cochranTest: Cochran _C_ Test

In l-ramirez-lopez/prospectr: Miscellaneous Functions for Processing and Sample Selection of Spectroscopic Data

Cochran C Test

Description

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Detects and removes replicate outliers in data series based on the Cochran C test for homogeneity in variance.

Usage

cochranTest(X, id, fun = 'sum', alpha = 0.05)

Arguments

X	a a numeric matrix (optionally a data frame that can be coerced to a numerical matrix).
id	factor of the replicate identifiers.
fun	function to aggregate data: 'sum' (default), 'mean', 'PC1' or 'PC2'.
alpha	p-value of the Cochran C test.

Details

The Cochran C test is test whether a single estimate of variance is significantly larger than a a group of variances. It can be computed as:

\mjdeqn

RMSD = \sqrt\frac1n \sum_i=1^n (y_i - \ddoty_i)^2RMSD = sqrt1/n sum (y_i - ddoty_i)^2

where \mjeqny_iy_i is the value of the side variable of the \mjeqniith sample, \mjeqn\ddoty_i\ddoty_i is the value of the side variable of the nearest neighbor of the \mjeqniith sample and \mjeqnnn is the total number of observations.

For multivariate data, the variance \mjeqnS_i^2S_i^2 can be computed on aggregated data, using a summary function (fun argument) such as sum, mean, or first principal components ('PC1' and 'PC2').

An observation is considered to have an outlying variance if the Cochran C statistic is higher than an upper limit critical value \mjeqnC_ULC_UL which can be evaluated with ('t Lam, 2010):

\mjdeqn

C_UL(\alpha, n, N) = 1 + [\fracN-1F_c(\alpha/N,(n-1),(N-1)(n-1))]^-1 C_UL(\alpha, n, N) = 1 + [\fracN-1F_c(\alpha/N,(n-1),(N-1)(n-1))]^-1

where \mjeqn\alpha\alpha is the p-value of the test, \mjeqnnn is the (average) number of replicates and \mjeqnF_cF_c is the critical value of the Fisher's \mjeqnFF ratio.

The replicates with outlying variance are removed and the test can be applied iteratively until no outlying variance is detected under the given p-value. Such iterative procedure is implemented in cochranTest, allowing the user to specify whether a set of replicates must be removed or not from the dataset by graphical inspection of the outlying replicates. The user has then the possibility to (i) remove all replicates at once, (ii) remove one or more replicates by giving their indices or (iii) remove nothing.

Value

a list with components:

• 'X': input matrix from which outlying observations (rows) have been removed

• 'outliers': numeric vector giving the row indices of the input data that have been flagged as outliers

Note

The test assumes a balanced design (i.e. data series have the same number of replicates).

Author(s)

Antoine Stevens

References

Centner, V., Massart, D.L., and De Noord, O.E., 1996. Detection of inhomogeneities in sets of NIR spectra. Analytica Chimica Acta 330, 1-17.

R.U.E. 't Lam (2010). Scrutiny of variance results for outliers: Cochran's test optimized. Analytica Chimica Acta 659, 68-84.

https://en.wikipedia.org/wiki/Cochran's_C_test

l-ramirez-lopez/prospectr documentation built on Feb. 18, 2024, 7:52 a.m.