npExact (version 0.2)

npMeanPaired: A test for the mean difference between two bounded random variables given matched pairs.

Description

This test requires that the user knows bounds before gathering the data such that the properties of the data generating process imply that all observations will be within these bounds. The data input consists of pairs of observations, each pair consisting of an observation of each random variable, different pairs being independently generated. No further distributional assumptions are made.

Usage

npMeanPaired(x1, x2, lower = 0, upper = 1, alpha = 0.05,
  alternative = "two.sided", epsilon = 1 * 10^(-6),
  iterations = 5000, max.iterations = 100000)

Arguments

x1, x2

the (non-empty) numerical data vectors which contain the variables to be tested. The first values of the vectors are assumed to be the first matched pair of observations, the second values the second matched pair and so on.

lower, upper

the theoretical lower and upper bounds on the data outcomes known ex-ante before gathering the data.

alpha

the type I error.

alternative

a character string describing the alternative hypothesis, can take values "greater", "less" or "two.sided".

epsilon

the tolerance in terms of probability of the Monte Carlo simulations.

iterations

the number of iterations used, should not be changed if the exact solution should be derived

max.iterations

the maximum number of iterations that should be carried out. This number could be increased to achieve greater accuracy in cases where the difference between the threshold probability and theta is small. Default: 10000

Value

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

method

a character string indicating the name and type of the test that was performed.

data.name

a character string giving the name(s) of the data.

alternative

a character string describing the alternative hypothesis.

estimate

the sample means of the given data.

probrej

numerical estimate of the rejection probability of the randomized test, derived by taking an average of iterations realizations of the rejection probability.

bounds

the lower and upper bounds of the variables.

null.value

the specified hypothesized value of the difference of the variable means.

alpha

the type I error.

theta

the parameter that minimizes the type II error.

pseudoalpha

theta*alpha, this is the level used when calculating the average rejection probability during the iterations.

rejection

logical indicator for whether or not the null hypothesis can be rejected.

iterations

the number of iterations that were performed.

Details

Under alternative = "greater", it is a test of the null hypothesis \(H_0: E(x_1) \le E(x_2)\) against the alternative hypothesis \(H_1: E(x_1) > E(x_2)\).

This test uses the known bounds of the variables to transform the data into [0, 1]. Then a random transformation is used to turn the data into binary-valued variables. On this variables the exact McNemar Test with level pseudoalpha is performed and the result recorded. The random transformation and the test are then repeated iterations times. If the average rejection probability probrej of the iterations is at least theta, then the null hypothesis is rejected. If however probrej is too close to the threshold theta then the number of iterations is increased. The algorithm keeps increasing the number of iterations until the bound on the mistake involved by running these iterations is below epsilon. This error epsilon is incorporated into the overall level alpha in order to maintain that the test is exact.

theta (and a value mu of the difference between the two means in the set of the alternative hypothesis) is found in an optimization procedure. theta and mu are chosen as to maximize the set of data generating processes belonging to the alternative hypothesis that yield type II error probability below 0.5. Please see the cited paper below for further information.

References

Schlag, Karl H. 2008, A New Method for Constructing Exact Tests without Making any Assumptions, Department of Economics and Business Working Paper 1109, Universitat Pompeu Fabra. Available at https://ideas.repec.org/p/upf/upfgen/1109.html.

See Also

https://homepage.univie.ac.at/karl.schlag/statistics.php

Examples

# NOT RUN {
## test whether pain after the surgery is less than before the surgery
data(pain)
npMeanPaired(pain$before, pain$after, lower = 0, upper = 100)

## when the computer was used in the surgery
before_pc <- pain[pain$pc == 1, "before"]
after_pc <- pain[pain$pc == 1, "after"]
npMeanPaired(before_pc, after_pc, lower = 0, upper = 100)

## test whether uncertainty decreased from the first to the second round
data(uncertainty)
npMeanPaired(uncertainty$w1, uncertainty$w2, upper = 60) ## or
with(uncertainty, npMeanPaired(w1, w2, upper = 60))

# }