R: Pearson's Chi-squared Test for Count Data

chisq.test {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)'!

[Package stats version 2.8.1 Index]