model uncertainty (model averaging) and R-Squared ($R^2$)

Is it possible to calculate r-squared for an "average model"? Lets say I have 4 different response variables that I want to model to a set (or subset) of 4 independent variables. I'd then like to ...
Dave M's user avatar
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2 votes
2 answers
2k views

What does it mean if I have a high F-stat but low $R^2$?

As far as I understand, a high F-stat leads to a high $R^2$, though the converse is not true. What does it mean if I have a high F-stat and a low $R^2$?
pythonuser's user avatar
40 votes
1 answer
27k views

Is there any difference between $r^2$ and $R^2$?

The correlation coefficient is usually written with a capital $R$ but sometimes not. I wonder if there really is a difference between $r^2$ and $R^2$? Can $r$ mean something else than a correlation ...
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26 votes
2 answers
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R-squared in quantile regression

I am using quantile regression to find predictors of 90th percentile of my data. I am doing this in R using the quantreg package. How can I determine $r^2$ for ...
rnso's user avatar
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24 votes
3 answers
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What is the relationship between R-squared and p-value in a regression?

tl;dr - for OLS regression, does a higher R-squared also imply a higher P-value? Specifically for a single explanatory variable (Y = a + bX + e) but would also be interested to know for n multiple ...
econometricstatsquestion's user avatar
19 votes
3 answers
4k views

Principal component analysis "backwards": how much variance of the data is explained by a given linear combination of the variables?

I have carried out a principal components analysis of six variables $A$, $B$, $C$, $D$, $E$ and $F$. If I understand correctly, unrotated PC1 tells me what linear combination of these variables ...
N26's user avatar
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18 votes
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Formula for 95% confidence interval for $R^2$

I googled and searched on stats.stackexchange but I cannot find the formula to calculate a 95% confidence interval for an $R^2$ value for a linear regression. Can anyone provide it? Even better, let'...
luciano's user avatar
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8 votes
3 answers
915 views

Justification for and optimality of $R^2_{adj.}$ as a model selection criterion

In a recent thread, use of adjusted $R^2$ ($R^2_{adj.}$) is mentioned in the context of model selection, e.g. The adjustment was invented as a solution to problems caused by variable selection ...
Richard Hardy's user avatar
6 votes
2 answers
1k views

Why is R2 not reported for GLMs based on last iteration of IRLS weighted least square regression with which it is fit

Given that GLMs are generally fit using iteratively reweighted least squares (based on a Fisher scoring algorithm to maximize the max likelihood objective, which is a variant of Newton-Raphson, see ...
Tom Wenseleers's user avatar
5 votes
4 answers
9k views

Can the coefficient of determination $R^2$ be more than one? What is its upper bound?

It is well known that if you add additional independent variables in a linear regression, the $R^2$ of the new model is at least as large as the previous model, so you obtain a lower bound for the $R^...
Irland's user avatar
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4 votes
1 answer
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Coefficient of determination for binary responses

D.R. Cox and Nanny Wermuth seem to suggest that the coefficient of determination (R squared) is misleading when you have binary responses, in fact if I am understanding well, they are saying that the ...
Cesare Camestre's user avatar
4 votes
3 answers
10k views

Multiple Correlation Coefficient with three or more independent variables

The formula for the multiple coefficient of correlation of two independent variables ($x_1$ and $x_2$) and an dependent variables ($y$) is this: $$R=\sqrt{\frac{r^2_{yx_1}+r^2_{yx_2}-2r_{yx_1}r_{yx_2}...
Michele Papasidero's user avatar
49 votes
2 answers
64k views

What is the adjusted R-squared formula in lm in R and how should it be interpreted?

What is the exact formula used in R lm() for the Adjusted R-squared? How can I interpret it? Adjusted r-squared formulas There seem to exist several formulas to ...
user avatar
25 votes
2 answers
18k views

Is a weighted $R^2$ in robust linear model meaningful for goodness of fit analysis?

I estimated a robust linear model in R with MM weights using the rlm() in the MASS package. `R`` does not provide an $R^2$ value ...
CraigMilligan's user avatar
20 votes
3 answers
232k views

What's the difference between multiple R and R squared?

In linear regression, we often get multiple R and R squared. What are the differences between them?
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15 votes
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Why is adjusted R-squared less than R-squared if adjusted R-squared predicts the model better?

As far as I understand, $R^2$ explains how well the model predicts the observation. Adjusted $R^2$ is the one that takes into account more observations (or degrees of freedom). So, Adjusted $R^2$ ...
user59756's user avatar
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14 votes
3 answers
22k views

Which is better: r-squared or adjusted r-squared?

I just started to learn about the following statistical measures, r-squared and adjusted r-squared and was wondering why can't we use adjusted r-squared for every regression model considering the ...
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