estimate_var_artifacts | R Documentation |
Taylor series approximations to estimate the variances of artifacts that have been estimated from other artifacts.
These functions are implemented internally in the create_ad
function and related functions, but are useful as general tools for manipulating artifact distributions.
Available functions include:
estimate_var_qxi
Estimate the variance of a qxi distribution from a qxa distribution and a distribution of u ratios.
estimate_var_rxxi
Estimate the variance of an rxxi distribution from an rxxa distribution and a distribution of u ratios.
estimate_var_qxa
Estimate the variance of a qxa distribution from a qxi distribution and a distribution of u ratios.
estimate_var_rxxa
Estimate the variance of an rxxa distribution from an rxxi distribution and a distribution of u ratios.
estimate_var_ut
Estimate the variance of a true-score u ratio distribution from an observed-score u ratio distribution and a reliability distribution.
estimate_var_ux
Estimate the variance of an observed-score u ratio distribution from a true-score u ratio distribution and a reliability distribution.
estimate_var_qyi
Estimate the variance of a qyi distribution from the following distributions: qya, rxyi, and ux.
estimate_var_ryyi
Estimate the variance of an ryyi distribution from the following distributions: ryya, rxyi, and ux.
estimate_var_qya
Estimate the variance of a qya distribution from the following distributions: qyi, rxyi, and ux.
estimate_var_ryya
Estimate the variance of an ryya distribution from the following distributions: ryyi, rxyi, and ux.
estimate_var_qxi( qxa, var_qxa = 0, ux, var_ux = 0, cor_qxa_ux = 0, ux_observed = TRUE, indirect_rr = TRUE, qxa_type = "alpha" ) estimate_var_qxa( qxi, var_qxi = 0, ux, var_ux = 0, cor_qxi_ux = 0, ux_observed = TRUE, indirect_rr = TRUE, qxi_type = "alpha" ) estimate_var_rxxi( rxxa, var_rxxa = 0, ux, var_ux = 0, cor_rxxa_ux = 0, ux_observed = TRUE, indirect_rr = TRUE, rxxa_type = "alpha" ) estimate_var_rxxa( rxxi, var_rxxi = 0, ux, var_ux = 0, cor_rxxi_ux = 0, ux_observed = TRUE, indirect_rr = TRUE, rxxi_type = "alpha" ) estimate_var_ut( rxx, var_rxx = 0, ux, var_ux = 0, cor_rxx_ux = 0, rxx_restricted = TRUE, rxx_as_qx = FALSE ) estimate_var_ux( rxx, var_rxx = 0, ut, var_ut = 0, cor_rxx_ut = 0, rxx_restricted = TRUE, rxx_as_qx = FALSE ) estimate_var_ryya( ryyi, var_ryyi = 0, rxyi, var_rxyi = 0, ux, var_ux = 0, cor_ryyi_rxyi = 0, cor_ryyi_ux = 0, cor_rxyi_ux = 0 ) estimate_var_qya( qyi, var_qyi = 0, rxyi, var_rxyi = 0, ux, var_ux = 0, cor_qyi_rxyi = 0, cor_qyi_ux = 0, cor_rxyi_ux = 0 ) estimate_var_qyi( qya, var_qya = 0, rxyi, var_rxyi = 0, ux, var_ux = 0, cor_qya_rxyi = 0, cor_qya_ux = 0, cor_rxyi_ux = 0 ) estimate_var_ryyi( ryya, var_ryya = 0, rxyi, var_rxyi = 0, ux, var_ux = 0, cor_ryya_rxyi = 0, cor_ryya_ux = 0, cor_rxyi_ux = 0 )
qxa |
Square-root of applicant reliability estimate. |
var_qxa |
Variance of square-root of applicant reliability estimate. |
ux |
Observed-score u ratio. |
var_ux |
Variance of observed-score u ratio. |
cor_qxa_ux |
Correlation between qxa and ux. |
ux_observed |
Logical vector determining whether u ratios are observed-score u ratios ( |
indirect_rr |
Logical vector determining whether reliability values are associated with indirect range restriction ( |
qxi |
Square-root of incumbent reliability estimate. |
var_qxi |
Variance of square-root of incumbent reliability estimate. |
cor_qxi_ux |
Correlation between qxi and ux. |
rxxa |
Incumbent reliability value. |
var_rxxa |
Variance of incumbent reliability values. |
cor_rxxa_ux |
Correlation between rxxa and ux. |
rxxi |
Incumbent reliability value. |
var_rxxi |
Variance of incumbent reliability values. |
cor_rxxi_ux |
Correlation between rxxi and ux. |
rxxi_type, rxxa_type, qxi_type, qxa_type |
String vector identifying the types of reliability estimates supplied (e.g., "alpha", "retest", "interrater_r", "splithalf"). See the documentation for |
rxx |
Generic argument for a reliability estimate (whether this is a reliability or the square root of a reliability is clarified by the |
var_rxx |
Generic argument for the variance of reliability estimates (whether this pertains to reliabilities or the square roots of reliabilities is clarified by the |
cor_rxx_ux |
Correlation between rxx and ux. |
rxx_restricted |
Logical vector determining whether reliability estimates were incumbent reliabilities ( |
rxx_as_qx |
Logical vector determining whether the reliability estimates were reliabilities ( |
ut |
True-score u ratio. |
var_ut |
Variance of true-score u ratio. |
cor_rxx_ut |
Correlation between rxx and ut. |
ryyi |
Incumbent reliability value. |
var_ryyi |
Variance of incumbent reliability values. |
rxyi |
Incumbent correlation between X and Y. |
var_rxyi |
Variance of incumbent correlations. |
cor_ryyi_rxyi |
Correlation between ryyi and rxyi. |
cor_ryyi_ux |
Correlation between ryyi and ux. |
cor_rxyi_ux |
Correlation between rxyi and ux. |
qyi |
Square-root of incumbent reliability estimate. |
var_qyi |
Variance of square-root of incumbent reliability estimate. |
cor_qyi_rxyi |
Correlation between qyi and rxyi. |
cor_qyi_ux |
Correlation between qyi and ux. |
qya |
Square-root of applicant reliability estimate. |
var_qya |
Variance of square-root of applicant reliability estimate. |
cor_qya_rxyi |
Correlation between qya and rxyi. |
cor_qya_ux |
Correlation between qya and ux. |
ryya |
Applicant reliability value. |
var_ryya |
Variance of applicant reliability values. |
cor_ryya_rxyi |
Correlation between ryya and rxyi. |
cor_ryya_ux |
Correlation between ryya and ux. |
#### Partial derivatives to estimate the variance of qxa using ux ####
Indirect range restriction:
b_ux = ((qxi^2 - 1) * ux) / sqrt((qxi^2 - 1) * ux^2 + 1)
b_qxi = (qxi * ux^2) / sqrt((qxi^2 - 1) * ux^2 + 1)
Direct range restriction:
b_ux = (qxi^2 * (qxi^2 - 1) * ux) / (sqrt(-qxi^2 / (qxi^2 * (ux^2 - 1) - ux^2)) * (qxi^2 * (ux^2 - 1) - ux^2)^2)
b_qxi = (qxi * ux^2) / (sqrt(-qxi^2 / (qxi^2 * (ux^2 - 1) - ux^2)) * (qxi^2 * (ux^2 - 1) - ux^2)^2)
#### Partial derivatives to estimate the variance of rxxa using ux ####
Indirect range restriction:
b_ux = 2 * (rxxi - 1) * ux
b_rxxi = ux^2
Direct range restriction:
b_ux = (2 * (rxxi - 1) * rxxi * ux) / (-rxxi * ux^2 + rxxi + ux^2)^2
b_rxxi = ux^2 / (-rxxi * ux^2 + rxxi + ux^2)^2
#### Partial derivatives to estimate the variance of rxxa using ut ####
b_ut = (2 * (rxxi - 1) * rxxi * ut) / (-rxxi * ut^2 + rxxi + ut^2)^2
b_rxxi = ut^2 / (-rxxi * ut^2 + rxxi + ut^2)^2
#### Partial derivatives to estimate the variance of qxa using ut ####
b_ut = (qxi^2 * (qxi^2 - 1) * ut) / (sqrt(-qxi^2 / (qxi^2 * (ut^2 - 1) - ut^2)) * (qxi^2 * (ut^2 - 1) - ut^2)^2)
b_qxi = (qxi * ut^2) / (sqrt(qxi^2 / (ut^2 - qxi^2 * (ut^2 - 1))) * (ut^2 - qxi^2 * (ut^2 - 1))^2)
#### Partial derivatives to estimate the variance of qxi using ux ####
Indirect range restriction:
b_ux = (1 - qxa^2) / (ux^3 * sqrt((qxa^2 + ux^2 - 1) / ux^2))
b_qxa = qxa / (ux^2 * sqrt((qxa^2 - 1) / ux^2 + 1))
Direct range restriction:
b_ux = -(qxa^2 * (qxa^2 - 1) * ux) / (sqrt((qxa^2 * ux^2) / (qxa^2 * (ux^2 - 1) + 1)) * (qxa^2 * (ux^2 - 1) + 1)^2)
b_qxa = (qxa * ux^2) / (sqrt((qxa^2 * ux^2) / (qxa^2 * (ux^2 - 1) + 1)) * (qxa^2 * (ux^2 - 1) + 1)^2)
#### Partial derivatives to estimate the variance of rxxi using ux ####
Indirect range restriction:
b_ux = (2 - 2 * rxxa) / ux^3
b_rxxa = 1 / ux^2
Direct range restriction:
b_ux = -(2 * (rxxa - 1) * rxxa * ux) / (rxxa * (ux^2 - 1) + 1)^2
b_rxxa = ux^2 / (rxxa * (ux^2 - 1) + 1)^2
#### Partial derivatives to estimate the variance of rxxi using ut ####
u_{T}: b_ut = -(2 * (rxxa - 1) * rxxa * ut) / (rxxa * (ut^2 - 1) + 1)^2
b_rxxa = ut^2 / (rxxa * (ut^2 - 1) + 1)^2
#### Partial derivatives to estimate the variance of qxi using ut ####
b_ut = -((qxa - 1) * qxa^2 * (qxa + 1) * ut) / (sqrt((qxa^2 * ut^2) / (qxa^2 * ut^2 - qxa^2 + 1)) * (qxa^2 * ut^2 - qxa^2 + 1)^2)
b_qxa = (qxa * ut^2) / (sqrt((qxa^2 * ut^2) / (qxa^2 * ut^2 - qxa^2 + 1)) * (qxa^2 * ut^2 - qxa^2 + 1)^2)
#### Partial derivatives to estimate the variance of ut using qxi ####
b_ux = (qxi^2 * ux) / (sqrt((qxi^2 * ux^2) / ((qxi^2 - 1) * ux^2 + 1)) * ((qxi^2 - 1) * ux^2 + 1)^2)
b_qxi = -((ux^2 - 1) * ((qxi^2 * ux^2) / ((qxi^2 - 1) * ux^2 + 1))^(3/2)) / (qxi^3 * ux^2)
#### Partial derivatives to estimate the variance of ut using rxxi ####
b_ux = (rxxi * ux) / (sqrt((rxxi * ux^2) / ((rxxi - 1) * ux^2 + 1)) * ((rxxi - 1) * ux^2 + 1)^2)
b_rxxi = -(ux^2 * (ux^2 - 1)) / (2 * sqrt((rxxi * ux^2) / ((rxxi - 1) * ux^2 + 1)) * ((rxxi - 1) * ux^2 + 1)^2)
#### Partial derivatives to estimate the variance of ut using qxa ####
b_ux = ux / (qxa^2 * sqrt((qxa^2 + ux^2 - 1) / qxa^2))
b_qxa = (1 - ux^2) / (qxa^3 * sqrt((qxa^2 + ux^2 - 1) / qxa^2))
#### Partial derivatives to estimate the variance of ut using rxxa ####
b_ux = ux / (rxxa * sqrt((rxxa + ux^2 - 1) / rxxa))
b_rxxa = (1 - ux^2) / (2 * rxxa^2 * sqrt((rxxa + ux^2 - 1) / rxxa))
#### Partial derivatives to estimate the variance of ux using qxi ####
b_ut = (qxi^2 * ut) / (sqrt(ut^2 / (ut^2 - qxi^2 * (ut^2 - 1))) * (ut^2 - qxi^2 * (ut^2 - 1))^2)
b_qxi = (qxi * (ut^2 - 1) * (ut^2 / (ut^2 - qxi^2 * (ut^2 - 1)))^(3/2)) / ut^2
#### Partial derivatives to estimate the variance of ux using rxxi ####
b_ut = (rxxi * ut) / (sqrt(ut^2 / (-rxxi * ut^2 + rxxi + ut^2)) * (-rxxi * ut^2 + rxxi + ut^2)^2)
b_rxxi = ((ut^2 - 1) * (ut^2 / (-rxxi * ut^2 + rxxi + ut^2))^1.5) / (2 * ut^2)
#### Partial derivatives to estimate the variance of ux using qxa ####
b_ut = (qxa^2 * ut) / sqrt(qxa^2 * (ut^2 - 1) + 1)
b_qxa = (qxa * (ut^2 - 1)) / sqrt(qxa^2 * (ut^2 - 1) + 1)
#### Partial derivatives to estimate the variance of ux using rxxa ####
b_ut = (rxxa * ut) / sqrt(rxxa * (ut^2 - 1) + 1)
b_rxxa = (ut^2 - 1) / (2 * sqrt(rxxa * (ut^2 - 1) + 1))
#### Partial derivatives to estimate the variance of ryya ####
b_ryyi = 1 / (rxyi^2 * (1 / ux^2 - 1) + 1)
b_ux = (2 * (ryyi - 1) * rxyi^2 * ux) / (ux^2 - rxyi^2 * (ux^2 - 1))^2
b_rxyi = (2 * (ryyi - 1) * rxyi * ux^2 * (ux^2 - 1)) / (ux^2 - rxyi^2 * (ux^2 - 1))^2
#### Partial derivatives to estimate the variance of qya ####
b_qxi = qyi / ((1 - rxyi^2 * (1 - 1 / ux^2)) * sqrt(1 - (1 - qyi^2) / (1 - rxyi^2 * (1 - 1 / ux^2))))
b_ux = -((1 - qyi^2) * rxyi^2) / (ux^3 * (1 - rxyi^2 * (1 - 1 / ux^2))^2 * sqrt(1 - (1 - qyi^2) / (1 - rxyi^2 * (1 - 1 / ux^2))))
b_rxyi = -((1 - qyi^2) * rxyi * (1 - 1 / ux^2)) / ((1 - rxyi^2 * (1 - 1 / ux^2))^2 * sqrt(1 - (1 - qyi^2) / (1 - rxyi^2 * (1 - 1 / ux^2))))
#### Partial derivatives to estimate the variance of ryyi ####
b_ryya = rxyi^2 * (1 / ux^2 - 1) + 1
u_{X}: b_ux = -(2 * (ryya - 1) * rxyi^2) / ux^3
b_rxyi = -(2 * (ryya - 1) * rxyi * (ux^2 - 1)) / ux^2
#### Partial derivatives to estimate the variance of qyi ####
b_qya = (qya * (1 - rxyi^2 * (1 - 1 / ux^2))) / sqrt(1 - (1 - qya^2) * (1 - rxyi^2 * (1 - 1 / ux^2)))
b_ux = ((1 - qya^2) * rxyi * (1 - 1 / ux^2)) / sqrt(1 - (1 - qya^2) * (1 - rxyi^2 * (1 - 1 / ux^2)))
b_rxyi = ((1 - qya^2) * rxyi^2) / (ux^3 * sqrt(1 - (1 - qya^2) * (1 - rxyi^2 * (1 - 1 / ux^2))))
estimate_var_qxi(qxa = c(.8, .85, .9, .95), var_qxa = c(.02, .03, .04, .05), ux = .8, var_ux = 0, ux_observed = c(TRUE, TRUE, FALSE, FALSE), indirect_rr = c(TRUE, FALSE, TRUE, FALSE)) estimate_var_qxa(qxi = c(.8, .85, .9, .95), var_qxi = c(.02, .03, .04, .05), ux = .8, var_ux = 0, ux_observed = c(TRUE, TRUE, FALSE, FALSE), indirect_rr = c(TRUE, FALSE, TRUE, FALSE)) estimate_var_rxxi(rxxa = c(.8, .85, .9, .95), var_rxxa = c(.02, .03, .04, .05), ux = .8, var_ux = 0, ux_observed = c(TRUE, TRUE, FALSE, FALSE), indirect_rr = c(TRUE, FALSE, TRUE, FALSE)) estimate_var_rxxa(rxxi = c(.8, .85, .9, .95), var_rxxi = c(.02, .03, .04, .05), ux = .8, var_ux = 0, ux_observed = c(TRUE, TRUE, FALSE, FALSE), indirect_rr = c(TRUE, FALSE, TRUE, FALSE)) estimate_var_ut(rxx = c(.8, .85, .9, .95), var_rxx = 0, ux = c(.8, .8, .9, .9), var_ux = c(.02, .03, .04, .05), rxx_restricted = c(TRUE, TRUE, FALSE, FALSE), rxx_as_qx = c(TRUE, FALSE, TRUE, FALSE)) estimate_var_ux(rxx = c(.8, .85, .9, .95), var_rxx = 0, ut = c(.8, .8, .9, .9), var_ut = c(.02, .03, .04, .05), rxx_restricted = c(TRUE, TRUE, FALSE, FALSE), rxx_as_qx = c(TRUE, FALSE, TRUE, FALSE)) estimate_var_ryya(ryyi = .9, var_ryyi = .04, rxyi = .4, var_rxyi = 0, ux = .8, var_ux = 0) estimate_var_ryya(ryyi = .9, var_ryyi = .04, rxyi = .4, var_rxyi = 0, ux = .8, var_ux = 0) estimate_var_qyi(qya = .9, var_qya = .04, rxyi = .4, var_rxyi = 0, ux = .8, var_ux = 0) estimate_var_ryyi(ryya = .9, var_ryya = .04, rxyi = .4, var_rxyi = 0, ux = .8, var_ux = 0)
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