Sequential Probability Ratio Test
Instead of using a fixed sample size to perform a statistical test, is tested in the Sequential Probability Ratio Test ( SPRT ) after each observation made based on all data recorded so far, whether a decision can be made for or against the null hypothesis. If this is not the case, the observation can be continued until the decision made .
History
It was developed by A. Wald SPRT 1942 in the United States. Application found it primarily in the defense industry, so that a generally available publication took place only in 1947.
Definition
Examines the realization of a random variable with the distribution and the unknown parameters. It is tested the null hypothesis against the alternative hypothesis. It should be rejected with a maximum and with at most as probability of error.
The SPRT is defined by the likelihood ratio as a test statistic and decision boundaries A and B.
For the test statistic following decision rules:
- Continuing to monitor if:
- Acceptance of, if:
- Acceptance of, if:
The decision boundaries
The definition of A and B must be designed in such a way that will and respected. This is the case if:
To pass to reach the lower limit or the probability is given by the operating characteristic function. The probability to accept the alternative hypothesis, thus exceeding the upper limit is described by the power function. Where the.
Example
As an example, the derivation of the SPRT for a 1 - sample comparison with binary data is intended.
In a clinical study, a new drug is being tested in a phase II study. The study will be terminated as soon as the proportion of patients with renal failure within the first 24 hours ≥ 25%. A share of 10% is normal and acceptable. The specified error probabilities are and.
After the i-th patient, are y observations with observations before and iy without renal failure. According to the binomial coefficient is.
The return portion of the SPRT is now obtained by taking the logarithm and reshape:
For arises as a continuation area.