# pearson.test: Pearson chi-square test for normality In nortest: Tests for Normality

## Description

Performs the Pearson chi-square test for the composite hypothesis of normality, see e.g. Thode (2002, Sec. 5.2).

## Usage

 `1` ```pearson.test(x, n.classes = ceiling(2 * (n^(2/5))), adjust = TRUE) ```

## Arguments

 `x` a numeric vector of data values. Missing values are allowed. `n.classes` The number of classes. The default is due to Moore (1986). `adjust` logical; if `TRUE` (default), the p-value is computed from a chi-square distribution with `n.classes`-3 degrees of freedom, otherwise from a chi-square distribution with `n.classes`-1 degrees of freedom.

## Details

The Pearson test statistic is P=∑ (C_{i} - E_{i})^{2}/E_{i}, where C_{i} is the number of counted and E_{i} is the number of expected observations (under the hypothesis) in class i. The classes are build is such a way that they are equiprobable under the hypothesis of normality. The p-value is computed from a chi-square distribution with `n.classes`-3 degrees of freedom if `adjust` is `TRUE` and from a chi-square distribution with `n.classes`-1 degrees of freedom otherwise. In both cases this is not (!) the correct p-value, lying somewhere between the two, see also Moore (1986).

## Value

A list with class “htest” containing the following components:

 `statistic` the value of the Pearson chi-square statistic. `p.value ` the p-value for the test. `method` the character string “Pearson chi-square normality test”. `data.name` a character string giving the name(s) of the data. `n.classes` the number of classes used for the test. `df` the degress of freedom of the chi-square distribution used to compute the p-value.

## Note

The Pearson chi-square test is usually not recommended for testing the composite hypothesis of normality due to its inferior power properties compared to other tests. It is common practice to compute the p-value from the chi-square distribution with `n.classes` - 3 degrees of freedom, in order to adjust for the additional estimation of two parameters. (For the simple hypothesis of normality (mean and variance known) the test statistic is asymptotically chi-square distributed with `n.classes` - 1 degrees of freedom.) This is, however, not correct as long as the parameters are estimated by `mean(x)` and `var(x)` (or `sd(x)`), as it is usually done, see Moore (1986) for details. Since the true p-value is somewhere between the two, it is suggested to run `pearson.test` twice, with `adjust = TRUE` (default) and with `adjust = FALSE`. It is also suggested to slightly change the default number of classes, in order to see the effect on the p-value. Eventually, it is suggested not to rely upon the result of the test.

The function call `pearson.test(x)` essentially produces the same result as the S-PLUS function call `chisq.gof((x-mean(x))/sqrt(var(x)), n.param.est=2)`.

Juergen Gross

## References

Moore, D.S. (1986): Tests of the chi-squared type. In: D'Agostino, R.B. and Stephens, M.A., eds.: Goodness-of-Fit Techniques. Marcel Dekker, New York.

Thode Jr., H.C. (2002): Testing for Normality. Marcel Dekker, New York.

## See Also

`shapiro.test` for performing the Shapiro-Wilk test for normality. `ad.test`, `cvm.test`, `lillie.test`, `sf.test` for performing further tests for normality. `qqnorm` for producing a normal quantile-quantile plot.

## Examples

 ```1 2``` ```pearson.test(rnorm(100, mean = 5, sd = 3)) pearson.test(runif(100, min = 2, max = 4)) ```

nortest documentation built on May 29, 2017, 10:02 a.m.