ADVANCED STATISTICS Cochran’s Q Test Kristine Joey PalenciaMP-Industrial Psychology
Definition• A nonparametric procedure for categorical data employed in a hypothesis testing situation involving a design with k=2 or more dependent samples• Cochrans 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.• Cochrans Q tests whether the probability of a target response is equal across all conditions; verify if k treatments have identical effects
Cochrans Q test isH0: The treatments are equally effective.Ha: There is a difference in effectivenessamong 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
Cochrans Q test is based on the following assumptions:•A large sample approximation; in particular, itassumes that b is "large".•The blocks (rows) were randomly selected fromthe population of all possible blocks.•The outcomes of the treatments can be coded asbinary responses (i.e., a "0" or "1") in a way that iscommon to all treatments within each block.
Example The researcher who had collected the Pet Shop datawanted to examine whether pet stores displayed differenttypes of reptiles during different times of the year. So, theresearcher visited each of the 12 stores four times during thenext year that were chosen because of their proximity toholidays, Valentine’s Day, July 4, Halloween and Christmas.During each visit, the researcher recorded if the shopdisplayed only snakes or lizards (coded = 0) or both types ofreptiles (coded = 1). In this analysis the one variable is the time of the year andthe 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, 1Research Hypothesis: The researcher hypothesized that pet shops would be more likely to display both reptiles to Christmas than during the other times of the yearHO = Stores are equally likely to display both types of reptiles during all parts of the year
Obtain the Q value and critical Chi-square value Q = 13.287 X2 = 7.82If the obtained Q is less than the critical X2, then retain thenull hypothesisIf the obtained Q is greater than critical X2, then reject thenull 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
To validate the null hypothesis:Pet Shop Valentines Day July 4 Halloween ChristmasN = 12 G1 = 4 G2 = 3 G3 = 2 G4 = 10Mean % 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.