# DrugTargetInspector

#### Assistance tool for patient treatment stratification

## P-value adjustments

In an enrichment analysis multiple categories are tested simultaneously.
For each individual test the same *significance threshold* $\alpha$ is used to judge if a category is significant.
This means $\alpha$ is the probability to make a *false positive prediction (Type-I-Error)*.
Subsequently, each test has probability $\alpha$ to make a Type-I-Error.
The problem with multiple testing is that this probability is accumulated.

For k tested hypotheses this probability is defined as:

$$ P(\text{at least one significant result}) = 1-(1-\alpha)^k $$Multiple testing procedures adjust p-values derived from multiple statistical tests to correct for the number of false positive predictions (Type-I-Error).

### Familywise error rate controlling p-value adjustments

When performing multiple hypotheses tests, the *familywise error rate (FWER)* is the probability of making at least one false positive prediction,
or Type-I-Error, among all the tested null hypotheses [1].

#### Bonferroni

The Bonferroni method [2], [3], [4] adjusts all p-values with the number of tested null hypotheses. The Bonferroni test is conservative and always controls the familywise error rate [1].

$$\tilde p_{i}\ =\ np_{i} $$#### Sidak

The Sidak method [2], [5] is slightly less conservative than the corresponding Bonferroni adjustment [6]. This adjustment is guaranteed to control the familywise error rate when all of the p-values are uniformly distributed and independent [7], [1].

#### Step-down methods

Step-down methods were first introduced by Holm [8]. These procedures examine p-values in order, from smallest to largest. If, after correction, a p-value is smaller than its predecessor it obtains the value of its predecessor. This ensures the monotonicity in the order. The benefit of using step-down methods is that the tests are made more powerful (smaller adjusted p-values) while, in most cases, maintaining strong control of the familywise error rate [1].

$$\tilde p_{i}\ =\ 1 - ( 1 - p_{i})^{n} $$##### Holm

The Holm adjustment [8] is a step-down approach for the Bonferroni method.

$$\tilde p_{i}\ =\ \begin{cases} np_{i} & \text{for } i=1\\ max \left( \tilde p_{(i-1)}, (n - i +1) p_{i} \right) & \text{for }i=2,...,n \end{cases}$$##### Holm-Sidak

The Holm-Sidak adjustment [8], [5] is a step-down approach for the Sidak method.

$$\tilde p_{i}\ =\begin{cases} 1-(1-p_{i})^{(n)} & \text{for } i=1\\ max \left( \tilde p_{(i-1)}, 1-(1-p_{i})^{(n-i+1)} \right) & \text{for }i=2 ,...,n \end{cases}$$##### Finner

The Finner method [9], [10] is a step-down approach for a slightly adapted Sidak method.

$$\tilde p_{i}\ =\ \begin{cases} np_{i} & \text{for } i=1\\ max \left( \tilde p_{(i-1)}, 1-(1-p_{i})^{(\frac{n}{i})} \right) & \text{for }i=2 ,...,n \end{cases}$$#### Step-up methods

In step-up methods p-values are examined in order, from largest to smallest. If, after correction, a p-value is bigger than its predecessor it obtains the value of its predecessor. This ensures the monotonicity in the order.

##### Hochberg

The Hochberg adjustment [11] is a step-up approach for the Bonferroni method. Hochberg showed that Holm's step-down adjustments also control the familywise error rate even when calculated in step-up fashion. Since p-values adjusted by Hochberg's method are always smaller than or equal to p-values adjusted by Holm's method, the Hochberg method is more powerful [1].

$$\tilde p_{i}\ =\ \begin{cases} p_{i} & \text{for } i=n\\ min \left( \tilde p_{(i-1)}, (n-i+1)p_{i} \right) & \text{for }i=n-1,...,1 \end{cases}$$### False discovery rate controlling p-value adjustments

FDR-controlling adjustments are less conservative than adjustments controlling the familywise error rate [12], [1].

$$FDR=E(\frac{FP}{FP+TN})\text{, where }\frac{FP}{FP+TN}=0\text{, when }FP=TN=0$$#### Benjamini-Hochberg

The Benjamini-Hochberg method [13], [12] is a step-up approach to control the false discovery rate. It assumes all p-values to be independent.

$$\tilde p_{i}\ =\ \begin{cases} p_{i} & \text{for } i=n\\ min \left( \tilde p_{(i-1)}, \frac{n}{i}p_{i} \right) & \text{for }i=n-1 ,...,1 \end{cases}$$#### Benjamini-Yekutieli

The Benjamini-Yekutieli method is an extension of the Benjamini-Hochberg adjustment that can also be applied when p-values are dependent [14]. This method always controls the false discovery rate, but is thus quite conservative [1].

$$\gamma = \sum_{i=1}^{n} \frac{1}{i} $$ $$\tilde p_{i}\ =\ \begin{cases} \gamma p_{i} & \text{for } i=n\\ min \left( \tilde p_{(i-1)}, \gamma \frac{n}{i}p_{i} \right) & \text{for }i=n-1 ,...,1 \end{cases}$$### Biblibgraphy

- p-Value Adjustments - SAS/STAT(R) 9.22 User's Guide, (View online)
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