estimate_var_artifacts {psychmeta}

Taylor series approximations for the variances of estimates artifact distributions.

Description

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.

Usage

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
)

Arguments

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 (TRUE) or true-score u ratios (FALSE).

indirect_rr

Logical vector determining whether reliability values are associated with indirect range restriction (TRUE) or direct range restriction (FALSE).

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 ma_r for a full list of acceptable reliability types.

rxx

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).

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 rxx_as_qx argument).

cor_rxx_ux

Correlation between rxx and ux.

rxx_restricted

Logical vector determining whether reliability estimates were incumbent reliabilities (TRUE) or applicant reliabilities (FALSE).

rxx_as_qx

Logical vector determining whether the reliability estimates were reliabilities (TRUE) or square-roots of reliabilities (FALSE).

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.

Details

#### 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]}}

Examples

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)