Analysis of Deviance for GKw Regression Models

Computes an analysis of deviance table for one or more fitted Generalized Kumaraswamy (GKw) regression model objects. When multiple models are provided, likelihood ratio tests are performed to compare nested models.

Usage

# S3 method for class 'gkwreg'
anova(object, ..., test = c("Chisq", "none"))

Arguments

object: An object of class "gkwreg", typically obtained from gkwreg.
...: Additional objects of class "gkwreg" for model comparison. Models must be nested and fitted to the same dataset.
test: A character string specifying the test statistic to use. Currently only "Chisq" (default) is supported, which performs likelihood ratio tests. Can also be "none" for no tests.

Value

An object of class c("anova.gkwreg", "anova", "data.frame"), with the following columns:

Resid. Df: Residual degrees of freedom
Resid. Dev: Residual deviance (-2 × log-likelihood)
Df: Change in degrees of freedom (for model comparisons)
Deviance: Change in deviance (for model comparisons)
Pr(>Chi): P-value from the chi-squared test (if test = "Chisq")

Details

When a single model is provided, the function returns a table showing the residual degrees of freedom and deviance.

When multiple models are provided, the function compares them using likelihood ratio tests (LRT). Models are automatically ordered by their complexity (degrees of freedom). The LRT statistic is computed as: $$LRT = 2(\ell_1 - \ell_0)$$ where $\ell_1$ is the log-likelihood of the more complex model and $\ell_0$ is the log-likelihood of the simpler (nested) model. Under the null hypothesis that the simpler model is adequate, the LRT statistic follows a chi-squared distribution with degrees of freedom equal to the difference in the number of parameters between the models.

Important: This method assumes that the models being compared are nested (i.e., one model is a special case of the other) and fitted to the same data. Comparing non-nested models or models fitted to different datasets will produce unreliable results. Use AIC or BIC for comparing non-nested models.

The deviance is defined as $-2 \times \text{log-likelihood}$. For models fitted by maximum likelihood, smaller (more negative) deviances indicate better fit. Note that deviance can be negative when the log-likelihood is positive, which occurs when density values exceed 1 (common in continuous distributions on bounded intervals). What matters for inference is the change in deviance between models, which should be positive when the more complex model fits better.

References

Wilks, S. S. (1938). The large-sample distribution of the likelihood ratio for testing composite hypotheses. The Annals of Mathematical Statistics, 9(1), 60–62. doi:10.1214/aoms/1177732360

Pawitan, Y. (2001). In All Likelihood: Statistical Modelling and Inference Using Likelihood. Oxford University Press.

Author

Lopes, J. E.

Examples

# \donttest{
# Load example data
data(GasolineYield)

# Fit a series of nested models
fit1 <- gkwreg(yield ~ 1, data = GasolineYield, family = "kw")
fit2 <- gkwreg(yield ~ temp, data = GasolineYield, family = "kw")
fit3 <- gkwreg(yield ~ batch + temp, data = GasolineYield, family = "kw")
#> Warning: NaNs produced

# ANOVA table for single model
anova(fit3)
#> Analysis of Deviance Table
#> 
#> Model: gkwreg(formula = yield ~ batch + temp, data = GasolineYield, 
#> Model:     family = "kw")
#> 
#>   Resid. Df Resid. Dev
#> 1  20.00000 -193.93865

# Compare nested models using likelihood ratio tests
anova(fit1, fit2, fit3)
#> Analysis of Deviance Table
#> 
#> Model 1: yield ~ 1
#> Model 2: yield ~ temp
#> Model 3: yield ~ batch + temp
#> 
#>      Resid. Df Resid. Dev Df  Deviance Pr(>Chi)    
#> fit1  30.00000  -57.02258                          
#> fit2  29.00000  -80.90259  1  23.88001  < 1e-04 ***
#> fit3  20.00000 -193.93865  9 113.03606  < 1e-04 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> Model 1 vs 2: Adding temperature is highly significant (p < 0.001)
#> Model 2 vs 3: Adding batch is highly significant (p < 0.001)

# Compare two models
anova(fit2, fit3, test = "Chisq")
#> Analysis of Deviance Table
#> 
#> Model 1: yield ~ temp
#> Model 2: yield ~ batch + temp
#> 
#>      Resid. Df Resid. Dev Df  Deviance Pr(>Chi)    
#> fit2  29.00000  -80.90259                          
#> fit3  20.00000 -193.93865  9 113.03606  < 1e-04 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

# Suppress test statistics
anova(fit1, fit2, fit3, test = "none")
#> Analysis of Deviance Table
#> 
#> Model 1: yield ~ 1
#> Model 2: yield ~ temp
#> Model 3: yield ~ batch + temp
#> 
#>      Resid. Df Resid. Dev Df  Deviance
#> fit1  30.00000  -57.02258             
#> fit2  29.00000  -80.90259  1  23.88001
#> fit3  20.00000 -193.93865  9 113.03606
# }