LengthBiased Sampling: A Review of Applications

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Length-Biased SamplingA Review of Applications

• Termeh Shafie
• Department of Statistics
• Umeå University
• termeh.shafie_at_stat.umu.se

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Outline

• Length-Biased Sampling the Estimation-Problem
• Applications Suggested Solutions
• Simulation under Misspecified Sampling Inclusion
  Probabilities

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Length-Biased Sampling

• The probability of sample inclusion of a
  population unit is related to the value of the
  variable measured.
• Cox (1969) Textile fibre sampling
• A simple illustration of the problem when
  estimating the population mean

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The Estimation Problem

• Assume there is a population with elements
• The mean of the population is

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The Estimation Problem

• Suppose observations form a
  sample with sample mean
• where
• if individual i is sampled
• otherwise

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The Estimation Problem

• The expected value of the sample mean is
• where
• are the inclusion probabilities of the
  population units.

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The Estimation Problem

• Using simple random sampling
• and thus

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The Estimation Problem

• However in general is unknown and thus
• The sample mean becomes a biased estimator of
  the population mean.

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Cox (1969)

• Derived the length-biased or weighted pdf and
  looked at the estimation of the population mean
  from a length-biased sample.
• Assume is a random sample
  with pdf

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Cox (1969)

• It can be shown that
• An unbiased estimator of is

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Cox (1969)

• with variance
• Note
• N

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Cox (1969)

• Relation between the moments of g(x) and f(x)
• The relative bias is thus

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2. APPLICATIONSTechnical/Industrial Sampling

• Cox (1969) Sampling textile fibres and the
  estimation of fibre length distribution.

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Marketing

• Shopping Center Sampling Mall Intercept
  Surveys
• Keillor et al (2001) Global consumer tendencies.
• Sudman (1980) Quota sampling techniques and
  weighting procedures to correct for frequency
  bias.
• Nowell et al (1991) correction techniques for
  length-biased sampling in two situations when
  total length of stay is known or estimated and
  when only the recurrence time is known.

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Epidemiology

• Sampling procedure for the collection of
  positive-valued or lifetime data are
  length-biased (Simon 1980, Zelen et al. 1969)
• Wang (1996) statistical analysis of
  length-biased data under proportional hazards
  model. A pseudo-likelihood approach for
  estimation of the parameters from length-biased
  data is presented.

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Resource Economics

• On-site sampling
• Deriving demand functions for a recreational site
  (Bockstael 1990, Ovaskainen et al. 2001)
• Charting trip taking behavior (Bowker 1998)
• Travel cost models of recreational demand (Moons
  et al. 2001)
• Contingent valuation surveys for the elicitation
  of non-market goods (Cameron et al. 1987, Nowell
  et al. 1988)

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Resource Economics

• Shaw (1988) Three problems with on-site samples
  regression
• Non-negative integers
• Truncation
• Endogeneous Stratification

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Resource Economics

• Shaw (1988) recreational demand modeling under
  two assumptions about the dependent variables
  distribution
• Normal distribution
• Poisson distribution
• y1,2,

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Resource Economics

• Englin Shonkwiler (1995)
• The Negative Binomial Model
• The truncated, stratified model is
• y1,2,

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Resource Economics

• Nunes (2003) Binary Choice Models
• The count variable is described by a Poisson
  distribution with an unobservable heterogeneity
  term correlated with the error term in a probit
  binary choice model

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3. Misspecification of Sampling Probabilities A
Simulation

• Aim
• To see whether or not the effect of
  missepecified sampling probabilities is large or
  not
• What happens if time per visit is correlated with
  frequency of visits when estimating the expected
  number of visits?

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Misspecification of Sampling Probabilities A
Simulation

• Time is modeled as a function of frequency of
  visits when estimating the population mean.
• Poisson
• Exponential
• Gamma
• The inclusion probabilities are proportional to
  the time spent at the site

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Misspecification of Sampling Probabilities A
Simulation

• The three estimators used for the simulation are
  
• The sample mean
• Shaws estimator
• Coxs Estimator

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Simulation Results
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Summary

• If the probabilities of sample inclusion of
  population units are related to the values of the
  variable measured, the parameter estimates will
  be biased and inconsistent.
• Thus correctly specified sampling
• inclusion mechanisms should
• not be neglected!

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References

• Bockstael , N.E., Strand, I.E., McConnell, K.E.,
  Arsanjani, F., 1990. Sample Selection Bias in the
  Estimation of Recreational Demand FunctionsAn
  Application to Sportfishing. Land Economics,
  vol.66. No 1,40-49
• Bowker, J.M., Leeworthy, V.R., 1998. Accounting
  for Ethnicity in Recreation Demand A Flexible
  Count Data Approach.Journal of Leisure research
  30(1),64-78.
• Bush, A.J, Hair, J.F., 1985. An Assessment of the
  Mall Intercept as a Data Collection Method.
  Journal of Marketing Research 22, 158-67.
• Cameron, T. A., James, M.D., 1987. Efficient
  Estimation Methods for "Close-Ended" Contingent
  Valuation Surveys. The Review of Economics and
  Statistics 69, 269-276.
• Cox, D.R., 1969. "Some Sampling Problems in
  Technology" in New Developments in Survey
  Sampling, U. L. Johnson and H. Smith, eds. New
  York Wiley Interscience.
• Englin, J., Shonkwiler, J.S., 1995. Estimating
  Social Welfare Using Count Data Models An
  Application to Long-Run Recreation Demand under
  Conditions of Endogenous Stratifications and
  Truncation. Review of Economics and Statistic 77,
  104-112.
• Keillor, B.D., D'Amico, M., Horton, V., 2001.
  Global Consumer Tendencies, Psychology and
  Marketing 18, 1-19.
• Laitila, T., 1998. Estimation of Combined
  Site-Choice and Trip-Frequency Models of
  Recreational Demand using Choice-based and
  On-Site Samples. Economics Letters 64, 17-23.
• Moons, E., Loomis, J., Proost, S., Eggermont, K.,
  Hermy, M., 2001. Travel Cost and Time Measurement
  in Travel Cost Models. Faculty of Economics and
  Applied Economic Sciences, Working Paper series,
  no 2001-22.
• Nakanishi, M., 1978. Frequency Bias in Shopper
  Surveys, in Preceedings of the American Marketing
  Association Educators Conferenc. Chicago
  American Marketing Association, 67-70.
• Nowell, C., Evans, M.A., McDonald, L., 1988.
  Length-Biased Sampling in Contingent Valuation
  Studies. Land Economics 64 (November), 367-71.
• Nowell, C., Stanley, L.R., 1991. Length-Biased
  Sampling in Mall Intercept Surveys. Journal of
  Marketing Research 28, 1991, 475-479.
• Nunes, L.C., 2003. Estimating Binary Choice
  Models With On-Site Samples. Faculdade de
  Economia, Universidade Nova de Lisboa.
• Ovaskainen, V., Mikkola, J., Pouta, E., 2001.
  Estimating Recreation Demand with On-Site Data
  An Application of Truncated and Endogenously
  Stratified Count Data Models. Journal of Forest
  Economics 72, 125-144.
• Santos Silva, J.M.C., 1997. Unobservables in
  Count Data Models for On-Site Samples. Economics
  Letters 54, 217-220.
• Satten, G.A., Kong, F., Wright, D.J., Glynn,
  S.A., Schreiber, G.B., 2004. How Special is a
  'Special' Interval Modeling Departure from
  Length-Biased Sampling in Renewal Processes.
  Biostatistics 5, 1, 145-151.
• Shaw, D., 1988. On-Site Samples' Regression,
  Problems of Non-negative Integers, Truncation,
  and Endogenous Stratification. Journal of
  Econometrics 37, 211-223.
• Simon, R. 1980. Length-Biased Sampling in
  Etiological Studies. Am. J. Epidem. 111, 444-452.
• Sudman, S., 1980. Improving the Quality of
  Shopping Center Sampling. Journal of Marketing
  Research 17, 1980, 423-431.

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And finally she stops