chisq.test {stats}  R Documentation 
Pearson's Chisquared Test for Count Data
Description
chisq.test
performs chisquared contingency table tests
and goodnessoffit tests.
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. 
y 
a numeric vector; ignored if 
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 
p 
a vector of probabilities of the same length as 
rescale.p 
a logical scalar; if TRUE then 
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. The default
B = 2000
implies a minimum pvalue of about 0.0005 (1/(B+1)
).
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 "htest"
containing the following
components:
statistic 
the value the chisquared test statistic. 
parameter 
the degrees of freedom of the approximate
chisquared distribution of the test statistic, 
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,

stdres 
standardized residuals,

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. Journal of the Royal Statistical Society Series B, 30, 582–598. doi:10.1111/j.25176161.1968.tb00759.x.
Patefield, W. M. (1981). Algorithm AS 159: An efficient method of generating r x c tables with given row and column totals. Applied Statistics, 30, 91–97. doi:10.2307/2346669.
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
## 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)'!