Sequential likelihood ratio test

A sequential likelihood ratio test for short SLQT ( English Sequential Probability Ratio Test , for short SPRT or Sequential Likelihood Ratio Test , for short SLRT ), also called sequential plausibility quotient test , is a sequential hypothesis test in statistics . Instead of performing a statistical test with a fixed sample size , after each observation made, based on all the data recorded so far, a test is carried out to determine whether a decision can be made for or against the null hypothesis . If this is not the case, the observation is continued until this decision can be made.

history

The SLQT was developed by A. Wald in the USA in 1942. It was mainly used in the armaments industry, so that a generally accessible publication did not take place until 1947.

definition

The realization of a random variable with the distribution and the unknown parameter is examined . The null hypothesis is tested against the alternative hypothesis . It should be rejected with at most and with at most as probability of error . ${\displaystyle x_{i}}$ ${\displaystyle X}$ ${\displaystyle f(x_{i};\theta )}$ ${\displaystyle \theta }$ ${\displaystyle H_{0}:\theta =\theta _{0}}$ ${\displaystyle H_{1}:\theta =\theta _{1}}$ ${\displaystyle H_{0}}$ ${\displaystyle \alpha }$ ${\displaystyle H_{1}}$ ${\displaystyle \beta }$

For a fixed sample size with the observations , the test statistic is given as a likelihood quotient (quotient of two likelihood functions ) by ${\displaystyle n}$ ${\displaystyle x_{1},x_{2},\ldots ,x_{n}}$

${\displaystyle {\mathfrak {L}}=\prod _{i=1}^{n}{\frac {f(x_{i};\theta _{1})}{f(x_{i};\theta _{0})}}}$

If you now choose decision limits A and B, then the following decision rules apply to the acceptance of the hypotheses:

• Continuation of observation if: ${\displaystyle B<{\mathfrak {L}}<A}$
• Assumption of if: ${\displaystyle H_{1}}$ ${\displaystyle {\mathfrak {L}}\geq A}$
• Assumption of if: ${\displaystyle H_{0}}$ ${\displaystyle {\mathfrak {L}}\leq B}$

The decision limits

The definition of A and B must be designed in such a way that and are adhered to. This is the case if: ${\displaystyle \alpha }$ ${\displaystyle \beta }$

${\displaystyle A={\frac {1-\beta }{\alpha }}}$

${\displaystyle B={\frac {\beta }{1-\alpha }}}$

The probability of reaching or exceeding the lower limit is indicated by the operating characteristics . The probability of accepting the alternative hypothesis and thus exceeding the upper limit is described by the quality function . That is true . ${\displaystyle P_{1}(\theta )}$ ${\displaystyle P_{0}(\theta )}$ ${\displaystyle P_{0}(\theta )+P_{1}(\theta )=1}$

example

The derivation of the SLQT for a 1-sample comparison for binary data is intended as an example.

In a clinical study, a new drug is being tested in a phase II study. The study is to be discontinued as soon as the proportion of patients with kidney failure is ≥ 25% within the first 24 hours. A percentage of 10% is normal and acceptable. The given error probabilities are and . ${\displaystyle \alpha }$ ${\displaystyle \beta }$

After the i th patients are y observations and iy before without kidney failure observations. According to the binomial coefficient is . ${\displaystyle {\mathfrak {L}}={\frac {{\tau _{1}}^{y}(1-\tau _{1})^{i-y}}{{\tau _{0}}^{y}(1-\tau _{0})^{i-y}}}}$

The continuation area of ​​the SLQT is now obtained by taking the logarithm and reshaping:

${\displaystyle {\frac {\ln {({\frac {\beta }{1-\alpha }}})}{\ln {({\frac {\tau _{1}(1-\tau _{0})}{\tau _{0}(1-\tau _{1})}})}}}+{\frac {\ln {({\frac {1-\tau _{0}}{1-\tau _{1}}})}}{\ln {({\frac {\tau _{1}(1-\tau _{0})}{\tau _{0}(1-\tau _{1})}})}}}\cdot i<y<{\frac {\ln {({\frac {1-\beta }{\alpha }}})}{\ln {({\frac {\tau _{1}(1-\tau _{0})}{\tau _{0}(1-\tau _{1})}})}}}+{\frac {\ln {({\frac {1-\tau _{0}}{1-\tau _{1}}})}}{\ln {({\frac {\tau _{1}(1-\tau _{0})}{\tau _{0}(1-\tau _{1})}})}}}\cdot i}$

In , , , arises as a continuation area. ${\displaystyle \tau _{0}=0{,}1}$ ${\displaystyle \tau _{1}=0{,}25}$ ${\displaystyle \alpha =0{,}001}$ ${\displaystyle \beta =0{,}2}$ ${\displaystyle -1{,}46+0{,}1659\cdot i<y<6{,}08+0{,}1659\cdot i}$

literature

• Abraham Wald : Sequential Analysis John Wiley & Sons, New York NY et al. 1947.
• BK Ghosh : Sequential Tests of Statistical Hypotheses. Reading: Addison-Wesley 1970
• Peter Bauer, Viktor Scheiber, Franz X. Wohlzog: Sequential statistical methods. Fischer, Stuttgart et al. 1986, ISBN 3-437-20343-6 .
• Albrecht Irle : Sequential analysis: optimal sequential tests. Stuttgart: Teubner 1990
• Holger Wilker: Sequential Statistics in Practice , BoD, Norderstedt 2012, ISBN 978-3848232529 .