Estimate_Total_NHT: Narain-Horvitz-Thompson estimates for a total from survey...
In samplingEstimates: Sampling Estimates
Description
Usage
Arguments
Details
Value
References
Examples
Description
Produces estimates for population totals that are estimated using the Narain (1951); Horvitz-Thompson (1952) point estimator and from survey data obtained from a single-stage sampling design, i.e. direct element sampling.
Usage
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Arguments
MatY.s
matrix (dataframe or vector) with n rows (observations) and Q columns (variables of interest), where n is the overall sample size of elements. The argument MatY.s can also be a vector which is internally treated as a matrix with Q=1. There must not be any missing value.
VecWk.s
vector of the elements sampling weights; its length is equal to n, the sample size. Values in VecWk.s must be greater than or equal to one. Columns of MatY.s and length of VecWk.s must be the same. There must not be any missing value.
VarEst
string indicating the mathematical expression for estimating the variance. Available options are: "HT", "SYG" or "Hajek". If VarEst argument is omitted, the default is "SYG", which requires the provision of the matrix of joint inclusion probabilities MatPkl.s. In the case that the argument MatPkl.s is not provided, VarEst is set to "Hajek". When using VarEst="Hajek", it is assumed a high-entropy sampling design, see Hajek (1964); care should be taken with highly-stratified samples, e.g. Berger (2005).
MatPkl.s
matrix of the second-order inclusion probabilities; its number of rows and columns is equal to n, the overall sample size of observed elements. Values in MatPkl.s must be greater than zero and less than or equal to one. There must not be any missing value.
PopSize
population size N. This argument may be optional; if it is not provided the computations are made using \hat{N}=∑_{k\in s}w_k, which estimates the total of elements in the population.
VecStratLb.s
vector of the strata labels; its length is equal to n, the sample size. Values in the argument VecStratLb.s can be numeric (integers), strings or a factor. It does not need to be sorted, however, all other arguments (variables, vectors, matrices) of size n must follow the same order of VecStratLb.s correspondingly. This argument is optional, if it is not provided the computations are made assuming that there is no stratification. There must not be any missing value.
VecStratSize.s
vector of the strata population sizes; its length is equal to n, the sample size. This vector contains, for each of the n observations, the size of the stratum each observation belongs to. The argument VecStratSize.s does not need to be sorted, however, all other arguments (variables, vectors, matrices) of size n must follow the same order of VecStratSize.s correspondingly. There must not be any missing value.
ShowStrata
logical. If TRUE partial results from each stratum is displayed. This is an optional argument; default is FALSE.
VecDomainLb.s
vector of the domains (sub-groups) labels; its length is equal to n, the sample size. Values in the argument VecDomainLb.s can be numeric (integers) or strings (characters). They do not need to be sorted, however this variable (column) must follow the same order of VecStratLb.s correspondingly with the other variables, vectors or matrices of size n. This argument is optional and there must not be any missing value.
Details
For the population total of the variable y:
t = ∑_{k\in U} y_k
the unbiased Narain (1951); Horvitz-Thompson (1952) estimator of t is given by:
\hat{t}_{NHT} = ∑_{k\in s} w_k y_k
where w_k denotes the sampling weight of the k-th element in the sample s, w_k=1/π_k with π_k denoting the inclusion probability of the k-th element in the sample. Let π_{kl} denotes the joint-inclusion probabilities of the k-th and l-th elements in the sample s. The variance of \hat{t}_{NHT} is given by:
V(\hat{t}_{HT}) = ∑_{k\in U}∑_{l\in U} (π_{kl}-π_kπ_l)w_k y_k w_l y_l
which can therefore be estimated by the Horvitz-Thompson variance estimator (implemented by the current function if VarEst="HT"):
\hat{V}_{HT}(\hat{t}_{NHT}) = ∑_{k\in s}∑_{l\in s} \frac{π_{kl}-π_kπ_l}{π_{kl}}w_k y_k w_l y_l
If the utilised sampling design is of fixed-size, the variance V(\hat{t}_{NHT}) can be estimated by the Sen-Yates-Grundy variance estimator (implemented by the current function if VarEst="SYG"):
\hat{V}_{SYG}(\hat{t}_{NHT}) = \frac{-1}{2}∑_{k\in s}∑_{l\in s} \frac{π_{kl}-π_kπ_l}{π_{kl}}≤ft(w_k y_k - w_l y_l\right)^2
For large-entropy sampling designs, the variance of \hat{t}_{NHT} is approximated by the Hajek (1964) variance:
V_{Hajek}(\hat{t}_{NHT}) \doteq \frac{N}{N-1}≤ft[∑_{k\in U} w_k y_k^2≤ft(\frac{w_k-1}{w_k}\right)-dG^2\right]
with d=∑_{k\in U}w_k^{-2}(w_k-1) and G=d^{-1}∑_{k\in U}w_k^{-1}(w_k-1)y_k.
This approximate variance can therefore be estimated by the variance estimator (implemented by the current function if VarEst="Hajek"):
\hat{V}_{Hajek}(\hat{t}_{NHT}) = \frac{n}{n-1}≤ft[∑_{k\in s}w_k^2 y_k^2≤ft(\frac{w_k-1}{w_k}\right)-\hat{d}\hat{G}^2\right]
where \hat{d}=∑_{k\in s}w_k^{-1}(w_k-1) and \hat{G}=\hat{d}^{-1}∑_{k\in s}(w_k-1)y_k.
The Hajek (1964) variance approximation is designed for large-entropy sampling designs and large populations, i.e. care should be taken with highly-stratified samples, e.g. Berger (2005).
Value
The function returns a dataframe with Q rows (the number of variables of interest) and some columns depending on input information and used expressions in computations. The results in the returned columns are:
Statistic
the utilised point estimator.
VariableName
the name of the variable of interest.
Estimate
the point estimate obtained from evaluating the sample data.
Variance
the estimated variance of the point estimator.
StdErr
the estimated standard error of the point estimator.
AbsErr
the estimated absolute error of the point estimator.
LInfCI95
the lower limit of the 95 percent confidence interval.
LSupCI95
the upper limit of the 95 percent confidence interval.
Range95
the range (width) of the 95 percent confidence interval.
PctCVE
the estimated coefficient of variation (in percentage).
DEff
the estimated design effect.
n
the overall sample size.
Nhat
an estimate of the population size (number of elements in the population) \hat{N}=∑_{k\in s}w_k.
fhat
an estimate of the overall sampling fraction \hat{f}=n/\hat{N}.
N
the population size (total of elements in the population).
f
the overall sampling fraction.
If a stratified sampling design was specified and if ShowStrata=TRUE some further columns are displayed with partial results. Note that these per-stratum partial results are NOT returned by the function, they are only on-screen information.
h
stratum counter.
Stratum
stratum label (integer, character).
nh
the sample size for the stratum h.
Nh
the size of the stratum h (total of elements in the stratum h).
fh
the sampling fraction for the stratum h.
Wh
the relative weight of the stratum h among all strata W_{h}=n_{h}/N_{h}.
If domains of study were specified these extra columns are displayed. Note that these per-domain results are NOT returned by the function, they are only on-screen information.
d
domain counter.
Domain
domain label.
nd
the sample size in the domain d.
Ndhat
an estimate of the population size (number of elements) for the domain d.
fdhat
an estimate of the sampling fraction for the domain d.
Wdhat
an estimate of the relative weight of the domain d among all domains.
References
Berger, Y. G. (2005) Variance estimation with highly stratified sampling designs with unequal probabilities. Australian & New Zealand Journal of Statistics, 47, 365–373.
Hajek, J. (1964) Asymptotic theory of rejective sampling with varying probabilities from a finite population. The Annals of Mathematical Statistics, 35, 4, 1491–1523.
Horvitz, D. G. and Thompson, D. J. (1952) A generalization of sampling without replacement from a finite universe. Journal of the American Statistical Association, 47, 663–685.
Narain, R. D. (1951) On sampling without replacement with varying probabilities. Journal of the Indian Society of Agricultural Statistics, 3, 169–175.
Sen, A. R. (1953) On the estimate of the variance in sampling with varying probabilities. Journal of the Indian Society of Agricultural Statistics, 5, 119–127.
Yates, F. and Grundy, P. M. (1953) Selection without replacement from within strata with probability proportional to size. Journal of the Royal Statistical Society B, 15, 253–261.
Examples
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##################################
## Setting up data to run examples
##################################
data(Sample1) ## Loads a data frame with the sample to be used in examples
N <- 570 ## Defining the population size
## Approximating the 2nd order inclusion probabilities with sample based quantitites
## (Note: this approximation is only suitable for large-entropy sampling designs)
require(samplingVarEst) ## Loading the necessary package
Probs2Mat <- Pkl.Hajek.s(Sample1$InclProbs) ## function from samplingVarEst package
head(Sample1) ## Showing the first rows of the sample data to be used
############################################################
## Example 1: A variable of interest, without stratification
############################################################
Estimate.Total.NHT(MatY.s = Sample1$y1 ,
VecWk.s = Sample1$Weights)
Estimate.Total.NHT(MatY.s = Sample1$y1 ,
VecWk.s = Sample1$Weights,
VarEst = "HT" )
Estimate.Total.NHT(MatY.s = Sample1$y1 ,
VecWk.s = Sample1$Weights,
VarEst = "SYG" ,
MatPkl.s = Probs2Mat )
Estimate.Total.NHT(MatY.s = Sample1$y1 ,
VecWk.s = Sample1$Weights,
VarEst = "SYG" ,
MatPkl.s = Probs2Mat ,
PopSize = N )
###################################################################################
## Example 2: A matrix/dataframe of 2 variables of interest, without stratification
###################################################################################
Estimate.Total.NHT(MatY.s = Sample1[ ,c("y1","y2")],
VecWk.s = Sample1$Weights ,
VarEst = "SYG" ,
MatPkl.s = Probs2Mat ,
PopSize = N )
#########################################################
## Example 3: A variable of interest, with stratification
#########################################################
Estimate.Total.NHT(MatY.s = Sample1$y1 ,
VecWk.s = Sample1$Weights ,
VecStratLb.s = Sample1$CharStrataNames,
VecStratSize.s = Sample1$StrataSizes )
Estimate.Total.NHT(MatY.s = Sample1$y1 ,
VecWk.s = Sample1$Weights ,
VecStratLb.s = Sample1$CharStrataNames,
VecStratSize.s = Sample1$StrataSizes ,
ShowStrata = TRUE )
###############################################################################
## Example 4: A matrix/dataframe (2 variables of interest), with stratification
###############################################################################
Estimate.Total.NHT(MatY.s = Sample1[ ,c("y1","y2")],
VecWk.s = Sample1$Weights ,
VecStratLb.s = Sample1$CharStrataNames,
VecStratSize.s = Sample1$StrataSizes ,
ShowStrata = TRUE )
#################################################################################
## Example 5: A matrix/dataframe (2 variables), no strata, with unplanned domains
#################################################################################
Estimate.Total.NHT(MatY.s = Sample1[ ,c("y1","y2")],
VecWk.s = Sample1$Weights ,
VecDomainLb.s = Sample1$CharDoms )
Estimate.Total.NHT(MatY.s = Sample1[ ,c("y1","y2")],
VecWk.s = Sample1$Weights ,
VecDomainLb.s = Sample1$NumDoms )
###################################################################################
## Example 6: A matrix/dataframe (2 variables), with strata, with unplanned domains
###################################################################################
Estimate.Total.NHT(MatY.s = Sample1[ ,c("y1","y2")],
VecWk.s = Sample1$Weights ,
VecStratLb.s = Sample1$CharStrataNames,
VecStratSize.s = Sample1$StrataSizes ,
ShowStrata = TRUE ,
VecDomainLb.s = Sample1$CharDoms )
samplingEstimates documentation built on May 2, 2019, 2 a.m.
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Description Usage Arguments Details Value References Examples
Description
Produces estimates for population totals that are estimated using the Narain (1951); Horvitz-Thompson (1952) point estimator and from survey data obtained from a single-stage sampling design, i.e. direct element sampling.
Usage
1 2 3 4 5 6 7 8 9 |
Arguments
MatY.s |
matrix (dataframe or vector) with n rows (observations) and Q columns (variables of interest), where n is the overall sample size of elements. The argument |
VecWk.s |
vector of the elements sampling weights; its length is equal to n, the sample size. Values in |
VarEst |
string indicating the mathematical expression for estimating the variance. Available options are: |
MatPkl.s |
matrix of the second-order inclusion probabilities; its number of rows and columns is equal to n, the overall sample size of observed elements. Values in |
PopSize |
population size N. This argument may be optional; if it is not provided the computations are made using \hat{N}=∑_{k\in s}w_k, which estimates the total of elements in the population. |
VecStratLb.s |
vector of the strata labels; its length is equal to n, the sample size. Values in the argument |
VecStratSize.s |
vector of the strata population sizes; its length is equal to n, the sample size. This vector contains, for each of the n observations, the size of the stratum each observation belongs to. The argument |
ShowStrata |
logical. If |
VecDomainLb.s |
vector of the domains (sub-groups) labels; its length is equal to n, the sample size. Values in the argument |
Details
For the population total of the variable y:
t = ∑_{k\in U} y_k
the unbiased Narain (1951); Horvitz-Thompson (1952) estimator of t is given by:
\hat{t}_{NHT} = ∑_{k\in s} w_k y_k
where w_k denotes the sampling weight of the k-th element in the sample s, w_k=1/π_k with π_k denoting the inclusion probability of the k-th element in the sample. Let π_{kl} denotes the joint-inclusion probabilities of the k-th and l-th elements in the sample s. The variance of \hat{t}_{NHT} is given by:
V(\hat{t}_{HT}) = ∑_{k\in U}∑_{l\in U} (π_{kl}-π_kπ_l)w_k y_k w_l y_l
which can therefore be estimated by the Horvitz-Thompson variance estimator (implemented by the current function if VarEst="HT"):
\hat{V}_{HT}(\hat{t}_{NHT}) = ∑_{k\in s}∑_{l\in s} \frac{π_{kl}-π_kπ_l}{π_{kl}}w_k y_k w_l y_l
If the utilised sampling design is of fixed-size, the variance V(\hat{t}_{NHT}) can be estimated by the Sen-Yates-Grundy variance estimator (implemented by the current function if VarEst="SYG"):
\hat{V}_{SYG}(\hat{t}_{NHT}) = \frac{-1}{2}∑_{k\in s}∑_{l\in s} \frac{π_{kl}-π_kπ_l}{π_{kl}}≤ft(w_k y_k - w_l y_l\right)^2
For large-entropy sampling designs, the variance of \hat{t}_{NHT} is approximated by the Hajek (1964) variance:
V_{Hajek}(\hat{t}_{NHT}) \doteq \frac{N}{N-1}≤ft[∑_{k\in U} w_k y_k^2≤ft(\frac{w_k-1}{w_k}\right)-dG^2\right]
with d=∑_{k\in U}w_k^{-2}(w_k-1) and G=d^{-1}∑_{k\in U}w_k^{-1}(w_k-1)y_k.
This approximate variance can therefore be estimated by the variance estimator (implemented by the current function if VarEst="Hajek"):
\hat{V}_{Hajek}(\hat{t}_{NHT}) = \frac{n}{n-1}≤ft[∑_{k\in s}w_k^2 y_k^2≤ft(\frac{w_k-1}{w_k}\right)-\hat{d}\hat{G}^2\right]
where \hat{d}=∑_{k\in s}w_k^{-1}(w_k-1) and \hat{G}=\hat{d}^{-1}∑_{k\in s}(w_k-1)y_k.
The Hajek (1964) variance approximation is designed for large-entropy sampling designs and large populations, i.e. care should be taken with highly-stratified samples, e.g. Berger (2005).
Value
The function returns a dataframe with Q rows (the number of variables of interest) and some columns depending on input information and used expressions in computations. The results in the returned columns are:
Statistic |
the utilised point estimator. |
VariableName |
the name of the variable of interest. |
Estimate |
the point estimate obtained from evaluating the sample data. |
Variance |
the estimated variance of the point estimator. |
StdErr |
the estimated standard error of the point estimator. |
AbsErr |
the estimated absolute error of the point estimator. |
LInfCI95 |
the lower limit of the 95 percent confidence interval. |
LSupCI95 |
the upper limit of the 95 percent confidence interval. |
Range95 |
the range (width) of the 95 percent confidence interval. |
PctCVE |
the estimated coefficient of variation (in percentage). |
DEff |
the estimated design effect. |
n |
the overall sample size. |
Nhat |
an estimate of the population size (number of elements in the population) \hat{N}=∑_{k\in s}w_k. |
fhat |
an estimate of the overall sampling fraction \hat{f}=n/\hat{N}. |
N |
the population size (total of elements in the population). |
f |
the overall sampling fraction. |
If a stratified sampling design was specified and if ShowStrata=TRUE some further columns are displayed with partial results. Note that these per-stratum partial results are NOT returned by the function, they are only on-screen information.
h |
stratum counter. |
Stratum |
stratum label (integer, character). |
nh |
the sample size for the stratum h. |
Nh |
the size of the stratum h (total of elements in the stratum h). |
fh |
the sampling fraction for the stratum h. |
Wh |
the relative weight of the stratum h among all strata W_{h}=n_{h}/N_{h}. |
If domains of study were specified these extra columns are displayed. Note that these per-domain results are NOT returned by the function, they are only on-screen information.
d |
domain counter. |
Domain |
domain label. |
nd |
the sample size in the domain d. |
Ndhat |
an estimate of the population size (number of elements) for the domain d. |
fdhat |
an estimate of the sampling fraction for the domain d. |
Wdhat |
an estimate of the relative weight of the domain d among all domains. |
References
Berger, Y. G. (2005) Variance estimation with highly stratified sampling designs with unequal probabilities. Australian & New Zealand Journal of Statistics, 47, 365–373.
Hajek, J. (1964) Asymptotic theory of rejective sampling with varying probabilities from a finite population. The Annals of Mathematical Statistics, 35, 4, 1491–1523.
Horvitz, D. G. and Thompson, D. J. (1952) A generalization of sampling without replacement from a finite universe. Journal of the American Statistical Association, 47, 663–685.
Narain, R. D. (1951) On sampling without replacement with varying probabilities. Journal of the Indian Society of Agricultural Statistics, 3, 169–175.
Sen, A. R. (1953) On the estimate of the variance in sampling with varying probabilities. Journal of the Indian Society of Agricultural Statistics, 5, 119–127.
Yates, F. and Grundy, P. M. (1953) Selection without replacement from within strata with probability proportional to size. Journal of the Royal Statistical Society B, 15, 253–261.
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 | ##################################
## Setting up data to run examples
##################################
data(Sample1) ## Loads a data frame with the sample to be used in examples
N <- 570 ## Defining the population size
## Approximating the 2nd order inclusion probabilities with sample based quantitites
## (Note: this approximation is only suitable for large-entropy sampling designs)
require(samplingVarEst) ## Loading the necessary package
Probs2Mat <- Pkl.Hajek.s(Sample1$InclProbs) ## function from samplingVarEst package
head(Sample1) ## Showing the first rows of the sample data to be used
############################################################
## Example 1: A variable of interest, without stratification
############################################################
Estimate.Total.NHT(MatY.s = Sample1$y1 ,
VecWk.s = Sample1$Weights)
Estimate.Total.NHT(MatY.s = Sample1$y1 ,
VecWk.s = Sample1$Weights,
VarEst = "HT" )
Estimate.Total.NHT(MatY.s = Sample1$y1 ,
VecWk.s = Sample1$Weights,
VarEst = "SYG" ,
MatPkl.s = Probs2Mat )
Estimate.Total.NHT(MatY.s = Sample1$y1 ,
VecWk.s = Sample1$Weights,
VarEst = "SYG" ,
MatPkl.s = Probs2Mat ,
PopSize = N )
###################################################################################
## Example 2: A matrix/dataframe of 2 variables of interest, without stratification
###################################################################################
Estimate.Total.NHT(MatY.s = Sample1[ ,c("y1","y2")],
VecWk.s = Sample1$Weights ,
VarEst = "SYG" ,
MatPkl.s = Probs2Mat ,
PopSize = N )
#########################################################
## Example 3: A variable of interest, with stratification
#########################################################
Estimate.Total.NHT(MatY.s = Sample1$y1 ,
VecWk.s = Sample1$Weights ,
VecStratLb.s = Sample1$CharStrataNames,
VecStratSize.s = Sample1$StrataSizes )
Estimate.Total.NHT(MatY.s = Sample1$y1 ,
VecWk.s = Sample1$Weights ,
VecStratLb.s = Sample1$CharStrataNames,
VecStratSize.s = Sample1$StrataSizes ,
ShowStrata = TRUE )
###############################################################################
## Example 4: A matrix/dataframe (2 variables of interest), with stratification
###############################################################################
Estimate.Total.NHT(MatY.s = Sample1[ ,c("y1","y2")],
VecWk.s = Sample1$Weights ,
VecStratLb.s = Sample1$CharStrataNames,
VecStratSize.s = Sample1$StrataSizes ,
ShowStrata = TRUE )
#################################################################################
## Example 5: A matrix/dataframe (2 variables), no strata, with unplanned domains
#################################################################################
Estimate.Total.NHT(MatY.s = Sample1[ ,c("y1","y2")],
VecWk.s = Sample1$Weights ,
VecDomainLb.s = Sample1$CharDoms )
Estimate.Total.NHT(MatY.s = Sample1[ ,c("y1","y2")],
VecWk.s = Sample1$Weights ,
VecDomainLb.s = Sample1$NumDoms )
###################################################################################
## Example 6: A matrix/dataframe (2 variables), with strata, with unplanned domains
###################################################################################
Estimate.Total.NHT(MatY.s = Sample1[ ,c("y1","y2")],
VecWk.s = Sample1$Weights ,
VecStratLb.s = Sample1$CharStrataNames,
VecStratSize.s = Sample1$StrataSizes ,
ShowStrata = TRUE ,
VecDomainLb.s = Sample1$CharDoms )
|
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