Conclusion and Discussion

Chapter 5 Conclusion and Discussion

In the previous two chapters, we conducted both simulation and the empirical studies to evaluate the performances of different IV estimators when applied in analyzing data sets with binary outcome, treatment, and IV. In addition to the traditional one-stage GLM, 2SPS, and 2SRI, we also consider 2SRI-T, a version of 2SRI that intends to replace unmeasured confounders through the use of the first-order Taylor expansion term of the error term _D. In the simulation design, strengths of IV, levels of confounding, probabilities of receiving the treatment, and sample sizes were considered in altogether 16 combined scenarios. Bias, standard error, MSE, and coverage probability are the main tools to evaluate the performances of the four estimators. Subsequently, we investigated the effect of having children on one’s life satisfaction in the empirical study, using the WVS data from Survey

Research Data Archive of Center for Survey Research, Academia Sinica.

Contradictory to Terza et al. (2008), we found that 2SRI did not outperform 2SPS according to the simulation results. In fact, these two had almost the same performances. As far as bias is concerned, 2SPS and 2SRI outperformed the other two estimators, and the one-stage GLM had the worst performance. Somewhat beyond our expectation was that 2SRI-T did not perform as well as 2SRI. On the other hand, 2SPS and 2SRI suffered from larger variability, while 2SRI-T generally had smaller standard error. Therefore, 2SRI-T might be preferred if MSE was the main concern.

As for the empirical study, the results revealed that having children or not did not significantly impact one’s life satisfaction. The conclusion was agreed upon no matter which method was applied. Moreover, the results of the four approaches were consistent with what we observed in the simulation study. 2SPS and 2SRI again had similar performances with similar estimated treatment effect and standard error, and standard errors were larger than the other two estimators.

Before concluding this chapter, we need to emphasize that the usefulness of the results we provide in this study rests on the availability of an appropriate IV. However, this is also a problem associated with any IV analysis. Without an appropriate IV, any of the methods cannot be implemented. Furthermore, due to the binary nature of the variables, we focus only on odds ratio as the effect of treatment. However, in many

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studies, risk difference and risk ratio may also be the parameters of interest. It may be worthwhile to investigate the performance of these IV estimators on the estimation of risk difference and risk ratio as well.

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94(434): 444-455.

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7. Hernán MA, Robins JM. Instruments for causal inference: an epidemiologist’s dream? Epidemiology. 2006; 17(4): 360-372.

8. Johnston KM, Gustafson P, Levy AR, Grootendorst P. Use of

instrumental variables in the analysis of generalized linear models in the presence of unmeasured confounding with applications to epidemiological research.

Statistics in Medicine. 2008;27: 1539-1556.

9. Kennedy P. A guide to Econometrics. 5^th ed. Cambridge, MA: MIT Press; 2003.

10. McClellan M, McNeil BJ, Newhouse JP. Does more intensive treatment of acute myocardial infarction in the elderly reduce mortality? Analysis using instrumental variables. Journal of the American Medical Association. 1994; 272(11): 859-866.

11. Rassen JA, Schneeweiss S, Glynn RJ, Mittleman MA, Brookhart MA.

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169(3):273-284.

12. Schneeweiss S, Solomon DH, Wang PS, Rassen JA, Brookhart MA. Simultaneous assessment of short-term gastrointestinal benefits and cardiovascular risks of selective cyclooxygenase 2 inhibitors and nonselective nonsteroidal

antiinflammatory drugs: an instrumental variable analysis. Arthritis

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13. Small DS. Sensitivity analysis for instrumental variables regression with

overidentifying restrictions. Journal of the American Statistical Association. 2007;

102(479): 1049-1058.

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addressing endogeneity in health econometric modeling. Journal of Health

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2335-2341.

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Appendix A.

Programming Code of Simulation (Under a=0.5,



₀=0.91, and n=10,000) y3=as.vector(numeric(n)); y4=as.vector(numeric(n))

int=log(411/166) for (j in 1:n){

lambda1[j]=int+log(3)*D1[j]+log(.5)*C[j]+log(.75)*X[j]

lambda2[j]=int+log(3)*D2[j]+log(.5)*C[j]+log(.75)*X[j]

lambda3[j]=int+log(3)*D3[j]+log(.5)*C[j]+log(.75)*X[j]

lambda4[j]=int+log(3)*D4[j]+log(.5)*C[j]+log(.75)*X[j]

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#Form Coefficients and Their Standard Errors as Matrices#

tra.or.coef.1=matrix(temp.tra.or.1,2,1000);sps.or.coef.1=matrix(temp.sps.or.1,2,1000);sri.or.coef.1=mat

bias.tra.or.1 = mean(tra.or.coef.1[1,])-log(3); bias.sps.or.1 = mean(sps.or.coef.1[1,])-log(3) bias.sri.or.1 = mean(sri.or.coef.1[1,])-log(3); bias.tay.or.1 = mean(tay.or.coef.1[1,])-log(3) bias.tra.or.2 = mean(tra.or.coef.2[1,])-log(3); bias.sps.or.2 = mean(sps.or.coef.2[1,])-log(3) bias.sri.or.2 = mean(sri.or.coef.2[1,])-log(3); bias.tay.or.2 = mean(tay.or.coef.2[1,])-log(3) bias.tra.or.3 = mean(tra.or.coef.3[1,])-log(3); bias.sps.or.3 = mean(sps.or.coef.3[1,])-log(3) bias.sri.or.3 = mean(sri.or.coef.3[1,])-log(3); bias.tay.or.3 = mean(tay.or.coef.3[1,])-log(3) bias.tra.or.4 = mean(tra.or.coef.4[1,])-log(3); bias.sps.or.4 = mean(sps.or.coef.4[1,])-log(3) bias.sri.or.4 = mean(sri.or.coef.4[1,])-log(3); bias.tay.or.4 = mean(tay.or.coef.4[1,])-log(3)

bias.or=matrix(c(bias.tra.or.1,bias.sps.or.1,bias.sri.or.1,bias.tay.or.1,bias.tra.or.2,bias.sps.or.2,bias.sri.or.

2,bias.tay.or.2,bias.tra.or.3,bias.sps.or.3,bias.sri.or.3,bias.tay.or.3,bias.tra.or.4,bias.sps.or.4,bias.sri.or.4,b ias.tay.or.4),nrow=4,ncol=4,byrow=T)

colnames(bias.or)=c("Tra. GLM","2SPS","2SRI-L","2SRI-T"); rownames(bias.or)=c(0.5,1,2,5)

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se.tra.or.1 = sd(tra.or.coef.1[1,]); se.sps.or.1 = sd(sps.or.coef.1[1,]) se.sri.or.1 = sd(sri.or.coef.1[1,]); se.tay.or.1 = sd(tay.or.coef.1[1,]) se.tra.or.2 = sd(tra.or.coef.2[1,]); se.sps.or.2 = sd(sps.or.coef.2[1,]) se.sri.or.2 = sd(sri.or.coef.2[1,]); se.tay.or.2 = sd(tay.or.coef.2[1,]) se.tra.or.3 = sd(tra.or.coef.3[1,]); se.sps.or.3 = sd(sps.or.coef.3[1,]) se.sri.or.3 = sd(sri.or.coef.3[1,]); se.tay.or.3 = sd(tay.or.coef.3[1,]) se.tra.or.4 = sd(tra.or.coef.4[1,]); se.sps.or.4 = sd(sps.or.coef.4[1,]) se.sri.or.4 = sd(sri.or.coef.4[1,]); se.tay.or.4 = sd(tay.or.coef.4[1,])

se.or=matrix(c(se.tra.or.1,se.sps.or.1,se.sri.or.1,se.tay.or.1,se.tra.or.2,se.sps.or.2,se.sri.or.2,se.tay.or.2,se.t ra.or.3,se.sps.or.3,se.sri.or.3,se.tay.or.3,se.tra.or.4,se.sps.or.4,se.sri.or.4,se.tay.or.4),nrow=4,ncol=4,byro w=T)

colnames(se.or)=colnames(bias.or); rownames(se.or)=rownames(bias.or)

#Calculate MSE##

mse.tra.or.1 = var(tra.or.coef.1[1,])+ mean(tra.or.coef.1[1,]-log(3))^2 mse.sps.or.1 = var(sps.or.coef.1[1,])+ mean(sps.or.coef.1[1,]-log(3))^2 mse.sri.or.1 = var(sri.or.coef.1[1,])+ mean(sri.or.coef.1[1,]-log(3))^2 mse.tay.or.1 = var(tay.or.coef.1[1,])+ mean(tay.or.coef.1[1,]-log(3))^2 mse.tra.or.2 = var(tra.or.coef.2[1,])+ mean(tra.or.coef.2[1,]-log(3))^2 mse.sps.or.2 = var(sps.or.coef.2[1,])+ mean(sps.or.coef.2[1,]-log(3))^2 mse.sri.or.2 = var(sri.or.coef.2[1,])+ mean(sri.or.coef.2[1,]-log(3))^2 mse.tay.or.2 = var(tay.or.coef.2[1,])+ mean(tay.or.coef.2[1,]-log(3))^2 mse.tra.or.3 = var(tra.or.coef.3[1,])+ mean(tra.or.coef.3[1,]-log(3))^2 mse.sps.or.3 = var(sps.or.coef.3[1,])+ mean(sps.or.coef.3[1,]-log(3))^2 mse.sri.or.3 = var(sri.or.coef.3[1,])+ mean(sri.or.coef.3[1,]-log(3))^2 mse.tay.or.3 = var(tay.or.coef.3[1,])+ mean(tay.or.coef.3[1,]-log(3))^2 mse.tra.or.4 = var(tra.or.coef.4[1,])+ mean(tra.or.coef.4[1,]-log(3))^2 mse.sps.or.4 = var(sps.or.coef.4[1,])+ mean(sps.or.coef.4[1,]-log(3))^2 mse.sri.or.4 = var(sri.or.coef.4[1,])+ mean(sri.or.coef.4[1,]-log(3))^2

‧

cp.tra.or.1 = sum( tra.or.coef.1[1,]+1.96*tra.or.coef.1[2,] >= log(3) &

tra.or.coef.1[1,]-1.96*tra.or.coef.1[2,] <= log(3) )/1000

cp.sps.or.1 = sum( sps.or.coef.1[1,]+1.96*sps.or.coef.1[2,] >= log(3) &

sps.or.coef.1[1,]-1.96*sps.or.coef.1[2,] <= log(3) )/1000

cp.sri.or.1 = sum( sri.or.coef.1[1,]+1.96*sri.or.coef.1[2,] >= log(3) &

sri.or.coef.1[1,]-1.96*sri.or.coef.1[2,] <= log(3) )/1000

cp.tay.or.1 = sum( tay.or.coef.1[1,]+1.96*tay.or.coef.1[2,] >= log(3) &

tay.or.coef.1[1,]-1.96*tay.or.coef.1[2,] <= log(3) )/1000

cp.tra.or.2 = sum( tra.or.coef.2[1,]+1.96*tra.or.coef.2[2,] >= log(3) &

tra.or.coef.2[1,]-1.96*tra.or.coef.2[2,] <= log(3) )/1000

cp.sps.or.2 = sum( sps.or.coef.2[1,]+1.96*sps.or.coef.2[2,] >= log(3) &

sps.or.coef.2[1,]-1.96*sps.or.coef.2[2,] <= log(3) )/1000

cp.sri.or.2 = sum( sri.or.coef.2[1,]+1.96*sri.or.coef.2[2,] >= log(3) &

sri.or.coef.2[1,]-1.96*sri.or.coef.2[2,] <= log(3) )/1000

cp.tay.or.2 = sum( tay.or.coef.2[1,]+1.96*tay.or.coef.2[2,] >= log(3) &

tay.or.coef.2[1,]-1.96*tay.or.coef.2[2,] <= log(3) )/1000

cp.tra.or.3 = sum( tra.or.coef.3[1,]+1.96*tra.or.coef.3[2,] >= log(3) &

tra.or.coef.3[1,]-1.96*tra.or.coef.3[2,] <= log(3) )/1000

cp.sps.or.3 = sum( sps.or.coef.3[1,]+1.96*sps.or.coef.3[2,] >= log(3) &

sps.or.coef.3[1,]-1.96*sps.or.coef.3[2,] <= log(3) )/1000

cp.sri.or.3 = sum( sri.or.coef.3[1,]+1.96*sri.or.coef.3[2,] >= log(3) &

sri.or.coef.3[1,]-1.96*sri.or.coef.3[2,] <= log(3) )/1000

cp.tay.or.3 = sum( tay.or.coef.3[1,]+1.96*tay.or.coef.3[2,] >= log(3) &

tay.or.coef.3[1,]-1.96*tay.or.coef.3[2,] <= log(3) )/1000

cp.tra.or.4 = sum( tra.or.coef.4[1,]+1.96*tra.or.coef.4[2,] >= log(3) &

tra.or.coef.4[1,]-1.96*tra.or.coef.4[2,] <= log(3) )/1000

cp.sps.or.4 = sum( sps.or.coef.4[1,]+1.96*sps.or.coef.4[2,] >= log(3) &

sps.or.coef.4[1,]-1.96*sps.or.coef.4[2,] <= log(3) )/1000

cp.sri.or.4 = sum( sri.or.coef.4[1,]+1.96*sri.or.coef.4[2,] >= log(3) &

sri.or.coef.4[1,]-1.96*sri.or.coef.4[2,] <= log(3) )/1000

cp.tay.or.4 = sum( tay.or.coef.4[1,]+1.96*tay.or.coef.4[2,] >= log(3) &

tay.or.coef.4[1,]-1.96*tay.or.coef.4[2,] <= log(3) )/1000

cp.or=matrix(c(cp.tra.or.1,cp.sps.or.1,cp.sri.or.1,cp.tay.or.1,cp.tra.or.2,cp.sps.or.2,cp.sri.or.2,cp.tay.or.2,c p.tra.or.3,cp.sps.or.3,cp.sri.or.3,cp.tay.or.3,cp.tra.or.4,cp.sps.or.4,cp.sri.or.4,cp.tay.or.4),nrow=4,ncol=4, byrow=T)

colnames(cp.or)=colnames(bias.or); rownames(cp.or)=rownames(bias.or)

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Appendix B.

Histograms of Estimated Coefficients under Different Values of a

Figure B.1 Histogram of Estimated Coefficients under a=0.5

Figure B.2 Histogram of Estimated Coefficients under a=1

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Figure B.3 Histogram of Estimated Coefficients under a=2

Figure B.4 Histogram of Estimated Coefficients under a=5

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Appendix C.

Tables of Simulation Results under Different



₀and n

Table C.1.1 Simulation Results as



₀= 0.71 and n = 1,000 under Weaker IV

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Table C.1.2 Simulation Results as



₀= 0.71 and n = 1,000 under Stronger IV

Bias a=2 a=5

Confounding Tra. GLM 2SPS-L 2SRI-L 2SRI-T Tra. GLM 2SPS-L 2SRI-L 2SRI-T 0.5 -0.3775 -0.1086 -0.0987 -0.2670 -0.2868 -0.1083 -0.1049 -0.1815

1 -0.6137 -0.1096 -0.1044 -0.4416 -0.4508 -0.1120 -0.1104 -0.2694 2 -0.9097 -0.1243 -0.1227 -0.7133 -0.7121 -0.1225 -0.1228 -0.4704 5 -1.0990 -0.0892 -0.0890 -0.9099 -1.0247 -0.1072 -0.1081 -0.6415

SE a=2 a=5

Confounding Tra. GLM 2SPS-L 2SRI-L 2SRI-T Tra. GLM 2SPS-L 2SRI-L 2SRI-T 0.5 0.2619 0.3889 0.3901 0.3191 0.2417 0.2995 0.3002 0.2971

1 0.2562 0.4351 0.4359 0.3201 0.2418 0.3188 0.3191 0.3033 2 0.2340 0.5690 0.5701 0.3286 0.2343 0.3670 0.3676 0.3101 5 0.1979 1.2574 1.2588 0.4853 0.2108 0.5847 0.5854 0.3986

MSE a=2 a=5

Confounding Tra. GLM 2SPS-L 2SRI-L 2SRI-T Tra. GLM 2SPS-L 2SRI-L 2SRI-T 0.5 0.2111 0.1630 0.1619 0.1731 0.1406 0.1014 0.1011 0.1212

1 0.4422 0.2013 0.2009 0.2975 0.2617 0.1142 0.1141 0.1646 2 0.8823 0.3392 0.3401 0.6167 0.5620 0.1497 0.1502 0.3175 5 1.2471 1.5890 1.5926 1.0633 1.0944 0.3534 0.3544 0.5704

Coverage

Probability a=2 a=5

Confounding Tra. GLM 2SPS-L 2SRI-L 2SRI-T Tra. GLM 2SPS-L 2SRI-L 2SRI-T 0.5 67.6% 94.1% 94.0% 85.4% 77.3% 93.9% 94.2% 89.1%

1 32.1% 93.6% 93.5% 65.8% 51.7% 93.7% 93.8% 80.6%

2 2.8% 93.3% 93.1% 35.5% 13.9% 94.8% 94.8% 69.1%

5 0.0% 94.2% 94.1% 48.2% 0.1% 95.6% 95.6% 55.5%

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Table C.2.1 Simulation Results as



₀= 0 and n = 1,000 under Weaker IV

Bias a=0.5 a=1

Confounding Tra. GLM 2SPS-L 2SRI-L 2SRI-T Tra. GLM 2SPS-L 2SRI-L 2SRI-T 0.5 -0.4284 -0.1473 -0.1413 -0.3306 -0.4302 -0.1497 -0.1432 -0.3331

1 -0.6677 -0.1480 -0.1438 -0.5116 -0.6600 -0.1529 -0.1483 -0.5092 2 -0.9528 -0.1486 -0.1482 -0.7891 -0.9431 -0.1485 -0.1477 -0.7750 5 -1.0998 -0.1199 -0.1274 -0.9751 -1.0985 -0.0945 -0.0985 -0.9495

SE a=0.5 a=1

Confounding Tra. GLM 2SPS-L 2SRI-L 2SRI-T Tra. GLM 2SPS-L 2SRI-L 2SRI-T 0.5 0.2233 0.3523 0.3540 0.2754 0.2179 0.3449 0.3464 0.2763

1 0.2156 0.3975 0.3982 0.2834 0.2129 0.3876 0.3882 0.2751 2 0.1979 0.5606 0.5612 0.2800 0.1931 0.5358 0.5363 0.2764 5 0.1687 1.7763 1.7766 0.4248 0.1680 1.5615 1.5620 0.4267

MSE a=0.5 a=1

Confounding Tra. GLM 2SPS-L 2SRI-L 2SRI-T Tra. GLM 2SPS-L 2SRI-L 2SRI-T 0.5 0.2334 0.1458 0.1453 0.1852 0.2325 0.1414 0.1405 0.1873

1 0.4923 0.1799 0.1793 0.3420 0.4809 0.1736 0.1727 0.3349 2 0.9471 0.3364 0.3369 0.7010 0.9267 0.3091 0.3094 0.6771 5 1.2380 3.1695 3.1725 1.1313 1.2349 2.4472 2.4497 1.0837

Coverage

Probability a=0.5 a=1

Confounding Tra. GLM 2SPS-L 2SRI-L 2SRI-T Tra. GLM 2SPS-L 2SRI-L 2SRI-T 0.5 50.0% 92.2% 92.0% 73.6% 49.3% 92.0% 92.1% 74.6%

1 12.9% 92.5% 92.7% 47.3% 13.0% 92.4% 92.4% 48.9%

2 0.3% 93.4% 93.4% 18.1% 0.2% 93.2% 93.3% 17.8%

5 0.0% 93.6% 93.6% 32.4% 0.0% 94.1% 94.1% 35.2%

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Table C.2.2 Simulation Results as



₀= 0 and n = 1,000 under Stronger IV

Bias a=2 a=5

Confounding Tra. GLM 2SPS-L 2SRI-L 2SRI-T Tra. GLM 2SPS-L 2SRI-L 2SRI-T 0.5 -0.4070 -0.1607 -0.1532 -0.3107 -0.3132 -0.1568 -0.1515 -0.2110

1 -0.6249 -0.1559 -0.1514 -0.4745 -0.4620 -0.1499 -0.1470 -0.2956 2 -0.9008 -0.1376 -0.1361 -0.7124 -0.7031 -0.1299 -0.1287 -0.4637 5 -1.0872 -0.0755 -0.0764 -0.8623 -1.0084 -0.0974 -0.0966 -0.6393

SE a=2 a=5

Confounding Tra. GLM 2SPS-L 2SRI-L 2SRI-T Tra. GLM 2SPS-L 2SRI-L 2SRI-T 0.5 0.2092 0.3186 0.3195 0.2617 0.1893 0.2408 0.2413 0.2310

1 0.2050 0.3526 0.3533 0.2632 0.1898 0.2534 0.2536 0.2414 2 0.1941 0.4605 0.4610 0.2719 0.1843 0.2926 0.2929 0.2582 5 0.1666 1.0349 1.0357 0.4027 0.1695 0.4730 0.4735 0.3185

MSE a=2 a=5

Confounding Tra. GLM 2SPS-L 2SRI-L 2SRI-T Tra. GLM 2SPS-L 2SRI-L 2SRI-T 0.5 0.2094 0.1273 0.1256 0.1650 0.1340 0.0826 0.0812 0.0979

1 0.4326 0.1486 0.1477 0.2944 0.2495 0.0867 0.0859 0.1457 2 0.8492 0.2310 0.2310 0.5815 0.5283 0.1025 0.1023 0.2816 5 1.2098 1.0768 1.0786 0.9057 1.0456 0.2332 0.2336 0.5102

Coverage

Probability a=2 a=5

Confounding Tra. GLM 2SPS-L 2SRI-L 2SRI-T Tra. GLM 2SPS-L 2SRI-L 2SRI-T 0.5 52.0% 91.4% 91.8% 76.4% 63.3% 89.6% 90.1% 82.9%

1 15.0% 92.1% 92.2% 52.4% 31.7% 90.6% 90.9% 70.7%

2 0.5% 93.7% 93.8% 21.1% 3.6% 92.0% 92.0% 46.0%

5 0.0% 95.1% 95.1% 40.8% 0.0% 94.4% 94.4% 41.3%

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Table C.3.1 Simulation Results as

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₀= 0 and n = 10,000 under Weaker IV

Bias a=0.5 a=1

Confounding Tra. GLM 2SPS-L 2SRI-L 2SRI-T Tra. GLM 2SPS-L 2SRI-L 2SRI-T 0.5 -0.4219 -0.1280 -0.1209 -0.3605 -0.4181 -0.1329 -0.1256 -0.3574

1 -0.6643 -0.1275 -0.1227 -0.5648 -0.6580 -0.1319 -0.1273 -0.5573 2 -0.9544 -0.1364 -0.1350 -0.8307 -0.9466 -0.1410 -0.1397 -0.8187 5 -1.1018 -0.1572 -0.1589 -0.9625 -1.0984 -0.1468 -0.1479 -0.9401

SE a=0.5 a=1

Confounding Tra. GLM 2SPS-L 2SRI-L 2SRI-T Tra. GLM 2SPS-L 2SRI-L 2SRI-T 0.5 0.0689 0.1098 0.1102 0.0817 0.0678 0.1050 0.1054 0.0791

1 0.0665 0.1241 0.1242 0.0839 0.0656 0.1195 0.1197 0.0839 2 0.0610 0.1771 0.1771 0.0865 0.0603 0.1693 0.1693 0.0874 5 0.0516 0.5577 0.5577 0.1203 0.0514 0.4793 0.4792 0.1192

MSE a=0.5 a=1

Confounding Tra. GLM 2SPS-L 2SRI-L 2SRI-T Tra. GLM 2SPS-L 2SRI-L 2SRI-T 0.5 0.1827 0.0284 0.0268 0.1366 0.1794 0.0287 0.0269 0.1340

1 0.4457 0.0316 0.0305 0.3260 0.4373 0.0317 0.0305 0.3176 2 0.9146 0.0500 0.0496 0.6976 0.8996 0.0485 0.0482 0.6779 5 1.2165 0.3357 0.3362 0.9409 1.2091 0.2512 0.2515 0.8981

Coverage

Probability a=0.5 a=1

Confounding Tra. GLM 2SPS-L 2SRI-L 2SRI-T Tra. GLM 2SPS-L 2SRI-L 2SRI-T 0.5 0.0% 79.2% 81.6% 0.8% 0.0% 78.5% 80.0% 0.7%

1 0.0% 82.6% 83.5% 0.0% 0.0% 81.5% 82.4% 0.0%

2 0.0% 87.7% 87.7% 0.0% 0.0% 86.5% 86.6% 0.0%

5 0.0% 92.9% 93.0% 0.0% 0.0% 91.9% 91.9% 0.0%

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Table C.3.2 Simulation Results as

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₀= 0 and n = 10,000 under Stronger IV

Bias a=2 a=5

Confounding Tra. GLM 2SPS-L 2SRI-L 2SRI-T Tra. GLM 2SPS-L 2SRI-L 2SRI-T 0.5 -0.3974 -0.1414 -0.1339 -0.3371 -0.2914 -0.1293 -0.1244 -0.2320

1 -0.6221 -0.1360 -0.1319 -0.5182 -0.4428 -0.1197 -0.1176 -0.3352 2 -0.9065 -0.1324 -0.1313 -0.7654 -0.6889 -0.1030 -0.1026 -0.5113 5 -1.0857 -0.1157 -0.1163 -0.8686 -1.0017 -0.0834 -0.0837 -0.6658

SE a=2 a=5

Confounding Tra. GLM 2SPS-L 2SRI-L 2SRI-T Tra. GLM 2SPS-L 2SRI-L 2SRI-T 0.5 0.0643 0.0952 0.0954 0.0746 0.0603 0.0734 0.0736 0.0688

1 0.0648 0.1062 0.1063 0.0804 0.0606 0.0783 0.0783 0.0777 2 0.0616 0.1447 0.1448 0.0875 0.0588 0.0925 0.0926 0.0862 5 0.0512 0.3167 0.3166 0.1166 0.0522 0.1503 0.1503 0.1018

MSE a=2 a=5

Confounding Tra. GLM 2SPS-L 2SRI-L 2SRI-T Tra. GLM 2SPS-L 2SRI-L 2SRI-T 0.5 0.1621 0.0291 0.0270 0.1192 0.0885 0.0221 0.0209 0.0586

1 0.3912 0.0298 0.0287 0.2750 0.1997 0.0205 0.0200 0.1184 2 0.8256 0.0385 0.0382 0.5934 0.4780 0.0192 0.0191 0.2688 5 1.1813 0.1137 0.1138 0.7680 1.0061 0.0295 0.0296 0.4536

Coverage

Probability a=2 a=5

Confounding Tra. GLM 2SPS-L 2SRI-L 2SRI-T Tra. GLM 2SPS-L 2SRI-L 2SRI-T 0.5 0.0% 71.6% 74.4% 0.7% 0.2% 60.7% 62.4% 7.7%

1 0.0% 78.4% 79.0% 0.0% 0.0% 67.3% 68.5% 0.5%

2 0.0% 84.5% 84.8% 0.0% 0.0% 80.4% 80.5% 0.0%

5 0.0% 92.2% 92.1% 0.0% 0.0% 90.4% 90.4% 0.0%

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Table C.4.1 Simulation Results as

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₀= 3 and n = 1,000 under Weaker IV

Bias a=0.5 a=1

Confounding Tra. GLM 2SPS-L 2SRI-L 2SRI-T Tra. GLM 2SPS-L 2SRI-L 2SRI-T 0.5 -0.3709 -0.0813 -0.0536 -0.2714 -0.3708 -0.0859 -0.0602 -0.2627

1 -0.6348 -0.1086 -0.0934 -0.4966 -0.6206 -0.0997 -0.0847 -0.4756 2 -0.9527 -0.1850 -0.1788 -0.8188 -0.9539 -0.1643 -0.1580 -0.7946 5 -1.1487 -0.5348 -0.5274 -1.0251 -1.1484 -0.4334 -0.4231 -0.9868

SE a=0.5 a=1

Confounding Tra. GLM 2SPS-L 2SRI-L 2SRI-T Tra. GLM 2SPS-L 2SRI-L 2SRI-T 0.5 0.5658 0.8959 0.9017 0.6834 0.5688 0.8927 0.8977 0.6853

1 0.5565 1.0116 1.0173 0.6784 0.5698 0.9968 1.0018 0.6757 2 0.5066 1.4220 1.4272 0.6524 0.5146 1.3636 1.3674 0.6683 5 0.4304 3.9676 3.9887 0.9607 0.4305 3.5243 3.5342 0.9461

MSE a=0.5 a=1

Confounding Tra. GLM 2SPS-L 2SRI-L 2SRI-T Tra. GLM 2SPS-L 2SRI-L 2SRI-T 0.5 0.4577 0.8092 0.8160 0.5407 0.4610 0.8043 0.8094 0.5387

1 0.7127 1.0351 1.0436 0.7069 0.7098 1.0036 1.0108 0.6827 2 1.1642 2.0564 2.0687 1.0960 1.1747 1.8864 1.8947 1.0781 5 1.5048 16.0281 16.1877 1.9737 1.5042 12.6085 12.6698 1.8689

Coverage

Probability a=0.5 a=1

Confounding Tra. GLM 2SPS-L 2SRI-L 2SRI-T Tra. GLM 2SPS-L 2SRI-L 2SRI-T 0.5 88.6% 94.3% 94.4% 92.7% 89.6% 94.1% 94.3% 93.7%

1 76.4% 94.4% 94.3% 88.6% 77.6% 94.7% 94.6% 88.9%

2 52.2% 93.8% 93.9% 77.3% 50.5% 94.5% 94.6% 79.2%

5 21.7% 93.9% 93.9% 86.3% 21.9% 94.8% 94.9% 85.8%

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Table C.4.2 Simulation Results as

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₀= 3 and n = 1,000 under Stronger IV

Bias a=2 a=5

Confounding Tra. GLM 2SPS-L 2SRI-L 2SRI-T Tra. GLM 2SPS-L 2SRI-L 2SRI-T 0.5 -0.3498 -0.0780 -0.0577 -0.2344 -0.2996 -0.1054 -0.1064 -0.1771

1 -0.5824 -0.0954 -0.0845 -0.4257 -0.4764 -0.1261 -0.1325 -0.3067 2 -0.9137 -0.1364 -0.1330 -0.7078 -0.7633 -0.1903 -0.2032 -0.5313 5 -1.1512 -0.2436 -0.2409 -0.8883 -1.0996 -0.2225 -0.2415 -0.6447

SE a=2 a=5

Confounding Tra. GLM 2SPS-L 2SRI-L 2SRI-T Tra. GLM 2SPS-L 2SRI-L 2SRI-T 0.5 0.5710 0.8526 0.8553 0.6776 0.5354 0.6998 0.7039 0.6255

1 0.5674 0.9331 0.9364 0.6544 0.5288 0.7389 0.7434 0.6108 2 0.5253 1.1999 1.2038 0.6653 0.5063 0.8844 0.8890 0.6209 5 0.4371 2.5591 2.5625 0.9139 0.4668 1.3909 1.3950 0.7873

MSE a=2 a=5

Confounding Tra. GLM 2SPS-L 2SRI-L 2SRI-T Tra. GLM 2SPS-L 2SRI-L 2SRI-T 0.5 0.4484 0.7330 0.7348 0.5140 0.3765 0.5008 0.5069 0.4226

1 0.6611 0.8799 0.8839 0.6095 0.5066 0.5619 0.5702 0.4671 2 1.1109 1.4582 1.4669 0.9436 0.8390 0.8184 0.8317 0.6678 5 1.5164 6.6083 6.6243 1.6244 1.4271 1.9841 2.0044 1.0355

Coverage

Probability a=2 a=5

Confounding Tra. GLM 2SPS-L 2SRI-L 2SRI-T Tra. GLM 2SPS-L 2SRI-L 2SRI-T 0.5 90.0% 94.1% 94.0% 93.5% 91.1% 94.5% 94.4% 93.9%

1 79.9% 94.3% 94.2% 89.8% 85.7% 94.7% 94.6% 93.2%

2 56.2% 95.1% 95.1% 81.8% 69.5% 95.0% 94.7% 87.5%

5 23.9% 95.3% 95.2% 85.1% 31.3% 94.8% 94.6% 87.8%

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Table C.5.1 Simulation Results as

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₀= 3 and n = 10,000 under Weaker IV

Bias a=0.5 a=1

Confounding Tra. GLM 2SPS-L 2SRI-L 2SRI-T Tra. GLM 2SPS-L 2SRI-L 2SRI-T 0.5 -0.3861 -0.0851 -0.0715 -0.3257 -0.3757 -0.0822 -0.0693 -0.3152

1 -0.6418 -0.0910 -0.0848 -0.5544 -0.6269 -0.0843 -0.0786 -0.5360 2 -0.9625 -0.0973 -0.0964 -0.8503 -0.9496 -0.0863 -0.0856 -0.8260 5 -1.1617 -0.0023 -0.0064 -0.9453 -1.1603 0.0048 0.0017 -0.9159

SE a=0.5 a=1

Confounding Tra. GLM 2SPS-L 2SRI-L 2SRI-T Tra. GLM 2SPS-L 2SRI-L 2SRI-T 0.5 0.1799 0.2849 0.2850 0.1934 0.1832 0.2780 0.2782 0.1958

1 0.1791 0.3202 0.3204 0.1955 0.1812 0.3118 0.3120 0.1978 2 0.1658 0.4455 0.4459 0.1895 0.1715 0.4265 0.4270 0.1984 5 0.1402 1.2849 1.2853 0.2919 0.1433 1.1137 1.1141 0.2887

MSE a=0.5 a=1

Confounding Tra. GLM 2SPS-L 2SRI-L 2SRI-T Tra. GLM 2SPS-L 2SRI-L 2SRI-T 0.5 0.1814 0.0884 0.0863 0.1435 0.1747 0.0840 0.0822 0.1377

1 0.4439 0.1108 0.1098 0.3456 0.4259 0.1043 0.1035 0.3264 2 0.9540 0.2079 0.2081 0.7590 0.9312 0.1894 0.1896 0.7217 5 1.3692 1.6510 1.6520 0.9787 1.3667 1.2404 1.2413 0.9222

Coverage

Probability a=0.5 a=1

Confounding Tra. GLM 2SPS-L 2SRI-L 2SRI-T Tra. GLM 2SPS-L 2SRI-L 2SRI-T 0.5 45.7% 94.6% 94.7% 64.3% 47.2% 94.8% 95.2% 66.9%

1 5.4% 94.4% 94.6% 21.5% 6.5% 94.3% 94.6% 24.4%

2 0.0% 94.4% 94.8% 0.8% 0.0% 94.4% 94.4% 1.4%

5 0.0% 94.4% 94.5% 10.4% 0.0% 94.9% 95.0% 13.5%

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Table C.5.2 Simulation Results as

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₀= 3 and n = 10,000 under Stronger IV

Bias a=2 a=5

Confounding Tra. GLM 2SPS-L 2SRI-L 2SRI-T Tra. GLM 2SPS-L 2SRI-L 2SRI-T 0.5 -0.3453 -0.0777 -0.0678 -0.2849 -0.2807 -0.0962 -0.0955 -0.2231

1 -0.5821 -0.0828 -0.0786 -0.4874 -0.4577 -0.1147 -0.1146 -0.3671 2 -0.9004 -0.0879 -0.0876 -0.7630 -0.7371 -0.1502 -0.1519 -0.5922 5 -1.1501 -0.0445 -0.0471 -0.8504 -1.0801 -0.1713 -0.1749 -0.6915

SE a=2 a=5

Confounding Tra. GLM 2SPS-L 2SRI-L 2SRI-T Tra. GLM 2SPS-L 2SRI-L 2SRI-T 0.5 0.1792 0.2613 0.2615 0.1936 0.1741 0.2238 0.2244 0.1891

1 0.1814 0.2892 0.2895 0.1975 0.1761 0.2379 0.2382 0.1909 2 0.1734 0.3764 0.3767 0.2004 0.1695 0.2801 0.2803 0.1927 5 0.1463 0.8074 0.8076 0.2924 0.1579 0.4452 0.4454 0.2726

MSE a=2 a=5

Confounding Tra. GLM 2SPS-L 2SRI-L 2SRI-T Tra. GLM 2SPS-L 2SRI-L 2SRI-T 0.5 0.1513 0.0743 0.0730 0.1187 0.1091 0.0594 0.0595 0.0856

1 0.3717 0.0905 0.0900 0.2766 0.2405 0.0698 0.0699 0.1712 2 0.8407 0.1494 0.1496 0.6224 0.5720 0.1010 0.1017 0.3879 5 1.3442 0.6540 0.6544 0.8087 1.1916 0.2276 0.2290 0.5524

Coverage

Probability a=2 a=5

Confounding Tra. GLM 2SPS-L 2SRI-L 2SRI-T Tra. GLM 2SPS-L 2SRI-L 2SRI-T 0.5 54.6% 94.2% 94.3% 71.6% 65.0% 92.2% 92.3% 78.7%

1 11.1% 93.9% 93.8% 31.6% 27.0% 91.8% 92.0% 49.7%

2 0.0% 94.4% 94.4% 3.5% 0.4% 91.1% 91.3% 13.2%

5 0.0% 95.1% 95.1% 18.4% 0.0% 93.0% 92.9% 27.5%

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Appendix D.

Questions Used in the WVS Questionnaire in the Empirical Analysis

Outcome Variable (y)

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V251. 在過去一年中，請問您家是有儲蓄、收支平衡、花掉一些積蓄，還是花掉積蓄而且還借錢？

(1)有儲蓄 (2)收支平衡 (3)花掉一些積蓄 (4)花掉積蓄而且還借錢

V252. 人們有時會把自己劃分到不同的階層中，請問您認為您自己是屬於哪一個階層？

(1)上階層 (2)中上階層 (3)中下階層 (4)勞工階層 (5)下階層

V241. 請問您現在是否有工作？

(01)全職（一週30小時或以上） (02)兼職（一週少於30小時）

(03)自己開業 (04)退休人員

(05)家庭主婦且無任何工作 (06)學生且無任何工作

(07)失業 (08)其他（請說明）：_________

V238. 這表示您的最高學歷？

(01)無 (02)國小肄業 (03)國小畢業

(04)國中肄業 (05)國中畢業 (06)高中肄業 (07)高中畢業 (08)高職肄業 (09)高職畢業 (10)專科肄業 (11)專科畢業 (12)大學（無學位）

(13)大學（有學位） (14)研究所（無學位） (15)研究所（有學位）

在文檔中兩階段工具變數估計量應用於二元反應變數之比較與實證研究 - 政大學術集成 (頁 29-52)

Chapter 5 Conclusion and Discussion

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Safety.2010; 19(6): 537-554.

International Journal of Epidemiology. 2000; 29: 722-729.

Statistics in Medicine. 2008;27: 1539-1556.

Rheumatism

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Economics. 2008; 27(3): 531-543.

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Appendix A.

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Appendix B.

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Appendix C.

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Bias a=2 a=5

SE a=2 a=5

MSE a=2 a=5

Coverage

Probability a=2 a=5

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Bias a=0.5 a=1

SE a=0.5 a=1

MSE a=0.5 a=1

Coverage

Probability a=0.5 a=1

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Bias a=2 a=5

SE a=2 a=5

MSE a=2 a=5

Coverage

Probability a=2 a=5

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