What do you need this normalization for? Your approach is Z-score-like and sounds a bit like the "batch correction" described in the paper Sims et al BMC Medical Genomics 2008, 1:42 - implemented eg in pamr.batchadjust() in R package pamr by Hastie and Tibshirani.

This is directed towards better comparability of samples between batches/labs/experiments etc for clustering and should preserve the fold changes (fingers crossed ;)). As far as I have seen - there is no precise statistical proof, but Sims et al shows on many exaples that it does the job of removing the batch effect.

Anyway - converting the data to Z-scores produces often a "visually nicer" heatmap - see the "scale" parameter in function heatmap.2() from gplots library. Just be careful if you cluster intensities (your one-channel case) or fold changes (your two-color) . In the second case it is harder to interpret the normalized data.

Still - if the array batch is homogenous (same lab, same RNA extraction) - you would not need such tricks.

I mainly agree with Michal, I would not normalize for the standard deviation, if you are not sure about comparison maybe try to change the signal (raw expression) by log natural or log 2 or play around, then go with the analysis. Remember to do the FDR after and proove the result with a PCR (somebody omit it but I still think it is necessary). Once you are sure about your data you can make them good looking and propose them as fold induction. I suggest you not to worry too much about strictly comparting the results from different labs, the main point is always the trend, there are too many biological and technical variables involved.

For one-channel microarrays may be useful, but for two channels I do not know. In the last case you can use techniques like loess (limma package of R) and BetWeen Slides normalizations, working with log-ratios. If you compare slides with the same biological condition in the green channel for example you could impose than the G quantiles distributions coincide...

Like most here, I would recommend against z-score transformation. Results (intensities) from a microarray experiment are already on the same basic scale, which defeats one of the largest uses of a z-score transformation. It can also amplifying small changes by increasing deviations to 1 and decreasing very large changes to deviations of 1. I have trouble seeing any upside.

z-normalization bases on that the population is normally distributed. Mean and SD will present the center and width of the distribution only if the distribution of the population is normal.

Logarithms of single channel data is not normal because of the presence of the background of the measurements (http://www.biomedcentral.com/1471-2105/5/5).

And, somehow logratios are not normal, also. (http://www.plosone.org/article/info%3Adoi%2F10.1371%2Fjournal.pone.0003555) Without subtraction of the background, the skew will become larger.

I agree with Michal not to normalize logratios. By dividing the logratios with their SD, the differences among different pairs of experiments will be equalized. This will prohibit comparing different experiments. Usually, finding correlated genes requires some sets of experiments.

I would recommend the three parameter normalization method for each channel data, and comparing multiple experiments. I also agree with Michal not to use clustering, since gene expression has a structure that is far from phylogenic trees. I would use PCA instead.

In general, I'd recommend building from the ground-up, theoretically, on any micro-array, PCR, or similar technologies. There are a lot of opportunities for breakdowns of the normality assumption, as Tomokazu pointed out. I would also be careful regarding bimodality, as some of these low-level effects can approximate all-or-nothing. While I don't tend to do any micro-array cluster analysis (by choice), I am familiar enough with cluster analysis to worry about less-obvious related issues in line toward your dendrogram/heatmap. More specifically, one should be very careful about a) the distance scaling chosen (e.g. Mahalanobis would approximate the z-score, but in multivariate space), and cluster formation (nearest neighbor, farthest neighbor, centroid). I haven't seen much careful discussion of these issue, but I'm not a dedicated reader of micro-array analysis. My cluster-analysis/multidimensional scaling experience stems from psychology where the scaling of metrics is a constant concern. I don't know if the micro-array field has gone through that process yet. I WOULD, however, pose that it's never a bad thing to make sure you've got a frim grasp on the implicit and explicit theoretical remificatinos of any such manipulations...particularly as you shift to hypothesis testing. A particular concern I have is the cross-sectional nature of the data in what is a dynamic system with feedback loops. Clusters in n-dimensional space good, but I'm guessing they're probably doing a fair amount of obscuring real mechanisms by virtue of "catching" state combinations of multiple mechanisms that may be indistinguishable cross-sectionally. It's certainly a good place to start, though. Same goes for PCA/FA...probably need to start with these, but my personal bias is that we should be moving toward a computational biology approach. OH! ...and watch out for regulation in your housekeepers, too! Yowsers that's been turning into a challenge, particularly as I try to convince my biologist colleagues who may not be as aware of the issue.

Basically, the normalisation is employed to correct the bias pertained in the microarray data. For single channel array, the quartile normalization is a popular method while for dual channel array, the loess normalisation is number choice, the other methods, for example the VSN, may also incorporated in. R package “limma” provides tools to perform microarray data normalisation any model based analysis. I suggest that you download and took a look of its menu. I do not think converting the raw data into z-score is meaningful normalization in microarray data analysis.

It is interesting to see how people think the issue of “data normalization” because this makes the bases of how we think data differences, which is the main purpose of our measurements. Many methodologies have been introduced, such as that Yongxiang pointed out.

However, as a view point of a scientist, the situation that various philosophies are leading various conclusions is not ideal, since we cannot integrate knowledge on different philosophies. We should carefully avoid such relativism, and a touchstone for this issue is falsifiability of the methodology; “Are the mathematical model and transformation appropriate for the data?”.

The three parameter method (or, “parametric normalization”) has the falsifiability and it acted well over a thousand of data. I do not know other methodologies that have been checked appropriateness of their mathematical models.

Also, I wish to introduce a model as what was recommended by Jason; functions of a cell that transcribe and maintain transcriptome from the genome has been described in words of thermodynamics. One may find why transcriptome is lognormally distributed, how we should evaluate differences in expressions, and why some genes are coincidentally regulated in the model.

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1298927/

Relationships among genes are similar to networks, because a factor regulates several genes simultaneously both up and down; by the factor, the genes are connected. Relationships among experiments may vary; time-course experiment, for example, will be like a chain. Hierarchical clustering makes a phylogenic tree. PCA just present the data structure as just it is. And, PCA finds relationships among both genes and experiments simultaneously in a connected manner. One will find which genes are actually separating which experiments.

Of course, the old-fashioned PCA has some weak points to apply for transcriptome data. An article that solves the weakness is under submitting; so please wait for a while.

I have been busy lately so I have just come across all of these answers. I like to thank you all very much. I might be back to discuss some points after reviewing the literature which many of you have kindly pointed out.

After reading all of your comments carefully and reading a good amount of the related literature. I have learnt new details and I believe that there are two distinct issues here:

(1) How to compensate for the non-biological variabilities in microarrays' experiments' design and implementation, such as labelling, hybridisation, scanning, background, non-specific binding ... etc. Most of the literature of normalisation and most of the answers here are around this issue. In other words: how to make the values given for the expression honestly reflect the mRNA concentration level (or the ratio of two levels) and to be comparable between different samples.

Though, my question was about another "normalisation" issue which is:

(2) Assuming that the given readings / values in the gene-sample matrix truly represent the biological meaning (expression level, ratio ... etc), how to prepare these expression profiles for clustering? If four genes A, B, C and D were measured at three time points to give the log-expression profiles of: A=(3, 5, 7), B=(5, 7, 9), C=(6, 10, 14), and D=(7, 11, 15). Are they all co-expressed? (B) if shifted it becomes (A), (C) if scaled it becomes (A), and (D) if shifted and scaled it becomes (A).

Based on genetics understanding, maybe based on some stoichiometry understanding of gene expression, which genes should be clustered together?

I mainly use Euclidean distance, if I use Pearson correlation then some "implicit normalisation" would be considered by the nature of this similarity metric.