Cochran's Q test is appropriate for random-blocks or repeated-measures data where the values are dichotomous (represented by zero or any non-zero numbers). The test computes a statistic Q which is compared to a critical value from the Chi-squared distribution. The Chi-squared distribution is an asymptotic approximation for the exact null distribution of Q. The null hypothesis (H0) is that the proportion of non-zero members is the same for all columns.

Consider the following data contained in wave1:

 Group1 Group2 Group3 Group4 Group5 Block1 0 0 0 0 1 Block2 0 0 0 0 1 Block3 0 0 0 1 1 Block4 0 1 0 1 1 Block5 0 1 0 1 0 Block6 1 1 0 1 0 Block7 1 1 0 0 0 Block8 1 0 0 0 0 Block9 1 0 0 0 0

The test statistic is smaller than the critical value

In this case there are 5 columns and 9 rows with a total of 45 entries which implies that the Chi-square approximation can be used (greater than 4 columns and 24 total entries). To run the test execute the following command:

`StatsCochranTest/T=1/Q  wave1`

The results are given in the Cochran Test table:

 Rows Columns Statistic Critical Conclusion P-value 9 5 5.56522 9.48773 1 0.234056

It may be somewhat perplexing that despite the fact that Group3 column is identically zero, the test does not suggest rejection of the null hypothesis where the proportion of "ON" blocks is the same across all groups.

Keeping Group3 column fixed, you can increase the proportion of "ON" blocks in all other columns until the Q value exceeds the critical value. This is done in wave2 shown below.

 Group1 Group2 Group3 Group4 Group5 Block1 0 0 0 0 1 Block2 0 0 0 0 1 Block3 0 0 0 1 1 Block4 0 1 0 1 1 Block5 1 1 0 1 1 Block6 1 1 0 1 1 Block7 1 1 0 1 0 Block8 1 1 0 0 0 Block9 1 0 0 0 0

To run the test on these data execute the following command:

`StatsCochranTest/T=1/Q  wave2`

The results are given in the Cochran Test table:

 Rows Columns Statistic Critical Conclusion P-value 9 5 10.3636 9.48773 0 0.0347281

As expected the test statistic has increased above the critical value and so the H0 must be rejected, i.e., the proportion of non-zero entries in the different columns is not the same.

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