# Reflect your Personnel Selection: R & Taylor-Russell Tables

February 16, 2012
By

(This article was first published on HR Blog » Rstat, and kindly contributed to R-bloggers)

Taylor-Russell tables (Taylor & Russell, 1939) are designed to estimate the percentage of future employees who will be successful on the job if a particular selection method (eg. test, assessment center, interview) is used. I have already described the kibit russell taylor-tool (in German).

Now I will show how the number of recruited candidates is calculated precisely with R. I use the manipulate package. Therefore, the code will only run in the RSudio IDE. The big advantage is that one can easily observe the influences of

1. the base rate of potentially suitable persons in the non-selected group of applicants as well as
2. the validity of the selection process (by moving the slider).

This little program is excellent for teaching.

The calculations are based on an article of Richard A. Mclellan in the journal “Personnel Decisions International” (1999): Theoretical Expectancies: Replacing Classic Utility Tables with Flexible, Accurate Computing Procedures.

And this is the result. Have fun trying.

```F1 <- function(P) {
SPLIT <- 0.42
A0 <- 2.50662823884
A1 <- -18.61500062529
A2 <- 41.391199773534
A3 <- -25.44106049637
B1 <- -8.4735109309
B2 <- 23.08336743743
B3 <- -21.06224101826
B4 <- 3.13082909833
C0 <- -2.78718931138
C1 <- -2.29796479134
C2 <- 4.85014127135
C3 <- 2.32121276858
D1 <- 3.54388924762
D2 <- 1.63706781897
Q <- P - 0.5
if (abs(Q) <= SPLIT) {
R <- Q*Q
PPN <- Q * (((A3 * R + A2) * R + A1) * R + A0) / ((((B4 * R + B3) * R + B2) * R + B1)*R +1.0)
return(PPN)
}
R <- P
if (Q > 0) {R =1.0-P}
if (R <= 0) {
print("You have entered a value that is not permitted. The result is false.")
return(0)
}
R <- sqrt(-log(R))
PPN <- (((C3 * R + C2) * R + C1) * R + C0) / ((D2 * R + D1) * R + 1.0)
if (Q < 0) {PPN =-PPN}
return(PPN)
}

F2 <- function(X) {
P1A <- 242.667955230532
P1B <- 21.97926616182942
P1C <- 6.996383488661914
P1D <- -3.5609843701815E-02
Q1A <- 215.058875869861
Q1B <- 91.1649054045149
Q1C <- 15.0827976304078
Q1D <- 1.0
P2A <- 300.459261020162
P2B <- 451.918953711873
P2C <- 339.320816734344
P2D <- 152.98928504694
P2E <- 43.1622272220567
P2F <- 7.21175825088309
P2G <- .564195517478994
P2H <- -1.36864857382717E-07
Q2A <- 300.459260956983
Q2B <- 790.950925327898
Q2C <- 931.35409485061
Q2D <- 638.980264465631
Q2E <- 277.585444743988
Q2F <- 77.0001529352295
Q2G <- 12.7827273196294
Q2H <- 1.0
P3A <- -2.99610707703542E-03
P3B <- -4.94730910623251E-02
P3C <- -.226956593539687
P3D <- -.278661308609648
P3E <- -2.23192459734185E-02
Q3A <- 1.06209230528468E-02
Q3B <- .19130892610783
Q3C <- 1.05167510706793
Q3D <- 1.98733201817135
Q3E <- 1.0
SQRT2 <- 1.4142135623731
SQRTPI <- 1.77245385090552

Y <- X/SQRT2
if (Y < 0) {
Y <- -Y
SN <- -1.0
} else {
SN <- 1.0
}
Y2 <- Y * Y
if (Y < 0.46875) {
R1 <- ((P1D * Y2 + P1C) * Y2 + P1B) * Y2 + P1A
R2 <- ((Q1D * Y2 + Q1C) * Y2 + Q1B) * Y2 + Q1A
ERFVAL <- Y * R1 / R2
if (SN == 1) LOAREA <- 0.5 + 0.5 * ERFVAL
else LOAREA <- 0.5 - 0.5 * ERFVAL
} else {
if (Y < 4.0) {
R1 <- ((((((P2H * Y + P2G) * Y + P2F) * Y + P2E) * Y + P2D) * Y + P2C) * Y + P2B) * Y + P2A
R2 <- ((((((Q2H * Y + Q2G) * Y + Q2F) * Y + Q2E) * Y + Q2D) * Y + Q2C) * Y + Q2B) * Y + Q2A
ERFCVAL <- exp(-Y2) * R1 / R2
} else {
Z <- Y2 * Y2
R1 <- (((P3E * Z + P3D) * Z + P3C) * Z + P3B) * Z + P3A
R2 <- (((Q3E * Z + Q3D) * Z + Q3C) * Z + Q3B) * Z + Q3A
ERFCVAL <- (exp(-Y2) / Y) * (1.0 / SQRTPI + R1 / (R2 * Y2))
}
if (SN == 1) LOAREA <- 1.0 - 0.5 * ERFCVAL
else LOAREA <- 0.5 * ERFCVAL
}
UPAREA <- 1.0 - LOAREA
return(UPAREA)
}

F3 <- function(H1, HK, R) {
X <- c(0.04691008, 0.23076534, 0.5, 0.76923466, 0.95308992)
W <- c(0.018854042, 0.038088059, 0.0452707394, 0.038088059, 0.018854042)
H2 <- HK
H12 <- (H1*H1 + H2*H2)/2.0
BV <- 0
if (abs(R) >= 0.7) {
R2 <- 1.0-R*R
R3 <- sqrt(R2)
if (R < 0) H2 <- -H2
H3 <- H1*H2
H7 <- exp(-H3 / 2.0)
if (R2 != 0) {
H6 <- abs(H1 - H2)
H5 <- H6 * H6 / 2.0
H6 <- H6 / R3
AA <- 0.5 - (H3 / 8.0)
AB <- 3.0 - (2.0 * AA * H5)
BV <- 0.13298076 * H6 * AB * F2(H6) - exp(-H5 / R2) * (AB + AA * R2) * 0.053051647
for (i in 1:5) {
R1 <- R3 * X[i]
RR <- R1 * R1
R2 <- sqrt( 1.0- RR)
BV <- BV - W[i] * exp(-H5 / RR) * (exp(-H3 / (1.0 + R2)) / R2 / H7 - 1.0 - AA	* RR)
}
}
if (R > 0 & H1 > H2) {
BV <- BV * R3 * H7 + F2(H1)
return(BV)
}
if (R > 0 & H1 <= H2) {
BV <- BV * R3 * H7 + F2(H2)
return(BV)
}
if (R < 0 & (F2(H1) - F2(H2)) < 0) {
BV <- 0 - BV * R3 * H7
return(BV)
}
if (R < 0 & (F2(H1) - F2(H2)) >= 0) {
BV <- (F2(H1) - F2(H2)) - BV * R3 * H7
return(BV)
}
}
H3 <- H1 * H2
for (i in 1:5)
{
R1 <- R * X[i]
RR2 <- 1.0 - R1 * R1
BV <- BV + W[i] * exp((R1 * H3 - H12) / RR2) / sqrt(RR2)
}
BV <- F2(H1) * F2(H2) + R * BV
return(BV)
}

true_positives <- function(N, ToSelect,BaseRate, Validity) {round(F3(F1(1.0-ToSelect/N), F1(1.0-BaseRate), Validity)/(ToSelect/N)*ToSelect,1)}
false_positives <- function(N, ToSelect,BaseRate, Validity) {round(ToSelect-F3(F1(1.0-ToSelect/N), F1(1.0-BaseRate), Validity)/(ToSelect/N)*ToSelect,1)}
false_negatives <- function(N, ToSelect,BaseRate, Validity) {round(N*BaseRate - F3(F1(1.0-ToSelect/N), F1(1.0-BaseRate), Validity)/(ToSelect/N)*ToSelect,1)}
true_negatives <- function(N, ToSelect,BaseRate, Validity) {N - true_positives(N, ToSelect,BaseRate, Validity) - false_positives(N, ToSelect,BaseRate, Validity) - false_negatives(N, ToSelect,BaseRate, Validity)}

library(manipulate)

manipulate(
barplot(
matrix(c(true_positives(Applicants, StaffRequirement, BaseRate, Validity),
false_positives(Applicants, StaffRequirement, BaseRate, Validity),
true_negatives(Applicants, StaffRequirement, BaseRate, Validity),
false_negatives(Applicants, StaffRequirement, BaseRate, Validity)),
nrow = 2, ncol=2, byrow=FALSE,
dimnames = list(c("rightly", "wrongly"), c("recruited", "rejected"))),
Applicants=slider(1,100, step=1, initial = 50),
StaffRequirement=slider(1,100, step=1, initial = 10),
BaseRate=slider(0,1, step=.01, initial = .25),
Validity=slider(0,1, step=.01, initial = .37))
```

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