Normalization Factor Determination

Supplementary Material S1

Assumptions:

1. Normalized Average of Log2(Ratios) = 0

2. Sum of Normalized Log2(Ratios) = 0

Premise:

N is the total number of genes used in the calculations

Tf is the fluorescent value at time f and T0 is the fluorescent value at time 0.

X is the sum of the log base 2 of the Ratios (i.e. a measure of the skew towards higher or lower values).

X = S(log2(Ratios)) = S(log2(Tf/T0)) = S(log2(Tf) - log2(T0)) = S(log2(Tf)) - S(log2(T0))

If you assume the real values are balanced (i.e. the assumptions above), but that the data is getting skewed by a uniform effect, then use X to find a correcting factor.

Proof:

Move the X onto the other side of the equation in order to get a sum of 0.

S(log2(Tf)) - S(log2(T0)) – X = 0

S(log2(Tf)) – (S(log2(T0)) + X) = 0

Move the X into the T0 summation by breaking it into N parts, i.e. X = S(X/N)

S(log2(Tf)) - S(log2(T0) + X/N) = 0

Now make a new variable called a which fits the equation a = 2X/N and therefore X/N = log2(a). Thus,

S(log2(Tf)) - S(log2(T0) + log2(a)) = 0

Combine the logs such that log2(T0) + log2(a) = log2(T0 × a), thus

S(log2(Tf)) - S(log2(T0 × a)) = 0

Recombining the summations and the logs give you

S(log2((Tf)/(T0 × a))) = 0

and

S(log2((Tf)/(T0) × (1/a))) = 0

So 1/a is the normalization factor that should be used to make the sum of the logs equal to 0.

Now if you remember,

X = S(log2(Ratios))

So if you divide X by N you get

X/N = (S(log2(Ratios)))/N = Average of the Log2(Ratios)

Thus, since a = 2X/N, then a = 2Average of the Log2(Ratios), so the Normalization Factor is the inverse of 2Average of the Log2(Ratios), which is multiplied against each Ratio (not the Log2(Ratio)).

This calculation leads us to what is known as the Geometric Mean, which is a method used in calculating the mean of ratios.

We can get to the formula for the Geometric Mean from a as follows:

a = 2S(log2(Ratios))/N

= 2(1/N)(S(log2(Ratios)))

= 2S((1/N)log2(Ratios))

Using the property of logs,

S((1/N)log2(Ratios)) = S(log2(Ratios)1/N)

Set b = (Ratios)1/N

a = 2S(log2(b))

= 2log2(b)1 × 2log2(b)2 × 2log2(b)3 × . . . × 2log2(b)N

= b1 × b2 × b3 × . . . × bN

Substituting for b gives

a = (Ratios)11/N × (Ratios)21/N × (Ratios)31/N × . . . × (Ratios)N1/N

= N√((Ratios)1 × (Ratios)2 × (Ratios)3 × . . . × (Ratios)N) = Geometric Mean

So a = 2Average of the Log2(Ratios) is also equal to the Nth-root of the product of all the Ratios.