Performs Cochran’s *Q* test for differences between related samples (D.A. Murray).

### Options

`PRINT` = string token |
Controls printed output (`test` ); default `test` |
---|---|

`METHOD` = string token |
Form of the test (`exact` , `chisquare` ); default `exac` for small samples, otherwise `chis` |

`GROUPS` = factor |
Defines the groups if there only one variable supplied for the `DATA` |

`STATISTIC` = scalar |
Scalar to save the Q value |

`PROBABILITY` = scalar |
Scalar to save the probability for the Q Test |

`MAXTIME` = scalar |
Defines a limit for the maximum time for calculating the exact test; default * i.e. no limit. |

### Parameter

`DATA` = variates |
List of related samples, or variate containing all the samples (the `GROUPS` option must then be set to indicate the variable recorded in each unit belongs) |
---|

### Description

Cochran’s *Q* test is an extension to the McNemar test for related samples that provides a method for testing for differences between three or more matched sets of frequencies or proportions. The matching samples can be based on *k* characteristics of *N* individuals that are associated with the response. Alternatively *N* individuals may be observed under *k* different treatments or conditions (e.g. different questions or one question at different times).

The data must be supplied as dichotomous variables containing 0 to represent failure (or absence), and 1 to represent success (or presence). The variables can be stored in separate variates and the `DATA`

parameter set to list them all. Alternatively, all the data can be stored in a single variate, and the `GROUPS`

option set to a factor to indicate which variable is recorded in each unit of the variate. (`QCOCHRAN`

then assumes that the individuals are recorded in the same order for each variable.)

In its original form, the test leads to a chi-square test (see the Method section). However, this may be inaccurate when there are small numbers of subjects or samples. Consequently `QCOCHRAN`

also provides an exact probability (based on the exact distribution of *Q* under a permutation model). The form of the test can be set to either chi-square or exact by using the `METHOD`

option. The default is to use the exact test if the number of values in the samples is less than 4 and the product of this value with the number of samples is less than 24, otherwise the chi-square method is used. The *Q* statistic can be saved using the `STATISTIC`

parameter, and the probability can be saved using the `PROBABILITY`

parameter.

Although `QCOCHRAN`

uses an efficient algorithm for calculating the exact probability, the time and memory required for this calculation can become impractible as the number of samples and values increases. Therefore, for large problems, the chi-square approximation should be used. However, for the exact calculation the `MAXTIME`

option can be used to supply the maximum amount of time (in seconds) that will be used to calculate the exact probabilty. If this is time exceeded, the computation is terminated.

The `PRINT`

option controls printed output, with settings:

`test` |
the Q value and probability (the default). |
---|

Options: `PRINT`

, `METHOD`

, `GROUPS`

, `STATISTIC`

, `PROBABILITY`

, `MAXTIME`

.

Parameter: `DATA`

.

### Method

The Cochran *Q* Test is calculated by:

*Q* = (*k* × (*k* – 1) × ∑_{j=1…k}{(*T _{j}* –

*Tbar*)

^{2}} ) / (

*k*× ∑

_{j=1…k}{

*u*} – ∑

_{i}_{j=1…k}{

*u*} )

_{i}^{2}where *k* is the number of samples, *T _{j}* is the sum of 1’s in the jth column,

*Tbar*is the mean of the

*T*‘s, and

_{j}*u*is the number of 1’s in the

_{i}*i*th row. Under the null hypothesis this has an approximate chi-square distribution with (

*k*-1) degrees of freedom.

The exact test is calculated using the permutation method of Patil (1975).

### Action with `RESTRICT`

If a parameter is restricted the statistics will be calculated using only those units included in the restriction.

### References

Patil K,D. (1975). Cochran’s Q Test: exact distribution. *Journal of the American Statistical Association*, 70, 186-189.

Siegel S. (1956). *Nonparametric Statistics for the Behavioural Sciences*. McGraw-Hill, New York.

### See also

Procedures: `CATRENDTEST`

, `MCNEMAR`

.

Commands for: Basic and nonparametric statistics.

### Example

CAPTION 'QCOCHRAN Example',\ !t('Responses by housewives under 3 types of interview.',\ 'Data from Siegel (1956), Nonparametric Statistics for',\ 'the Behavioural Sciences, page 164.'); STYLE=meta,plain VARIATE [VALUES=0,1,0,0,1,1,1,0,1,0,1,1,1,1,1,1,1,1] Response1 & [VALUES=0,1,1,0,0,1,1,1,0,0,1,1,1,1,1,1,1,1] Response2 & [VALUES=0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,1,0,0] Response3 QCOCHRAN Response1,Response2,Response3