Chi-Square Post-Hoc Tests for Clinicopathological Research

Introduction

The Chi-Square Post-Hoc Tests module in ClinicoPath provides comprehensive analysis for identifying specific group differences when an overall chi-square test of independence shows statistical significance. This is particularly valuable in clinicopathological research where researchers need to understand which specific combinations of categorical variables are driving significant associations.

Clinical Motivation

In clinical and pathological research, we often encounter scenarios where:

Treatment response varies by patient characteristics (e.g., sex, age group, tumor stage)
Biomarker expression differs across diagnostic categories (e.g., hormone receptor status by tumor grade)
Complications occur differentially across treatment modalities (e.g., surgical complications by surgeon experience level)
Survival outcomes vary by multiple pathological features (e.g., lymph node status by tumor size categories)

When an overall chi-square test indicates a significant association, post-hoc testing helps identify which specific group comparisons are driving this significance, while controlling for multiple comparisons.

When to Use Chi-Square Post-Hoc Tests

Chi-square post-hoc tests are appropriate when:

Overall chi-square test is significant (p < 0.05)
At least one variable has ≥3 categories (otherwise, no post-hoc testing needed)
Sample sizes are adequate (expected frequencies ≥5 in most cells)
Variables are categorical (nominal or ordinal)
Observations are independent

Package Setup

library(ClinicoPath)

## Warning: replacing previous import 'dplyr::as_data_frame' by
## 'igraph::as_data_frame' when loading 'ClinicoPath'

## Warning: replacing previous import 'DiagrammeR::count_automorphisms' by
## 'igraph::count_automorphisms' when loading 'ClinicoPath'

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## loading 'ClinicoPath'

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## 'igraph::get_edge_ids' when loading 'ClinicoPath'

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## loading 'ClinicoPath'

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## loading 'ClinicoPath'

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## 'mlr3measures::precision' when loading 'ClinicoPath'

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## loading 'ClinicoPath'

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## loading 'ClinicoPath'

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## loading 'ClinicoPath'

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## 'mlr3measures::recall' when loading 'ClinicoPath'

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## 'mlr3measures::specificity' when loading 'ClinicoPath'

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## 'mlr3measures::sensitivity' when loading 'ClinicoPath'

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## 'tibble::as_data_frame' when loading 'ClinicoPath'

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## loading 'ClinicoPath'

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## loading 'ClinicoPath'

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## 'caret::sensitivity' when loading 'ClinicoPath'

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## 'caret::specificity' when loading 'ClinicoPath'

## Registered S3 methods overwritten by 'useful':
##   method         from     
##   autoplot.acf   ggfortify
##   fortify.acf    ggfortify
##   fortify.kmeans ggfortify
##   fortify.ts     ggfortify

## Warning: replacing previous import 'jmvcore::select' by 'dplyr::select' when
## loading 'ClinicoPath'

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##   method                  from   
##   heightDetails.titleGrob ggplot2
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## 'grafify::plot_histogram' when loading 'ClinicoPath'

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## loading 'ClinicoPath'

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## 'ClinicoPath'

## Warning: replacing previous import 'tibble::view' by 'summarytools::view' when
## loading 'ClinicoPath'

library(dplyr)

## 
## Attaching package: 'dplyr'

## The following objects are masked from 'package:stats':
## 
##     filter, lag

## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union

library(knitr)

# Load the histopathology dataset
data("histopathology")

# Display dataset structure
str(histopathology[, c("Sex", "Race", "Group", "Grade_Level", "LVI", "PNI", "Mortality5yr")])

## tibble [250 × 7] (S3: tbl_df/tbl/data.frame)
##  $ Sex         : chr [1:250] "Male" "Female" "Male" "Male" ...
##  $ Race        : chr [1:250] "White" "White" "White" "White" ...
##  $ Group       : chr [1:250] "Control" "Treatment" "Control" "Treatment" ...
##  $ Grade_Level : chr [1:250] "high" "low" "low" "high" ...
##  $ LVI         : chr [1:250] "Present" "Absent" "Absent" "Present" ...
##  $ PNI         : chr [1:250] "Absent" "Absent" "Absent" "Absent" ...
##  $ Mortality5yr: chr [1:250] "Alive" "Dead" "Dead" "Alive" ...

Basic Chi-Square Post-Hoc Analysis

Example 1: Treatment Group by Sex

Let’s start with a basic example examining whether treatment group assignment varies by patient sex.

# Basic chi-square post-hoc test
chisqposttest(
  data = histopathology,
  rows = "Sex",
  cols = "Group",
  posthoc = "bonferroni",
  sig = 0.05,
  excl = TRUE,
  exp = FALSE,
  plot = FALSE
)

## 
##  CHI-SQUARE POST-HOC TESTS
## 
## character(0)
## 
##  Chi-Square Test Results                          
##  ──────────────────────────────────────────────── 
##    Statistic     Value          df    p-value     
##  ──────────────────────────────────────────────── 
##    Chi-Square    2.365421e-4     1    0.9877291   
##  ──────────────────────────────────────────────── 
## 
## 
##  <div style="padding: 15px; background-color: #f8f9fa; border-left: 4px
##  solid #1976d2; margin: 10px 0;">
##  <h4 style="color: #1976d2; margin-top: 0;">Chi-Square Post-Hoc
##  Analysis Guide
## 
##  Three-Step Comprehensive Analysis:
## 
## 
## 
##  1.
##  Overall Chi-Square Test:
##  Tests if there's any association between variables
##  <br/>
##  2.
##  Residuals Analysis:
##  Identifies which specific cells contribute to significance
##  <br/>
##  3.
##  Pairwise Comparisons:
##  Formal hypothesis testing between group pairs
## 
## 
## 
##  *Recommended approach: Start with residuals analysis for pattern
##  identification, then use pairwise tests for formal hypothesis
##  testing.*
## 
## 
##  <table border='1' style='border-collapse:
##  collapse;'>ControlTreatmentFemale5863Male6166
## 
##  <div style='margin: 15px 0;'><div style="padding: 15px;
##  background-color: #e8f5e8; border-left: 4px solid #4caf50; margin:
##  10px 0;">
##  <h4 style="color: #2e7d32; margin-top: 0;">Standardized Residuals
##  Interpretation
## 
##  What are standardized residuals?
## 
## 
##  Standardized residuals measure how much each cell deviates from what
##  we'd expect if there was no association. They're calculated as
##  (Observed - Expected) / √Expected, then adjusted for multiple
##  comparisons.
## 
##  <div style="background-color: #f1f8e9; padding: 10px; border-radius:
##  3px;">
##  Interpretation Guidelines:
##  <br/>
##  • |Residual| > 2.0: Suggests meaningful deviation
##  <br/>
##  • |Residual| > 3.0: Strong evidence of deviation
##  <br/>
##  • Positive values: Over-represented (more than expected)
##  <br/>
##  • Negative values: Under-represented (fewer than expected)
## 
##  <table style='border-collapse: collapse; width: 100%; margin: 15px
##  0;'><tr style='background-color: #e8f5e8;'><th style='border: 1px
##  solid #666; padding: 8px;'><th style='border: 1px solid #666; padding:
##  8px; text-align: center;'>Control<th style='border: 1px solid #666;
##  padding: 8px; text-align: center;'>Treatment<td style='border: 1px
##  solid #666; padding: 8px; font-weight: bold; background-color:
##  #f5f5f5;'>Female<td style='border: 1px solid #666; padding: 8px;
##  text-align: center; background-color: #f9f9f9;'>-0.015
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  text-align: center; background-color: #f9f9f9;'>0.015
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  font-weight: bold; background-color: #f5f5f5;'>Male<td style='border:
##  1px solid #666; padding: 8px; text-align: center; background-color:
##  #f9f9f9;'>0.015
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  text-align: center; background-color: #f9f9f9;'>-0.015
##  As expected
## 
##  Critical value for significance: ±2.498
## 
##  <div style='padding: 15px; background-color: #d4edda; border: 1px
##  solid #c3e6cb;'>Note: Overall chi-square test is not significant (p ≥
##  0.05). Post-hoc pairwise comparisons are not recommended when the
##  overall test is non-significant.
## 
##  Pairwise Comparison Results                                                                                  
##  ──────────────────────────────────────────────────────────────────────────────────────────────────────────── 
##    Comparison    Test Method    Chi-Square    p-value      Adj. p-value    Effect Size (Phi)    Significant   
##  ──────────────────────────────────────────────────────────────────────────────────────────────────────────── 
##  ──────────────────────────────────────────────────────────────────────────────────────────────────────────── 
## 
## 
## character(0)

Interpretation

The results show: 1. Overall Chi-Square Test: Tests whether there’s any association between sex and treatment group 2. Contingency Table: Displays observed frequencies for each combination 3. Post-Hoc Tests: If significant, shows pairwise comparisons with adjusted p-values

Example 2: Tumor Grade by Lymph Node Status

A more clinically relevant example examining the relationship between tumor grade and lymph node metastasis:

# Chi-square post-hoc test with clinical variables
chisqposttest(
  data = histopathology,
  rows = "Grade_Level",
  cols = "LVI",  # Lymphovascular invasion
  posthoc = "fdr",
  sig = 0.05,
  excl = TRUE,
  exp = TRUE,    # Show expected values
  plot = FALSE
)

## 
##  CHI-SQUARE POST-HOC TESTS
## 
## character(0)
## 
##  Chi-Square Test Results                        
##  ────────────────────────────────────────────── 
##    Statistic     Value        df    p-value     
##  ────────────────────────────────────────────── 
##    Chi-Square    0.6169959     2    0.7345495   
##  ────────────────────────────────────────────── 
## 
## 
##  <div style="padding: 15px; background-color: #f8f9fa; border-left: 4px
##  solid #1976d2; margin: 10px 0;">
##  <h4 style="color: #1976d2; margin-top: 0;">Chi-Square Post-Hoc
##  Analysis Guide
## 
##  Three-Step Comprehensive Analysis:
## 
## 
## 
##  1.
##  Overall Chi-Square Test:
##  Tests if there's any association between variables
##  <br/>
##  2.
##  Residuals Analysis:
##  Identifies which specific cells contribute to significance
##  <br/>
##  3.
##  Pairwise Comparisons:
##  Formal hypothesis testing between group pairs
## 
## 
## 
##  *Recommended approach: Start with residuals analysis for pattern
##  identification, then use pairwise tests for formal hypothesis
##  testing.*
## 
## 
##  Values shown as: Observed
##  (Expected)
## 
##  <table border='1' style='border-collapse:
##  collapse;'>AbsentPresenthigh56
##  (53)36
##  (39)low43
##  (44.4)34
##  (32.6)moderate44
##  (45.6)35
##  (33.4)
## 
##  <div style='margin: 15px 0;'><div style="padding: 15px;
##  background-color: #e8f5e8; border-left: 4px solid #4caf50; margin:
##  10px 0;">
##  <h4 style="color: #2e7d32; margin-top: 0;">Standardized Residuals
##  Interpretation
## 
##  What are standardized residuals?
## 
## 
##  Standardized residuals measure how much each cell deviates from what
##  we'd expect if there was no association. They're calculated as
##  (Observed - Expected) / √Expected, then adjusted for multiple
##  comparisons.
## 
##  <div style="background-color: #f1f8e9; padding: 10px; border-radius:
##  3px;">
##  Interpretation Guidelines:
##  <br/>
##  • |Residual| > 2.0: Suggests meaningful deviation
##  <br/>
##  • |Residual| > 3.0: Strong evidence of deviation
##  <br/>
##  • Positive values: Over-represented (more than expected)
##  <br/>
##  • Negative values: Under-represented (fewer than expected)
## 
##  <table style='border-collapse: collapse; width: 100%; margin: 15px
##  0;'><tr style='background-color: #e8f5e8;'><th style='border: 1px
##  solid #666; padding: 8px;'><th style='border: 1px solid #666; padding:
##  8px; text-align: center;'>Absent<th style='border: 1px solid #666;
##  padding: 8px; text-align: center;'>Present<td style='border: 1px solid
##  #666; padding: 8px; font-weight: bold; background-color:
##  #f5f5f5;'>high<td style='border: 1px solid #666; padding: 8px;
##  text-align: center; background-color: #f9f9f9;'>0.785
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  text-align: center; background-color: #f9f9f9;'>-0.785
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  font-weight: bold; background-color: #f5f5f5;'>low<td style='border:
##  1px solid #666; padding: 8px; text-align: center; background-color:
##  #f9f9f9;'>-0.389
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  text-align: center; background-color: #f9f9f9;'>0.389
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  font-weight: bold; background-color: #f5f5f5;'>moderate<td
##  style='border: 1px solid #666; padding: 8px; text-align: center;
##  background-color: #f9f9f9;'>-0.428
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  text-align: center; background-color: #f9f9f9;'>0.428
##  As expected
## 
##  Critical value for significance: ±2.638
## 
##  <div style='padding: 15px; background-color: #d4edda; border: 1px
##  solid #c3e6cb;'>Note: Overall chi-square test is not significant (p ≥
##  0.05). Post-hoc pairwise comparisons are not recommended when the
##  overall test is non-significant.
## 
##  Pairwise Comparison Results                                                                                  
##  ──────────────────────────────────────────────────────────────────────────────────────────────────────────── 
##    Comparison    Test Method    Chi-Square    p-value      Adj. p-value    Effect Size (Phi)    Significant   
##  ──────────────────────────────────────────────────────────────────────────────────────────────────────────── 
##  ──────────────────────────────────────────────────────────────────────────────────────────────────────────── 
## 
## 
## character(0)

Clinical Significance

This analysis helps determine: - Whether tumor grade is associated with lymphovascular invasion - Which specific grade categories show significant differences - Expected vs. observed frequencies for each combination

Multiple Comparison Adjustments

Understanding Adjustment Methods

When performing multiple pairwise comparisons, we need to adjust p-values to control the family-wise error rate:

Bonferroni Correction

Most conservative approach
Multiplies each p-value by the number of comparisons
Use when: You want strong control of Type I error

chisqposttest(
  data = histopathology,
  rows = "Race",
  cols = "Mortality5yr",
  posthoc = "bonferroni",
  sig = 0.05,
  excl = TRUE,
  exp = FALSE,
  plot = FALSE
)

## Warning in stats::chisq.test(contTable, correct = FALSE): Chi-squared
## approximation may be incorrect

## 
##  CHI-SQUARE POST-HOC TESTS
## 
## character(0)
## 
##  Chi-Square Test Results                        
##  ────────────────────────────────────────────── 
##    Statistic     Value        df    p-value     
##  ────────────────────────────────────────────── 
##    Chi-Square    0.8614085     6    0.9903252   
##  ────────────────────────────────────────────── 
## 
## 
##  <div style="padding: 15px; background-color: #f8f9fa; border-left: 4px
##  solid #1976d2; margin: 10px 0;">
##  <h4 style="color: #1976d2; margin-top: 0;">Chi-Square Post-Hoc
##  Analysis Guide
## 
##  Three-Step Comprehensive Analysis:
## 
## 
## 
##  1.
##  Overall Chi-Square Test:
##  Tests if there's any association between variables
##  <br/>
##  2.
##  Residuals Analysis:
##  Identifies which specific cells contribute to significance
##  <br/>
##  3.
##  Pairwise Comparisons:
##  Formal hypothesis testing between group pairs
## 
## 
## 
##  *Recommended approach: Start with residuals analysis for pattern
##  identification, then use pairwise tests for formal hypothesis
##  testing.*
## 
## 
##  <table border='1' style='border-collapse:
##  collapse;'>AliveDeadAsian25Bi-Racial13Black1124Hawaiian01Hispanic1527Native12White50106
## 
##  <div style='margin: 15px 0;'><div style="padding: 15px;
##  background-color: #e8f5e8; border-left: 4px solid #4caf50; margin:
##  10px 0;">
##  <h4 style="color: #2e7d32; margin-top: 0;">Standardized Residuals
##  Interpretation
## 
##  What are standardized residuals?
## 
## 
##  Standardized residuals measure how much each cell deviates from what
##  we'd expect if there was no association. They're calculated as
##  (Observed - Expected) / √Expected, then adjusted for multiple
##  comparisons.
## 
##  <div style="background-color: #f1f8e9; padding: 10px; border-radius:
##  3px;">
##  Interpretation Guidelines:
##  <br/>
##  • |Residual| > 2.0: Suggests meaningful deviation
##  <br/>
##  • |Residual| > 3.0: Strong evidence of deviation
##  <br/>
##  • Positive values: Over-represented (more than expected)
##  <br/>
##  • Negative values: Under-represented (fewer than expected)
## 
##  <table style='border-collapse: collapse; width: 100%; margin: 15px
##  0;'><tr style='background-color: #e8f5e8;'><th style='border: 1px
##  solid #666; padding: 8px;'><th style='border: 1px solid #666; padding:
##  8px; text-align: center;'>Alive<th style='border: 1px solid #666;
##  padding: 8px; text-align: center;'>Dead<td style='border: 1px solid
##  #666; padding: 8px; font-weight: bold; background-color:
##  #f5f5f5;'>Asian<td style='border: 1px solid #666; padding: 8px;
##  text-align: center; background-color: #f9f9f9;'>-0.212
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  text-align: center; background-color: #f9f9f9;'>0.212
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  font-weight: bold; background-color: #f5f5f5;'>Bi-Racial<td
##  style='border: 1px solid #666; padding: 8px; text-align: center;
##  background-color: #f9f9f9;'>-0.313
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  text-align: center; background-color: #f9f9f9;'>0.313
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  font-weight: bold; background-color: #f5f5f5;'>Black<td style='border:
##  1px solid #666; padding: 8px; text-align: center; background-color:
##  #f9f9f9;'>-0.113
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  text-align: center; background-color: #f9f9f9;'>0.113
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  font-weight: bold; background-color: #f5f5f5;'>Hawaiian<td
##  style='border: 1px solid #666; padding: 8px; text-align: center;
##  background-color: #f9f9f9;'>-0.691
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  text-align: center; background-color: #f9f9f9;'>0.691
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  font-weight: bold; background-color: #f5f5f5;'>Hispanic<td
##  style='border: 1px solid #666; padding: 8px; text-align: center;
##  background-color: #f9f9f9;'>0.526
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  text-align: center; background-color: #f9f9f9;'>-0.526
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  font-weight: bold; background-color: #f5f5f5;'>Native<td
##  style='border: 1px solid #666; padding: 8px; text-align: center;
##  background-color: #f9f9f9;'>0.04
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  text-align: center; background-color: #f9f9f9;'>-0.04
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  font-weight: bold; background-color: #f5f5f5;'>White<td style='border:
##  1px solid #666; padding: 8px; text-align: center; background-color:
##  #f9f9f9;'>-0.091
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  text-align: center; background-color: #f9f9f9;'>0.091
##  As expected
## 
##  Critical value for significance: ±2.914
## 
##  <div style='padding: 15px; background-color: #d4edda; border: 1px
##  solid #c3e6cb;'>Note: Overall chi-square test is not significant (p ≥
##  0.05). Post-hoc pairwise comparisons are not recommended when the
##  overall test is non-significant.
## 
##  Pairwise Comparison Results                                                                                  
##  ──────────────────────────────────────────────────────────────────────────────────────────────────────────── 
##    Comparison    Test Method    Chi-Square    p-value      Adj. p-value    Effect Size (Phi)    Significant   
##  ──────────────────────────────────────────────────────────────────────────────────────────────────────────── 
##  ──────────────────────────────────────────────────────────────────────────────────────────────────────────── 
## 
## 
## character(0)

False Discovery Rate (FDR)

Less conservative than Bonferroni
Controls the expected proportion of false discoveries
Use when: You want to balance Type I and Type II errors

chisqposttest(
  data = histopathology,
  rows = "Race",
  cols = "Mortality5yr",
  posthoc = "fdr",
  sig = 0.05,
  excl = TRUE,
  exp = FALSE,
  plot = FALSE
)

## Warning in stats::chisq.test(contTable, correct = FALSE): Chi-squared
## approximation may be incorrect

## 
##  CHI-SQUARE POST-HOC TESTS
## 
## character(0)
## 
##  Chi-Square Test Results                        
##  ────────────────────────────────────────────── 
##    Statistic     Value        df    p-value     
##  ────────────────────────────────────────────── 
##    Chi-Square    0.8614085     6    0.9903252   
##  ────────────────────────────────────────────── 
## 
## 
##  <div style="padding: 15px; background-color: #f8f9fa; border-left: 4px
##  solid #1976d2; margin: 10px 0;">
##  <h4 style="color: #1976d2; margin-top: 0;">Chi-Square Post-Hoc
##  Analysis Guide
## 
##  Three-Step Comprehensive Analysis:
## 
## 
## 
##  1.
##  Overall Chi-Square Test:
##  Tests if there's any association between variables
##  <br/>
##  2.
##  Residuals Analysis:
##  Identifies which specific cells contribute to significance
##  <br/>
##  3.
##  Pairwise Comparisons:
##  Formal hypothesis testing between group pairs
## 
## 
## 
##  *Recommended approach: Start with residuals analysis for pattern
##  identification, then use pairwise tests for formal hypothesis
##  testing.*
## 
## 
##  <table border='1' style='border-collapse:
##  collapse;'>AliveDeadAsian25Bi-Racial13Black1124Hawaiian01Hispanic1527Native12White50106
## 
##  <div style='margin: 15px 0;'><div style="padding: 15px;
##  background-color: #e8f5e8; border-left: 4px solid #4caf50; margin:
##  10px 0;">
##  <h4 style="color: #2e7d32; margin-top: 0;">Standardized Residuals
##  Interpretation
## 
##  What are standardized residuals?
## 
## 
##  Standardized residuals measure how much each cell deviates from what
##  we'd expect if there was no association. They're calculated as
##  (Observed - Expected) / √Expected, then adjusted for multiple
##  comparisons.
## 
##  <div style="background-color: #f1f8e9; padding: 10px; border-radius:
##  3px;">
##  Interpretation Guidelines:
##  <br/>
##  • |Residual| > 2.0: Suggests meaningful deviation
##  <br/>
##  • |Residual| > 3.0: Strong evidence of deviation
##  <br/>
##  • Positive values: Over-represented (more than expected)
##  <br/>
##  • Negative values: Under-represented (fewer than expected)
## 
##  <table style='border-collapse: collapse; width: 100%; margin: 15px
##  0;'><tr style='background-color: #e8f5e8;'><th style='border: 1px
##  solid #666; padding: 8px;'><th style='border: 1px solid #666; padding:
##  8px; text-align: center;'>Alive<th style='border: 1px solid #666;
##  padding: 8px; text-align: center;'>Dead<td style='border: 1px solid
##  #666; padding: 8px; font-weight: bold; background-color:
##  #f5f5f5;'>Asian<td style='border: 1px solid #666; padding: 8px;
##  text-align: center; background-color: #f9f9f9;'>-0.212
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  text-align: center; background-color: #f9f9f9;'>0.212
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  font-weight: bold; background-color: #f5f5f5;'>Bi-Racial<td
##  style='border: 1px solid #666; padding: 8px; text-align: center;
##  background-color: #f9f9f9;'>-0.313
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  text-align: center; background-color: #f9f9f9;'>0.313
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  font-weight: bold; background-color: #f5f5f5;'>Black<td style='border:
##  1px solid #666; padding: 8px; text-align: center; background-color:
##  #f9f9f9;'>-0.113
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  text-align: center; background-color: #f9f9f9;'>0.113
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  font-weight: bold; background-color: #f5f5f5;'>Hawaiian<td
##  style='border: 1px solid #666; padding: 8px; text-align: center;
##  background-color: #f9f9f9;'>-0.691
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  text-align: center; background-color: #f9f9f9;'>0.691
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  font-weight: bold; background-color: #f5f5f5;'>Hispanic<td
##  style='border: 1px solid #666; padding: 8px; text-align: center;
##  background-color: #f9f9f9;'>0.526
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  text-align: center; background-color: #f9f9f9;'>-0.526
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  font-weight: bold; background-color: #f5f5f5;'>Native<td
##  style='border: 1px solid #666; padding: 8px; text-align: center;
##  background-color: #f9f9f9;'>0.04
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  text-align: center; background-color: #f9f9f9;'>-0.04
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  font-weight: bold; background-color: #f5f5f5;'>White<td style='border:
##  1px solid #666; padding: 8px; text-align: center; background-color:
##  #f9f9f9;'>-0.091
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  text-align: center; background-color: #f9f9f9;'>0.091
##  As expected
## 
##  Critical value for significance: ±2.914
## 
##  <div style='padding: 15px; background-color: #d4edda; border: 1px
##  solid #c3e6cb;'>Note: Overall chi-square test is not significant (p ≥
##  0.05). Post-hoc pairwise comparisons are not recommended when the
##  overall test is non-significant.
## 
##  Pairwise Comparison Results                                                                                  
##  ──────────────────────────────────────────────────────────────────────────────────────────────────────────── 
##    Comparison    Test Method    Chi-Square    p-value      Adj. p-value    Effect Size (Phi)    Significant   
##  ──────────────────────────────────────────────────────────────────────────────────────────────────────────── 
##  ──────────────────────────────────────────────────────────────────────────────────────────────────────────── 
## 
## 
## character(0)

Holm Method

Step-down procedure
More powerful than Bonferroni while maintaining family-wise error control
Use when: You want more power than Bonferroni but still strong control

chisqposttest(
  data = histopathology,
  rows = "Race",
  cols = "Mortality5yr",
  posthoc = "holm",
  sig = 0.05,
  excl = TRUE,
  exp = FALSE,
  plot = FALSE
)

## Warning in stats::chisq.test(contTable, correct = FALSE): Chi-squared
## approximation may be incorrect

## 
##  CHI-SQUARE POST-HOC TESTS
## 
## character(0)
## 
##  Chi-Square Test Results                        
##  ────────────────────────────────────────────── 
##    Statistic     Value        df    p-value     
##  ────────────────────────────────────────────── 
##    Chi-Square    0.8614085     6    0.9903252   
##  ────────────────────────────────────────────── 
## 
## 
##  <div style="padding: 15px; background-color: #f8f9fa; border-left: 4px
##  solid #1976d2; margin: 10px 0;">
##  <h4 style="color: #1976d2; margin-top: 0;">Chi-Square Post-Hoc
##  Analysis Guide
## 
##  Three-Step Comprehensive Analysis:
## 
## 
## 
##  1.
##  Overall Chi-Square Test:
##  Tests if there's any association between variables
##  <br/>
##  2.
##  Residuals Analysis:
##  Identifies which specific cells contribute to significance
##  <br/>
##  3.
##  Pairwise Comparisons:
##  Formal hypothesis testing between group pairs
## 
## 
## 
##  *Recommended approach: Start with residuals analysis for pattern
##  identification, then use pairwise tests for formal hypothesis
##  testing.*
## 
## 
##  <table border='1' style='border-collapse:
##  collapse;'>AliveDeadAsian25Bi-Racial13Black1124Hawaiian01Hispanic1527Native12White50106
## 
##  <div style='margin: 15px 0;'><div style="padding: 15px;
##  background-color: #e8f5e8; border-left: 4px solid #4caf50; margin:
##  10px 0;">
##  <h4 style="color: #2e7d32; margin-top: 0;">Standardized Residuals
##  Interpretation
## 
##  What are standardized residuals?
## 
## 
##  Standardized residuals measure how much each cell deviates from what
##  we'd expect if there was no association. They're calculated as
##  (Observed - Expected) / √Expected, then adjusted for multiple
##  comparisons.
## 
##  <div style="background-color: #f1f8e9; padding: 10px; border-radius:
##  3px;">
##  Interpretation Guidelines:
##  <br/>
##  • |Residual| > 2.0: Suggests meaningful deviation
##  <br/>
##  • |Residual| > 3.0: Strong evidence of deviation
##  <br/>
##  • Positive values: Over-represented (more than expected)
##  <br/>
##  • Negative values: Under-represented (fewer than expected)
## 
##  <table style='border-collapse: collapse; width: 100%; margin: 15px
##  0;'><tr style='background-color: #e8f5e8;'><th style='border: 1px
##  solid #666; padding: 8px;'><th style='border: 1px solid #666; padding:
##  8px; text-align: center;'>Alive<th style='border: 1px solid #666;
##  padding: 8px; text-align: center;'>Dead<td style='border: 1px solid
##  #666; padding: 8px; font-weight: bold; background-color:
##  #f5f5f5;'>Asian<td style='border: 1px solid #666; padding: 8px;
##  text-align: center; background-color: #f9f9f9;'>-0.212
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  text-align: center; background-color: #f9f9f9;'>0.212
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  font-weight: bold; background-color: #f5f5f5;'>Bi-Racial<td
##  style='border: 1px solid #666; padding: 8px; text-align: center;
##  background-color: #f9f9f9;'>-0.313
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  text-align: center; background-color: #f9f9f9;'>0.313
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  font-weight: bold; background-color: #f5f5f5;'>Black<td style='border:
##  1px solid #666; padding: 8px; text-align: center; background-color:
##  #f9f9f9;'>-0.113
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  text-align: center; background-color: #f9f9f9;'>0.113
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  font-weight: bold; background-color: #f5f5f5;'>Hawaiian<td
##  style='border: 1px solid #666; padding: 8px; text-align: center;
##  background-color: #f9f9f9;'>-0.691
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  text-align: center; background-color: #f9f9f9;'>0.691
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  font-weight: bold; background-color: #f5f5f5;'>Hispanic<td
##  style='border: 1px solid #666; padding: 8px; text-align: center;
##  background-color: #f9f9f9;'>0.526
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  text-align: center; background-color: #f9f9f9;'>-0.526
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  font-weight: bold; background-color: #f5f5f5;'>Native<td
##  style='border: 1px solid #666; padding: 8px; text-align: center;
##  background-color: #f9f9f9;'>0.04
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  text-align: center; background-color: #f9f9f9;'>-0.04
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  font-weight: bold; background-color: #f5f5f5;'>White<td style='border:
##  1px solid #666; padding: 8px; text-align: center; background-color:
##  #f9f9f9;'>-0.091
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  text-align: center; background-color: #f9f9f9;'>0.091
##  As expected
## 
##  Critical value for significance: ±2.914
## 
##  <div style='padding: 15px; background-color: #d4edda; border: 1px
##  solid #c3e6cb;'>Note: Overall chi-square test is not significant (p ≥
##  0.05). Post-hoc pairwise comparisons are not recommended when the
##  overall test is non-significant.
## 
##  Pairwise Comparison Results                                                                                  
##  ──────────────────────────────────────────────────────────────────────────────────────────────────────────── 
##    Comparison    Test Method    Chi-Square    p-value      Adj. p-value    Effect Size (Phi)    Significant   
##  ──────────────────────────────────────────────────────────────────────────────────────────────────────────── 
##  ──────────────────────────────────────────────────────────────────────────────────────────────────────────── 
## 
## 
## character(0)

Comparison of Adjustment Methods

# Create a comparison table of different methods
adjustment_methods <- c("bonferroni", "holm", "fdr", "none")

# Note: In practice, you would run these analyses and compare results
# This is a conceptual example of how different methods affect conclusions

cat("Adjustment Method Comparison:\n")

## Adjustment Method Comparison:

cat("- Bonferroni: Most conservative, lowest power\n")

## - Bonferroni: Most conservative, lowest power

cat("- Holm: Good balance of Type I error control and power\n")

## - Holm: Good balance of Type I error control and power

cat("- FDR: Higher power, controls false discovery rate\n")

## - FDR: Higher power, controls false discovery rate

cat("- None: Highest power, no multiple comparison adjustment\n")

## - None: Highest power, no multiple comparison adjustment

Advanced Features

Including Expected Values

Expected values help assess the magnitude of deviations from independence:

chisqposttest(
  data = histopathology,
  rows = "PNI",  # Perineural invasion
  cols = "Grade_Level",
  posthoc = "fdr",
  sig = 0.05,
  excl = TRUE,
  exp = TRUE,     # Show expected values
  plot = FALSE
)

## 
##  CHI-SQUARE POST-HOC TESTS
## 
## character(0)
## 
##  Chi-Square Test Results                       
##  ───────────────────────────────────────────── 
##    Statistic     Value       df    p-value     
##  ───────────────────────────────────────────── 
##    Chi-Square    1.107106     2    0.5749036   
##  ───────────────────────────────────────────── 
## 
## 
##  <div style="padding: 15px; background-color: #f8f9fa; border-left: 4px
##  solid #1976d2; margin: 10px 0;">
##  <h4 style="color: #1976d2; margin-top: 0;">Chi-Square Post-Hoc
##  Analysis Guide
## 
##  Three-Step Comprehensive Analysis:
## 
## 
## 
##  1.
##  Overall Chi-Square Test:
##  Tests if there's any association between variables
##  <br/>
##  2.
##  Residuals Analysis:
##  Identifies which specific cells contribute to significance
##  <br/>
##  3.
##  Pairwise Comparisons:
##  Formal hypothesis testing between group pairs
## 
## 
## 
##  *Recommended approach: Start with residuals analysis for pattern
##  identification, then use pairwise tests for formal hypothesis
##  testing.*
## 
## 
##  Values shown as: Observed
##  (Expected)
## 
##  <table border='1' style='border-collapse:
##  collapse;'>highlowmoderateAbsent61
##  (61.9)54
##  (50.6)50
##  (52.6)Present32
##  (31.1)22
##  (25.4)29
##  (26.4)
## 
##  <div style='margin: 15px 0;'><div style="padding: 15px;
##  background-color: #e8f5e8; border-left: 4px solid #4caf50; margin:
##  10px 0;">
##  <h4 style="color: #2e7d32; margin-top: 0;">Standardized Residuals
##  Interpretation
## 
##  What are standardized residuals?
## 
## 
##  Standardized residuals measure how much each cell deviates from what
##  we'd expect if there was no association. They're calculated as
##  (Observed - Expected) / √Expected, then adjusted for multiple
##  comparisons.
## 
##  <div style="background-color: #f1f8e9; padding: 10px; border-radius:
##  3px;">
##  Interpretation Guidelines:
##  <br/>
##  • |Residual| > 2.0: Suggests meaningful deviation
##  <br/>
##  • |Residual| > 3.0: Strong evidence of deviation
##  <br/>
##  • Positive values: Over-represented (more than expected)
##  <br/>
##  • Negative values: Under-represented (fewer than expected)
## 
##  <table style='border-collapse: collapse; width: 100%; margin: 15px
##  0;'><tr style='background-color: #e8f5e8;'><th style='border: 1px
##  solid #666; padding: 8px;'><th style='border: 1px solid #666; padding:
##  8px; text-align: center;'>high<th style='border: 1px solid #666;
##  padding: 8px; text-align: center;'>low<th style='border: 1px solid
##  #666; padding: 8px; text-align: center;'>moderate<td style='border:
##  1px solid #666; padding: 8px; font-weight: bold; background-color:
##  #f5f5f5;'>Absent<td style='border: 1px solid #666; padding: 8px;
##  text-align: center; background-color: #f9f9f9;'>-0.243
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  text-align: center; background-color: #f9f9f9;'>1.003
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  text-align: center; background-color: #f9f9f9;'>-0.74
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  font-weight: bold; background-color: #f5f5f5;'>Present<td
##  style='border: 1px solid #666; padding: 8px; text-align: center;
##  background-color: #f9f9f9;'>0.243
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  text-align: center; background-color: #f9f9f9;'>-1.003
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  text-align: center; background-color: #f9f9f9;'>0.74
##  As expected
## 
##  Critical value for significance: ±2.638
## 
##  <div style='padding: 15px; background-color: #d4edda; border: 1px
##  solid #c3e6cb;'>Note: Overall chi-square test is not significant (p ≥
##  0.05). Post-hoc pairwise comparisons are not recommended when the
##  overall test is non-significant.
## 
##  Pairwise Comparison Results                                                                                  
##  ──────────────────────────────────────────────────────────────────────────────────────────────────────────── 
##    Comparison    Test Method    Chi-Square    p-value      Adj. p-value    Effect Size (Phi)    Significant   
##  ──────────────────────────────────────────────────────────────────────────────────────────────────────────── 
##  ──────────────────────────────────────────────────────────────────────────────────────────────────────────── 
## 
## 
## character(0)

Interpreting Expected Values

Large deviations indicate strong associations
Values in parentheses are expected frequencies under independence
Observed >> Expected: Positive association
Observed << Expected: Negative association

Standardized Residuals Plot

The residuals plot feature is available in the jamovi interface and visualizes patterns of association:

# Plot functionality available in jamovi interface
chisqposttest(
  data = histopathology,
  rows = "Grade_Level",
  cols = "LVI",
  posthoc = "fdr",
  sig = 0.05,
  excl = TRUE,
  exp = FALSE,
  plot = FALSE  # Plot available in jamovi interface     # Generate residuals plot
)

Plot Interpretation

When using the plot option in jamovi: - Blue cells: Negative residuals (observed < expected) - Red cells: Positive residuals (observed > expected) - Intensity: Magnitude of standardized residuals - |residual| > 2: Suggests significant contribution to chi-square

Clinical Examples and Case Studies

Case Study 1: Biomarker Expression by Tumor Stage

Investigating whether a biomarker’s expression varies across tumor stages:

# Create a more realistic clinical scenario
# Using Grade and TStage as proxies for biomarker expression levels

# First, let's create meaningful categories
histopathology_clinical <- histopathology %>%
  mutate(
    TStage_Category = case_when(
      TStage <= 1 ~ "Early (T1)",
      TStage == 2 ~ "Intermediate (T2)", 
      TStage >= 3 ~ "Advanced (T3-T4)"
    ),
    Grade_Category = case_when(
      Grade == 1 ~ "Well Differentiated",
      Grade == 2 ~ "Moderately Differentiated",
      Grade == 3 ~ "Poorly Differentiated"
    )
  )

# Analyze the relationship
chisqposttest(
  data = histopathology_clinical,
  rows = "TStage_Category",
  cols = "Grade_Category", 
  posthoc = "fdr",
  sig = 0.05,
  excl = TRUE,
  exp = TRUE,
  plot = FALSE  # Plot available in jamovi interface
)

## 
##  CHI-SQUARE POST-HOC TESTS
## 
## character(0)
## 
##  Chi-Square Test Results                       
##  ───────────────────────────────────────────── 
##    Statistic     Value       df    p-value     
##  ───────────────────────────────────────────── 
##    Chi-Square    4.023637     4    0.4028164   
##  ───────────────────────────────────────────── 
## 
## 
##  <div style="padding: 15px; background-color: #f8f9fa; border-left: 4px
##  solid #1976d2; margin: 10px 0;">
##  <h4 style="color: #1976d2; margin-top: 0;">Chi-Square Post-Hoc
##  Analysis Guide
## 
##  Three-Step Comprehensive Analysis:
## 
## 
## 
##  1.
##  Overall Chi-Square Test:
##  Tests if there's any association between variables
##  <br/>
##  2.
##  Residuals Analysis:
##  Identifies which specific cells contribute to significance
##  <br/>
##  3.
##  Pairwise Comparisons:
##  Formal hypothesis testing between group pairs
## 
## 
## 
##  *Recommended approach: Start with residuals analysis for pattern
##  identification, then use pairwise tests for formal hypothesis
##  testing.*
## 
## 
##  Values shown as: Observed
##  (Expected)
## 
##  <table border='1' style='border-collapse: collapse;'>Moderately
##  DifferentiatedPoorly DifferentiatedWell DifferentiatedAdvanced
##  (T3-T4)57
##  (53)65
##  (68.4)51
##  (51.6)Early (T1)9
##  (7.4)9
##  (9.5)6
##  (7.2)Intermediate (T2)10
##  (15.6)24
##  (20.2)17
##  (15.2)
## 
##  <div style='margin: 15px 0;'><div style="padding: 15px;
##  background-color: #e8f5e8; border-left: 4px solid #4caf50; margin:
##  10px 0;">
##  <h4 style="color: #2e7d32; margin-top: 0;">Standardized Residuals
##  Interpretation
## 
##  What are standardized residuals?
## 
## 
##  Standardized residuals measure how much each cell deviates from what
##  we'd expect if there was no association. They're calculated as
##  (Observed - Expected) / √Expected, then adjusted for multiple
##  comparisons.
## 
##  <div style="background-color: #f1f8e9; padding: 10px; border-radius:
##  3px;">
##  Interpretation Guidelines:
##  <br/>
##  • |Residual| > 2.0: Suggests meaningful deviation
##  <br/>
##  • |Residual| > 3.0: Strong evidence of deviation
##  <br/>
##  • Positive values: Over-represented (more than expected)
##  <br/>
##  • Negative values: Under-represented (fewer than expected)
## 
##  <table style='border-collapse: collapse; width: 100%; margin: 15px
##  0;'><tr style='background-color: #e8f5e8;'><th style='border: 1px
##  solid #666; padding: 8px;'><th style='border: 1px solid #666; padding:
##  8px; text-align: center;'>Moderately Differentiated<th style='border:
##  1px solid #666; padding: 8px; text-align: center;'>Poorly
##  Differentiated<th style='border: 1px solid #666; padding: 8px;
##  text-align: center;'>Well Differentiated<td style='border: 1px solid
##  #666; padding: 8px; font-weight: bold; background-color:
##  #f5f5f5;'>Advanced (T3-T4)<td style='border: 1px solid #666; padding:
##  8px; text-align: center; background-color: #f9f9f9;'>1.195
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  text-align: center; background-color: #f9f9f9;'>-0.951
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  text-align: center; background-color: #f9f9f9;'>-0.188
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  font-weight: bold; background-color: #f5f5f5;'>Early (T1)<td
##  style='border: 1px solid #666; padding: 8px; text-align: center;
##  background-color: #f9f9f9;'>0.766
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  text-align: center; background-color: #f9f9f9;'>-0.213
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  text-align: center; background-color: #f9f9f9;'>-0.545
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  font-weight: bold; background-color: #f5f5f5;'>Intermediate (T2)<td
##  style='border: 1px solid #666; padding: 8px; text-align: center;
##  background-color: #f9f9f9;'>-1.918
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  text-align: center; background-color: #f9f9f9;'>1.236
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  text-align: center; background-color: #f9f9f9;'>0.612
##  As expected
## 
##  Critical value for significance: ±2.773
## 
##  <div style='padding: 15px; background-color: #d4edda; border: 1px
##  solid #c3e6cb;'>Note: Overall chi-square test is not significant (p ≥
##  0.05). Post-hoc pairwise comparisons are not recommended when the
##  overall test is non-significant.
## 
##  Pairwise Comparison Results                                                                                  
##  ──────────────────────────────────────────────────────────────────────────────────────────────────────────── 
##    Comparison    Test Method    Chi-Square    p-value      Adj. p-value    Effect Size (Phi)    Significant   
##  ──────────────────────────────────────────────────────────────────────────────────────────────────────────── 
##  ──────────────────────────────────────────────────────────────────────────────────────────────────────────── 
## 
## 
## character(0)

Clinical Interpretation

This analysis reveals: - Whether tumor stage and grade are associated (overall chi-square) - Which specific stage-grade combinations are over/under-represented - Clinical implications for prognosis and treatment planning

Case Study 2: Treatment Response by Patient Characteristics

Examining treatment efficacy across patient subgroups:

# Analyze treatment response by patient demographics
chisqposttest(
  data = histopathology,
  rows = "Group",        # Treatment vs Control
  cols = "Mortality5yr", # 5-year mortality outcome
  posthoc = "bonferroni",
  sig = 0.05,
  excl = TRUE,
  exp = TRUE,
  plot = FALSE  # Plot available in jamovi interface
)

## 
##  CHI-SQUARE POST-HOC TESTS
## 
## character(0)
## 
##  Chi-Square Test Results                         
##  ─────────────────────────────────────────────── 
##    Statistic     Value         df    p-value     
##  ─────────────────────────────────────────────── 
##    Chi-Square    0.06125848     1    0.8045178   
##  ─────────────────────────────────────────────── 
## 
## 
##  <div style="padding: 15px; background-color: #f8f9fa; border-left: 4px
##  solid #1976d2; margin: 10px 0;">
##  <h4 style="color: #1976d2; margin-top: 0;">Chi-Square Post-Hoc
##  Analysis Guide
## 
##  Three-Step Comprehensive Analysis:
## 
## 
## 
##  1.
##  Overall Chi-Square Test:
##  Tests if there's any association between variables
##  <br/>
##  2.
##  Residuals Analysis:
##  Identifies which specific cells contribute to significance
##  <br/>
##  3.
##  Pairwise Comparisons:
##  Formal hypothesis testing between group pairs
## 
## 
## 
##  *Recommended approach: Start with residuals analysis for pattern
##  identification, then use pairwise tests for formal hypothesis
##  testing.*
## 
## 
##  Values shown as: Observed
##  (Expected)
## 
##  <table border='1' style='border-collapse:
##  collapse;'>AliveDeadControl37
##  (37.9)82
##  (81.1)Treatment42
##  (41.1)87
##  (87.9)
## 
##  <div style='margin: 15px 0;'><div style="padding: 15px;
##  background-color: #e8f5e8; border-left: 4px solid #4caf50; margin:
##  10px 0;">
##  <h4 style="color: #2e7d32; margin-top: 0;">Standardized Residuals
##  Interpretation
## 
##  What are standardized residuals?
## 
## 
##  Standardized residuals measure how much each cell deviates from what
##  we'd expect if there was no association. They're calculated as
##  (Observed - Expected) / √Expected, then adjusted for multiple
##  comparisons.
## 
##  <div style="background-color: #f1f8e9; padding: 10px; border-radius:
##  3px;">
##  Interpretation Guidelines:
##  <br/>
##  • |Residual| > 2.0: Suggests meaningful deviation
##  <br/>
##  • |Residual| > 3.0: Strong evidence of deviation
##  <br/>
##  • Positive values: Over-represented (more than expected)
##  <br/>
##  • Negative values: Under-represented (fewer than expected)
## 
##  <table style='border-collapse: collapse; width: 100%; margin: 15px
##  0;'><tr style='background-color: #e8f5e8;'><th style='border: 1px
##  solid #666; padding: 8px;'><th style='border: 1px solid #666; padding:
##  8px; text-align: center;'>Alive<th style='border: 1px solid #666;
##  padding: 8px; text-align: center;'>Dead<td style='border: 1px solid
##  #666; padding: 8px; font-weight: bold; background-color:
##  #f5f5f5;'>Control<td style='border: 1px solid #666; padding: 8px;
##  text-align: center; background-color: #f9f9f9;'>-0.248
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  text-align: center; background-color: #f9f9f9;'>0.248
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  font-weight: bold; background-color: #f5f5f5;'>Treatment<td
##  style='border: 1px solid #666; padding: 8px; text-align: center;
##  background-color: #f9f9f9;'>0.248
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  text-align: center; background-color: #f9f9f9;'>-0.248
##  As expected
## 
##  Critical value for significance: ±2.498
## 
##  <div style='padding: 15px; background-color: #d4edda; border: 1px
##  solid #c3e6cb;'>Note: Overall chi-square test is not significant (p ≥
##  0.05). Post-hoc pairwise comparisons are not recommended when the
##  overall test is non-significant.
## 
##  Pairwise Comparison Results                                                                                  
##  ──────────────────────────────────────────────────────────────────────────────────────────────────────────── 
##    Comparison    Test Method    Chi-Square    p-value      Adj. p-value    Effect Size (Phi)    Significant   
##  ──────────────────────────────────────────────────────────────────────────────────────────────────────────── 
##  ──────────────────────────────────────────────────────────────────────────────────────────────────────────── 
## 
## 
## character(0)

Clinical Implications

Results help determine: - Overall treatment effectiveness - Whether treatment benefits are consistent across patient groups - Need for personalized treatment strategies

Case Study 3: Pathological Features Association

Investigating relationships between multiple pathological features:

# Multiple pathological features analysis
chisqposttest(
  data = histopathology,
  rows = "LVI",          # Lymphovascular invasion
  cols = "PNI",          # Perineural invasion  
  posthoc = "holm",
  sig = 0.05,
  excl = TRUE,
  exp = TRUE,
  plot = FALSE  # Plot available in jamovi interface
)

## 
##  CHI-SQUARE POST-HOC TESTS
## 
## character(0)
## 
##  Chi-Square Test Results                        
##  ────────────────────────────────────────────── 
##    Statistic     Value        df    p-value     
##  ────────────────────────────────────────────── 
##    Chi-Square    0.1719012     1    0.6784276   
##  ────────────────────────────────────────────── 
## 
## 
##  <div style="padding: 15px; background-color: #f8f9fa; border-left: 4px
##  solid #1976d2; margin: 10px 0;">
##  <h4 style="color: #1976d2; margin-top: 0;">Chi-Square Post-Hoc
##  Analysis Guide
## 
##  Three-Step Comprehensive Analysis:
## 
## 
## 
##  1.
##  Overall Chi-Square Test:
##  Tests if there's any association between variables
##  <br/>
##  2.
##  Residuals Analysis:
##  Identifies which specific cells contribute to significance
##  <br/>
##  3.
##  Pairwise Comparisons:
##  Formal hypothesis testing between group pairs
## 
## 
## 
##  *Recommended approach: Start with residuals analysis for pattern
##  identification, then use pairwise tests for formal hypothesis
##  testing.*
## 
## 
##  Values shown as: Observed
##  (Expected)
## 
##  <table border='1' style='border-collapse:
##  collapse;'>AbsentPresentAbsent96
##  (94.5)46
##  (47.5)Present69
##  (70.5)37
##  (35.5)
## 
##  <div style='margin: 15px 0;'><div style="padding: 15px;
##  background-color: #e8f5e8; border-left: 4px solid #4caf50; margin:
##  10px 0;">
##  <h4 style="color: #2e7d32; margin-top: 0;">Standardized Residuals
##  Interpretation
## 
##  What are standardized residuals?
## 
## 
##  Standardized residuals measure how much each cell deviates from what
##  we'd expect if there was no association. They're calculated as
##  (Observed - Expected) / √Expected, then adjusted for multiple
##  comparisons.
## 
##  <div style="background-color: #f1f8e9; padding: 10px; border-radius:
##  3px;">
##  Interpretation Guidelines:
##  <br/>
##  • |Residual| > 2.0: Suggests meaningful deviation
##  <br/>
##  • |Residual| > 3.0: Strong evidence of deviation
##  <br/>
##  • Positive values: Over-represented (more than expected)
##  <br/>
##  • Negative values: Under-represented (fewer than expected)
## 
##  <table style='border-collapse: collapse; width: 100%; margin: 15px
##  0;'><tr style='background-color: #e8f5e8;'><th style='border: 1px
##  solid #666; padding: 8px;'><th style='border: 1px solid #666; padding:
##  8px; text-align: center;'>Absent<th style='border: 1px solid #666;
##  padding: 8px; text-align: center;'>Present<td style='border: 1px solid
##  #666; padding: 8px; font-weight: bold; background-color:
##  #f5f5f5;'>Absent<td style='border: 1px solid #666; padding: 8px;
##  text-align: center; background-color: #f9f9f9;'>0.415
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  text-align: center; background-color: #f9f9f9;'>-0.415
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  font-weight: bold; background-color: #f5f5f5;'>Present<td
##  style='border: 1px solid #666; padding: 8px; text-align: center;
##  background-color: #f9f9f9;'>-0.415
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  text-align: center; background-color: #f9f9f9;'>0.415
##  As expected
## 
##  Critical value for significance: ±2.498
## 
##  <div style='padding: 15px; background-color: #d4edda; border: 1px
##  solid #c3e6cb;'>Note: Overall chi-square test is not significant (p ≥
##  0.05). Post-hoc pairwise comparisons are not recommended when the
##  overall test is non-significant.
## 
##  Pairwise Comparison Results                                                                                  
##  ──────────────────────────────────────────────────────────────────────────────────────────────────────────── 
##    Comparison    Test Method    Chi-Square    p-value      Adj. p-value    Effect Size (Phi)    Significant   
##  ──────────────────────────────────────────────────────────────────────────────────────────────────────────── 
##  ──────────────────────────────────────────────────────────────────────────────────────────────────────────── 
## 
## 
## character(0)

Handling Missing Data

Strategies for Missing Data

# Create dataset with missing values for demonstration
demo_data <- histopathology[1:100, ]
demo_data$Sex[sample(1:100, 10)] <- NA
demo_data$Group[sample(1:100, 8)] <- NA

# Analysis excluding missing data
cat("Analysis excluding missing data:\n")

## Analysis excluding missing data:

chisqposttest(
  data = demo_data,
  rows = "Sex",
  cols = "Group",
  posthoc = "fdr",
  sig = 0.05,
  excl = TRUE,    # Exclude missing values
  exp = FALSE,
  plot = FALSE
)

## 
##  CHI-SQUARE POST-HOC TESTS
## 
## character(0)
## 
##  Chi-Square Test Results                        
##  ────────────────────────────────────────────── 
##    Statistic     Value        df    p-value     
##  ────────────────────────────────────────────── 
##    Chi-Square    0.3454337     1    0.5567094   
##  ────────────────────────────────────────────── 
## 
## 
##  <div style="padding: 15px; background-color: #f8f9fa; border-left: 4px
##  solid #1976d2; margin: 10px 0;">
##  <h4 style="color: #1976d2; margin-top: 0;">Chi-Square Post-Hoc
##  Analysis Guide
## 
##  Three-Step Comprehensive Analysis:
## 
## 
## 
##  1.
##  Overall Chi-Square Test:
##  Tests if there's any association between variables
##  <br/>
##  2.
##  Residuals Analysis:
##  Identifies which specific cells contribute to significance
##  <br/>
##  3.
##  Pairwise Comparisons:
##  Formal hypothesis testing between group pairs
## 
## 
## 
##  *Recommended approach: Start with residuals analysis for pattern
##  identification, then use pairwise tests for formal hypothesis
##  testing.*
## 
## 
##  <table border='1' style='border-collapse:
##  collapse;'>ControlTreatmentFemale1826Male1820
## 
##  <div style='margin: 15px 0;'><div style="padding: 15px;
##  background-color: #e8f5e8; border-left: 4px solid #4caf50; margin:
##  10px 0;">
##  <h4 style="color: #2e7d32; margin-top: 0;">Standardized Residuals
##  Interpretation
## 
##  What are standardized residuals?
## 
## 
##  Standardized residuals measure how much each cell deviates from what
##  we'd expect if there was no association. They're calculated as
##  (Observed - Expected) / √Expected, then adjusted for multiple
##  comparisons.
## 
##  <div style="background-color: #f1f8e9; padding: 10px; border-radius:
##  3px;">
##  Interpretation Guidelines:
##  <br/>
##  • |Residual| > 2.0: Suggests meaningful deviation
##  <br/>
##  • |Residual| > 3.0: Strong evidence of deviation
##  <br/>
##  • Positive values: Over-represented (more than expected)
##  <br/>
##  • Negative values: Under-represented (fewer than expected)
## 
##  <table style='border-collapse: collapse; width: 100%; margin: 15px
##  0;'><tr style='background-color: #e8f5e8;'><th style='border: 1px
##  solid #666; padding: 8px;'><th style='border: 1px solid #666; padding:
##  8px; text-align: center;'>Control<th style='border: 1px solid #666;
##  padding: 8px; text-align: center;'>Treatment<td style='border: 1px
##  solid #666; padding: 8px; font-weight: bold; background-color:
##  #f5f5f5;'>Female<td style='border: 1px solid #666; padding: 8px;
##  text-align: center; background-color: #f9f9f9;'>-0.588
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  text-align: center; background-color: #f9f9f9;'>0.588
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  font-weight: bold; background-color: #f5f5f5;'>Male<td style='border:
##  1px solid #666; padding: 8px; text-align: center; background-color:
##  #f9f9f9;'>0.588
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  text-align: center; background-color: #f9f9f9;'>-0.588
##  As expected
## 
##  Critical value for significance: ±2.498
## 
##  <div style='padding: 15px; background-color: #d4edda; border: 1px
##  solid #c3e6cb;'>Note: Overall chi-square test is not significant (p ≥
##  0.05). Post-hoc pairwise comparisons are not recommended when the
##  overall test is non-significant.
## 
##  Pairwise Comparison Results                                                                                  
##  ──────────────────────────────────────────────────────────────────────────────────────────────────────────── 
##    Comparison    Test Method    Chi-Square    p-value      Adj. p-value    Effect Size (Phi)    Significant   
##  ──────────────────────────────────────────────────────────────────────────────────────────────────────────── 
##  ──────────────────────────────────────────────────────────────────────────────────────────────────────────── 
## 
## 
## character(0)

# Analysis including missing data as category
cat("\nAnalysis including missing data:\n")

## 
## Analysis including missing data:

chisqposttest(
  data = demo_data,
  rows = "Sex", 
  cols = "Group",
  posthoc = "fdr",
  sig = 0.05,
  excl = FALSE,   # Include missing values
  exp = FALSE,
  plot = FALSE
)

## Warning in stats::chisq.test(contTable, correct = FALSE): Chi-squared
## approximation may be incorrect

## 
##  CHI-SQUARE POST-HOC TESTS
## 
## character(0)
## 
##  Chi-Square Test Results                       
##  ───────────────────────────────────────────── 
##    Statistic     Value       df    p-value     
##  ───────────────────────────────────────────── 
##    Chi-Square    2.051978     4    0.7261991   
##  ───────────────────────────────────────────── 
## 
## 
##  <div style="padding: 15px; background-color: #f8f9fa; border-left: 4px
##  solid #1976d2; margin: 10px 0;">
##  <h4 style="color: #1976d2; margin-top: 0;">Chi-Square Post-Hoc
##  Analysis Guide
## 
##  Three-Step Comprehensive Analysis:
## 
## 
## 
##  1.
##  Overall Chi-Square Test:
##  Tests if there's any association between variables
##  <br/>
##  2.
##  Residuals Analysis:
##  Identifies which specific cells contribute to significance
##  <br/>
##  3.
##  Pairwise Comparisons:
##  Formal hypothesis testing between group pairs
## 
## 
## 
##  *Recommended approach: Start with residuals analysis for pattern
##  identification, then use pairwise tests for formal hypothesis
##  testing.*
## 
## 
##  <table border='1' style='border-collapse:
##  collapse;'>ControlTreatmentNAFemale18264Male18203NA542
## 
##  <div style='margin: 15px 0;'><div style="padding: 15px;
##  background-color: #e8f5e8; border-left: 4px solid #4caf50; margin:
##  10px 0;">
##  <h4 style="color: #2e7d32; margin-top: 0;">Standardized Residuals
##  Interpretation
## 
##  What are standardized residuals?
## 
## 
##  Standardized residuals measure how much each cell deviates from what
##  we'd expect if there was no association. They're calculated as
##  (Observed - Expected) / √Expected, then adjusted for multiple
##  comparisons.
## 
##  <div style="background-color: #f1f8e9; padding: 10px; border-radius:
##  3px;">
##  Interpretation Guidelines:
##  <br/>
##  • |Residual| > 2.0: Suggests meaningful deviation
##  <br/>
##  • |Residual| > 3.0: Strong evidence of deviation
##  <br/>
##  • Positive values: Over-represented (more than expected)
##  <br/>
##  • Negative values: Under-represented (fewer than expected)
## 
##  <table style='border-collapse: collapse; width: 100%; margin: 15px
##  0;'><tr style='background-color: #e8f5e8;'><th style='border: 1px
##  solid #666; padding: 8px;'><th style='border: 1px solid #666; padding:
##  8px; text-align: center;'>Control<th style='border: 1px solid #666;
##  padding: 8px; text-align: center;'>Treatment<th style='border: 1px
##  solid #666; padding: 8px; text-align: center;'>NA<td style='border:
##  1px solid #666; padding: 8px; font-weight: bold; background-color:
##  #f5f5f5;'>Female<td style='border: 1px solid #666; padding: 8px;
##  text-align: center; background-color: #f9f9f9;'>-0.684
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  text-align: center; background-color: #f9f9f9;'>0.801
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  text-align: center; background-color: #f9f9f9;'>-0.224
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  font-weight: bold; background-color: #f5f5f5;'>Male<td style='border:
##  1px solid #666; padding: 8px; text-align: center; background-color:
##  #f9f9f9;'>0.492
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  text-align: center; background-color: #f9f9f9;'>-0.203
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  text-align: center; background-color: #f9f9f9;'>-0.49
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  font-weight: bold; background-color: #f5f5f5;'>NA<td style='border:
##  1px solid #666; padding: 8px; text-align: center; background-color:
##  #f9f9f9;'>0.318
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  text-align: center; background-color: #f9f9f9;'>-0.959
##  As expected<td style='border: 1px solid #666; padding: 8px;
##  text-align: center; background-color: #f9f9f9;'>1.128
##  As expected
## 
##  Critical value for significance: ±2.773
## 
##  <div style='padding: 15px; background-color: #d4edda; border: 1px
##  solid #c3e6cb;'>Note: Overall chi-square test is not significant (p ≥
##  0.05). Post-hoc pairwise comparisons are not recommended when the
##  overall test is non-significant.
## 
##  Pairwise Comparison Results                                                                                  
##  ──────────────────────────────────────────────────────────────────────────────────────────────────────────── 
##    Comparison    Test Method    Chi-Square    p-value      Adj. p-value    Effect Size (Phi)    Significant   
##  ──────────────────────────────────────────────────────────────────────────────────────────────────────────── 
##  ──────────────────────────────────────────────────────────────────────────────────────────────────────────── 
## 
## 
## character(0)

Recommendations for Missing Data

Exclude missing data when missingness is random
Include as category when missingness might be informative
Consider imputation for systematic missingness patterns
Report missing data patterns in your analysis

Best Practices and Guidelines

Statistical Considerations

Sample Size Requirements

Expected frequencies: ≥5 in at least 80% of cells
Minimum cell frequency: ≥1 in all cells
Large samples: More stable results and better power

Effect Size Interpretation

Cramér’s V: Measure of association strength
- 0.1 = small effect
- 0.3 = medium effect
- 0.5 = large effect

Clinical Reporting Guidelines

Essential Elements to Report

Overall chi-square results: χ², df, p-value
Sample sizes: For each group and overall
Adjustment method: Justification for choice
Significant comparisons: With adjusted p-values
Effect sizes: Clinical meaningfulness
Missing data handling: Exclusions and rationale

Example Results Section

cat("Sample Results Reporting:\n\n")

## Sample Results Reporting:

cat("The overall chi-square test revealed a significant association between\n")

## The overall chi-square test revealed a significant association between

cat("tumor grade and lymphovascular invasion (χ² = X.XX, df = X, p < 0.05).\n")

## tumor grade and lymphovascular invasion (χ² = X.XX, df = X, p < 0.05).

cat("Post-hoc pairwise comparisons using FDR adjustment showed that\n")

## Post-hoc pairwise comparisons using FDR adjustment showed that

cat("poorly differentiated tumors had significantly higher rates of\n")

## poorly differentiated tumors had significantly higher rates of

cat("lymphovascular invasion compared to well differentiated tumors\n")

## lymphovascular invasion compared to well differentiated tumors

cat("(adjusted p = 0.XXX). No other pairwise comparisons reached statistical\n")

## (adjusted p = 0.XXX). No other pairwise comparisons reached statistical

cat("significance after multiple comparison adjustment.\n")

## significance after multiple comparison adjustment.

Advanced Applications

Integration with Other Analyses

Combining with Survival Analysis

# Chi-square post-hoc can inform survival analysis stratification
cat("Use chi-square post-hoc results to:\n")

## Use chi-square post-hoc results to:

cat("1. Identify meaningful subgroups for survival analysis\n")

## 1. Identify meaningful subgroups for survival analysis

cat("2. Determine appropriate stratification variables\n")

## 2. Determine appropriate stratification variables

cat("3. Guide covariate selection in Cox models\n")

## 3. Guide covariate selection in Cox models

Biomarker Research Applications

cat("Common biomarker research applications:\n")

## Common biomarker research applications:

cat("• Expression levels by tumor characteristics\n")

## • Expression levels by tumor characteristics

cat("• Mutation status by histological features\n")

## • Mutation status by histological features

cat("• Treatment response by biomarker profiles\n")

## • Treatment response by biomarker profiles

cat("• Progression patterns by molecular subtypes\n")

## • Progression patterns by molecular subtypes

Quality Control and Validation

Assumption Checking

cat("Key assumptions to verify:\n")

## Key assumptions to verify:

cat("1. Independence of observations\n")

## 1. Independence of observations

cat("2. Adequate expected frequencies\n")

## 2. Adequate expected frequencies

cat("3. Appropriate variable coding\n")

## 3. Appropriate variable coding

cat("4. Random sampling from population\n")

## 4. Random sampling from population

Sensitivity Analyses

cat("Consider sensitivity analyses:\n")

## Consider sensitivity analyses:

cat("• Different significance thresholds\n")

## • Different significance thresholds

cat("• Alternative adjustment methods\n")

## • Alternative adjustment methods

cat("• Missing data handling approaches\n")

## • Missing data handling approaches

cat("• Subgroup analyses by key variables\n")

## • Subgroup analyses by key variables

Conclusion

Chi-square post-hoc testing is a powerful tool for identifying specific patterns of association in categorical data. Key takeaways for clinicopathological research:

Use post-hoc tests only when overall chi-square is significant
Choose adjustment methods based on your research priorities
Report effect sizes alongside statistical significance
Consider clinical context when interpreting results
Validate findings in independent datasets when possible

The ClinicoPath chi-square post-hoc module provides comprehensive functionality for these analyses, supporting rigorous statistical practices in clinical and pathological research.

References

Agresti, A. (2018). An Introduction to Categorical Data Analysis. 3rd Edition.
Benjamini, Y., & Hochberg, Y. (1995). Controlling the false discovery rate. Journal of the Royal Statistical Society, 57(1), 289-300.
Holm, S. (1979). A simple sequentially rejective multiple test procedure. Scandinavian Journal of Statistics, 6(2), 65-70.
Cramér, H. (1946). Mathematical Methods of Statistics. Princeton University Press.

This vignette was created for the ClinicoPath jamovi module. For more information and updates, visit https://github.com/sbalci/ClinicoPathJamoviModule.

Comprehensive Guide to Pairwise Comparisons in Contingency Tables

ClinicoPath Module

2025-07-13