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. Author manuscript; available in PMC: 2012 Nov 7.
Published in final edited form as: J Am Stat Assoc. 2012 Jan 1;105(491):1135–1146. doi: 10.1198/jasa.2010.tm08463

Table 2.

Simulation results of the relative biases, SEs and MISEs of the IPW and AIPW estimates of θ(·) using π̂ inconsistent for π0 and/or δ = Ê(Y|Z, U) inconsistent for E(Y|Z, U), based on 500 replications (in parenthesis are the Monte Carlo SEs)

n = 500
n = 300
Relative bias1 EMP SE2 EST SE3 EMP MISE4 Relative bias EMP SE EST SE EMP MISE
Normal case
AIPW (π wrong) 0.017 (0.004) 0.481 (0.016) 0.475 (0.002) 0.251 (0.017) 0.018 (0.005) 0.635 (0.020) 0.629 (0.008) 0.423 (0.028)
AIPW (E[Y|Z, U] wrong) 0.023 (0.004) 0.640 (0.018) 0.636 (0.007) 0.442 (0.021) 0.021 (0.010) 0.832 (0.034) 0.825 (0.019) 0.728 (0.042)
AIPW (both wrong) 0.068 (0.003) 0.641 (0.012) 0.638 (0.004) 0.835 (0.019) 0.066 (0.004) 0.841 (0.022) 0.837 (0.011) 1.125 (0.065)
IPW (π wrong) 0.105 (0.004) 0.471 (0.011) 0.462 (0.003) 0.522 (0.027) 0.108 (0.006) 0.632 (0.021) 0.629 (0.003) 0.723 (0.044)
Logistic case
AIPW (π wrong) 0.052 (0.021) 0.254 (0.007) 0.251 (0.001) 0.066 (0.003) 0.102 (0.026) 0.295 (0.012) 0.289 (0.001) 0.092 (0.008)
AIPW (E[Y|Z, U] wrong) 0.056 (0.022) 0.236 (0.006) 0.233 (0.001) 0.057 (0.003) 0.095 (0.027) 0.276 (0.007) 0.271 (0.001) 0.080 (0.005)
AIPW (both wrong) 0.975 (0.058) 0.249 (0.008) 0.250 (0.001) 0.111 (0.005) 0.978 (0.064) 0.286 (0.010) 0.281 (0.001) 0.136 (0.008)
IPW (π wrong) 0.662 (0.047) 0.229 (0.007) 0.223 (0.001) 0.075 (0.004) 0.667 (0.051) 0.267 (0.011) 0.263 (0.001) 0.099 (0.008)
1

Relative bias is defined as ∫∣bias^{θ^(z)}/θ(z)∣dF(z).

2

EMP SE is the empirical SE, defined as ∫SE^EMP{θ^(z)}dF(z), where SE^EMP{θ^(z)} is the sampling SE of the replicated θ̂(z).

3

EST SE is the estimated SE, defined as ∫SE^EST{θ^(z)}dF(z), where SE^EST{θ^(z)} is the sampling average of the replicated sandwich estimates SE^{θ^(z)}.

4

EMP MISE is the empirical MISE, defined as ∫{θ̂(z) − θ(z)}2 dF(z).