Let **X**
^{(i)}, *i* = 1, ..., *n*, represent the intensity values of the *i*-th of *n* chips, each consisting of *m* × *m* (e.g., 650 × 650) cells
. Assuming that biological systems respond to relative, rather than absolute differences in gene expression, for each pair of chips a matrix of pointwise (log) ratios is defined as

Given that the intensity at each cell is highly determined by the sequence of the probe [8], the spatial distribution of differences in log-intensities should have no identifiable features, except for probes belonging to probe sets related to the genes that are differentially expressed under the conditions the samples were taken. Here, we assume that the proportion of differentially expressed genes is small. Thus, since probes belonging to a probe set are (more or less) randomly distributed across the chip, cells of related genes are rarely located next to each other, so that no obvious pattern should be discernable. If, however, chip **X**
^{(i) }has a localized 'defect', this should result in a similar pattern across all **R**
^{(i,i'≠i) }in the region of the defect. To allow for visual inspection of such pattern, we draw on the fact that the distribution of differences in log-intensities should be (more or less) symmetrical, except for outliers caused by rare events affecting small areas in particular chips. Probe-wise outliers (due to both differential expression and defects) can be identified by comparing each chip to a measure of central tendency derived from all other chips. Although other measures of central tendency will be discussed below, we start our discussion with the special case of the arithmetic mean, which is known to be optimal in the classical linear model ([21])

Let **R**
^{(i,i') }= Δ^{(i,i') }+ **D**
^{(i) }- **D**
^{(i') }+ ε where Δ^{(i,i') }indicates the random contribution from the differentially expressed genes, **D**
^{(i) }describes the defects of the *i*-th chip, and ε other random errors. Then, **D**
^{(i) }contributes not only to
(bars indicating the average over the index replaced by dot), but also, albeit with only 1/*n* of the intensity, to each of the other
as a 'negative shadow' or 'ghost' image. As the number of chips *n* increases, however, the law of large numbers allows for approximating the linear equation system (1), with hats indicating estimators, as

From (2), we get the linear equation system:

where **I** = (δ_{
j = j'})_{
j,j' = 1...n
}and **J** = (1)_{
j,j' = 1...n
}. A system
has the trivial solution **Y** = **D** whenever column sums are zero (**JY** = **0**). As (2) guarantees that
, setting
yields the solution

as the linear model estimate for the deviation of the *i*-th chip from the other chips. As the number of chips increases, ghosting reduces, so that any discernable pattern in
in the limit would suggest a defect.

The above justification for obtaining residuals within the linear model by subtracting the average is well known. Still, spelling out and justifying the individual steps above helps in two ways. First, we can fine tune the method for the particular situation we are faced with and, second, we can provide numerical examples comparing the proposed non-parametric with the traditional parametric approach. The justification for the choice of the arithmetic mean (average) as the measure of central tendency in linear models relies either on the law of large numbers and the central limit theorem or on the assumption that the distribution of errors is symmetrical, in general, and Gaussian, in particular. Neither assumption is easily justified for the errors caused by defects on a chip.

The arithmetic mean is known to be relatively sensitive to outliers. Thus, to discriminate outliers from observations close to the centre of the non-outliers, one would need either a very large number of chips or a measure of central tendency that is less likely to be affected by the outliers themselves. While microarray 'experiments' now typically consist of more than a single chip, the number of chips analyzed under comparable conditions is still too small to rely on the central limit theorem for outlier detection. With the number of chips in the single digits, even 'Winsorization' may not be feasible. Moreover, the need for choosing some Winsorization cut-off points adds an undesirable level of arbitrariness to the results. The median, as the most robust form of Winsorization, provides for a simple alternative measure of central tendency: