Practical Applications of qDEA
In qDEA: Quantile Data Envelopment Analysis

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>",
  fig.width = 7,
  fig.height = 5
)
library(qDEA)

Introduction

This vignette demonstrates practical applications of qDEA through real-world examples and workflows. We'll cover:

Hospital efficiency analysis
Retail store performance
Handling outliers and noisy data
Benchmarking and target setting
Reporting results

Case Study 1: Hospital Efficiency Analysis

The Problem

A hospital administrator wants to evaluate the efficiency of 12 hospitals using:

Inputs: Number of doctors and nurses
Outputs: Outpatient and inpatient treatments

The administrator suspects that 1-2 hospitals may have data quality issues or operate under exceptional circumstances.

Analysis

# Load hospital data
data(CST22)

# Examine the data
print(CST22)

# Prepare inputs and outputs
X <- as.matrix(CST22[, c("DOCTORS", "NURSES")])
Y <- as.matrix(CST22[, c("OUT_PATIENTS", "IN_PATIENTS")])

# Summary statistics
cat("Input summary:\n")
summary(X)
cat("\nOutput summary:\n")
summary(Y)

Step 1: Standard DEA Analysis

# Run standard DEA (no outliers allowed)
dea_result <- qDEA(X = X, Y = Y, 
                   orient = "in", 
                   RTS = "VRS", 
                   qout = 0,
                   getproject = TRUE)

# Create results table
results_dea <- data.frame(
  Hospital = CST22$HOSPITAL,
  Efficiency = round(dea_result$effvals, 3),
  Rank = rank(-dea_result$effvals, ties.method = "min")
)

print(results_dea)

cat("\nEfficient hospitals:", 
    sum(dea_result$effvals >= 0.99), "out of", nrow(X))

Step 2: Robust qDEA Analysis

# Run qDEA allowing 10% outliers (≈1 hospital)
qdea_result <- qDEA(X = X, Y = Y,
                    orient = "in",
                    RTS = "VRS",
                    qout = 0.10,
                    getproject = TRUE)

# Compare DEA and qDEA results
results_comparison <- data.frame(
  Hospital = CST22$HOSPITAL,
  DEA_Eff = round(dea_result$effvals, 3),
  qDEA_Eff = round(qdea_result$effvalsq, 3),
  Change = round(qdea_result$effvalsq - dea_result$effvals, 3),
  DEA_Rank = rank(-dea_result$effvals, ties.method = "min"),
  qDEA_Rank = rank(-qdea_result$effvalsq, ties.method = "min")
)

print(results_comparison)

Step 3: Target Setting

# Calculate targets for inefficient hospitals
targets <- data.frame(
  Hospital = CST22$HOSPITAL,
  Current_Doctors = X[,1],
  Target_Doctors = round(qdea_result$PROJ_DATA$X0HATq[,1], 1),
  Doctor_Reduction = round(X[,1] - qdea_result$PROJ_DATA$X0HATq[,1], 1),
  Current_Nurses = X[,2],
  Target_Nurses = round(qdea_result$PROJ_DATA$X0HATq[,2], 1),
  Nurse_Reduction = round(X[,2] - qdea_result$PROJ_DATA$X0HATq[,2], 1),
  Efficiency = round(qdea_result$effvalsq, 3)
)

# Show only inefficient hospitals
inefficient <- targets[targets$Efficiency < 0.99, ]
print(inefficient)

# Calculate total potential savings
cat("\nTotal potential reductions:\n")
cat("Doctors:", sum(targets$Doctor_Reduction), "\n")
cat("Nurses:", sum(targets$Nurse_Reduction), "\n")

Step 4: Peer Benchmarks

# Identify peer hospitals for benchmarking
peers <- qdea_result$PEER_DATA$PEERSq

# Show peers for an inefficient hospital (e.g., Hospital D)
cat("Benchmark hospitals for Hospital D:\n")
hospital_d_peers <- peers[peers$dmu0 == "D", ]
print(hospital_d_peers[order(-hospital_d_peers$z), ])

Management Report

# Create executive summary
cat("=" , rep("=", 50), "\n", sep="")
cat("HOSPITAL EFFICIENCY ANALYSIS - EXECUTIVE SUMMARY\n")
cat("=" , rep("=", 50), "\n", sep="")

cat("\nDATA: 12 hospitals\n")
cat("INPUTS: Doctors, Nurses\n")
cat("OUTPUTS: Outpatients, Inpatients\n")
cat("METHOD: qDEA with VRS, 10% outlier allowance\n")

cat("\n--- EFFICIENCY RESULTS ---\n")
cat("Mean efficiency:", round(mean(qdea_result$effvalsq), 3), "\n")
cat("Median efficiency:", round(median(qdea_result$effvalsq), 3), "\n")
cat("Efficient hospitals:", sum(qdea_result$effvalsq >= 0.99), "\n")
cat("Inefficient hospitals:", sum(qdea_result$effvalsq < 0.99), "\n")

cat("\n--- IMPROVEMENT POTENTIAL ---\n")
cat("If all hospitals achieve target efficiency:\n")
cat("  Doctor reduction:", sum(targets$Doctor_Reduction), 
    "(", round(100*sum(targets$Doctor_Reduction)/sum(X[,1]), 1), "%)\n")
cat("  Nurse reduction:", sum(targets$Nurse_Reduction),
    "(", round(100*sum(targets$Nurse_Reduction)/sum(X[,2]), 1), "%)\n")

cat("\n--- TOP PERFORMERS ---\n")
top3 <- head(results_comparison[order(-results_comparison$qDEA_Eff), ], 3)
print(top3[, c("Hospital", "qDEA_Eff")])

cat("\n--- NEEDS IMPROVEMENT ---\n")
bottom3 <- head(results_comparison[order(results_comparison$qDEA_Eff), ], 3)
print(bottom3[, c("Hospital", "qDEA_Eff")])

Case Study 2: Retail Store Performance

The Problem

A retail chain wants to evaluate store performance with potential outliers due to: - Special events or temporary factors - Data entry errors - Unique local market conditions

# Load retail data
data(CST21)
print(CST21)

# Prepare data
X <- as.matrix(CST21[, c("EMPLOYEES", "FLOOR_AREA")])
Y <- as.matrix(CST21$SALES)

Sensitivity Analysis: How Many Outliers?

# Test different outlier proportions
qout_values <- c(0, 0.05, 0.10, 0.15, 0.20)

sensitivity_results <- data.frame(
  Store = CST21$STORE
)

for (q in qout_values) {
  result <- qDEA(X = X, Y = Y, orient = "out", RTS = "CRS", qout = q)
  col_name <- paste0("qout_", sprintf("%.2f", q))
  sensitivity_results[[col_name]] <- round(result$effvalsq, 3)
}

print(sensitivity_results)

# Calculate how efficiency changes with qout
sensitivity_results$Range <- apply(
  sensitivity_results[, -1], 1, 
  function(x) max(x) - min(x)
)

cat("\nStores most sensitive to outlier allowance:\n")
print(sensitivity_results[order(-sensitivity_results$Range), 
                          c("Store", "Range")])

Recommended Approach

# Use moderate outlier allowance
result_retail <- qDEA(X = X, Y = Y,
                      orient = "out",
                      RTS = "VRS",
                      qout = 0.10,
                      getproject = TRUE)

# Performance report
performance <- data.frame(
  Store = CST21$STORE,
  Employees = X[,1],
  Floor_Area = X[,2],
  Actual_Sales = Y[,1],
  Target_Sales = round(result_retail$PROJ_DATA$Y0HATq[,1], 0),
  Sales_Gap = round(result_retail$PROJ_DATA$Y0HATq[,1] - Y[,1], 0),
  Efficiency = round(result_retail$effvalsq, 3)
)

print(performance)

# Classify stores
performance$Category <- ifelse(
  performance$Efficiency >= 0.95, "Excellent",
  ifelse(performance$Efficiency >= 0.85, "Good",
         ifelse(performance$Efficiency >= 0.75, "Needs Improvement",
                "Critical"))
)

cat("\nStore Classification:\n")
table(performance$Category)

Case Study 3: Dealing with Outliers

Identifying Outliers

data(CST11)
X <- as.matrix(CST11$EMPLOYEES)
Y <- as.matrix(CST11$SALES_EJOR)

# Run with very restrictive outlier allowance
strict <- qDEA(X = X, Y = Y, orient = "out", RTS = "CRS", qout = 0.01)

# Run with moderate outlier allowance  
moderate <- qDEA(X = X, Y = Y, orient = "out", RTS = "CRS", qout = 0.15)

# Stores with big efficiency changes are likely outliers
outlier_check <- data.frame(
  Store = CST11$STORE,
  Strict = round(strict$effvalsq, 3),
  Moderate = round(moderate$effvalsq, 3),
  Change = round(moderate$effvalsq - strict$effvalsq, 3)
)

print(outlier_check)

# Flag potential outliers (large efficiency changes)
outlier_check$Potential_Outlier <- outlier_check$Change > 0.10

cat("\nPotential outliers identified:\n")
print(outlier_check[outlier_check$Potential_Outlier, ])

Impact of Outlier Removal

# Compare DEA vs qDEA to see impact of outlier allowance
impact <- data.frame(
  Store = CST11$STORE,
  DEA = round(strict$effvals, 3),
  qDEA = round(moderate$effvalsq, 3),
  Difference = round(moderate$effvalsq - strict$effvals, 3)
)

print(impact)

cat("\nMean efficiency:\n")
cat("DEA (no outliers):", round(mean(strict$effvals), 3), "\n")
cat("qDEA (15% outliers):", round(mean(moderate$effvalsq), 3), "\n")

Workflow: Complete Analysis Template

Here's a complete workflow you can adapt:

# ==========================================
# COMPLETE qDEA ANALYSIS WORKFLOW
# ==========================================

# 1. Load and examine data
data(CST22)  # Replace with your data
X <- as.matrix(CST22[, c("DOCTORS", "NURSES")])
Y <- as.matrix(CST22[, c("OUT_PATIENTS", "IN_PATIENTS")])

# Check data quality
summary(X)
summary(Y)
# Look for: missing values, extreme values, data entry errors

# 2. Run standard DEA baseline
baseline <- qDEA(X = X, Y = Y, 
                 orient = "in",    # Choose: in, out, inout
                 RTS = "VRS",      # Choose: CRS, VRS, DRS, IRS
                 qout = 0)

# 3. Run robust qDEA
robust <- qDEA(X = X, Y = Y,
               orient = "in",
               RTS = "VRS",
               qout = 0.10,       # Adjust based on expected outliers
               nqiter = 3,        # Iterative refinement
               getproject = TRUE) # Get targets

# 4. Compare results
comparison <- data.frame(
  Unit = rownames(X),
  DEA = round(baseline$effvals, 3),
  qDEA = round(robust$effvalsq, 3),
  Change = round(robust$effvalsq - baseline$effvals, 3)
)

# 5. Identify outliers
potential_outliers <- comparison$Unit[abs(comparison$Change) > 0.10]

# 6. Calculate targets
targets <- data.frame(
  Unit = rownames(X),
  Efficiency = round(robust$effvalsq, 3),
  Current_Input1 = X[,1],
  Target_Input1 = round(robust$PROJ_DATA$X0HATq[,1], 2),
  Current_Input2 = X[,2],
  Target_Input2 = round(robust$PROJ_DATA$X0HATq[,2], 2)
)

# 7. Generate report
# Export to CSV
write.csv(comparison, "efficiency_comparison.csv", row.names = FALSE)
write.csv(targets, "efficiency_targets.csv", row.names = FALSE)

# 8. Optional: Bootstrap for confidence intervals
boot_result <- qDEA(X = X, Y = Y,
                    orient = "in",
                    RTS = "VRS",
                    qout = 0.10,
                    nboot = 1000,
                    seedval = 12345)

boot_ci <- data.frame(
  Unit = rownames(X),
  Efficiency = round(boot_result$effvalsq, 3),
  BC_Efficiency = round(boot_result$BOOT_DATA$effvalsq.bc, 3),
  Bias = round(boot_result$effvalsq - boot_result$BOOT_DATA$effvalsq.bc, 3)
)

Best Practices Summary

1. Data Preparation

✓ Check for missing values
✓ Verify all values are positive
✓ Look for extreme outliers or data entry errors
✓ Ensure comparable units (scale if necessary)
✓ Document data sources and definitions

2. Model Selection

✓ Choose orientation based on managerial control
✓ Use VRS unless scale efficiency is of interest
✓ Start with qout = 0.10 (10% outliers)
✓ Test sensitivity to qout selection

3. Interpretation

✓ Efficiency scores are relative, not absolute
✓ Compare units within same analysis only
✓ Consider context (outliers may be legitimate)
✓ Verify targets are achievable
✓ Use peers for benchmarking

4. Reporting

✓ Document methodology clearly
✓ Report both DEA and qDEA results
✓ Explain outlier allowance rationale
✓ Provide actionable recommendations
✓ Include sensitivity analysis

5. Common Mistakes to Avoid

✗ Comparing efficiency across different analyses
✗ Using CRS when scale varies significantly
✗ Setting qout too high (> 0.25)
✗ Ignoring data quality issues
✗ Over-interpreting small efficiency differences

Exporting Results

To CSV

# Prepare comprehensive results
results_export <- data.frame(
  Unit = CST22$HOSPITAL,
  Input1 = X[,1],
  Input2 = X[,2],
  Output1 = Y[,1],
  Output2 = Y[,2],
  DEA_Efficiency = round(baseline$effvals, 4),
  qDEA_Efficiency = round(robust$effvalsq, 4),
  Target_Input1 = round(robust$PROJ_DATA$X0HATq[,1], 2),
  Target_Input2 = round(robust$PROJ_DATA$X0HATq[,2], 2)
)

# Export
write.csv(results_export, "qDEA_results.csv", row.names = FALSE)

To Excel (requires openxlsx package)

library(openxlsx)

# Create workbook
wb <- createWorkbook()

# Add worksheets
addWorksheet(wb, "Efficiency Scores")
addWorksheet(wb, "Targets")
addWorksheet(wb, "Peers")

# Write data
writeData(wb, "Efficiency Scores", comparison)
writeData(wb, "Targets", targets)
writeData(wb, "Peers", robust$PEER_DATA$PEERSq)

# Save
saveWorkbook(wb, "qDEA_analysis.xlsx", overwrite = TRUE)

Visualization Examples

Efficiency Distribution

data(CST22)
X <- as.matrix(CST22[, c("DOCTORS", "NURSES")])
Y <- as.matrix(CST22[, c("OUT_PATIENTS", "IN_PATIENTS")])

result <- qDEA(X = X, Y = Y, orient = "in", RTS = "VRS", qout = 0.10)

# Histogram
hist(result$effvalsq, 
     breaks = 10,
     col = "lightblue",
     border = "white",
     main = "Distribution of Efficiency Scores",
     xlab = "Efficiency",
     ylab = "Frequency")
abline(v = mean(result$effvalsq), col = "red", lwd = 2, lty = 2)
legend("topleft", 
       legend = paste("Mean =", round(mean(result$effvalsq), 3)),
       col = "red", lty = 2, lwd = 2)

Efficiency Comparison

# Compare DEA and qDEA
dea <- qDEA(X = X, Y = Y, orient = "in", RTS = "VRS", qout = 0)
qdea <- qDEA(X = X, Y = Y, orient = "in", RTS = "VRS", qout = 0.10)

# Scatter plot
plot(dea$effvals, qdea$effvalsq,
     xlim = c(0.4, 1.2), ylim = c(0.4, 1.2),
     xlab = "DEA Efficiency",
     ylab = "qDEA Efficiency",
     main = "DEA vs qDEA Efficiency Scores",
     pch = 19, col = "blue")
abline(0, 1, col = "red", lty = 2)  # 45-degree line
text(dea$effvals, qdea$effvalsq, 
     labels = CST22$HOSPITAL, 
     pos = 3, cex = 0.8)
grid()

Conclusion

This vignette has demonstrated practical applications of qDEA including:

Complete hospital and retail efficiency analyses
Outlier detection and handling
Target setting and benchmarking
Sensitivity analysis
Result interpretation and reporting

For more details on the underlying methodology, see the main package vignette.

qDEA
Quantile Data Envelopment Analysis

Practical Applications of qDEA
In qDEA: Quantile Data Envelopment Analysis

Introduction

Case Study 1: Hospital Efficiency Analysis

The Problem

Analysis

Step 1: Standard DEA Analysis

Step 2: Robust qDEA Analysis

Step 3: Target Setting

Step 4: Peer Benchmarks

Management Report

Case Study 2: Retail Store Performance

The Problem

Sensitivity Analysis: How Many Outliers?

Recommended Approach

Case Study 3: Dealing with Outliers

Identifying Outliers

Impact of Outlier Removal

Workflow: Complete Analysis Template

Best Practices Summary

1. Data Preparation

2. Model Selection

3. Interpretation

4. Reporting

5. Common Mistakes to Avoid

Exporting Results

To CSV

To Excel (requires openxlsx package)

Visualization Examples

Efficiency Distribution

Efficiency Comparison

Conclusion

Further Reading

Try the qDEA package in your browser

R Package Documentation

Browse R Packages

We want your feedback!

qDEA Quantile Data Envelopment Analysis

Practical Applications of qDEA In qDEA: Quantile Data Envelopment Analysis

Introduction

Case Study 1: Hospital Efficiency Analysis

The Problem

Analysis

Step 1: Standard DEA Analysis

Step 2: Robust qDEA Analysis

Step 3: Target Setting

Step 4: Peer Benchmarks

Management Report

Case Study 2: Retail Store Performance

The Problem

Sensitivity Analysis: How Many Outliers?

Recommended Approach

Case Study 3: Dealing with Outliers

Identifying Outliers

Impact of Outlier Removal

Workflow: Complete Analysis Template

Best Practices Summary

1. Data Preparation

2. Model Selection

3. Interpretation

4. Reporting

5. Common Mistakes to Avoid

Exporting Results

To CSV

To Excel (requires openxlsx package)

Visualization Examples

Efficiency Distribution

Efficiency Comparison

Conclusion

Further Reading

Try the qDEA package in your browser

R Package Documentation

Browse R Packages

We want your feedback!

qDEA
Quantile Data Envelopment Analysis

Practical Applications of qDEA
In qDEA: Quantile Data Envelopment Analysis