Ratio estimation under SRS

1
Ratio estimation under SRS

• Assume
• Absence of nonsampling error
• SRS of size n from a pop of size N
• Ratio estimation is alternative to under SRS,
  uses auxiliary information (X )
• Sample data observe yi and xi
• Population information
• Have yi and xi on all individual units, or
• Have summary statistics from the population
  distribution of X, such as population mean, total
  of X
• Ratio estimation is also used to estimate
  population parameter called a ratio (B )

2
Uses

• Estimate a ratio
• Tree volume or bushels per acre
• Per capita income
• Liability to asset ratio
• More precise estimator of population parameters
• If X and Y are correlated, can improve upon
• Estimating totals when pop size N is unknown
• Avoids need to know N in formula for
• Domain estimation
• Obtaining estimates of subsamples
• Incorporate known information into estimates
• Postratification
• Adjust for nonresponse

3
Estimating a ratio, B

• Population parameter for the ratio B
• Examples
• Number of bushels harvested (y) per acre (x)
• Number of children (y) per single-parent
  household (x)
• Total usable weight (y) relative to total
  shipment weight (x) for chickens

4
Estimating a ratio

• SRS of n observation units
• Collect data on y and x for each OU
• Natural estimator for B ?

5
Estimating a ratio -2

• Estimator for B
• is a biased estimator for B
• is a ratio of random variables

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Bias of
7
Bias of 2

• Bias is small if
• Sample size n is large
• Sample fraction n/N is large
• is large
• is small (pop std deviation for x)
• High positive correlation between X and Y
• (see Lohr p. 67)

8
Estimated variance of estimator for B

• Estimator for
• If is unknown?

9
Variance of

• Variance is small if
• sample size n is large
• sample fraction n/N is large
• deviations about line e y ? Bx are small
• correlation between X and Y close to ?1
• is large

10
Ag example 1

• Frame 1987 Agricultural Census
• Take SRS of 300 counties from 3078 counties to
  estimate conditions in 1992
• Collect data on y , have data on x for sample
• Existing knowledge about the population

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Ag example 2

• Estimate

0.9866 farm acres in 1992 relative to 1987 farm
acres
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Ag example 3

• Need to calculate variance of ei s

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Ag example 4

• For each county i, calculate
• Coffee Co, AL example
• Sum of squares for ei

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Ag example 5
15
Estimating proportions

• If denominator variable is random, use ratio
  estimator to estimate the proportion p
• Example (p. 72)
• 10 plots under protected oak trees used to assess
  effect of feral pigs on native vegetation on
  Santa Cruz Island, CA
• Count live seedlings y and total number of
  seedlings x per plot
• Y and X correlated due to common environmental
  factors
• Estimate proportion of live seedlings to total
  number of seedlings

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Estimating population mean

• Estimator for
• Adjustment factor for sample mean
• A measure of discrepancy between sample and
  population information, and
• Improves precision if X and Y are correlated

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Underlying model

• with B gt 0
• B is a slope
• B gt 0 indicates X and Y are positively
  correlated
• Absence of intercept implies line must go
  through origin (0, 0)

0
18
Using population mean of X to adjust sample mean

• Discrepancy between sample pop info for X is
  viewed as evidence that same relative discrepancy
  exists between

19
Bias of

• Ratio estimator for the population mean is biased
• Rules of thumb for bias of apply

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Estimator for variance of

• Estimator for variance of

21
Ag example 6

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Ag example - 8

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Ag example 9

• Expect a linear relationship between X and Y
  (Figure 3.1)
• Note that sample mean is not equal to population
  mean for X

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MSE under ratio estimation

• Recall
• MSE Variance Bias2
• SRS estimators are unbiased so
• MSE Variance
• Ratio estimators are biased so
• MSE gt Variance
• Use MSE to compare design/estimation strategies
• EX compare sample mean under SRS with ratio
  estimator for pop mean under SRS

25
Sample mean vs. ratio estimator of mean

• is smaller than
  if and only if
• For example, if and
• ratio estimation will be better than SRS

26
Estimating the MSE

• Estimate MSE with sample estimates of bias and
  variance of estimator
• This tends to underestimate MSE
• and are approximations
• Estimated MSE is less biased if
• is small (see earlier slide)
• Large sample size or sampling fraction
• High correlation for X and Y
• is a precise estimate (small CV for )
• We have a reasonably large sample size (n gt 30)

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Ag example 10

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Estimating population total t

• Estimator for t
• Is biased?
• Estimator for

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Ag example 11

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Summary of ratio estimation

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Summary of ratio estn 2

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Regression estimation

• What if relationship between y and x is linear,
  but does NOT pass through the origin
• Better model in this case is

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Regression estimation 2

• New estimator is a regression estimator
• To estimate , is predicted value
  from regression of y on x at
• Adjustment factor for sample mean is linear,
  rather than multiplicative

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Estimating population mean

• Regression estimator
• Estimating regression parameters

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Estimating pop mean 2

• Sample variances, correlation, covariance

36
Bias in regression estimator
37
Estimating variance

• Note This is a different residual than ratio
  estimation (predicted values differ)

38
Estimating the MSE

• Plugging sample estimates into Lohr, equation
  3.13

39
Estimating population total t

• Is regression estimator for t unbiased?

40
Tree example

• Goal obtain a precise estimate of number of
  dead trees in an area
• Sample
• Select n 25 out of N 100 plots
• Make field determination of number of dead trees
  per plot, yi
• Population
• For all N 100 plots, have photo determination
  on number of dead trees per plot, xi
• Calculate 11.3 dead trees per plot

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Tree example 2

• Lohr, p. 77-78
• Data
• Plot of y vs. x
• Output from PROC REG
• Components for calculating estimators and
  estimating the variance of the estimators
• We will use PROC SURVEYREG, which will give you
  the correct output for regression estimators

42
Tree example 3

• Estimated mean number of dead trees/plot
• Estimated total number of dead trees

43
Tree example 4

• Due to small sample size, Lohr uses t
  -distribution w/ n ? 2 degrees of freedom
• Half-width for 95 CI
• Approx 95 CI for ty is (1115, 1283) dead trees

44
Related estimators

• Ratio estimator
• B0 0 ? ratio model
• Ratio estimator ? regression estimator with no
  intercept
• Difference estimation
• B1 1 ? slope is assumed to be 1

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Domain estimation under SRS

• Usually interested in estimates and inferences
  for subpopulations, called domains
• If we have not used stratification to set the
  sample size for each domain, then we should use
  domain estimation
• We will assume SRS for this discussion
• If we use stratified sampling with strata
  domains, then use stratum estimators (Ch 4)
• To use stratification, need to know domain
  assignment for each unit in the sampling frame
  prior to sampling

46
Stratification vs. domain estimation

• In stratified random sampling
• Define sample size in each stratum before
  collecting data
• Sample size in stratum h is fixed, or known
• In other words, the sample size nh is the same
  for each sample selected under the specified
  design
• In domain estimation
• nd sample size in domain d is random
• Dont know nd until after the data have been
  collected
• The value of nd changes from sample to sample

47
Population partitioned into domains

• Recall U index set for population 1, 2, , N
  
• Domain index set for domain d 1, 2, , D
• Ud 1, 2, , Nd where Nd number of OUs in
  domain d in the population
• In sample of size n
• nd number of sample units from domain d are
  in the sample
• Sd index set for sample belonging to domain d

Domain D
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Boat owner example

• Population
• N 400,000 boat owners (currently licensed)
• Sample
• n 1,500 owners selected using SRS
• Divide universe (population) into 2 domains
• d 1 own open motor boat gt 16 ft. (large boat)
• d 2 do not own this type of boat
• Of the n 1500 sample owners
• n1 472 owners of open motor boat gt 16 ft.
• n2 1028 owners do not own this kind of boat

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New population parameters

• Domain mean
• Domain total

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Boat owner example - 2

• Estimate population domain mean
• Estimate the average number of children for boat
  owners from domain 1
• Estimate proportion of boat owners from domain 1
  who have children
• Estimate population domain total
• Estimate the total number of children for large
  boat owners (domain 1)

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New population parameter 2

• Ratio form of population mean
• Numerator variable
• Denominator variable

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Boat owner example - 3

• Estimate mean number of children for owners from
  domain 1

Applies to whole pop
Zero values for OUs that are not in domain 1
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Boat example 4
54

Estimator for population domain mean
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Boat example 5

• Domain 1 data

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Boat example 6

• Domain 1 and domain 2 data combined

1104 zeros 76 zeros from domain 1 1028
zeros from domain 2
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Boat example 7

• Two ways of estimating mean

Whole data set
Domain 1 data only
58

Estimator for variance of
59
Boat example 8
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Boat example 9
61
Approximation for estimator of variance of
Domain 1 data only
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Estimated variance of

• Estimator for
• Domain variance estimator is directly related

63

Relationship to estimating a ratio with

• Population mean of X
• Residual

64

Relationship to estimating a ratio with - 2

• Residual variance

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Estimator for variance of
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Estimating a population domain total

• If we know the domain sizes, Nd

67
Estimating a population domain total - 2

• If we do NOT know the domain sizes

Standard SRS estimator using u as the variable
68
Boat example 10

• Do not know the domain size, N1

69
Comparing 2 domain means

• Suppose we want to test the hypothesis that two
  domain means are equal
• Construct a z-test with Type 1 error rate ? (for
  falsely rejecting null hypothesis)
• Test statistic
• Critical value z?/2
• Reject H0 if z gt z?/2

70
Boat example - 10

• Large boat owners (d 1)
• Other boat owners (d 2)

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Boat example - 11

• Test whether domain means are equal at ? 0.05
• Calculate z-statistic
• Critical value z?/2 z0.25 1.96
• Apply rejection rule
• z -1.041.04 lt 1.96 z0.25
• Fail to reject H0

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Overview

• Population parameters
• Mean
• Total
• Proportion (w/ fixed denom)
• Ratio
• Includes proportion w/ random denominator
• Domain mean
• Domain total

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Overview 2

• Estimation strategies
• No auxiliary information
• Auxiliary information X, no intercept
• Y and X positively correlated
• Linear relationship passes through origin
• Auxiliary information X, intercept
• Y and X positively correlated
• Linear relationship does not pass through origin

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Overview 3

• Make a table of population parameters (rows) by
  estimation strategy (columns)
• In each cell, write down
• Estimator for population parameter
• Estimator for variance of estimated parameter
• Residual ei
• Notes
• Some cells will be blank
• Look for relationship between mean and total, and
  mean and proportion
• Look at how the variance formulas for many of the
  estimators are essentially the same form