estimate_var_rho_tsa: Taylor Series Approximation of variance of rho corrected for...
In psychmeta: Psychometric Meta-Analysis Toolkit
estimate_var_rho_tsa R Documentation
Taylor Series Approximation of variance of ρ corrected for psychometric artifacts
Description
Functions to estimate the variance of ρ corrected for psychometric artifacts.
These functions use Taylor series approximations (i.e., the delta method) to estimate the variance in observed effect sizes predictable from the variance in artifact distributions based on the partial derivatives.
The available Taylor-series functions include:
estimate_var_rho_tsa_meas
Variance of ρ corrected for measurement error only
estimate_var_rho_tsa_uvdrr
Variance of ρ corrected for univariate direct range restriction (i.e., Case II) and measurement error
estimate_var_rho_tsa_bvdrr
Variance of ρ corrected for bivariate direct range restriction and measurement error
estimate_var_rho_tsa_uvirr
Variance of ρ corrected for univariate indirect range restriction (i.e., Case IV) and measurement error
estimate_var_rho_tsa_bvirr
Variance of ρ corrected for bivariate indirect range restriction (i.e., Case V) and measurement error
estimate_var_rho_tsa_rb1
Variance of ρ corrected using Raju and Burke's TSA1 correction for direct range restriction and measurement error
estimate_var_rho_tsa_rb2
Variance of ρ corrected using Raju and Burke's TSA2 correction for direct range restriction and measurement error. Note that a typographical error in Raju and Burke's article has been corrected in this function so as to compute appropriate partial derivatives.
Usage
estimate_var_rho_tsa_meas(
mean_rtp,
var_rxy,
var_e,
mean_qx = 1,
var_qx = 0,
mean_qy = 1,
var_qy = 0,
...
)
estimate_var_rho_tsa_uvdrr(
mean_rtpa,
var_rxyi,
var_e,
mean_ux = 1,
var_ux = 0,
mean_qxa = 1,
var_qxa = 0,
mean_qyi = 1,
var_qyi = 0,
...
)
estimate_var_rho_tsa_bvdrr(
mean_rtpa,
var_rxyi,
var_e = 0,
mean_ux = 1,
var_ux = 0,
mean_uy = 1,
var_uy = 0,
mean_qxa = 1,
var_qxa = 0,
mean_qya = 1,
var_qya = 0,
...
)
estimate_var_rho_tsa_uvirr(
mean_rtpa,
var_rxyi,
var_e,
mean_ut = 1,
var_ut = 0,
mean_qxa = 1,
var_qxa = 0,
mean_qyi = 1,
var_qyi = 0,
...
)
estimate_var_rho_tsa_bvirr(
mean_rtpa,
var_rxyi,
var_e = 0,
mean_ux = 1,
var_ux = 0,
mean_uy = 1,
var_uy = 0,
mean_qxa = 1,
var_qxa = 0,
mean_qya = 1,
var_qya = 0,
sign_rxz = 1,
sign_ryz = 1,
...
)
estimate_var_rho_tsa_rb1(
mean_rtpa,
var_rxyi,
var_e,
mean_ux = 1,
var_ux = 0,
mean_rxx = 1,
var_rxx = 0,
mean_ryy = 1,
var_ryy = 0,
...
)
estimate_var_rho_tsa_rb2(
mean_rtpa,
var_rxyi,
var_e,
mean_ux = 1,
var_ux = 0,
mean_qx = 1,
var_qx = 0,
mean_qy = 1,
var_qy = 0,
...
)
Arguments
mean_rtp
Mean corrected correlation.
var_rxy
Variance of observed correlations.
var_e
Error variance of observed correlations
mean_qx
Mean square root of reliability for X.
var_qx
Variance of square roots of reliability estimates for X.
mean_qy
Mean square root of reliability for Y.
var_qy
Variance of square roots of reliability estimates for Y.
...
Additional arguments.
mean_rtpa
Mean corrected correlation.
var_rxyi
Variance of observed correlations.
mean_ux
Mean observed-score u ratio for X.
var_ux
Variance of observed-score u ratios for X.
mean_qxa
Mean square root of unrestricted reliability for X.
var_qxa
Variance of square roots of unrestricted reliability estimates for X.
mean_qyi
Mean square root of restricted reliability for Y.
var_qyi
Variance of square roots of restricted reliability estimates for Y.
mean_uy
Mean observed-score u ratio for Y.
var_uy
Variance of observed-score u ratios for Y.
mean_qya
Mean square root of unrestricted reliability for Y.
var_qya
Variance of square roots of unrestricted reliability estimates for Y.
mean_ut
Mean true-score u ratio for X.
var_ut
Variance of true-score u ratios for X.
sign_rxz
Sign of the relationship between X and the selection mechanism.
sign_ryz
Sign of the relationship between Y and the selection mechanism.
mean_rxx
Mean reliability for X.
var_rxx
Variance of reliability estimates for X.
mean_ryy
Mean reliability for Y.
var_ryy
Variance of reliability estimates for Y.
Details
######## Measurement error only ########
The attenuation formula for measurement error is
rxy = rtp * qx * qy
where rxy is an observed correlation, rtp is a true-score correlation, and qx and qy are the square roots of reliability coefficients for X and Y, respectively.
The Taylor series approximation of the variance of rtp can be computed using the following linear equation,
var_rtp ~= (var_rxy - var_e - (b1^2 * var_qx + b2^2 * var_qy)) / b3^2
where b1, b2, and b3 are first-order partial derivatives of the attenuation formula with respect to qx, qy, and rtp, respectively.
The first-order partial derivatives of the attenuation formula are:
b1 = rtp * qy
b2 = rtp * qx
b3 = qx * qy
######## Univariate direct range restriction (UVDRR; i.e., Case II) ########
The UVDRR attenuation procedure may be represented as
rxyi = ux * rxpa * qxa / sqrt((ux^2 - 1) * rxpa^2 * qxa^2 + 1) * qyi
The attenuation formula can also be represented as:
rxyi = qxa * qyi * rtpa * ux * A
where
A = 1 / sqrt(rtpa^2 * qxa^2* (ux^2 - 1) + 1)
The Taylor series approximation of the variance of rtpa can be computed using the following linear equation,
var_rtpa ~= (var_rxyi - var_e - (b1^2 * var_qxa + b2^2 * var_qyi + b3^2 * var_ux)) / b4^2
where b1, b2, b3, and b4 are first-order partial derivatives of the attenuation formula with respect to qxa, qyi, ux, and rtpa, respectively.
The first-order partial derivatives of the attenuation formula are:
b1 = qyi * rtpa * ux * A^3
b2 = qxa * qyi * rtpa * ux * A / qyi
b3 = -(qyi * rtpa * qxa * (rtpa^2 * qxa^2 - 1)) * A^3
b4 = (qyi * qxa * ux) * A^3
######## Univariate indirect range restriction (UVIRR; i.e., Case IV) ########
Under univariate indirect range restriction, the attenuation formula yielding rxyi is:
rxyi = (ut * qxa) / (sqrt(ut^2 * qxa^2 + 1 - qxa^2)) * (ut * rtpa) / (sqrt(ut^2 * rtpa^2 + 1 - rtpa^2))
The attenuation formula can also be represented as:
rxyi = qxa * qyi * rtpa * ut^2 * A * B
where
A = 1 / sqrt(ut^2 * rtpa^2 - rtpa^2 + 1)
and
B = 1 / sqrt(ut^2 * qxa^2 - qxa^2 + 1)
The Taylor series approximation of the variance of rtpa can be computed using the following linear equation,
var_rtpa ~= (var_rxyi - var_e - (b1^2 * var_qxa + b2^2 * var_qyi + b3^2 * var_ut)) / b4^2
where b1, b2, b3, and b4 are first-order partial derivatives of the attenuation formula with respect to qxa, qyi, ut, and rtpa, respectively.
The first-order partial derivatives of the attenuation formula are:
b1 = rxyi / qxa - rxyi * qxa * B^2 * (ut^2 - 1)
b2 = rxyi / qyi
b3 = (2 * rxyi) / ut - rxyi * ut * qxa^2 * B^2 - rxyi * ut * rtpa^2 * A^2
b4 = rxyi / rtpa - rxyi * rtpa * A^2 * (ut^2 - 1)
######## Bivariate direct range restriction (BVDRR) ########
Under bivariate direct range restriction, the attenuation formula yielding rxyi is:
rxyi = (sqrt((1/(qya * qxa) - rtpa^2 * qya * qxa)^2 + 4 * rtpa^2 * ux^2 * uy^2) + rtpa^2 * qya * qxa - 1/(qya * qxa))/(2 * rtpa * ux * uy)
where
A = sqrt((1/(qya * qxa) - qya * rtpa^2 * qxa)^2 + 4 * rtpa^2 * ux^2 * uy^2)
The Taylor series approximation of the variance of rtpa can be computed using the following linear equation,
var_rtpa ~= (var_rxyi - var_e - (b1^2 * var_qxa + b2^2 * var_qya + b3^2 * var_ux + b4^2 * var_uy)) / b5^2
where b1, b2, b3, b4, and b5 are first-order partial derivatives of the attenuation formula with respect to qxa, qya, ux, uy, and rtpa, respectively.
First, we define terms to simplify the computation of partial derivatives:
B = (qya^2 * rtpa^2 * qxa^2 + qya * qxa * A - 1)
C = 2 * qya^2 * rtpa * qxa^2 * ux * uy * sqrt((1/(qya * qxa) - qya * rtpa^2 * qxa)^2 + 4 * rtpa^2 * ux^2 * uy^2)
The first-order partial derivatives of the attenuation formula are:
b1 = ((rtpa^2 * qxa^2 * qya^2 + 1) * B) / (qxa * C)
b2 = ((rtpa^2 * qxa^2 * qya^2 + 1) * B) / (qya * C)
b3 = -((qya * rtpa * qxa - 1) * (qya * rtpa * qxa + 1) * B) / (ux * C)
b4 = -((qya * rtpa * qxa - 1) * (qya * rtpa * qxa + 1) * B) / (uy * C)
b5 = ((rtpa^2 * qxa^2 * qya^2 + 1) * B) / (rtpa * C)
######## Bivariate indirect range restriction (BVIRR; i.e., Case V) ########
Under bivariate indirect range restriction, the attenuation formula yielding rxyi is:
rxyi = (rtpa * qxa * qya - lambda * sqrt(abs(1 - ux^2) * abs(1 - uy^2))) / (uy * ux)
The Taylor series approximation of the variance of rtpa can be computed using the following linear equation,
var_rtpa ~= (var_rxyi - var_e - (b1^2 * var_qxa + b2^2 * var_qya + b3^2 * var_ux + b4^2 * var_uy)) / b5^2
where b1, b2, b3, b4, and b5 are first-order partial derivatives of the attenuation formula with respect to qxa, qya, ux, uy, and rtpa, respectively.
First, we define terms to simplify the computation of partial derivatives:
b1 = rtpa * qya / (ux * uy)
b2 = rtpa * qxa / (ux * uy)
b3 = (lambda * (1 - ux^2) * sqrt(abs(1 - uy^2))) / (uy * abs(1 - ux^2)^1.5) - rxyi / ux
b4 = (lambda * (1 - uy^2) * sqrt(abs(1 - ux^2))) / (ux * abs(1 - uy^2)^1.5) - rxyi / uy
b5 = (qxa * qya) / (ux * uy)
######## Raju and Burke's TSA1 procedure ########
Raju and Burke's attenuation formula may be represented as
rxyi = (rtpa * ux * sqrt(ryya * rxxa)) / sqrt(rtpa^2 * ryya * rxxa * ux^2 - rtpa^2 * ryya * rxxa + 1)
The Taylor series approximation of the variance of rtpa can be computed using the following linear equation,
var_rtpa ~= (var_rxyi - var_e - (B^2 * var_ryya + C^2 * var_rxxa + D^2 * var_ux)) / A^2
where A, B, C, and D are first-order partial derivatives of the attenuation formula with respect to rtpa, rxxa, ryya, and ux, respectively.
The first-order partial derivatives of the attenuation formula are:
A = rxyi / rtpa + (rxyi^3 * (1 - ux^2)) / (rtpa * ux^2)
B = .5 * (rxyi / ryya + (rxyi^3 * (1 - ux^2)) / (ryya * ux^2))
C = .5 * (rxyi / rxxa + (rxyi^3 * (1 - ux^2)) / (rxxa * ux^2))
D = (rxyi - rxyi^3) / ux
######## Raju and Burke's TSA2 procedure ########
Raju and Burke's attenuation formula may be represented as
rxyi = (rtpa * qya * qxa * ux) / sqrt(rtpa^2 * qya^2 * qxa^2 * ux^2 - rtpa^2 * qya^2 * qxa^2 + 1)
The Taylor series approximation of the variance of rtpa can be computed using the following linear equation,
var_rtpa ~= (var_rxyi - var_e - (F^2 * var_qya + G^2 * var_qxa + H^2 * var_ux)) / E^2
where E, F, G, and H are first-order partial derivatives of the attenuation formula with respect to rtpa, qxa, qya, and ux, respectively.
The first-order partial derivatives of the attenuation formula (with typographic errors in the original article corrected) are:
E = rxyi / rtpa + (rxyi^3 * (1 - ux^2)) / (rtpa * ux^2)
F = (rxyi / qya + (rxyi^3 * (1 - ux^2)) / (qya * ux^2))
G = (rxyi / qxa + (rxyi^3 * (1 - ux^2)) / (qxa * ux^2))
H = (rxyi - rxyi^3) / ux
Value
Vector of meta-analytic variances estimated via Taylor series approximation.
Notes
A typographical error in Raju and Burke's article has been corrected in estimate_var_rho_tsa_rb2 so as to compute appropriate partial derivatives.
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. doi: 10.1177/1094428119859398
Hunter, J. E., Schmidt, F. L., & Le, H. (2006).
Implications of direct and indirect range restriction for meta-analysis methods and findings.
Journal of Applied Psychology, 91(3), 594–612. doi: 10.1037/0021-9010.91.3.594
Raju, N. S., & Burke, M. J. (1983). Two new procedures for studying validity generalization.
Journal of Applied Psychology, 68(3), 382–395. doi: 10.1037/0021-9010.68.3.382
Examples
estimate_var_rho_tsa_meas(mean_rtp = .5, var_rxy = .02, var_e = .01,
mean_qx = .8, var_qx = .005,
mean_qy = .8, var_qy = .005)
estimate_var_rho_tsa_uvdrr(mean_rtpa = .5, var_rxyi = .02, var_e = .01,
mean_ux = .8, var_ux = .005,
mean_qxa = .8, var_qxa = .005,
mean_qyi = .8, var_qyi = .005)
estimate_var_rho_tsa_bvdrr(mean_rtpa = .5, var_rxyi = .02, var_e = .01,
mean_ux = .8, var_ux = .005,
mean_uy = .8, var_uy = .005,
mean_qxa = .8, var_qxa = .005,
mean_qya = .8, var_qya = .005)
estimate_var_rho_tsa_uvirr(mean_rtpa = .5, var_rxyi = .02, var_e = .01,
mean_ut = .8, var_ut = .005,
mean_qxa = .8, var_qxa = .005,
mean_qyi = .8, var_qyi = .005)
estimate_var_rho_tsa_bvirr(mean_rtpa = .5, var_rxyi = .02, var_e = .01,
mean_ux = .8, var_ux = .005,
mean_uy = .8, var_uy = .005,
mean_qxa = .8, var_qxa = .005,
mean_qya = .8, var_qya = .005,
sign_rxz = 1, sign_ryz = 1)
estimate_var_rho_tsa_rb1(mean_rtpa = .5, var_rxyi = .02, var_e = .01,
mean_ux = .8, var_ux = .005,
mean_rxx = .8, var_rxx = .005,
mean_ryy = .8, var_ryy = .005)
estimate_var_rho_tsa_rb2(mean_rtpa = .5, var_rxyi = .02, var_e = .01,
mean_ux = .8, var_ux = .005,
mean_qx = .8, var_qx = .005,
mean_qy = .8, var_qy = .005)
psychmeta documentation built on Aug. 26, 2022, 5:14 p.m.
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| estimate_var_rho_tsa | R Documentation |
Taylor Series Approximation of variance of ρ corrected for psychometric artifacts
Description
Functions to estimate the variance of ρ corrected for psychometric artifacts. These functions use Taylor series approximations (i.e., the delta method) to estimate the variance in observed effect sizes predictable from the variance in artifact distributions based on the partial derivatives.
The available Taylor-series functions include:
estimate_var_rho_tsa_meas
Variance of ρ corrected for measurement error onlyestimate_var_rho_tsa_uvdrr
Variance of ρ corrected for univariate direct range restriction (i.e., Case II) and measurement errorestimate_var_rho_tsa_bvdrr
Variance of ρ corrected for bivariate direct range restriction and measurement errorestimate_var_rho_tsa_uvirr
Variance of ρ corrected for univariate indirect range restriction (i.e., Case IV) and measurement errorestimate_var_rho_tsa_bvirr
Variance of ρ corrected for bivariate indirect range restriction (i.e., Case V) and measurement errorestimate_var_rho_tsa_rb1
Variance of ρ corrected using Raju and Burke's TSA1 correction for direct range restriction and measurement errorestimate_var_rho_tsa_rb2
Variance of ρ corrected using Raju and Burke's TSA2 correction for direct range restriction and measurement error. Note that a typographical error in Raju and Burke's article has been corrected in this function so as to compute appropriate partial derivatives.
Usage
estimate_var_rho_tsa_meas( mean_rtp, var_rxy, var_e, mean_qx = 1, var_qx = 0, mean_qy = 1, var_qy = 0, ... ) estimate_var_rho_tsa_uvdrr( mean_rtpa, var_rxyi, var_e, mean_ux = 1, var_ux = 0, mean_qxa = 1, var_qxa = 0, mean_qyi = 1, var_qyi = 0, ... ) estimate_var_rho_tsa_bvdrr( mean_rtpa, var_rxyi, var_e = 0, mean_ux = 1, var_ux = 0, mean_uy = 1, var_uy = 0, mean_qxa = 1, var_qxa = 0, mean_qya = 1, var_qya = 0, ... ) estimate_var_rho_tsa_uvirr( mean_rtpa, var_rxyi, var_e, mean_ut = 1, var_ut = 0, mean_qxa = 1, var_qxa = 0, mean_qyi = 1, var_qyi = 0, ... ) estimate_var_rho_tsa_bvirr( mean_rtpa, var_rxyi, var_e = 0, mean_ux = 1, var_ux = 0, mean_uy = 1, var_uy = 0, mean_qxa = 1, var_qxa = 0, mean_qya = 1, var_qya = 0, sign_rxz = 1, sign_ryz = 1, ... ) estimate_var_rho_tsa_rb1( mean_rtpa, var_rxyi, var_e, mean_ux = 1, var_ux = 0, mean_rxx = 1, var_rxx = 0, mean_ryy = 1, var_ryy = 0, ... ) estimate_var_rho_tsa_rb2( mean_rtpa, var_rxyi, var_e, mean_ux = 1, var_ux = 0, mean_qx = 1, var_qx = 0, mean_qy = 1, var_qy = 0, ... )
Arguments
mean_rtp |
Mean corrected correlation. |
var_rxy |
Variance of observed correlations. |
var_e |
Error variance of observed correlations |
mean_qx |
Mean square root of reliability for X. |
var_qx |
Variance of square roots of reliability estimates for X. |
mean_qy |
Mean square root of reliability for Y. |
var_qy |
Variance of square roots of reliability estimates for Y. |
... |
Additional arguments. |
mean_rtpa |
Mean corrected correlation. |
var_rxyi |
Variance of observed correlations. |
mean_ux |
Mean observed-score u ratio for X. |
var_ux |
Variance of observed-score u ratios for X. |
mean_qxa |
Mean square root of unrestricted reliability for X. |
var_qxa |
Variance of square roots of unrestricted reliability estimates for X. |
mean_qyi |
Mean square root of restricted reliability for Y. |
var_qyi |
Variance of square roots of restricted reliability estimates for Y. |
mean_uy |
Mean observed-score u ratio for Y. |
var_uy |
Variance of observed-score u ratios for Y. |
mean_qya |
Mean square root of unrestricted reliability for Y. |
var_qya |
Variance of square roots of unrestricted reliability estimates for Y. |
mean_ut |
Mean true-score u ratio for X. |
var_ut |
Variance of true-score u ratios for X. |
sign_rxz |
Sign of the relationship between X and the selection mechanism. |
sign_ryz |
Sign of the relationship between Y and the selection mechanism. |
mean_rxx |
Mean reliability for X. |
var_rxx |
Variance of reliability estimates for X. |
mean_ryy |
Mean reliability for Y. |
var_ryy |
Variance of reliability estimates for Y. |
Details
######## Measurement error only ########
The attenuation formula for measurement error is
rxy = rtp * qx * qy
where rxy is an observed correlation, rtp is a true-score correlation, and qx and qy are the square roots of reliability coefficients for X and Y, respectively.
The Taylor series approximation of the variance of rtp can be computed using the following linear equation,
var_rtp ~= (var_rxy - var_e - (b1^2 * var_qx + b2^2 * var_qy)) / b3^2
where b1, b2, and b3 are first-order partial derivatives of the attenuation formula with respect to qx, qy, and rtp, respectively. The first-order partial derivatives of the attenuation formula are:
b1 = rtp * qy
b2 = rtp * qx
b3 = qx * qy
######## Univariate direct range restriction (UVDRR; i.e., Case II) ########
The UVDRR attenuation procedure may be represented as
rxyi = ux * rxpa * qxa / sqrt((ux^2 - 1) * rxpa^2 * qxa^2 + 1) * qyi
The attenuation formula can also be represented as:
rxyi = qxa * qyi * rtpa * ux * A
where
A = 1 / sqrt(rtpa^2 * qxa^2* (ux^2 - 1) + 1)
The Taylor series approximation of the variance of rtpa can be computed using the following linear equation,
var_rtpa ~= (var_rxyi - var_e - (b1^2 * var_qxa + b2^2 * var_qyi + b3^2 * var_ux)) / b4^2
where b1, b2, b3, and b4 are first-order partial derivatives of the attenuation formula with respect to qxa, qyi, ux, and rtpa, respectively. The first-order partial derivatives of the attenuation formula are:
b1 = qyi * rtpa * ux * A^3
b2 = qxa * qyi * rtpa * ux * A / qyi
b3 = -(qyi * rtpa * qxa * (rtpa^2 * qxa^2 - 1)) * A^3
b4 = (qyi * qxa * ux) * A^3
######## Univariate indirect range restriction (UVIRR; i.e., Case IV) ########
Under univariate indirect range restriction, the attenuation formula yielding rxyi is:
rxyi = (ut * qxa) / (sqrt(ut^2 * qxa^2 + 1 - qxa^2)) * (ut * rtpa) / (sqrt(ut^2 * rtpa^2 + 1 - rtpa^2))
The attenuation formula can also be represented as:
rxyi = qxa * qyi * rtpa * ut^2 * A * B
where
A = 1 / sqrt(ut^2 * rtpa^2 - rtpa^2 + 1)
and
B = 1 / sqrt(ut^2 * qxa^2 - qxa^2 + 1)
The Taylor series approximation of the variance of rtpa can be computed using the following linear equation,
var_rtpa ~= (var_rxyi - var_e - (b1^2 * var_qxa + b2^2 * var_qyi + b3^2 * var_ut)) / b4^2
where b1, b2, b3, and b4 are first-order partial derivatives of the attenuation formula with respect to qxa, qyi, ut, and rtpa, respectively. The first-order partial derivatives of the attenuation formula are:
b1 = rxyi / qxa - rxyi * qxa * B^2 * (ut^2 - 1)
b2 = rxyi / qyi
b3 = (2 * rxyi) / ut - rxyi * ut * qxa^2 * B^2 - rxyi * ut * rtpa^2 * A^2
b4 = rxyi / rtpa - rxyi * rtpa * A^2 * (ut^2 - 1)
######## Bivariate direct range restriction (BVDRR) ########
Under bivariate direct range restriction, the attenuation formula yielding rxyi is:
rxyi = (sqrt((1/(qya * qxa) - rtpa^2 * qya * qxa)^2 + 4 * rtpa^2 * ux^2 * uy^2) + rtpa^2 * qya * qxa - 1/(qya * qxa))/(2 * rtpa * ux * uy)
where
A = sqrt((1/(qya * qxa) - qya * rtpa^2 * qxa)^2 + 4 * rtpa^2 * ux^2 * uy^2)
The Taylor series approximation of the variance of rtpa can be computed using the following linear equation,
var_rtpa ~= (var_rxyi - var_e - (b1^2 * var_qxa + b2^2 * var_qya + b3^2 * var_ux + b4^2 * var_uy)) / b5^2
where b1, b2, b3, b4, and b5 are first-order partial derivatives of the attenuation formula with respect to qxa, qya, ux, uy, and rtpa, respectively. First, we define terms to simplify the computation of partial derivatives:
B = (qya^2 * rtpa^2 * qxa^2 + qya * qxa * A - 1)
C = 2 * qya^2 * rtpa * qxa^2 * ux * uy * sqrt((1/(qya * qxa) - qya * rtpa^2 * qxa)^2 + 4 * rtpa^2 * ux^2 * uy^2)
The first-order partial derivatives of the attenuation formula are:
b1 = ((rtpa^2 * qxa^2 * qya^2 + 1) * B) / (qxa * C)
b2 = ((rtpa^2 * qxa^2 * qya^2 + 1) * B) / (qya * C)
b3 = -((qya * rtpa * qxa - 1) * (qya * rtpa * qxa + 1) * B) / (ux * C)
b4 = -((qya * rtpa * qxa - 1) * (qya * rtpa * qxa + 1) * B) / (uy * C)
b5 = ((rtpa^2 * qxa^2 * qya^2 + 1) * B) / (rtpa * C)
######## Bivariate indirect range restriction (BVIRR; i.e., Case V) ########
Under bivariate indirect range restriction, the attenuation formula yielding rxyi is:
rxyi = (rtpa * qxa * qya - lambda * sqrt(abs(1 - ux^2) * abs(1 - uy^2))) / (uy * ux)
The Taylor series approximation of the variance of rtpa can be computed using the following linear equation,
var_rtpa ~= (var_rxyi - var_e - (b1^2 * var_qxa + b2^2 * var_qya + b3^2 * var_ux + b4^2 * var_uy)) / b5^2
where b1, b2, b3, b4, and b5 are first-order partial derivatives of the attenuation formula with respect to qxa, qya, ux, uy, and rtpa, respectively. First, we define terms to simplify the computation of partial derivatives:
b1 = rtpa * qya / (ux * uy)
b2 = rtpa * qxa / (ux * uy)
b3 = (lambda * (1 - ux^2) * sqrt(abs(1 - uy^2))) / (uy * abs(1 - ux^2)^1.5) - rxyi / ux
b4 = (lambda * (1 - uy^2) * sqrt(abs(1 - ux^2))) / (ux * abs(1 - uy^2)^1.5) - rxyi / uy
b5 = (qxa * qya) / (ux * uy)
######## Raju and Burke's TSA1 procedure ########
Raju and Burke's attenuation formula may be represented as
rxyi = (rtpa * ux * sqrt(ryya * rxxa)) / sqrt(rtpa^2 * ryya * rxxa * ux^2 - rtpa^2 * ryya * rxxa + 1)
The Taylor series approximation of the variance of rtpa can be computed using the following linear equation,
var_rtpa ~= (var_rxyi - var_e - (B^2 * var_ryya + C^2 * var_rxxa + D^2 * var_ux)) / A^2
where A, B, C, and D are first-order partial derivatives of the attenuation formula with respect to rtpa, rxxa, ryya, and ux, respectively. The first-order partial derivatives of the attenuation formula are:
A = rxyi / rtpa + (rxyi^3 * (1 - ux^2)) / (rtpa * ux^2)
B = .5 * (rxyi / ryya + (rxyi^3 * (1 - ux^2)) / (ryya * ux^2))
C = .5 * (rxyi / rxxa + (rxyi^3 * (1 - ux^2)) / (rxxa * ux^2))
D = (rxyi - rxyi^3) / ux
######## Raju and Burke's TSA2 procedure ########
Raju and Burke's attenuation formula may be represented as
rxyi = (rtpa * qya * qxa * ux) / sqrt(rtpa^2 * qya^2 * qxa^2 * ux^2 - rtpa^2 * qya^2 * qxa^2 + 1)
The Taylor series approximation of the variance of rtpa can be computed using the following linear equation,
var_rtpa ~= (var_rxyi - var_e - (F^2 * var_qya + G^2 * var_qxa + H^2 * var_ux)) / E^2
where E, F, G, and H are first-order partial derivatives of the attenuation formula with respect to rtpa, qxa, qya, and ux, respectively. The first-order partial derivatives of the attenuation formula (with typographic errors in the original article corrected) are:
E = rxyi / rtpa + (rxyi^3 * (1 - ux^2)) / (rtpa * ux^2)
F = (rxyi / qya + (rxyi^3 * (1 - ux^2)) / (qya * ux^2))
G = (rxyi / qxa + (rxyi^3 * (1 - ux^2)) / (qxa * ux^2))
H = (rxyi - rxyi^3) / ux
Value
Vector of meta-analytic variances estimated via Taylor series approximation.
Notes
A typographical error in Raju and Burke's article has been corrected in estimate_var_rho_tsa_rb2 so as to compute appropriate partial derivatives.
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. doi: 10.1177/1094428119859398
Hunter, J. E., Schmidt, F. L., & Le, H. (2006). Implications of direct and indirect range restriction for meta-analysis methods and findings. Journal of Applied Psychology, 91(3), 594–612. doi: 10.1037/0021-9010.91.3.594
Raju, N. S., & Burke, M. J. (1983). Two new procedures for studying validity generalization. Journal of Applied Psychology, 68(3), 382–395. doi: 10.1037/0021-9010.68.3.382
Examples
estimate_var_rho_tsa_meas(mean_rtp = .5, var_rxy = .02, var_e = .01,
mean_qx = .8, var_qx = .005,
mean_qy = .8, var_qy = .005)
estimate_var_rho_tsa_uvdrr(mean_rtpa = .5, var_rxyi = .02, var_e = .01,
mean_ux = .8, var_ux = .005,
mean_qxa = .8, var_qxa = .005,
mean_qyi = .8, var_qyi = .005)
estimate_var_rho_tsa_bvdrr(mean_rtpa = .5, var_rxyi = .02, var_e = .01,
mean_ux = .8, var_ux = .005,
mean_uy = .8, var_uy = .005,
mean_qxa = .8, var_qxa = .005,
mean_qya = .8, var_qya = .005)
estimate_var_rho_tsa_uvirr(mean_rtpa = .5, var_rxyi = .02, var_e = .01,
mean_ut = .8, var_ut = .005,
mean_qxa = .8, var_qxa = .005,
mean_qyi = .8, var_qyi = .005)
estimate_var_rho_tsa_bvirr(mean_rtpa = .5, var_rxyi = .02, var_e = .01,
mean_ux = .8, var_ux = .005,
mean_uy = .8, var_uy = .005,
mean_qxa = .8, var_qxa = .005,
mean_qya = .8, var_qya = .005,
sign_rxz = 1, sign_ryz = 1)
estimate_var_rho_tsa_rb1(mean_rtpa = .5, var_rxyi = .02, var_e = .01,
mean_ux = .8, var_ux = .005,
mean_rxx = .8, var_rxx = .005,
mean_ryy = .8, var_ryy = .005)
estimate_var_rho_tsa_rb2(mean_rtpa = .5, var_rxyi = .02, var_e = .01,
mean_ux = .8, var_ux = .005,
mean_qx = .8, var_qx = .005,
mean_qy = .8, var_qy = .005)
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