estimate_artifacts {psychmeta}

Estimation of applicant and incumbent reliabilities and of true- and observed-score u ratios

Description

Functions to estimate the values of artifacts from other artifacts. These functions allow for reliability estimates to be corrected/attenuated for range restriction and allow u ratios to be converted between observed-score and true-score metrics. Some functions also allow for the extrapolation of an artifact from other available information.

Available functions include:

• estimate_rxxa: Estimate the applicant reliability of variable X from X's incumbent reliability value and X's observed-score or true-score u ratio.

• estimate_rxxa_u: Estimate the applicant reliability of variable X from X's observed-score and true-score u ratios.

• estimate_rxxi: Estimate the incumbent reliability of variable X from X's applicant reliability value and X's observed-score or true-score u ratio.

• estimate_rxxi_u: Estimate the incumbent reliability of variable X from X's observed-score and true-score u ratios.

• estimate_ux: Estimate the true-score u ratio for variable X from X's reliability coefficient and X's observed-score u ratio.

• estimate_uy: Estimate the observed-score u ratio for variable X from X's reliability coefficient and X's true-score u ratio.

• estimate_ryya: Estimate the applicant reliability of variable Y from Y's incumbent reliability value, Y's correlation with X, and X's u ratio.

• estimate_ryyi: Estimate the incumbent reliability of variable Y from Y's applicant reliability value, Y's correlation with X, and X's u ratio.

• estimate_uy: Estimate the observed-score u ratio for variable Y from Y's applicant and incumbent reliability coefficients.

• estimate_up: Estimate the true-score u ratio for variable Y from Y's applicant and incumbent reliability coefficients.

Usage

estimate_rxxa(
  rxxi,
  ux,
  ux_observed = TRUE,
  indirect_rr = TRUE,
  rxxi_type = "alpha"
)

estimate_rxxi(
  rxxa,
  ux,
  ux_observed = TRUE,
  indirect_rr = TRUE,
  rxxa_type = "alpha"
)

estimate_ut(ux, rxx, rxx_restricted = TRUE)

estimate_ux(ut, rxx, rxx_restricted = TRUE)

estimate_ryya(
  ryyi,
  rxyi,
  ux,
  rxx = 1,
  rxx_restricted = FALSE,
  ux_observed = TRUE,
  indirect_rr = TRUE,
  rxx_type = "alpha"
)

estimate_ryyi(
  ryya,
  rxyi,
  ux,
  rxx = 1,
  rxx_restricted = FALSE,
  ux_observed = TRUE,
  indirect_rr = TRUE,
  rxx_type = "alpha"
)

estimate_uy(ryyi, ryya, indirect_rr = TRUE, ryy_type = "alpha")

estimate_up(ryyi, ryya)

estimate_rxxa_u(ux, ut)

estimate_rxxi_u(ux, ut)

Arguments

rxxi

Vector of incumbent reliability estimates for X.

ux

Vector of observed-score u ratios for X (if used in the context of estimating a reliability value, a true-score u ratio may be supplied by setting ux_observed to FALSE).

ux_observed

Logical vector determining whether each element of ux is an observed-score u ratio (TRUE) or a true-score u ratio (FALSE).

indirect_rr

Logical vector determining whether each reliability value is associated with indirect range restriction (TRUE) or direct range restriction (FALSE). Note #1: For estimate_ryya and estimate_ryyi, this argument refers to whether X is indirectly or directly range restricted (Y is assumed to always be indirectly range restricted via selection on X or another variable). Note #2: When rxxi_type, rxxa_type, or rxx_type refers to an internal consistency reliability method, the corresponding reliability estimates will be treated as being impacted by indirect range restriction because, even when X is directly range restricted, the inter-item relations used to evaluate internal consistency reliability are indirectly range restricted via selection on X's total scores.

rxxi_type, rxxa_type, rxx_type, ryy_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.

rxxa

Vector of applicant reliability estimates for X.

rxx

Vector of reliability estimates for X (used in the context of estimating ux and ut - specify that reliability is an incumbent value by setting rxx_restricted to FALSE).

rxx_restricted

Logical vector determining whether each element of rxx is an incumbent reliability (TRUE) or an applicant reliability (FALSE).

ut

Vector of true-score u ratios for X.

ryyi

Vector of incumbent reliability estimates for Y.

rxyi

Vector of observed-score incumbent correlations between X and Y.

ryya

Vector of applicant reliability estimates for Y.

Details

#### Formulas to estimate rxxa ####

Formulas for indirect range restriction:

\rho_{XX_{a}}=1-u_{X}^{2}\left(1-\rho_{XX_{i}}\right)

\rho_{XX_{a}}=\frac{\rho_{XX_{i}}}{\rho_{XX_{i}}+u_{T}^{2}-\rho_{XX_{i}}u_{T}^{2}}

Formula for direct range restriction:

\rho_{XX_{a}}=\frac{\rho_{XX_{i}}}{u_{X}^{2}\left[1+\rho_{XX_{i}}\left(\frac{1}{u_{X}^{2}}-1\right)\right]}

#### Formulas to estimate rxxi ####

Formulas for indirect range restriction:

\rho_{XX_{i}}=1-\frac{1-\rho_{XX_{a}}}{u_{X}^{2}}

\rho_{XX_{i}}=1-\frac{1-\rho_{XX_{a}}}{\rho_{XX_{a}}\left[u_{T}^{2}-\left(1-\frac{1}{\rho_{XX_{a}}}\right)\right]}

Formula for direct range restriction:

\rho_{XX_{i}}=\frac{\rho_{XX_{i}}u_{X}^{2}}{1+\rho_{XX_{i}}\left(u_{X}^{2}-1\right)}

#### Formulas to estimate ut ####

u_{T}=\sqrt{\frac{\rho_{XX_{i}}u_{X}^{2}}{1+\rho_{XX_{i}}u_{X}^{2}-u_{X}^{2}}}

u_{T}=\sqrt{\frac{u_{X}^{2}-\left(1-\rho_{XX_{a}}\right)}{\rho_{XX_{a}}}}

#### Formulas to estimate ux ####

u_{X}=\sqrt{\frac{u_{T}^{2}}{\rho_{XX_{i}}\left(1+\frac{u_{T}^{2}}{\rho_{XX_{i}}}-u_{T}^{2}\right)}}

u_{X}=\sqrt{\rho_{XX_{a}}\left[u_{T}^{2}-\left(1-\frac{1}{\rho_{XX_{a}}}\right)\right]}

#### Formulas to estimate ryya #### Formula for direct range restriction (i.e., when selection is based on X):

\rho_{YY_{a}}=1-\frac{1-\rho_{YY_{i}}}{1-\rho_{XY_{i}}^{2}\left(1-\frac{1}{u_{X}^{2}}\right)}

Formula for indirect range restriction (i.e., when selection is based on a variable other than X):

\rho_{YY_{a}}=1-\frac{1-\rho_{YY_{i}}}{1-\rho_{TY_{i}}^{2}\left(1-\frac{1}{u_{T}^{2}}\right)}

#### Formulas to estimate ryyi #### Formula for direct range restriction (i.e., when selection is based on X):

\rho_{YY_{i}}=1-\left(1-\rho_{YY_{a}}\right)\left[1-\rho_{XY_{i}}^{2}\left(1-\frac{1}{u_{X}^{2}}\right)\right]

Formula for indirect range restriction (i.e., when selection is based on a variable other than X):

\rho_{YY_{i}}=1-\left(1-\rho_{YY_{a}}\right)\left[1-\rho_{TY_{i}}^{2}\left(1-\frac{1}{u_{X}^{2}}\right)\right]

#### Formula to estimate uy ####

u_{Y}=\sqrt{\frac{1-\rho_{YY_{a}}}{1-\rho_{YY_{i}}}}

#### Formula to estimate up ####

u_{P}=\sqrt{\frac{\frac{1-\rho_{YY_{a}}}{1-\rho_{YY_{i}}}-\left(1-\rho_{YY_{a}}\right)}{\rho_{YY_{a}}}}

Value

A vector of estimated artifact values.

References

Schmidt, F. L., & Hunter, J. E. (2015). Methods of meta-analysis: Correcting error and bias in research findings (3rd ed.). Sage. doi:10.4135/9781483398105 p. 127.

Le, H., & Schmidt, F. L. (2006). Correcting for indirect range restriction in meta-analysis: Testing a new meta-analytic procedure. Psychological Methods, 11(4), 416–438. doi:10.1037/1082-989X.11.4.416

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

Le, H., Oh, I.-S., Schmidt, F. L., & Wooldridge, C. D. (2016). Correction for range restriction in meta-analysis revisited: Improvements and implications for organizational research. Personnel Psychology, 69(4), 975–1008. doi:10.1111/peps.12122

Examples

estimate_rxxa(rxxi = .8, ux = .8, ux_observed = TRUE)
estimate_rxxi(rxxa = .8, ux = .8, ux_observed = TRUE)
estimate_ut(ux = .8, rxx = .8, rxx_restricted = TRUE)
estimate_ux(ut = .8, rxx = .8, rxx_restricted = TRUE)
estimate_ryya(ryyi = .8, rxyi = .3, ux = .8)
estimate_ryyi(ryya = .8, rxyi = .3, ux = .8)
estimate_uy(ryyi = c(.5, .7), ryya = c(.7, .8))
estimate_up(ryyi = c(.5, .7), ryya = c(.7, .8))
estimate_rxxa_u(ux = c(.7, .8), ut = c(.65, .75))
estimate_rxxi_u(ux = c(.7, .8), ut = c(.65, .75))