ContingencyTests {coin} | R Documentation |

Testing the independence of two nominal or ordered factors.

## S3 method for class 'formula' chisq_test(formula, data, subset = NULL, weights = NULL, ...) ## S3 method for class 'table' chisq_test(object, ...) ## S3 method for class 'IndependenceProblem' chisq_test(object, ...) ## S3 method for class 'formula' cmh_test(formula, data, subset = NULL, weights = NULL, ...) ## S3 method for class 'table' cmh_test(object, ...) ## S3 method for class 'IndependenceProblem' cmh_test(object, ...) ## S3 method for class 'formula' lbl_test(formula, data, subset = NULL, weights = NULL, ...) ## S3 method for class 'table' lbl_test(object, ...) ## S3 method for class 'IndependenceProblem' lbl_test(object, ...)

`formula` |
a formula of the form |

`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 |

`weights` |
an optional formula of the form |

`object` |
an object inheriting from classes |

`...` |
further arguments to be passed to |

`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.

An object inheriting from class `"IndependenceTest"`

.

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).

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. doi: 10.2307/3001775

Cochran, W.G. (1954). Some methods for strengthening the common *χ^2*
tests. *Biometrics* **10**(4), 417–451. doi: 10.2307/3001616

Davis, L. J. (1986). Exact tests for *2 x 2* contingency
tables. *The American Statistician* **40**(2), 139–141.
doi: 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. doi: 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. doi: 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. doi: 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.
doi: 10.1093/biomet/35.1-2.176

## 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)

[Package *coin* version 1.4-1 Index]