chisq.test
Pearson's Chisquared Test for Count Data
chisq.test
performs chisquared contingency table tests
and goodnessoffit tests.
 Keywords
 distribution, htest
Usage
chisq.test(x, y = NULL, correct = TRUE, p = rep(1/length(x), length(x)), rescale.p = FALSE, simulate.p.value = FALSE, B = 2000)
Arguments
 x
 a numeric vector or matrix.
x
andy
can also both be factors.  y
 a numeric vector; ignored if
x
is a matrix. Ifx
is a factor,y
should be a factor of the same length.  correct
 a logical indicating whether to apply continuity
correction when computing the test statistic for 2 by 2 tables: one
half is subtracted from all $O  E$ differences; however, the
correction will not be bigger than the differences themselves. No correction
is done if
simulate.p.value = TRUE
.  p
 a vector of probabilities of the same length of
x
. An error is given if any entry ofp
is negative.  rescale.p
 a logical scalar; if TRUE then
p
is rescaled (if necessary) to sum to 1. Ifrescale.p
is FALSE, andp
does not sum to 1, an error is given.  simulate.p.value
 a logical indicating whether to compute pvalues by Monte Carlo simulation.
 B
 an integer specifying the number of replicates used in the Monte Carlo test.
Details
If x
is a matrix with one row or column, or if x
is a
vector and y
is not given, then a goodnessoffit test
is performed (x
is treated as a onedimensional
contingency table). The entries of x
must be nonnegative
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 twodimensional contingency table: the entries of x
must be nonnegative 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 Pearson's chisquared test is
performed of the null hypothesis that the joint distribution of the
cell counts in a 2dimensional contingency table is the product of the
row and column marginals.
If simulate.p.value
is FALSE
, the pvalue is computed
from the asymptotic chisquared distribution of the test statistic;
continuity correction is only used in the 2by2 case (if correct
is TRUE
, the default). Otherwise the pvalue is computed for a
Monte Carlo test (Hope, 1968) with B
replicates.
In the contingency table case simulation is done by random sampling from the set of all contingency tables with given marginals, and works only if the marginals are strictly positive. Continuity correction is never used, and the statistic is quoted without it. Note that this is not the usual sampling situation assumed for the chisquared test but rather that for Fisher's exact test.
In the goodnessoffit case simulation is done by random sampling from
the discrete distribution specified by p
, each sample being
of size n = sum(x)
. This simulation is done in R and may be
slow.
Value

A list with class
 statistic
 the value the chisquared test statistic.
 parameter
 the degrees of freedom of the approximate
chisquared distribution of the test statistic,
NA
if the pvalue is computed by Monte Carlo simulation.  p.value
 the pvalue for the test.
 method
 a character string indicating the type of test performed, and whether Monte Carlo simulation or continuity correction was used.
 data.name
 a character string giving the name(s) of the data.
 observed
 the observed counts.
 expected
 the expected counts under the null hypothesis.
 residuals
 the Pearson residuals,
(observed  expected) / sqrt(expected)
.  stdres
 standardized residuals,
(observed  expected) / sqrt(V)
, whereV
is the residual cell variance (Agresti, 2007, section 2.4.5 for the case wherex
is a matrix,n * p * (1  p)
otherwise).
"htest"
containing the following
components:
Source
The code for Monte Carlo simulation is a C translation of the Fortran algorithm of Patefield (1981).
References
Hope, A. C. A. (1968) A simplified Monte Carlo significance test procedure. J. Roy, Statist. Soc. B 30, 582598.
Patefield, W. M. (1981) Algorithm AS159. An efficient method of generating r x c tables with given row and column totals. Applied Statistics 30, 9197.
Agresti, A. (2007) An Introduction to Categorical Data Analysis, 2nd ed., New York: John Wiley & Sons. Page 38.
See Also
For goodnessoffit testing, notably of continuous distributions,
ks.test
.
Examples
library(stats)
## From Agresti(2007) p.39
M < as.table(rbind(c(762, 327, 468), c(484, 239, 477)))
dimnames(M) < list(gender = c("F", "M"),
party = c("Democrat","Independent", "Republican"))
(Xsq < chisq.test(M)) # Prints test summary
Xsq$observed # observed counts (same as M)
Xsq$expected # expected counts under the null
Xsq$residuals # Pearson residuals
Xsq$stdres # standardized residuals
## Effect of simulating pvalues
x < matrix(c(12, 5, 7, 7), ncol = 2)
chisq.test(x)$p.value # 0.4233
chisq.test(x, simulate.p.value = TRUE, B = 10000)$p.value
# around 0.29!
## Testing for population probabilities
## Case A. Tabulated data
x < c(A = 20, B = 15, C = 25)
chisq.test(x)
chisq.test(as.table(x)) # the same
x < c(89,37,30,28,2)
p < c(40,20,20,15,5)
try(
chisq.test(x, p = p) # gives an error
)
chisq.test(x, p = p, rescale.p = TRUE)
# 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.
chisq.test(x, p = p) # maybe doubtful, but is ok!
chisq.test(x, p = p, simulate.p.value = TRUE)
## Case B. Raw data
x < trunc(5 * runif(100))
chisq.test(table(x)) # NOT 'chisq.test(x)'!