gofcat (version 0.1.2)

hosmerlem: Hosmer-Lemeshow Test for Categorical Response Models

Description

This is a post estimation goodness-of-fit test for categorical response models. It currently supports the binary, multinomial and ordinal logistic regression models.

Usage

hosmerlem(model, group = 10, tables = FALSE, customFreq = NULL)

Arguments

model

a model object or data.frame of observed and estimated values. The following class of objects can be directly passed to the function: glm(), vglm(), serp(), polr(), clm(), and mlogit(). Other class of objects require providing a data.frame of observed and predicted values.

group

Default to 10. The number of groups to be formed from the original observations.

tables

Default to FALSE. When TRUE, both the observed and the expected frequency tables are printed alongside test results.

customFreq

a vector of custom group frequencies to be used instead of the default equal group frequencies. The vector should, however, sum up to the total number of original observations.

Value

chi.sq

realized value of the chi-square statistic.

df

the degrees of freedom.

p.value

the p-value of the test.

observed

table of observed frequencies.

expected

table of estimated frequencies.

rho

percentage of estimated frequencies greater than one.

type

a character vector indicating the type of HL-test conducted.

tables

TRUE or FALSE logical vector.

Details

The implemented tests are discussed in Hosmer and Lemeshow (1980) (for the binary case), Fagerland, Hosmer, and Bofin (2008); Fagerland and Hosmer (2012) (for the multinomial case) and Fagerland and Hosmer (2013, 2016, 2017) (for the ordinal case). In each of the three settings, one makes a split of the original observations into k groups (10 by default), after ranking the observations by ordinal scores (OS). Ties are separated following an additional ranking based on the observed values. A Pearson chi-squared statistic is thereafter obtained from a (k X c) expected and observed frequency tables. The Hosmerlem() function automatically dictates which of the three HL tests to run based on the type of model supplied or by inspecting the class/level of response category in the supplied data.frame of predicted and observed values.

On the choice of k-groups, Fagerland and Hosmer (2013, 2016) recommend using ten groups. It is believed that number of groups below six adversely affects the heterogeneity within groups thereby resulting in a test with low power. Also having too many groups could lead to a sparsely populated contingency tables, affecting the distributional assumptions of the test statistic. The test statistic follows the chi-squared distribution with (k - 2)(r - 1) + (r - 2) degrees of freedom. Where k and r are respectively the number of groups and response category. For chi-square estimation to hold, it is recommended to have the percentage of estimated frequencies greater than 80 percent of all estimated frequencies (Fagerland and Hosmer, 2017).

Finally, it is particularly recommended to compare the results of the ordinal Hosmer-Lemeshow test with the lipsitz and the Pulkstenis-Robinson tests (Fagerland and Hosmer, 2016; 2017).

References

Hosmer, D. W. and Lemeshow, S. (1980). Goodness of fit tests for the multiple logistic regression model. Communications in Statistics-Theory and Methods, 9, 1043-1069.

Fagerland, M. W., Hosmer, D. W. and Bofin, A. M. (2008). Multinomial goodness-of-fit tests for logistic regression models. Statistics in Medicine, 27, 4238-4253.

Fagerland, M. W. and Hosmer, D. W. (2012). A generalized Hosmer-Lemeshow goodness-of-fit test for multinomial logistic regression models. Stata Journal, 12, 447-453.

Fagerland, M. W. and Hosmer, D. W. (2013). A goodness-of-fit test for the proportional odds regression model. Statistics in Medicine, 32, 2235-2249.

Fagerland, M. W. and Hosmer, D. W. (2016). Tests for goodness of fit in ordinal logistic regression models. Journal of Statistical Computation and Simulation, 86, 3398-3418.

Fagerland, M. W. and Hosmer, D. W. (2017). How to test for goodness of fit in ordinal logistic regression models. Stata Journal, 17, 668-686.

See Also

lipsitz, pulkroben, brant.test, LR.test

Examples

# NOT RUN {
require(VGAM)

# Binary L-H test
set.seed(1)
gm <- glm(rbinom(100,1,0.5) ~ rnorm(100), family=binomial)
hosmerlem(gm, group = 10, customFreq = NULL, tables = TRUE)

# multinomial L-H test
vg <- vglm(RET ~ DIAB + GH + BP, model = TRUE,
           family = multinomial(parallel = TRUE), data = retinopathy)
hosmerlem(vg)

# ordinal L-H test
## proportional odds model
cu <- update(vg, family = cumulative(link = 'logitlink', parallel = TRUE))
hosmerlem(cu, tables=TRUE)
hosmerlem(cu, group = 5, tables=TRUE, customFreq = c(rep(100,5), 113))

## adjacent category model
ac <- update(vg, family = acat(parallel = TRUE))
hosmerlem(ac)

## continuation ratio model
cr <- update(vg, family = cratio(parallel = TRUE))
hosmerlem(cr)

## using data.frame
y <- ordered(retinopathy$RET)
df <- data.frame(y, fitted.values(cr))
hosmerlem(df, tables = TRUE)

# }