chisq.test               package:stats               R Documentation

Pearson's Chi-squared Test for Count Data

Description:

     'chisq.test' performs chi-squared contingency table tests and
     goodness-of-fit 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 vector or matrix.

       y: a vector; ignored if 'x' is a matrix.

 correct: a logical indicating whether to apply continuity correction
          when computing the test statistic for 2x2 tables: one half is
          subtracted from all |O-E| differences.  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 of 'p' is negative.

rescale.p: a logical scalar; if TRUE then 'p' is rescaled (if
          necessary) to sum to 1.  If 'rescale.p' is FALSE, and 'p'
          does not sum to 1, an error is given.

simulate.p.value: a logical indicating whether to compute p-values 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 _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.

     If 'x' is a matrix with at least two rows and columns, it is taken
     as a two-dimensional contingency table.  Again, the entries of 'x'
     must be non-negative integers.  Otherwise, 'x' and 'y' must be
     vectors or factors of the same length; incomplete cases are
     removed, the objects are coerced into factor objects, and the
     contingency table is computed from these.  Then, Pearson's
     chi-squared test 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 is performed.

     If 'simulate.p.value' is '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
     'correct' is 'TRUE', the default).  Otherwise the p-value 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. 
     (A C translation of the algorithm of Patefield (1981) is used.) 
     Continuity correction is never used, and the statistic is quoted
     without it.  Note that this is not the usual sampling situation
     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
     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 chi-squared test statistic.

parameter: the degrees of freedom of the approximate chi-squared
          distribution of the test statistic, 'NA' if 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)'.

References:

     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.

Examples:

     ## Not really a good example
     chisq.test(InsectSprays$count > 7, InsectSprays$spray)
                                     # Prints test summary
     chisq.test(InsectSprays$count > 7, InsectSprays$spray)$observed
                                     # Counts observed
     chisq.test(InsectSprays$count > 7, InsectSprays$spray)$expected
                                     # Counts expected under the null

     ## 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)'!

