estimate_var_rho_tsa {psychmeta}

Taylor Series Approximation of variance of \rho corrected for psychometric artifacts

Description

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.

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

\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}}

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)