cubic.calib.gof: Cubic Calibration Likelihood-Ratio Test for Logistic...

In ebrahim.gof: Goodness-of-Fit and Calibration Tests for Logistic Regression

Cubic Calibration Likelihood-Ratio Test for Logistic Regression

Description

cubic.calib.gof() asks whether the calibration curve of a binary risk model bends on the logit scale. It fits two logistic regressions of the outcome on the logit of the model's own predicted risk, \hat\eta: one with \hat\eta alone, and one that adds \hat\eta^2 and \hat\eta^3. Twice the drop in deviance is referred to a chi-squared distribution on 2 degrees of freedom.

Usage

cubic.calib.gof(object, predicted_probs = NULL)

Arguments

object	Either a fitted glm with family = binomial and a 0/1 response, or a numeric 0/1 outcome vector, in which case predicted_probs is required.
predicted_probs	Predicted probabilities for object when it is an outcome vector, for example from an external model. Ignored when object is a glm.

Details

Where the test comes from. This is not a new test, and the package does not claim it. Testing a fitted model by adding powers of its own fitted values goes back to Tukey's (1949) one degree of freedom for non-additivity, which adds the square of the fitted values, and Ramsey (1969) made it a general specification test for linear regression, RESET, by adding their square and cube. Pregibon (1980) carried the idea to generalized linear models as a goodness-of-link test: refit the model with constructed variables computed from its fitted linear predictor, and read the change in deviance. Morgan (1985) studied the logistic model whose linear predictor is a cubic polynomial, the cubic logistic model, whose extra terms bend the two tails of the logistic curve. cubic.calib.gof() is this family's test for a logistic model: the constructed variables are \hat\eta^2 and \hat\eta^3, and the test is the likelihood-ratio test of their two coefficients. The GiViTI calibration belt (Finazzi et al., 2011; Nattino et al., 2014) fits the same kind of polynomial in the logit of the predicted risk, choosing its degree from the data and testing all of its coefficients; this function fixes the degree at three and tests only the curvature.

What it tests. When the model was fitted by maximum likelihood to the same data, the logistic regression of y on \hat\eta reproduces the fit exactly (intercept 0, slope 1), so the test asks only whether the calibration curve is curved on the logit scale: the quadratic term picks up an asymmetric bend, as from a complementary log-log truth fitted by a logit, and the cubic term a symmetric one, as from tails that are too heavy or too light. With the predictions of an external model (predicted_probs) the intercept and slope are free as well, so the test still asks only about curvature; calibration-in-the-large and the calibration slope need their own tests (Cox, 1958).

What to know before using it. The test weighs every record by its own linear predictor. A few records with corrupted covariates, whose predictions are pushed to the extremes, can therefore drive its false-alarm rate far above the nominal level; the grouped directed test edge.gof is protected against that. The polynomial is fitted on the standardized \hat\eta, which spans the same polynomials and leaves the statistic unchanged but keeps the fit well conditioned.

Value

A one-row data.frame with columns Test ("Cubic-LR"), Test_Statistic (the likelihood-ratio statistic), df (2, or fewer if a column cannot be estimated), Method ("likelihood ratio") and p_value. A fit that fails or does not converge gives an NA p-value with a warning.

Author(s)

Ebrahim Khaled Ebrahim [e-mail]

References

Cox, D. R. (1958). Two further applications of a model for binary regression. Biometrika, 45(3-4), 562-565. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1093/biomet/45.3-4.562")}

Finazzi, S., Poole, D., Luciani, D., Cogo, P. E. and Bertolini, G. (2011). Calibration belt for quality-of-care assessment based on dichotomous outcomes. PLoS ONE, 6(2), e16110. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1371/journal.pone.0016110")}

Morgan, B. J. T. (1985). The cubic logistic model for quantal assay data. Journal of the Royal Statistical Society, Series C (Applied Statistics), 34(2), 105-113. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.2307/2347362")}

Nattino, G., Finazzi, S. and Bertolini, G. (2014). A new calibration test and a reappraisal of the calibration belt for the assessment of prediction models based on dichotomous outcomes. Statistics in Medicine, 33(14), 2390-2407. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1002/sim.6100")}

Pregibon, D. (1980). Goodness of link tests for generalized linear models. Journal of the Royal Statistical Society, Series C (Applied Statistics), 29(1), 15-23. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.2307/2346405")}

Ramsey, J. B. (1969). Tests for specification errors in classical linear least-squares regression analysis. Journal of the Royal Statistical Society, Series B, 31(2), 350-371. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1111/j.2517-6161.1969.tb00796.x")}

Tukey, J. W. (1949). One degree of freedom for non-additivity. Biometrics, 5(3), 232. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.2307/3001938")}

See Also

edge.gof for the grouped directed test, run.all.gof, where this test is the row "Cubic-LR".

Examples

set.seed(1)
x <- runif(1000, -3, 3)
y <- rbinom(1000, 1, plogis(0.6 * x))
fit <- glm(y ~ x, family = binomial())
cubic.calib.gof(fit)                             # correct model

y2 <- rbinom(1000, 1, 1 - exp(-exp(0.6 * x)))       # complementary log-log truth
cubic.calib.gof(glm(y2 ~ x, family = binomial()))   # fitted by a logit: the curve bends

## frozen predictions, e.g. an external validation
cubic.calib.gof(y, predicted_probs = fitted(fit))

ebrahim.gof documentation built on Oct. 11, 2026, 5:07 p.m.