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Ratio estimation under SRS

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Title: 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

6
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

11
Ag example 2
  • Estimate

0.9866 farm acres in 1992 relative to 1987 farm
acres
12
Ag example 3
  • Need to calculate variance of ei s

13
Ag example 4
  • For each county i, calculate
  • Coffee Co, AL example
  • Sum of squares for ei

14
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

16
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

17
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

20
Estimator for variance of
  • Estimator for variance of

21
Ag example 6

22
Ag example - 8

23
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

24
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)

27
Ag example 10

28
Estimating population total t
  • Estimator for t
  • Is biased?
  • Estimator for

29
Ag example 11

30
Summary of ratio estimation

31
Summary of ratio estn 2

32
Regression estimation
  • What if relationship between y and x is linear,
    but does NOT pass through the origin
  • Better model in this case is

33
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

34
Estimating population mean
  • Regression estimator
  • Estimating regression parameters

35
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

41
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

45
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
48
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

49
New population parameters
  • Domain mean
  • Domain total

50
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)

51
New population parameter 2
  • Ratio form of population mean
  • Numerator variable
  • Denominator variable

52
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
53
Boat example 4
54

Estimator for population domain mean
55
Boat example 5
  • Domain 1 data

56
Boat example 6
  • Domain 1 and domain 2 data combined

1104 zeros 76 zeros from domain 1 1028
zeros from domain 2
57
Boat example 7
  • Two ways of estimating mean

Whole data set
Domain 1 data only
58

Estimator for variance of
59
Boat example 8
60
Boat example 9
61
Approximation for estimator of variance of
Domain 1 data only
62
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

65

Estimator for variance of
66
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)

71
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

72
Overview
  • Population parameters
  • Mean
  • Total
  • Proportion (w/ fixed denom)
  • Ratio
  • Includes proportion w/ random denominator
  • Domain mean
  • Domain total

73
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

74
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
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