Benjamin N Hochberg Test How to Calculate

The Benjamin N Hochberg test is a step-down procedure for controlling the false discovery rate (FDR) in multiple hypothesis testing. It provides a way to identify which hypotheses are likely true while controlling the expected proportion of false discoveries.

What is the Hochberg Test?

The Hochberg test is a multiple testing procedure that controls the false discovery rate (FDR), which is the expected proportion of false discoveries among the rejected hypotheses. It's particularly useful when dealing with many hypotheses where the family-wise error rate (FWER) control might be too conservative.

Key Concepts

False Discovery Rate (FDR): The expected proportion of false discoveries among the rejected hypotheses.
Step-down procedure: The test examines hypotheses in order of their p-values, starting with the most significant.
Critical values: The test uses critical values based on the desired FDR level and the number of hypotheses.

The Hochberg test is less conservative than the Bonferroni correction and more powerful than the Holm procedure for controlling the FDR.

How to Calculate the Hochberg Test

The Hochberg test involves the following steps:

Order all p-values from smallest to largest.
Calculate the critical values using the formula:

Critical value for hypothesis i = (i/m) × α

Where: i = hypothesis rank, m = total number of hypotheses, α = desired FDR level
Compare each p-value to its corresponding critical value.
Reject all hypotheses with p-values less than or equal to their critical values.

Assumptions

The test assumes that the p-values are independent.
The hypotheses are exchangeable (no ordering is imposed).
The test controls the FDR but not the FWER.

Example Calculation

Consider testing 5 hypotheses with the following p-values: 0.01, 0.02, 0.03, 0.04, 0.05. We want to control the FDR at 0.05.

Hypothesis	P-value	Rank	Critical Value	Decision
H1	0.01	1	0.01	Reject
H2	0.02	2	0.02	Reject
H3	0.03	3	0.03	Reject
H4	0.04	4	0.04	Reject
H5	0.05	5	0.05	Do not reject

In this example, all hypotheses except H5 would be rejected at the 0.05 FDR level.

Interpreting Results

The Hochberg test provides several key pieces of information:

Rejected hypotheses: These are the hypotheses that are declared significant based on the FDR control.
Non-rejected hypotheses: These are the hypotheses that do not meet the significance threshold.
FDR estimate: The proportion of false discoveries among the rejected hypotheses.

Remember that the Hochberg test controls the FDR, not the FWER. This means that while the expected proportion of false discoveries is controlled, there may still be some false positives among the rejected hypotheses.

FAQ

What is the difference between the Hochberg test and the Bonferroni correction?

The Hochberg test controls the false discovery rate (FDR), which is the expected proportion of false discoveries among the rejected hypotheses. The Bonferroni correction controls the family-wise error rate (FWER), which is the probability of making at least one false discovery. The Hochberg test is less conservative and more powerful than the Bonferroni correction.

When should I use the Hochberg test instead of the Holm procedure?

The Hochberg test is generally more powerful than the Holm procedure for controlling the FDR. However, the Holm procedure is more conservative and may be preferred in situations where the FDR control is not critical.

Can the Hochberg test be used with dependent p-values?

The Hochberg test assumes that the p-values are independent. If the p-values are dependent, the test may not control the FDR properly. In such cases, alternative methods that account for dependence should be considered.