Results from MathSciNet: Mathematical Reviews on the Webc Copyright American Mathematical Society 2023
MR4571186 62G05 62D05 62G20
Che, Menglu (1-YALE-DBH) ; Han, Peisong (1-MI-BPH) ;Lawless, Jerald F. (3-WTRL-S)
Improving estimation efficiency for two-phase, outcome-dependent samplingstudies. (English. English summary)Electron. J. Stat. 17 (2023), no. 1, 1043–1073.

When some covariates are expensive or difficult to measure, two-phase outcome dependent sampling (ODS) is often employed. To make the argument more concrete, denotethe outcome, inexpensive-to-measure covariates, and expensive-to-measure covariates
by Y , X, and Z, respectively. In two-phase ODS, (Y, X) are taken from all individuals
in Phase 1, and then Z is taken from selected individuals in Phase 2.
In this article, the authors study the problem of estimating the parameter β in aparametric model of the conditional density f (Y |X, Z; β ) under two-phase ODS. The
conditional maximum likelihood (CML) method is popularly chosen, because it does
not need to model f (Z|X) unlike the full maximum likelihood method and it can handle
zero selection probabilities in Phase 2. However, CML leads to efficient loss due todiscarding the information on those who do not enter Phase 2.
To pursue efficiency gain over CML, the authors summarize the information on Phase
1 through a parametric model of the conditional density f (Y |X; θ ). More specifically,
they investigate the maximum empirical likelihood (MEL) estimation of β , where theinformation is explicitly incorporated as some zero expectation restriction implied bythis model, as well as the zero score restriction from CML.
Their analysis starts from the case with strictly positive selection probabilities inPhase 2 for all individuals. Under the oracle scenario where the selection probabilities
are known, the MEL estimator for β is shown to attain efficiency gain over CML in thesense that the asymptotic variance of the former is smaller than the one for the latter.
The analysis is further extended to the cases where positive selection probabilities areparametrically modeled (and estimated) and zero selection probabilities are allowed.
Efficiency gain of MEL over CML in finite samples is confirmed via Monte Carlo simulations, and a real data analysis from the National Health and Nutrition Examination
Survey is also conducted.
Masayuki Hirukawa[References]

1. Baksi, A. J., Treibel, T. A., Davies, J. E., Hadjiloizou, N., Foale, R. A.,
Parker, K. H., Francis, D. P., Mayet, J. and Hughes, A. D. (2009). A metaanalysis of the mechanism of blood pressure change with aging. Journal of theAmerican College of Cardiology 54 2087–2092.
2. Barnett, I. J., Lee, S. and Lin, X. (2013). Detecting rare variant effects using extreme phenotype sampling in sequencing association studies. Genetic Epidemiology
37 142–151.
3. Bjørnland, T., Bye, A., Ryeng, E., Wisløff, U. and Langaas, M. (2018).
Powerful extreme phenotype sampling designs and score tests for genetic association
studies. Statistics in Medicine 37 4234–4251. MR3879425 MR3879425
4. Breslow, N. and Cain, K. (1988). Logistic regression for two-stage case-control
data. Biometrika 75 11–20. MR0932812 MR0932812
5. Breslow, N. E. and Holubkov, R. (1997). Maximum likelihood estimationof logistic regression parameters under two-phase, outcome-dependent sampling.
Journal of the Royal Statistical Society: Series B (Statistical Methodology) 59 447–461. MR1440590 MR1440590
6. Breslow, N. E. and Holubkov, R. (1997). Weighted likelihood, pseudolikelihood

Results from MathSciNet: Mathematical Reviews on the Webc Copyright American Mathematical Society 2023
and maximum likelihood methods for logistic regression analysis of two-stage data.
Statistics in Medicine 16 103–116.
7. Chatterjee, N., Chen, Y.-H. and Breslow, N. E. (2003). A pseudoscoreestimator for regression problems with two-phase sampling. Journal of the American
Statistical Association 98 158–168. MR1965682 MR1965682
8. Chatterjee, N., Chen, Y.-H., Maas, P. and Carroll, R. J. (2016). Constrained maximum likelihood estimation for model calibration using summary-level
information from external big data sources. Journal of the American Statistical
Association 111 107–117. MR3494641 MR3494641
9. Che, M., Lawless, J. F. and Han, P. (2020). Empirical and Conditional Likelihoods for Two-Phase Studies. Canadian Journal of Statistics
doi.org/10.1002/cjs.11566. MR4267924 MR4267924
10. Derkach, A., Lawless, J. F. and Sun, L. (2015). Score tests for association
under response-dependent sampling designs for expensive covariates. Biometrika
102 988–994. MR3431569 MR3431569
11. Espin-Garcia, O., Craiu, R. V. and Bull, S. B. (2018). Two-phase designs forjoint quantitative-trait-dependent and genotype-dependent sampling in post-GWAS
regional sequencing. Genetic epidemiology 42 104–116.
12. Han, P. and Lawless, J. F. (2016). Comment on “Constrained Maximum Likelihood Estimation for Model Calibration Using Summary-level Information from
External Big Data Source”. Journal of the American Statistical Association 111
118–121. MR3494642 MR3494642
13. Han, P. and Lawless, J. F. (2019). Empirical likelihood estimation using auxiliarysummary information with different covariate distributions. Statistica Sinica 29
1321–1342. MR3932520 MR3932520
14. Han, P., Taylor, J. M. G. and Mukherjee, B. (2022). Integrating risk prediction models with no model details. Canadian Journal of Statistics 00 To appear.
MR4595233
15. Hermansen, K. (2000). Diet, blood pressure and hypertension. British Journal ofNutrition 83 S113–S119.
16. Horvitz, D. G. and Thompson, D. J. (1952). A generalization of sampling without
replacement from a finite universe. Journal of the American Statistical Association
47 663–685. MR0053460 MR0053460
17. Huang, C.-Y., Qin, J. and Tsai, H.-T. (2016). Efficient estimation of the Coxmodel with auxiliary subgroup survival information. Journal of the American Statistical Association 111 787–799. MR3538705 MR3538705
18. Imbens, G. W. (2002). Generalized method of moments and empirical likelihood.
Journal of Business & Economic Statistics 20 493–506. MR1973800 MR1973800
19. Keogh, R. H. and Cox, D. R. (2014). Case-control studies 4. Cambridge UniversityPress, Cambridge, UK. MR3443808 MR3443808
20. Kim, S., Cai, J. and Lu, W. (2013). More efficient estimators for case-cohort
studies. Biometrika 100 695–708. MR3094446 MR3094446
21. Kulich, M. and Lin, D. (2004). Improving the efficiency of relative-risk estimationin case-cohort studies. Journal of the American Statistical Association 99 832–844.
MR2090916 MR2090916
22. Lawless, J., Kalbfleisch, J. and Wild, C. (1999). Semiparametric methods
for response-selective and missing data problems in regression. Journal of theRoyal Statistical Society: Series B (Statistical Methodology) 61 413–438. MR1680310
MR1680310
23. Lin, D.-Y., Zeng, D. and Tang, Z.-Z. (2013). Quantitative trait analysis insequencing studies under trait-dependent sampling. Proceedings of the National

Results from MathSciNet: Mathematical Reviews on the Webc Copyright American Mathematical Society 2023
Academy of Sciences of the United States of America 110 12247–12252. MR3105371
MR3105371
24. Little, R. J. and Rubin, D. B. (2019). Statistical analysis with missing data 793.
John Wiley & Sons, Hoboken, New Jersey, USA. MR1925014 MR1925014
25. Owen, A. B. (2001). Empirical likelihood. Chapman and Hall/CRC, Boca Raton,FL, USA.
26. Pepe, M. S. and Fleming, T. R. (1991). A nonparametric method for dealing
with mismeasured covariate data. Journal of the American Statistical Association
86 108–113. MR1137103 MR1137103
27. Piegorsch, W. W., Weinberg, C. R. and Taylor, J. A. (1994). Nonhierarchical logistic models and case-only designs for assessing susceptibility inpopulation-based case-control studies. Statistics in Medicine 13 153–162.
28. Qin, J. (2000). Combining parametric and empirical likelihoods. Biometrika 87484–490. MR1782493 MR1782493
29. Qin, J. and Lawless, J. (1994). Empirical likelihood and general estimating
equations. The Annals of Statistics 300–325. MR1272085 MR1272085
30. Qin, J., Zhang, B. and Leung, D. H. (2009). Empirical likelihood in missing
data problems. Journal of the American Statistical Association 104 1492–1503.
MR2750574 MR2750574
31. Reilly, M. and Pepe, M. S. (1995). A mean score method for missing andauxiliary covariate data in regression models. Biometrika 82 299–314. MR1354230
MR1354230
32. Rivera-Rodriguez, C., Haneuse, S., Wang, M. and Spiegelman, D. (2020).
Augmented pseudo-likelihood estimation for two-phase studies. Statistical Methods
in Medical Research 29 344–358. MR4064512 MR4064512
33. Robins, J. M., Rotnitzky, A. and Zhao, L. P. (1994). Estimation of regression
coefficients when some regressors are not always observed. Journal of the American
Statistical Association 89 846–866. MR1294730 MR1294730
34. Schaid, D. J., Jenkins, G. D., Ingle, J. N. and Weinshilboum, R. M. (2013).
Two-phase designs to follow-up genome-wide association signals with DNA resequencing studies. Genetic epidemiology 37 229–238.
35. Scott, A. J. and Wild, C. J. (1997). Fitting regression models to case-control
data by maximum likelihood. Biometrika 84 57–71. MR1450191 MR1450191
36. Scott, A. J. and Wild, C. J. (2011). Fitting regression models with responsebiased samples. Canadian Journal of Statistics 39 519–536. MR2842429 MR2842429
37. Tao, R., Zeng, D. and Lin, D.-Y. (2017). Efficient Semiparametric Inference Under Two-Phase Sampling, With Applications to Genetic Association Studies. Journal of the American Statistical Association 112 1468–1476. MR3750869 MR3750869
38. Tsiatis, A. (2007). Semiparametric theory and missing data. Springer Science &Business Media, New York, NY, USA. MR2233926 MR2233926
39. Weaver, M. A. and Zhou, H. (2005). An estimated likelihood method for continuous outcome regression models with outcome-dependent sampling. Journal of theAmerican Statistical Association 100 459–469. MR2160550 MR2160550
40. Zhang, Z. and Rockette, H. E. (2006). Semiparametric maximum likelihood for
missing covariates in parametric regression. Annals of the Institute of Statistical
Mathematics 58 687–706. MR2345163 MR2345163
41. Zhao, Y., Lawless, J. F. and McLeish, D. L. (2009). Likelihood methods forregression models with expensive variables missing by design. Biometrical Journal: Journal of Mathematical Methods in Biosciences 51 123–136. MR2667516
MR2667516
42. Zhou, H., Weaver, M. A., Qin, J., Longnecker, M. and Wang, M. C. (2002).

Results from MathSciNet: Mathematical Reviews on the Webc Copyright American Mathematical Society 2023
A semiparametric empirical likelihood method for data from an outcome-dependent
sampling scheme with a continuous outcome. Biometrics 58 413–421. MR1908182
MR1908182
Note: This list reflects references listed in the original paper as accurately as possible
with no attempt to correct errors.