SPRT.default: Wald's Sequential Probability Ratio Test

In SPRT: Wald's Sequential Probability Ratio Test

Description

Perform Wald's Sequential Probability Ratio Test on variables with a Normal, Bernoulli, Exponential or Poisson distribution. Returns an object with support for print and plot methods.

Usage

## Default S3 method:
SPRT(distribution = "bernoulli", type1 = 0.05, type2 = 0.2, h0, h1,
values = NULL, n = NULL, k = NULL)

Arguments

distribution	a character string specifying the distribution. Must be one of "bernoulli" (default), "normal", "exponential" or "poisson".
type1	the type I error. A number between 0 and 1.
type2	the type II error. A number between 0 and 1.
h0	the expected value of the random variable under the null hypothesis.
h1	the expected value of the random variable under the alternative hypothesis.
values	an optional vector containing values of the random variable. A logical vector when distribution is "bernoulli" and a numerical vector otherwise.
n	an optional numerical scalar for the number of observations of the random variable. n is optional and can be used as an alternative to values.
k	an optional numerical scalar for the cumulative sum of the random variable. k is optional and can be used as an alternative to values.

Details

Perform Wald's Sequential Probability Test on a simple hypothesis test of the null against the alternative.

The null hypothesis tested is that the expected value of the random variable is equal to h0. The alternative hypothesis tested is that the expected value of the random variable is equal to h1.

The expected value of the variable is the probability of success of a Bernoulli variable, the mean of a Normal variable, the mean (as well as the variance) of a Poisson variable, and the mean (as well as the standard deviation and scale parameter) of an Exponential distribution.

Optionally, specify values, a vector with observations of the random variable in the order in which they occurred. values is a logical vector of TRUE or FALSE observations in the case of a Bernoulli variable, and a numerical vector otherwise.

Or specify n and k as an alternative to values. n is the number of observations, and k is the cumulative sum of the random variable across observations (or the number of successes in the case of a Bernoulli variable). When values is given, SPRT infers n and k, and removes any NA values in the process.

Value

A list with class "SPRT" containing the following components:

distribution	equal to distribution.
n	equal to n if given, or the length of the values vector otherwise.
k	equal to k if given, or the cumulative sum of the values vector otherwise.
h0	equal to h0.
h1	equal to h1.
wald.A	the natural logarithm of Wald's A boundary (see waldBoundary).
wald.B	the natural logarithm of Wald's B boundary (see waldBoundary).
k.boundaries	a numerical 2x2 matrix with the slope and intercept of H0 and H1 acceptance regions under the null and alternative hypotheses.
llr.coefficients	a numerical vector of the n and k coefficients behind the random variable's log-likelihood ratio.
llr	a numerical scalar of the random variable's log-likelihood ratio.
decision	the outcome of the Sequential Probability Ratio Test. Returns FALSE when llr >= wald.A, TRUE when llr <= wald.B and NA otherwise.
interpretation	a character vector interpreting the outcome of the Sequential Probability Ratio Test. Returns "Accept H1" when decision is FALSE, "Accept H0" when decision is TRUE and "Continue testing" otherwise.
data.llr	a data frame comparing the random variable's log-likelihood ratio against the natural logarithm of Wald's A and B boundaries. The data frame's columns are named n, values, k, wald.B, wald.A and llr. It contains a row for every observation of the random variable. This output is only available when you specify values.
data.sum	a data frame comparing the random variable's cumulative sum, k, against acceptance boundaries for the null and alternative hypotheses. The data frame's columns are named n, values, k, h0 and h1. It contains a row for every observation of the random variable. This output is only available when you specify values.
llr.fn	a function that returns the value of the random variable's log-likelihood ratio for different n and k. This function is a closure; it encapsulates your distribution, type1, type2, h0 and h1 settings.
h0.fn	a function that returns the k acceptance boundary for the null hypothesis for different n. This function is a closure; it encapsulates your distribution, type1, type2, h0 and h1 settings.
h1.fn	a function that returns the k acceptance boundary for the alternative hypothesis for different n. This function is a closure; it encapsulates your distribution, type1, type2, h0 and h1 settings.

Note

This function returns an object with support for print and plot methods.

Author(s)

Stephane Mikael Bottine

References

Ghosh, B.K. and Sen, P.K. (1991). Handbook of Sequential Analysis, Marcel Dekker, New York. Wald, A. (1947). Sequential Analysis, Dover, New York.

See Also

plot.SPRT, print.SPRT, waldBoundary

Examples

# SPRT on a normally distributed random variable
set.seed(123)
test <- SPRT(distribution = "normal", type1 = 0.05, type2 = 0.20,
  h0 = 0, h1 = 1, values = rnorm(10))

# Test outcome
test

# Cumulative sum of the random variable vs H0 and H1 boundaries
test$data.sum
plot(test)

# Sequential log-likelihood ratio vs Wald's A and B constants
test$data.llr
plot(test, log = "y")

# Calculate the log-likelihood ratio across scalars or vectors
set.seed(123)
test$llr.fn(n = 10, sum(rnorm(10)))

set.seed(123)
test$llr.fn(n = seq(1,10,1), k = cumsum(rnorm(10)))

# Calculate H0 and H1 boundaries
test$h0.fn(n = seq(1,10,1))
test$h1.fn(n = seq(1,10,1))

Example output

Wald's Sequential Probability Ratio Test (SPRT)

Decision:
Accept H0

Distribution: normal 
n: 10, k: 0.7462564
h0: 0, h1: 1

Wald boundaries (log):
> B boundary:       -1.558 
> A boundary:        2.773 
> Likelihood ratio:  -4.254 

Preview k boundaries:
 n  values      k      h0   h1
 1 -0.5605 -0.560 -1.0581 3.27
 2 -0.2302 -0.791 -0.5581 3.77
 3  1.5587  0.768 -0.0581 4.27
 4  0.0705  0.839  0.4419 4.77
 5  0.1293  0.968  0.9419 5.27
    n      values          k          h0       h1
1   1 -0.56047565 -0.5604756 -1.05814462 3.272589
2   2 -0.23017749 -0.7906531 -0.55814462 3.772589
3   3  1.55870831  0.7680552 -0.05814462 4.272589
4   4  0.07050839  0.8385636  0.44185538 4.772589
5   5  0.12928774  0.9678513  0.94185538 5.272589
6   6  1.71506499  2.6829163  1.44185538 5.772589
7   7  0.46091621  3.1438325  1.94185538 6.272589
8   8 -1.26506123  1.8787713  2.44185538 6.772589
9   9 -0.68685285  1.1919184  2.94185538 7.272589
10 10 -0.44566197  0.7462564  3.44185538 7.772589
    n      values          k    wald.B   wald.A        llr
1   1 -0.56047565 -0.5604756 -1.558145 2.772589 -1.0604756
2   2 -0.23017749 -0.7906531 -1.558145 2.772589 -1.7906531
3   3  1.55870831  0.7680552 -1.558145 2.772589 -0.7319448
4   4  0.07050839  0.8385636 -1.558145 2.772589 -1.1614364
5   5  0.12928774  0.9678513 -1.558145 2.772589 -1.5321487
6   6  1.71506499  2.6829163 -1.558145 2.772589 -0.3170837
7   7  0.46091621  3.1438325 -1.558145 2.772589 -0.3561675
8   8 -1.26506123  1.8787713 -1.558145 2.772589 -2.1212287
9   9 -0.68685285  1.1919184 -1.558145 2.772589 -3.3080816
10 10 -0.44566197  0.7462564 -1.558145 2.772589 -4.2537436
[1] -4.253744
 [1] -1.0604756 -1.7906531 -0.7319448 -1.1614364 -1.5321487 -0.3170837
 [7] -0.3561675 -2.1212287 -3.3080816 -4.2537436
 [1] -1.05814462 -0.55814462 -0.05814462  0.44185538  0.94185538  1.44185538
 [7]  1.94185538  2.44185538  2.94185538  3.44185538
 [1] 3.272589 3.772589 4.272589 4.772589 5.272589 5.772589 6.272589 6.772589
 [9] 7.272589 7.772589

SPRT documentation built on May 2, 2019, 6:35 a.m.