How to interpret standardized residuals (chi square independence analysis)?

Question

Firstly, when and why is it appropriate to use standardized residuals to accept either the null hypothesis or alternative hypothesis? When is not appropriate?

Secondly, how can standardized residuals be interpreted in terms of the hypotheses (whether the two variables are associated or not)? What do the standardized residuals indicate about whether there is an association or not?

Thank you.

Answer 1

When and Why to Use Standardized Residuals

Appropriate Use:

1. After Conducting Chi-Square Test for Independence:
2. Purpose: To determine if there is a significant association between two categorical variables.
3. Null Hypothesis (H₀): The two variables are independent.
4. Alternative Hypothesis (H_A): The two variables are associated.
5. To Understand Specific Cell Contributions:
6. Purpose: Standardized residuals help identify which specific cells in a contingency table contribute most to the overall chi-square statistic.
7. When the Overall Test is Significant: If the chi-square test indicates a significant association, standardized residuals can pinpoint where the significant differences lie.

Inappropriate Use:

1. Without a Significant Chi-Square Test:
2. If the overall chi-square test is not significant, examining standardized residuals is not meaningful as it indicates there is no significant association to investigate further.
3. For Continuous Data:
4. Chi-square tests and their residuals are designed for categorical data. They are not suitable for continuous variables without appropriate categorization.

Interpreting Standardized Residuals

Interpretation in Terms of Hypotheses:

1. Null Hypothesis (H₀​): The standardized residuals should be close to zero, indicating the observed frequencies are close to the expected frequencies under independence.
2. Alternative Hypothesis (H_A​): Large absolute values of standardized residuals suggest deviations from what would be expected under the null hypothesis, indicating an association between the variables.

Magnitude of Standardized Residuals (note that these are just rules of thumb):

1. Residuals ~ 0: Suggest the observed count is close to the expected count, supporting the null hypothesis of independence.
2. Residuals ±2: Indicate a mild deviation from the expected count, suggesting a possible but not strong association.
3. Residuals ±3 or more: Indicate a strong deviation from the expected count, suggesting a significant association between the variables.

In summary, standardized residuals are a tool for understanding the specific contributions of cells in a contingency table to the overall chi-square statistic. They provide insights into which cells show significant deviations from expected counts, helping to clarify the nature of the association between categorical variables.

Hope this was helpful.

Answer 2

Hey Ferena!

I'll be taking a more hands-on approach here with some analogies and examples to add to the above explanation.

Appropriate Use:

1. Purpose: Imagine you want to find out if there's a relationship between two variables, like gender (male/female) and preference for a new product (like/dislike). You use a chi-square test for this.
2. When: After the chi-square test tells you there’s a significant relationship, you look at standardized residuals to see which specific combinations (like males liking the product) are contributing to this relationship.
3. Why: The chi-square test gives a big picture (is there a relationship?), while standardized residuals give the details (where exactly is the relationship?).

Inappropriate Use:

1. If the chi-square test says there’s no significant relationship, then looking at standardized residuals won’t be helpful or meaningful.

Interpreting Standardized Residuals

Interpreting Residuals:

1. Think of standardized residuals as "surprise factors" for each cell in your table. They tell you how surprising (different) the actual counts are compared to what you’d expect if there was no relationship.

Example and Analogy:

1. Chi-Square Test Analogy: Imagine you’re at a party and you want to know if people’s favorite drinks (like soda or juice) depend on their age (kids or adults).
2. Step 1: You check the overall vibe (chi-square test) and find that yes, there’s something going on with drink preferences and age.
3. Step 2: You then talk to specific groups (standardized residuals) to see where the surprises are. Maybe you find out that way more kids prefer soda than you expected, and fewer adults prefer juice than you thought.

What Standardized Residuals Tell You:

1. Big Positive Residual: If a cell has a big positive residual, like kids preferring soda way more than expected, it means there's a strong, surprising preference in that group.
2. Big Negative Residual: If a cell has a big negative residual, like adults preferring juice way less than expected, it means there's a strong, surprising lack of preference in that group.

Thresholds for Surprise:

1. Residuals larger than about 2 (positive or negative) are considered surprising. This means the observed counts are significantly different from the expected counts.