Decoding Chi-Square Test Results: A Comprehensive Guide

Key Highlights of Chi-Square Test Interpretation

P-value: The p-value helps determine the statistical significance of the test. A p-value less than the significance level (typically 0.05) suggests rejecting the null hypothesis, indicating a significant association between the variables.
Degrees of Freedom (df): Degrees of freedom influence the chi-square distribution and are calculated based on the number of categories in the variables being analyzed.
Expected vs. Observed Counts: Comparing observed and expected counts helps identify which categories contribute most to the chi-square statistic, indicating potential dependencies between variables.

Understanding the Chi-Square Test

The Chi-Square test is a statistical method used to determine if there is a significant association between two categorical variables. It compares the observed frequencies of categories with the frequencies one would expect if there were no association between the variables. Essentially, it assesses how well the observed data fits with the expected data under the assumption of independence.

The chi-square (Χ²) statistic is a test that measures how a model compares to actual observed data. The data used in calculating a chi-square statistic must be random, raw, mutually exclusive, drawn from independent variables, and drawn from a large enough sample. The Chi-Square test is suitable for analyzing categorical data and examining relationships or differences between variables. The chi-square test is a statistical test that assesses the independence or goodness of fit between observed and expected frequencies in categorical data.

The Core Idea Behind the Test

At its heart, the chi-square test compares observed values in a dataset to expected values, which represent what would be seen if the null hypothesis is true. The null hypothesis typically states that there is no association between the variables being studied. By comparing these values, the test determines whether any difference between the observed and expected values is due to random chance or a genuine relationship between the variables.

An example of a data chart useful for performing statistical analysis.

Types of Chi-Square Tests

There are two primary types of Chi-square tests:

Chi-Square Test of Independence: This test is used to determine if there is a statistically significant relationship between two categorical variables. For example, it can be used to assess whether there is an association between smoking behavior and gender.
Chi-Square Goodness of Fit Test: This test is used to determine if the observed distribution of a single categorical variable differs from a hypothesized distribution. For example, it can be used to test if the distribution of jelly bean colors in a bin matches a pre-defined expected distribution.

Both tests involve variables that divide data into categories. The test statistic involves finding the squared difference between actual and expected data values, and dividing that difference by the expected data values.

Interpreting the Provided Chi-Square Test Results

The results you've provided include several key components: the Chi-Square statistic, degrees of freedom (df), the significance level (Asymp. Sig. or p-value), and some warnings about expected cell counts. Let's break down each of these elements to provide a clear interpretation.

Understanding the Output Table

Here is the table you provided:

Test	Value	df	Asymp. Sig. (2-sided)
Pearson Chi-Square	380.000	361	0.236
Likelihood Ratio	119.829	361	1.000
Linear-by-Linear Association	18.851	1	0.000
N of Valid Cases	20

Key Components and Their Interpretation

1. Pearson Chi-Square

Value: 380.000
df (degrees of freedom): 361
Asymp. Sig. (2-sided) or p-value: 0.236

The Pearson Chi-Square test is used to determine if there is a statistically significant difference between the expected frequencies and the observed frequencies in one or more categories of a contingency table. In this case, the chi-square statistic is 380, with 361 degrees of freedom, yielding a p-value of 0.236. A p-value of 0.236 is greater than the typical significance level of 0.05, suggesting that you fail to reject the null hypothesis. This means there is not enough evidence to conclude that there is a significant association between the variables being tested.

2. Likelihood Ratio

Value: 119.829
df (degrees of freedom): 361
Asymp. Sig. (2-sided) or p-value: 1.000

The Likelihood Ratio test is another test for independence between categorical variables, often used when sample sizes are small. Here, the chi-square statistic is 119.829, with 361 degrees of freedom, resulting in a p-value of 1.000. A p-value of 1.000 indicates that there is no evidence to reject the null hypothesis. This extremely high p-value strongly suggests that the variables are independent.

3. Linear-by-Linear Association

Value: 18.851
df (degrees of freedom): 1
Asymp. Sig. (2-sided) or p-value: 0.000

The Linear-by-Linear Association test is used when both variables are ordinal to test for a linear trend between them. In this instance, the chi-square statistic is 18.851, with 1 degree of freedom, and a p-value of 0.000. A p-value of 0.000 is less than the typical significance level of 0.05, leading to the rejection of the null hypothesis. This indicates that there is a significant linear association between the two ordinal variables.

4. N of Valid Cases

Value: 20

This indicates that the analysis is based on only 20 valid cases. This is a small sample size, which can affect the reliability of the chi-square test results.

5. Warning about Expected Cell Counts

The warning "400 cells (100.0%) have expected count less than 5. The minimum expected count is .05" is critical. The Chi-Square test is approximate and becomes more accurate as the counts in the cells of the table get larger. A general rule is that chi-square tests should be used only when the expected cell counts are 5 or more in at least 80% of the cells. When expected cell counts are low, the test may not be valid. In this case, all cells have expected counts less than 5, which raises serious concerns about the validity of the Pearson Chi-Square and Likelihood Ratio test results.

Practical Implications of Low Expected Counts

Having low expected cell counts can lead to an overestimation of the chi-square statistic, potentially resulting in a Type I error (incorrectly rejecting the null hypothesis). The chi-square test is an approximate method that becomes more accurate as the counts in the cells of the table get larger. Therefore, it is important to ensure that the expected counts are sufficiently large to trust the results.

Addressing the Issue of Low Expected Cell Counts

Given the warning about low expected cell counts, here are several strategies to consider:

Collect More Data: Increasing the sample size can increase the expected cell counts.
Combine Categories: If it is meaningful to do so, combine categories to increase the expected cell counts in the remaining categories.
Use an Alternative Test: Consider using Fisher's exact test, which is more appropriate for small samples and can provide more accurate results when expected cell counts are low.

This video explains the basics of the Chi-Square test. It provides a clear explanation of how the test works, including the hypotheses, the test statistic, and the interpretation of the results.

Summary of Findings

Based on the provided output:

The Pearson Chi-Square test and the Likelihood Ratio test results are likely unreliable due to the extremely low expected cell counts.
The Linear-by-Linear Association test suggests a significant linear association between the ordinal variables, but this should be interpreted cautiously given the small sample size.
The small sample size (N=20) and the warning about low expected cell counts indicate that the results should be interpreted with caution, and further investigation with a larger sample or alternative statistical methods may be necessary.

In summary, while the Linear-by-Linear Association test indicates a significant linear trend, the overall chi-square analysis is compromised by the small sample size and low expected cell counts. It is essential to address these issues before drawing firm conclusions about the relationships between the variables.

FAQ

What is a Chi-Square test used for?

A Chi-Square test is used to determine whether there is a statistically significant association between categorical variables. It compares the observed frequencies with the expected frequencies under the assumption of no association.

What does the p-value tell you in a Chi-Square test?

The p-value indicates the probability of obtaining test results as extreme as, or more extreme than, the results actually observed, assuming that the null hypothesis is correct. A small p-value (typically ≤ 0.05) suggests that the null hypothesis should be rejected.

What are degrees of freedom (df) and how are they calculated?

Degrees of freedom (df) refer to the number of independent pieces of information used to calculate a statistic. For a Chi-Square test of independence, the degrees of freedom are calculated as (number of rows - 1) * (number of columns - 1).

What is the significance level (alpha)?

The significance level (alpha), often set at 0.05, is the threshold used to determine whether the p-value is small enough to reject the null hypothesis. If the p-value is less than or equal to alpha, the null hypothesis is rejected.

What does it mean if the expected cell counts are too low?

If the expected cell counts are too low (typically less than 5), the Chi-Square test results may not be reliable. Low expected cell counts can lead to an overestimation of the Chi-Square statistic and inaccurate p-values. In such cases, alternative tests like Fisher's exact test or combining categories may be more appropriate.