G-Test for Likelihood-Ratio Testing of Count Data

A goodness-of-fit or test of independence based on the log-likelihood ratio (G-statistic), serving as an asymptotically equivalent alternative to the chi-squared test.

Usage

gTest(
  x,
  y = NULL,
  correct = c("none", "williams", "yates"),
  p = rep(1/length(x), length(x)),
  rescaleP = FALSE
)

Arguments

x

a numeric vector or matrix. x and y can also both be factors.

y

a numeric vector; ignored if x is a matrix. If x is a factor, y should be a factor of the same length.

correct

the correction to be applied, one of "none" (default), "williams" or "yates". See the Details.

p

a vector of probabilities of the same length as x (goodness-of-fit test only). An error is given if any entry of p is negative.

rescaleP

logical; if TRUE then p is rescaled (if necessary) to sum to 1. If rescaleP is FALSE, and p does not sum to 1, an error is given.

Value

A list with class "htest" containing the following components:

statistic

the value of the G test statistic.

parameter

the degrees of freedom of the approximate chi-squared distribution of the test statistic.

p.value

the p-value of the test.

method

a character string indicating the type of test performed, and whether a correction was used.

data.name

a character string giving the name(s) of the data.

observed

the observed counts (before any continuity correction).

expected

the expected counts under the null hypothesis.

Details

gTest performs log-likelihood ratio contingency table tests and goodness-of-fit tests.

The G-test is also called "Likelihood Ratio Test" and is asymptotically equivalent to the Pearson chi-squared test but not usually used when analyzing 2x2 tables. It is used in logistic regression and loglinear modeling which involves contingency tables.

If x is a matrix with one row or column, or if x is a vector and y is not given, then a goodness-of-fit test is performed (x is treated as a one-dimensional contingency table). The entries of x must be non-negative integers. In this case, the hypothesis tested is whether the population probabilities equal those in p, or are all equal if p is not given.

If x is a matrix with at least two rows and columns, it is taken as a two-dimensional contingency table: the entries of x must be non-negative integers. Otherwise, x and y must be vectors or factors of the same length; cases with missing values are removed, the objects are coerced to factors, and the contingency table is computed from these. Then the G-test is performed on the null hypothesis that the joint distribution of the cell counts in a 2-dimensional contingency table is the product of the row and column marginals.

Williams' correction (Williams, 1976) divides the statistic by a factor \(q > 1\) and can be used for both test types. Yates' continuity correction is only defined for 2x2 tables (independence) resp. two data values (goodness-of-fit).

Note

Based on code by Pete Hurd, adapted to conform to package standards.

References

Agresti, A. (2007) An Introduction to Categorical Data Analysis, 2nd ed., New York: John Wiley & Sons. Page 38.

Sokal, R. R. and Rohlf, F. J. (2012) Biometry: The Principles and Practice of Statistics in Biological Research, 4th ed., New York: W. H. Freeman and Co.

Williams, D. A. (1976) Improved likelihood ratio tests for complete contingency tables. Biometrika, 63, 33-37.

See also

chisq.test()

Other test.categorical: barnardTest(), bhapkarTest(), breslowDayTest(), cochranQTest(), lehmacherTest(), mantelTrendTest(), stuartMaxwellTest(), woolfTest()

Examples

## From Agresti (2007), p. 39
M <- as.table(rbind(c(762, 327, 468), c(484, 239, 477)))
dimnames(M) <- list(gender = c("M", "F"),
                    party  = c("Democrat", "Independent", "Republican"))

(Xsq <- gTest(M))   # Prints test summary
#> 
#> 	Log likelihood ratio (G-test) test of independence without correction
#> 
#> data:  M
#> G = 30.017, df = 2, p-value = 3.034e-07
#> 

Xsq$observed        # observed counts (same as M)
#>       party
#> gender Democrat Independent Republican
#>      M      762         327        468
#>      F      484         239        477
Xsq$expected        # expected counts under the null
#>   Democrat Independent Republican
#> M 703.6714    319.6453   533.6834
#> F 542.3286    246.3547   411.3166


## Testing for population probabilities
## Case A. Tabulated data
x <- c(A = 20, B = 15, C = 25)
gTest(x)
#> 
#> 	Log likelihood ratio (G-test) goodness of fit test
#> 
#> data:  x
#> G = 2.5267, df = 2, p-value = 0.2827
#> 
gTest(as.table(x))             # the same
#> 
#> 	Log likelihood ratio (G-test) goodness of fit test
#> 
#> data:  as.table(x)
#> G = 2.5267, df = 2, p-value = 0.2827
#> 
x <- c(89, 37, 30, 28, 2)
p <- c(40, 20, 20, 15, 5)
try(
gTest(x, p = p)                # gives an error
)
#> Error in gTest(x, p = p) : probabilities must sum to 1
# works
p <- c(0.40, 0.20, 0.20, 0.19, 0.01)
# Expected count in category 5
# is 1.86 < 5 ==> chi square approx.
gTest(x, p = p)                # maybe doubtful, but is ok!
#> 
#> 	Log likelihood ratio (G-test) goodness of fit test
#> 
#> data:  x
#> G = 5.8414, df = 4, p-value = 0.2113
#> 

## Case B. Raw data
x <- trunc(5 * runif(100))
gTest(table(x))                # NOT 'gTest(x)'!
#> 
#> 	Log likelihood ratio (G-test) goodness of fit test
#> 
#> data:  table(x)
#> G = 2.5267, df = 4, p-value = 0.6399
#>