In official economic statistics, the goal of surveys is often to estimate the mean or the total of a heteroskedastic log-normal variable. For this purpose, ratio imputation has been often used to treat missing values. However, there are three competing ratio estimators in the literature: ordinary least squares (OLS); ratio of means (RoM); and mean of ratios(MoR). It is not quite obvious which of the estimators is best, thus leading to a gap between theory and practice. The objective of this article is to fill in this gap by unifying ratio imputation models and by proposing a novel estimation strategy for selecting a ratio imputation model based on the magnitude of heteroskedasticity. The proposed method estimates the magnitude of heteroskedasticity under the framework of weighted least squares (WLS). Using the 135,000 simulated datasets that cover different parameter settings, the results in the Monte Carlo simulation give a strong support for the proposed method. If the estimated magnitude of heteroskedasticity is moderate, RoM should be used. If the estimated magnitude is severe, MoR should be used. If the estimated magnitude is mild, either OLS or RoM may be used.
BaraldiA.N. and EndersC.K., An introduction to modern missing data analyses, Journal of School Psychology48 (2010), 5-37.
3.
CheemaJ.R., Some general guidelines for choosing missing data handling methods in educational research, Journal of Modern Applied Statistical Methods13 (2014), 53-75.
4.
CochranW.G., Sampling techniques, 3rd ed., John Wiley and Sons, 1977.
5.
DeGrootM.H. and SchervishM.J., Probability and statistics, 3rd ed., Addison-Wesley, 2002.
6.
de WaalT., PannekoekJ. and ScholtusS., Handbook of statistical data editing and imputation, John Wiley and Sons, 2011.
7.
Di ZioM. and GuarneraU., Contamination model for selective editing, Journal of Official Statistics29 (2013), 539-555.
8.
DondersA.R.T., van der HeijdenG.J.M.G., StijnenT. and MoonsK.G.M., Review: A gentle introduction to imputation of missing values, Journal of Clinical Epidemiology59 (2006), 1087-1091.
9.
EggheL., Averages of ratios compared to ratios of averages: Mathematical results, Journal of Informetrics6 (2012), 307-317.
10.
EisenhauerJ.G., Regression through the origin, Teaching Statistics25 (2003), 76-80.
11.
FeenstraR.C., InklaarR. and TimmerM.P., Penn World Table 9.0, 2016 [cited November 3, 2016]. Available from: http://wwwrug.nl/research/ggdc/data/pwt/pwt-9.0.
GuptaA.K. and KabeD.G., Theory of sample surveys, World Scientific, 2011.
14.
HoenigJ.M., JonesC.M., PollockK.H., RobsonD.S. and WadeD.L., Calculation of catch rate and total catch in roving surveys of anglers, Biometrics53 (1997), 306-317.
15.
HothornT., ZeileisA., FarebrotherR.W., CumminsC., MilloG. and MitchellD., Package `lmtest', 2015 [cited July 6, 2016]. Available from: https://cran.r-project.org/web/pack-ages/lmtest/lmtest.pdf.
16.
HuM.,SalvucciS. and SalvucciR., A Study of Imputation Algorithms, Working Paper No. 2001-2017. U.S. Department of Education. National Center for Education Statistics. 2001 [cited July 6, 2016]. Available from: http://ncesed.gov/pubs2001/200117.pdf.
17.
KingG., HonakerJ., JosephA. and ScheveK., Analyzing incomplete political science data: An alternative algorithm for multiple imputation, American Political Science Review, 95 (2001), 49-69.
18.
LarivièreV. and GingrasY., Averages of ratios vs. ratios of averages: An empirical analysis of four levels of aggregation, Journal of Informetrics5 (2011), 392-399.
19.
LiangH., SuH. and ZouG., Confidence intervals for a common mean with missing data with applications in AIDS study, Computational Statistics and Data Analysis53 (2008), 546-553.
20.
LittleR.J.A. and RubinD.B., Statistical analysis with missing data, second ed., John Wiley and Sons, 2002.
21.
LiuL., YujuanT., YingfuL. and ZouG., Imputation for missing data and variance estimation when auxiliary information is incomplete, Model Assisted Statistics and Applications1 (2005), 83-94.
22.
Office for National Statistics, Change to imputation method used for the turnover question in monthly business surveys, Guidance and Methodology: Retail Sales. 2014 [cited July 6, 2016]. Available from: http://www.ons.gov.uk/ons/guide- method/method-quality/specific/economy/retail-sales/index. html.
23.
RaoT.J., Mean of ratios or ratio of means or both? Journal of Statistical Planning and Inference102 (2002), 129-138.
24.
RoyallR.M., On finite population sampling theory under certain linear regression models, Biometrika57 (1970), 377-387.
25.
RubinD.B., Multiple imputation for nonresponse in surveys, John Wiley and Sons, 1987.
26.
SchaferJ.L., Analysis of incomplete multivariate data, Chapman and Hall/CRC, 1997.
27.
ShaoJ., Cold deck and ratio imputation, Survey Methodology26 (2000), 79-85.
28.
SnowdonP., Ratio methods for estimating forest biomass, New Zealand Journal of Forestry Science22 (1992), 54-62.
29.
ThompsonK.J. and WashingtonK.T., A response propensity based evaluation of the treatment of unit nonresponse for selected business surveys, Federal Committee on Statistical Methodology 2012 Research Conference, 2012 [cited July 6, 2016]. Available from: https://fcsmsites.usa.gov/files/ 2014/05/Thompson_2012FCSM_III-B.pdf.
30.
von HippelP.T., Should a normal imputation model be modified to impute skewed variables? Sociological Methods Research42 (2013), 105-138.
31.
WooldridgeJ.M., Introductory econometrics: A modern approach, 4th ed., South-Western, 2009.
32.
ZarnochS.J. and BechtoldW.A., Estimating mapped-plot forest attributes with ratios of means, Canadian Journal of Forest Research30 (2000), 688-697.
33.
ZouG.H., LiY.F., ZhuR. and GuanZ., Imputation of mean of ratios for missing data and its application to PPSWR sampling, Acta Mathematica Sinica, English Series26 (2010), 863-874.
