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
)
Square-root of applicant reliability estimate.
Variance of square-root of applicant reliability estimate.
Observed-score u ratio.
Variance of observed-score u ratio.
Correlation between qxa and ux.
Logical vector determining whether u ratios are observed-score u ratios (TRUE) or true-score u ratios (FALSE).
Logical vector determining whether reliability values are associated with indirect range restriction (TRUE) or direct range restriction (FALSE).
Square-root of incumbent reliability estimate.
Variance of square-root of incumbent reliability estimate.
Correlation between qxi and ux.
Incumbent reliability value.
Variance of incumbent reliability values.
Correlation between rxxa and ux.
Incumbent reliability value.
Variance of incumbent reliability values.
Correlation between rxxi and ux.
String vector identifying the types of reliability estimates supplied (e.g., "alpha", "retest", "interrater_r", "splithalf"). See the documentation for ma_r for a full list of acceptable reliability types.
Generic argument for a reliability estimate (whether this is a reliability or the square root of a reliability is clarified by the rxx_as_qx argument).
Generic argument for the variance of reliability estimates (whether this pertains to reliabilities or the square roots of reliabilities is clarified by the rxx_as_qx argument).
Correlation between rxx and ux.
Logical vector determining whether reliability estimates were incumbent reliabilities (TRUE) or applicant reliabilities (FALSE).
Logical vector determining whether the reliability estimates were reliabilities (TRUE) or square-roots of reliabilities (FALSE).
True-score u ratio.
Variance of true-score u ratio.
Correlation between rxx and ut.
Incumbent reliability value.
Variance of incumbent reliability values.
Incumbent correlation between X and Y.
Variance of incumbent correlations.
Correlation between ryyi and rxyi.
Correlation between ryyi and ux.
Correlation between rxyi and ux.
Square-root of incumbent reliability estimate.
Variance of square-root of incumbent reliability estimate.
Correlation between qyi and rxyi.
Correlation between qyi and ux.
Square-root of applicant reliability estimate.
Variance of square-root of applicant reliability estimate.
Correlation between qya and rxyi.
Correlation between qya and ux.
Applicant reliability value.
Variance of applicant reliability values.
Correlation between ryya and rxyi.
Correlation between ryya and ux.
#### Partial derivatives to estimate the variance of qxa using ux ####
Indirect range restriction: $$b_{u_{X}}=\frac{(q_{X_{i}}^{2}-1)u_{X}}{\sqrt{(q_{X_{i}}^{2}-1)u_{X}^{2}+1}}$$ $$b_{q_{X_{i}}}=\frac{q_{X_{i}}^{2}u_{X}^{2}}{\sqrt{(q_{X_{i}}^{2}-1)u_{X}^{2}+1}}$$
Direct range restriction: $$b_{u_{X}}=\frac{q_{X_{i}}^{2}(q_{X_{i}}^{2}-1)u_{X}}{\sqrt{-\frac{q_{X_{i}}^{2}}{q_{X_{i}}^{2}(u_{X}^{2}-1)-u_{X}^{2}}}(q_{X_{i}}^{2}(u_{X}^{2}-1)-u_{X}^{2})^{2}}$$ $$b_{q_{X_{i}}}=\frac{q_{X_{i}}u_{X}^{2}}{\sqrt{-\frac{q_{X_{i}}^{2}}{q_{X_{i}}^{2}(u_{X}^{2}-1)-u_{X}^{2}}}(q_{X_{i}}^{2}(u_{X}^{2}-1)-u_{X}^{2})^{2}}$$
#### Partial derivatives to estimate the variance of rxxa using ux ####
Indirect range restriction: $$b_{u_{X}}=2\left(\rho_{XX_{i}}-1\right)u_{X}$$ $$\rho_{XX_{i}}: b_{\rho_{XX_{i}}}=u_{X}^{2}$$
Direct range restriction: $$b_{u_{X}}=\frac{2(\rho_{XX_{i}}-1)\rho_{XX_{i}}u_{X}}{(-\rho_{XX_{i}}u_{X}^{2}+\rho_{XX_{i}}+u_{X}^{2})^{2}}$$ $$b_{\rho_{XX_{i}}}=\frac{u_{X}^{2}}{(-\rho_{XX_{i}}u_{X}^{2}+\rho_{XX_{i}}+u_{X}^{2})^{2}}$$
#### Partial derivatives to estimate the variance of rxxa using ut ####
$$b_{u_{T}}=\frac{2(\rho_{XX_{i}}-1)*\rho_{XX_{i}}u_{T}}{(-\rho_{XX_{i}}u_{T}^{2}+\rho_{XX_{i}}+u_{T}^{2})^{2}}$$ $$b_{\rho_{XX_{i}}}=\frac{u_{T}^{2}}{(-\rho_{XX_{i}}u_{T}^{2}+\rho_{XX_{i}}+u_{T}^{2})^{2}}$$
#### Partial derivatives to estimate the variance of qxa using ut ####
$$b_{u_{T}}=\frac{q_{X_{i}}^{2}(q_{X_{i}}^{2}-1)u_{T}}{\sqrt{\frac{-q_{X_{i}}^{2}}{q_{X_{i}}^{2}*(u_{T}^{2}-1)-u_{T}^{2}}}(q_{X_{i}}^{2}(u_{T}^{2}-1)-u_{T}^{2})^{2}}$$ $$b_{q_{X_{i}}}=\frac{q_{X_{i}}u_{T}^{2}}{\sqrt{\frac{q_{X_{i}}^{2}}{u_{T}^{2}-q_{X_{i}}^{2}(u_{T}^{2}-1)}}(u_{T}^{2}-q_{X_{i}}^{2}(u_{T}^{2}-1))^{2}}$$
#### Partial derivatives to estimate the variance of qxi using ux ####
Indirect range restriction: $$b_{u_{X}}=\frac{1-qxa^{2}}{u_{X}^{3}\sqrt{\frac{q_{X_{a}}^{2}+u_{X}^{2}-1}{u_{X}^{2}}}}$$ $$b_{q_{X_{a}}}=\frac{q_{X_{a}}}{u_{X}^{2}\sqrt{\frac{q_{X_{a}-1}^{2}}{u_{X}^{2}}+1}}$$
Direct range restriction: $$b_{u_{X}}=-\frac{q_{X_{a}}^{2}(q_{X_{a}}^{2}-1)u_{X}}{\sqrt{\frac{q_{X_{a}}^{2}u_{X}^{2}}{q_{X_{a}}^{2}(u_{X}^{2}-1)+1}}(q_{X_{a}}^{2}(u_{X}^{2}-1)+1)^{2}}$$ $$b_{q_{X_{a}}}=\frac{q_{X_{a}}u_{X}^{2}}{\sqrt{\frac{q_{X_{a}}^{2}u_{X}^{2}}{q_{X_{a}}^{2}(u_{X}^{2}-1)+1}}(q_{X_{a}}^{2}(u_{X}^{2}-1)+1)^{2}}$$
#### Partial derivatives to estimate the variance of rxxi using ux ####
Indirect range restriction: $$b_{u_{X}}=\frac{2-2\rho_{XX_{a}}}{u_{X}^{3}}$$ $$b_{\rho_{XX_{a}}}=\frac{1}{u_{X}^{2}}$$
Direct range restriction: $$b_{u_{X}}=-\frac{2(\rho_{XX_{a}}-1)\rho_{XX_{a}}u_{X}}{(\rho_{XX_{a}}(u_{X}^{2}-1)+1)^{2}}$$ $$b_{\rho_{XX_{a}}}=\frac{u_{X}^{2}}{(\rho_{XX_{a}}(u_{X}^{2}-1)+1)^{2}}$$
#### Partial derivatives to estimate the variance of rxxi using ut ####
$$u_{T}: b_{u_{T}}=-\frac{2(\rho_{XX_{a}}-1)\rho_{XX_{a}}u_{T}}{(\rho_{XX_{a}}(u_{T}^{2}-1)+1)^{2}}$$ $$b_{\rho_{XX_{a}}}=\frac{u_{T}^{2}}{(\rho_{XX_{a}}(u_{T}^{2}-1)+1)^{2}}$$
#### Partial derivatives to estimate the variance of qxi using ut ####
$$b_{u_{T}}=-\frac{(q_{X_{a}}-1)q_{X_{a}}^{2}(q_{X_{a}}+1)u_{T}}{\sqrt{\frac{q_{X_{a}}^{2}u_{T}^{2}}{q_{X_{a}}^{2}u_{T}^{2}-q_{X_{a}}^{2}+1}}(q_{X_{a}}^{2}u_{T}^{2}-q_{X_{a}}^{2}+1)^{2}}$$ $$b_{q_{X_{a}}}=\frac{q_{X_{a}}u_{T}^{2}}{\sqrt{\frac{q_{X_{a}}^{2}u_{T}^{2}}{q_{X_{a}}^{2}u_{T}^{2}-q_{X_{a}}^{2}+1}}(q_{X_{a}}^{2}u_{T}^{2}-q_{X_{a}}^{2}+1)^{2}}$$
#### Partial derivatives to estimate the variance of ut using qxi ####
$$b_{u_{X}}=\frac{q_{X_{i}}^{2}u_{X}}{\sqrt{\frac{q_{X_{i}}^{2}u_{X}^{2}}{(q_{X_{i}}^{2}-1)u_{X}^{2}+1}}((q_{X_{i}}^{2}-1)u_{X}^{2}+1)^{2}}$$ $$b_{q_{X_{i}}}=-\frac{u_{X}^{2}(u_{X}^{2}-1)}{\sqrt{\frac{q_{X_{i}}^{2}u_{X}^{2}}{(q_{X_{i}}^{2}-1)u_{X}^{2}+1}}((q_{X_{i}}^{2}-1)u_{X}^{2}+1)^{2}}$$
#### Partial derivatives to estimate the variance of ut using rxxi ####
$$b_{u_{X}}=\frac{\rho_{XX_{i}}u_{X}}{\sqrt{\frac{\rho_{XX_{i}}u_{X}^{2}}{(\rho_{XX_{i}}-1)u_{X}^{2}+1}}((\rho_{XX_{i}}-1)u_{X}^{2}+1)^{2}}$$ $$b_{\rho_{XX_{i}}}=-\frac{u_{X}^{2}(u_{X}^{2}-1)}{2\sqrt{\frac{\rho_{XX_{i}}u_{X}^{2}}{(\rho_{XX_{i}}-1)u_{X}^{2}+1}}((\rho_{XX_{i}}-1)u_{X}^{2}+1)^{2}}$$
#### Partial derivatives to estimate the variance of ut using qxa ####
$$b_{u_{X}}=\frac{u_{X}}{q_{X_{a}}^{2}\sqrt{\frac{q_{X_{a}}^{2}+u_{X}^{2}-1}{q_{X_{a}}^{2}}}}$$ $$b_{q_{X_{a}}}=\frac{1-u_{X}^{2}}{q_{X_{a}}^{3}\sqrt{\frac{q_{X_{a}}^{2}+u_{X}^{2}-1}{q_{X_{a}}^{2}}}}$$
#### Partial derivatives to estimate the variance of ut using rxxa ####
$$b_{u_{X}}=\frac{u_{X}}{\rho_{XX_{a}}\sqrt{\frac{\rho_{XX_{a}}+u_{X}^{2}-1}{\rho_{XX_{a}}}}}$$ $$b_{\rho_{XX_{a}}}=\frac{1-u_{X}^{2}}{2\rho_{XX_{a}}^{2}\sqrt{\frac{\rho_{XX_{a}}+u_{X}^{2}-1}{\rho_{XX_{a}}}}}$$
#### Partial derivatives to estimate the variance of ux using qxi ####
$$b_{u_{T}}=\frac{q_{X_{i}}^{2}u_{T}}{\sqrt{\frac{u_{T}^{2}}{u_{T}^{2}-q_{X_{i}}^{2}(u_{T}^{2}-1)}}(u_{T}^{2}-q_{X_{i}}^{2}(u_{T}^{2}-1))^{2}}$$ $$b_{q_{X_{i}}}=\frac{q_{X_{i}}(u_{T}^{2}-1)\left(\frac{u_{T}^{2}}{u_{T}^{2}-q_{X_{i}}^{2}(u_{T}^{2}-1)}\right)^{1.5}}{u_{T}^{2}}$$
#### Partial derivatives to estimate the variance of ux using rxxi ####
$$b_{u_{T}}=\frac{\rho_{XX_{i}}u_{T}}{\sqrt{\frac{u_{T}^{2}}{-\rho_{XX_{i}}u_{T}^{2}+\rho_{XX_{i}}+u_{T}^{2}}}(-\rho_{XX_{i}}u_{T}^{2}+\rho_{XX_{i}}+u_{T}^{2})^{2}}$$ $$b_{\rho_{XX_{i}}}=\frac{(u_{T}^{2}-1)\left(\frac{u_{T}^{2}}{-\rho_{XX_{i}}u_{T}^{2}+\rho_{XX_{i}}+u_{T}^{2}}\right)^{1.5}}{2u_{T}^{2}}$$
#### Partial derivatives to estimate the variance of ux using qxa ####
$$b_{u_{T}}=\frac{q_{X_{a}}^{2}u_{T}}{\sqrt{q_{X_{a}}^{2}(u_{T}^{2}-1)+1}}$$ $$b_{q_{X_{a}}}=\frac{q_{X_{a}}(u_{T}-1)}{\sqrt{q_{X_{a}}^{2}(u_{T}^{2}-1)+1}}$$
#### Partial derivatives to estimate the variance of ux using rxxa ####
$$b_{u_{T}}=\frac{\rho_{XX_{a}}u_{T}}{\sqrt{\rho_{XX_{a}}(u_{T}^{2}-1)+1}}$$ $$b_{\rho_{XX_{a}}}=\frac{u_{T}^{2}-1}{2\sqrt{\rho_{XX_{a}}(u_{T}^{2}-1)+1}}$$
#### Partial derivatives to estimate the variance of ryya ####
$$b_{\rho_{YY_{i}}}=\frac{1}{\rho_{XY_{i}}^{2}\left(\frac{1}{u_{X}^{2}}-1\right)+1}$$ $$b_{u_{X}}=\frac{2(\rho_{YY_{i}}-1)\rho_{XY_{i}}^{2}u_{X}}{(u_{X}^{2}-\rho_{XY_{i}}^{2}(u_{X}^{2}-1))^{2}}$$ $$b_{\rho_{XY_{i}}}=\frac{2(\rho_{YY_{i}}-1)\rho_{XY_{i}}u_{X}^{2}(u_{X}^{2}-1)}{(u_{X}^{2}-\rho_{XY_{i}}^{2}(u_{X}^{2}-1))^{2}}$$
#### Partial derivatives to estimate the variance of qya ####
$$b_{q_{Y_{i}}}=\frac{q_{Y_{i}}}{\left[1-\rho_{XY_{i}}^{2}\left(1-\frac{1}{u_{X}^{2}}\right)\right]\sqrt{1-\frac{1-q_{Y_{i}}^{2}}{1-\rho_{XY_{i}}^{2}\left(1-\frac{1}{u_{X}^{2}}\right)}}}$$ $$b_{u_{X}}=-\frac{(1-q_{Y_{i}}^{2})\rho_{XY_{i}}^{2}}{u_{X}^{3}\left[1-\rho_{XY_{i}}^{2}\left(1-\frac{1}{u_{X}^{2}}\right)\right]\sqrt{1-\frac{1-q_{Y_{i}}^{2}}{1-\rho_{XY_{i}}^{2}\left(1-\frac{1}{u_{X}^{2}}\right)}}}$$ $$b_{\rho_{XY_{i}}}=-\frac{(1-q_{Y_{i}}^{2})\rho_{XY_{i}}\left(1-\frac{1}{u_{X}^{2}}\right)}{\left[1-\rho_{XY_{i}}^{2}\left(1-\frac{1}{u_{X}^{2}}\right)\right]\sqrt{1-\frac{1-q_{Y_{i}}^{2}}{1-\rho_{XY_{i}}^{2}\left(1-\frac{1}{u_{X}^{2}}\right)}}}$$
#### Partial derivatives to estimate the variance of ryyi ####
$$\rho_{YY_{a}}: b_{\rho_{YY_{a}}}=\rho_{XY_{i}}^{2}\left(\frac{1}{u_{X}^{2}}-1\right)+1$$ $$b_{u_{X}}=-\frac{2(\rho_{YY_{a}}-1)\rho_{XY_{i}}^{2}}{u_{X}^{3}}$$ $$b_{\rho_{XY_{i}}}=-\frac{2(\rho_{YY_{a}}-1)\rho_{XY_{i}}(u_{X}^{2}-1)}{u_{X}^{2}}$$
#### Partial derivatives to estimate the variance of qyi ####
$$b_{q_{Y_{a}}}=\frac{q_{Y_{a}}\left[1-\rho_{XY_{i}}^{2}\left(1-\frac{1}{u_{X}^{2}}\right)\right]}{\sqrt{1-\left(1-q_{Y_{a}}\right)\left[1-\rho_{XY_{i}}^{2}\left(1-\frac{1}{u_{X}^{2}}\right)\right]}}$$ $$b_{u_{X}}=\frac{(1-q_{Y_{a}}^{2})\rho_{XY_{i}}\left(1-\frac{1}{u_{X}^{2}}\right)}{\sqrt{1-\left(1-q_{Y_{a}}\right)\left[1-\rho_{XY_{i}}^{2}\left(1-\frac{1}{u_{X}^{2}}\right)\right]}}$$ $$b_{\rho_{XY_{i}}}=\frac{(1-q_{Y_{a}}^{2})\rho_{XY_{i}}^{2}}{u_{X}^{3}\sqrt{1-\left(1-q_{Y_{a}}\right)\left[1-\rho_{XY_{i}}^{2}\left(1-\frac{1}{u_{X}^{2}}\right)\right]}}$$
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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