Log2RG log2RG cA log2RkiAG

where the ci(A) are loess smoothers fitted individually to data from each print tip. This method is recommended for routine use because it deals with both intensity-dependent effects and subarray variation.

There are many other features of the data that could be used in the normalization process. However, further normalization should be applied only when there is clear evidence from diagnostic plots indicating the need for such normalization. Unnecessary estimation of such effects and trend removal may add noise to the data. One variation, after using print-tip loess normalization, is to further standardize the M values from each print tip to have the same scale. In particular, it is assumed that the variance of the M from each print-tip group is given by a,c2.

We can robustly estimate a, = MAD, /^H1MADi where MAD is the median absolute deviation. The scale-normalized values for grid i are then given by M, = M,/a,. This normalization is typically not required except in cases in which the arrays are extremely noisy. An example in which such normalization might be required is shown in Fig. 3, where we see that the variability of M from the fourth row of grids is significantly larger than that for the other grids. Applying the scale normalization removes this difference.

Usually all the spots on the array are used in the normalization methods described previously, because this provides the most stability in terms of the

Fig. 2. An MA-plot after global loess normalization (top) shows print tip effects, which are eliminated using the print-tip specific normalization (bottom). (See Color Insert.)

number of spots and the flexibility to operate in a print-tip—specific manner. However, sometimes the expression profiles in the biological samples are more divergent than has been assumed in the cases mentioned. The previous strategies

Fig. 3. Box plots by print-tip group of M after print-tip normalization indicate a need for further scale normalization. (See Color Insert.)

can be employed if a suitable set of control spots that are known to be not differentially expressed are printed on the array. Ideally these would span the range of possible concentrations. One such method is to use a microarray sample pool (MSP) titration series in which the entire clone library is pooled and then titrated at different concentrations. Because, in theory, all labeled cDNA sequences should hybridize to this series, it should not be subject to sample-specific biases. Differential genes should not bias a loess curve through the control spots.

Sometimes it is useful to combine the MSP normalization and the print-tip—specific normalization. It is suggested (Yang et al., 2002) that one take a weighted average of the print-tip—specific adjustment and the MSP normalization, in which the weights are dependent on the intensity. Define

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