Standardized Residuals

Definition

Standardized residuals are the residuals (the difference between the observed and predicted values) divided by their standard error. They are used to assess the fit of a statistical model and identify potential outliers or influential observations.

5 Must Know Facts For Your Next Test

1. Standardized residuals follow a standard normal distribution (mean 0, standard deviation 1) when the model assumptions are met.
2. Large standardized residuals (typically greater than 2 or 3 in absolute value) may indicate potential outliers or model misspecification.
3. Standardized residuals are used to check the assumption of homogeneity of variance in regression models.
4. In the context of the chi-square test for homogeneity, standardized residuals can help identify which cells contribute the most to the overall chi-square statistic.
5. Comparing the standardized residuals between different models can help determine which model provides a better fit to the data.

Review Questions

• Explain how standardized residuals are calculated and their purpose in assessing model fit.
  • Standardized residuals are calculated by dividing the residuals (the difference between the observed and predicted values) by their standard error. This standardization allows for the comparison of residuals on a common scale, with a mean of 0 and a standard deviation of 1 when the model assumptions are met. Standardized residuals are used to identify potential outliers or influential observations, as well as to check the assumption of homogeneity of variance in regression models. Large standardized residuals (typically greater than 2 or 3 in absolute value) may indicate that the model is not adequately fitting the data and that further investigation is needed.
• Describe how standardized residuals are used in the context of the chi-square test for homogeneity.
  • In the chi-square test for homogeneity, standardized residuals can be used to identify which cells in the contingency table contribute the most to the overall chi-square statistic. Cells with large standardized residuals (in absolute value) indicate that the observed and expected frequencies in those cells differ significantly, suggesting that the null hypothesis of homogeneity may not be true. By examining the pattern of standardized residuals, researchers can gain insights into the specific areas where the data deviates from the expected homogeneous distribution, which can inform further analysis and interpretation.
• Explain how comparing standardized residuals can be used to determine which statistical model provides a better fit to the data.
  • Comparing the standardized residuals between different statistical models can help determine which model provides a better fit to the data. Models with smaller standardized residuals, in terms of both the magnitude and the number of large residuals, are generally considered to fit the data better. This is because the standardized residuals reflect the degree of deviation between the observed and predicted values, and models with a closer fit will have smaller, more evenly distributed standardized residuals. By comparing the standardized residuals across different models, researchers can assess the relative performance of the models and select the one that best represents the underlying relationships in the data.

Related terms

Residuals:

The difference between the observed and predicted values in a statistical model.

Standard Error:

A measure of the variability or uncertainty in an estimate, calculated as the standard deviation of the sampling distribution of the estimate.

Outliers:

Observations that are significantly different from the majority of the data, which can have a disproportionate influence on the statistical analysis.