๐ Understanding Expected Frequencies and Marginal Sums
In goodness-of-fit tests, both expected frequencies and marginal sums play crucial roles, but they represent different aspects of the data and the hypothesis being tested. Let's clarify each concept:
๐ Definition of Expected Frequencies
Expected frequencies are the values you would expect to see in each category of your data if the null hypothesis is true. They are calculated based on the theoretical distribution you are comparing your observed data to.
๐ Definition of Marginal Sums
Marginal sums (also sometimes called marginal totals) are the sums of the values in the rows or columns of a contingency table. They represent the total counts for each category of a single variable, irrespective of the other variable.
๐ Comparison Table
| Feature |
Expected Frequencies |
Marginal Sums |
| Definition |
Frequencies expected under the null hypothesis. |
Sums of row or column values in a contingency table. |
| Calculation |
Calculated based on the theoretical distribution. For example, if testing if a die is fair, each number (1-6) would have an expected frequency of $\frac{total \, observations}{6}$. |
Directly calculated from the observed data by summing the values. |
| Role in GOF Test |
Used to compare with observed frequencies to calculate the test statistic (e.g., Chi-square). |
Used to understand the distribution of individual variables and are often part of the process of calculating expected frequencies. |
| Example |
If you roll a die 60 times and are testing if it's fair, the expected frequency for each number (1-6) is 10. |
In a table showing gender vs. favorite color, the marginal sum for 'Female' would be the total number of females in the sample, regardless of their favorite color. |
๐ Key Takeaways
- ๐ Expected frequencies are theoretical values, while marginal sums are calculated from observed data.
- ๐งช Expected frequencies are directly used in goodness-of-fit test calculations (e.g., calculating the chi-square statistic).
- ๐ก Marginal sums help understand the distribution of individual variables and contribute to calculating expected frequencies in some cases.
- ๐ Understanding the difference is essential for correctly performing and interpreting goodness-of-fit tests.
- ๐ Consider the example: When testing whether observed data matches expected (theoretical) distributions, we often use the Chi-square test: $\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}$, where $O_i$ is observed frequency and $E_i$ is expected frequency.