dixonTest {dixonTest}

Dixons Outlier Test (Q-Test)

Description

Performs Dixons single outlier test.

Usage

dixonTest(x, alternative = c("two.sided", "greater", "less"), refined = FALSE)

Arguments

x

a numeric vector of data

alternative

the alternative hypothesis. Defaults to "two.sided"

refined

logical indicator, whether the refined version or the Q-test shall be performed. Defaults to FALSE

Details

Let X denote an identically and independently distributed normal variate. Further, let the increasingly ordered realizations denote x_1 \le x_2 \le \ldots \le x_n. Dixon (1950) proposed the following ratio statistic to detect an outlier (two sided):

r_{j,i-1} = \max\left\{\frac{x_n - x_{n-j}}{x_n - x_i}, \frac{x_{1+j} - x_1}{x_{n-i} - x_1}\right\}

The null hypothesis, no outlier, is tested against the alternative, at least one observation is an outlier (two sided). The subscript j on the r symbol indicates the number of outliers that are suspected at the upper end of the data set, and the subscript i indicates the number of outliers suspected at the lower end. For r_{10} it is also common to use the statistic Q.

The statistic for a single maximum outlier is:

r_{j,i-1} = \left(x_n - x_{n-j} \right) / \left(x_n - x_i\right)

The null hypothesis is tested against the alternative, the maximum observation is an outlier.

For testing a single minimum outlier, the test statistic is:

r_{j,i-1} = \left(x_{1+j} - x_1 \right) / \left(x_{n-i} - x_1 \right)

The null hypothesis is tested against the alternative, the minimum observation is an outlier.

Apart from the earlier Dixons Q-test (i.e. r_{10}), a refined version that was later proposed by Dixon can be performed with this function, where the statistic r_{j,i-1} depends on the sample size as follows:

r_{10}:	3 \le n \le 7
r_{11}:	8 \le n \le 10
r_{21};	11 \le n \le 13
r_{22}:	14 \le n \le 30

The p-value is computed with the function pdixon.

References

Dixon, W. J. (1950) Analysis of extreme values. Ann. Math. Stat. 21, 488–506. doi:10.1214/aoms/1177729747.

Dean, R. B., Dixon, W. J. (1951) Simplified statistics for small numbers of observation. Anal. Chem. 23, 636–638. doi:10.1021/ac60052a025.

McBane, G. C. (2006) Programs to compute distribution functions and critical values for extreme value ratios for outlier detection. J. Stat. Soft. 16. doi:10.18637/jss.v016.i03.

Examples

## example from Dean and Dixon 1951, Anal. Chem., 23, 636-639.
x <- c(40.02, 40.12, 40.16, 40.18, 40.18, 40.20)
dixonTest(x, alternative = "two.sided")

## example from the dataplot manual of NIST
x <- c(568, 570, 570, 570, 572, 578, 584, 596)
dixonTest(x, alternative = "greater", refined = TRUE)