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Judging Political Promotion of Judges: Survival Analysis, Split Population Model and Matching Method
| |
| | Unformatted Document Text:
9
assumption may not be valid, though we can not be sure. In this sense, we will have model dependent results from application of any parametric model to raw data (As for strict causal inference in political science, see Diamond and Sekhon (2004), Ho et al. (2004) and King and Zeng (2006)).
If we take causal inference seriously, the effect of YJL on latent promotion time T*
should be a difference between Ti* of judge i who would belong to YJL and T
i
* of the same
judge i who would not belong to YJL. Let X be a dummy variable of YJL (which is called treatment in the literature of causal inference), M be a vector of all measured covariates. That is, realized causal effect for judge i is
∆
T
i
*(M
i
) = T
i
*(X
i
=1, M
i
) - T
i
*(X
i
=0, M
i
).
If we assume that no omitted variables are associated with X
i
and affect T
i
*, this
difference is solely caused by a difference of X
i
’s value. Nonetheless, “the fundamental
problem of causal inference” (Holland 1986) arises since we cannot observe both T
i
*(
Ｘ
i
=1) and
T
i
*(
Ｘ
i
=0) but at most either of the two. If X
i
=1 and an event of interest occurs (E
i
=1), we
observe T
i
*(X
i
=1, M
i
)= T
i
but not T
i
*(X
i
=0, M
i
). If X
i
=0 and E
i
=1, we observe T
i
*(X
i
=0, M
i
) =
T
i
but not T
i
*(X
i
=1, M
i
). Otherwise, we observe neither of the two.
However, we are interested not in the realized effect on value Ti* of judge i but in a
potential effect on a random variable T* of judge population. One of the quantities of interest for most scholars is called “average treatment effect” in the causal inference literature, and is expressed as
∑
=
n
i
n
1
1
E(T
i
*| X
i
=1, M
i
) - E(T
i
*| X
i
=0, M
i
).
Suppose that we assume E(T*|X, M) has a functional form g(X, M) such as a linear
function or logistic link. After we estimate parameters in g(.) by using a conventional survival model, we can estimate an average treatment effect by calculating
∑
=
n
i
n
1
1
g(X
i
=1, M
i
) - g(X
i
=0, M
i
).
A problem is that the model misspecification arises if E(T*|X, M) is not equal to g(X, M)
and thus the estimate differs from the estimand.
Matching. One way to avoid this problem is matching. Given the assumption of no omitted variable, T
i
* of a hypothetical non-YJL judge i (but, in fact, X
i
=1 is observed) should be equal to
that of an observed non-YJL judge j (X
j
=0) whose covariates M have the same values as those of
judge i,
T
i
*(X
i
=0, M
i
) = T
j
*(X
j
=0, M
j
=M
i
).
We can estimate unobserved T
i
*(X
i
=0, M
i
) by T
j
*(X
j
=0, M
j
=M
i
) = T
j
if the latter is
observed, namely, E
j
=1. As often the case with survival analysis, we assume censoring is
|
| | Authors: Fukumoto, Kentaro. |
|
| |
|
|
9
assumption may not be valid, though we can not be sure. In this sense, we will have model
dependent results from application of any parametric model to raw data (As for strict causal
inference in political science, see Diamond and Sekhon (2004), Ho et al. (2004) and King and
Zeng (2006)).
If we take causal inference seriously, the effect of YJL on latent promotion time T*
should be a difference between Ti* of judge i who would belong to YJL and T
i
* of the same
judge i who would not belong to YJL. Let X be a dummy variable of YJL (which is called
treatment in the literature of causal inference), M be a vector of all measured covariates. That
is, realized causal effect for judge i is
∆
T
i
*(M
i
) = T
i
*(X
i
=1, M
i
) - T
i
*(X
i
=0, M
i
).
If we assume that no omitted variables are associated with X
i
and affect T
i
*, this
difference is solely caused by a difference of X
i
’s value. Nonetheless, “the fundamental
problem of causal inference” (Holland 1986) arises since we cannot observe both T
i
*(
Ｘ
i
=1) and
T
i
*(
Ｘ
i
=0) but at most either of the two. If X
i
=1 and an event of interest occurs (E
i
=1), we
observe T
i
*(X
i
=1, M
i
)= T
i
but not T
i
*(X
i
=0, M
i
). If X
i
=0 and E
i
=1, we observe T
i
*(X
i
=0, M
i
) =
T
i
but not T
i
*(X
i
=1, M
i
). Otherwise, we observe neither of the two.
However, we are interested not in the realized effect on value Ti* of judge i but in a
potential effect on a random variable T* of judge population. One of the quantities of interest
for most scholars is called “average treatment effect” in the causal inference literature, and is
expressed as
∑
=
n
i
n
1
1
E(T
i
*| X
i
=1, M
i
) - E(T
i
*| X
i
=0, M
i
).
Suppose that we assume E(T*|X, M) has a functional form g(X, M) such as a linear
function or logistic link. After we estimate parameters in g(.) by using a conventional survival
model, we can estimate an average treatment effect by calculating
∑
=
n
i
n
1
1
g(X
i
=1, M
i
) - g(X
i
=0, M
i
).
A problem is that the model misspecification arises if E(T*|X, M) is not equal to g(X, M)
and thus the estimate differs from the estimand.
Matching. One way to avoid this problem is matching. Given the assumption of no omitted
variable, T
i
* of a hypothetical non-YJL judge i (but, in fact, X
i
=1 is observed) should be equal to
that of an observed non-YJL judge j (X
j
=0) whose covariates M have the same values as those of
judge i,
T
i
*(X
i
=0, M
i
) = T
j
*(X
j
=0, M
j
=M
i
).
We can estimate unobserved T
i
*(X
i
=0, M
i
) by T
j
*(X
j
=0, M
j
=M
i
) = T
j
if the latter is
observed, namely, E
j
=1. As often the case with survival analysis, we assume censoring is
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