Hosmer-Lemeshow Goodness of Fit Test

The HLtest function computes the classical Hosmer-Lemeshow (1980) goodness of fit test for a binomial glm object in logistic regression

The general idea is to assesses whether or not the observed event rates match expected event rates in subgroups of the model population. The Hosmer-Lemeshow test specifically identifies subgroups as the deciles of fitted event values, or other quantiles as determined by the g argument. Given these subgroups, a simple chisquare test on g-2 df is used.

In addition to print and summary methods, a plot method is supplied to visualize the discrepancies between observed and fitted frequencies.

Usage

HosmerLemeshow(model, g = 10)

HLtest(model, g = 10)

# S3 method for class 'HLtest'
print(x, ...)
# S3 method for class 'HLtest'
summary(object, ...)
# S3 method for class 'HLtest'
plot(x, ...)
# S3 method for class 'HLtest'
rootogram(x, ...)

Arguments

model: A glm model object in the binomial family
g: Number of groups used to partition the fitted values for the GOF test.
x, object: A HLtest object
...: Other arguments passed down to methods

Value

A class HLtest object with the following components:

table: A data.frame describing the results of partitioning the data into g groups with the following columns: cut, total, obs, exp, chi
chisq: The chisquare statistics
df: Degrees of freedom
p.value: p value
groups: Number of groups
call: model call

References

Hosmer, David W., Lemeshow, Stanley (1980). A goodness-of-fit test for multiple logistic regression model. Communications in Statistics, Series A, 9, 1043-1069.

Hosmer, David W., Lemeshow, Stanley (2000). Applied Logistic Regression, New York: Wiley, ISBN 0-471-61553-6

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.

Author

Michael Friendly

Examples


data(birthwt, package="MASS")
# how to do this without attach?
attach(birthwt)
#> The following object is masked from package:ca:
#> 
#>     smoke
  race = factor(race, labels = c("white", "black", "other"))
  ptd = factor(ptl > 0)
  ftv = factor(ftv)
  levels(ftv)[-(1:2)] = "2+"
  bwt <- data.frame(low = factor(low), age, lwt, race,
      smoke = (smoke > 0), ptd, ht = (ht > 0), ui = (ui > 0), ftv)
detach(birthwt)

options(contrasts = c("contr.treatment", "contr.poly"))
BWmod <- glm(low ~ ., family=binomial, data=bwt)

(hlt <- HLtest(BWmod))
#> Hosmer and Lemeshow Goodness-of-Fit Test 
#> 
#> Call:
#> glm(formula = low ~ ., family = binomial, data = bwt)
#>  ChiSquare df   P_value
#>   5.981786  8 0.6492722
str(hlt)
#> List of 6
#>  $ table  :'data.frame':	10 obs. of  5 variables:
#>   ..$ cut  : chr [1:10] "[0.0161,0.0788]" "(0.0788,0.119]" "(0.119,0.189]" "(0.189,0.229]" ...
#>   ..$ total: num [1:10] 19 19 19 19 19 18 19 19 19 19
#>   ..$ obs  : num [1:10] 19 17 15 13 16 15 11 10 9 5
#>   ..$ exp  : num [1:10] 18 17.2 16.2 15.1 14.3 ...
#>   ..$ chi  : num [1:10] 0.242 -0.0374 -0.3036 -0.5317 0.4611 ...
#>  $ chisq  : num 5.98
#>  $ df     : num 8
#>  $ p.value: num 0.649
#>  $ groups : num 10
#>  $ call   : language glm(formula = low ~ ., family = binomial, data = bwt)
#>  - attr(*, "class")= chr "HLtest"
summary(hlt)
#> Partition for Hosmer and Lemeshow Goodness-of-Fit Test 
#> 
#>                cut total obs       exp         chi
#> 1  [0.0161,0.0788]    19  19 17.974124  0.24197519
#> 2   (0.0788,0.119]    19  17 17.154854 -0.03738777
#> 3    (0.119,0.189]    19  15 16.222714 -0.30357311
#> 4    (0.189,0.229]    19  13 15.063836 -0.53174994
#> 5    (0.229,0.268]    19  16 14.259007  0.46105470
#> 6    (0.268,0.312]    18  15 12.757257  0.62791490
#> 7    (0.312,0.402]    19  11 12.545096 -0.43623285
#> 8    (0.402,0.493]    19  10 10.593612 -0.18238140
#> 9    (0.493,0.618]    19   9  8.552849  0.15289685
#> 10   (0.618,0.904]    19   5  4.876650  0.05585725
#> Hosmer and Lemeshow Goodness-of-Fit Test 
#> 
#> Call:
#> glm(formula = low ~ ., family = binomial, data = bwt)
#>  ChiSquare df   P_value
#>   5.981786  8 0.6492722
plot(hlt)


# basic model
BWmod0 <- glm(low ~ age, family=binomial, data=bwt)
(hlt0 <- HLtest(BWmod0))
#> Hosmer and Lemeshow Goodness-of-Fit Test 
#> 
#> Call:
#> glm(formula = low ~ age, family = binomial, data = bwt)
#>  ChiSquare df   P_value
#>   9.307311  8 0.3170385
str(hlt0)
#> List of 6
#>  $ table  :'data.frame':	10 obs. of  5 variables:
#>   ..$ cut  : chr [1:10] "[0.128,0.231]" "(0.231,0.26]" "(0.26,0.29]" "(0.29,0.301]" ...
#>   ..$ total: num [1:10] 20 23 26 13 13 25 18 16 22 13
#>   ..$ obs  : num [1:10] 17 19 14 8 8 18 10 13 15 8
#>   ..$ exp  : num [1:10] 15.76 17.23 18.6 9.09 8.95 ...
#>   ..$ chi  : num [1:10] 0.311 0.426 -1.066 -0.361 -0.317 ...
#>  $ chisq  : num 9.31
#>  $ df     : num 8
#>  $ p.value: num 0.317
#>  $ groups : num 10
#>  $ call   : language glm(formula = low ~ age, family = binomial, data = bwt)
#>  - attr(*, "class")= chr "HLtest"
summary(hlt0)
#> Partition for Hosmer and Lemeshow Goodness-of-Fit Test 
#> 
#>              cut total obs       exp         chi
#> 1  [0.128,0.231]    20  17 15.764147  0.31126591
#> 2   (0.231,0.26]    23  19 17.229899  0.42643885
#> 3    (0.26,0.29]    26  14 18.599113 -1.06641920
#> 4   (0.29,0.301]    13   8  9.088499 -0.36106223
#> 5  (0.301,0.312]    13   8  8.947209 -0.31666633
#> 6  (0.312,0.334]    25  18 16.793786  0.29434049
#> 7  (0.334,0.346]    18  10 11.779364 -0.51844629
#> 8  (0.346,0.357]    16  13 10.284015  0.84692725
#> 9  (0.357,0.381]    22  15 13.736398  0.34093679
#> 10 (0.381,0.418]    13   8  7.777571  0.07975704
#> Hosmer and Lemeshow Goodness-of-Fit Test 
#> 
#> Call:
#> glm(formula = low ~ age, family = binomial, data = bwt)
#>  ChiSquare df   P_value
#>   9.307311  8 0.3170385
plot(hlt0)