1. Definition
- The Sequential Probability Ratio Test (SPRT) is a sequential hypothesis test that evaluates data as it is collected, instead of fixing the sample size in advance (like traditional/fixed-horizon tests).
- At each step, it calculates a likelihood ratio comparing two hypotheses:
$H_0: \theta = \theta_0 \quad \text{vs} \quad H_1: \theta = \theta_1$
- Based on this ratio, the test decides:
- Accept H₀,
- Accept H₁, or
- Continue sampling.
2. Likelihood Ratio
At each observation, compute the cumulative likelihood ratio:
$\Lambda_n = \frac{L(\text{data up to n} \mid H_1)}{L(\text{data up to n} \mid H_0)}$
Where $L$ is the likelihood function.
3. Decision Rules
Two thresholds are set using Type I (α) and Type II (β) error rates:
$A = \frac{1-\beta}{\alpha}, \quad B = \frac{\beta}{1-\alpha}$
- If $\Lambda_n \geq A$ → accept H₁.
- If $\Lambda_n \leq B$ → accept H₀.
- If $B < \Lambda_n < A$ → continue sampling.
This is why it’s called a “ratio test” → the decision is based on comparing likelihood ratios to boundaries.
4. Example
Suppose:
- H₀: Conversion rate p = 0.10
- H₁: Conversion rate p = 0.12
- α = 0.05, β = 0.20
Thresholds:
$A = \frac{1-0.20}{0.05} = 16, \quad B = \frac{0.20}{0.95} \approx 0.21$
- As you observe conversions sequentially, update likelihood ratio $\Lambda_n$.
- If $\Lambda_n > 16$ → conclude H₁ (p = 0.12).
- If $\Lambda_n < 0.21$ → conclude H₀ (p = 0.10).
- Otherwise → keep sampling.
5. Advantages
- Efficiency: On average, requires fewer samples than fixed-horizon tests.
- Early stopping: Can stop as soon as enough evidence accumulates.
- Strong error control: Maintains chosen α and β.
6. Limitations
- Only designed for simple hypotheses (fixed H₀ and H₁ values).
- Not as straightforward for composite hypotheses (e.g., H₀: p ≤ 0.10).
- Requires continuous monitoring and real-time updating → more complex in practice.
7. Applications
- Originally used in WWII quality control (e.g., accept/reject military equipment shipments with fewer samples).
- Today used in:
- Clinical trials (to stop early if treatment is clearly effective/ineffective).
- A/B testing (sequential evaluation of conversion rates).
- Industrial quality control.
8. Comparison with Other Methods
| Method | Sample Size | Decision Timing | Typical Use |
|---|---|---|---|
| Fixed-Horizon Test | Fixed (pre-planned) | At the end only | Classical A/B testing |
| Group Sequential Test | Pre-planned interim looks | At fixed checkpoints | Clinical trials |
| SPRT | Variable (stops as soon as evidence is strong enough) | Continuous | Quality control, sequential A/B tests |
9. Key Takeaway
- The SPRT is a sequential testing method that compares likelihood ratios at every step.
- If the evidence strongly favors H₀ or H₁, you stop early.
- It’s more sample-efficient than fixed-horizon tests, but requires careful planning and monitoring.
In short:
The Sequential Probability Ratio Test (SPRT) is a sequential hypothesis testing method that checks data continuously. At each step, it compares the likelihood ratio to two boundaries (A and B). If the ratio is extreme, it accepts H₀ or H₁; otherwise, it keeps sampling. This often saves sample size while controlling α and β.