Interpreting Regression Coefficients for Log-Transformed Variables









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15 may 2014 Thus the log-transformation actually exacerbated the problem of skewness in this particular example. In general


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Interpreting Regression Coefficients for Log-Transformed Variables

A log transformation is often useful for data which exhibit right skewness (positively skewed) and for data where the variability of residuals increases for 
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213292 Interpreting Regression Coefficients for Log-Transformed Variables

Cornell Statistical Consulting Unit

Interpreting Regression Coefficients for Log-

Transformed Variables

Statnews #83

Cornell Statistical Consulting Unit

Created June 2012. Last updated September 2020

Introduction

Log transformations are one of the most commonly used transformations, but interpreting results of an analysis with log-transformed data may be challenging. This newsletter focuses on how to obtain estimated parameters of interest and how to interpret the coefficients in a regression model involving log-transformed variables. A log transformation is often useful for data which exhibit right skewness (positively skewed), and for data where the variability of residuals increases for larger values of the dependent variable. When some variables are log-transformed, estimating parameters of interest based on the model may involve more calculation than simply taking the anti-log of certain regression coefficients.

The log-normal distribution

To properly back transform into the original scale we need to understand some details about the log-normal distribution. In probability theory, a log-normal distribution is the distribution of the think of ܻ distribution. So it will be helpful to understand the behavior of ܻ true:

The mean of ܻ

The median of ܻ

The variance of ܻ

Suppose we fit a linear regression model with predictors ݔଵǡǥǡݔ௣ and log-transformed response

Cornell Statistical Consulting Unit

is important to note that exponentiating this predicted value does not provide an estimate of the mean of ܻ. Given the three facts stated above, an estimate of the mean of ܻ where ߪ

Coefficient interpretation

Interpreting parameter estimates in a linear regression when some variables are log-transformed expected value of the response variable, holding all other predictors constant. The interpretation of regression coefficients when one or more variables are log-transformed depends on whether the dependent variable, independent variable, or both are transformed. To understand each of these cases, consider an example in which weight is the dependent variable and height is the only independent variable.

Only the dependent variable is transformed

Linear change in the independent variable is associated with multiplicative change in the dependent variable.

Suppose the fitted model is

Explanation

ఉൌݕȀݕ, and is the percent change in ݕ associated with a one-unit increase in ݔ.

Only the independent variable is transformed

Multiplicative change in the independent variable is associated with linear change in the dependent variable.

Fitted model:

Cornell Statistical Consulting Unit

Explanation

Then Both the independent and dependent variable are transformed Multiplicative change in the independent variable is associated with multiplicative change in the dependent variable.

Fitted model:

Explanation

so that the percent change in ݕ associated with a one-percent increase in ݔ is As always if you would like assistance with this topic or any other statistical consulting question, feel free to contact statistical consultants at CSCU.

Author: Jing Yang

Cornell Statistical Consulting Unit

Interpreting Regression Coefficients for Log-

Transformed Variables

Statnews #83

Cornell Statistical Consulting Unit

Created June 2012. Last updated September 2020

Introduction

Log transformations are one of the most commonly used transformations, but interpreting results of an analysis with log-transformed data may be challenging. This newsletter focuses on how to obtain estimated parameters of interest and how to interpret the coefficients in a regression model involving log-transformed variables. A log transformation is often useful for data which exhibit right skewness (positively skewed), and for data where the variability of residuals increases for larger values of the dependent variable. When some variables are log-transformed, estimating parameters of interest based on the model may involve more calculation than simply taking the anti-log of certain regression coefficients.

The log-normal distribution

To properly back transform into the original scale we need to understand some details about the log-normal distribution. In probability theory, a log-normal distribution is the distribution of the think of ܻ distribution. So it will be helpful to understand the behavior of ܻ true:

The mean of ܻ

The median of ܻ

The variance of ܻ

Suppose we fit a linear regression model with predictors ݔଵǡǥǡݔ௣ and log-transformed response

Cornell Statistical Consulting Unit

is important to note that exponentiating this predicted value does not provide an estimate of the mean of ܻ. Given the three facts stated above, an estimate of the mean of ܻ where ߪ

Coefficient interpretation

Interpreting parameter estimates in a linear regression when some variables are log-transformed expected value of the response variable, holding all other predictors constant. The interpretation of regression coefficients when one or more variables are log-transformed depends on whether the dependent variable, independent variable, or both are transformed. To understand each of these cases, consider an example in which weight is the dependent variable and height is the only independent variable.

Only the dependent variable is transformed

Linear change in the independent variable is associated with multiplicative change in the dependent variable.

Suppose the fitted model is

Explanation

ఉൌݕȀݕ, and is the percent change in ݕ associated with a one-unit increase in ݔ.

Only the independent variable is transformed

Multiplicative change in the independent variable is associated with linear change in the dependent variable.

Fitted model:

Cornell Statistical Consulting Unit

Explanation

Then Both the independent and dependent variable are transformed Multiplicative change in the independent variable is associated with multiplicative change in the dependent variable.

Fitted model:

Explanation

so that the percent change in ݕ associated with a one-percent increase in ݔ is As always if you would like assistance with this topic or any other statistical consulting question, feel free to contact statistical consultants at CSCU.

Author: Jing Yang