var_error_r_bvirr: Taylor series approximation of the sampling variance of...

In psychmeta/psychmeta: Psychometric Meta-Analysis Toolkit

Taylor series approximation of the sampling variance of correlations corrected using the bivariate indirect range restriction correction (Case V)

Description

This function propagates error in the bivariate indirect range-restriction correction formula to allow for the computation of a pseudo compound attenuation factor in individual-correction meta-analysis. Traditional methods for estimating compound attenuation factors (i.e., dividing the observed correlation by the corrected correlation) do not work with the BVIRR correction because BVIRR has an additive term that makes the corrected correlation inappropriate for use in estimating the effect of the correction on the variance of the sampling distribution of correlations. The equation-implied adjustment for the BVIRR correction (i.e., the first derivative of the correction equation with respect to the observed correlation) underestimates the error of corrected correlations, so this function helps to account for that additional error.

Usage

var_error_r_bvirr(
  rxyi,
  var_e = NULL,
  ni,
  na = NA,
  ux = rep(1, length(rxyi)),
  ux_observed = rep(TRUE, length(rxyi)),
  uy = rep(1, length(rxyi)),
  uy_observed = rep(TRUE, length(rxyi)),
  qx = rep(1, length(rxyi)),
  qx_restricted = rep(TRUE, length(rxyi)),
  qx_type = rep("alpha", length(rxyi)),
  k_items_x = rep(NA, length(rxyi)),
  qy = rep(1, length(rxyi)),
  qy_restricted = rep(TRUE, length(rxyi)),
  qy_type = rep("alpha", length(rxyi)),
  k_items_y = rep(NA, length(rxyi)),
  mean_rxyi = NULL,
  mean_ux = NULL,
  mean_uy = NULL,
  mean_qxa = NULL,
  mean_qya = NULL,
  var_rxyi = NULL,
  var_ux = NULL,
  var_uy = NULL,
  var_qxa = NULL,
  var_qya = NULL,
  cor_rxyi_ux = 0,
  cor_rxyi_uy = 0,
  cor_rxyi_qxa = 0,
  cor_rxyi_qya = 0,
  cor_ux_uy = 0,
  cor_ux_qxa = 0,
  cor_ux_qya = 0,
  cor_uy_qxa = 0,
  cor_uy_qya = 0,
  cor_qxa_qya = 0,
  sign_rxz = 1,
  sign_ryz = 1,
  r_deriv_only = FALSE
)

Arguments

rxyi	Vector of observed correlations.
var_e	Vector of estimated sampling variances for rxyi values.
ni	Vector of incumbent sample sizes (necessary when variances of correlations/artifacts are not supplied).
na	Optional vector of applicant sample sizes (for estimating error variance of u ratios and applicant reliabilities).
ux	Vector of observed-score u ratios for X.
ux_observed	Logical vector in which each entry specifies whether the corresponding ux value is an observed-score u ratio (TRUE) or a true-score u ratio. All entries are TRUE by default.
uy	Vector of observed-score u ratios for Y.
uy_observed	Logical vector in which each entry specifies whether the corresponding uy value is an observed-score u ratio (TRUE) or a true-score u ratio. All entries are TRUE by default.
qx	Vector of square roots of reliability estimates for X.
qx_restricted	Logical vector determining whether each element of qx is derived from an incumbent reliability (TRUE) or an applicant reliability (FALSE).
qx_type, qy_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.
k_items_x, k_items_y	Numeric vector identifying the number of items in each scale.
qy	Vector of square roots of reliability estimates for X.
qy_restricted	Logical vector determining whether each element of qy is derived from an incumbent reliability (TRUE) or an applicant reliability (FALSE).
mean_rxyi	Mean observed correlation.
mean_ux	Mean observed-score u ratio for X (for use in estimating sampling errors in the context of a meta-analysis).
mean_uy	Mean observed-score u ratio for Y (for use in estimating sampling errors in the context of a meta-analysis).
mean_qxa	Mean square-root applicant reliability estimate for X (for use in estimating sampling errors in the context of a meta-analysis).
mean_qya	Mean square-root applicant reliability estimate for Y (for use in estimating sampling errors in the context of a meta-analysis).
var_rxyi	Optional pre-specified variance of correlations.
var_ux	Optional pre-specified variance of observed-score u ratios for X.
var_uy	Optional pre-specified variance of observed-score u ratios for Y.
var_qxa	Optional pre-specified variance of square-root applicant reliability estimate for X.
var_qya	Optional pre-specified variance of square-root applicant reliability estimate for Y.
cor_rxyi_ux	Correlation between rxyi and ux (zero by default).
cor_rxyi_uy	Correlation between rxyi and uy (zero by default).
cor_rxyi_qxa	Correlation between rxyi and qxa (zero by default).
cor_rxyi_qya	Correlation between rxyi and qya (zero by default).
cor_ux_uy	Correlation between ux and uy (zero by default).
cor_ux_qxa	Correlation between ux and qxa (zero by default).
cor_ux_qya	Correlation between ux and qya (zero by default).
cor_uy_qxa	Correlation between uy and qxa (zero by default).
cor_uy_qya	Correlation between uy and qya (zero by default).
cor_qxa_qya	Correlation between qxa and qya (zero by default).
sign_rxz	Sign of the relationship between X and the selection mechanism.
sign_ryz	Sign of the relationship between Y and the selection mechanism.
r_deriv_only	Logical scalar determining whether to use the partial derivative with respect to rxyi only (TRUE) or a full Taylor series approximation of the disattenuation formula (FALSE).

Details

Per the principles of propagation of uncertainty and assuming that q_{X_{a}}, q_{Y_{a}}, u_{X}, u_{Y}, and \rho_{XY_{i}}, are independent, we can derive a linear approximation of the sampling error of \rho_{TP_{a}}. We begin with the bivariate indirect range restriction formula,

\rho_{TP_{a}}=\frac{\rho_{XY_{i}}u_{X}u_{Y}+\lambda\sqrt{\left|1-u_{X}^{2}\right|\left|1-u_{Y}^{2}\right|}}{q_{X_{a}}q_{Y_{a}}}

which implies the following linear approximation of the sampling variance of \rho_{TP_{a}}:

SE_{\rho_{TP_{a}}}^{2}=b_{1}^{2}SE_{q_{X_{a}}}^{2}+b_{2}^{2}SE_{q_{Y_{a}}}^{2}+b_{3}^{2}SE_{u_{X}}^{2}+b_{4}^{2}SE_{u_{Y}}^{2}+b_{5}^{2}SE_{\rho_{XY_{i}}}^{2}

where b_{1}, b_{2}, b_{3}, b_{4}, and b_{5} are the first-order partial derivatives of the disattenuation formula with respect to q_{X_{a}}, q_{Y_{a}}, u_{X}, u_{Y}, and \rho_{XY_{i}}, respectively. These partial derivatives are computed as follows:

b_{1}=\frac{\partial\rho_{TP_{a}}}{\partial q_{X_{a}}}=-\frac{\rho_{TP_{a}}}{q_{X_{a}}}

b_{2}=\frac{\partial\rho_{TP_{a}}}{\partial q_{Y_{a}}}=-\frac{\rho_{TP_{a}}}{q_{Y_{a}}}

b_{3}=\frac{\partial\rho_{TP_{a}}}{\partial u_{X}}=\left[\rho_{XY_{i}}u_{Y}-\frac{\lambda u_{X}\left(1-u_{X}^{2}\right)\sqrt{\left|1-u_{Y}^{2}\right|}}{\left|1-u_{X}^{2}\right|^{1.5}}\right]/\left(q_{X_{a}}q_{Y_{a}}\right)

b_{4}=\frac{\partial\rho_{TP_{a}}}{\partial u_{Y}}=\left[\rho_{XY_{i}}u_{X}-\frac{\lambda u_{Y}\left(1-u_{Y}^{2}\right)\sqrt{\left|1-u_{X}^{2}\right|}}{\left|1-u_{Y}^{2}\right|^{1.5}}\right]/\left(q_{X_{a}}q_{Y_{a}}\right)

b_{5}=\frac{\partial\rho_{TP_{a}}}{\partial\rho_{XY_{i}}}=\frac{u_{X}u_{Y}}{q_{X_{a}}q_{Y_{a}}}

Value

A vector of corrected correlations' sampling-error variances.

References

Dahlke, J. A., & Wiernik, B. M. (2020). Not restricted to selection research: Accounting for indirect range restriction in organizational research. Organizational Research Methods, 23(4), 717–749. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1177/1094428119859398")}

Examples

var_error_r_bvirr(rxyi = .3, var_e = var_error_r(r = .3, n = 100), ni = 100,
                ux = .8, uy = .8,
                qx = .9, qx_restricted = TRUE,
                qy = .9, qy_restricted = TRUE,
                sign_rxz = 1, sign_ryz = 1)

psychmeta/psychmeta documentation built on Feb. 12, 2024, 1:21 a.m.