Raw data is light intensity (luminescence) for *Control* and *Treatment* \[C, T\]

We work with the logarithm (base 2) of these values \[LC=\log_2(C)\\LT=\log_2(T)\]

Then **Average expression** is \[AvgExp = \frac{LT + LC}{2}\] and **Fold change** is \[logFC = LT - LC=\log_2\left(\frac{T}{C}\right)\]

Classical statistics. We have two scenarios, called *hypothesis*

- H
_{0}: Nothing happens - H
_{1}: Something happens

But our experiment cannot tell us directly which one is true

Every experiment has 4 contributions

- Biology (or Nature)
- Natural variability
- Instrument
- Noise

We want to test the hypothesis “black horses are taller or shorter than white horses”

- We want to compare
*average height*of black horses v/s white horses - Each particular horse can be short or tall
- We use a measuring tape
- Horses do not stay put

We may have bad luck. Maybe black and white horses have the same average height, but

- we got only the small black horses and the tall white ones
- the horses move a lot and we cannot measure the correct values

Therefore, we cannot be 100% sure that our results correspond to reality

But we can have a *degree of confidence* that we are not far away

In terms of hypothesis test, we have

- H
_{0}: the real averages are the same. We only see variability and noise. - H
_{1}: the real averages are different. The measurements are too different to be only noise.

The *p*-value is the probability of observing the experimental data \(X\), assuming that H_{0} is true \[ℙ(X|H_0)\]

Notice that \[ℙ(X|H_0)≠ℙ(H_0|X)\] In other words, the *p*-value **is not** the probability that the null hypothesis is true, given the experimental result

Ideally we will like to know this last probability, but it is hard to do so

Under the null hypothesis, the height difference is 0

If we can also assume that the noise and variability follow a Normal distribution \(N(0,σ^2),\) we have

We have another problem. We do not know σ²

A clever biologist found a solution

We measure the *variance* in our data and we use it

But we have to pay a price: *We have less confidence*

Traditionally, 5% and 1% are used as *p*-value thresholds

But there is nothing to decide that these are good values

Indeed, in Gene Expression, these value are usually too big

There are several approaches

- Family Wise Error Rate control (Bonferroni)
- False Discovery Rate control (Hochberg)
- Others

The basic difference is the trade-off between *False Positives* and *False Negatives*

In every hypothesis test, we can be wrong in two ways

- Type 1: False positive
- putting innocents in the jail

- Type 2: False negatives
- freeing criminals

Usually improving one means worsening of the other

Family Wise Error Rate multiplies each *p*-value by the number of cases \[p.adj = p.value ⋅ N\]

It reduces *False Positives* and increases *False Negatives*

Sometimes we get nothing significant

FDR sorts the *p*-values and multiplies each by an increasing value \[p.adj = p.value \cdot\frac{i}{N}\]

If we get \(p.adj<0.05\) then the probability that it is a false positive is 5%