In microarray normalization, why is the normalization factor this?

Question

I am working on analysis of a huge number of microarray files. I was trying to understand the need for normalization in microarray data and was going through this paper by John Quackenbush(2002). In the paper, the author mentions that

There are a number of reasons why data must be normalized, including unequal quantities of starting RNA, differences in labeling or detection efficiencies between the fluorescent dyes used, and systematic biases in the measured expression levels

Then he talks about a simple normalization techniques. Assuming that the total hybridization intensities summed over all elements in the arrays should be the same for each sample, he defines a normalization factor is calculated by summing the measured intensities in both channels-

where G_i and R_i are the measured intensities for the 'i'th array element (for example, the green and red (or experimental and control) intensities in a two-color microarray assay) and N_array is the total number of elements represented in the microarray.

Then the author says this -

This is the part I don't understand. What is the need to introduce $G_k^{'}$, $R_k^{'}$ and why are they what they are? Most importantly, why is $T_i$ equal to $R_i/G_i$ first and then ($1/N_{total}$)*($R_i/G_i$)?

Any ideas?

Answer 1

$G_k^{'}$ and $R_k^{'}$ are normalized values of $G_k$ and $R_k$.

Take say G as $[1,2,3,4]$ and R as $[100,150,200,400]$ as your values and you want to normalize them. This is scaling one of them onto the other and bringing them on an equal level to compare. So in your case the factor is $85$ units. So a unit of R amounts to $85$ units in G.

So to scale G to R, multiply G by $85$ or you can scale R to the level of G by dividing R by $85$. So the values are either $[85,170,..]$ and $[100,150,200,400]$ or $[1,2,3,4]$ and $[1.17,1.76,..]$ from our example.

I think it should be mentioned as $T^{'} = R_i/(G_i*N_{total})$ as in log ratio from the statement in your question.