cochranQTest

Performs a Cochran Q test for related observations.

Synopsis

cochranQTest (x)

Required Arguments

float x[[]] (Input)
Array of size nObservations × nVariables containing the matrix of dichotomized data. There are nObservations readings of zero or one on each of the nVariables treatments.

Return Value

The p-value, pValue, for the Cochran Q statistic.

Optional Arguments

qStatistic (Output)
Cochran’s Q statistic.

Description

Function cochranQTest computes the Cochran Q test statistic that may be used to determine whether or not M matched sets of responses differ significantly among themselves. The data may be thought of as arising out of a randomized block design in which the outcome variable must be success or failure, coded as 1.0 and 0.0, respectively. Within each block, a multivariate vector of 1’s or 0’s is observed. The hypothesis is that the probability of success within a block does not depend upon the treatment.

Assumptions

  1. The blocks are a random sample from the population of all possible blocks.
  2. The outcome of each treatment is dichotomous.

Hypothesis

The hypothesis being tested may be stated in at least two ways.

  1. \(H_0\) : All treatments have the same effect.

    \(H_1\) : The treatments do not all have the same effect.

  2. Let \(p_{ij}\) denote the probability of outcome 1.0 in block i, treatment j.

    \(H_0 : p_{i1}=p_{i2}=\ldots=p_{ic}\) for each i.

    \(H_1 : p_{ij}\neq p_{ik}\) for some i, and some \(j\neq k\).

    where c (equal to nVariables) is the number of treatments.

The null hypothesis is rejected if Cochrans’s Q statistic is too large.

Remarks

  1. The input data must consist of zeros and ones only. For example, the data may be pass-fail information on nVariables questions asked of nObservations people or the test responses of nObservations individuals to nVariables different conditions.
  2. The resulting statistic is distributed approximately as chi-squared with nVariables − 1 degrees of freedom if nObservations is not too small. nObservations greater than or equal to 5 × nVariables is a conservative recommendation.

Example

The following example is taken from Siegal (1956, p. 164). It measures the responses of 18 women to 3 types of interviews.

from __future__ import print_function
from numpy import *
from pyimsl.stat.cochranQTest import cochranQTest

x = array([
    [0.0, 0.0, 0.0],
    [1.0, 1.0, 0.0],
    [0.0, 1.0, 0.0],
    [0.0, 0.0, 0.0],
    [1.0, 0.0, 0.0],
    [1.0, 1.0, 0.0],
    [1.0, 1.0, 0.0],
    [0.0, 1.0, 0.0],
    [1.0, 0.0, 0.0],
    [0.0, 0.0, 0.0],
    [1.0, 1.0, 1.0],
    [1.0, 1.0, 1.0],
    [1.0, 1.0, 0.0],
    [1.0, 1.0, 0.0],
    [1.0, 1.0, 0.0],
    [1.0, 1.0, 1.0],
    [1.0, 1.0, 0.0],
    [1.0, 1.0, 0.0]])

pq = cochranQTest(x)
print("pq = %9.5f" % pq)

Output

pq =   0.00024

Warning Errors

IMSLS_ALL_0_OR_1 “x” consists of either all ones or all zeros. “q” is set to NaN (not a number). “pq” is set to 1.0.

Fatal Errors

IMSLS_INVALID_X_VALUES “x[#][#]” = #. “x” must consist of zeros and ones only.