Standard Categorical Tests
Executive Summary

Executive Summary

Are Contract and Churn related?

Observations
7043
Table Size
3 × 2
Chi-Square
1184.6
P Value
5.86e-258
Cramér's V
0.41
Association
associated (moderate)
Across 7,043 observations in a 3 × 2 table: Contract and Churn ARE related: the chi-square test rejects independence (p = 5.86e-258), and Cramér's V = 0.41 grades the association as moderate. The combination driving it most is Month-to-month × Yes, which appears 1655 times vs 1028.3 expected under independence; Month-to-month × No is correspondingly under-represented (2220 vs 2846.7 expected).
What this means— written from the computed numbers · yours to edit

Contract and Churn are associated: the chi-square test rejects independence with p = 5.86e-258, and Cramér's V = 0.41 indicates a moderate relationship. Month-to-month contracts drive the association—1,655 customers with this contract type churned (vs. 1,028.3 expected under independence), while 2,220 did not churn (vs. 2,846.7 expected), showing Month-to-month customers churn at much higher rates than longer-term contracts.

Overview

Analysis Overview

Association test between Contract and Churn across 7,043 observations.

N Observations7043
N Levels A3
N Levels B2
N Cells6
What this means— written from the computed numbers · yours to edit

Contract type and churn are strongly associated across 7,043 observations. The chi-square test detects a relationship far stronger than chance would produce—a pattern real enough to act on directionally. Cramér's V quantifies this strength on a 0–1 scale; the exact value appears in The detail below. Importantly, association means the two variables move together; it does not mean contract type causes churn. A third factor—such as customer tenure, service satisfaction, or pricing sensitivity—could drive both.

Data Preparation

Data Quality

Missing-value recoding, category lumping, and sparse-count handling.

Initial Rows7043
Final Rows7043
Rows Removed0
Pct Cells Low Expected0
What this means— written from the computed numbers · yours to edit

All 7,043 rows were retained; no rows were dropped and no blanks required recoding to Missing. Both columns were complete with 8 or fewer categories, eliminating the need for rare-category lumping. Expected cell counts ranged from 390.9 to 2,846.7, well above the minimum threshold of 5, so the standard chi-square test was appropriate and no sparse-count correction was needed. The dataset required no further cleaning.

Visualization

Contingency Table

Counts of every Contract × Churn combination.

What this means— written from the computed numbers · yours to edit

The largest cell is Month-to-month × No with 2,220 observations (31.5% of 7,043 rows). The data concentrates in non-churn rows across all contract types, but the distribution shifts sharply by contract: Month-to-month has 1,655 churners, while Two year has only 48 and One year has 166. This concentration pattern—high churn in Month-to-month, low in Two year—is the visual signature of the association. The color gradient would show Month-to-month rows as warm (high churn) and Two year rows as cool (low churn), revealing the Contract–Churn relationship at a glance.

Visualization

Composition by Group

Share of each Churn category within every Contract level (rows sum to 100%).

What this means— written from the computed numbers · yours to edit

Within each contract type, the share of churners varies dramatically: Month-to-month shows 42.7% churn, One year shows 11.3%, and Two year shows only 2.8%. Conversely, the share of non-churners rises from 57.3% (Month-to-month) to 97.2% (Two year). This is the pattern the association test detects—the Churn profile is not constant across Contract levels. The steepest swing is in the Two year group, where non-churn dominates at 97.2%, contrasting sharply with Month-to-month's 57.3% non-churn rate.

Data Table

Statistical Tests

Chi-square, Fisher's exact (when run), and Cramér's V.

TestStatisticDfP ValueInterpretation
Chi-square test118525.86e-258Significant association at p < 0.05.
Cramér's V (effect size)0.41——Association strength is moderate (0 = none, 1 = perfect; <0.1 negligible, 0.1–0.3 weak, 0.3–0.5 moderate, >0.5 strong).
What this means— written from the computed numbers · yours to edit

The chi-square statistic is 1,184.6 on 2 degrees of freedom, yielding p = 5.86e-258, far below the 0.05 threshold and indicating a highly significant association. Cramér's V = 0.41 grades the strength as moderate (scale: <0.1 negligible, 0.1–0.3 weak, 0.3–0.5 moderate, >0.5 strong). The enormous p-value reflects both a real relationship and a large sample; the moderate V confirms the relationship is substantial enough to matter practically, not merely a statistical artifact of sample size.

Data Table

Largest Deviations from Independence

The cells that depart most from the counts independence would predict.

CellObservedExpectedStd Residual
Month-to-month × Yes1655102834
Month-to-month × No22202847-34
Two year × No1647124525.37
Two year × Yes48449.8-25.37
One year × No1307108214.92
One year × Yes166390.9-14.92
What this means— written from the computed numbers · yours to edit

Month-to-month × Yes is the most over-represented cell: 1,655 observed versus 1,028.3 expected, with a standardized residual of 34—the strongest deviation in the table. Its complement, Month-to-month × No, is the most under-represented: 2,220 observed versus 2,846.7 expected (residual −34). Two year × Yes is the second-largest departure (48 observed vs. 449.8 expected; residual −25.37), while Two year × No is correspondingly over-represented (1,647 vs. 1,245.2; residual 25.37). One year shows a smaller but consistent pattern: 166 churners observed versus 390.9 expected (residual −14.92). These deviations explain the entire association: Month-to-month and Two year are the primary drivers.

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The code that did it

Categorical Association — Chi-Square & Fisher

Tests whether two categorical columns are related: contingency table, chi-square test of independence (Fisher's exact test when counts are sparse), Cramér's V effect size, and the cells that deviate most from independence.

Why This Method?

The chi-square test is the standard answer to "are these two categories related?" — one p-value for the whole table. Fisher's exact test covers the sparse-count cases where chi-square is unreliable, and Cramér's V turns the verdict into a 0-1 strength score so significance is never confused with importance.

What This Analysis Covers

  • Contingency table heatmap (all combinations, including zeros)
  • Row-share bars: composition of one column within each level of the other
  • Chi-square + Fisher exact + Cramér's V, tabulated with interpretations
  • The most over- and under-represented combinations (standardized residuals)

Standard Library

Platform standard-library module (LAT-1441): runs on ANY dataset via the semantic mapping {var_a, var_b}. All narrative is derived from the user's own column names and computed values.

suppressPackageStartupMessages(library(DT))
suppressPackageStartupMessages(library(htmlwidgets))
suppressPackageStartupMessages(library(arrow))
suppressPackageStartupMessages(library(knitr))
suppressPackageStartupMessages(library(rmarkdown))
suppressPackageStartupMessages(library(dplyr))
suppressPackageStartupMessages(library(tidyr))
suppressPackageStartupMessages(library(ggplot2))
suppressPackageStartupMessages(library(stringr))
suppressPackageStartupMessages(library(lubridate))
suppressPackageStartupMessages(library(broom))
suppressPackageStartupMessages(library(Matrix))
suppressPackageStartupMessages(library(cluster))
suppressPackageStartupMessages(library(data.table))

Core Analysis Pipeline

compute_shared <- function(df, params, col_map = list()) {
  # === SHARED EXPORTS ===
  #   initial_rows/final_rows/rows_removed  $ row accounting
  #   name_a / name_b     $ humanized user names of var_a / var_b
  #   levels_a / levels_b $ ordered level labels used in the table
  #   n_missing_a/_b      $ blanks/NA recoded to "Missing"
  #   n_lumped_a/_b       $ levels folded into "Other" (beyond 8)
  #   tab                 $ contingency table (var_a rows x var_b cols)
  #   contingency_df      $ long format incl. zeros: level_a, level_b, count
  #   proportions_df      $ level_a, share_pct, level_b (row shares x100)
  #   test_results_df     $ test, statistic, df, p_value, interpretation
  #   deviations_df       $ cell, observed, expected, std_residual (top 10)
  #   chi_stat/chi_df/chi_p $ chi-square results
  #   fisher_p            $ Fisher exact p (NA when not run)
  #   fisher_ran          $ logical — was Fisher run
  #   pct_low_expected    $ % of cells with expected count < 5
  #   cramers_v / strength_band $ effect size + verdict band
  #   p_effective / test_used   $ the p-value the verdict rests on
  #   significant         $ logical — p_effective < 0.05
  #   top_over / top_under $ 1-row data.frames (may be NULL) — extreme cells
  #   metrics / json_output
  # === /SHARED EXPORTS ===

Step 1: Discover the two mapped columns

initial_rows <- nrow(df)
  for (k in c("var_a", "var_b")) {
    if (!k %in% names(df)) {
      stop(sprintf("column_mapping must map &#x27;%s' to a categorical column.", k))
    }
  }
  name_a <- humanize_semantic("var_a", col_map)
  name_b <- humanize_semantic("var_b", col_map)

Step 2: Clean both columns — character, "Missing", lump beyond 8 levels

prep_categorical <- function(v, max_levels = 8) {
    x <- trimws(as.character(v))
    n_missing <- sum(is.na(x) | x == "")
    x[is.na(x) | x == ""] <- "Missing"
    freq <- sort(table(x), decreasing = TRUE)
    lumped <- character(0)
    if (length(freq) > max_levels) {
      keep <- names(freq)[seq_len(max_levels)]
      lumped <- setdiff(names(freq), keep)
      x[x %in% lumped] <- "Other"
    }

Levels ordered by frequency; "Other"/"Missing" pushed last for readability

freq2 <- sort(table(x), decreasing = TRUE)
    lv <- names(freq2)
    special <- intersect(c("Other", "Missing"), lv)
    lv <- c(setdiff(lv, special), special)
    list(x = factor(x, levels = lv), n_missing = n_missing,
         n_lumped = length(lumped))
  }
  pa <- prep_categorical(df[["var_a"]])
  pb <- prep_categorical(df[["var_b"]])

  if (nlevels(pa$x) < 2) {
    stop(sprintf("&#x27;%s' has only one distinct value (%s) — an association test needs at least two categories in each column.",
                 name_a, levels(pa$x)[1]))
  }
  if (nlevels(pb$x) < 2) {
    stop(sprintf("&#x27;%s' has only one distinct value (%s) — an association test needs at least two categories in each column.",
                 name_b, levels(pb$x)[1]))
  }

  final_rows <- length(pa$x)
  rows_removed <- initial_rows - final_rows
  if (final_rows < 20) {
    stop(sprintf("Only %d usable rows — the association test between &#x27;%s' and '%s' needs at least 20.",
                 final_rows, name_a, name_b))
  }

Step 3: Contingency table + chi-square test

tab <- table(pa$x, pb$x)
  n <- sum(tab)
  is_2x2 <- nrow(tab) == 2 && ncol(tab) == 2

2x2 keeps the default Yates continuity correction; larger tables use the uncorrected statistic. Low-expected-count warnings are suppressed here and narrated (Fisher covers that case below).

chi <- suppressWarnings(
    if (is_2x2) chisq.test(tab) else chisq.test(tab, correct = FALSE)
  )
  chi_stat <- as.numeric(chi$statistic)
  chi_df   <- as.numeric(chi$parameter)
  chi_p    <- as.numeric(chi$p.value)
  expected <- chi$expected
  pct_low_expected <- 100 * mean(expected < 5)

Range of the EXPECTED counts matrix (not the observed table) — narrated in Data Quality so any "expected cell counts ranged from X to Y" claim is computed from the right matrix.

min_expected <- round(min(expected), 1)
  max_expected <- round(max(expected), 1)

Step 4: Fisher's exact test when chi-square is unreliable (or 2x2)

fisher_ran <- FALSE
  fisher_p <- NA_real_
  fisher_or <- NA_real_
  if (is_2x2 || pct_low_expected > 20) {
    fr <- tryCatch(
      if (is_2x2) fisher.test(tab)
      else fisher.test(tab, simulate.p.value = TRUE, B = 10000),
      error = function(e) NULL
    )
    if (!is.null(fr)) {
      fisher_ran <- TRUE
      fisher_p <- as.numeric(fr$p.value)
      if (is_2x2 && !is.null(fr$estimate)) fisher_or <- as.numeric(fr$estimate)
    }
  }

The verdict rests on Fisher when it ran (it is the reliable test in exactly the situations that triggered it), else on chi-square.

if (fisher_ran && !is.na(fisher_p)) {
    p_effective <- fisher_p
    test_used <- "Fisher&#x27;s exact test"
  } else {
    p_effective <- chi_p
    test_used <- "chi-square test"
  }
  significant <- !is.na(p_effective) && p_effective < 0.05

Step 5: Cramér's V effect size

cramers_v <- sqrt(chi_stat / (n * (min(dim(tab)) - 1)))
  strength_band <- if (is.na(cramers_v)) "unknown"
    else if (cramers_v < 0.1) "negligible"
    else if (cramers_v < 0.3) "weak"
    else if (cramers_v < 0.5) "moderate"
    else "strong"

Step 6: Standardized residuals — which cells drive the association

obs_long <- as.data.frame(tab, stringsAsFactors = FALSE)
  names(obs_long) <- c("level_a", "level_b", "count")
  contingency_df <- obs_long
  contingency_df$level_a <- as.character(contingency_df$level_a)
  contingency_df$level_b <- as.character(contingency_df$level_b)

  exp_long <- as.data.frame(as.table(expected), stringsAsFactors = FALSE)
  std_long <- as.data.frame(as.table(chi$stdres), stringsAsFactors = FALSE)
  dev_all <- data.frame(
    cell = paste0(as.character(obs_long$level_a), " × ",
                  as.character(obs_long$level_b)),
    observed = obs_long$count,
    expected = round(exp_long$Freq, 1),
    std_residual = round(std_long$Freq, 2),
    stringsAsFactors = FALSE
  )
  dev_all <- dev_all[is.finite(dev_all$std_residual), , drop = FALSE]

Ties in |residual| (guaranteed in 2-column tables) list the over-represented cell first — it reads better in the narrative.

dev_all <- dev_all[order(-abs(dev_all$std_residual), -dev_all$std_residual), , drop = FALSE]
  rownames(dev_all) <- NULL
  deviations_df <- head(dev_all, 10)

Most over/under-represented cells (guard: never index into empty/all-NA)

top_over <- if (nrow(dev_all) > 0 && max(dev_all$std_residual) > 0)
    dev_all[which.max(dev_all$std_residual), , drop = FALSE] else NULL
  top_under <- if (nrow(dev_all) > 0 && min(dev_all$std_residual) < 0)
    dev_all[which.min(dev_all$std_residual), , drop = FALSE] else NULL

Step 7: Row shares — composition of var_b within each var_a level

prop <- prop.table(tab, 1) * 100
  prop_long <- as.data.frame(as.table(prop), stringsAsFactors = FALSE)
  proportions_df <- data.frame(
    level_a = as.character(prop_long$Var1),
    share_pct = round(prop_long$Freq, 1),
    level_b = as.character(prop_long$Var2),
    stringsAsFactors = FALSE
  )

Step 8: Test-results table

chi_name <- if (is_2x2) "Chi-square test(Yates-corrected)" else "Chi-square test"
  chi_interp <- paste0(
    if (!is.na(chi_p) && chi_p < 0.05) "Significant association at p < 0.05."
    else "No significant association at p < 0.05.",
    if (pct_low_expected > 20) paste0(" Caution: ", round(pct_low_expected),
                                      "% of cells have expected counts below 5 — prefer Fisher&#x27;s row.")
    else ""
  )
  test_rows <- list(data.frame(
    test = chi_name, statistic = round(chi_stat, 2), df = chi_df,
    p_value = signif(chi_p, 3), interpretation = chi_interp,
    stringsAsFactors = FALSE
  ))
  if (fisher_ran) {
    test_rows[[length(test_rows) + 1]] <- data.frame(
      test = if (is_2x2) "Fisher&#x27;s exact test" else "Fisher's exact test (simulated, B=10,000)",
      statistic = if (is_2x2 && !is.na(fisher_or)) round(fisher_or, 2) else NA_real_,
      df = NA_real_, p_value = signif(fisher_p, 3),
      interpretation = paste0(
        if (!is.na(fisher_p) && fisher_p < 0.05) "Significant association at p < 0.05."
        else "No significant association at p < 0.05.",
        if (is_2x2 && !is.na(fisher_or)) sprintf(" Statistic shown is the odds ratio(%.2f).", fisher_or)
        else " Exact test — reliable with sparse cell counts."
      ),
      stringsAsFactors = FALSE
    )
  }
  test_rows[[length(test_rows) + 1]] <- data.frame(
    test = "Cramér&#x27;s V (effect size)", statistic = round(cramers_v, 3),
    df = NA_real_, p_value = NA_real_,
    interpretation = paste0("Association strength is ", strength_band,
                            " (0 = none, 1 = perfect; <0.1 negligible, 0.1–0.3 weak, 0.3–0.5 moderate, >0.5 strong)."),
    stringsAsFactors = FALSE
  )
  test_results_df <- do.call(rbind, test_rows)
  rownames(test_results_df) <- NULL

  metrics <- list(
    `Observations`       = as.integer(n),
    `Table Size`         = paste0(nrow(tab), " × ", ncol(tab)),
    `Chi-Square`         = round(chi_stat, 2),
    `P Value`            = signif(p_effective, 3),
    `Cramér&#x27;s V`         = round(cramers_v, 3),
    `Association`        = if (significant) paste0("associated(", strength_band, ")") else "not associated"
  )

  json_output <- list(
    answer = paste0(
      name_a, " and ", name_b,
      if (significant) " are statistically associated(" else " show no statistically significant association(",
      test_used, " p = ", signif(p_effective, 3),
      "; Cramér&#x27;s V = ", round(cramers_v, 3), ", ", strength_band, ") across ",
      format(n, big.mark = ","), " rows.",
      if (significant && !is.null(top_over)) paste0(
        " The most over-represented combination is ", top_over$cell,
        " (", top_over$observed, " observed vs ", top_over$expected, " expected).")
      else ""
    ),
    cards = lapply(
      c("tldr", "overview", "preprocessing", "contingency_heatmap",
        "proportions_bar", "test_results", "largest_deviations"),
      function(cid) list(id = cid, metrics = metrics)
    )
  )

  list(
    initial_rows = initial_rows, final_rows = final_rows,
    rows_removed = rows_removed,
    name_a = name_a, name_b = name_b,
    levels_a = levels(pa$x), levels_b = levels(pb$x),
    n_missing_a = pa$n_missing, n_missing_b = pb$n_missing,
    n_lumped_a = pa$n_lumped, n_lumped_b = pb$n_lumped,
    tab = tab, contingency_df = contingency_df,
    proportions_df = proportions_df, test_results_df = test_results_df,
    deviations_df = deviations_df,
    chi_stat = chi_stat, chi_df = chi_df, chi_p = chi_p,
    fisher_p = fisher_p, fisher_ran = fisher_ran, is_2x2 = is_2x2,
    pct_low_expected = pct_low_expected,
    min_expected = min_expected,
    max_expected = max_expected,
    cramers_v = cramers_v, strength_band = strength_band,
    p_effective = p_effective, test_used = test_used,
    significant = significant,
    top_over = top_over, top_under = top_under,
    metrics = metrics, json_output = json_output
  )
}

Find the level_b whose share varies most across level_a (the visual story)

spread_by_b <- tapply(pdf$share_pct, pdf$level_b, function(s) max(s) - min(s))
  spread_by_b <- spread_by_b[is.finite(spread_by_b)]
  spread_txt <- ""
  if (length(spread_by_b) > 0) {
    b_star <- names(spread_by_b)[which.max(spread_by_b)]
    sub <- pdf[pdf$level_b == b_star, , drop = FALSE]
    hi <- sub[which.max(sub$share_pct), ]
    lo <- sub[which.min(sub$share_pct), ]
    spread_txt <- paste0(
      " The share of \"", b_star, "\" varies most: from ", lo$share_pct,
      "% of ", lo$level_a, " to ", hi$share_pct, "% of ", hi$level_a, ".",
      if (shared$significant) " That spread is what the association test is detecting."
      else " Even this largest spread is within what chance would produce."
    )
  }
  list(
    title = "Composition by Group",
    description = paste0("Share of each ", shared$name_b, " category within every ",
                         shared$name_a, " level(rows sum to 100%)."),
    text = paste0(
      "Each group of bars is one ", shared$name_a, " level, split into the ",
      "percentage of rows falling in each ", shared$name_b, " category. If the ",
      "two columns were independent, every group would show the same profile.",
      spread_txt
    ),
    chart_labels = list(level_a = shared$name_a, share_pct = "Share(%)"),
    data = list(proportions = pdf)
  )
}

# Card: test_results (table)
card_test_results <- function(shared, df, params) {
  list(
    title = "Statistical Tests",
    description = "Chi-square, Fisher&#x27;s exact (when run), and Cramér's V.",
    text = paste0(
      "The chi-square statistic is ", round(shared$chi_stat, 2), " on ",
      shared$chi_df, " degrees of freedom(p = ", signif(shared$chi_p, 3), ").",
      if (shared$fisher_ran) paste0(
        " Fisher&#x27;s exact test",
        if (shared$is_2x2) "" else " (p-value simulated with 10,000 replicates)",
        " gives p = ", signif(shared$fisher_p, 3),
        if (shared$pct_low_expected > 20) paste0(" and is the reliable verdict here, since ",
          round(shared$pct_low_expected), "% of cells have expected counts under 5.")
        else ".") else "",
      " Cramér&#x27;s V = ", round(shared$cramers_v, 3), " grades the strength as ",
      shared$strength_band, ": a tiny p-value on a large dataset can still mean ",
      "a practically ", if (shared$cramers_v < 0.3) "small" else "substantial",
      " relationship, so read significance and strength together."
    ),
    data = list(test_results = shared$test_results_df)
  )
}

# Card: largest_deviations (table)
card_largest_deviations <- function(shared, df, params) {
  over_txt <- if (!is.null(shared$top_over) && shared$top_over$std_residual > 2) {
    paste0(shared$top_over$cell, " is the most over-represented combination(",
           shared$top_over$observed, " observed vs ", shared$top_over$expected,
           " expected; standardized residual ", shared$top_over$std_residual, ")")
  } else NULL
  under_txt <- if (!is.null(shared$top_under) && shared$top_under$std_residual < -2) {
    paste0(shared$top_under$cell, " is the most under-represented(",
           shared$top_under$observed, " vs ", shared$top_under$expected,
           " expected; residual ", shared$top_under$std_residual, ")")
  } else NULL
  hot_txt <- if (!is.null(over_txt) || !is.null(under_txt)) {
    paste0(paste(c(over_txt, under_txt), collapse = ", and "), ". ")
  } else {
    "No individual cell deviates strongly from independence(all standardized residuals are within ±2). "
  }
  list(
    title = "Largest Deviations from Independence",
    description = "The cells that depart most from the counts independence would predict.",
    text = paste0(
      "Standardized residuals measure how far each cell&#x27;s observed count sits ",
      "from its expected count, in standard-deviation units — beyond ±2 is a ",
      "meaningful departure. ", hot_txt,
      "These cells are where the ", shared$name_a, "–", shared$name_b,
      " relationship actually lives, and the natural place to focus follow-up."
    ),
    data = list(largest_deviations = shared$deviations_df)
  )
}
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