Question
A company randomizes access to a new training programme, recording \(T=1\) for invited workers and \(T=0\) otherwise. It finds that invited workers practise more. Suppose the data are generated by these equations:
Practice hours: \(M=2T+U\).
Performance improvement: \(Y=T+4M+\varepsilon\).
Here \(T\) is independent of the mean-zero disturbances \(U\) and \(\varepsilon\), and \(U\) and \(\varepsilon\) are independent of each other. Both \(T\) and \(U\) have positive variance. An evaluator fits one population linear regression of \(Y\) on \(T\) alone, then another of Y on both \(T\) and \(M\). The evaluator sees a much smaller programme coefficient after including practice hours and claims the programme barely helped.
What are the two coefficients on \(T\)? Separate the programme's direct, mediated and total causal effects, and explain which regression answers the company's question about the programme's overall benefit.
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Hint 1
First substitute the practice-hours equation into the performance equation.
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Solution
Substitute \(M=2T+U\) into \(Y=T+4M+\varepsilon\):
\[Y=T+4(2T+U)+\varepsilon=9T+4U+\varepsilon\].
Because \(T\) is randomized independently of both noise terms, \(\operatorname{Cov}(T,4U+\varepsilon)=0\). In the population regression of \(Y\) on an intercept and \(T\) alone, the coefficient on \(T\) is \(\frac{\operatorname{Cov}(Y,T)}{\operatorname{Var}(T)}=9\). It measures the mean change in outcome when programme access moves from zero to one, allowing practice hours to respond naturally.
For the second regression, the original structural equation is \(Y=T+4M+\varepsilon\). By the stipulated independence, \(\varepsilon\) is uncorrelated with \(T\) and \(M\) (\(M\) is a function of \(T\) and \(U\)). The population coefficients of \(Y\) regressed on \(T\) and \(M\) are therefore 1 and 4. Positive variance of \(U\) ensures \(M\) is not a perfect multiple of \(T\), so the individual coefficients are identified.
Holding \(M\) fixed while changing \(T\) gives the direct effect 1. The programme also increases \(M\) by 2, which raises \(Y\) by \(4\times2=8\) through the mediator. \(\text{Total causal effect}=\text{direct }1+\text{mediated }8=9\). For evaluating overall benefit of assigning workers to the programme, the unadjusted randomized comparison (9) is relevant. The adjusted coefficient (1) answers the different question, 'What effect remains if practice hours are held fixed?' It is not an unbiased estimate of the programme's total effect in this model. This result is a property of the explicitly assumed causal structure, not a general instruction to omit all controls.
Assumptions
\(T\) is an independently randomized binary treatment; \(U\) and \(\varepsilon\) are centered, mutually independent and independent of \(T\).
The structural equations represent causal changes under intervention, with no unmodelled backdoor path in the stipulated model.
Positive variances of T and U ensure both population regression coefficients are identifiable.
Final Answer
Unadjusted \(T\) coefficient = 9; adjusted \(T\) coefficient = 1. \(\text{Direct effect}=1\), \(\text{mediated effect}=8\), \(\text{total effect}=9\) performance units.