Package 'Frames2'

Bankier-Kalton-Anderson estimator

Description

Produces estimates for population total and mean using the Bankier-Kalton-Anderson estimator from survey data obtained from a dual frame sampling design. Confidence intervals are also computed, if required.

Usage

BKA(ysA, ysB, pi_A, pi_B, pik_ab_B, pik_ba_A, domains_A, domains_B, 
conf_level = NULL)

Arguments

ysA	A numeric vector of length $n_A$ or a numeric matrix or data frame of dimensions $n_A$ x $c$ containing information about variable(s) of interest from $s_A$.
ysB	A numeric vector of length $n_B$ or a numeric matrix or data frame of dimensions $n_B$ x $c$ containing information about variable(s) of interest from $s_B$.
pi_A	A numeric vector of length $n_A$ or a square numeric matrix of dimension $n_A$ containing first order or first and second order inclusion probabilities for units included in $s_A$.
pi_B	A numeric vector of length $n_B$ or a square numeric matrix of dimension $n_B$ containing first order or first and second order inclusion probabilities for units included in $s_B$.
pik_ab_B	A numeric vector of size $n_A$ containing first order inclusion probabilities according to sampling design in frame B for units belonging to overlap domain that have been selected in $s_A$.
pik_ba_A	A numeric vector of size $n_B$ containing first order inclusion probabilities according to sampling design in frame A for units belonging to overlap domain that have been selected in $s_B$.
domains_A	A character vector of size $n_A$ indicating the domain each unit from $s_A$ belongs to. Possible values are "a" and "ab".
domains_B	A character vector of size $n_B$ indicating the domain each unit from $s_B$ belongs to. Possible values are "b" and "ba".
conf_level	(Optional) A numeric value indicating the confidence level for the confidence intervals, if desired.

Details

BKA estimator of population total is given by

$\hat{Y}_{BKA} = \sum_{i \in s_A}\tilde{d}_i^Ay_i + \sum_{i \in s_B}\tilde{d}_i^By_i$

where $\tilde{d}_i^A =\left\{\begin{array}{lcc} d_i^A & \textrm{if } i \in a\\ (1/d_i^A + 1/d_i^B)^{-1} & \textrm{if } i \in ab \end{array} \right.$ and $\tilde{d}_i^B =\left\{\begin{array}{lcc} d_i^B & \textrm{if } i \in b\\ (1/d_i^A + 1/d_i^B)^{-1} & \textrm{if } i \in ba \end{array} \right.$ being $d_i^A$ and $d_i^B$ the design weights, obtained as the inverse of the first order inclusion probabilities, that is, $d_i^A = 1/\pi_i^A$ and $d_i^B = 1/\pi_i^B$.

To estimate variance of this estimator, one uses following approach proposed by Rao and Skinner (1996)

$\hat{V}(\hat{Y}_{BKA}) = \hat{V}(\sum_{i \in s_A}\tilde{z}_i^A) + \hat{V}(\sum_{i \in s_B}\tilde{z}_i^B)$

with $\tilde{z}_i^A = \delta_i(a)y_i + (1 - \delta_i(a))y_i\pi_i^A/(\pi_i^A + \pi_i^B)$ and $\tilde{z}_i^B = \delta_i(b)y_i + (1 - \delta_i(b))y_i\pi_i^B/(\pi_i^A + \pi_i^B)$, being $\delta_i(a)$ and $\delta_i(b)$ the indicator variables for domain $a$ and domain $b$, respectively. If both first and second order probabilities are known, variances and covariances involved in calculation of $\hat{\beta}$ and $\hat{V}(\hat{Y}_{FB})$ are estimated using functions VarHT and CovHT, respectively. If only first order probabilities are known, variances are estimated using Deville's method and covariances are estimated using following expression

$\widehat{Cov}(\hat{X}, \hat{Y}) = \frac{\hat{V}(X + Y) - \hat{V}(X) - \hat{V}(Y)}{2}$

Value

BKA returns an object of class "EstimatorDF" which is a list with, at least, the following components:

Call	the matched call.
Est	total and mean estimation for main variable(s).
VarEst	variance estimation for main variable(s).

If parameter conf_level is different from NULL, object includes component

ConfInt

total and mean estimation and confidence intervals for main variables(s).

In addition, components TotDomEst and MeanDomEst are available when estimator is based on estimators of the domains. Component Param shows value of parameters involded in calculation of the estimator (if any). By default, only Est component (or ConfInt component, if parameter conf_level is different from NULL) is shown. It is possible to access to all the components of the objects by using function summary.

References

Bankier, M. D. (1986) Estimators Based on Several Stratified Samples With Applications to Multiple Frame Surveys. Journal of the American Statistical Association, Vol. 81, 1074 - 1079.

Kalton, G. and Anderson, D. W. (1986) Sampling Rare Populations. Journal of the Royal Statistical Society, Ser. A, Vol. 149, 65 - 82.

Rao, J. N. K. and Skinner, C. J. (1996) Estimation in Dual Frame Surveys with Complex Designs. Proceedings of the Survey Method Section, Statistical Society of Canada, 63 - 68.

Skinner, C. J. and Rao, J. N. K. (1996) Estimation in Dual Frame Surveys with Complex Designs. Journal of the American Statistical Association, Vol. 91, 433, 349 - 356.

See Also

JackBKA

Examples

data(DatA)
data(DatB)
data(PiklA)
data(PiklB)

#Let calculate BKA estimator for population total for variable Leisure
BKA(DatA$Lei, DatB$Lei, PiklA, PiklB, DatA$ProbB, DatB$ProbA, 
DatA$Domain, DatB$Domain)

#Now, let calculate BKA estimator and a 90% confidence interval for population 
#total for variable Feeding considering only first order inclusion probabilities
BKA(DatA$Feed, DatB$Feed, DatA$ProbA, DatB$ProbB, DatA$ProbB, 
DatB$ProbA, DatA$Domain, DatB$Domain, 0.90)

---

DF calibration estimator

Description

Produces estimates for population totals and means using the DF calibration estimator from survey data obtained from a dual frame sampling design. Confidence intervals are also computed, if required.

Usage

CalDF(ysA, ysB, pi_A, pi_B, domains_A, domains_B, N_A = NULL, N_B = NULL, 
N_ab = NULL, xsAFrameA = NULL, xsBFrameA = NULL, xsAFrameB = NULL, xsBFrameB = NULL, 
xsT = NULL, XA = NULL, XB = NULL, X = NULL, met = "linear", conf_level = NULL)

Arguments

ysA	A numeric vector of length $n_A$ or a numeric matrix or data frame of dimensions $n_A$ x $c$ containing information about variable(s) of interest from $s_A$.
ysB	A numeric vector of length $n_B$ or a numeric matrix or data frame of dimensions $n_B$ x $c$ containing information about variable(s) of interest from $s_B$.
pi_A	A numeric vector of length $n_A$ or a square numeric matrix of dimension $n_A$ containing first order or first and second order inclusion probabilities for units included in $s_A$.
pi_B	A numeric vector of length $n_B$ or a square numeric matrix of dimension $n_B$ containing first order or first and second order inclusion probabilities for units included in $s_B$.
domains_A	A character vector of length $n_A$ indicating the domain each unit from $s_A$ belongs to. Possible values are "a" and "ab".
domains_B	A character vector of length $n_B$ indicating the domain each unit from $s_B$ belongs to. Possible values are "b" and "ba".
N_A	(Optional) A numeric value indicating the size of frame A.
N_B	(Optional) A numeric value indicating the size of frame B.
N_ab	(Optional) A numeric value indicating the size of the overlap domain.
xsAFrameA	(Optional) A numeric vector of length $n_A$ or a numeric matrix or data frame of dimensions $n_A$ x $m_A$, with $m_A$ the number of auxiliary variables in frame A, containing auxiliary information in frame A for units included in $s_A$.
xsBFrameA	(Optional) A numeric vector of length $n_B$ or a numeric matrix or data frame of dimensions $n_B$ x $m_A$, with $m_A$ the number of auxiliary variables in frame A, containing auxiliary information in frame A for units included in $s_B$. For units in domain $b$, these values are 0.
xsAFrameB	(Optional) A numeric vector of length $n_A$ or a numeric matrix or data frame of dimensions $n_A$ x $m_B$, with $m_B$ the number of auxiliary variables in frame B, containing auxiliary information in frame B for units included in $s_A$. For units in domain $a$, these values are 0.
xsBFrameB	(Optional) A numeric vector of length $n_B$ or a numeric matrix or data frame of dimensions $n_B$ x $m_B$, with $m_B$ the number of auxiliary variables in frame B, containing auxiliary information in frame B for units included in $s_B$.
xsT	(Optional) A numeric vector of length $n$ or a numeric matrix or data frame of dimensions $n$ x $m_T$, with $m_T$ the number of auxiliary variables in both frames, containing auxiliary information for all units in the entire sample $s = s_A \cup s_B$.
XA	(Optional) A numeric value or vector of length $m_A$, with $m_A$ the number of auxiliary variables in frame A, indicating the population totals for the auxiliary variables considered in frame A.
XB	(Optional) A numeric value or vector of length $m_B$, with $m_B$ the number of auxiliary variables in frame B, indicating the population totals for the auxiliary variables considered in frame B.
X	(Optional) A numeric value or vector of length $m_T$, with $m_T$ the number of auxiliary variables in both frames, indicating the population totals for the auxiliary variables considered in both frames.
met	(Optional) A character vector indicating the distance that must be used in calibration process. Possible values are "linear", "raking" and "logit". Default is "linear".
conf_level	(Optional) A numeric value indicating the confidence level for the confidence intervals, if desired.

Details

DF calibration estimator of population total is given by

$\hat{Y}_{CalDF} = \hat{Y}_a + \hat{\eta}\hat{Y}_{ab} + \hat{Y}_b + (1 - \hat{\eta})\hat{Y}_{ba}$

where $\hat{Y}_a = \sum_{i \in s_a}\tilde{d}_i y_i, \hat{Y}_{ab} = \sum_{i \in s_{ab}}\tilde{d}_i y_i$, $\hat{Y}_b = \sum_{i \in s_b}\tilde{d}_i y_i$ and $\hat{Y}_{ba} = \sum_{i \in s_{ba}}\tilde{d}_i y_i$, with $\tilde{d}_i$ calibration weights which are calculated having into account a different set of constraints, depending on the case. For instance, if $N_A, N_B$ and $N_{ab}$ are all known and no other auxiliary information is available, calibration constraints are

$\sum_{i \in s_a}\tilde{d}_i = N_a, \sum_{i \in s_{ab}}\tilde{d}_i = N_{ab}, \sum_{i \in s_{ba}}\tilde{d}_i = N_{ba}, \sum_{i \in s_b}\tilde{d}_i = N_b$

Optimal value for $\hat{\eta}$ to minimice variance of the estimator is given by $\hat{V}(\hat{N}_{ba})/(\hat{V}(\hat{N}_{ab}) + \hat{V}(\hat{N}_{ba}))$. If both first and second order probabilities are known, variances are estimated using function VarHT. If only first order probabilities are known, variances are estimated using Deville's method.

Function covers following scenarios:

• There is not any additional auxiliary variable

  • $N_A, N_B$ and $N_{ab}$ unknown

  • $N_A$ and $N_B$ known and $N_{ab}$ unknown

  • $N_{ab}$ known and $N_A$ and $N_B$ unknown

  • $N_A, N_B$ and $N_{ab}$ known

• At least, information about one additional auxiliary variable is available

  • $N_A$ and $N_B$ known and $N_{ab}$ unknown

  • $N_{ab}$ known and $N_A$ and $N_B$ unknown

  • $N_A, N_B$ and $N_{ab}$ known

To obtain an estimator of the variance for this estimator, one can use Deville's expression

$\hat{V}(\hat{Y}_{CalDF}) = \frac{1}{1-\sum_{k\in s} a_k^2}\sum_{k\in s}(1-\pi_k)\left(\frac{e_k}{\pi_k} - \sum_{l\in s} a_{l} \frac{e_l}{\pi_l}\right)^2$

where $a_k=(1-\pi_k)/\sum_{l\in s} (1-\pi_l)$ and $e_k$ are the residuals of the regression with auxiliary variables as regressors.

Value

CalDF returns an object of class "EstimatorDF" which is a list with, at least, the following components:

Call	the matched call.
Est	total and mean estimation for main variable(s).
VarEst	variance estimation for main variable(s).

If parameter conf_level is different from NULL, object includes component

ConfInt

total and mean estimation and confidence intervals for main variables(s).

In addition, components TotDomEst and MeanDomEst are available when estimator is based on estimators of the domains. Component Param shows value of parameters involded in calculation of the estimator (if any). By default, only Est component (or ConfInt component, if parameter conf_level is different from NULL) is shown. It is possible to access to all the components of the objects by using function summary.

References

Ranalli, M. G., Arcos, A., Rueda, M. and Teodoro, A. (2013) Calibration estimation in dual frame surveys. arXiv:1312.0761 [stat.ME]

Deville, J. C., Sarndal, C. E. (1992) Calibration estimators in survey sampling. Journal of the American Statistical Association, 87, 376 - 382

See Also

JackCalDF

Examples

data(DatA)
data(DatB)
data(PiklA)
data(PiklB)

#Let calculate DF calibration estimator for variable Feeding, without
#considering any auxiliary information
CalDF(DatA$Feed, DatB$Feed, PiklA, PiklB, DatA$Domain, DatB$Domain)

#Now, let calculate DF calibration estimator for variable Clothing when the frame
#sizes and the overlap domain size are known
CalDF(DatA$Clo, DatB$Clo, PiklA, PiklB, DatA$Domain, DatB$Domain, 
N_A = 1735, N_B = 1191, N_ab = 601)

#Finally, let calculate DF calibration estimator and a 90% confidence interval
#for population total for variable Feeding, considering Income as auxiliary variable in 
#frame A and Metres2 as auxiliary variable in frame B and with frame sizes and overlap 
#domain size known.
CalDF(DatA$Feed, DatB$Feed, PiklA, PiklB, DatA$Domain, DatB$Domain, 
N_A = 1735, N_B =  1191, N_ab = 601, xsAFrameA = DatA$Inc, xsBFrameA = DatB$Inc, 
xsAFrameB = DatA$M2, xsBFrameB = DatB$M2, XA = 4300260, XB = 176553, 
conf_level = 0.90)

---

SF calibration estimator

Description

Produces estimates for population totals and means using the SF calibration estimator from survey data obtained from a dual frame sampling design. Confidence intervals are also computed, if required.

Usage

CalSF(ysA, ysB, pi_A, pi_B, pik_ab_B, pik_ba_A, domains_A, domains_B, N_A = NULL,
N_B = NULL, N_ab = NULL, xsAFrameA = NULL, xsBFrameA = NULL, xsAFrameB = NULL, 
xsBFrameB = NULL, xsT = NULL, XA = NULL, XB = NULL, X = NULL, met = "linear", 
conf_level = NULL)

Arguments

ysA	A numeric vector of length $n_A$ or a numeric matrix or data frame of dimensions $n_A$ x $c$ containing information about variable(s) of interest from $s_A$.
ysB	A numeric vector of length $n_B$ or a numeric matrix or data frame of dimensions $n_B$ x $c$ containing information about variable(s) of interest from $s_B$.
pi_A	A numeric vector of length $n_A$ or a square numeric matrix of dimension $n_A$ containing first order or first and second order inclusion probabilities for units included in $s_A$.
pi_B	A numeric vector of length $n_B$ or a square numeric matrix of dimension $n_B$ containing first order or first and second order inclusion probabilities for units included in $s_B$.
pik_ab_B	A numeric vector of size $n_A$ containing first order inclusion probabilities according to sampling desing in frame B for units belonging to overlap domain that have been selected in $s_A$.
pik_ba_A	A numeric vector of size $n_B$ containing first order inclusion probabilities according to sampling desing in frame A for units belonging to overlap domain that have been selected in $s_B$.
domains_A	A character vector of size $n_A$ indicating the domain each unit from $s_A$ belongs to. Possible values are "a" and "ab".
domains_B	A character vector of size $n_B$ indicating the domain each unit from $s_B$ belongs to. Possible values are "b" and "ba".
N_A	(Optional) A numeric value indicating the size of frame A
N_B	(Optional) A numeric value indicating the size of frame B
N_ab	(Optional) A numeric value indicating the size of the overlap domain
xsAFrameA	(Optional) A numeric vector of length $n_A$ or a numeric matrix or data frame of dimensions $n_A$ x $m_A$, with $m_A$ the number of auxiliary variables in frame A, containing auxiliary information in frame A for units included in $s_A$.
xsBFrameA	(Optional) A numeric vector of length $n_B$ or a numeric matrix or data frame of dimensions $n_B$ x $m_A$, with $m_A$ the number of auxiliary variables in frame A, containing auxiliary information in frame A for units included in $s_B$. For units in domain $b$, these values are 0.
xsAFrameB	(Optional) A numeric vector of length $n_A$ or a numeric matrix or data frame of dimensions $n_A$ x $m_B$, with $m_B$ the number of auxiliary variables in frame B, containing auxiliary information in frame B for units included in $s_A$. For units in domain $a$, these values are 0.
xsBFrameB	(Optional) A numeric vector of length $n_B$ or a numeric matrix or data frame of dimensions $n_B$ x $m_B$, with $m_B$ the number of auxiliary variables in frame B, containing auxiliary information in frame B for units included in $s_B$.
xsT	(Optional) A numeric vector of length $n$ or a numeric matrix or data frame of dimensions $n$ x $m_T$, with $m_T$ the number of auxiliary variables in both frames, containing auxiliary information for all units in the entire sample $s = s_A \cup s_B$.
XA	(Optional) A numeric value or vector of length $m_A$, with $m_A$ the number of auxiliary variables in frame A, indicating the population totals for the auxiliary variables considered in frame A.
XB	(Optional) A numeric value or vector of length $m_B$, with $m_B$ the number of auxiliary variables in frame B, indicating the population totals for the auxiliary variables considered in frame B.
X	(Optional) A numeric value or vector of length $m_T$, with $m_T$ the number of auxiliary variables in both frames, indicating the population totals for the auxiliary variables considered in both frames.
met	(Optional) A character vector indicating the distance that must be used in calibration process. Possible values are "linear", "raking" and "logit". Default is "linear".
conf_level	(Optional) A numeric value indicating the confidence level for the confidence intervals, if desired.

Details

SF calibration estimator of population total is given by

$\hat{Y}_{CalSF} = \hat{Y}_a + \hat{Y}_{ab} + \hat{Y}_b$

where $\hat{Y}_a = \sum_{i \in s_a}\tilde{d}_i y_i, \hat{Y}_{ab} = \sum_{i \in (s_{ab} \cup s_{ba})}\tilde{d}_i y_i$ and $\hat{Y}_b = \sum_{i \in s_b} \tilde{d}_i y_i$, with $\tilde{d}_i$ calibration weights which are calculated having into account a different set of constraints, depending on the case. For instance, if $N_A, N_B$ and $N_{ab}$ are known and no other auxiliary information is available, calibration constraints are

$\sum_{i \in s_a}\tilde{d}_i = N_a, \sum_{i \in s_{ab} \cup s_{ba}}\tilde{d}_i = N_{ab}, \sum_{i \in s_{ba}}\tilde{d}_i = N_{ba}$

Function covers following scenarios:

• There is not any additional auxiliary variable

  • $N_A, N_B$ and $N_{ab}$ unknown

  • $N_{ab}$ known and $N_A$ and $N_B$ unknown

  • $N_A$ and $N_B$ known and $N_{ab}$ unknown

  • $N_A, N_B$ and $N_{ab}$ known

• At least, information about one additional auxiliary variable is available

  • $N_{ab}$ known and $N_A$ and $N_B$ unknown

  • $N_A$ and $N_B$ known and $N_{ab}$ unknown

  • $N_A, N_B$ and $N_{ab}$ known

To obtain an estimator of the variance for this estimator, one can use Deville's expression

$\hat{V}(\hat{Y}_{CalSF}) = \frac{1}{1-\sum_{k\in s} a_k^2}\sum_{k\in s}(1-\pi_k)\left(\frac{e_k}{\pi_k} - \sum_{l\in s} a_{l} \frac{e_l}{\pi_l}\right)^2$

where $a_k=(1-\pi_k)/\sum_{l\in s} (1-\pi_l)$ and $e_k$ are the residuals of the regression with auxiliary variables as regressors.

Value

CalSF returns an object of class "EstimatorDF" which is a list with, at least, the following components:

Call	the matched call.
Est	total and mean estimation for main variable(s).
VarEst	variance estimation for main variable(s).

If parameter conf_level is different from NULL, object includes component

ConfInt

total and mean estimation and confidence intervals for main variables(s).

In addition, components TotDomEst and MeanDomEst are available when estimator is based on estimators of the domains. Component Param shows value of parameters involded in calculation of the estimator (if any). By default, only Est component (or ConfInt component, if parameter conf_level is different from NULL) is shown. It is possible to access to all the components of the objects by using function summary.

References

Ranalli, M. G., Arcos, A., Rueda, M. and Teodoro, A. (2013) Calibration estimation in dual frame surveys. arXiv:1312.0761 [stat.ME]

Deville, J. C., Sarndal, C. E. (1992) Calibration estimators in survey sampling. Journal of the American Statistical Association, 87, 376 - 382

See Also

JackCalSF

Examples

data(DatA)
data(DatB)
data(PiklA)
data(PiklB)

#Let calculate SF calibration estimator for variable Clothing, without
#considering any auxiliary information
CalSF(DatA$Clo, DatB$Clo, PiklA, PiklB, DatA$ProbB, DatB$ProbA, 
DatA$Domain, DatB$Domain)

#Now, let calculate SF calibration estimator for variable Leisure when the frame
#sizes and the overlap domain size are known
CalSF(DatA$Lei, DatB$Lei, PiklA, PiklB, DatA$ProbB, DatB$ProbA, DatA$Domain, 
DatB$Domain, N_A = 1735, N_B = 1191, N_ab = 601)

#Finally, let calculate SF calibration estimator and a 90% confidence interval
#for population total for variable Feeding, considering Income and Metres2 as auxiliary 
#variables and with frame sizes and overlap domain size known.
CalSF(DatA$Feed, DatB$Feed, PiklA, PiklB, DatA$ProbB, DatB$ProbA, DatA$Domain, 
DatB$Domain, N_A = 1735, N_B =  1191, N_ab = 601, xsAFrameA = DatA$Inc, 
xsBFrameA = DatB$Inc, xsAFrameB = DatA$M2, xsBFrameB = DatB$M2, 
XA = 4300260, XB = 176553, conf_level = 0.90)

---

Summary of estimators

Description

Returns all possible estimators that can be computed according to the information provided

Usage

Compare(ysA, ysB, pi_A, pi_B, domains_A, domains_B, pik_ab_B = NULL, pik_ba_A = NULL, 
N_A = NULL, N_B = NULL, N_ab = NULL, xsAFrameA = NULL, xsBFrameA = NULL,  
xsAFrameB = NULL, xsBFrameB = NULL, XA = NULL, XB = NULL, met = "linear", 
conf_level = NULL)

Arguments

ysA	A numeric vector of length $n_A$ or a numeric matrix or data frame of dimensions $n_A$ x $c$ containing information about variable(s) of interest from $s_A$.
ysB	A numeric vector of length $n_B$ or a numeric matrix or data frame of dimensions $n_B$ x $c$ containing information about variable(s) of interest from $s_B$.
pi_A	A numeric vector of length $n_A$ or a square numeric matrix of dimension $n_A$ containing first order or first and second order inclusion probabilities for units included in $s_A$.
pi_B	A numeric vector of length $n_B$ or a square numeric matrix of dimension $n_B$ containing first order or first and second order inclusion probabilities for units included in $s_B$.
domains_A	A character vector of length $n_A$ indicating the domain each unit from $s_A$ belongs to. Possible values are "a" and "ab".
domains_B	A character vector of length $n_B$ indicating the domain each unit from $s_B$ belongs to. Possible values are "b" and "ba".
pik_ab_B	(Optional) A numeric vector of size $n_A$ containing first order inclusion probabilities according to sampling desing in frame B for units belonging to overlap domain that have been selected in $s_A$.
pik_ba_A	(Optional) A numeric vector of size $n_B$ containing first order inclusion probabilities according to sampling desing in frame A for units belonging to overlap domain that have been selected in $s_B$.
N_A	(Optional) A numeric value indicating the size of frame A.
N_B	(Optional) A numeric value indicating the size of frame B.
N_ab	(Optional) A numeric value indicating the size of the overlap domain.
xsAFrameA	(Optional) A numeric vector of length $n_A$ or a numeric matrix or data frame of dimensions $n_A$ x $m_A$, with $m_A$ the number of auxiliary variables in frame A, containing auxiliary information in frame A for units included in $s_A$.
xsBFrameA	(Optional) A numeric vector of length $n_B$ or a numeric matrix or data frame of dimensions $n_B$ x $m_A$, with $m_A$ the number of auxiliary variables in frame A, containing auxiliary information in frame A for units included in $s_B$. For units in domain $b$, these values are 0.
xsAFrameB	(Optional) A numeric vector of length $n_A$ or a numeric matrix or data frame of dimensions $n_A$ x $m_B$, with $m_B$ the number of auxiliary variables in frame B, containing auxiliary information in frame B for units included in $s_A$. For units in domain $a$, these values are 0.
xsBFrameB	(Optional) A numeric vector of length $n_B$ or a numeric matrix or data frame of dimensions $n_B$ x $m_B$, with $m_B$ the number of auxiliary variables in frame B, containing auxiliary information in frame B for units included in $s_B$.
XA	(Optional) A numeric value or vector of length $m_A$, with $m_A$ the number of auxiliary variables in frame A, indicating the population totals for the auxiliary variables considered in frame A.
XB	(Optional) A numeric value or vector of length $m_B$, with $m_B$ the number of auxiliary variables in frame B, indicating the population totals for the auxiliary variables considered in frame B.
met	(Optional) A character vector indicating the distance that must be used in calibration process. Possible values are "linear", "raking" and "logit". Default is "linear".
conf_level	(Optional) A numeric value indicating the confidence level for the confidence intervals, if desired.

Examples

data(DatA)
data(DatB)
data(PiklA)
data(PiklB)

Compare(DatA$Feed, DatB$Feed, PiklA, PiklB, DatA$Domain, DatB$Domain)

---

Covariance estimator between two Horvitz - Thompson estimators

Description

Computes the covariance estimator between two Horvitz - Thompson estimators of population total from survey data obtained from a single stage sampling design

Usage

CovHT(y, x, pikl)

Arguments

y	A numeric vector of size n containing information about first variable of interest in the sample
x	A numeric vector of size n containing information about second variable of interest in the sample
pikl	A square numeric matrix of dimension n containing first and second order inclusion probabilities for units included in the sample

Details

Covariance estimator between two Horvitz - Thompson estimators of population total is given by

$\hat{Cov}(\hat{Y}_{HT}, \hat{X}_{HT}) = \sum_{k \in s}\sum_{l \in s} \frac{\pi_{kl} - \pi_k \pi_l}{\pi_{kl}}\frac{y_k}{\pi_k}\frac{x_l}{\pi_l}$

Value

A numeric value representing covariance estimator between two Horvitz - Thompson estimators for population total for considered values

References

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 @references Sarndal, C. E., Swensson, B. and Wretman, J. (1992) Model Assisted Survey Sampling. Springer-Verlag. New York.

See Also

HT VarHT

Examples

##########   Example 1   ##########
Indicators <- c(1, 2, 3, 4, 5)
X <- c(13, 18, 20, 14, 9)
Y <- c(2, 0.5, 1.2, 3.3, 2)
#Let draw two simple random samples without replacement of size 2
s <- sample(Indicators, 2)
sX <- X[s]
sY <- Y[s]
#Now, let calculate the associated probability matrix with first and
#second order inclusion probabilities
Ps <- matrix(c(0.4,0.2, 0.2,0.4), 2, 2)
CovHT(sX, sY, Ps)

##########   Example 2   ##########
data(DatA)
attach(DatA)
data(PiklA)
#Let calculate Horvitz - Thompson estimator for total of variable Clothing in Frame A.
HT(Clo, ProbA)
#Let calculate Horvitz - Thompson estimator for total of variable Feeding in Frame A.
HT(Feed, ProbA)
#And now, let compute the covariance between the previous estimators
CovHT(Clo, Feed, PiklA)

---

Joint sample database

Description

This dataset contains some variables coming from a real dual frame survey conducted in 2013 in Andalusia (Spain) by a scientific institute specialized in social topics. With this dataset it is intented to show how to properly split a joint dual frame sample into subsamples, so functions of Frame2 can be used.

Usage

Dat

Format

Drawnby

Indicates whether individual was selected in the landline sample(1) or in the cell phone sample(2).

Stratum

Indicates the stratum each individual belongs to. For individuals selected in cell phone sample, value of this variable is NA.

Opinion

Response of the individual to the question: Do you think that immigrants currently living in Andalusia are quite a lot? 1 represents "yes" and 0 represents "no".

Landline

Indicates whether individual has a landline (1) or not (0).

Cell

Indicates whether individual has a cell phone(1) or not(0).

ProbLandline

First order inclusion probability of reaching the individual by landline.

ProbCell

First order inclusion probability of reaching the individual by cell phone.

Income

Monthly income (in euros) of the individual.

Details

The survey was based on two frames: a landline frame and a cell phone frame. Landline frame was stratified by province and simple random sampling without replacement was considered in cell phone frame. The size of the whole sample was $n = 2402$. Total of the variable Income in the whole population is $X_{Income} = 12686232063$.

Examples

data(Dat)
attach(Dat)

#We are going to split dataset Dat into two new datasets, each 
#one corresponding to a frame: frame containing individuals
#using landline and frame containing individuals using cell phone.

FrameLandline <- Dat[Landline == 1,]
FrameCell <- Dat[Cell == 1,]

#Equally, we can split the original dataset in three new different 
#datasets, each one corresponding to one domain: first domain containing
#individuals using only landline, second domain containing individuals
#using only cell phone and the third domain containing individuals
#using both landline and cell phone.

DomainLandline <- Dat[Landline == 1 & Cell == 0,]
DomainCell <- Dat[Landline == 0 & Cell == 1,]
DomainBoth <- Dat[Landline == 1 & Cell == 1,]

#From the domain datasets, we can build frame datasets

FrameLandline <- rbind(DomainLandline, DomainBoth)
FrameCell <- rbind(DomainCell, DomainBoth)

---

Database of household expenses for frame A

Description

This dataset contains some variables regarding household expenses for a sample of 105 households selected from a list of landline phones (let say, frame A) in a particular city in a specific month.

Usage

DatA

Format

Domain

A string indicating the domain each household belongs to. Possible values are "a" if household belongs to domain a or "ab" if household belongs to overlap domain.

Feed

Feeding expenses (in euros) at the househould

Clo

Clothing expenses (in euros) at the household

Lei

Leisure expenses (in euros) at the household

Inc

Household income (in euros). Values for this variable are only available for households included in frame A. For households included in domain b, value of this variable is set to 0.

Tax

Household municipal taxes (in euros) paid. Values for this variable are only available for households included in frame A. For households included in domain b, value of this variable is set to 0.

M2

Square meters of the house. Values for this variable are only available for households included in frame B. For households included in domain a, value of this variable is set to 0.

Size

Household size. Values for this variable are only available for households included in frame B. For households included in domain a, value of this variable is set to 0.

ProbA

First order inclusion probability in frame A. This probability is 0 for households included in domain b.

ProbB

First order inclusion probability in frame B. This probability is 0 for households included in domain a.

Stratum

A numeric value indicating the stratum each household belongs to.

Details

The sample, of size $n_A = 105$, has been drawn from a population of $N_A = 1735$ households with landline phone according to a stratified random sampling. Population units were divided in 6 different strata. Population sizes of these strata are $N_A^h = (727, 375, 113, 186, 115, 219)$. $N_{ab} = 601$ of the households composing the population have, also, mobile phone. On the other hand, frame totals for auxiliary variables in this frame are $X_{Income}^A = 4300260$ and $X_{Taxes}^A = 215577$.

See Also

PiklA

Examples

data(DatA)
attach(DatA)
#Let perform a brief descriptive analysis for the three main variables
param <- data.frame(Feed, Clo, Lei)
summary (param)
hist (Feed)
hist (Clo)
hist (Lei)

---

Database of household expenses for frame B

Description

This dataset contains some variables regarding household expenses for a sample of 135 households selected from a list of mobile phones (let say, frame B) in a particular city in a specific month.

Usage

DatB

Format

Domain

A string indicating the domain each household belongs to. Possible values are "b" if household belongs to domain b or "ba" if household belongs to overlap domain.

Feed

Feeding expenses (in euros) at the househould

Clo

Clothing expenses (in euros) at the household

Lei

Leisure expenses (in euros) at the household

Inc

Household income (in euros). Values for this variable are only available for households included in frame A. For households included in domain b, value of this variable is set to 0.

Tax

Household municipal taxes (in euros) paid. Values for this variable are only available for households included in frame A. For households included in domain b, value of this variable is set to 0.

M2

Square meters of the house. Values for this variable are only available for households included in frame B. For households included in domain a, value of this variable is set to 0.

Size

Household size. Values for this variable are only available for households included in frame B. For households included in domain a, value of this variable is set to 0.

ProbA

First order inclusion probability in frame A. This probability is 0 for households included in domain b.

ProbB

First order inclusion probability in frame B. This probability is 0 for households included in domain a.

Details

The sample, of size $n_B = 135$, has been drawn from a population of $N_B = 1191$ households with mobile phone according to a simple random sampling without replacement design. $N_{ab} = 601$ of these households have, also, landline phone. On the other hand, frame totals for auxiliary variables in this frame are $X_{Metres2}^B = 176553$ and $X_{Size}^B = 3529$

See Also

PiklB

Examples

data(DatB)
attach(DatB)
#Let perform a brief descriptive analysis for the three main variables
param <- data.frame(Feed, Clo, Lei)
summary (param)
hist (Feed)
hist (Clo)
hist (Lei)

---

Database of students' program choice for frame A

Description

This dataset contains some variables regarding the program choice for a sample of 180 students included in the sampling frame A.

Usage

DatMA

Format

Id_Pop

An integer from 1 to $N$, with $N$ the number of students in the whole population, identifying the student within the population.

Id_Frame

An integer from 1 to $N_A$, with $N_A$ the number of students in the frame, identifying the student within the frame.

Prog

A factor with three categories (academic, general and vocation) indicating the program choice of the student.

Ses

An ordinal factor with three categories (low, middle and high) indicating the socio-economical status of the student.

Read

A number indicating the mark of the student in a reading test.

Write

A number indicating the mark of the student in a writing test.

Sch_Size

A number indicating the size of the school the students belongs to.

Domain

A string indicating the domain each student belongs to. Possible values are "a" if student belongs to domain a or "ab" if student belongs to overlap domain.

ProbA

First order inclusion probability in frame A.

ProbB

First order inclusion probability in frame B. This probability is 0 for students included in domain a.

Details

The sample, of size $n_A = 180$, has been drawn from a population of $N_A = 5500$ students according to a proportional-to-size sampling desing according to the size of the school. So, students attending bigger schools have a higher probability of being selected in the sample. $N_{ab} = 2000$ of the students composing the population belongs also to frame B.

See Also

DatPopM

Examples

data(DatMA)
attach(DatMA)
#Let perform a brief descriptive analysis for the main variable
summary (Prog)
#And let do the same for the numerical auxiliary variables Read and Write
summary(Read)
summary(Write)

---

Database of students' program choice for frame B

Description

This dataset contains some variables regarding the program choice for a sample of 232 students included in the sampling frame B.

Usage

DatMB

Format

Id_Pop

An integer from 1 to $N$, with $N$ the number of students in the whole population, identifying the student within the population.

Id_Frame

An integer from 1 to $N_B$, with $N_B$ the number of students in the frame, identifying the student within the frame.

Prog

A factor with three categories (academic, general and vocation) indicating the program choice of the student.

Ses

An ordinal factor with three categories (low, middle and high) indicating the socio-economical status of the student.

Read

A number indicating the mark of the student in a reading test.

Write

A number indicating the mark of the student in a writing test.

Sch_Size

A number indicating the size of the school the students belongs to.

Domain

A string indicating the domain each student belongs to. Possible values are "b" if student belongs to domain b or "ba" if student belongs to overlap domain.

ProbA

First order inclusion probability in frame A. This probability is 0 for students included in domain b.

ProbB

First order inclusion probability in frame B.

Details

The sample, of size $n_B = 232$, has been drawn from a population of $N_B = 6500$ students according to a simple random sampling design. $N_{ab} = 2000$ of the students composing the population belongs also to frame A.

See Also

DatPopM

Examples

data(DatMB)
attach(DatMB)
#Let perform a brief descriptive analysis for the main variable
summary (Prog)
#And let do the same for the numerical auxiliary variables Read and Write
summary(Read)
summary(Write)

---

Database of auxiliary information for the whole population of students

Description

This dataset contains population information about the auxiliary variables of the population of students

Usage

DatPopM

Format

Ses

An ordinal factor with three categories (low, middle and high) indicating the socio-economical status of the student.

Read

A number indicating the mark of the student in a reading test.

Write

A number indicating the mark of the student in a writing test.

Domain

A string indicating the domain each student belongs to. Possible values are "a" if student belongs to domain a, "b" if student belongs to domain b or "ab" if student belongs to overlap domain.

Details

The population size is $N = 10000$.

See Also

DatMA DatMB

Examples

data(DatPopM)
attach(DatPopM)
#Let perform a brief descriptive analysis for the three auxiliary variables
summary (Ses)
summary(Read)
summary(Write)

---

Domains

Description

Given a main vector, an auxiliary one and a value of the latter, identifies positions of the auxiliary vector corresponding to values other than the given one. Then, turns zero values of the main vector corresponding to these positions.

Usage

Domains (y, domains, value)

Arguments

y	A numeric main vector of size n
domains	A numeric/character/logic auxiliary vector of size n
value	A value of the auxiliary vector

Value

A numeric vector, copy of y, with some values turned zero depending on values of domains and value

Examples

##########   Example 1   ##########
U <- c(13, 18, 20, 14, 9)
#Let build an auxiliary vector indicating whether values in U are above or below the mean.
aux <- c("Below", "Above", "Above", "Below", "Below")
#Now, only values below the mean remain, the other ones are turned zero.
Domains (U, aux, "Below")

##########   Example 2   ##########
data(DatA)
attach(DatA)
#Let calculate total feeding expenses corresponding to households in domain a.
sum (Domains (Feed, Domain, "a"))

---

Fuller-Burmeister estimator

Description

Produces estimates for population totals and means using the Fuller - Burmeister estimator from survey data obtained from a dual frame sampling desing. Confidence intervals are also computed, if required.

Usage

FB(ysA, ysB, pi_A, pi_B, domains_A, domains_B, conf_level = NULL)

Arguments

ysA	A numeric vector of length $n_A$ or a numeric matrix or data frame of dimensions $n_A$ x $c$ containing information about variable of interest from $s_A$.
ysB	A numeric vector of length $n_B$ or a numeric matrix or data frame of dimensions $n_B$ x $c$ containing information about variable of interest from $s_B$.
pi_A	A numeric vector of length $n_A$ or a square numeric matrix of dimension $n_A$ containing first order or first and second order inclusion probabilities for units included in $s_A$.
pi_B	A numeric vector of length $n_B$ or a square numeric matrix of dimension $n_B$ containing first order or first and second order inclusion probabilities for units included in $s_B$.
domains_A	A character vector of size $n_A$ indicating the domain each unit from $s_A$ belongs to. Possible values are "a" and "ab".
domains_B	A character vector of size $n_B$ indicating the domain each unit from $s_B$ belongs to. Possible values are "b" and "ba".
conf_level	(Optional) A numeric value indicating the confidence level for the confidence intervals.

Details

Fuller-Burmeister estimator of population total is given by

$\hat{Y}_{FB} = \hat{Y}_a^A + \hat{\beta_1}\hat{Y}_{ab}^A + (1 - \hat{\beta_1})\hat{Y}_{ab}^B + \hat{Y}_b^B + \hat{\beta_2}(\hat{N}_{ab}^A - \hat{N}_{ab}^B)$

where optimal values for $\hat{\beta}$ to minimize variance of the estimator are:

$\left( \begin{array}{c} \hat{\beta}_1\\ \hat{\beta}_2 \end{array} \right) = - \left( \begin{array}{cc} \hat{V}(\hat{Y}_{ab}^A - \hat{Y}_{ab}^B) & \widehat{Cov}(\hat{Y}_{ab}^A - \hat{Y}_{ab}^B, \hat{N}_{ab}^A - \hat{N}_{ab}^B)\\ \widehat{Cov}(\hat{Y}_{ab}^A - \hat{Y}_{ab}^B, \hat{N}_{ab}^A - \hat{N}_{ab}^B) & \hat{V}(\hat{N}_{ab}^A - \hat{N}_{ab}^B) \end{array} \right)^{-1} \times$

$\left( \begin{array}{c} \widehat{Cov}(\hat{Y}_a^A + \hat{Y}_b^B + \hat{Y}_{ab}^B, \hat{Y}_{ab}^A - \hat{Y}_{ab}^B)\\ \widehat{Cov}(\hat{Y}_a^A + \hat{Y}_b^B + \hat{Y}_{ab}^B, \hat{N}_{ab}^A - \hat{N}_{ab}^B) \end{array} \right)$

Due to Fuller-Burmeister estimator is not defined for estimating population sizes, estimation of the mean is computed as $\hat{Y}_{FB} / \hat{N}_H$, where $\hat{N}_H$ is the estimation of the population size using Hartley estimator. Estimated variance for the Fuller-Burmeister estimator can be obtained through expression

$\hat{V}(\hat{Y}_{FB}) = \hat{V}(\hat{Y}_a^A) + \hat{V}(\hat{Y}^B) + \hat{\beta}_1[\widehat{Cov}(\hat{Y}_a^A, \hat{Y}_{ab}^A) - \widehat{Cov}(\hat{Y}^B, \hat{Y}_{ab}^B)]$

$+ \hat{\beta}_2[\widehat{Cov}(\hat{Y}_a^A, \hat{N}_{ab}^A) - \widehat{Cov}(\hat{Y}^B, \hat{N}_{ab}^B)]$

If both first and second order probabilities are known, variances and covariances involved in calculation of $\hat{\beta}$ and $\hat{V}(\hat{Y}_{FB})$ are estimated using functions VarHT and CovHT, respectively. If only first order probabilities are known, variances are estimated using Deville's method and covariances are estimated using following expression

$\widehat{Cov}(\hat{X}, \hat{Y}) = \frac{\hat{V}(X + Y) - \hat{V}(X) - \hat{V}(Y)}{2}$

Value

FB returns an object of class "EstimatorDF" which is a list with, at least, the following components:

Call	the matched call.
Est	total and mean estimation for main variable(s).
VarEst	variance estimation for main variable(s).

If parameter conf_level is different from NULL, object includes component

ConfInt

total and mean estimation and confidence intervals for main variables(s).

In addition, components TotDomEst and MeanDomEst are available when estimator is based on estimators of the domains. Component Param shows value of parameters involded in calculation of the estimator (if any). By default, only Est component (or ConfInt component, if parameter conf_level is different from NULL) is shown. It is possible to access to all the components of the objects by using function summary.

References

Fuller, W.A. and Burmeister, L.F. (1972). Estimation for Samples Selected From Two Overlapping Frames ASA Proceedings of the Social Statistics Sections, 245 - 249.

See Also

Hartley JackFB

Examples

data(DatA)
data(DatB)
data(PiklA)
data(PiklB)

#Let calculate Fuller-Burmeister estimator for variable Clothing
FB(DatA$Clo, DatB$Clo, PiklA, PiklB, DatA$Domain, DatB$Domain)

#Now, let calculate Fuller-Burmeister estimator and a 90% confidence interval
#for variable Leisure, considering only first order inclusion probabilities
FB(DatA$Lei, DatB$Lei, DatA$ProbA, DatB$ProbB, DatA$Domain, 
DatB$Domain, 0.90)

---

Hartley estimator

Description

Produces estimates for population totals and means using Hartley estimator from survey data obtained from a dual frame sampling design. Confidence intervals are also computed, if required.

Usage

Hartley(ysA, ysB, pi_A, pi_B, domains_A, domains_B, conf_level = NULL)

Arguments

ysA	A numeric vector of length $n_A$ or a numeric matrix or data frame of dimensions $n_A$ x $c$ containing information about variable of interest from $s_A$.
ysB	A numeric vector of length $n_B$ or a numeric matrix or data frame of dimensions $n_B$ x $c$ containing information about variable of interest from $s_B$.
pi_A	A numeric vector of length $n_A$ or a square numeric matrix of dimension $n_A$ containing first order or first and second order inclusion probabilities for units included in $s_A$.
pi_B	A numeric vector of length $n_B$ or a square numeric matrix of dimension $n_B$ containing first order or first and second order inclusion probabilities for units included in $s_B$.
domains_A	A character vector of size $n_A$ indicating the domain each unit from $s_A$ belongs to. Possible values are "a" and "ab".
domains_B	A character vector of size $n_B$ indicating the domain each unit from $s_B$ belongs to. Possible values are "b" and "ba".
conf_level	(Optional) A numeric value indicating the confidence level for the confidence intervals.

Details

Hartley estimator of population total is given by

$\hat{Y}_H = \hat{Y}_a^A + \hat{\theta}\hat{Y}_{ab}^A + (1 - \hat{\theta})\hat{Y}_{ab}^B + \hat{Y}_b^B$

where $\hat{\theta} \in [0, 1]$. Optimum value for $\hat{\theta}$ to minimize variance of the estimator is

$\hat{\theta}_{opt} = \frac{\hat{V}(\hat{Y}_{ab}^B) + \widehat{Cov}(\hat{Y}_b^B, \hat{Y}_{ab}^B) - \widehat{Cov}(\hat{Y}_a^A, \hat{Y}_{ab}^A)}{\hat{V}(\hat{Y}_{ab}^A) + \hat{V}(\hat{Y}_{ab}^B)}$

Taking into account the independence between $s_A$ and $s_B$, an estimator for the variance of the Hartley estimator can be obtained as follows:

$\hat{V}(\hat{Y}_H) = \hat{V}(\hat{Y}_a^A + \hat{\theta}\hat{Y}_{ab}^A) + \hat{V}((1 - \hat{\theta})\hat{Y}_{ab}^B + \hat{Y}_b^B)$

If both first and second order probabilities are known, variances and covariances involved in calculation of $\hat{\theta}_{opt}$ and $\hat{V}(\hat{Y}_H)$ are estimated using functions VarHT and CovHT, respectively. If only first order probabilities are known, variances are estimated using Deville's method and covariances are estimated using following expression

$\widehat{Cov}(\hat{X}, \hat{Y}) = \frac{\hat{V}(X + Y) - \hat{V}(X) - \hat{V}(Y)}{2}$

Value

Hartley returns an object of class "EstimatorDF" which is a list with, at least, the following components:

Call	the matched call.
Est	total and mean estimation for main variable(s).
VarEst	variance estimation for main variable(s).

If parameter conf_level is different from NULL, object includes component

ConfInt

total and mean estimation and confidence intervals for main variables(s).

In addition, components TotDomEst and MeanDomEst are available when estimator is based on estimators of the domains. Component Param shows value of parameters involded in calculation of the estimator (if any). By default, only Est component (or ConfInt component, if parameter conf_level is different from NULL) is shown. It is possible to access to all the components of the objects by using function summary.

References

Hartley, H. O. (1962) Multiple Frames Surveys. Proceedings of the American Statistical Association, Social Statistics Sections, 203 - 206.

Hartley, H. O. (1974) Multiple frame methodology and selected applications. Sankhya C, Vol. 36, 99 - 118.

See Also

JackHartley

Examples

data(DatA)
data(DatB)
data(PiklA)
data(PiklB)

#Let calculate Hartley estimator for variable Feeding
Hartley(DatA$Feed, DatB$Feed, PiklA, PiklB, DatA$Domain, DatB$Domain)

#Now, let calculate Hartley estimator and a 90% confidence interval
#for variable Leisure, considering only first order inclusion probabilities
Hartley(DatA$Lei, DatB$Lei, DatA$ProbA, DatB$ProbB, DatA$Domain, 
DatB$Domain, 0.90)

---

Horvitz - Thompson estimator

Description

Computes the Horvitz - Thompson estimator

Usage

HT(y, pik)

Arguments

y	A numeric vector of size n containing information about variable of interest
pik	A numeric vector of size n containing first order inclusion probabilities for units included in y

Details

Horvitz - Thompson estimator of population total is given by

$\hat{Y}_{HT} = \sum_{k \in s} \frac{y_k}{\pi_k}$

Value

A numeric value representing Horvitz - Thompson estimator for population total for considered values

References

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

See Also

VarHT

Examples

##########   Example 1   ##########
U <- c(13, 18, 20, 14, 9)
#A simple random sample of size 2 without replacement is drawn from population
s <- sample(U, 2)
ps <- c(0.4, 0.4)
HT(s, ps)

##########   Example 2   ##########
data(DatA)
attach(DatA)
#Let estimate population total for variable Feeding in frame A
HT(Feed, ProbA)

---

Confidence intervals for Bankier-Kalton-Anderson estimator based on jackknife method

Description

Calculates confidence intervals for Bankier-Kalton-Anderson estimator using jackknife procedure

Usage

JackBKA(ysA, ysB, piA, piB, pik_ab_B, pik_ba_A, domainsA, domainsB, 
conf_level, sdA = "srs", sdB = "srs", strA = NULL, strB = NULL, clusA = NULL,
clusB = NULL, fcpA = FALSE, fcpB = FALSE)

Arguments

ysA	A numeric vector of length $nA$ or a numeric matrix or data frame of dimensions $nA$ x $c$ containing information about variable of interest from $s_A$.
ysB	A numeric vector of length $nB$ or a numeric matrix or data frame of dimensions $nB$ x $c$ containing information about variable of interest from $s_B$.
piA	A numeric vector of length $nA$ or a square numeric matrix of dimension $nA$ containing first order or first and second order inclusion probabilities for units included in $s_A$.
piB	A numeric vector of length $nB$ or a square numeric matrix of dimension $nB$ containing first order or first and second order inclusion probabilities for units included in $s_B$.
pik_ab_B	A numeric vector of size $nA$ containing first order inclusion probabilities according to sampling design in frame B for units belonging to overlap domain that have been selected in $s_A$.
pik_ba_A	A numeric vector of size $nB$ containing first order inclusion probabilities according to sampling design in frame A for units belonging to overlap domain that have been selected in $s_B$.
domainsA	A character vector of size $nA$ indicating the domain each unit from $s_A$ belongs to. Possible values are "a" and "ab".
domainsB	A character vector of size $nB$ indicating the domain each unit from $s_B$ belongs to. Possible values are "b" and "ba".
conf_level	A numeric value indicating the confidence level for the confidence intervals.
sdA	(Optional) A character vector indicating the sampling design considered in frame A. Possible values are "srs" (simple random sampling without replacement), "pps" (probabilities proportional to size sampling), "str" (stratified sampling), "clu" (cluster sampling) and "strclu" (stratified cluster sampling). Default is "srs".
sdB	(Optional) A character vector indicating the sampling design considered in frame B. Possible values are "srs" (simple random sampling without replacement), "pps" (probabilities proportional to size sampling), "str" (stratified sampling), "clu" (cluster sampling) and "strclu" (stratified cluster sampling). Default is "srs".
strA	(Optional) A numeric vector indicating the stratum each unit in frame A belongs to, if a stratified sampling or a stratified cluster sampling has been considered in frame A.
strB	(Optional) A numeric vector indicating the stratum each unit in frame B belongs to, if a stratified sampling or a stratified cluster sampling has been considered in frame B.
clusA	(Optional) A numeric vector indicating the cluster each unit in frame A belongs to, if a cluster sampling or a stratified cluster sampling has been considered in frame A.
clusB	(Optional) A numeric vector indicating the cluster each unit in frame B belongs to, if a cluster sampling or a stratified cluster sampling has been considered in frame B.
fcpA	(Optional) A logic value indicating if a finite population correction factor should be considered in frame A. Default is FALSE.
fcpB	(Optional) A logic value indicating if a finite population correction factor should be considered in frame B. Default is FALSE.

Details

Let suppose a non stratified sampling design in frame A and a stratified sampling design in frame B where frame has been divided into L strata and a sample of size $n_{Bl}$ from the $N_{Bl}$ composing the l-th stratum is selected In this context, jackknife variance estimator of a estimator $\hat{Y}_c$ is given by

$v_J(\hat{Y}_c) = \frac{n_{A}-1}{n_{A}}\sum_{i\in s_A} (\hat{Y}_{c}^{A}(i) -\overline{Y}_{c}^{A})^2 + \sum_{l=1}^{L}\frac{n_{Bl}-1}{n_{Bl}} \sum_{i\in s_{Bl}} (\hat{Y}_{c}^{B}(lj) -\overline{Y}_{c}^{Bl})^2$

with $\hat{Y}_c^A(i)$ the value of estimator $\hat{Y}_c$ after dropping $i-th$ unit from ysA and $\overline{Y}_{c}^{A}$ the mean of values $\hat{Y}_c^A(i)$. Similarly, $\hat{Y}_c^B(lj)$ is the value taken by $\hat{Y}_c$ after dropping j-th unit of l-th from sample ysB and $\overline{Y}_{c}^{Bl}$ is the mean of values $\hat{Y}_c^B(lj)$. If needed, a finite population correction factor can be included in frames by replacing $\hat{Y}_{c}^{A}(i)$ or $\hat{Y}_{c}^{B}(lj)$ with $\hat{Y}_{c}^{A*}(i)= \hat{Y}_{c}+\sqrt{1-\overline{\pi}_A} (\hat{Y}_{c}^{A}(i) -\hat{Y}_{c})$ or $\hat{Y}_{c}^{B*}(lj)= \hat{Y}_{c}+\sqrt{1-\overline{\pi}_B} (\hat{Y}_{c}^{B}(lj) -\hat{Y}_{c})$, where $\overline{\pi}_A = \sum_{i \in s_A}\pi_{iA}/nA$ and $\overline{\pi}_B = \sum_{j \in s_B}\pi_{jB}/nB$ A confidence interval for any parameter of interest, $Y$ can be calculated, then, using the pivotal method.

Value

A numeric matrix containing estimations of population total and population mean and their corresponding confidence intervals obtained through jackknife method.

References

Wolter, K. M. (2007) Introduction to Variance Estimation. 2nd Edition. Springer, Inc., New York.

See Also

BKA

Examples

data(DatA)
data(DatB)

#Let obtain a 95% jackknife confidence interval for variable Clothing,
#supposing a stratified sampling in frame A and a simple random sampling without
#replacement  in frame B with no finite population correction factor in any frame.
JackBKA(DatA$Feed, DatB$Feed, DatA$ProbA, DatB$ProbB, DatA$ProbB,
DatB$ProbA, DatA$Domain, DatB$Domain, 0.95, "str", "srs",
strA = DatA$Stratum)

#Let check how interval estimation varies when a finite 
#population correction factor is considered in both frames.
JackBKA(DatA$Feed, DatB$Feed, DatA$ProbA, DatB$ProbB, DatA$ProbB,
DatB$ProbA, DatA$Domain, DatB$Domain, 0.95, "str", "srs", 
strA = DatA$Stratum, fcpA = TRUE, fcpB = TRUE)

---