People Analytics
2026-04-24
mydata <- read.csv2("People analytics.csv")
head(mydata)## ID Track Econ Comm Analyt Finance
## 1 1 Operational track 4.6 4.8 3.5 3.1
## 2 2 Operational track 3.7 3.4 2.7 3.1
## 3 3 Operational track 4.4 4.8 3.3 3.3
## 4 4 Operational track 4.5 4.7 2.5 3.4
## 5 5 Operational track 3.9 3.5 3.4 3.3
## 6 6 Operational track 3.7 3.0 3.4 4.0Description of the variables:
- ID: Unique employee identifier.
- Track: Categorical variable indicating the employee’s assigned development track (Operational, Analytical, or Strategic).
- Econ: Competency score measuring the employee’s level of business and economic reasoning (scale 1–6).
- Comm: Competency score reflecting professional and strategic communication skills (scale 1–6).
- Analyt: Competency score capturing analytical and data‑driven problem‑solving ability (scale 1–6).
- Finance: Competency score measuring the employee’s ability to make sound financial decisions (scale 1–6).
mydata$Track <- factor(mydata$Track)mydata_DA <- mydata[c(3:6)]
desc_stats <- data.frame(
N = apply(mydata_DA, 2, function(x) sum(!is.na(x))),
Missing = apply(mydata_DA, 2, function(x) sum(is.na(x))),
Mean = apply(mydata_DA, 2, mean, na.rm = TRUE),
Variance = apply(mydata_DA, 2, var, na.rm = TRUE),
Min = apply(mydata_DA, 2, min, na.rm = TRUE),
Max = apply(mydata_DA, 2, max, na.rm = TRUE)
)
round(desc_stats, 2)## N Missing Mean Variance Min Max
## Econ 310 0 4.29 0.48 2.0 6.0
## Comm 310 0 4.26 0.55 2.3 6.0
## Analyt 310 0 4.23 0.62 2.4 6.0
## Finance 310 0 4.03 0.54 2.2 5.6table(mydata$Track)##
## Analytical track Operational track Strategic track
## 105 95 110# Multiple ANOVAs by Track
fit <- aov(cbind(Econ, Comm, Analyt, Finance) ~ Track, data = mydata)
summary(fit)## Response Econ :
## Df Sum Sq Mean Sq F value Pr(>F)
## Track 2 11.07 5.5351 12.497 6.058e-06 ***
## Residuals 307 135.98 0.4429
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Response Comm :
## Df Sum Sq Mean Sq F value Pr(>F)
## Track 2 76.494 38.247 127.34 < 2.2e-16 ***
## Residuals 307 92.208 0.300
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Response Analyt :
## Df Sum Sq Mean Sq F value Pr(>F)
## Track 2 86.461 43.230 125.19 < 2.2e-16 ***
## Residuals 307 106.017 0.345
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Response Finance :
## Df Sum Sq Mean Sq F value Pr(>F)
## Track 2 55.559 27.7797 77.554 < 2.2e-16 ***
## Residuals 307 109.966 0.3582
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1# Multicolinearity
vars <- c("Econ", "Comm", "Analyt", "Finance")
vif_all(mydata, vars)## $Econ
## Comm Analyt Finance
## 1.001786 1.274940 1.274983
##
## $Comm
## Econ Analyt Finance
## 1.061216 1.314375 1.280724
##
## $Analyt
## Econ Comm Finance
## 1.032028 1.004366 1.030999
##
## $Finance
## Econ Comm Analyt
## 1.059498 1.004668 1.058407# Box’s M test for homogeneity of covariance matrices across Track groups
library(biotools)
boxM(mydata[, c("Econ", "Comm", "Analyt", "Finance")],
mydata$Track)##
## Box's M-test for Homogeneity of Covariance Matrices
##
## data: mydata[, c("Econ", "Comm", "Analyt", "Finance")]
## Chi-Sq (approx.) = 22.064, df = 20, p-value = 0.3371# Computes squared Mahalanobis distance to identify multivariate outliers based on variable covariance structure.
mydata <- mahal_distance(
data = mydata,
vars = c("Econ", "Comm", "Analyt", "Finance"),
group = "Track",
prob = 0.99)
outliers <- mydata %>%
filter(outlier)
outliers## # A tibble: 3 × 9
## Track ID Econ Comm Analyt Finance md cutoff outlier
## <fct> <int> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <lgl>
## 1 Analytical tr… 199 3 5.1 5.8 3.4 13.7 13.3 TRUE
## 2 Operational t… 41 5.5 4.7 5.2 4.4 13.5 13.3 TRUE
## 3 Strategic tra… 290 5.6 3.8 4.4 2.3 15.2 13.3 TRUE# Creates a Chi-square Q-Q plot to compare observed squared Mahalanobis distances with theoretical values for assessing multivariate normality.
n <- length(mydata$md)
theoretical <- qchisq(ppoints(n), df = 4)
observed <- sort(mydata$md)
plot(theoretical, observed,
main = "Chi-square Q-Q plot (Squared Mahalanobis distance)",
xlab = "Theoretical Chi-square quantiles",
ylab = "Observed squared Mahalanobis distance",
pch = 20)
abline(0, 1, col = "darkgreen", lwd = 2)
# Creates separate Chi-square Q-Q plots to assess multivariate normality within each Track group
md_by_group <- split(mydata$md, mydata$Track)
par(mfrow = c(1, 3))
for (g in names(md_by_group)) {
observed <- sort(md_by_group[[g]])
n <- length(observed)
theoretical <- qchisq(ppoints(n), df = 4)
plot(
theoretical, observed,
main = g,
xlab = "Theoretical Chi-square quantiles",
ylab = "Observed squared Mahalanobis distance",
pch = 20,
asp = 1
)
abline(0, 1, col = "darkgreen", lwd = 2)
}
par(mfrow = c(1, 1))#Linear discriminannt analysis
lda_model <- lda(Track ~ Econ + Comm + Analyt + Finance,
data = mydata)lda_model## Call:
## lda(Track ~ Econ + Comm + Analyt + Finance, data = mydata)
##
## Prior probabilities of groups:
## Analytical track Operational track Strategic track
## 0.3387097 0.3064516 0.3548387
##
## Group means:
## Econ Comm Analyt Finance
## Analytical track 4.027619 3.994286 4.908571 4.562857
## Operational track 4.466316 3.789474 3.606316 3.514737
## Strategic track 4.380000 4.920909 4.127273 3.960909
##
## Coefficients of linear discriminants:
## LD1 LD2
## Econ -0.2880613 0.04679253
## Comm -0.2016790 1.81157732
## Analyt 1.2518192 0.05949666
## Finance 0.9100927 0.10336903
##
## Proportion of trace:
## LD1 LD2
## 0.5963 0.4037# Extract eigenvalues to assess the importance of LDA discriminant functions
eigenvalues <- lda_eigenvalues(
mydata = mydata,
group = "Track",
variables = c("Econ", "Comm", "Analyt", "Finance")
)
eigenvalues## [1] 1.2264305 0.8303744#Canonical correlations and squared canonical correlations
round(canonical_corr_from_eigen(eigenvalues), 3)## CanonicalCorr SqCanonicalCorr
## LD1 0.742 0.551
## LD2 0.674 0.454# Multivariate ANOVA (MANOVA) testing Track effects on all outcomes jointly
man <- manova(cbind(Econ, Comm, Analyt, Finance) ~ Track,
data = mydata)# Canonical discriminant analysis based on the MANOVA model
library(candisc)
can <- candisc(man)
can##
## Canonical Discriminant Analysis for Track:
##
## CanRsq Eigenvalue Difference Percent Cumulative
## 1 0.55085 1.22643 0.39606 59.628 59.628
## 2 0.45366 0.83037 0.39606 40.372 100.000
##
## Test of H0: The canonical correlations in the
## current row and all that follow are zero
##
## LR test stat approx F numDF denDF Pr(> F)
## 1 0.24539 77.422 8 608 < 2.2e-16 ***
## 2 0.54634 84.421 3 305 < 2.2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1# Raw (unstandardized) canonical discriminant coefficients (weights)
can$coeffs.raw## Can1 Can2
## Econ 0.2880613 0.04679253
## Comm 0.2016790 1.81157732
## Analyt -1.2518192 0.05949666
## Finance -0.9100927 0.10336903# Standardized canonical discriminant coefficients (beta weights)
can$coeffs.std## Can1 Can2
## Econ 0.1917124 0.03114166
## Comm 0.1105289 0.99282312
## Analyt -0.7356302 0.03496315
## Finance -0.5446863 0.06186589# Structure matrix (canonical loadings)
DA_structure_matrix(mydata = mydata,
group = "Track",
lda_model = lda_model,
variables = c("Econ", "Comm", "Analyt", "Finance"))## LD1 LD2
## Econ -0.25534686 0.04173591
## Comm -0.06028717 0.99683350
## Analyt 0.81204226 0.09059543
## Finance 0.63710169 0.09461488# Computes the discriminant scores (LD1, LD2, ...) by projecting each observation onto the linear discriminant functions
lda_scores <- predict(lda_model)$x
# Compute group centroids (mean LDA scores) for each Track
centroids <- data.frame(
Track = mydata$Track,
lda_scores) %>%
dplyr::group_by(Track) %>%
dplyr::summarise(across(everything(), mean))
centroids## # A tibble: 3 × 3
## Track LD1 LD2
## <fct> <dbl> <dbl>
## 1 Analytical track 1.46 -0.399
## 2 Operational track -1.21 -0.935
## 3 Strategic track -0.353 1.19# Scatterplot of LDA discriminant scores by Track
data_lda <- bind_cols(mydata, as.data.frame(lda_scores))
ggplot(data_lda, aes(LD1, LD2, color = Track, shape = Track)) +
geom_point(size = 2.0, alpha = 0.7) +
stat_ellipse(type = "t", level = 0.68) +
labs(
title = "Canonical Discriminant Functions",
x = "Discriminant Function 1",
y = "Discriminant Function 2"
) +
theme_minimal() +
theme(legend.position = "right")
# Confusion matrix comparing actual Track membership with LDA classifications
pred <- predict(lda_model)
table(Actual = mydata$Track,
Predicted = pred$class)## Predicted
## Actual Analytical track Operational track
## Analytical track 87 5
## Operational track 7 79
## Strategic track 14 11
## Predicted
## Actual Strategic track
## Analytical track 13
## Operational track 9
## Strategic track 85# LDA classification accuracy
mean(pred$class == mydata$Track)## [1] 0.8096774# Cross-validated LDA classification using leave-one-out validation
lda_cv <- lda(Track ~ Econ + Comm + Analyt + Finance,
data = mydata,
CV = TRUE)
table(Actual = mydata$Track,
Predicted = lda_cv$class)## Predicted
## Actual Analytical track Operational track
## Analytical track 85 5
## Operational track 7 79
## Strategic track 15 11
## Predicted
## Actual Strategic track
## Analytical track 15
## Operational track 9
## Strategic track 84# Cross-validated classification accuracy
mean(lda_cv$class == mydata$Track)## [1] 0.8# Cross-validated classification accuracy by group
confusin_matrix <- table(Actual = mydata$Track, Predicted = lda_cv$class)
group_accuracy <- diag(prop.table(confusin_matrix, 1))
group_accuracy## Analytical track Operational track Strategic track
## 0.8095238 0.8315789 0.7636364# Add discriminant scores (LD1, LD2, …) and predicted group to mydata
mydata <- cbind(
mydata,
as.data.frame(pred$x), # D1, D2 (LD1, LD2)
Predicted_Track = pred$class # predicted group membership
)# Add posterior probabilities
mydata <- cbind(
mydata,
round(as.data.frame(pred$posterior), 3)
)head(mydata, 3)## Track ID Econ Comm Analyt Finance md cutoff
## 1 Analytical track 96 3.6 3.0 6.0 4.5 8.420470 13.2767
## 2 Analytical track 97 5.0 3.9 4.6 4.0 2.995015 13.2767
## 3 Analytical track 98 5.3 4.3 4.3 4.2 4.711675 13.2767
## outlier LD1 LD2 Predicted_Track Analytical track
## 1 FALSE 3.09450498 -2.1613645 Analytical track 0.999
## 2 FALSE 0.30211480 -0.6004153 Analytical track 0.529
## 3 FALSE -0.05850241 0.1410782 Strategic track 0.244
## Operational track Strategic track
## 1 0.001 0.000
## 2 0.290 0.181
## 3 0.235 0.521new_person <- data.frame(
Econ = 2.0,
Comm = 2.7,
Analyt = 4.2,
Finance = 2.9
)pred_new <- predict(lda_model, newdata = new_person)pred_new$class # Assigned group## [1] Operational track
## Levels: Analytical track Operational track Strategic trackpred_new$x # LD1 and LD2 scores (centered and comparable)## LD1 LD2
## 1 -0.09351622 -3.05219pred_new$posterior # Posterior probabilities## Analytical track Operational track Strategic track
## 1 0.1454469 0.8524696 0.002083432