Functions to estimate the variance of \(\rho\) 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 \(\rho\) corrected for measurement error only
estimate_var_rho_tsa_uvdrr Variance of \(\rho\) corrected for univariate direct range restriction (i.e., Case II) and measurement error
estimate_var_rho_tsa_bvdrr Variance of \(\rho\) corrected for bivariate direct range restriction and measurement error
estimate_var_rho_tsa_uvirr Variance of \(\rho\) corrected for univariate indirect range restriction (i.e., Case IV) and measurement error
estimate_var_rho_tsa_bvirr Variance of \(\rho\) corrected for bivariate indirect range restriction (i.e., Case V) and measurement error
estimate_var_rho_tsa_rb1 Variance of \(\rho\) corrected using Raju and Burke's TSA1 correction for direct range restriction and measurement error
estimate_var_rho_tsa_rb2 Variance of \(\rho\) 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,
...)
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.
Vector of meta-analytic variances estimated via Taylor series approximation.
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
$$\rho_{XY}=\rho_{TP}q_{X}q_{Y}$$ where \(\rho_{XY}\) is an observed correlation, \(\rho_{TP}\) is a true-score correlation, and \(q_{X}\) and \(q_{Y}\) are the square roots of reliability coefficients for X and Y, respectively.
The Taylor series approximation of the variance of \(\rho_{TP}\) can be computed using the following linear equation,
$$var_{\rho_{TP}} \approx \left[var_{r_{XY}}-var_{e}-\left(b_{1}^{2}var_{q_{X}}+b_{2}^{2}var_{q_{Y}}\right)\right]/b_{3}^{2}$$
where \(b_{1}\), \(b_{2}\), and \(b_{3}\) are first-order partial derivatives of the attenuation formula with respect to \(q_{X}\), \(q_{Y}\), and \(\rho_{TP}\), respectively. The first-order partial derivatives of the attenuation formula are:
$$b_{1}=\frac{\partial\rho_{XY}}{\partial q_{X}}=\rho_{TP}q_{Y}$$ $$b_{2}=\frac{\partial\rho_{XY}}{\partial q_{Y}}=\rho_{TP}q_{X}$$ $$b_{3}=\frac{\partial\rho_{XY}}{\partial\rho_{TP}}=q_{X}q_{Y}$$
######## Univariate direct range restriction (UVDRR; i.e., Case II) ########
The UVDRR attenuation procedure may be represented as
$$\rho_{XY_{i}}=\frac{\rho_{TP_{a}}q_{Y_{i}}q_{X_{a}}u_{X}}{\sqrt{\rho_{TP_{a}}^{2}q_{X_{a}}^{2}\left(u_{X}^{2}-1\right)+1}}$$
The attenuation formula can also be represented as:
$$\rho_{XY_{i}}=\rho_{TP_{a}}q_{Y_{i}}q_{X_{a}}u_{X}A$$
where
$$A=\frac{1}{\sqrt{\rho_{TP_{a}}^{2}q_{X_{a}}^{2}\left(u_{X}^{2}-1\right)+1}}$$
The Taylor series approximation of the variance of \(\rho_{TP_{a}}\) can be computed using the following linear equation,
$$var_{\rho_{TP_{a}}} \approx \left[var_{r_{XY_{i}}}-var_{e}-\left(b_{1}^{2}var_{q_{X_{a}}}+b_{2}^{2}var_{q_{Y_{i}}}+b_{3}^{2}var_{u_{X}}\right)\right]/b_{4}^{2}$$
where \(b_{1}\), \(b_{2}\), \(b_{3}\), and \(b_{4}\) are first-order partial derivatives of the attenuation formula with respect to \(q_{X_{a}}\), \(q_{Y_{i}}\), \(u_{X}\), and \(\rho_{TP_{a}}\), respectively. The first-order partial derivatives of the attenuation formula are:
$$b_{1}=\frac{\partial\rho_{XY_{i}}}{\partial q_{X_{a}}}=\rho_{TP_{a}}q_{Y_{i}}u_{X}A^{3}$$ $$b_{2}=\frac{\partial\rho_{XY_{i}}}{\partial q_{Y_{i}}}=\frac{\rho_{XY_{i}}}{q_{Y_{i}}}$$ $$b_{3}=\frac{\partial\rho_{XY_{i}}}{\partial u_{X}}=-\rho_{TP_{a}}q_{Y_{i}}q_{X_{a}}\left(\rho_{TP_{a}}^{2}q_{X_{a}}^{2}-1\right)A^{3}$$ $$b_{4}=\frac{\partial\rho_{XY_{i}}}{\partial\rho_{TP_{a}}}=q_{Y_{i}}q_{X_{a}}u_{X}A^{3}$$
######## Univariate indirect range restriction (UVIRR; i.e., Case IV) ########
Under univariate indirect range restriction, the attenuation formula yielding \(\rho_{XY_{i}}\) is:
$$\rho_{XY_{i}}=\frac{u_{T}q_{X_{a}}}{\sqrt{u_{T}^{2}q_{X_{a}}^{2}+1-q_{X_{a}}^{2}}}\frac{u_{T}\rho_{TP_{a}}}{\sqrt{u_{T}^{2}\rho_{TP_{a}}^{2}+1-\rho_{TP_{a}}^{2}}}$$
The attenuation formula can also be represented as:
$$\rho_{XY_{i}}=q_{X_{a}}q_{Y_{i}}\rho_{TP_{a}}u_{T}^{2}AB$$
where
$$A=\frac{1}{\sqrt{u_{T}^{2}q_{X_{a}}^{2}+1-q_{X_{a}}^{2}}}$$
and
$$B=\frac{1}{\sqrt{u_{T}^{2}\rho_{TP_{a}}^{2}+1-\rho_{TP_{a}}^{2}}}$$
The Taylor series approximation of the variance of \(\rho_{TP_{a}}\) can be computed using the following linear equation,
$$var_{\rho_{TP_{a}}} \approx \left[var_{r_{XY_{i}}}-var_{e}-\left(b_{1}^{2}var_{q_{X_{a}}}+b_{2}^{2}var_{q_{Y_{i}}}+b_{3}^{2}var_{u_{T}}\right)\right]/b_{4}^{2}$$
where \(b_{1}\), \(b_{2}\), \(b_{3}\), and \(b_{4}\) are first-order partial derivatives of the attenuation formula with respect to \(q_{X_{a}}\), \(q_{Y_{i}}\), \(u_{T}\), and \(\rho_{TP_{a}}\), respectively. The first-order partial derivatives of the attenuation formula are:
$$b_{1}=\frac{\partial\rho_{XY_{i}}}{\partial q_{X_{a}}}=\frac{\rho_{XY_{i}}}{q_{X_{a}}}-\rho_{XY_{i}}q_{X_{a}}B^{2}\left(u_{T}^{2}-1\right)$$ $$b_{2}=\frac{\partial\rho_{XY_{i}}}{\partial q_{Y_{i}}}=\frac{\rho_{XY_{i}}}{q_{Y_{i}}}$$ $$b_{3}=\frac{\partial\rho_{XY_{i}}}{\partial u_{T}}=\frac{2\rho_{XY_{i}}}{u_{T}}-\rho_{XY_{i}}u_{T}q_{X_{a}}^{2}B^{2}-\rho_{XY_{i}}u_{T}\rho_{TP_{a}}^{2}A^{2}$$ $$b_{4}=\frac{\partial\rho_{XY_{i}}}{\partial\rho_{TP_{a}}}=\frac{\rho_{XY_{i}}}{\rho_{TP_{a}}}-\rho_{XY_{i}}\rho_{TP_{a}}A^{2}\left(u_{T}^{2}-1\right)$$
######## Bivariate direct range restriction (BVDRR) ########
Under bivariate direct range restriction, the attenuation formula yielding \(\rho_{XY_{i}}\) is:
$$\rho_{XY_{i}}=\frac{A+\rho_{TP_{a}}^{2}q_{X_{a}}q_{Y_{a}}-\frac{1}{q_{X_{a}}q_{Y_{a}}}}{2\rho_{TP_{a}}u_{X}u_{Y}}$$
where
$$A=\sqrt{\left(\frac{1}{q_{X_{a}}q_{Y_{a}}}-\rho_{TP_{a}}^{2}q_{X_{a}}q_{Y_{a}}\right)^{2}+4\rho_{TP_{a}}u_{X}^{2}u_{Y}^{2}}$$
The Taylor series approximation of the variance of \(\rho_{TP_{a}}\) can be computed using the following linear equation,
$$var_{\rho_{TP_{a}}} \approx \left[var_{r_{XY_{i}}}-var_{e}-\left(b_{1}^{2}var_{q_{X_{a}}}+b_{2}^{2}var_{q_{Y_{i}}}+b_{3}^{2}var_{u_{X}}+b_{4}^{2}var_{u_{Y}}\right)\right]/b_{5}^{2}$$
where \(b_{1}\), \(b_{2}\), \(b_{3}\), \(b_{4}\), and \(b_{5}\) are first-order partial derivatives of the attenuation formula with respect to \(q_{X_{a}}\), \(q_{Y_{a}}\), \(u_{X}\), \(u_{Y}\), and \(\rho_{TP_{a}}\), respectively. First, we define terms to simplify the computation of partial derivatives:
$$B=\left(\rho_{TP_{a}}^{2}q_{X_{a}}^{2}q_{Y_{a}}^{2}+q_{X_{a}}q_{Y_{a}}A-1\right)$$
$$C=2\rho_{TP_{a}}q_{X_{a}}^{2}q_{Y_{a}}^{2}u_{X}u_{Y}A$$
The first-order partial derivatives of the attenuation formula are:
$$b_{1}=\frac{\partial\rho_{XY_{i}}}{\partial q_{X_{a}}}=\frac{\left(\rho_{TP_{a}}^{2}q_{X_{a}}^{2}q_{Y_{a}}^{2}+1\right)B}{q_{X_{a}}C}$$ $$b_{2}=\frac{\partial\rho_{XY_{i}}}{\partial q_{Y_{i}}}=\frac{\left(\rho_{TP_{a}}^{2}q_{X_{a}}^{2}q_{Y_{a}}^{2}+1\right)B}{q_{Y_{a}}C}$$ $$b_{3}=\frac{\partial\rho_{XY_{i}}}{\partial u_{X}}=-\frac{\left(\rho_{TP_{a}}q_{X_{a}}q_{Y_{a}}-1\right)\left(\rho_{TP_{a}}q_{X_{a}}q_{Y_{a}}+1\right)B}{u_{X}C}$$ $$b_{4}=\frac{\partial\rho_{XY_{i}}}{\partial u_{Y}}=-\frac{\left(\rho_{TP_{a}}q_{X_{a}}q_{Y_{a}}-1\right)\left(\rho_{TP_{a}}q_{X_{a}}q_{Y_{a}}+1\right)B}{u_{Y}C}$$ $$b_{5}=\frac{\partial\rho_{XY_{i}}}{\partial\rho_{TP_{a}}}=\frac{\left(\rho_{TP_{a}}^{2}q_{X_{a}}^{2}q_{Y_{a}}^{2}+1\right)B}{\rho_{TP_{a}}C}$$
######## Bivariate indirect range restriction (BVIRR; i.e., Case V) ########
Under bivariate indirect range restriction, the attenuation formula yielding \(\rho_{XY_{i}}\) is:
$$\rho_{XY_{i}}=\frac{\rho_{TP_{a}}q_{X_{a}}q_{Y_{a}}-\lambda\sqrt{\left|1-u_{X}^{2}\right|\left|1-u_{Y}^{2}\right|}}{u_{X}u_{Y}}$$
The Taylor series approximation of the variance of \(\rho_{TP_{a}}\) can be computed using the following linear equation,
$$var_{\rho_{TP_{a}}} \approx \left[var_{r_{XY_{i}}}-var_{e}-\left(b_{1}^{2}var_{q_{X_{a}}}+b_{2}^{2}var_{q_{Y_{i}}}+b_{3}^{2}var_{u_{X}}+b_{4}^{2}var_{u_{Y}}\right)\right]/b_{5}^{2}$$
where \(b_{1}\), \(b_{2}\), \(b_{3}\), \(b_{4}\), and \(b_{5}\) are first-order partial derivatives of the attenuation formula with respect to \(q_{X_{a}}\), \(q_{Y_{a}}\), \(u_{X}\), \(u_{Y}\), and \(\rho_{TP_{a}}\), respectively. First, we define terms to simplify the computation of partial derivatives:
$$b_{1}=\frac{\partial\rho_{XY_{i}}}{\partial q_{X_{a}}}=\frac{\rho_{TP_{a}}q_{Y_{a}}}{u_{X}u_{Y}}$$ $$b_{2}=\frac{\partial\rho_{XY_{i}}}{\partial q_{Y_{i}}}=\frac{\rho_{TP_{a}}q_{X_{a}}}{u_{X}u_{Y}}$$ $$b_{3}=\frac{\partial\rho_{XY_{i}}}{\partial u_{X}}=\frac{\lambda\left(1-u_{X}^{2}\right)\sqrt{\left|1-u_{Y}^{2}\right|}}{u_{Y}\left|1-u_{X}^{2}\right|^{1.5}}-\frac{\rho_{XY_{i}}}{u_{X}}$$ $$b_{4}=\frac{\partial\rho_{XY_{i}}}{\partial u_{Y}}=\frac{\lambda\left(1-u_{Y}^{2}\right)\sqrt{\left|1-u_{X}^{2}\right|}}{u_{X}\left|1-u_{Y}^{2}\right|^{1.5}}-\frac{\rho_{XY_{i}}}{u_{Y}}$$ $$b_{5}=\frac{\partial\rho_{XY_{i}}}{\partial\rho_{TP_{a}}}=\frac{q_{X_{a}}q_{Y_{a}}}{u_{X}u_{Y}}$$
######## Raju and Burke's TSA1 procedure ########
Raju and Burke's attenuation formula may be represented as
$$\rho_{XY_{i}}=\frac{\rho_{TP_{a}}u_{X}\sqrt{\rho_{XX_{a}}\rho_{YY_{a}}}}{\sqrt{\rho_{TP_{a}}^{2}\rho_{XX_{a}}\rho_{YY_{a}}u_{X}^{2}-\rho_{TP_{a}}^{2}\rho_{XX_{a}}\rho_{YY_{a}}+1}}$$
The Taylor series approximation of the variance of \(\rho_{TP_{a}}\) can be computed using the following linear equation,
$$var_{\rho_{TP_{a}}} \approx \left[var_{r_{XY_{i}}}-var_{e}-\left(B^{2}var_{\rho_{YY_{a}}}+C^{2}var_{\rho_{XX_{a}}}+D^{2}var_{u_{X}}\right)\right]/A^{2}$$
where A, B, C, and D are first-order partial derivatives of the attenuation formula with respect to \(\rho_{TP_{a}}\), \(\rho_{XX_{a}}\), \(\rho_{YY_{a}}\), and \(u_{X}\), respectively. The first-order partial derivatives of the attenuation formula are:
$$A=\frac{\partial\rho_{XY_{i}}}{\partial\rho_{TP_{a}}}=\frac{\rho_{XY_{i}}}{\rho_{TP_{a}}}+\frac{\rho_{XY_{i}\left(1-u_{X}^{2}\right)}^{3}}{\rho_{TP_{a}}u_{X}^{2}}$$ $$B=\frac{\partial\rho_{XY_{i}}}{\partial\rho_{YY_{a}}}=\frac{1}{2}\left(\frac{\rho_{XY_{i}}}{\rho_{YY_{a}}}+\frac{\rho_{XY_{i}\left(1-u_{X}^{2}\right)}^{3}}{\rho_{YY_{a}}u_{X}^{2}}\right)$$ $$C=\frac{\partial\rho_{XY_{i}}}{\partial\rho_{XX_{a}}}=\frac{1}{2}\left(\frac{\rho_{XY_{i}}}{\rho_{XX_{a}}}+\frac{\rho_{XY_{i}\left(1-u_{X}^{2}\right)}^{3}}{\rho_{XX_{a}}u_{X}^{2}}\right)$$ $$D=\frac{\partial\rho_{XY_{i}}}{\partial u_{X}}=\frac{\rho_{XY_{i}}-\rho_{XY_{i}}^{3}}{u_{X}}$$
######## Raju and Burke's TSA2 procedure ########
Raju and Burke's attenuation formula may be represented as
$$\rho_{XY_{i}}=\frac{\rho_{TP_{a}}q_{X_{a}}q_{Y_{a}}u_{X}}{\sqrt{\rho_{TP_{a}}^{2}q_{X_{a}}^{2}q_{Y_{a}}^{2}u_{X}^{2}-\rho_{TP_{a}}^{2}q_{X_{a}}^{2}q_{Y_{a}}^{2}+1}}$$
The Taylor series approximation of the variance of \(\rho_{TP_{a}}\) can be computed using the following linear equation,
$$var_{\rho_{TP_{a}}} \approx \left[var_{r_{XY_{i}}}-var_{e}-\left(F^{2}var_{q_{Y_{a}}}+G^{2}var_{q_{X_{a}}}+H^{2}var_{u_{X}}\right)\right]/E^{2}$$
where E, F, G, and H are first-order partial derivatives of the attenuation formula with respect to \(\rho_{TP_{a}}\), \(q_{X_{a}}\), \(q_{Y_{a}}\), and \(u_{X}\), respectively. The first-order partial derivatives of the attenuation formula (with typographic errors in the original article corrected) are:
$$E=\frac{\partial\rho_{XY_{i}}}{\partial\rho_{TP_{a}}}=\frac{\rho_{XY_{i}}}{\rho_{TP_{a}}}+\frac{\rho_{XY_{i}\left(1-u_{X}^{2}\right)}^{3}}{\rho_{TP_{a}}u_{X}^{2}}$$ $$F=\frac{\partial\rho_{XY_{i}}}{\partial q_{Y_{a}}}=\frac{\rho_{XY_{i}}}{q_{Y_{a}}}+\frac{\rho_{XY_{i}\left(1-u_{X}^{2}\right)}^{3}}{q_{Y_{a}}u_{X}^{2}}$$ $$G=\frac{\partial\rho_{XY_{i}}}{\partial q_{X_{a}}}=\frac{\rho_{XY_{i}}}{q_{X_{a}}}+\frac{\rho_{XY_{i}\left(1-u_{X}^{2}\right)}^{3}}{q_{X_{a}}u_{X}^{2}}$$ $$H=\frac{\partial\rho_{XY_{i}}}{\partial u_{X}}=\frac{\rho_{XY_{i}}-\rho_{XY_{i}}^{3}}{u_{X}}$$
Dahlke, J. A., & Wiernik, B. M. (2018). One of these artifacts is not like the others: Accounting for indirect range restriction in organizational and psychological research. Manuscript submitted for review.
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<U+2013>612. https://doi.org/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<U+2013>395. https://doi.org/10.1037/0021-9010.68.3.382
# NOT RUN {
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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