Hypothesis Testing > Sequential Probability Ratio Test

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## What is a Sequential Probability Ratio Test?

A **sequential probability ratio test (SPRT) **is a hypothesis test for sequential samples.

Sequential sampling works in a very non-traditional way; instead of a fixed sample size, you choose one item (or a few) at a time, and then test your hypothesis. You can either:

- Reject the null hypothesis (H
_{0}) in favor of the alternate hypothesis (H_{1}) and stop, - Keep the null hypothesis and stop,
- Fail to reach either conclusion and continue sampling.

If you fail to reach a conclusion, you repeat the sampling and then the hypothesis test. You keep on repeating this process until you have a sound conclusion, so *you don’t know the how big your sample will be until you’re finished testing. *

## About the SPRT

Sequential analysis hypothesis testing generally **enables a researcher to come to a conclusion with a minimum amount of data. **With Wald’s SPRT, the amount of data points required to come to a conclusion can be defined by a random variable, called the **sample number N _{s}**. The boundary of the decision region depends on the expected value of this random variable, called the

**Average Sample Number (ASN)**. The ASN for the SPRT is lower than all other sequential tests and is usually lower than traditional, fixed-size sampling methods.

SPRT is based on the likelihood ratio statistic λ^{n}. Likelihood ratio tests are extremely difficult to perform by hand, and so software is necessary. However, you do need to specify some conditions, including two constants, A and B (where A > B). These define the conditions under which the null hypothesis will be rejected or not:

- Accept H
_{0}: λ^{(n)}≤ B - No conclusion (resample): B ≤ λ
^{(n)}< A - Reject the null hypothesis in favor of H
_{1}: λ^{(n)}≥ A

These three conditions are represented as **decision regions** (accept/no decision/reject) in the above image.

A and B are relatively simple to calculate with the following formulas:

**Where**:

- α is the alpha level for the test,
- Β is the beta level for the test.

## Disadvantages

A major issue with the SPRT is that the optimality of the test only applies to simple hypotheses (e.g. H_{0} < 10; H_{1} ≥ 10).

Unless an upper bound is specified, the ASN can become much larger than the amount of data available. A modified SPRT, called the *Truncated Sequential Probability Ratio Test* (TSPRT), addresses this issue. The test is the same, except a decision is made at a certain maximum sample size. The ASN can also be large if there is a mismatch between the data and the H_{0} and H_{1} models.