Testing the independence of two nominal or ordered factors.
# S3 method for formula
chisq_test(formula, data, subset = NULL, weights = NULL, …)
# S3 method for table
chisq_test(object, …)
# S3 method for IndependenceProblem
chisq_test(object, …)# S3 method for formula
cmh_test(formula, data, subset = NULL, weights = NULL, …)
# S3 method for table
cmh_test(object, …)
# S3 method for IndependenceProblem
cmh_test(object, …)
# S3 method for formula
lbl_test(formula, data, subset = NULL, weights = NULL, …)
# S3 method for table
lbl_test(object, …)
# S3 method for IndependenceProblem
lbl_test(object, …)
a formula of the form y ~ x | block where y and x are
factors and block is an optional factor for stratification.
an optional data frame containing the variables in the model formula.
an optional vector specifying a subset of observations to be used. Defaults
to NULL.
an optional formula of the form ~ w defining integer valued case
weights for each observation. Defaults to NULL, implying equal
weight for all observations.
further arguments to be passed to independence_test.
chisq_test, cmh_test and lbl_test provide the Pearson
chi-squared test, the generalized Cochran-Mantel-Haenszel test and the
linear-by-linear association test. A general description of these methods is
given by Agresti (2002).
The null hypothesis of independence, or conditional independence given
block, between y and x is tested.
If y and/or x are ordered factors, the default scores,
1:nlevels(y) and 1:nlevels(x) respectively, can be altered using
the scores argument (see independence_test); this
argument can also be used to coerce nominal factors to class "ordered".
(lbl_test coerces to class "ordered" under any circumstances.)
If both y and x are ordered factors, a linear-by-linear
association test is computed and the direction of the alternative hypothesis
can be specified using the alternative argument. For the Pearson
chi-squared test, this extension was given by Yates (1948) who also discussed
the situation when either the response or the covariate is an ordered factor;
see also Cochran (1954) and Armitage (1955) for the particular case when
y is a binary factor and x is ordered. The Mantel-Haenszel
statistic (Mantel and Haenszel, 1959) was similarly extended by Mantel (1963)
and Landis, Heyman and Koch (1978).
The conditional null distribution of the test statistic is used to obtain
distribution = "asymptotic"). Alternatively, the
distribution can be approximated via Monte Carlo resampling or computed
exactly for univariate two-sample problems by setting distribution to
"approximate" or "exact" respectively. See
asymptotic, approximate and exact
for details.
Agresti, A. (2002). Categorical Data Analysis, Second Edition. Hoboken, New Jersey: John Wiley & Sons.
Armitage, P. (1955). Tests for linear trends in proportions and frequencies. Biometrics 11(3), 375--386. 10.2307/3001775
Cochran, W.G. (1954). Some methods for strengthening the common
Davis, L. J. (1986). Exact tests for
Landis, J. R., Heyman, E. R. and Koch, G. G. (1978). Average partial association in three-way contingency tables: a review and discussion of alternative tests. International Statistical Review 46(3), 237--254. 10.2307/1402373
Mantel, N. and Haenszel, W. (1959). Statistical aspects of the analysis of data from retrospective studies of disease. Journal of the National Cancer Institute 22(4), 719--748. 10.1093/jnci/22.4.719
Mantel, N. (1963). Chi-square tests with one degree of freedom: extensions of the Mantel-Haenszel procedure. Journal of the American Statistical Association 58(303), 690--700. 10.1080/01621459.1963.10500879
Yates, F. (1948). The analysis of contingency tables with groupings based on quantitative characters. Biometrika 35(1/2), 176--181. 10.1093/biomet/35.1-2.176
# NOT RUN {
## Example data
## Davis (1986, p. 140)
davis <- matrix(
c(3, 6,
2, 19),
nrow = 2, byrow = TRUE
)
davis <- as.table(davis)
## Asymptotic Pearson chi-squared test
chisq_test(davis)
chisq.test(davis, correct = FALSE) # same as above
## Approximative (Monte Carlo) Pearson chi-squared test
ct <- chisq_test(davis,
distribution = approximate(nresample = 10000))
pvalue(ct) # standard p-value
midpvalue(ct) # mid-p-value
pvalue_interval(ct) # p-value interval
size(ct, alpha = 0.05) # test size at alpha = 0.05 using the p-value
## Exact Pearson chi-squared test (Davis, 1986)
## Note: disagrees with Fisher's exact test
ct <- chisq_test(davis,
distribution = "exact")
pvalue(ct) # standard p-value
midpvalue(ct) # mid-p-value
pvalue_interval(ct) # p-value interval
size(ct, alpha = 0.05) # test size at alpha = 0.05 using the p-value
fisher.test(davis)
## Laryngeal cancer data
## Agresti (2002, p. 107, Tab. 3.13)
cancer <- matrix(
c(21, 2,
15, 3),
nrow = 2, byrow = TRUE,
dimnames = list(
"Treatment" = c("Surgery", "Radiation"),
"Cancer" = c("Controlled", "Not Controlled")
)
)
cancer <- as.table(cancer)
## Exact Pearson chi-squared test (Agresti, 2002, p. 108, Tab. 3.14)
## Note: agrees with Fishers's exact test
(ct <- chisq_test(cancer,
distribution = "exact"))
midpvalue(ct) # mid-p-value
pvalue_interval(ct) # p-value interval
size(ct, alpha = 0.05) # test size at alpha = 0.05 using the p-value
fisher.test(cancer)
## Homework conditions and teacher's rating
## Yates (1948, Tab. 1)
yates <- matrix(
c(141, 67, 114, 79, 39,
131, 66, 143, 72, 35,
36, 14, 38, 28, 16),
byrow = TRUE, ncol = 5,
dimnames = list(
"Rating" = c("A", "B", "C"),
"Condition" = c("A", "B", "C", "D", "E")
)
)
yates <- as.table(yates)
## Asymptotic Pearson chi-squared test (Yates, 1948, p. 176)
chisq_test(yates)
## Asymptotic Pearson-Yates chi-squared test (Yates, 1948, pp. 180-181)
## Note: 'Rating' and 'Condition' as ordinal
(ct <- chisq_test(yates,
alternative = "less",
scores = list("Rating" = c(-1, 0, 1),
"Condition" = c(2, 1, 0, -1, -2))))
statistic(ct)^2 # chi^2 = 2.332
## Asymptotic Pearson-Yates chi-squared test (Yates, 1948, p. 181)
## Note: 'Rating' as ordinal
chisq_test(yates,
scores = list("Rating" = c(-1, 0, 1))) # Q = 3.825
## Change in clinical condition and degree of infiltration
## Cochran (1954, Tab. 6)
cochran <- matrix(
c(11, 7,
27, 15,
42, 16,
53, 13,
11, 1),
byrow = TRUE, ncol = 2,
dimnames = list(
"Change" = c("Marked", "Moderate", "Slight",
"Stationary", "Worse"),
"Infiltration" = c("0-7", "8-15")
)
)
cochran <- as.table(cochran)
## Asymptotic Pearson chi-squared test (Cochran, 1954, p. 435)
chisq_test(cochran) # X^2 = 6.88
## Asymptotic Cochran-Armitage test (Cochran, 1954, p. 436)
## Note: 'Change' as ordinal
(ct <- chisq_test(cochran,
scores = list("Change" = c(3, 2, 1, 0, -1))))
statistic(ct)^2 # X^2 = 6.66
## Change in size of ulcer crater for two treatment groups
## Armitage (1955, Tab. 2)
armitage <- matrix(
c( 6, 4, 10, 12,
11, 8, 8, 5),
byrow = TRUE, ncol = 4,
dimnames = list(
"Treatment" = c("A", "B"),
"Crater" = c("Larger", "< 2/3 healed",
">= 2/3 healed", "Healed")
)
)
armitage <- as.table(armitage)
## Approximative (Monte Carlo) Pearson chi-squared test (Armitage, 1955, p. 379)
chisq_test(armitage,
distribution = approximate(nresample = 10000)) # chi^2 = 5.91
## Approximative (Monte Carlo) Cochran-Armitage test (Armitage, 1955, p. 379)
(ct <- chisq_test(armitage,
distribution = approximate(nresample = 10000),
scores = list("Crater" = c(-1.5, -0.5, 0.5, 1.5))))
statistic(ct)^2 # chi_0^2 = 5.26
## Relationship between job satisfaction and income stratified by gender
## Agresti (2002, p. 288, Tab. 7.8)
## Asymptotic generalized Cochran-Mantel-Haenszel test (Agresti, p. 297)
(ct <- cmh_test(jobsatisfaction)) # CMH = 10.2001
## The standardized linear statistic
statistic(ct, type = "standardized")
## The standardized linear statistic for each block
statistic(ct, type = "standardized", partial = TRUE)
## Asymptotic generalized Cochran-Mantel-Haenszel test (Agresti, p. 297)
## Note: 'Job.Satisfaction' as ordinal
cmh_test(jobsatisfaction,
scores = list("Job.Satisfaction" = c(1, 3, 4, 5))) # L^2 = 9.0342
## Asymptotic linear-by-linear association test (Agresti, p. 297)
## Note: 'Job.Satisfaction' and 'Income' as ordinal
(lt <- lbl_test(jobsatisfaction,
scores = list("Job.Satisfaction" = c(1, 3, 4, 5),
"Income" = c(3, 10, 20, 35))))
statistic(lt)^2 # M^2 = 6.1563
## The standardized linear statistic
statistic(lt, type = "standardized")
## The standardized linear statistic for each block
statistic(lt, type = "standardized", partial = TRUE)
# }
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