Understanding and Creating AHP Matrices

Introduction

The Analytic Hierarchy Process (AHP) is a structured technique for organizing and analyzing complex decisions. In the context of soil quality assessment, AHP helps assign weights to different soil indicators based on expert judgment about their relative importance.

This vignette explains:

The fundamentals of AHP methodology
How to create pairwise comparison matrices
How to interpret the Consistency Ratio
Strategies for improving inconsistent matrices
Practical examples for soil quality assessment

library(soilquality)

Why Use AHP for Soil Quality?

Different soil properties have different impacts on soil quality depending on:

Land use: Agricultural vs. forest vs. urban soils
Management goals: Crop production vs. carbon sequestration vs. erosion control
Regional context: Tropical vs. temperate, arid vs. humid
Soil type: Sandy vs. clayey, acidic vs. alkaline

AHP provides a systematic way to incorporate expert knowledge about these context-specific priorities into the SQI calculation.

Equal Weights vs. AHP Weights

data(soil_ucayali)

# Define indicators
indicators <- c("pH", "OM", "N", "P")

# Calculate SQI with equal weights (default)
result_equal <- compute_sqi_properties(
  data = soil_ucayali,
  properties = indicators,
  id_column = "SampleID"
)

cat("Equal weights (assigned to MDS indicators selected by PCA):\n")
#> Equal weights (assigned to MDS indicators selected by PCA):
print(round(result_equal$weights, 3))
#>     N    pH    OM 
#> 0.333 0.333 0.333
cat("\nMDS indicators selected:", paste(result_equal$mds, collapse = ", "), "\n")
#> 
#> MDS indicators selected: N, pH, OM

# To use AHP weights, create a pairwise matrix for the indicators
# Note: The pairwise matrix should match the MDS indicators selected by PCA
# For demonstration, let's calculate AHP weights manually
pairwise <- matrix(c(
  1,   1/3, 1/5, 1/3,  # pH
  3,   1,   1/2, 1,    # OM
  5,   2,   1,   3,    # N
  3,   1,   1/3, 1     # P
), nrow = 4, byrow = TRUE)
colnames(pairwise) <- rownames(pairwise) <- indicators

ahp_result <- ahp_weights(pairwise, indicators = indicators)

cat("\nAHP weights (for all indicators):\n")
#> 
#> AHP weights (for all indicators):
print(round(ahp_result$weights, 3))
#>    pH    OM     N     P 
#> 0.080 0.227 0.486 0.207
cat("\nConsistency Ratio:", round(ahp_result$CR, 4), "\n")
#> 
#> Consistency Ratio: 0.0126

cat("\nMean SQI with equal weights:", 
    round(mean(result_equal$results$SQI), 3), "\n")
#> 
#> Mean SQI with equal weights: 0.432

The Saaty Scale

AHP uses the Saaty scale to express relative importance in pairwise comparisons:

Value	Meaning	Explanation
1	Equal importance	Both indicators contribute equally
3	Moderate importance	Experience slightly favors one indicator
5	Strong importance	Experience strongly favors one indicator
7	Very strong importance	One indicator is strongly favored
9	Extreme importance	Evidence favoring one indicator is overwhelming
2, 4, 6, 8	Intermediate values	When compromise is needed

Reciprocals (1/3, 1/5, etc.) are used when the second indicator is more important than the first.

Example: Comparing pH and Organic Matter

Question: “How much more important is organic matter compared to pH for overall soil quality in tropical agricultural soils?”

Possible answers:

1: They are equally important
3: OM is moderately more important (typical answer)
5: OM is strongly more important
1/3: pH is moderately more important than OM

Creating Pairwise Comparison Matrices

Method 1: Interactive Creation

The easiest way to create an AHP matrix is interactively:

# Define your indicators
indicators <- c("pH", "OM", "N", "P", "K")

# Create matrix interactively
# You'll be prompted to compare each pair
ahp_matrix <- create_ahp_matrix(indicators, mode = "interactive")

# View results
print(ahp_matrix)

# The matrix is stored in ahp_matrix$matrix
# Use it in SQI calculation
result <- compute_sqi_properties(
  data = soil_ucayali,
  properties = indicators,
  pairwise_matrix = ahp_matrix$matrix,
  id_column = "SampleID"
)

Method 2: Manual Matrix Construction

For reproducible analysis or when working with pre-determined expert judgments:

# Define indicators
indicators <- c("pH", "OM", "N", "P")

# Create the pairwise comparison matrix
# Each cell [i,j] represents: "How much more important is indicator i than j?"
pairwise <- matrix(c(
  # pH   OM    N     P
    1,   1/3,  1/5,  1/3,   # pH row
    3,   1,    1/2,  1,     # OM row
    5,   2,    1,    3,     # N row
    3,   1,    1/3,  1      # P row
), nrow = 4, byrow = TRUE)

colnames(pairwise) <- rownames(pairwise) <- indicators

# View the matrix
print(pairwise)
#>    pH        OM         N         P
#> pH  1 0.3333333 0.2000000 0.3333333
#> OM  3 1.0000000 0.5000000 1.0000000
#> N   5 2.0000000 1.0000000 3.0000000
#> P   3 1.0000000 0.3333333 1.0000000

# Calculate weights
ahp_result <- ahp_weights(pairwise, indicators = indicators)

cat("\nIndicator weights:\n")
#> 
#> Indicator weights:
print(round(ahp_result$weights, 3))
#>    pH    OM     N     P 
#> 0.080 0.227 0.486 0.207
cat("\nConsistency Ratio:", round(ahp_result$CR, 4), "\n")
#> 
#> Consistency Ratio: 0.0126

Interpretation: In this example:

N is 5x more important than pH
OM is 3x more important than pH
N is 2x more important than OM
N is 3x more important than P

Method 3: Using Importance Ratios

A simpler approach when you know relative importance:

# Specify relative importance scores
# Example: pH=1, OM=2, N=3, P=2, K=1.5
indicators <- c("pH", "OM", "N", "P", "K")
importance <- c(1, 2, 3, 2, 1.5)

# Convert to pairwise matrix
pairwise_from_ratios <- ratio_to_saaty(importance)
colnames(pairwise_from_ratios) <- rownames(pairwise_from_ratios) <- indicators

# Calculate weights
ahp_from_ratios <- ahp_weights(indicators, pairwise = pairwise_from_ratios)

cat("Weights from importance ratios:\n")
#> Weights from importance ratios:
print(round(ahp_from_ratios$weights, 3))
#>    pH    OM     N     P     K 
#> 0.105 0.211 0.316 0.211 0.158
cat("\nConsistency Ratio:", round(ahp_from_ratios$CR, 4), "\n")
#> 
#> Consistency Ratio: 0

Understanding the Consistency Ratio

The Consistency Ratio (CR) measures the logical consistency of your pairwise comparisons.

What is Consistency?

Consistent judgments follow transitive logic:

If A is 3x more important than B
And B is 2x more important than C
Then A should be 6x more important than C (3 × 2 = 6)

Inconsistent judgments violate this logic:

If A is 3x more important than B
And B is 2x more important than C
But you say A is only 2x more important than C (inconsistent!)

Interpreting CR Values

CR Value	Interpretation	Action
< 0.10	Acceptable consistency	Use the weights
0.10 - 0.20	Marginal consistency	Review and consider revising
> 0.20	Unacceptable consistency	Revise judgments

# Example 1: Consistent matrix (CR < 0.1)
consistent_matrix <- matrix(c(
  1,   3,   5,
  1/3, 1,   2,
  1/5, 1/2, 1
), nrow = 3, byrow = TRUE)

indicators_3 <- c("A", "B", "C")
colnames(consistent_matrix) <- rownames(consistent_matrix) <- indicators_3

result_consistent <- ahp_weights(consistent_matrix, indicators = indicators_3)
cat("Consistent matrix - CR:", round(result_consistent$CR, 4), "\n")
#> Consistent matrix - CR: 0.0032

# Example 2: Inconsistent matrix (CR > 0.1)
inconsistent_matrix <- matrix(c(
  1,   3,   2,
  1/3, 1,   5,
  1/2, 1/5, 1
), nrow = 3, byrow = TRUE)

colnames(inconsistent_matrix) <- rownames(inconsistent_matrix) <- indicators_3

result_inconsistent <- ahp_weights(inconsistent_matrix, indicators = indicators_3)
#> Warning in ahp_weights(inconsistent_matrix, indicators = indicators_3):
#> Consistency Ratio (0.4037) exceeds 0.1. Consider revising judgments.
cat("Inconsistent matrix - CR:", round(result_inconsistent$CR, 4), "\n")
#> Inconsistent matrix - CR: 0.4037

Improving Inconsistent Matrices

If your CR exceeds 0.10, here are strategies to improve consistency:

Strategy 1: Identify Problematic Comparisons

# Create an inconsistent matrix
indicators <- c("pH", "OM", "N", "P")
problematic <- matrix(c(
  1,   2,   4,   3,
  1/2, 1,   5,   2,
  1/4, 1/5, 1,   1/3,
  1/3, 1/2, 3,   1
), nrow = 4, byrow = TRUE)

colnames(problematic) <- rownames(problematic) <- indicators

# Calculate weights and CR
result_prob <- ahp_weights(problematic, indicators = indicators)
cat("Initial CR:", round(result_prob$CR, 4), "\n\n")
#> Initial CR: 0.0379

# Check for major inconsistencies
# Compare direct vs. indirect comparisons
cat("Checking consistency:\n")
#> Checking consistency:
cat("Direct: pH vs N =", problematic[1, 3], "\n")
#> Direct: pH vs N = 4
cat("Indirect: pH vs OM × OM vs N =", 
    problematic[1, 2] * problematic[2, 3], "\n")
#> Indirect: pH vs OM × OM vs N = 10
cat("Ratio:", problematic[1, 3] / (problematic[1, 2] * problematic[2, 3]), "\n\n")
#> Ratio: 0.4

# This shows a major inconsistency!
# pH vs N is 4, but pH vs OM (2) × OM vs N (5) = 10

Strategy 2: Revise Extreme Comparisons

# Revise the problematic comparison
improved <- problematic
improved[2, 3] <- 2  # Change OM vs N from 5 to 2
improved[3, 2] <- 1/2  # Update reciprocal

result_improved <- ahp_weights(improved, indicators = indicators)
cat("Improved CR:", round(result_improved$CR, 4), "\n")
#> Improved CR: 0.046

Strategy 3: Use Fewer Extreme Values

# Original: uses values 1-5
moderate_matrix <- matrix(c(
  1,   2,   3,   2,
  1/2, 1,   2,   1,
  1/3, 1/2, 1,   1/2,
  1/2, 1,   2,   1
), nrow = 4, byrow = TRUE)

colnames(moderate_matrix) <- rownames(moderate_matrix) <- indicators

result_moderate <- ahp_weights(moderate_matrix, indicators = indicators)
cat("Moderate values CR:", round(result_moderate$CR, 4), "\n")
#> Moderate values CR: 0.0038

Strategy 4: Simplify the Hierarchy

If you have many indicators (>7), consider grouping them:

# Instead of comparing 10 indicators directly:
# indicators <- c("pH", "OM", "N", "P", "K", "Ca", "Mg", "CEC", "BD", "EC")

# Group into categories and use two-level AHP:
# Level 1: Compare categories
categories <- c("Chemical", "Fertility", "Physical")
category_pairwise <- matrix(c(
  1,   2,   3,
  1/2, 1,   2,
  1/3, 1/2, 1
), nrow = 3, byrow = TRUE)

# Level 2: Compare indicators within each category
chemical_indicators <- c("pH", "EC")
fertility_indicators <- c("OM", "N", "P", "K")
physical_indicators <- c("BD", "Sand", "Clay")

# Then combine weights: category_weight × indicator_weight_within_category

Practical Examples

Example 1: Tropical Agricultural Soils

Focus on nutrient availability and organic matter:

# Indicators for tropical agriculture
tropical_indicators <- c("pH", "OM", "N", "P", "CEC")

# Expert judgment: OM and N are most critical
# pH and CEC are moderately important
# P is less critical (often fixed by fertilizer)
tropical_pairwise <- matrix(c(
  # pH   OM    N     P     CEC
    1,   1/3,  1/3,  2,    1,     # pH
    3,   1,    1,    5,    2,     # OM
    3,   1,    1,    5,    2,     # N
    1/2, 1/5,  1/5,  1,    1/3,   # P
    1,   1/2,  1/2,  3,    1      # CEC
), nrow = 5, byrow = TRUE)

colnames(tropical_pairwise) <- rownames(tropical_pairwise) <- tropical_indicators

tropical_ahp <- ahp_weights(tropical_pairwise, indicators = tropical_indicators)

cat("Tropical agriculture weights:\n")
#> Tropical agriculture weights:
print(round(tropical_ahp$weights, 3))
#>    pH    OM     N     P   CEC 
#> 0.124 0.328 0.328 0.061 0.158
cat("\nCR:", round(tropical_ahp$CR, 4), "\n")
#> 
#> CR: 0.0056

# Calculate SQI (note: pairwise matrix would need to match MDS indicators)
result_tropical <- compute_sqi_properties(
  data = soil_ucayali,
  properties = tropical_indicators,
  id_column = "SampleID"
)

cat("\nMDS indicators selected:", paste(result_tropical$mds, collapse = ", "), "\n")
#> 
#> MDS indicators selected: N, pH, OM
cat("Mean SQI:", round(mean(result_tropical$results$SQI), 3), "\n")
#> Mean SQI: 0.432

Example 2: Erosion Risk Assessment

Focus on physical properties:

# Indicators for erosion risk
erosion_indicators <- c("Sand", "Clay", "OM", "BD")

# Expert judgment: OM and BD are most critical for erosion
# Texture (Sand, Clay) is moderately important
erosion_pairwise <- matrix(c(
  # Sand  Clay  OM    BD
    1,    2,    1/3,  1/2,   # Sand
    1/2,  1,    1/3,  1/2,   # Clay
    3,    3,    1,    1,     # OM
    2,    2,    1,    1      # BD
), nrow = 4, byrow = TRUE)

colnames(erosion_pairwise) <- rownames(erosion_pairwise) <- erosion_indicators

erosion_ahp <- ahp_weights(erosion_pairwise, indicators = erosion_indicators)

cat("Erosion risk weights:\n")
#> Erosion risk weights:
print(round(erosion_ahp$weights, 3))
#>  Sand  Clay    OM    BD 
#> 0.172 0.122 0.389 0.317
cat("\nCR:", round(erosion_ahp$CR, 4), "\n")
#> 
#> CR: 0.0301

Example 3: Carbon Sequestration Potential

Focus on organic matter and related properties:

# Indicators for carbon sequestration
carbon_indicators <- c("OM", "SOC", "N", "Clay", "CEC")

# Expert judgment: OM and SOC are most critical
# Clay and CEC support carbon stabilization
# N indicates decomposition rate
carbon_pairwise <- matrix(c(
  # OM   SOC   N     Clay  CEC
    1,   1,    3,    2,    2,     # OM
    1,   1,    3,    2,    2,     # SOC
    1/3, 1/3,  1,    1/2,  1/2,   # N
    1/2, 1/2,  2,    1,    1,     # Clay
    1/2, 1/2,  2,    1,    1      # CEC
), nrow = 5, byrow = TRUE)

colnames(carbon_pairwise) <- rownames(carbon_pairwise) <- carbon_indicators

carbon_ahp <- ahp_weights(carbon_pairwise, indicators = carbon_indicators)

cat("Carbon sequestration weights:\n")
#> Carbon sequestration weights:
print(round(carbon_ahp$weights, 3))
#>    OM   SOC     N  Clay   CEC 
#> 0.298 0.298 0.089 0.158 0.158
cat("\nCR:", round(carbon_ahp$CR, 4), "\n")
#> 
#> CR: 0.003

Best Practices for AHP in Soil Quality

1. Involve Multiple Experts

# Collect judgments from multiple experts
expert1_matrix <- create_ahp_matrix(indicators, mode = "interactive")
expert2_matrix <- create_ahp_matrix(indicators, mode = "interactive")
expert3_matrix <- create_ahp_matrix(indicators, mode = "interactive")

# Aggregate using geometric mean
n_experts <- 3
aggregated_matrix <- (expert1_matrix$matrix * 
                      expert2_matrix$matrix * 
                      expert3_matrix$matrix)^(1/n_experts)

# Calculate final weights
final_ahp <- ahp_weights(aggregated_matrix, indicators = indicators)

2. Document Your Rationale

Always document why you assigned specific importance values:

# Example documentation
rationale <- list(
  "OM vs pH" = "OM rated 3x more important because it affects multiple 
                soil functions (nutrient supply, water retention, structure)",
  "N vs P" = "N rated 2x more important because it's more limiting in 
              tropical soils and harder to supplement",
  "pH vs P" = "Equal importance (1) because both are easily managed 
               through amendments"
)

# Include this in your analysis report

3. Perform Sensitivity Analysis

Test how results change with different weights:

# Compare different AHP weight scenarios
indicators <- c("pH", "OM", "N", "P")

# Scenario 1: Equal importance
equal_weights <- rep(1/4, 4)
names(equal_weights) <- indicators

# Scenario 2: OM-focused
om_focused <- matrix(c(
  1,   1/5, 1/3, 1/2,
  5,   1,   2,   3,
  3,   1/2, 1,   2,
  2,   1/3, 1/2, 1
), nrow = 4, byrow = TRUE)
colnames(om_focused) <- rownames(om_focused) <- indicators

ahp_om <- ahp_weights(om_focused, indicators = indicators)

# Scenario 3: N-focused
n_focused <- matrix(c(
  1,   1/3, 1/5, 1/2,
  3,   1,   1/2, 2,
  5,   2,   1,   3,
  2,   1/2, 1/3, 1
), nrow = 4, byrow = TRUE)
colnames(n_focused) <- rownames(n_focused) <- indicators

ahp_n <- ahp_weights(n_focused, indicators = indicators)

# Compare weights
cat("Equal weights:\n")
#> Equal weights:
print(round(equal_weights, 3))
#>   pH   OM    N    P 
#> 0.25 0.25 0.25 0.25

cat("\nOM-focused weights (CR =", round(ahp_om$CR, 4), "):\n")
#> 
#> OM-focused weights (CR = 0.0054 ):
print(round(ahp_om$weights, 3))
#>    pH    OM     N     P 
#> 0.088 0.483 0.272 0.157

cat("\nN-focused weights (CR =", round(ahp_n$CR, 4), "):\n")
#> 
#> N-focused weights (CR = 0.0054 ):
print(round(ahp_n$weights, 3))
#>    pH    OM     N     P 
#> 0.088 0.272 0.483 0.157

# Calculate SQI with equal weights to show the approach
result_sqi <- compute_sqi_properties(
  data = soil_ucayali,
  properties = indicators,
  id_column = "SampleID"
)

cat("\nMean SQI with equal weights:", round(mean(result_sqi$results$SQI), 3), "\n")
#> 
#> Mean SQI with equal weights: 0.432
cat("Note: Different weight scenarios would produce different SQI values\n")
#> Note: Different weight scenarios would produce different SQI values
cat("when applied to the MDS indicators selected by PCA.\n")
#> when applied to the MDS indicators selected by PCA.

4. Validate with Field Data

When possible, validate AHP weights against field observations:

# Example: Compare SQI with crop yield data
# Assuming you have yield data
yield_data <- data.frame(
  SampleID = soil_ucayali$SampleID,
  Yield = rnorm(50, 3000, 500)  # Example yield data
)

# Merge with SQI results
merged <- merge(result_ahp$results, yield_data, by = "SampleID")

# Check correlation
cor_sqi_yield <- cor(merged$SQI, merged$Yield)
cat("Correlation between SQI and yield:", round(cor_sqi_yield, 3), "\n")

# If correlation is low, reconsider your weights

Summary

This vignette covered:

AHP fundamentals: Using expert judgment to weight soil indicators
Saaty scale: The 1-9 scale for pairwise comparisons
Creating matrices: Interactive, manual, and ratio-based methods
Consistency Ratio: Measuring and improving logical consistency
Practical examples: Context-specific weighting for different goals
Best practices: Multiple experts, documentation, sensitivity analysis

Key takeaways:

Use AHP when indicator importance varies by context
Aim for CR < 0.10 for reliable weights
Document your rationale for transparency
Test sensitivity to different weight scenarios
Validate results with field data when possible

References

Saaty, T. L. (1980). The Analytic Hierarchy Process: Planning, Priority Setting, Resource Allocation. McGraw-Hill.
Saaty, T. L. (2008). Decision making with the analytic hierarchy process. International Journal of Services Sciences, 1(1), 83-98.
Rezaei-Moghaddam, K., & Karami, E. (2008). A multiple criteria evaluation of sustainable agricultural development models using AHP. Environment, Development and Sustainability, 10(4), 407-426.
Masto, R. E., Chhonkar, P. K., Singh, D., & Patra, A. K. (2008). Alternative soil quality indices for evaluating the effect of intensive cropping, fertilisation and manuring for 31 years in the semi-arid soils of India. Environmental Monitoring and Assessment, 136(1-3), 419-435.

soilquality package

2025-12-26