Quantile Regression Quantile Regression

Quantile Regression

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What is Quantile Regression?

Before we can delve into quantile regression, let us first understand what exactly a quantile is. A quantile is a point in a distribution that relates to its rank order of values within that distribution. The middle value within the sample is known as the median, or middle quantile.

Quantile regression allows us to quantify the association of explanatory variables with a conditional quantile of a dependent variable even when the data does not satisfy the assumption of normality (when the distribution of sample means across independent samples is normal). Quantile regression is generally used when the data does not satisfy the assumptions for linear regression. 

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When is Quantile Regression Used?

Quantile regression is often used for the following reasons: 

  • If there are outliers present in the data.
  • If the residuals don’t follow a normal distribution.
  • If the key assumptions of linear regression are not satisfied. 
  • If an increase in the error variance of the data is occurring with an increase in the outcome variable. 

Quantile Regression Formula

The quantile regression model equation is as follows:

Where, 

p = the number of regressor variables

n = the number of data points

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Linear Regression vs. Quantile Regression

  • Outcome: Linear regression predicts the conditional mean while quantile regression predicts conditional quantities. 
  • Applicability: Linear regression can be applied even when n is small. Quantile regression, on the other hand, requires sufficient data to be applied. 
  • Assumption of Normality: Linear regression makes the assumption of normality while quantile regression is distribution agnostic. 
  • Outliers: Linear regression is sensitive to outliers while quantile regression is robust to outliers.
  • Computations: Linear regression is computationally acute while quantile regression is computationally intensive.

Advantages of Quantile Regression

These are a few advantages of quantile regression:

  • Facilitates the understanding of relationships between outliers: Quantile regression can be used to understand the relationship between variables that are outside the mean of the data. This helps us understand outcomes that are not normally distributed and have non-linear relationships with predictor variables. 
  • Helps Discover Useful Predictive Relationships in Ecology: Quantile regression is used in ecology to uncover the predictive relationships between variables even when there is no relationship or a weak relationship between their means. 
  • Can be Used to Identify Key Determinants: Quantile regressions allows us to discard the assumption that variables operate at the upper tails of the distribution so that we can uncover the factors that are significant determinants of the variables.

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FAQs on Quantile Regression

A quantile refers to a point in a distribution that relates to its rank order of values within that distribution.

Quantile regression is a kind of regression analysis used in statistics to estimate the conditional median of the response variable.

Quantile regression is generally used when the data does not satisfy the assumptions of linear regression.

A few strengths of quantile regression are; 

  • Can be used to understand the relationship between outliers. 
  • Can be used to uncover predictive relationships in ecology even when there is no relationship or a weak relationship between the variables.

The key differences between quantile regression and linear regression are; 

  • Linear regression makes the assumption of normality while quantile regression is distribution agnostic and can be used even when the data does not satisfy the assumptions of linear regression. 
  • Linear regression is sensitive to outliers while quantile regression is robust to outliers.

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