Hosmer-Lemeshow Goodness of Fit Test for Assessing Logistic Regression Calibration

Computes the Hosmer-Lemeshow C or H goodness-of-fit test for a logistic regression model, assessing whether observed event rates match predicted probabilities across grouped subsets of the data.

Usage

hosmerLemeshowTest(x, ...)

# S3 method for class 'glm'
hosmerLemeshowTest(x, nGroups = 10, type = c("C", "H"), ...)

# Default S3 method
hosmerLemeshowTest(x, obs, nGroups = 10, type = c("C", "H"), ...)

# S3 method for class 'HosmerLemeshowTest'
print(x, digits = 4, details = FALSE, ...)

Arguments

x: either a numeric vector of fitted probabilities, each in \([0, 1]\) and without missing values, or a fitted binomial glm() object, from which fitted probabilities and observed outcomes are extracted.
...: further arguments passed to methods.
nGroups: integer, the number of groups (default is 10). Must be at least 3.
type: the type of statistic, one of "C" (default, quantile-based groups) or "H" (equal-width groups on \([0, 1]\)).
obs: a numeric vector of observed binary outcomes (0 or 1) of the same length as x, without missing values; unused for the glm method.
digits: number of significant digits to display.
details: logical; if TRUE, prints observed and expected counts for both outcome classes by group. Default is FALSE.

Value

An object of class c("HosmerLemeshowTest", "htest"), which is a list with components:

statistic: the chi-squared test statistic.
parameter: the degrees of freedom (number of groups used minus 2).
p.value: the p-value of the test.
method: a character string describing the test.
type: the type of statistic computed ("C" or "H").
nGroups: the number of groups actually used (may be less than requested if quantile breaks coincide or groups are empty).
observed: matrix of observed counts for both outcome classes per group.
expected: matrix of expected counts for both outcome classes per group.
data.name: a character string giving the names of the data.

The print method accepts a details argument; if TRUE, observed and expected counts for both outcome classes are printed by group.

Details

The C statistic groups observations by quantiles of the fitted probabilities (equal-frequency bins), while the H statistic uses equal-width intervals on \([0, 1]\) (e.g. 0–0.1, 0.1–0.2, ... for nGroups = 10), as defined in Lemeshow and Hosmer (1982). Groups that contain no observations are dropped with a warning, and the degrees of freedom are based on the number of groups actually used.

Both statistics are asymptotically compared with a chi-squared distribution with \(g - 2\) degrees of freedom (\(g\) the number of groups used) under the null hypothesis of adequate fit. This is the convention for a model evaluated on its development data; for an external validation sample, \(g\) degrees of freedom would be appropriate. The approximation may be unreliable in small samples; a warning is issued when expected cell counts fall below 5.

References

Lemeshow, S. and Hosmer, D. W. (1982) A review of goodness of fit statistics for use in the development of logistic regression models. American Journal of Epidemiology, 115(1), 92–106.

Hosmer, D. W., Lemeshow, S. and Sturdivant, R. X. (2013) Applied Logistic Regression, 3rd ed., New York: Wiley.

Examples

set.seed(111)
x1  <- factor(sample(1:3, 50, replace = TRUE))
x2  <- rnorm(50)
obs <- sample(c(0, 1), 50, replace = TRUE)

model <- glm(obs ~ x1 + x2, family = binomial)

# glm method: probabilities and outcomes are extracted from the model
hosmerLemeshowTest(model)
#> Warning: some expected counts are less than 5; the chi-squared approximation may be unreliable
#> 
#>   Hosmer-Lemeshow C statistic 
#> 
#> data:  obs ~ x1 + x2 
#> X-squared = 4.85 , df = 8 , p-value = 0.7735 
#> Number of groups: 10 
#> 

# equivalent call with explicit vectors
res <- hosmerLemeshowTest(fitted(model), obs, type = "C")
#> Warning: some expected counts are less than 5; the chi-squared approximation may be unreliable
res
#> 
#>   Hosmer-Lemeshow C statistic 
#> 
#> data:  fitted(model) and obs 
#> X-squared = 4.85 , df = 8 , p-value = 0.7735 
#> Number of groups: 10 
#> 

print(res, details = TRUE)
#> 
#>   Hosmer-Lemeshow C statistic 
#> 
#> data:  fitted(model) and obs 
#> X-squared = 4.85 , df = 8 , p-value = 0.7735 
#> Number of groups: 10 
#> 
#> Observed vs Expected counts by group:
#>               Obs 0s Exp 0s Obs 1s Exp 1s
#> [0.262,0.325]      3 3.5250      2 1.4750
#> (0.325,0.374]      4 3.2600      1 1.7400
#> (0.374,0.514]      2 2.7142      3 2.2858
#> (0.514,0.526]      3 2.4012      2 2.5988
#> (0.526,0.539]      4 2.3274      1 2.6726
#> (0.539,0.551]      2 2.2662      3 2.7338
#>  (0.551,0.56]      2 2.2145      3 2.7855
#>  (0.56,0.567]      2 2.1814      3 2.8186
#> (0.567,0.584]      1 2.1236      4 2.8764
#> (0.584,0.616]      2 1.9866      3 3.0134
#>