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Quantile Regression: A Flexible Alternative to Linear Regression

What is Quantile regression?

Imagine you have a set of numbers and want to find the middle value or the median. In regular regression, you try to find the average or mean value. But what if you want to find other values, like the lower or upper values or values that split the numbers into quarters?

Quantile regression helps you with that. It is a special way of analyzing data that allows you to find different parts of the numbers, not just the average. It tells you how different factors or variables affect different parts of the data.

Let’s say you have a graph with points scattered around. Each point represents a person, and you have their height and weight. In regular regression, you would find how average height and weight are related. But with quantile regression, you can find out how height and weight are related for shorter, taller, or people in between.

Quantile regression is useful when the data is not all the same, and there might be some unusual or extreme values. It helps you understand how things change at different points of the data, not just in the middle.

It’s like having a more complete picture of the data, looking at different parts instead of just the average. It helps us understand how different factors affect different parts of the data and gives us a better understanding of the whole story.

Quantile regression visualisation with GAUSS 23.

Quantile regression history

Quantile regression is a statistical modeling technique that has gained increasing attention in recent years. It allows the estimation of conditional quantiles of a response variable given a set of predictor variables.

Unlike ordinary least squares regression, which estimates the mean of the response variable, quantile regression provides a more flexible approach by estimating the conditional distribution of the response variable. This makes it particularly useful when the distribution of the response variable is non-normal or when there are outliers in the data.

Quantile regression has its roots in the work of mathematicians and statisticians dating as far back as 1760 (Ruđer Josip Bošković developed the first evidence of the least absolute criterion) and Laplace (1789, “Méthode de situation”). However, it was not until the 1970s and 80s, and the contributions of Koenker and Basset, that the method gained popularity in the statistical community. The development of computing power and the availability of large datasets facilitated the use of quantile regression in a wide range of applications, from economics and finance to environmental science and engineering.

Today, quantile regression is a widely used tool in predictive analytics, providing a valuable alternative to traditional regression methods.

Estimated quantiles with Statgraphics 19.

When to use Quantile regression?

Quantile regression provides a more comprehensive analysis by considering the entire distribution of the response variable. It is particularly useful when dealing with data that exhibits heteroscedasticity (varying levels of spread) or when the response variable is skewed or has outliers.

Quantile regression can be more robust and flexible than traditional regression methods, especially when focused on extreme values or tail events. Quantile regression has applications in various fields, including economics, finance, environmental sciences, health research, and social sciences. It allows researchers to analyze the relationship between variables across different quantiles, providing a more nuanced understanding of the data and capturing the conditional distributional effects.

Quantile regression can be used in a wide range of applications in predictive analytics, such as finance, healthcare, and marketing.

• In finance, it can be used to estimate the Value at Risk (VaR) of a portfolio of assets given a set of predictors.
• In healthcare, it can be used to estimate the probability of a patient developing a certain condition, given their medical history and other relevant factors.
• In marketing, it can be used to estimate the probability of a customer purchasing a product, given their demographic profile and past purchasing behavior.

Quantile Regression with Statgraphics 19

Quantile Regression Webinar by Statgraphics Technologies, Inc.

This webinar, led by Neil Palhemus, an expert in the field, delves into the fundamental concepts and practical applications of quantile regression. Discover the application of quantile regression using Statgraphics 19.

The webinar explores the basic model structure, provides examples, and discusses analysis options and methods. Learn about the model construction process and the relationship between predictor variables and quantiles. The presentation highlights the differences between quantile regression and ordinary least squares regression, emphasizing that quantile regression focuses on predicting selected quantiles rather than the mean.

The webinar demonstrates the usage of Statgraphics 19 for implementing quantile regression and explains how to install and configure R, a statistical computing language required for quantile regression in Statgraphics.

Whether you are new to quantile regression or seeking to enhance your understanding of its applications, this webinar provides valuable insights and practical examples.

Takeaways

• Quantile regression estimates conditional quantiles of response variables given predictor variables, providing a more flexible approach than ordinary least squares regression.
• It can handle heteroscedasticity and outliers in data and can include interaction terms, nonlinear functions, and time-varying coefficients.
• Quantile regression is widely used in predictive analytics and has applications in healthcare, finance, environmental science, and engineering.
• It provides a more flexible and powerful tool for analyzing the relationship between variables and is useful for non-normal distribution and outliers in data.

FAQ

How does quantile regression compare to other linear and logistic regression models?

Quantile regression differs from linear and logistic regression by estimating the mean of the response variable and its distribution at different quantiles. It allows for a more detailed analysis of the relationship between predictors and the response variable.

What are the assumptions and limitations of quantile regression?

Assumptions of quantile regression include linearity of the conditional distribution of the response variable and homoscedasticity of the error term. Limitations include difficulty interpreting results, sensitivity to outliers, and misspecification of the conditional distribution.

How do you select the appropriate quantile level for a given dataset?

The appropriate quantile level for a given dataset can be selected by identifying the specific research question and the distribution of the data. A range of quantile levels may be examined to determine the optimal fit for the regression model.

Can quantile regression be used for non-linear relationships between variables?

Quantile regression can be used for non-linear relationships between variables by incorporating polynomial terms or splines in the model. It provides estimates of conditional quantiles and allows for the examination of the heterogeneity of effects across the distribution of the response variable.

How do you interpret the results of a quantile regression analysis?

The interpretation of quantile regression analysis results involves examining the coefficients of the independent variables at various quantiles of the dependent variable. This approach provides a more nuanced understanding of the relationship between the variables and can identify differences in the effects across the distribution.