Quantile Regression

5Quantile Regressions with Two Numerical Predictors

5.1 Introduction

As the extension of the quantile regressions (QRs) with a numerical predictor presented in previous chapter, this chapter presents examples of QRs with two numerical predictors. Figure 5.1 presents alternative up‐and‐down or causal relationships between three numerical variables Y1, X1, and X2.

Figure 5.1a shows that both X1 and X2 have direct effects on Y1, and they do not have a causal relationship. This represents an additive model of Y1 on X1 and X2, which is the simplest model in a three‐dimensional space. Figure 5.1b shows that X1 has a direct effect on Y1, and X2 has an indirect effect on Y1 through X1. This represents a two‐way interaction model. Even though there is no arrow from X2 to Y1, X2 is an upper or cause factor or variable of Y2. In other words, X2 has a partial direct effect on Y1. And Figure 5.1c also represents a two‐way interaction model, with the addition that X2 is defined to have a direct effect on Y1.

5.2 Alternative QRs Based on Data_Faad.wf1

As an extension of the alternative models of Y1 on a numerical variable X1, presented in previous chapter, this section will present selected QRs of Y1 on two numerical predictors X1 and X2.

5.2.1 Alternative QRs Based on (X1,X2,Y1)

5.2.1.1 Additive QR

Based on any set of the numerical variables, Y1, X1, and X2, the additive linear regression, which is the simplest linear regression in a three‐dimensional space, can be represented using ...