Thesequential probability ratio test (SPRT) is a specificsequential hypothesis test, developed byAbraham Wald^[1] and later proven to be optimal by Wald andJacob Wolfowitz.^[2]Neyman and Pearson's 1933 result inspired Wald to reformulate it as a sequential analysis problem. The Neyman-Pearson lemma, by contrast, offers arule of thumb for when all the data is collected (and its likelihood ratio known).

While originally developed for use inquality control studies in the realm of manufacturing, SPRT has been formulated for use in the computerized testing of human examinees as a termination criterion.^[3][4][5]

As in classicalhypothesis testing, SPRT starts with a pair of hypotheses, say${\displaystyle H_0}$ and${\displaystyle H_1}$ for thenull hypothesis andalternative hypothesis respectively. They must be specified as follows:

${\displaystyle H_0:p=p_0}$

${\displaystyle H_1:p=p_1}$

The next step is to calculate the cumulative sum of the log-likelihood ratio,${\displaystyle \log {\mathrm{\Lambda }}_i}$, as new data arrive: with${\displaystyle S_0=0}$, then, for${\displaystyle i}$=1,2,...,

${\displaystyle S_i=S_{i-1}+\log {\mathrm{\Lambda }}_i}$

Thestopping rule is a simple thresholding scheme:

• : continue monitoring (critical inequality)
• ${\displaystyle S_i\ge b}$: Accept${\displaystyle H_1}$
• ${\displaystyle S_i\le a}$: Accept${\displaystyle H_0}$

where${\displaystyle a}$ and${\displaystyle b}$ () depend on the desiredtype I and type II errors,${\displaystyle \alpha }$ and${\displaystyle \beta }$. They may be chosen as follows:

${\displaystyle a\approx \log \frac{\beta }{1-\alpha }}$ and${\displaystyle b\approx \log \frac{1-\beta }{\alpha }}$

In other words,${\displaystyle \alpha }$ and${\displaystyle \beta }$ must be decided beforehand in order to set the thresholds appropriately. The numerical value will depend on the application. The reason for being only an approximation is that, in the discrete case, the signal may cross the threshold between samples. Thus, depending on the penalty of making an error and thesampling frequency, one might set the thresholds more aggressively. The exact bounds are correct in the continuous case.

A textbook example isparameter estimation of aprobability distribution function. Consider theexponential distribution:

${\displaystyle f_{\theta }(x)={\theta }^{-1}{e}^{-\frac{x}{\theta }},x,\theta >0}$

The hypotheses are

${\displaystyle \{\begin{array}{l}{H}_0:\theta ={\theta }_0\\ H_1:\theta ={\theta }_1\end{array}{\theta }_1>{\theta }_0.}$

Then the log-likelihood function (LLF) for one sample is

The cumulative sum of the LLFs for allx is

${\displaystyle S_n=\sum _{i=1}^n\log \mathrm{\Lambda }(x_i)=-n\log \left(\frac{{\theta }_1}{{\theta }_0}\right)+\left(\frac{{\theta }_1-{\theta }_0}{{\theta }_0{\theta }_1}\right)\sum _{i=1}^n{x}_i}$

Accordingly, the stopping rule is:

After re-arranging we finally find

The thresholds are simply twoparallel lines withslope${\displaystyle \log ({\theta }_1/{\theta }_0)}$. Sampling should stop when the sum of the samples makes an excursion outside thecontinue-sampling region.

The test is done on the proportion metric, and tests that a variablep is equal to one of two desired points,p₁ orp₂. The region between these two points is known as theindifference region (IR). For example, suppose you are performing a quality control study on a factory lot of widgets. Management would like the lot to have 3% or less defective widgets, but 1% or less is the ideal lot that would pass with flying colors. In this example,p₁ = 0.01 andp₂ = 0.03 and the region between them is the IR because management considers these lots to be marginal and is OK with them being classified either way. Widgets would be sampled one at a time from the lot (sequential analysis) until the test determines, within an acceptable error level, that the lot is ideal or should be rejected.

The SPRT is currently the predominant method of classifying examinees in a variable-lengthcomputerized classification test (CCT)^{[citation needed]}. The two parameters arep₁ andp₂ are specified by determining a cutscore (threshold) for examinees on the proportion correct metric, and selecting a point above and below that cutscore. For instance, suppose the cutscore is set at 70% for a test. We could selectp₁ = 0.65 andp₂ = 0.75 . The test then evaluates the likelihood that an examinee's true score on that metric is equal to one of those two points. If the examinee is determined to be at 75%, they pass, and they fail if they are determined to be at 65%.

These points are not specified completely arbitrarily. A cutscore should always be set with a legally defensible method, such as amodified Angoff procedure. Again, the indifference region represents the region of scores that the test designer is OK with going either way (pass or fail). The upper parameterp₂ is conceptually the highest level that the test designer is willing to accept for a Fail (because everyone below it has a good chance of failing), and the lower parameterp₁ is the lowest level that the test designer is willing to accept for a pass (because everyone above it has a decent chance of passing). While this definition may seem to be a relatively small burden, consider thehigh-stakes case of a licensing test for medical doctors: at just what point should we consider somebody to be at one of these two levels?

While the SPRT was first applied to testing in the days ofclassical test theory, as is applied in the previous paragraph, Reckase (1983) suggested thatitem response theory be used to determine thep₁ andp₂ parameters. The cutscore and indifference region are defined on the latent ability (theta) metric, and translated onto the proportion metric for computation. Research on CCT since then has applied this methodology for several reasons:

1. Large item banks tend to be calibrated with IRT
2. This allows more accurate specification of the parameters
3. By using the item response function for each item, the parameters are easily allowed to vary between items.

Spiegelhalter et al.^[6] have shown that SPRT can be used to monitor the performance of doctors, surgeons and other medical practitioners in such a way as to give early warning of potentially anomalous results. They showed how it could have helped identifyHarold Shipman as a murderer well before he was actually identified.^[6]

More recently, in 2011, an extension of the SPRT method called Maximized Sequential Probability Ratio Test (MaxSPRT)^[7] was introduced. The salient feature of MaxSPRT is the allowance of a composite, one-sided alternative hypothesis, and the introduction of an upper stopping boundary. The method has been used in several medical research studies.^[8]

• CUSUM
• Computerized classification test
• Wald test
• Likelihood-ratio test

• Ghosh, Bhaskar Kumar (1970).Sequential Tests of Statistical Hypotheses. Reading:Addison-Wesley.
• Holger Wilker:Sequential-Statistik in der Praxis, BoD, Norderstedt 2012,ISBN 978-3848232529.

• Wald's Sequential Probability Ratio Test forR by Stéphane Bottine
• Wald's Sequential Probability Ratio Test forPython by Zhenning Yu

• Statistical tests
• Sequential methods
• Mathematical psychology

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