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What does it mean when I add a new variable to my linear model and the R^2 stays the same?

In what cases (if any) does r^2 remain unchanged on adding a new variable?How can I predict values from new inputs of a linear model in R?Does a stepwise approach produce the highest $R^2$ model?F test and t test in linear regression modelCompare linear regression models (same and different response variable)In linear model, if you add one more variable, then what happens to the constant?Getting estimate and CI for dummy variable in linear modelCircularity in Linear Regression: Independent variable used as dependent in the same modelWhat is the difference between generalized linear models and generalized least squaresPCA without response variable to get linearly dependent set of linear (mixed) model inputswhy does adding new variables to a regression model keep R squared unchanged

4

1

$begingroup$

I'm inclined to think that the new variable is not correlated to the response. But could the new variable be correlated to another variable in the model?

linear-model r-squared

share|cite|improve this question

asked Mar 7 at 19:46

Chance113Chance113

362

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  It depends, could you provide us with some reduced data lines or output from your linear models. Without more information it's hard to assist you
  $endgroup$
  – OliverFishCode
  Mar 7 at 19:54
  

  
  
  
  


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  It shouldn't stay exactly the same unless it is perfectly orthogonal to your response, or is a linear combination of the variables already included. It may be that the change is smaller than the number of decimal places displayed.
  $endgroup$
  – gung♦
  Mar 7 at 20:02
  

  
  
  
  


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  @gung What you can infer is that the new variable is orthogonal to the response modulo the subspace generated by the other variables. That's more general than the two options you mention.
  $endgroup$
  – whuber♦
  Mar 7 at 20:12
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @whuber, yes, I suppose so.
  $endgroup$
  – gung♦
  Mar 7 at 20:17
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Test your variables for multicollinearity en.wikipedia.org/wiki/Multicollinearity probably some features are linearly connected. Use caret package and vif() in R sthda.com/english/articles/39-regression-model-diagnostics/…
  $endgroup$
  – Tom Zinger
  Mar 7 at 22:33
  
  

  
  
  
  

 | 
show 2 more comments

4

1

$begingroup$

I'm inclined to think that the new variable is not correlated to the response. But could the new variable be correlated to another variable in the model?

linear-model r-squared

share|cite|improve this question

asked Mar 7 at 19:46

Chance113Chance113

362

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  It depends, could you provide us with some reduced data lines or output from your linear models. Without more information it's hard to assist you
  $endgroup$
  – OliverFishCode
  Mar 7 at 19:54
  

  
  
  
  


• 
  
  
  

  
  5
  

  
  
  
  
  
  

  
  $begingroup$
  It shouldn't stay exactly the same unless it is perfectly orthogonal to your response, or is a linear combination of the variables already included. It may be that the change is smaller than the number of decimal places displayed.
  $endgroup$
  – gung♦
  Mar 7 at 20:02
  

  
  
  
  


• 
  
  
  

  
  5
  

  
  
  
  
  
  

  
  $begingroup$
  @gung What you can infer is that the new variable is orthogonal to the response modulo the subspace generated by the other variables. That's more general than the two options you mention.
  $endgroup$
  – whuber♦
  Mar 7 at 20:12
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @whuber, yes, I suppose so.
  $endgroup$
  – gung♦
  Mar 7 at 20:17
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Test your variables for multicollinearity en.wikipedia.org/wiki/Multicollinearity probably some features are linearly connected. Use caret package and vif() in R sthda.com/english/articles/39-regression-model-diagnostics/…
  $endgroup$
  – Tom Zinger
  Mar 7 at 22:33
  
  

  
  
  
  

 | 
show 2 more comments

$begingroup$

I'm inclined to think that the new variable is not correlated to the response. But could the new variable be correlated to another variable in the model?

linear-model r-squared

share|cite|improve this question

asked Mar 7 at 19:46

Chance113Chance113

362

$endgroup$

share|cite|improve this question

asked Mar 7 at 19:46

Chance113Chance113

362

share|cite|improve this question

asked Mar 7 at 19:46

Chance113Chance113

362

asked Mar 7 at 19:46

Chance113Chance113

362

asked Mar 7 at 19:46

Chance113Chance113

362

2 Answers
2

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oldest

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5

$begingroup$

Seeing little to no change in $R^2$ when you add a variable to a linear model means that the variable has little to no additional explanatory power to the response over what is already in your model. As you note, this can be either because it tells you almost nothing about the response or it explains the same variation in the response as the variables already in the model.

share|cite|improve this answer

answered Mar 7 at 19:55

TrynnaDoStatTrynnaDoStat

5,57211335

$endgroup$

add a comment | 

1

$begingroup$

As others have alluded, seeing no change in $R^2$ when you add a variable to your regression is unusual. In finite samples, this should only happen when your new variable is a linear combination of variables already present. In this case, most standard regression routines simply exclude that variable from the regression, and your $R^2$ will remain unchanged because the model was effectively unchanged.

As you notice, this does not mean the variable is unimportant, but rather that you are unable to distinguish its effect from that of the other variables in your model.

More broadly however, I (and many here at Cross Validated) would caution against using R^2 for model selection and interpretation. What I've discussed above is how the $R^2$ could not change and the variable still be important. Worse yet, the $R^2$ could change somewhat (or even dramatically) when you include an irrelevant variable. Broadly, using $R^2$ for model selection fell out of favor in the 70s, when it was dropped in favor of AIC (and its contemporaries). Today -- a typical statistician would recommend using cross validation (see the site name) for your model selection.

In general, adding a variable increases $R^2$ -- so using $R^2$ to determine a variables importance is a bit of a wild goose chase. Even when trying to understand simple situations you will end up with a completely absurd collection of variables.

share|cite|improve this answer

answered Mar 7 at 20:13

user5957401user5957401

29727

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Could you elaborate on the $R^2$ could change somewhat (or even dramatically) when you include an irrelevant variable, specifically on the case of a dramatical change? In which sense would the variable then be irrelevant?
  $endgroup$
  – Richard Hardy
  Mar 7 at 21:38
  
  

  
  
  
  

add a comment | 

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5

$begingroup$

Seeing little to no change in $R^2$ when you add a variable to a linear model means that the variable has little to no additional explanatory power to the response over what is already in your model. As you note, this can be either because it tells you almost nothing about the response or it explains the same variation in the response as the variables already in the model.

share|cite|improve this answer

answered Mar 7 at 19:55

TrynnaDoStatTrynnaDoStat

5,57211335

$endgroup$

add a comment | 

5

$begingroup$

Seeing little to no change in $R^2$ when you add a variable to a linear model means that the variable has little to no additional explanatory power to the response over what is already in your model. As you note, this can be either because it tells you almost nothing about the response or it explains the same variation in the response as the variables already in the model.

share|cite|improve this answer

answered Mar 7 at 19:55

TrynnaDoStatTrynnaDoStat

5,57211335

$endgroup$

add a comment | 

$begingroup$

Seeing little to no change in $R^2$ when you add a variable to a linear model means that the variable has little to no additional explanatory power to the response over what is already in your model. As you note, this can be either because it tells you almost nothing about the response or it explains the same variation in the response as the variables already in the model.

share|cite|improve this answer

answered Mar 7 at 19:55

TrynnaDoStatTrynnaDoStat

5,57211335

$endgroup$

share|cite|improve this answer

answered Mar 7 at 19:55

TrynnaDoStatTrynnaDoStat

5,57211335

answered Mar 7 at 19:55

TrynnaDoStatTrynnaDoStat

5,57211335

answered Mar 7 at 19:55

TrynnaDoStatTrynnaDoStat

5,57211335

1

$begingroup$

As others have alluded, seeing no change in $R^2$ when you add a variable to your regression is unusual. In finite samples, this should only happen when your new variable is a linear combination of variables already present. In this case, most standard regression routines simply exclude that variable from the regression, and your $R^2$ will remain unchanged because the model was effectively unchanged.

As you notice, this does not mean the variable is unimportant, but rather that you are unable to distinguish its effect from that of the other variables in your model.

More broadly however, I (and many here at Cross Validated) would caution against using R^2 for model selection and interpretation. What I've discussed above is how the $R^2$ could not change and the variable still be important. Worse yet, the $R^2$ could change somewhat (or even dramatically) when you include an irrelevant variable. Broadly, using $R^2$ for model selection fell out of favor in the 70s, when it was dropped in favor of AIC (and its contemporaries). Today -- a typical statistician would recommend using cross validation (see the site name) for your model selection.

In general, adding a variable increases $R^2$ -- so using $R^2$ to determine a variables importance is a bit of a wild goose chase. Even when trying to understand simple situations you will end up with a completely absurd collection of variables.

share|cite|improve this answer

answered Mar 7 at 20:13

user5957401user5957401

29727

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Could you elaborate on the $R^2$ could change somewhat (or even dramatically) when you include an irrelevant variable, specifically on the case of a dramatical change? In which sense would the variable then be irrelevant?
  $endgroup$
  – Richard Hardy
  Mar 7 at 21:38
  
  

  
  
  
  

add a comment | 

1

$begingroup$

As others have alluded, seeing no change in $R^2$ when you add a variable to your regression is unusual. In finite samples, this should only happen when your new variable is a linear combination of variables already present. In this case, most standard regression routines simply exclude that variable from the regression, and your $R^2$ will remain unchanged because the model was effectively unchanged.

As you notice, this does not mean the variable is unimportant, but rather that you are unable to distinguish its effect from that of the other variables in your model.

More broadly however, I (and many here at Cross Validated) would caution against using R^2 for model selection and interpretation. What I've discussed above is how the $R^2$ could not change and the variable still be important. Worse yet, the $R^2$ could change somewhat (or even dramatically) when you include an irrelevant variable. Broadly, using $R^2$ for model selection fell out of favor in the 70s, when it was dropped in favor of AIC (and its contemporaries). Today -- a typical statistician would recommend using cross validation (see the site name) for your model selection.

In general, adding a variable increases $R^2$ -- so using $R^2$ to determine a variables importance is a bit of a wild goose chase. Even when trying to understand simple situations you will end up with a completely absurd collection of variables.

share|cite|improve this answer

answered Mar 7 at 20:13

user5957401user5957401

29727

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Could you elaborate on the $R^2$ could change somewhat (or even dramatically) when you include an irrelevant variable, specifically on the case of a dramatical change? In which sense would the variable then be irrelevant?
  $endgroup$
  – Richard Hardy
  Mar 7 at 21:38
  
  

  
  
  
  

add a comment | 

$begingroup$

As others have alluded, seeing no change in $R^2$ when you add a variable to your regression is unusual. In finite samples, this should only happen when your new variable is a linear combination of variables already present. In this case, most standard regression routines simply exclude that variable from the regression, and your $R^2$ will remain unchanged because the model was effectively unchanged.

As you notice, this does not mean the variable is unimportant, but rather that you are unable to distinguish its effect from that of the other variables in your model.

More broadly however, I (and many here at Cross Validated) would caution against using R^2 for model selection and interpretation. What I've discussed above is how the $R^2$ could not change and the variable still be important. Worse yet, the $R^2$ could change somewhat (or even dramatically) when you include an irrelevant variable. Broadly, using $R^2$ for model selection fell out of favor in the 70s, when it was dropped in favor of AIC (and its contemporaries). Today -- a typical statistician would recommend using cross validation (see the site name) for your model selection.

In general, adding a variable increases $R^2$ -- so using $R^2$ to determine a variables importance is a bit of a wild goose chase. Even when trying to understand simple situations you will end up with a completely absurd collection of variables.

share|cite|improve this answer

answered Mar 7 at 20:13

user5957401user5957401

29727

$endgroup$

share|cite|improve this answer

answered Mar 7 at 20:13

user5957401user5957401

29727

answered Mar 7 at 20:13

user5957401user5957401

29727

answered Mar 7 at 20:13

user5957401user5957401

29727