Overview
The Sequential Probability Ratio Test (SPRT),
proposed by Abraham Wald (1945), is a method of hypothesis testing that
evaluates data sequentially rather than fixing the
sample size in advance.
It is widely used in quality control, clinical trials, and
agricultural research where early stopping can save both time
and resources.
Theoretical Background
We test simple hypotheses:
\[
H_0: \theta = \theta_0 \quad \text{vs.} \quad H_1: \theta = \theta_1
\]
After \(n\) observations \((X_1, X_2, \ldots, X_n)\), the
likelihood ratio is:
\[
\Lambda_n = \prod_{i=1}^n \frac{f(X_i; \theta_1)}{f(X_i; \theta_0)}
\]
or equivalently,
\[
\log \Lambda_n = \sum_{i=1}^n \log \left( \frac{f(X_i; \theta_1)}{f(X_i;
\theta_0)} \right).
\]
Derivation of Decision Boundaries
To control Type I error (\(\alpha\))
and Type II error (\(\beta\)), Wald
proposed comparing the likelihood ratio \(\Lambda_n\) with two thresholds.
- Accept \(H_0\) if \(\Lambda_n \leq B\)
- Accept \(H_1\) if \(\Lambda_n \geq A\)
- Continue sampling otherwise
where
\[
A = \frac{1-\beta}{\alpha},
\qquad
B = \frac{\beta}{1-\alpha}.
\]
Why these thresholds?
Type I error control: Probability of wrongly
rejecting \(H_0\) should not exceed
\(\alpha\).
This sets the upper boundary \(A\).
Type II error control: Probability of wrongly
rejecting \(H_1\) should not exceed
\(\beta\).
This sets the lower boundary \(B\).
Thus, the SPRT is designed so that:
\[
P(\text{Reject } H_0 | H_0 \text{ true}) \leq \alpha, \qquad
P(\text{Reject } H_1 | H_1 \text{ true}) \leq \beta.
\]
This guarantees the desired error rates in the
sequential framework.
Example 1: Binomial Data
Suppose we want to test whether the probability of success is \(p_0 = 0.1\) vs \(p_1 = 0.3\).
library(SPRT)
# Simulated binary outcomes (1 = success, 0 = failure)
x <- c(0,0,1,0,1,1,1,0,0,1,0,0)
# Run SPRT
res <- sprt(x, alpha = 0.05, beta = 0.1, p0 = 0.1, p1 = 0.3)
# Print results
res
## $decision
## [1] "Reject H0"
##
## $n_decision
## [1] 7
##
## $logL
## [1] -0.2513144 -0.5026289 0.5959834 0.3446690 1.4432813 2.5418936 3.6405059
##
## $A
## [1] 2.890372
##
## $B
## [1] -2.251292
# Plot sequential test path
sprt_plot(res)
# Observations from a Normal distribution
x1 <- c(52, 55, 58, 63, 66, 70, 74)
result1 <- sprt(
x1,
alpha = 0.05,
beta = 0.1,
p0 = 50,
p1 = 65,
dist = "normal",
sigma = 10
)
result1
## $decision
## [1] "Reject H0"
##
## $n_decision
## [1] 7
##
## $logL
## [1] -0.825 -1.200 -1.125 -0.300 0.975 2.850 5.325
##
## $A
## [1] 2.890372
##
## $B
## [1] -2.251292
# Yields from a fertilizer trial (kg/plot)
yield <- c(47, 50, 52, 49, 58, 61, 63, 54, 57)
fert_test <- sprt(
yield,
alpha = 0.05,
beta = 0.1,
p0 = 45,
p1 = 55,
dist = "normal",
sigma = 8
)
fert_test
## $decision
## [1] "Reject H0"
##
## $n_decision
## [1] 7
##
## $logL
## [1] -0.46875 -0.46875 -0.15625 -0.31250 0.93750 2.65625 4.68750
##
## $A
## [1] 2.890372
##
## $B
## [1] -2.251292
