Understanding Heteroskedasticity and Transformations in Linear Regression Analysis

Linear regression is a widely used statistical method for predicting outcomes based on input variables. However, analyzing the results of a linear regression model can be complicated, particularly when there is heteroskedasticity or a violation of the assumption of homoscedasticity. This can lead to incorrect or unreliable predictions and can be challenging to diagnose and address. In this blog post, we’ll explore the concept of heteroskedasticity in linear regression analysis and how to identify and correct it using transformations. I know this can be a challenging topic, but I’ll do my best to explain it in a way that is clear and easy to understand. By the end of this post, you’ll have a better understanding of how to diagnose and correct heteroskedasticity in linear regression models, and you’ll be equipped with the tools you need to make more accurate predictions. So, let’s get started!

Table of Contents

Linear Regression

Linear regression is a valuable statistical tool for establishing a linear connection between one or more predictors and the target.

Simple linear regression and multiple linear regression are the two types of linear regression.

Simple linear regression entails only two variables: the independent variable, also known as the predictor, and the dependent variable, also known as the response. If there are more than two independent variables, this is known as multiple linear regression, in which several independent variables are linked to a single dependent variable.

A statistical relationship refers to a scenario in which there is no formula or equation to establish a connection between two variables. For instance, there is no formula that can compare a person’s height and weight. If there is a fixed formula, it is a deterministic relationship.

Homoscedasticity & Heteroskedasticity

Homoscedasticity: This assumption of the classical linear regression model entails that the variation of the error term should be consistent for all observations. The Error Term should be Homoscedastic (it should have a constant variance).

The presence of non-constant variance in the error terms results in heteroskedasticity.

Goldfeld-Quandt Test to test for heteroscedasticity

We can use the Goldfeld-Quandt Test to test for heteroscedasticity.

The test splits the data into two groups and tests to see if the variances of the residuals are similar across the groups. The Goldfeld-Quandt Test is a statistical method used to check if there is heteroscedasticity in a regression model. It does this by dividing the data into two groups and comparing the variances of the residuals between the groups. If the variances are significantly different, then it suggests that the model has heteroscedasticity. The test is useful when the plot of residuals versus fitted values shows a funnel shape pattern, which is a common indication of heteroscedasticity.

Plotting the residuals versus fitted value graphs to test for heteroscedasticity

We can test for heteroscedasticity by plotting the residuals versus fitted value graphs.

The scatter plot is a good way to check whether the data are homoscedastic (meaning the residuals are equal across the regression line).

The following scatter plots show examples of data that are not homoscedastic (i.e. heteroscedastic):

Linear Regression, heteroskedasticity & myths of transformations

If the plot exhibits a pattern, we need to worry.

The funnel shape pattern means heteroskedasticity. If the plot of residuals against fitted values shows a funnel-shaped pattern, it suggests that there is heteroscedasticity in the regression model. This means that the variability of residuals is not constant across all levels of the independent variable, violating the assumption of homoscedasticity. To correct this, techniques like log or square root transformations can be used or weighted least squares regression can be applied instead of ordinary least squares regression.

Remedy: A non-linear correction might fix the problem. Transformation of the response variable such as log(Y) or √Y

The parabolic shape pattern means the model didn’t capture non-linear effects. A parabolic or curved pattern in the plot of residuals against fitted values means that the regression model failed to capture the non-linear effects in the data. This can lead to inaccurate predictions of the dependent variable, and more complex models with non-linear terms may be required. It is important to identify the type of non-linearity present and choose the appropriate modeling technique to address it.

Remedy: This case requires a non-linear transformation of predictors such as log (X), √X

It is important to consider that non-constant variance can also occur due to the presence of outliers in the data.

A general idea of transformations

Linear regression is effective even if the input variables have highly non-normal distributions. The crucial factor is the correlation between the inputs and outputs, not the distribution of the inputs alone. You do not need to change the input variables simply because their distribution is skewed. Rather, you modify them so that the linear pattern that the model is attempting to draw through your data is sensible.

Let’s talk about log transformations.

logarithmic transformations may adjust data distribution to a less skewed — hopefully, Gaussian-like distribution.

You can see that the center case (y) has been transformed into symmetry, while the more mildly right skew case (x) is now somewhat left skew. On the other hand, the most skew variable (z) is still (slightly) right skew, even after taking logs. If we wanted our distributions to look more normal, the transformation definitely improved the second and third cases. We can see that this might help.

Log transformations can be used in regression models to deal with heteroscedasticity. It works by taking the logarithm of the dependent variable which compresses larger values, reducing the impact of outliers and making the residuals approximately homoscedastic. However, it should only be used when it makes sense for the data and research question.

Other transformations like the square root, will also pull large values like that. Square root transformations can be used in regression models to deal with heteroscedasticity. It works by taking the square root of the dependent variable, which reduces the influence of extreme values and makes the residuals approximately homoscedastic. However, as with log transformations, square root transformations should be used with caution and only when they make sense in the context of the data and the research question.

The purpose of a transformations

The goal of a transformation is not to hide or disguise outliers. Outliers are data points that do not conform to the general nature or description of the data. Prioritizing the adjustment of the data to accommodate outliers is often an incorrect approach. The best approach is to obtain a scientifically valid and statistically accurate description of the data and then examine any outliers separately. Outliers should not dictate the description of the rest of the data.

The aim of a transformation is to produce residuals that are symmetrically distributed around zero. The spread of the residuals changes in a systematic manner with variations in the dependent variable (known as “heteroscedasticity”). Therefore, the transformation’s objective is to eliminate this systematic spread variation, leading to approximate “homoscedasticity.” The transformation also aims to linearize the relationship between variables.

Frequently Asked Questions

Why heteroscedasticity is a problem?

Heteroscedasticity occurs when the variability of the residuals in a regression model is not constant across all values of the independent variable. This violates the assumption of constant variance and can lead to biased and inefficient estimates of the regression coefficients and standard errors, resulting in inaccurate predictions and invalid inferences. Detecting and correcting heteroscedasticity is crucial to ensure the reliability and validity of the regression analysis.

What causes heteroscedasticity?

Heteroscedasticity in a regression model can result from outliers, omitted variables, incorrect functional form, measurement errors, different scales of measurement, unobserved variables, or sample selection bias. Identifying the underlying causes of heteroscedasticity is essential for selecting the appropriate corrective measures.

What is heteroscedasticity with example?

Heteroscedasticity can occur in a regression model when the variability of the residuals is not constant across all values of the independent variable. For example, in a housing price prediction model based on square footage, if the variability of the errors is higher for larger houses than smaller ones, the model’s predictions are less precise for larger houses. Similarly, in a salary prediction model based on years of experience, if the variability of the errors is higher for more experienced employees, the model’s predictions are less precise for highly experienced employees.

Conclusion

The article “Understanding Heteroskedasticity and Transformations in Linear Regression Analysis” discusses the issue of heteroskedasticity in linear regression models and how to address it through transformations of the data. The article begins by defining heteroskedasticity and explaining how it can lead to biased estimates of the regression coefficients and incorrect inferences about the statistical significance of the predictors. Next, the article explains how to address heteroskedasticity through transformations of the data. The article emphasizes the importance of addressing heteroskedasticity in linear regression models to ensure accurate and reliable results. While transformations can be effective in addressing heteroskedasticity, they should be used judiciously and with careful consideration of their impact on the interpretation of the results.

Hope this article “Understanding Heteroskedasticity and Transformations in Linear Regression Analysis” helped you in gaining a new perspective.

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