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.
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)
A vector of estimated artifact values.
Vector of incumbent reliability estimates for X.
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).
Logical vector determining whether each element of ux is an observed-score u ratio (TRUE) or a true-score u ratio (FALSE).
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.
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.
Vector of applicant reliability estimates for X.
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).
Logical vector determining whether each element of rxx is an incumbent reliability (TRUE) or an applicant reliability (FALSE).
Vector of true-score u ratios for X.
Vector of incumbent reliability estimates for Y.
Vector of observed-score incumbent correlations between X and Y.
Vector of applicant reliability estimates for Y.
#### 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 #### $$u_{P}=\sqrt{\frac{\frac{1-\rho_{YY_{a}}}{1-\rho_{YY_{i}}}-\left(1-\rho_{YY_{a}}\right)}{\rho_{YY_{a}}}}$$
Schmidt, F. L., & Hunter, J. E. (2015). Methods of meta-analysis: Correcting error and bias in research findings (3rd ed.). Sage. tools:::Rd_expr_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. tools:::Rd_expr_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. tools:::Rd_expr_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. tools:::Rd_expr_doi("10.1111/peps.12122")
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))
Run the code above in your browser using DataLab