Functions to estimate the variance of 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 corrected for measurement error only
estimate_var_rho_tsa_uvdrr
Variance of corrected for univariate direct range restriction (i.e., Case II) and measurement error
estimate_var_rho_tsa_bvdrr
Variance of corrected for bivariate direct range restriction and measurement error
estimate_var_rho_tsa_uvirr
Variance of corrected for univariate indirect range restriction (i.e., Case IV) and measurement error
estimate_var_rho_tsa_bvirr
Variance of corrected for bivariate indirect range restriction (i.e., Case V) and measurement error
estimate_var_rho_tsa_rb1
Variance of corrected using Raju and Burke's TSA1 correction for direct range restriction and measurement error
estimate_var_rho_tsa_rb2
Variance of 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.
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,
...
)
Vector of meta-analytic variances estimated via Taylor series approximation.
Mean corrected correlation.
Variance of observed correlations.
Error variance of observed correlations
Mean square root of reliability for X.
Variance of square roots of reliability estimates for X.
Mean square root of reliability for Y.
Variance of square roots of reliability estimates for Y.
Additional arguments.
Mean corrected correlation.
Variance of observed correlations.
Mean observed-score u ratio for X.
Variance of observed-score u ratios for X.
Mean square root of unrestricted reliability for X.
Variance of square roots of unrestricted reliability estimates for X.
Mean square root of restricted reliability for Y.
Variance of square roots of restricted reliability estimates for Y.
Mean observed-score u ratio for Y.
Variance of observed-score u ratios for Y.
Mean square root of unrestricted reliability for Y.
Variance of square roots of unrestricted reliability estimates for Y.
Mean true-score u ratio for X.
Variance of true-score u ratios for X.
Sign of the relationship between X and the selection mechanism.
Sign of the relationship between Y and the selection mechanism.
Mean reliability for X.
Variance of reliability estimates for X.
Mean reliability for Y.
Variance of reliability estimates for Y.
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.
######## Measurement error only ########
The attenuation formula for measurement error is
_XY=_TPq_Xq_Yrxy = rtp * qx * qy where _XYrxy is an observed correlation, _TPrtp is a true-score correlation, and q_Xqx and q_Yqy are the square roots of reliability coefficients for X and Y, respectively.
The Taylor series approximation of the variance of _TPrtp can be computed using the following linear equation,
var__TP [var_r_XY-var_e-(b_1^2var_q_X+b_2^2var_q_Y)]/b_3^2var_rtp ~= (var_rxy - var_e - (b1^2 * var_qx + b2^2 * var_qy)) / b3^2
where b_1b1, b_2b2, and b_3b3 are first-order partial derivatives of the attenuation formula with respect to q_Xqx, q_Yqy, and _TPrtp, respectively. The first-order partial derivatives of the attenuation formula are:
b_1=_XY q_X=_TPq_Yb1 = rtp * qy b_2=_XY q_Y=_TPq_Xb2 = rtp * qx b_3=_XY_TP=q_Xq_Yb3 = qx * qy
######## Univariate direct range restriction (UVDRR; i.e., Case II) ########
The UVDRR attenuation procedure may be represented as
_XY_i=_TP_aq_Y_iq_X_au_X_TP_a^2q_X_a^2(u_X^2-1)+1rxyi = ux * rxpa * qxa / sqrt((ux^2 - 1) * rxpa^2 * qxa^2 + 1) * qyi
The attenuation formula can also be represented as:
_XY_i=_TP_aq_Y_iq_X_au_XArxyi = qxa * qyi * rtpa * ux * A
where
A=1_TP_a^2q_X_a^2(u_X^2-1)+1A = 1 / sqrt(rtpa^2 * qxa^2* (ux^2 - 1) + 1)
The Taylor series approximation of the variance of _TP_artpa can be computed using the following linear equation,
var__TP_a [var_r_XY_i-var_e-(b_1^2var_q_X_a+b_2^2var_q_Y_i+b_3^2var_u_X)]/b_4^2var_rtpa ~= (var_rxyi - var_e - (b1^2 * var_qxa + b2^2 * var_qyi + b3^2 * var_ux)) / b4^2
where b_1b1, b_2b2, b_3b3, and b_4b4 are first-order partial derivatives of the attenuation formula with respect to q_X_aqxa, q_Y_iqyi, u_Xux, and _TP_artpa, respectively. The first-order partial derivatives of the attenuation formula are:
b_1=_XY_i q_X_a=_TP_aq_Y_iu_XA^3b1 = qyi * rtpa * ux * A^3 b_2=_XY_i q_Y_i=_XY_iq_Y_ib2 = qxa * qyi * rtpa * ux * A / qyi b_3=_XY_i u_X=-_TP_aq_Y_iq_X_a(_TP_a^2q_X_a^2-1)A^3b3 = -(qyi * rtpa * qxa * (rtpa^2 * qxa^2 - 1)) * A^3 b_4=_XY_i_TP_a=q_Y_iq_X_au_XA^3b4 = (qyi * qxa * ux) * A^3
######## Univariate indirect range restriction (UVIRR; i.e., Case IV) ########
Under univariate indirect range restriction, the attenuation formula yielding _XY_irxyi is:
_XY_i=u_Tq_X_au_T^2q_X_a^2+1-q_X_a^2u_T_TP_au_T^2_TP_a^2+1-_TP_a^2rxyi = (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:
_XY_i=q_X_aq_Y_i_TP_au_T^2ABrxyi = qxa * qyi * rtpa * ut^2 * A * B
where
A=1u_T^2q_X_a^2+1-q_X_a^2A = 1 / sqrt(ut^2 * rtpa^2 - rtpa^2 + 1)
and
B=1u_T^2_TP_a^2+1-_TP_a^2B = 1 / sqrt(ut^2 * qxa^2 - qxa^2 + 1)
The Taylor series approximation of the variance of _TP_artpa can be computed using the following linear equation,
var__TP_a [var_r_XY_i-var_e-(b_1^2var_q_X_a+b_2^2var_q_Y_i+b_3^2var_u_T)]/b_4^2var_rtpa ~= (var_rxyi - var_e - (b1^2 * var_qxa + b2^2 * var_qyi + b3^2 * var_ut)) / b4^2
where b_1b1, b_2b2, b_3b3, and b_4b4 are first-order partial derivatives of the attenuation formula with respect to q_X_aqxa, q_Y_iqyi, u_Tut, and _TP_artpa, respectively. The first-order partial derivatives of the attenuation formula are:
b_1=_XY_i q_X_a=_XY_iq_X_a-_XY_iq_X_aB^2(u_T^2-1)b1 = rxyi / qxa - rxyi * qxa * B^2 * (ut^2 - 1) b_2=_XY_i q_Y_i=_XY_iq_Y_ib2 = rxyi / qyi b_3=_XY_i u_T=2_XY_iu_T-_XY_iu_Tq_X_a^2B^2-_XY_iu_T_TP_a^2A^2b3 = (2 * rxyi) / ut - rxyi * ut * qxa^2 * B^2 - rxyi * ut * rtpa^2 * A^2 b_4=_XY_i_TP_a=_XY_i_TP_a-_XY_i_TP_aA^2(u_T^2-1)b4 = rxyi / rtpa - rxyi * rtpa * A^2 * (ut^2 - 1)
######## Bivariate direct range restriction (BVDRR) ########
Under bivariate direct range restriction, the attenuation formula yielding _XY_irxyi is:
_XY_i=A+_TP_a^2q_X_aq_Y_a-1q_X_aq_Y_a2_TP_au_Xu_Yrxyi = (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=(1q_X_aq_Y_a-_TP_a^2q_X_aq_Y_a)^2+4_TP_au_X^2u_Y^2A = sqrt((1/(qya * qxa) - qya * rtpa^2 * qxa)^2 + 4 * rtpa^2 * ux^2 * uy^2)
The Taylor series approximation of the variance of _TP_artpa can be computed using the following linear equation,
var__TP_a [var_r_XY_i-var_e-(b_1^2var_q_X_a+b_2^2var_q_Y_i+b_3^2var_u_X+b_4^2var_u_Y)]/b_5^2var_rtpa ~= (var_rxyi - var_e - (b1^2 * var_qxa + b2^2 * var_qya + b3^2 * var_ux + b4^2 * var_uy)) / b5^2
where b_1b1, b_2b2, b_3b3, b_4b4, and b_5b5 are first-order partial derivatives of the attenuation formula with respect to q_X_aqxa, q_Y_aqya, u_Xux, u_Yuy, and _TP_artpa, respectively. First, we define terms to simplify the computation of partial derivatives:
B=(_TP_a^2q_X_a^2q_Y_a^2+q_X_aq_Y_aA-1)B = (qya^2 * rtpa^2 * qxa^2 + qya * qxa * A - 1)
C=2_TP_aq_X_a^2q_Y_a^2u_Xu_YAC = 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:
b_1=_XY_i q_X_a=(_TP_a^2q_X_a^2q_Y_a^2+1)Bq_X_aCb1 = ((rtpa^2 * qxa^2 * qya^2 + 1) * B) / (qxa * C) b_2=_XY_i q_Y_i=(_TP_a^2q_X_a^2q_Y_a^2+1)Bq_Y_aCb2 = ((rtpa^2 * qxa^2 * qya^2 + 1) * B) / (qya * C) b_3=_XY_i u_X=-(_TP_aq_X_aq_Y_a-1)(_TP_aq_X_aq_Y_a+1)Bu_XCb3 = -((qya * rtpa * qxa - 1) * (qya * rtpa * qxa + 1) * B) / (ux * C) b_4=_XY_i u_Y=-(_TP_aq_X_aq_Y_a-1)(_TP_aq_X_aq_Y_a+1)Bu_YCb4 = -((qya * rtpa * qxa - 1) * (qya * rtpa * qxa + 1) * B) / (uy * C) b_5=_XY_i_TP_a=(_TP_a^2q_X_a^2q_Y_a^2+1)B_TP_aCb5 = ((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 _XY_irxyi is:
_XY_i=_TP_aq_X_aq_Y_a-|1-u_X^2||1-u_Y^2|u_Xu_Yrxyi = (rtpa * qxa * qya - lambda * sqrt(abs(1 - ux^2) * abs(1 - uy^2))) / (uy * ux)
The Taylor series approximation of the variance of _TP_artpa can be computed using the following linear equation,
var__TP_a [var_r_XY_i-var_e-(b_1^2var_q_X_a+b_2^2var_q_Y_i+b_3^2var_u_X+b_4^2var_u_Y)]/b_5^2var_rtpa ~= (var_rxyi - var_e - (b1^2 * var_qxa + b2^2 * var_qya + b3^2 * var_ux + b4^2 * var_uy)) / b5^2
where b_1b1, b_2b2, b_3b3, b_4b4, and b_5b5 are first-order partial derivatives of the attenuation formula with respect to q_X_aqxa, q_Y_aqya, u_Xux, u_Yuy, and _TP_artpa, respectively. First, we define terms to simplify the computation of partial derivatives:
b_1=_XY_i q_X_a=_TP_aq_Y_au_Xu_Yb1 = rtpa * qya / (ux * uy) b_2=_XY_i q_Y_i=_TP_aq_X_au_Xu_Yb2 = rtpa * qxa / (ux * uy) b_3=_XY_i u_X=(1-u_X^2)|1-u_Y^2|u_Y|1-u_X^2|^1.5-_XY_iu_Xb3 = (lambda * (1 - ux^2) * sqrt(abs(1 - uy^2))) / (uy * abs(1 - ux^2)^1.5) - rxyi / ux b_4=_XY_i u_Y=(1-u_Y^2)|1-u_X^2|u_X|1-u_Y^2|^1.5-_XY_iu_Yb4 = (lambda * (1 - uy^2) * sqrt(abs(1 - ux^2))) / (ux * abs(1 - uy^2)^1.5) - rxyi / uy b_5=_XY_i_TP_a=q_X_aq_Y_au_Xu_Yb5 = (qxa * qya) / (ux * uy)
######## Raju and Burke's TSA1 procedure ########
Raju and Burke's attenuation formula may be represented as
_XY_i=_TP_au_X_XX_a_YY_a_TP_a^2_XX_a_YY_au_X^2-_TP_a^2_XX_a_YY_a+1rxyi = (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 _TP_artpa can be computed using the following linear equation,
var__TP_a [var_r_XY_i-var_e-(B^2var__YY_a+C^2var__XX_a+D^2var_u_X)]/A^2var_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 _TP_artpa, _XX_arxxa, _YY_aryya, and u_Xux, respectively. The first-order partial derivatives of the attenuation formula are:
A=_XY_i_TP_a=_XY_i_TP_a+_XY_i(1-u_X^2)^3_TP_au_X^2A = rxyi / rtpa + (rxyi^3 * (1 - ux^2)) / (rtpa * ux^2) B=_XY_i_YY_a=12(_XY_i_YY_a+_XY_i(1-u_X^2)^3_YY_au_X^2)B = .5 * (rxyi / ryya + (rxyi^3 * (1 - ux^2)) / (ryya * ux^2)) C=_XY_i_XX_a=12(_XY_i_XX_a+_XY_i(1-u_X^2)^3_XX_au_X^2)C = .5 * (rxyi / rxxa + (rxyi^3 * (1 - ux^2)) / (rxxa * ux^2)) D=_XY_i u_X=_XY_i-_XY_i^3u_XD = (rxyi - rxyi^3) / ux
######## Raju and Burke's TSA2 procedure ########
Raju and Burke's attenuation formula may be represented as
_XY_i=_TP_aq_X_aq_Y_au_X_TP_a^2q_X_a^2q_Y_a^2u_X^2-_TP_a^2q_X_a^2q_Y_a^2+1rxyi = (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 _TP_artpa can be computed using the following linear equation,
var__TP_a [var_r_XY_i-var_e-(F^2var_q_Y_a+G^2var_q_X_a+H^2var_u_X)]/E^2var_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 _TP_artpa, q_X_aqxa, q_Y_aqya, and u_Xux, respectively. The first-order partial derivatives of the attenuation formula (with typographic errors in the original article corrected) are:
E=_XY_i_TP_a=_XY_i_TP_a+_XY_i(1-u_X^2)^3_TP_au_X^2E = rxyi / rtpa + (rxyi^3 * (1 - ux^2)) / (rtpa * ux^2) F=_XY_i q_Y_a=_XY_iq_Y_a+_XY_i(1-u_X^2)^3q_Y_au_X^2F = (rxyi / qya + (rxyi^3 * (1 - ux^2)) / (qya * ux^2)) G=_XY_i q_X_a=_XY_iq_X_a+_XY_i(1-u_X^2)^3q_X_au_X^2G = (rxyi / qxa + (rxyi^3 * (1 - ux^2)) / (qxa * ux^2)) H=_XY_i u_X=_XY_i-_XY_i^3u_XH = (rxyi - rxyi^3) / ux
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. tools:::Rd_expr_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. tools:::Rd_expr_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. tools:::Rd_expr_doi("10.1037/0021-9010.68.3.382")
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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