🎰 Sequential Probability Ratio Test (SPRT)

Sequential Probability Ratio Test (SPRT)#

A sequential test that accumulates the likelihood ratio and stops when it crosses preset bounds.

Important

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What it is#

The sequential probability ratio test (SPRT), due to Abraham Wald, is a hypothesis test that evaluates data as it arrives rather than fixing the sample size in advance. It compares two simple hypotheses,

\[H_0 : \theta = \theta_0 \quad \text{vs} \quad H_1 : \theta = \theta_1,\]

and after each observation decides to accept \(H_0\), accept \(H_1\), or keep sampling.

The running likelihood ratio#

After \(n\) observations, accumulate the likelihood ratio

\[\Lambda_n = \frac{L(\text{data}_{1:n} \mid H_1)}{L(\text{data}_{1:n} \mid H_0)}.\]

Decision boundaries#

Two boundaries are set from the desired Type I (\(\alpha\)) and Type II (\(\beta\)) error rates:

\[A = \frac{1 - \beta}{\alpha}, \qquad B = \frac{\beta}{1 - \alpha}.\]

Then stop when \(\Lambda_n \ge A\) (accept \(H_1\)) or \(\Lambda_n \le B\) (accept \(H_0\)); while \(B < \Lambda_n < A\), continue.

Worked example#

For conversion rates \(H_0 : p = 0.10\) vs \(H_1 : p = 0.12\) with \(\alpha = 0.05, \beta = 0.20\),

\[A = \frac{1 - 0.20}{0.05} = 16, \qquad B = \frac{0.20}{0.95} \approx 0.21.\]

Update \(\Lambda_n\) as conversions come in; cross 16 → conclude \(H_1\), drop below 0.21 → conclude \(H_0\), otherwise keep going.

Strengths and limits#

On average the SPRT needs fewer samples than a fixed-horizon test, stops as soon as evidence is decisive, and holds the chosen \(\alpha\) and \(\beta\). The catch: it’s built for simple hypotheses (fixed \(\theta_0, \theta_1\)), is awkward for composite ones (e.g. \(p \le 0.10\)), and needs real-time monitoring. Wald introduced it for WWII quality control (accepting or rejecting munitions lots with fewer inspections); today it appears in clinical trials, sequential A/B testing and industrial QC. Versus group-sequential designs (which look at fixed checkpoints), the SPRT is genuinely continuous.

Theme: Sequential Methods & Bandits · All terminology

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Mind map — connected ideas

Likelihood Ratio (LR) · Bayesian Sequential Testing · Stopping Rules · Sequential Settings · Frequentist · Group Sequential Testing

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