Dixon Type (Q) Tests

Local Algorithm - One-Dimensional Algorithm

Dixon Q-test algorithm uses the ratio of the gap between the possible a value outside the region of interest and the next closest value to it to the total range of the data set. This value is compared to a set of statistically determined values for a variety of confidence intervals. The basic formula is as follows:

$$(1) \quad Q^{\text{test}} = \frac{Y_N - Y_{N-1}}{Y_N - Y_1} \, \text{,}$$

and

$$(2) \quad Q^{\text{test}} = \frac{Y_2 - Y_1}{Y_N - Y_1} \, \text{,}$$

where (1) is used to test whether the largest value is a value outside the region of interest, and (2) is used to test whether the smallest value in the data set of length $N$ is a value outside the region of interest. If $Q^{\text{test}} > Q^{\text{table}}$, the value in question is a value outside the region of interest.

Input Parameters

Parameter	Type	Constraint	Description	Remarks
$Y$	$Y \in \mathbb R^N$	$N \in \mathbb{N}$	Input data sequenceof length $N$
$Q^{\text{table}}$			A table of calculated $Q^{\text{table}}$ values for a specific confidence level, e.g. $90$ %, $95$ % or $99$ %

Output Parameters

Parameter	Type	Constraint	Description	Remarks
$\hat{Y}$	$\hat{Y} \in \mathbb R^N$		Values in the $Y$ list which are outside the region of interest are marked

Single Steps using the Algorithm

• Outlier Detection with Dixon-type (Q) tests

References

• D.B. Rorabacher, Statistical Treatment for Rejection of Deviant Values: Critical Values of Dixon Q Parameter and Related Subrange Ratios at the 95 percent Confidence Level, Anal. Chem., vol. 63(2), pp. 139-146, 1991.

  http://pubs.acs.org/doi/pdf/10.1021/ac00002a010

• S. Walfish, A review of statistical outlier methods. Pharmaceutical Technology, 2006. Retrieved from www.pharmtech.com.