Expected frequencies

Definition

Expected frequencies are the theoretical counts of occurrences in each category of a contingency table under the assumption of independence between the variables. These values are calculated based on the overall total and the distribution of the marginal totals, serving as a foundation for various statistical tests, particularly in log-linear models. In multi-way contingency tables, expected frequencies help to assess the fit of the model by comparing them to observed frequencies.

5 Must Know Facts For Your Next Test

1. Expected frequencies are calculated using the formula: $$E_{ij} = \frac{(Row\ Total_i)(Column\ Total_j)}{Grand\ Total}$$ for each cell in a contingency table.
2. In multi-way contingency tables, expected frequencies serve as a benchmark to determine whether observed data fits well with the proposed model.
3. If any expected frequency is less than 5, it can affect the validity of statistical tests like the Chi-square test, leading to unreliable results.
4. Log-linear models utilize expected frequencies to estimate parameters and assess interactions between multiple categorical variables.
5. A good model fit is indicated when the observed frequencies closely match the expected frequencies across all categories in a contingency table.

Review Questions

• How do you calculate expected frequencies for a two-way contingency table, and why is this calculation important?
  • To calculate expected frequencies for a two-way contingency table, you use the formula: $$E_{ij} = \frac{(Row\ Total_i)(Column\ Total_j)}{Grand\ Total}$$. This calculation is important because it provides a baseline against which we can compare observed frequencies to assess if there is a significant relationship between the variables. By evaluating how closely observed counts align with expected counts, we can determine if any deviations are due to chance or indicate a true association.
• Discuss the implications of having low expected frequencies when performing a Chi-square test in the context of multi-way contingency tables.
  • When performing a Chi-square test, having low expected frequencies—typically less than 5—can undermine the validity of the test results. In multi-way contingency tables, low expected frequencies indicate that some categories may have insufficient data to draw reliable conclusions about relationships between variables. This situation could lead to an increased likelihood of Type I or Type II errors. Researchers may need to consider combining categories or using alternative statistical methods that are more appropriate for small sample sizes.
• Evaluate how expected frequencies influence model fitting in log-linear models and what this means for interpreting relationships among categorical variables.
  • In log-linear models, expected frequencies play a crucial role in estimating parameters and fitting the model to data. By analyzing how well observed frequencies match expected frequencies across different levels of categorical variables, researchers can determine if there are significant interactions or effects within their data. A strong fit suggests that the relationships among categorical variables are accurately captured by the model, allowing for valid interpretations and predictions about how these variables behave in different contexts.

Related terms

Observed frequencies: Observed frequencies are the actual counts recorded in each category of a contingency table, representing the real-world data collected from an experiment or survey.

Chi-square test: The Chi-square test is a statistical method used to determine if there is a significant association between categorical variables by comparing observed and expected frequencies.

Log-linear model: A log-linear model is a statistical model used to analyze the relationship between categorical variables by examining the logarithm of expected cell counts in contingency tables.