Pearson's Chi-squared Test for Count Data
chisq.test performs chi-squared contingency table tests
and goodness-of-fit tests.
chisq.test(x, y = NULL, correct = TRUE, p = rep(1/length(x), length(x)), rescale.p = FALSE, simulate.p.value = FALSE, B = 2000)
- a numeric vector or matrix.
ycan also both be factors.
- a numeric vector; ignored if
xis a matrix. If
xis a factor,
yshould be a factor of the same length.
- 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.
- a vector of probabilities of the same length of
x. An error is given if any entry of
- a logical scalar; if TRUE then
pis rescaled (if necessary) to sum to 1. If
rescale.pis FALSE, and
pdoes not sum to 1, an error is given.
- a logical indicating whether to compute p-values by Monte Carlo simulation.
- an integer specifying the number of replicates used in the Monte Carlo test.
x is a matrix with one row or column, or if
x is a
y is not given, then a goodness-of-fit test
is performed (
x is treated as a one-dimensional
contingency table). The entries of
x must be non-negative
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.
x is a matrix with at least two rows and columns, it is
taken as a two-dimensional contingency table: the entries of
must be non-negative integers. Otherwise,
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 chi-squared test is
performed of the null hypothesis that the joint distribution of the
cell counts in a 2-dimensional contingency table is the product of the
row and column marginals.
FALSE, the p-value is computed
from the asymptotic chi-squared distribution of the test statistic;
continuity correction is only used in the 2-by-2 case (if
TRUE, the default). Otherwise the p-value is computed for a
Monte Carlo test (Hope, 1968) with
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 chi-squared test but rather that for Fisher's exact test.
In the goodness-of-fit case simulation is done by random sampling from
the discrete distribution specified by
p, each sample being
n = sum(x). This simulation is done in Rand may be
- A list with class
"htest"containing the following components:
statistic the value the chi-squared test statistic. parameter the degrees of freedom of the approximate chi-squared distribution of the test statistic,
NAif the p-value is computed by Monte Carlo simulation.
p.value the p-value 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), where
Vis the residual cell variance (Agresti, 2007, section 2.4.5 for the case where
xis a matrix,
n * p * (1 - p)otherwise).
The code for Monte Carlo simulation is a C translation of the Fortran algorithm of Patefield (1981).
Hope, A. C. A. (1968) A simplified Monte Carlo significance test procedure. J. Roy, Statist. Soc. B 30, 582--598.
Patefield, W. M. (1981) Algorithm AS159. An efficient method of generating r x c tables with given row and column totals. Applied Statistics 30, 91--97.
Agresti, A. (2007) An Introduction to Categorical Data Analysis, 2nd ed., New York: John Wiley & Sons. Page 38.
For goodness-of-fit testing, notably of continuous distributions,
## 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 p-values 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)'!