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

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 = NULL,
  model = NULL,
  m = NULL,
  G = 10,
  method = c("chisq", "normal")
)

Arguments

y: A fitted binary logistic glm (then predicted_probs is taken from it automatically), or a numeric vector of binary responses (0/1) for binary data / counts of successes for grouped data.
predicted_probs: Numeric vector of predicted probabilities from the logistic regression model. Must be same length as y.
model: Optional glm object. Required only for the original Farrington test with grouped data (when m is provided and G is NULL).
m: Optional numeric vector of trial counts for each observation (for grouped data). If NULL, data is assumed to be binary.
G: Optional integer specifying the number of groups for binary data grouping. Default is 10. If NULL, no grouping is performed and m must be provided.
method: Reference distribution for the grouped EF statistic: "chisq" (default) refers $T_{EF}$ to a $\chi^2_{G-2}$ distribution; "normal" uses the standardized $Z_{EF}$ (the behaviour of package versions <= 1.0.0).

Value

A data frame with the following columns:

Test: Character string identifying the test performed
Test_Statistic: Numeric value of the standardized test statistic
p_value: Numeric p-value for the test

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.

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 and G set to NULL): Use the original Farrington test, which requires the fitted model object. Leaving G at its default keeps the automatic grouping, and m and model are then ignored.
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.

References

Farrington CP (1996). "On Assessing Goodness of Fit of Generalized Linear Models to Sparse Data." Journal of the Royal Statistical Society, Series B, 58(2), 349-360. doi:10.1111/j.2517-6161.1996.tb02086.x

Hosmer DW, Lemeshow S (1980). "A goodness-of-fit test for the multiple logistic regression model." Communications in Statistics - Theory and Methods, 9(10), 1043-1069. doi:10.1080/03610928008827941

Ebrahim EK, El-Kotory A (2026). "A Directional Hosmer-Lemeshow Goodness-of-Fit Test for Sparse Logistic Regression." arXiv:2607.15454 [stat.ME]. doi:10.48550/arXiv.2607.15454

Ebrahim EK (2026). "Goodness-of-Fit Tests and Calibration Machine-Learning Algorithms for Logistic Regression with Sparse Data." M.Sc. thesis, Alexandria University. arXiv:2608.11140 [stat.ME]. doi:10.48550/arXiv.2608.11140

Author

Ebrahim Khaled Ebrahim ebrahimkhaled@alexu.edu.eg

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)
#>                 Test Test_Statistic   p_value
#> 1 Ebrahim-Farrington      -1.250567 0.9344997

# 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)
set.seed(456)
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. G = NULL is required: left at its default of 10 the
# call takes the automatic-grouping branch instead, which ignores 'model' and 'm'
# and refers binomial counts to the binary statistic.
result_grouped <- ef.gof(y_grouped, predicted_probs_grouped,
                         model = model_grouped, m = m_trials,
                         G = NULL)
print(result_grouped)
#>                  Test Test_Statistic   p_value
#> 1 Farrington-Original     -0.2010413 0.5796669