A data scientist is developing a linear regression model to predict the annual income of individuals based on several predictor variables, including years of experience. A preliminary analysis of the target variable, Annual_Income, reveals that its distribution is strongly right-skewed. Furthermore, after fitting an initial model, an examination of the residual vs. fitted values plot shows a distinct cone shape, where the variance of the residuals increases as the predicted income increases. Which of the following data transformation techniques is the most direct and appropriate method to address both the right-skewness and the observed heteroscedasticity in this scenario?
Standardize both the target variable and the predictor variables.
Apply an exponential transformation to the Annual_Income variable.
Apply a Box-Cox transformation to the Annual_Income variable.
Apply a logarithmic transformation to the Annual_Income variable.
The correct answer is to apply a logarithmic transformation to the Annual_Income variable. A logarithmic transformation is highly effective at correcting strong right-skewness by compressing the scale of larger values more than smaller values. This process often results in a more symmetric, normal-like distribution. Additionally, this transformation can stabilize the variance, which is a common remedy for the type of heteroscedasticity where the error variance is proportional to the mean of the dependent variable, as indicated by the cone-shaped residual plot. While a Box-Cox transformation could also be used to find an optimal power transformation, the logarithmic transformation is a more direct, standard, and interpretable first choice for financial data like income, which often exhibits exponential growth patterns. Standardization does not alter the shape of a variable's distribution and thus will not correct skewness. An exponential transformation would exacerbate the existing right-skewness, making the problem worse.
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What is a logarithmic transformation and why does it help address skewness?
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How does a Box-Cox transformation differ from a logarithmic transformation?