Standardized residuals are the residuals from a statistical model that have been standardized, or transformed, to have a mean of 0 and a standard deviation of 1. This standardization allows for easier interpretation and comparison of the magnitude of the residuals across different models or variables.
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Standardized residuals are used to identify outliers in a dataset, as values greater than 2 or 3 in absolute value are considered potential outliers.
In the context of the test of independence (Section 11.3), standardized residuals can be used to identify cells in a contingency table that contribute the most to the overall chi-square statistic.
For the test of homogeneity (Section 11.4), standardized residuals can help determine which categories or groups are driving any significant differences found in the test.
Standardized residuals follow a standard normal distribution (mean 0, standard deviation 1) under the null hypothesis, which allows for the use of z-scores and p-values in their interpretation.
Examining the pattern and magnitude of standardized residuals can provide insights into the underlying assumptions and fit of the statistical model being used.
Review Questions
Explain how standardized residuals can be used to assess the goodness of fit of a statistical model.
Standardized residuals are a useful tool for evaluating the fit of a statistical model because they provide a standardized measure of the difference between the observed and predicted values. By transforming the raw residuals to have a mean of 0 and a standard deviation of 1, standardized residuals allow for easier identification of outliers and patterns in the data. Residuals with large absolute values (greater than 2 or 3) indicate poor model fit, as they suggest the model is not accurately capturing the relationship between the variables. Analyzing the distribution and trends in the standardized residuals can help identify violations of model assumptions, such as non-normality or heteroscedasticity, and guide further model refinement.
Describe how standardized residuals can be used in the context of the test of independence (Section 11.3).
In the test of independence, standardized residuals can be used to identify the specific cells in a contingency table that are contributing the most to the overall chi-square statistic. By examining the standardized residuals for each cell, you can determine which cells have observed counts that deviate significantly from the expected counts under the null hypothesis of independence. Cells with large positive or negative standardized residuals (typically greater than 2 or 3 in absolute value) indicate that the observed and expected counts in that cell differ substantially, suggesting a lack of independence between the variables. This information can help you pinpoint the specific areas of the contingency table driving any significant results in the overall test of independence.
Explain how standardized residuals can provide insights into the results of the test for homogeneity (Section 11.4).
In the test for homogeneity, standardized residuals can be used to identify the specific categories or groups that are contributing the most to any significant differences found in the test. By examining the standardized residuals for each category, you can determine which categories have observed counts that deviate significantly from the expected counts under the null hypothesis of homogeneity. Categories with large positive or negative standardized residuals (typically greater than 2 or 3 in absolute value) indicate that the observed and expected counts in that category differ substantially, suggesting a lack of homogeneity across the groups. This information can help you understand the nature and source of the significant differences found in the overall test for homogeneity, allowing you to draw more nuanced conclusions about the underlying patterns in the data.