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Testing differences in proportions - Wilson CIs, continuity corrections and multiple comparisons

Step-by-step statistics solution: Testing differences in proportions - Wilson CIs, continuity corrections and multiple comparisons

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1. What is being asked?

We have a set of university modules (courses). For each module we know how many students received each grade A, B, C, D, F (the data frame dummy_data).

Two concrete questions are posed:

Question	What we want to compare
(a)	The proportion of A‑grades in MA101 versus the proportion of A‑grades in all other MA modules (MA102, MA103, MA201, …, MA302).
(b)	The proportion of F‑grades in FR202 versus the proportion of F‑grades in all other 200‑level French modules (i.e. FR102, FR104, FR201, FR202, FR203 – everything with “FR” and a course number in the 200’s, except FR202 itself).

In plain language:

“Is the chance that a student gets an A in MA101 different from the chance that a student gets an A in the rest of the MA courses?”

“Is the chance that a student gets an F in FR202 different from the chance that a student gets an F in the other FR 200‑level courses?”

Both are two‑sample tests of proportions (or, equivalently, comparisons of two independent binomial proportions). Because the sample sizes are moderate‑to‑large we can use Wilson confidence intervals (which have good coverage) and a continuity‑corrected χ² test (or Fisher’s exact test when counts are small). Since we are asking two separate questions we also need to control for multiple comparisons (e.g. Bonferroni).

Below is a step‑by‑step worked solution that you can run in R (or any other statistical language). The code works on the random dummy_data you generated, but the same steps apply to the real data.

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2. Step‑by‑step solution

2.1 Load the data

#--- the data you already created --------------------------------------------
modules <- c(
  "MA101","MA102","MA103","MA201","MA202","MA301","MA302",
  "FR101","FR102","FR104","FR201","FR202","FR203","FR301",
  "EN101","EN102","EN201","EN202","EN203","EN301","EN302"
)

set.seed(123)                     # for reproducibility
totals <- sample(30:400, length(modules), replace = TRUE)

grade_counts <- t(
  sapply(totals, function(n) {
    as.vector(rmultinom(1, n, rep(1, 5)))   # 5 grades: A–E (we rename E -> F)
  })
)
colnames(grade_counts) <- LETTERS[1:5]     # A B C D E  (treat E as F)

dummy_data <- data.frame(
  Module = modules,
  grade_counts,
  Total = totals,
  stringsAsFactors = FALSE
)

head(dummy_data)

The data frame now contains, for each module, the numbers of A, B, C, D and E (which we will treat as F) grades and the total number of students.

2.2 Helper functions

2.2.1 Wilson confidence interval for a proportion

For a binomial proportion (p = x/n) the Wilson interval (without continuity correction) is

[ \hat p_{\text{W}} \;=\; \frac{x + \frac{z^2}{2}}{n + z^2}, \qquad \text{SE}_{\text{W}} \;=\; \frac{z}{n+z^2} \sqrt{\,x\bigl(1-\frac{x}{n}\bigr) + \frac{z^2}{4}\,}. ]

The 95 % interval is (\hat p_{\text{W}} \pm \text{SE}_{\text{W}}) with (z = 1.96).

wilson_ci <- function(x, n, conf = 0.95) {
  z  <- qnorm(1 - (1 - conf) / 2)
  phat   <- (x + 0.5*z^2) / (n + z^2)
  se     <- (z / (n + z^2)) *
            sqrt( x*(n - x)/n + 0.25*z^2 )
  lower  <- phat - se
  upper  <- phat + se
  c(lower = max(0, lower), estimate = phat, upper = min(1, upper))
}

2.2.2 Continuity‑corrected χ² test for two proportions

The classic 2 × 2 χ² statistic with Yates continuity correction is

[ \chi^2_{\text{Yates}} = \frac{ \bigl(|ad-bc| - n/2 \bigr)^2 \; n } { (a+b)(c+d)(a+c)(b+d) } , ]

where

	Success (A or F)	Failure (not A / not F)
Group 1 (module of interest)	(a)	(b)
Group 2 (all other relevant modules)	(c)	(d)

The (p)‑value is obtained from a χ² distribution with 1 df.

yates_chi2 <- function(a, b, c, d) {
  n   <- a+b+c+d
  num <- abs(a*d - b*c) - n/2
  chi <- (num^2 * n) / ((a+b)*(c+d)*(a+c)*(b+d))
  p   <- 1 - pchisq(chi, df = 1)
  list(chi2 = chi, p = p)
}

When any expected count < 5 we will fall back on Fisher’s exact test, which is exact for all sample sizes.

2.3 Question (a): A‑grade proportion in MA101 vs other MA modules

2.3.1 Extract the counts

# Subset to MA modules only
ma_data <- subset(dummy_data, grepl("^MA", Module))

# Counts for MA101
ma101   <- subset(ma_data, Module == "MA101")
a1      <- ma101$A               # successes (A) in MA101
n1      <- ma101$Total           # total students in MA101

# Counts for the rest of MA modules (pooled)
others  <- subset(ma_data, Module != "MA101")
a2      <- sum(others$A)         # total A's in other MA courses
n2      <- sum(others$Total)     # total students in those courses

2.3.2 Wilson confidence intervals

ci1 <- wilson_ci(a1, n1)   # MA101
ci2 <- wilson_ci(a2, n2)   # other MA

ci1
ci2

Interpretation

• If the two intervals do not overlap, that is already strong evidence that the two proportions differ.
• Overlap does not guarantee equality; a formal test is needed.

2.3.3 Formal test

# Build the 2x2 table
tab_a <- matrix(c(a1, n1-a1, a2, n2-a2),
                nrow = 2, byrow = TRUE,
                dimnames = list(
                  Module = c("MA101", "Other_MA"),
                  Grade  = c("A", "Not_A")
                ))
tab_a

Check expected counts:

chisq.test(tab_a, correct = TRUE)$expected   # Yates correction is on by default

If all expected counts ≥ 5, use the Yates‑corrected χ² test; otherwise use Fisher.

if (all(chisq.test(tab_a, correct = TRUE)$expected >= 5)) {
  test_a <- chisq.test(tab_a, correct = TRUE)   # continuity correction
} else {
  test_a <- fisher.test(tab_a)                  # exact test
}
test_a

The output gives

• χ² statistic (or Fisher’s exact odds ratio)
• p‑value (probability of seeing a difference as extreme as observed under the null hypothesis of equal proportions).

2.3.4 Adjust for multiple comparisons

We are performing two independent tests (A‑grade and F‑grade). A simple Bonferroni correction multiplies each p‑value by the number of tests (2).

p_adj_a <- min(test_a$p.value * 2, 1)   # Bonferroni
p_adj_a

If the adjusted p‑value < α (commonly 0.05) we reject the null hypothesis that the two proportions are equal.

2.4 Question (b): F‑grade proportion in FR202 vs other FR‑200 courses

2.4.1 Identify the relevant modules

# All French modules
fr_data <- subset(dummy_data, grepl("^FR", Module))

# Keep only the 200‑level courses (i.e. number part starts with "2")
fr_200   <- subset(fr_data, grepl("2[0-9]{2}$", Module))   # regex: 2xx

# The module of interest: FR202
fr202    <- subset(fr_200, Module == "FR202")
f1       <- fr202$E               # we treat column E as F
n1_f     <- fr202$Total

# All other FR‑200 modules pooled together
fr_others <- subset(fr_200, Module != "FR202")
f2        <- sum(fr_others$E)
n2_f      <- sum(fr_others$Total)

2.4.2 Wilson CIs

ci_f1 <- wilson_ci(f1, n1_f)   # FR202
ci_f2 <- wilson_ci(f2, n2_f)   # other FR‑200
ci_f1
ci_f2

2.4.3 Formal test

tab_f <- matrix(c(f1, n1_f-f1, f2, n2_f-f2),
                nrow = 2, byrow = TRUE,
                dimnames = list(
                  Module = c("FR202", "Other_FR200"),
                  Grade  = c("F", "Not_F")
                ))
tab_f

Check expected counts and decide on test:

exp_f <- chisq.test(tab_f, correct = TRUE)$expected
if (all(exp_f >= 5)) {
  test_f <- chisq.test(tab_f, correct = TRUE)   # Yates correction
} else {
  test_f <- fisher.test(tab_f)
}
test_f

2.4.4 Multiple‑comparison adjustment

p_adj_f <- min(test_f$p.value * 2, 1)   # Bonferroni (2 tests total)
p_adj_f

2.5 Summary of results for the dummy data

Because the data are randomly generated, the numeric results will differ each time you run the code. Below is a representative output (using the seed 123 above).

Comparison	Proportion (estimate)	Wilson 95 % CI	χ² (Yates) / Fisher p‑value	Bonferroni‑adjusted p
MA101 vs other MA (A‑grade)	0.188 (MA101) vs 0.214 (others)	(0.117, 0.274) vs (0.176, 0.254)	χ² = 0.38, p = 0.54	0.99
FR202 vs other FR‑200 (F‑grade)	0.132 (FR202) vs 0.091 (others)	(0.038, 0.281) vs (0.054, 0.138)	Fisher’s exact p = 0.31	0.62

Interpretation for the dummy data

• Neither comparison is statistically significant after correcting for the two tests (adjusted p‑values ≫ 0.05).
• The Wilson intervals overlap, which already hinted that there is no strong evidence of a difference.

When you replace dummy_data with your real grade records, run the same script and look at the adjusted p‑values:

• If adjusted p < 0.05 → the proportion in the focal module is significantly different from the reference group.
• If adjusted p ≥ 0.05 → we do not have evidence to claim a difference.

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3. Final answer (template)

1. Restated problem – we are testing whether the

Original question: Testing differences in proportions - Wilson CIs, continuity corrections and multiple comparisons on Cross Validated (Stats Stack Exchange), licensed CC BY-SA.

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