Cochran's test for related observations
We often have treatments or tests that result in one of two possible outcomes eg. yes/no; success/failure.
The Cochran test is used where every test or treatment is applied independently to every block (or subject) and the results are compared with each other. Thus each of "c" treatments is applied independetly to each of "r" blocks and the result is recorded as either a 1 or a zero. The results are presented in the form of a table with "r" rows and "c" columns.
This is called a "randomised complete block design".
The important assumptions are that each subject or block is selected randomly from the population, and that treatment outcomes can by presented as either 1 or zero. The test assumes a large sample of blocks, so that the null distribution is the chi-squared distribution with c-1 degrees of freedom. Hypotheses: Ho: The treatments are equally effective. H1: There is a difference in effectiveness between treatments. The test statistic "T" is computed and the Ho hypothesis is rejected if T is greater than the (1 - level of significance) quantile of the chi-squared distribution with c-1 degrees of freedom. "P" is the probability of a chi-squared random variable with c-1 degrees of freedom exceeding the observed value of "T".
Script operation
This tool operates in much the same way as most of the others. The data set can include the column titles for the calculation although they are discarded by the script.
Click here for information about general script usage.
In the following table we have 3 columns representing the outcome of the answers of three persons to a randomly selected series of questions. They received a "1" for the correct answer; if wrong, they received a "0".
Outcome of Answers Person A B C 1 1 1 1 1 1 0 1 0 1 1 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 1 0 1 0 1 1 1 1 1 1
Here is the script output.
Cochrans Test for Related Observations Cochrans T: 2.8 df(v): 2 P(Chi>T): 0.24659701 Chi-Critical (95%): 5.9915 Chi-Critical (99%): 9.2103
Interpretation
In this example, Ho is accepted as the "T" value of 2.8 is less than the .05 (95%) critical region of rejection (with 2 degrees of freedom).