Question
A scholarship committee records two strengths for each applicant: strong grades \(G\in\{0,1\}\) and strong volunteering \(V\in\{0,1\}\). Across all applicants, G and V are independent, and each equals 1 with probability 1/2.
The committee awards a scholarship to anyone who has at least one strength: \(G+V\ge1\). A researcher then studies only scholarship recipients and fits an ordinary linear regression of \(V\) on \(G\), including an intercept. Among recipients, what are \(\operatorname{Cov}(G,V)\) and the regression slope? Compare them with the full applicant population. Does a negative selected-sample slope imply that having good grades causes less volunteering?
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Hint 1
List the four possible (G,V) pairs in the population. All have probability 1/4, but (0,0) never receives a scholarship.
Check your answer
Solution
Before selection, independent fair binary strengths yield the four states (0,0), (0,1), (1,0), (1,1), each with probability 1/4. The scholarship criterion excludes only (0,0), so \(P(\text{selected})=3/4\), and the other three states each have conditional probability 1/3.
Among recipients,
\[\mathbb{E}[G\mid S]=2/3,\qquad\mathbb{E}[V\mid S]=2/3,\qquad\mathbb{E}[GV\mid S]=1/3\].
The conditional covariance is therefore
\(\operatorname{Cov}(G,V\mid S)=1/3-(2/3)(2/3)=\)\(-1/9\).
Because \(G\) is Bernoulli with success probability \(2/3\) among recipients, \(\operatorname{Var}(G\mid S)=(2/3)(1/3)=2/9\). For linear regression with an intercept, the population least-squares slope is
\(\beta_S=\operatorname{Cov}(G,V\mid S)/\operatorname{Var}(G\mid S)=(-1/9)/(2/9)=\)\(-1/2\).
A direct crosscheck is \(\mathbb{E}[V\mid G=0,S]=1\) (such recipients must have \(V=1\)) and \(\mathbb{E}[V\mid G=1,S]=1/2\), so the difference in group means is \(-1/2\).
In the full applicant pool, independence gives \(\operatorname{Cov}(G,V)=0\) and slope 0 because \(\operatorname{Var}(G)=1/4>0\). Conditioning on scholarship receipt selects on a common consequence of \(G\) and \(V\) (a collider). Once a recipient has weak grades, strong volunteering is guaranteed, whereas strong-grade recipients can have either volunteering status. The resulting negative association is a selection artifact; this model contains no causal effect of grades on volunteering.
Assumptions
G and V are independent fair Bernoulli indicators in the applicant population.
Scholarship selection is deterministic given \(G+V\ge1\); no additional admissions discretion or missing data exist.
Regression is an intercept-including population least-squares projection within the selected group, not a finite-sample noisy estimate.
Final Answer
Among recipients, \(\operatorname{Cov}(G,V)=-1/9\) and the regression slope of \(V\) on \(G\) is \(-1/2\). In the full population both are zero. The negative association is created by selecting on \(G+V\ge1\), not by a causal trade-off.