Cochran’s Q TestKristine Joey Palencia

MP-Industrial Psychology

ADVANCED STATISTICS

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

Cochran's Q test is

H0: The treatments are equally effective.Ha: There is a difference in effectiveness among treatments.

Subject(case)

Treatment A

Treatment B

Treatment C

Treatment 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

•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.

Cochran's Q test is based on the following assumptions:

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.

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

Pet Shop Valentine’s Day

July 4 Halloween Christmas

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

Pet Shop

Valentine’s Day

July 4 Halloween Christmas L L2

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

G or Tj = sum for each column

L or Ui = sum for each row or block

k = number of cases/treatment

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

Critical values of

Chi-square

Q = 13.287 X2 = 7.82

Obtain the Q value and critical Chi-square value

If the obtained Q is less than the critical X2, then retain the null hypothesisIf 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 sampleThere were more stores displaying both types of reptiles during Christmas buying season than during the other times of the year

Pet Shop Valentine's Day July 4 Halloween Christmas

N = 12 G1 = 4 G2 = 3 G3 = 2 G4 = 10

Mean % 33% 25% 17% 83%

To validate the null hypothesis:

As hypothesized, there were more stores displaying both types of reptiles during the Christmas buying season than during the other times of the year.

Application on SPSS

Application on XLSTAT (Excel)

Thank you!Thank you!

End of Report