# Statistical analysis of ratio estimators and their estimators of variances when the auxiliary variate is measured with error

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## Abstract

Forest inventory relies heavily on sampling strategies. Ratio estimators use information of an auxiliary variable (*x*) to improve the estimation of a parameter of a target variable (*y*). We evaluated the effect of measurement error (ME) in the auxiliary variate on the statistical performance of three ratio estimators of the target parameter total τ_{ y }. The analyzed estimators are: the ratio-of-means, mean-of-ratios, and an unbiased ratio estimator. Monte Carlo simulations were conducted over a population of more than 14,000 loblolly pine (*Pinus taeda* L.) trees, using tree volume (*v*) and diameter at breast height (*d*) as the target and auxiliary variables, respectively. In each simulation three different sample sizes were randomly selected. Based on the simulations, the effect of different types (systematic and random) and levels (low to high) of MEs in *x* on the bias, variance, and mean square error of three ratio estimators was assessed. We also assessed the estimators of the variance of the ratio estimators. The ratio-of-means estimator had the smallest root mean square error. The mean-of-ratios estimator was found quite biased (20%). When the MEs are random, neither the accuracy (i.e. bias) of any of the ratio estimators is greatly affected by type and level of ME nor its precision (i.e. variance). Positive systematic MEs decrease the bias but increase the variance of all the ratio estimators. Only the variance estimator of the ratio-of-means estimator is biased, being especially large for the smallest sample size, and larger for negative MEs, mainly if they are systematic.

## Keywords

Sampling Forest inventory Design-based inference Variance estimators Bias## Notes

### Acknowledgments

We gratefully acknowledge Roy C. Beltz, U.S. Forest Service, Forestry Sciences Lab, Starkville, Mississippi for providing the population data used in our study.

## References

- Bay J, Stefanski LA (2000) Adjusting data for measurement error to reduce bias when estimating coefficients of a quadratic model. In: Proceedings of the Survey Research Methods Section, American Statistical Association, pp 731–733Google Scholar
- Canavan SJ, Hann DW (2004) The two-stage method for measurement error characterization. For Sci 50(6):743–756Google Scholar
- Chandhok PK (1988) Stratified sampling under measurement error. In: Proceedings of the Survey Research Methods Section, American Statistical Association, pp 508–510Google Scholar
- Cochran WG (1977) Sampling techniques, 3rd edn. Wiley, New York, USA, 428 ppGoogle Scholar
- Cunia T (1965) Some theories on reliability of volume estimates in a forest inventory sample. For Sci 11(1):115–127Google Scholar
- Dryver AL, Chao CT (2007) Ratio estimators in adaptive cluster sampling. Environmetrics 18:607–620CrossRefGoogle Scholar
- Ek AR (1971) A comparison of some estimators in forest sampling. For Sci 17(1):2–13Google Scholar
- Frayer WE, Furnival GM (1999) Forest survey sampling designs: a history. J For 97(12):4–10Google Scholar
- Fuller WA (1987) Measurement error models. Wiley, USA, 464 ppGoogle Scholar
- Gertner GZ (1988) Regressor variable errors and the estimation and prediction with linear and nonlinear models. In: Sloboda B (ed) Biometric models and simulation techniques for process of research and applications in forestry. Schriften aus der Forslichen Fakultat der Universität Göttingen, Göttingen, Germany, Band No. 160, pp 54–65Google Scholar
- Gertner GZ (1990) The sensitivity of measurement error in stand volume estimation. Can J For Res 20(6):800–804CrossRefGoogle Scholar
- Goodman LA, Hartley HO (1958) The precision of unbiased ratio-type estimators. J Am Stat Assoc 53(282):491–508CrossRefGoogle Scholar
- Goodman LA, Hartley HO (1969) Corrigenda: the precision of unbiased ratio-type estimators. J Am Stat Assoc 64(328):1700CrossRefGoogle Scholar
- Gregoire TG (1998) Design-based and model-based inference in survey sampling: appreciating the difference. Can J For Res 28(10):1429–1447CrossRefGoogle Scholar
- Gregoire TG, Salas C (2009) Ratio estimation with measurement error in the auxiliary variate. Biometrics 62(2). doi: 10.1111/j.1541-0420.2008.01110.x
- Gregoire TG, Schabenberger O (1999) Sampling-skewed bilogical populations: behavior of confidence intervals for the population total. Ecology 80(3):1056–1065CrossRefGoogle Scholar
- Gregoire TG, Valentine HT (2008) Sampling strategies for natural resources and the environment. Chapman & Hall/CRC, New York, 474 ppGoogle Scholar
- Gregoire TG, Williams M (1992) Identifying and evaluating the components of non-measurement error in the application of standard volume equations. Statistician 41(5):509–518CrossRefGoogle Scholar
- Grosenbaugh LR (1964) Some suggestions for better sample-tree measurement. In: Anon (ed) Proceedings. Society of American Foresters, Boston, MA, USA, pp 36–42Google Scholar
- Hansen MH, Hurwitz WN, Marks ES, Mauldin WP (1951) Response errors in surveys. J Am Stat Assoc 46(254):147–190CrossRefGoogle Scholar
- Hartley HO, Ross A (1954) Unbiased ratio estimators. Nature 174(4423):270–271CrossRefGoogle Scholar
- Hordo M, Kiviste A, Sims A, Lang M (2008) Outliers and/or measurement errors on the permanent sample plot data. In: Reynolds KM (ed) Proceedings of the sustainable forestry in theory and practice: recent advances in inventory and monitoring, statistics and modeling, information and knowledge management, and policy science. USDA For Serv Gen Tech Rep, PNW-688. Portland, OR, USA, p 15Google Scholar
- Horvitz DG, Thompson DJ (1952) A generalization of sampling without replacement from a finite universe. J Am Stat Assoc 47(260):663–685CrossRefGoogle Scholar
- Hutchison MC (1971) A monte carlo comparison of some ratio estimators. Biometrika 58(2):313–321CrossRefGoogle Scholar
- Kangas A (1996) On the bias and variance in tree volume predictions due to model and measurement errors. Scand J For Res 11:281–290CrossRefGoogle Scholar
- Kangas A (1998) Effects of errors-in-variables on coefficients of a growth model on prediction of growth. For Ecol Manag 102(2):203–212CrossRefGoogle Scholar
- Kangas AS, Kangas J (1999) Optimization bias in forest management planning solutions due to errors in forest variables. Silva Fenn 33(4):303–315Google Scholar
- Koop JC (1968) An exercise in ratio estimation. Ann Math Stat 22(1):29–30Google Scholar
- Magnussen S (2001) Saddlepoint approximations for statistical inference of PPP sample estimates. Scand J For Res 16:180–192CrossRefGoogle Scholar
- Mickey MR (1959) Some finite population unbiased ratio and regression estimators. J Am Stat Assoc 59(287):594–612CrossRefGoogle Scholar
- Myers RH (1990) Classical and modern regression with applications, 2nd edn. Duxbury, Pacific Grove, 488 ppGoogle Scholar
- Poso S, Wang G, Tuominen S (1999) Weighting alternative estimates when using multi-source auxiliary data for forest inventory. Silva Fenn 33(1):41–50Google Scholar
- R Development Core Team (2007) R: a language and environment for statistical computing. Available from http://www.R-project.org [version 2.5.0]. R Foundation for Statistical Computing, Vienna, Austria
- Raj D (1964) A note on the variance of ratio estimate. J Am Stat Assoc 59(307):895–898CrossRefGoogle Scholar
- Rao JNK (1968) Some small sample results in ratio and regression estimation. J Ind Stat Assoc 6:160–168Google Scholar
- Rice JA (1988) Mathematical statistics and data analysis. Wadsworth, Pacific Grove, 595 ppGoogle Scholar
- Robinson AP, Hamlin DC, Fairweather SE (1999) Improving forest inventories: three ways to incorporate auxiliary information. J For 97(12):38–42Google Scholar
- Royall RM, Cumberland WG (1981) An empirical study of the ratio estimator and estimators of its variance. J Am Stat Assoc 76(373):66–77CrossRefGoogle Scholar
- Scali J, Testa V, Kahr M, Strudler M (2005) Measuring nonsampling error in the statistics of income individual tax return study. In: Proceedings of the survey research methods section, American Statistical Association, pp 3520–3525Google Scholar
- Stage AR, Wykoff WR (1998) Adapting distance-independent forest growth models to represent spatial variability: effects of sampling design on model coefficients. For Sci 44(2):224–238Google Scholar
- Sukhatme PV, Sukhatme BV (1970) Sampling theory of surveys with applications, 2nd edn. Iowa State University Press, Ames, 452 ppGoogle Scholar
- Tin M (1965) Comparison of some ratio estimators. J Am Stat Assoc 60(309):294–307CrossRefGoogle Scholar