Cochran's Q test is a nonparametric test used to compare dependent categorical variables across three or more conditions. It is an extension of the McNemar test for related samples. The test compares the probability of a target response across different conditions to see if treatments have identical effects. It assumes a large sample size and that blocks were randomly selected. An example uses data from pet stores to test if they are equally likely to display both reptiles (snakes and lizards) during different times of the year. The Cochran's Q test rejects the null hypothesis, finding stores were more likely to display both during Christmas than other times.

1. ADVANCED STATISTICS
Cochran’s Q Test
Kristine Joey Palencia
MP-Industrial Psychology

2. Definition
• A nonparametric procedure for categorical data employed in a
hypothesis testing situation involving a design with k=2 or
more dependent samples
• 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.
• Cochran's Q tests whether the probability of a target response
is equal across all conditions; verify if k treatments have
identical effects

3. Cochran's Q test is
H0: The treatments are equally effective.
Ha: There is a difference in effectiveness
among treatments.
Subject Treatment Treatment Treatment Treatment
(case) A B C D
1 1 1 0 0
2 1 1 0 1
3 1 0 0 0
4 1 1 1 0
5 1 1 0 1
6 1 1 0 1

4. Cochran's Q test is based on the
following assumptions:
•A large sample approximation; in particular, it
assumes that b is "large".
•The blocks (rows) were randomly selected from
the population of all possible blocks.
•The outcomes of the treatments can be coded as
binary responses (i.e., a "0" or "1") in a way that is
common to all treatments within each block.

5. Example
The researcher who had collected the Pet Shop data
wanted to examine whether pet stores displayed different
types of reptiles during different times of the year. So, the
researcher visited each of the 12 stores four times during the
next year that were chosen because of their proximity to
holidays, Valentine’s Day, July 4, Halloween and Christmas.
During each visit, the researcher recorded if the shop
displayed only snakes or lizards (coded = 0) or both types of
reptiles (coded = 1).
In this analysis the one variable is the time of the year and
the response variable is the type of reptile(s) displayed.

6. Data from the 12 stores:
0, 0, 0, 1 0, 0, 0, 1 0, 0, 0, 1 1, 1, 1, 1
1, 0, 0, 1 0, 1, 0, 1 1, 0, 0, 1 0, 0, 0, 1
0, 1, 0, 0 0, 0, 0, 0 1, 0, 0, 1 0, 0, 1, 1
Research Hypothesis:
The researcher hypothesized that pet shops
would be more likely to display both reptiles to
Christmas than during the other times of the year
HO = Stores are equally likely to display both types of
reptiles during all parts of the year

7. Pet Shop Valentine’s July 4 Halloween Christmas
Day
1 0 0 0 1
2 0 0 0 1
3 0 0 0 1
4 1 1 1 1
5 1 0 0 1
6 0 1 0 1
7 1 0 0 1
8 0 0 0 1
9 0 1 0 0
10 0 0 0 0
11 1 0 0 1
12 0 0 1 1
N = 12 G1 = 4 G2 = 3 G3 = 2 G4 = 10
Snakes or lizards = 0 ; snakes and lizards = 1

8. Pet Valentine’s July 4 Halloween Christmas L L2
Shop Day
1 0 0 0 1 1 1
2 0 0 0 1 1 1
3 0 0 0 1 1 1
4 1 1 1 1 4 16
5 1 0 0 1 2 4
6 0 1 0 1 2 4
7 1 0 0 1 2 4
8 0 0 0 1 1 1
9 0 1 0 0 1 1
10 0 0 0 0 0 0
11 1 0 0 1 2 4
12 0 0 1 1 2 4
N = 12 G1 = 4 G2 = 3 G3 = 2 G4 = 10 L= 19 L2 = 41
Number of conditions (k) = 4

9. k = number of cases/treatment
G or Tj = sum for each column
L or Ui = sum for each row or
block

11. Use the Chi-square table to determine the
critical X2 value for df = k – 1 and p = . 05
X2 (df = 3, p = .05) = 7.82

12. Critical
values of
Chi-square

13. Obtain the Q value and critical Chi-square value
Q = 13.287 X2 = 7.82
If the obtained Q is less than the critical X2, then retain the
null hypothesis
If the obtained Q is greater than critical X2, then reject the
null hypothesis
REJECT THE NULL HYPOTHESIS
There is a relationship between the subject’s values on
one categorical variable and their values on the other
categorical variable, in the population represented by the
sample
There were more stores displaying both types of reptiles
during Christmas buying season than during the other times
of the year

14. To validate the null hypothesis:
Pet Shop Valentine's Day July 4 Halloween Christmas
N = 12 G1 = 4 G2 = 3 G3 = 2 G4 = 10
Mean % 33% 25% 17% 83%
As hypothesized, there were more stores
displaying both types of reptiles during the
Christmas buying season than during the
other times of the year.

15. Application on SPSS

16. Application on XLSTAT (Excel)

17. End of Report
Thank you!