Understanding the Sampling Bias: A Case Study on NBA Drafts

Abstract

In several real data applications a biased sample arises naturally from the selection procedure. Recently, Economou et al. (Biom J 62: 238–249, 2020) used the concept of bivariate weighted distributions and proposed four different families of weight functions to describe cases in which the bias in a bivariate sample is caused by adopting sampling schemes that result in over- or under-representation of individuals with specific properties in the sample. The current paper focuses on revealing the contribution of each variable to the bias in the bivariate sample. More specifically, under the Bayesian perspective, Approximate Bayesian Computation methods are used to sample approximately from the posterior distribution, and the Deviance Information Criterion is employed to compare the fit of the models obtained by using different weight functions. The proposed method is illustrated to a real data set concerning NBA draft players.

Appendix

In this Appendix the posterior density is reported for the general case and in detail for the special case of the application.

The likelihood function of a biased bivariate sample $D = (x_j, y_j), j=1,\ldots ,n$ from a parent population with known pdf $f(x,y;\theta )$ where $\theta$ unknown parameters’ vector, when the bias in the sample is described by the weight function $w_{i}(x,y;\theta ,\gamma _X,\gamma _Y)$ is

$$\begin{aligned} \prod _{j=1}^{n}f_{w_{i}}(x_j,y_j;\theta ,\gamma _X,\gamma _Y)=\prod _{j=1}^{n}f_{w_{i}}(x_j,y_j;\zeta )= \frac{\prod _{j=1}^{n} w_{i}(x_j, y_j;\zeta )f(x_j, y_j;\theta )}{E_{f}^n[w_i(X,Y;\zeta )]}. \end{aligned}$$

Let $\pi (\zeta )$ be the joint prior density of the parameters of the model, where $\zeta = (\theta , \gamma _X, \gamma _Y)$. Then, the posterior density of the model has the form:

$$\begin{aligned} \pi (\zeta |{\mathrm{data}})\propto & {} \frac{\prod _{j=1}^{n} w_{i}(x_j, y_j;\zeta )f(x_j, y_j;\theta )}{E_{f}^n[w_i(X,Y;\zeta )]} \cdot \pi (\zeta ). \end{aligned}$$

Based on the discussion of Sect. 4.2, the joint distribution of height and the vertical jump in the population of interest is a bivariate normal. Moreover, independence of the parameters of the model is assumed and a prior distribution is adopted for each parameter $\mu _X$, $\mu _Y$, $\sigma ^2_{X}$, $\sigma ^2_{Y}$, $\rho$, $\gamma _X$ and $\gamma _Y$. Then, the posterior density takes the form:

$$\begin{aligned} \pi (\zeta |data)\propto & {} \frac{\prod _{j=1}^{n} w_{i}(x_j, y_j;\zeta )f(x_j, y_j;\theta )}{E_{f}^n[w_i(X,Y;\zeta )]}\\&\pi (\mu _X) \pi (\mu _Y) \pi (\sigma ^2_X) \pi (\sigma ^2_Y) \pi (\rho ) \pi (\gamma _X)\pi (\gamma _Y). \end{aligned}$$

Using the priors described in Sect. 4.2 the following relation is obtained:

$$\begin{aligned} \pi (\zeta |data)\propto & {} \frac{\prod _{j=1}^{n} w_{i}(x_j, y_j;\zeta )}{E_{f}^n[w_i(X,Y;\zeta )]} \cdot \\&\exp \left[ -\frac{1}{2(1-\rho ^2)} \sum _{j=1}^n \left[ \frac{(x_j-\mu _X)^2}{\sigma ^2_X}\right. \right. +\\&\left. \left. \frac{(y_j-\mu _Y)^2}{\sigma ^2_Y}-2\rho \frac{(x_j-\mu _X)(y_j-\mu _Y)}{\sigma _X\sigma _Y}\right] \right] \\&\exp \left[ -\frac{1}{2}\left( \frac{(\mu _X-76.5)^2}{4.167^2}+ \frac{(\mu _Y-30)^2}{4^2}\right) \right] \\&\cdot (1+\rho )^{25-1} (1-\rho )^{30-1} (1-\rho ^2)^{-n/2}\\&\left( \frac{1}{\sigma ^2_X}\right) ^{2+1+n/2}\exp \left[ -\frac{4.167^2}{\sigma ^2_X}\right] \left( \frac{1}{\sigma ^2_Y}\right) ^{2+1+n/2}\exp \left[ -\frac{4^2}{\sigma ^2_Y}\right] \cdot \\&\exp \left[ -\frac{1}{2}\left( \frac{(\gamma _X-1)^2}{10}+\frac{(\gamma _Y-1)^2}{10}\right) \right] I(\gamma _X>0) \cdot I(\gamma _Y>0) \end{aligned}$$

which can be expressed equivalently as

$$\begin{aligned} \pi (\zeta |data)\propto & {} \frac{\prod _{j=1}^{n} w_{i}(x_j, y_j;\zeta )}{E_{f}^n[w_i(X,Y;\zeta )]} \cdot \\&\exp \left[ -\frac{1}{2(1-\rho ^2)} \sum _{j=1}^n \left[ \frac{(x_j-\mu _X)^2}{\sigma ^2_X}\right. \right. \\&\left. \left. + \frac{(y_j-\mu _Y)^2}{\sigma ^2_Y}-2\rho \frac{(x_j-\mu _X)(y_j-\mu _Y)}{\sigma _X\sigma _Y}\right] \right] \\&\exp \left[ -\frac{1}{2}\left( \frac{(\mu _X-76.5)^2}{4.167^2}+ \frac{(\mu _Y-30)^2}{4^2}\right) \right] \cdot \\&(1+\rho )^{24-n/2} (1-\rho )^{29-n/2}\\&\left( \frac{1}{\sigma ^2_X\sigma ^2_Y}\right) ^{3+n/2} \exp \left[ -\frac{4.167^2}{\sigma ^2_X}-\frac{4^2}{\sigma ^2_Y}\right] \cdot \\&\exp \left[ -\frac{1}{2}\left( \frac{(\gamma _X-1)^2}{10}+\frac{(\gamma _Y-1)^2}{10}\right) \right] I(\gamma _X>0) \cdot I(\gamma _Y>0). \end{aligned}$$

For the model $\mathcal {M}_{1f}$, i.e., $i=1$ and $\gamma _X, \ \gamma _Y$ strictly positive, the posterior density has the form

$$\begin{aligned} \pi (\zeta |data)\propto & {} \frac{\prod _{j=1}^{n} \left( 1 - \left( 1-\Phi \left( \frac{x_j-\mu _X}{\sigma _X}\right) ^{\gamma _X} \right) \left( 1-\Phi \left( \frac{y_j-\mu _Y}{\sigma _Y}\right) ^{\gamma _Y} \right) \right) }{E_{f}^n\left[ \left( 1 - \left( 1-\Phi \left( \frac{X-\mu _X}{\sigma _X}\right) ^{\gamma _X} \right) \left( 1-\Phi \left( \frac{Y-\mu _Y}{\sigma _Y}\right) ^{\gamma _Y} \right) \right) \right] } \cdot \\&\exp \left[ -\frac{1}{2(1-\rho ^2)} \sum _{j=1}^n \left[ \frac{(x_j-\mu _X)^2}{\sigma ^2_X}\right. \right. \\&\left. \left. +\frac{(y_j-\mu _Y)^2}{\sigma ^2_Y}-2\rho \frac{(x_j-\mu _X)(y_j-\mu _Y)}{\sigma _X\sigma _Y}\right] \right] \\&\exp \left[ -\frac{1}{2}\left( \frac{(\mu _X-76.5)^2}{4.167^2}+ \frac{(\mu _Y-30)^2}{4^2}\right) \right] \cdot \\&(1+\rho )^{24-n/2} (1-\rho )^{29-n/2}\\&\left( \frac{1}{\sigma ^2_X\sigma ^2_Y}\right) ^{3+n/2} \exp \left[ -\frac{4.167^2}{\sigma ^2_X}-\frac{4^2}{\sigma ^2_Y}\right] \cdot \\&\exp \left[ -\frac{1}{2}\left( \frac{(\gamma _X-1)^2}{10}+\frac{(\gamma _Y-1)^2}{10}\right) \right] I(\gamma _X>0) \cdot I(\gamma _Y>0). \end{aligned}$$

Due to the posterior’s form direct sampling from it or even sampling from a standard MCMC method is not an easy task. Thus, ABC methods are used.