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Interpret a multiple linear regression when Y is log transformed [duplicate]

Interpretation of log transformed predictor and/or responseHow to interpret logarithmically transformed coefficients in linear regression?Does the first line of interaction output from lmer contain all levels of variables?Interpretation of log transformed coefficients, OLS regressionDummy variables in multiple regressionCoefficients linear and log-linear regression modelHow to find y values given x values and summary statistics on yInteraction effects with three dummy variables - interpretationInterpretation of variable in a probit modelinterpreting HLM coefficientsInterpretation of interaction term in a probit estimation

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4

$begingroup$

This question already has an answer here:

• 
  Interpretation of log transformed predictor and/or response
  
  3 answers
  
  

I have the following multiple linear regression model:
Log(y) = B0 + B1X1 + B2X2 + B3x3 + e.
X1 is a dummy that can take 0 = male and 1 = female and
X2 and X3 are continuous variables.

I am not entirely sure on how to interpret the coefficients for the variables.
The coefficient for the dummy variable is 0,20. Does that mean, that changing from male to female (male is baseline) the Y will increase by an average of 20%. Is it directly translated into percentage?

And for the continuous variables, the coefficient for X2 is 0,1. Does that mean that increasing X2 with 1 unit increases Y with an average of 10%? Again, is it directly translated into percentage?

regression data-transformation interpretation logarithm

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edited Apr 27 at 17:42

Richard Hardy

28.6k644131

asked Apr 27 at 14:37

maSmaS

311

New contributor

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marked as duplicate by COOLSerdash, Richard Hardy, Siong Thye Goh, kjetil b halvorsen, Michael Chernick Apr 27 at 20:49

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

add a comment | 

4

$begingroup$

This question already has an answer here:

• 
  Interpretation of log transformed predictor and/or response
  
  3 answers
  
  

I have the following multiple linear regression model:
Log(y) = B0 + B1X1 + B2X2 + B3x3 + e.
X1 is a dummy that can take 0 = male and 1 = female and
X2 and X3 are continuous variables.

I am not entirely sure on how to interpret the coefficients for the variables.
The coefficient for the dummy variable is 0,20. Does that mean, that changing from male to female (male is baseline) the Y will increase by an average of 20%. Is it directly translated into percentage?

And for the continuous variables, the coefficient for X2 is 0,1. Does that mean that increasing X2 with 1 unit increases Y with an average of 10%? Again, is it directly translated into percentage?

regression data-transformation interpretation logarithm

share|cite|improve this question

edited Apr 27 at 17:42

Richard Hardy

28.6k644131

asked Apr 27 at 14:37

maSmaS

311

New contributor

maS is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

marked as duplicate by COOLSerdash, Richard Hardy, Siong Thye Goh, kjetil b halvorsen, Michael Chernick Apr 27 at 20:49

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

add a comment | 

$begingroup$

This question already has an answer here:

• 
  Interpretation of log transformed predictor and/or response
  
  3 answers
  
  

I have the following multiple linear regression model:
Log(y) = B0 + B1X1 + B2X2 + B3x3 + e.
X1 is a dummy that can take 0 = male and 1 = female and
X2 and X3 are continuous variables.

I am not entirely sure on how to interpret the coefficients for the variables.
The coefficient for the dummy variable is 0,20. Does that mean, that changing from male to female (male is baseline) the Y will increase by an average of 20%. Is it directly translated into percentage?

And for the continuous variables, the coefficient for X2 is 0,1. Does that mean that increasing X2 with 1 unit increases Y with an average of 10%? Again, is it directly translated into percentage?

regression data-transformation interpretation logarithm

share|cite|improve this question

edited Apr 27 at 17:42

Richard Hardy

28.6k644131

asked Apr 27 at 14:37

maSmaS

311

New contributor

maS is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

share|cite|improve this question

edited Apr 27 at 17:42

Richard Hardy

28.6k644131

asked Apr 27 at 14:37

maSmaS

311

New contributor

maS is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

share|cite|improve this question

edited Apr 27 at 17:42

Richard Hardy

28.6k644131

asked Apr 27 at 14:37

maSmaS

311

New contributor

maS is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

edited Apr 27 at 17:42

Richard Hardy

28.6k644131

edited Apr 27 at 17:42

Richard Hardy

28.6k644131

asked Apr 27 at 14:37

maSmaS

311

New contributor

maS is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

asked Apr 27 at 14:37

maSmaS

311

1 Answer
1

active

oldest

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4

$begingroup$

Positive coefficients somehow indicate a positive effect, but they don't simply turn into percentages. There is a transformation. Let's say your model is $log y = b_0+b_1x_1$; this means $y=e^b_0+b_1x_1=A_0e^b_1x_1$. So, dummy or not, if $x_1$ increases by $1$ unit, $y$ increases by $e^b_1$, i.e. if $b_1=0.2$, $y$ increases by $e^0.2approx 1.22$, i.e. $22%$. The case is similar for your continuous variable.

share|cite|improve this answer

answered Apr 27 at 15:20

gunesgunes

8,5091418

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I have read that if the natural log is used it is approximately translated into a percentage change, if the change in x is small. Is that correct?
  $endgroup$
  – maS
  Apr 27 at 15:45
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Yes, but approximate is the key term here, because Taylor expansion of $e^x$ is: $e^x=1+x+x^2/2!,...approx 1+x$ when $x$ is small. Which means $x%$ increase in something. Here, to execute this idea, your coefficients should be small.
  $endgroup$
  – gunes
  Apr 27 at 15:47
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thanks gunes. Would you recommend using natural log then as my log transformation or what base should i use? cant seem to find a solid explanation of the choice
  $endgroup$
  – maS
  Apr 27 at 16:08
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Also: inference on transformed variables $ne$ inference on un-transformed variables.
  $endgroup$
  – Alexis
  Apr 27 at 16:57
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @maS using base N results in model $y=e^b_0log N+b_1log N x_1$, where you solve for $a_i=b_ilog N$. It's one to one.
  $endgroup$
  – gunes
  Apr 27 at 17:36
  

  
  
  
  

add a comment | 

4

$begingroup$

Positive coefficients somehow indicate a positive effect, but they don't simply turn into percentages. There is a transformation. Let's say your model is $log y = b_0+b_1x_1$; this means $y=e^b_0+b_1x_1=A_0e^b_1x_1$. So, dummy or not, if $x_1$ increases by $1$ unit, $y$ increases by $e^b_1$, i.e. if $b_1=0.2$, $y$ increases by $e^0.2approx 1.22$, i.e. $22%$. The case is similar for your continuous variable.

share|cite|improve this answer

answered Apr 27 at 15:20

gunesgunes

8,5091418

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I have read that if the natural log is used it is approximately translated into a percentage change, if the change in x is small. Is that correct?
  $endgroup$
  – maS
  Apr 27 at 15:45
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Yes, but approximate is the key term here, because Taylor expansion of $e^x$ is: $e^x=1+x+x^2/2!,...approx 1+x$ when $x$ is small. Which means $x%$ increase in something. Here, to execute this idea, your coefficients should be small.
  $endgroup$
  – gunes
  Apr 27 at 15:47
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thanks gunes. Would you recommend using natural log then as my log transformation or what base should i use? cant seem to find a solid explanation of the choice
  $endgroup$
  – maS
  Apr 27 at 16:08
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Also: inference on transformed variables $ne$ inference on un-transformed variables.
  $endgroup$
  – Alexis
  Apr 27 at 16:57
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @maS using base N results in model $y=e^b_0log N+b_1log N x_1$, where you solve for $a_i=b_ilog N$. It's one to one.
  $endgroup$
  – gunes
  Apr 27 at 17:36
  

  
  
  
  

add a comment | 

4

$begingroup$

Positive coefficients somehow indicate a positive effect, but they don't simply turn into percentages. There is a transformation. Let's say your model is $log y = b_0+b_1x_1$; this means $y=e^b_0+b_1x_1=A_0e^b_1x_1$. So, dummy or not, if $x_1$ increases by $1$ unit, $y$ increases by $e^b_1$, i.e. if $b_1=0.2$, $y$ increases by $e^0.2approx 1.22$, i.e. $22%$. The case is similar for your continuous variable.

share|cite|improve this answer

answered Apr 27 at 15:20

gunesgunes

8,5091418

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I have read that if the natural log is used it is approximately translated into a percentage change, if the change in x is small. Is that correct?
  $endgroup$
  – maS
  Apr 27 at 15:45
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Yes, but approximate is the key term here, because Taylor expansion of $e^x$ is: $e^x=1+x+x^2/2!,...approx 1+x$ when $x$ is small. Which means $x%$ increase in something. Here, to execute this idea, your coefficients should be small.
  $endgroup$
  – gunes
  Apr 27 at 15:47
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thanks gunes. Would you recommend using natural log then as my log transformation or what base should i use? cant seem to find a solid explanation of the choice
  $endgroup$
  – maS
  Apr 27 at 16:08
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Also: inference on transformed variables $ne$ inference on un-transformed variables.
  $endgroup$
  – Alexis
  Apr 27 at 16:57
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @maS using base N results in model $y=e^b_0log N+b_1log N x_1$, where you solve for $a_i=b_ilog N$. It's one to one.
  $endgroup$
  – gunes
  Apr 27 at 17:36
  

  
  
  
  

add a comment | 

$begingroup$

Positive coefficients somehow indicate a positive effect, but they don't simply turn into percentages. There is a transformation. Let's say your model is $log y = b_0+b_1x_1$; this means $y=e^b_0+b_1x_1=A_0e^b_1x_1$. So, dummy or not, if $x_1$ increases by $1$ unit, $y$ increases by $e^b_1$, i.e. if $b_1=0.2$, $y$ increases by $e^0.2approx 1.22$, i.e. $22%$. The case is similar for your continuous variable.

share|cite|improve this answer

answered Apr 27 at 15:20

gunesgunes

8,5091418

$endgroup$

share|cite|improve this answer

answered Apr 27 at 15:20

gunesgunes

8,5091418

answered Apr 27 at 15:20

gunesgunes

8,5091418

answered Apr 27 at 15:20

gunesgunes

8,5091418