The Cochran Q test is a non-parametric analysis method designed to evaluate whether there are significant differences between two or more dependent samples originating from populations with identical or different distributions. This test can be considered an extension of the McNemar Test, which is specifically used for two dependent samples. When the assumption of normality is not met in paired data analysis, the Cochran Q test serves as a suitable alternative to the Paired T-test.
Case Example
Taken from: Handbook of Parametric and Nonparametric Statistical Procedures, Fifth Edition By David J. Sheskin
Example 26.1
A market researcher asks 12 female subjects whether they would buy cars produced by three different companies. Specifically, subjects were asked if they would purchase cars produced by the following car manufacturers: Chenesco, Howasaki, and Gemini. Here are the responses from 12 subjects: Subject 1 said she would buy Chenesco and Howasaki but not Gemini; Subject 2 said she would only buy Howasaki; Subject 3 said she would buy all three car brands; Subject 4 said she would only buy Howasaki; Subject 5 said she would only buy Howasaki; Subject 6 said she would buy Howasaki and Gemini but not Chenesco; Subject 7 said she would not buy any car; Subject 8 said she would only buy Howasaki; Subject 9 said she would buy Chenesco and Howasaki but not Gemini; Subject 10 said she would only buy Howasaki; Subject 11 said she would not buy any car; and Subject 12 said she would only buy Gemini. Can the market researcher conclude that there is a difference in car preferences based on subject responses?
Analysis Steps
Here are the steps for the Cochran Q test and PostHoc test using SmartstatXL, Excel Add-in:
- Activate the worksheet (Sheet) to be analyzed.
- Place the cursor on the dataset (to create a dataset, see the Data Preparation method). The dataset can be arranged in two layouts:
- Grouped by level (comparisons between levels)
- Grouped by variable (comparisons between variables)
- If the active cell is not on the dataset, SmartstatXL will automatically try to determine the dataset.
- Activate the SmartstatXL Tab
- Click the Non-Parametric Menu. SmartstatXL will display a dialog box to ensure whether the dataset is correct or not (usually the dataset cell address is automatically selected correctly).
- If correct, click the Next Button
- The following Non-Parametric Test Dialog Box will then appear:
- If the data layout used is based on variable comparisons, the following dialog box will appear:
- Next, press the "OK" button
Analysis Results
The following is the Analysis Output for the Cochran Q Test:
Statistical Summary
From the analysis using the Cochran Q Test, the research objective is to determine whether there is a difference in car purchase preferences between the brands Chenesco, Gemini, and Howasaki based on the responses from 12 female subjects.
Based on the analysis results, the obtained Q statistic value is 8.000 with degrees of freedom (df) of 2. The p-value obtained from the test is 0.018, which is less than the significance level \( \alpha = 0.05 \) (p < 0.05).
Therefore, we can reject the null hypothesis (H0) stating that there is no preference difference between the three car brands. This means there is a significant difference in car purchase preferences between at least two of the three brands studied, at a significance level of 5%.
In other words, the responses from the subjects indicate that there is a car brand that is more or less preferred compared to the other brands among the three options provided.
Average Difference Comparison:
From the analysis of average value comparison, we can see the subjects' preferences towards the three car brands.
- Chenesco: Out of 12 subjects, on average 0.25 (or 25%) stated that they would buy a Chenesco brand car. This falls under subset 'a', indicating that the preference level for Chenesco is the lowest or at least equal to other brands that are also in the same subset.
- Gemini: Like Chenesco, 25% of subjects stated they would buy a Gemini brand car. This also falls under subset 'a'.
- Howasaki: Out of 12 subjects, on average 0.75 (or 75%) stated that they would buy a Howasaki brand car. This falls under subset 'b', indicating that the preference level for Howasaki is the highest among the three brands.
The Critical Distance (CD-Q) is 0.49. This is the minimum difference between two proportions required to state that the difference is significant. In this case, the difference between Howasaki (0.75) and Chenesco (0.25) and Gemini (0.25) is 0.50, which is greater than the critical distance. This further confirms that the preference for Howasaki is significantly higher compared to the other two brands.
Therefore, based on the provided data, subjects have a higher preference for purchasing cars of the Howasaki brand compared to Chenesco or Gemini.
