Ebrahim-Farrington Goodness-of-Fit Test for Logistic Regression

Description

Performs the Ebrahim-Farrington goodness-of-fit test for logistic regression models. This test is particularly effective for binary data and sparse datasets, providing an improved alternative to the traditional Hosmer-Lemeshow test.

Usage

ef.gof(y, predicted_probs, model = NULL, m = NULL, G = 10)

Arguments

Details

The Ebrahim-Farrington test is based on Farrington's (1996) theoretical framework but simplified for practical implementation with binary data. The test uses a modified Pearson chi-square statistic with data-dependent grouping, where observations are grouped by their predicted probabilities.

For binary data (when G is specified), the test automatically groups observations into G groups based on predicted probabilities and applies the simplified Ebrahim-Farrington statistic:

Z_{EF} = \frac{T_{EF} - (G - 2)}{\sqrt{2(G-2)}}

where T_{EF} is the modified Pearson chi-square statistic, and G is the number of groups.

For grouped data (when m is provided), the test applies the original Farrington test with full variance calculations.

Value

A data frame with the following columns:

Note

• For binary data with automatic grouping (G specified): Use the Ebrahim-Farrington test which is computationally efficient and doesn't require the model specification.

• For grouped data (m provided): Use the original Farrington test which requires the fitted model object.

• The test statistic follows a standard normal distribution under the null hypothesis of adequate model fit.

• For binary data with m=1 for all observations and no grouping, the test is not applicable and will return a p-value of 1.

Author(s)

Ebrahim Khaled Ebrahim ebrahimkhaled@alexu.edu.eg

References

Farrington, C. P. (1996). On Assessing Goodness of Fit of Generalized Linear Models to Sparse Data. *Journal of the Royal Statistical Society. Series B (Methodological)*, 58(2), 349-360. Ebrahim, K. E. (2025). Goodness-of-Fits Tests and Calibration Machine Learning Algorithms for Logistic Regression Model with Sparse Data. *Master's Thesis*, Alexandria University. Hosmer, D. W., & Lemeshow, S. (1980). A goodness-of-fit test for the multiple logistic regression model. *Communications in Statistics - Theory and Methods*, 9(10), 1043–1069. https://doi.org/10.1080/03610928008827941

See Also

hoslem.test for the Hosmer-Lemeshow test

Examples

# Example 1: Binary data with automatic grouping (Ebrahim-Farrington test)
set.seed(123)
n <- 500
x <- rnorm(n)
linpred <- 0.5 + 1.2 * x
prob <- 1 / (1 + exp(-linpred))
y <- rbinom(n, 1, prob)

# Fit logistic regression
model <- glm(y ~ x, family = binomial())
predicted_probs <- fitted(model)

# Perform Ebrahim-Farrington test with 10 groups
result <- ef.gof(y, predicted_probs, G = 10)
print(result)

# Example 2: Compare with different number of groups
result_4 <- ef.gof(y, predicted_probs, G = 4)
result_20 <- ef.gof(y, predicted_probs, G = 20)

# Example 3: Grouped data (original Farrington test)
# Note: This requires actual grouped data with trials > 1
# Simulated grouped data
n_groups <- 50
m_trials <- sample(5:20, n_groups, replace = TRUE)
x_grouped <- rnorm(n_groups)
linpred_grouped <- -0.5 + 1.0 * x_grouped
prob_grouped <- 1 / (1 + exp(-linpred_grouped))
y_grouped <- rbinom(n_groups, m_trials, prob_grouped)

# Fit model for grouped data
data_grouped <- data.frame(successes = y_grouped, trials = m_trials, x = x_grouped)
model_grouped <- glm(cbind(successes, trials - successes) ~ x, 
                     data = data_grouped, family = binomial())
predicted_probs_grouped <- fitted(model_grouped)

# Original Farrington test
result_grouped <- ef.gof(y_grouped, predicted_probs_grouped, 
                         model = model_grouped, m = m_trials)
print(result_grouped)