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Test Whether a Proportion Matches an Expected Value

Source: R/binomial_test.R
binomial_test.Rd

binomial_test() tests whether the observed proportion of a binary variable differs from a hypothesized proportion. It uses the exact binomial test, making it valid for any sample size.

Think of it as:

  • Testing whether a coin is fair (proportion of heads = 50 percent)

  • Checking if your sample's gender ratio matches the population

  • Verifying if satisfaction rates meet a target proportion

The test tells you:

  • Whether the observed proportion differs significantly from the expected

  • The exact p-value (based on the binomial distribution)

  • A confidence interval for the true proportion

Usage

binomial_test(data, ..., p = 0.5, weights = NULL, conf.level = 0.95)

Arguments

data

Your survey data (a data frame or tibble)

...

One or more binary variables to test. Each must have exactly 2 categories (e.g., Yes/No, Male/Female, 0/1, TRUE/FALSE)

p

The hypothesized proportion to test against (Default: 0.50 = 50 percent). It refers to Group 1, the category of the first valid case in the data (as in SPSS). With 0.5 the test is two-tailed; with any other value it is one-tailed in the direction of the observed proportion, as SPSS reports it.

weights

Optional survey weights for population-representative results. Give a column name (unquoted or as a string), an expression such as sampling_weight * 2, or a numeric vector with one weight per row.

conf.level

Confidence level for intervals (Default: 0.95 = 95 percent)

Value

Test results showing whether the proportion differs from expected, including:

  • Category counts and observed proportions

  • Test proportion (null hypothesis)

  • Exact p-value (two-tailed for p = 0.5, otherwise one-tailed; column alternative: "two.sided", "less" or "greater")

  • Confidence interval for the true proportion of Group 1 (Clopper-Pearson)

Details

Understanding the Results

P-value: If p < 0.05, the proportion differs significantly from the test proportion

  • p < 0.001: Very strong evidence the proportion differs

  • p < 0.01: Strong evidence the proportion differs

  • p < 0.05: Moderate evidence the proportion differs

  • p > 0.05: No significant difference from expected proportion

Observed Proportion: The actual proportion in your data. Compare this with the test proportion to see the direction of any difference.

Confidence Interval: The range likely to contain the true population proportion. If the test proportion falls outside this range, the result is significant.

When to Use This

Use the binomial test when:

  • You want to compare an observed proportion to a known value

  • Your variable has exactly 2 categories

  • You need an exact test (not relying on normal approximation)

  • Sample size is small (where chi-square may not be reliable)

Relationship to Other Tests

  • For testing association between two categorical variables: Use chi_square() instead

  • For comparing proportions between groups: Use chi-square or z-test for proportions

  • For larger samples with normal approximation: results will be very similar to a one-sample z-test for proportions

Group 1 and one-tailed tests

As in SPSS NPAR TESTS /BINOMIAL, Group 1 is the category of the first valid case in the data (per group with group_by()), and p is its hypothesized proportion. For p = 0.5 the p-value is two-tailed. For any other p SPSS reports a one-tailed p-value in the direction of the observed proportion ("the proportion of cases in the first group < p"); binomial_test() does the same. To test the other category, put a case of it first or use 1 - p.

Weighted analyses

Each case weight is rounded to a whole number before the categories are counted. With all weights equal to 1 the weighted test reproduces the unweighted one exactly. Weighted results are not compared with SPSS output; see vignette("spss-compatibility").

References

Conover, W. J. (1999). Practical nonparametric statistics (3rd ed.). John Wiley & Sons.

See also

binom.test for the base R exact binomial test.

chi_square for testing associations between categorical variables.

Other hypothesis_tests: ancova(), chi_square(), chisq_gof(), factorial_anova(), fisher_test(), friedman_test(), kruskal_wallis(), mann_whitney(), mcnemar_test(), oneway_anova(), t_test(), wilcoxon_test()

Examples

# Load required packages and data
library(dplyr)
data(survey_data)

# Test whether gender split is 50/50
survey_data %>%
  binomial_test(gender, p = 0.50)
#> Binomial Test: gender
#>   Group 1 (Female): prop = 0.522 vs 0.500, p = 0.026 *, N = 2500
#> Use summary() for detailed output.

# Test whether East region proportion is 50%
survey_data %>%
  binomial_test(region, p = 0.50)
#> Binomial Test: region
#>   Group 1 (East): prop = 0.194 vs 0.500, p < 0.001 ***, N = 2500
#> Use summary() for detailed output.

# Multiple variables at once
survey_data %>%
  binomial_test(gender, region, p = 0.50)
#> Binomial Test: gender
#>   Group 1 (Female): prop = 0.522 vs 0.500, p = 0.026 *, N = 2500
#> Binomial Test: region
#>   Group 1 (East): prop = 0.194 vs 0.500, p < 0.001 ***, N = 2500
#> Use summary() for detailed output.

# Weighted analysis
survey_data %>%
  binomial_test(gender, p = 0.50, weights = sampling_weight)
#> Binomial Test: gender [Weighted]
#>   Group 1 (Female): prop = 0.522 vs 0.500, p = 0.026 *, N = 2500
#> Use summary() for detailed output.

# Grouped analysis (separate test per region)
survey_data %>%
  group_by(region) %>%
  binomial_test(gender, p = 0.50)
#> [region = East]
#> Binomial Test: gender
#>   Group 1 (Female): prop = 0.509 vs 0.500, p = 0.716, N = 485
#> [region = West]
#> Binomial Test: gender
#>   Group 1 (Male): prop = 0.474 vs 0.500, p = 0.023 *, N = 2015
#> Use summary() for detailed output.