# ContingencyTests

##### Tests of Independence in Two- or Three-Way Contingency Tables

Testing the independence of two nominal or ordered factors.

- Keywords
- htest

##### Usage

```
# 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, …)

##### Arguments

- formula
a formula of the form

`y ~ x | block`

where`y`

and`x`

are factors and`block`

is an optional factor for stratification.- data
an optional data frame containing the variables in the model formula.

- subset
an optional vector specifying a subset of observations to be used. Defaults to

`NULL`

.- weights
an optional formula of the form

`~ w`

defining integer valued case weights for each observation. Defaults to`NULL`

, implying equal weight for all observations.- object
an object inheriting from classes

`"table"`

or`"'>IndependenceProblem"`

.- …
further arguments to be passed to

`independence_test`

.

##### Details

`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
\(p\)-values and an asymptotic approximation of the exact distribution is
used by default (`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.

##### Value

##### Note

The exact versions of the Pearson chi-squared test and the generalized Cochran-Mantel-Haenszel test do not necessarily result in the same \(p\)-value as Fisher's exact test (Davis, 1986).

##### References

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 \(\chi^2\)
tests. *Biometrics* **10**(4), 417--451. 10.2307/3001616

Davis, L. J. (1986). Exact tests for \(2 \times 2\) contingency
tables. *The American Statistician* **40**(2), 139--141.
10.1080/00031305.1986.10475377

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

##### Examples

```
# 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)
cmh_test(jobsatisfaction) # CMH = 10.2001
## 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
# }
```

*Documentation reproduced from package coin, version 1.3-1, License: GPL-2*