The IMSL_COCHRANQ function performs a Cochran Q test for related observations.

This routine requires an IDL Advanced Math and Stats license. For more information, contact your sales or technical support representative.

The IMSL_COCHRANQ function 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 of 0’s is observed. The hypothesis is that the probability of success within a block does not depend upon the treatment.

Assumptions

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

Hypothesis

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

1. H₀ : All treatments have the same effect.

   H₁ : 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.

   H₀:p_i1 = p_i2 = ... = pic for each i.

   H₁:p_ij ≠ p_ik for some i, and some j ≠ k.

   where c (equal to N_ELEMENTS(x(0, *))) is the number of treatments.

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

Remarks

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1. The input data must consist of zeros and ones only. For example, let n_variables = N_ELEMENTS(x(0, *)) and n_observations = N_ELEMENTS(x(*, 0)), then the data may be pass-fail information on n_variables questions asked of n_observations people or the test responses of n_observations individuals to n_variables different conditions.
2. The resulting statistic is distributed approximately as chi-squared with n_variables − 1 degrees of freedom if n_observations is not too small. N_observations greater than or equal to 5 x n_variables is a conservative recommendation.

Example

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

x = TRANSPOSE([[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	=	IMSL_COCHRANQ(x)

PRINT, 'pq =', pq

IDL prints:

pq =	0.000240266

Errors

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Warning Errors

STAT_ALL_0_OR_1: “x” consists of either all ones or all zeros. “q” is set to NaN (not a number). “Result” is set to 1.0.

Fatal Errors

STAT_INVALID_X_VALUES: “x(#, #)” = #. “x” must consist of zeros and ones only.

Syntax

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Result = IMSL_COCHRANQ(X [, /DOUBLE] [, Q=variable])

Return Value

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The p-value for the Cochran Q statistic.

Arguments

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X

Two-dimensional array containing the matrix of dichotomized data.

Keywords

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DOUBLE (optional)

If present and nonzero, then double precision is used.

Q (optional)

Named variable into which the Cochran’s Q statistic is stored.

Version History

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6.4	Introduced