Hypothesis Tests for Goodness of Fit

Introduction

Key Concepts

Understanding Goodness of Fit

Goodness of fit tests evaluate whether sample data fit a distribution from a certain population. It assesses the discrepancy between observed frequencies and expected frequencies under a specific hypothesis, typically the null hypothesis.

Types of Goodness of Fit Tests

The most common goodness of fit test is the Chi-Square ($\chi^2$) test. There are also other tests like the Kolmogorov-Smirnov test and the Anderson-Darling test, but the Chi-Square test is predominantly used in categorical data analysis.

Chi-Square Goodness of Fit Test

The Chi-Square Goodness of Fit test evaluates whether the distribution of observed categorical data matches an expected distribution. It is particularly useful for testing hypotheses about the distribution of frequencies across different categories.

Hypotheses in Goodness of Fit Tests

• Null Hypothesis ($H_0$): Assumes that there is no significant difference between the observed and expected frequencies.
• Alternative Hypothesis ($H_a$): Suggests that there is a significant difference between the observed and expected frequencies.

Test Statistic

The test statistic for the Chi-Square Goodness of Fit test is calculated using the formula: $$ \chi^2 = \sum \frac{(O_i - E_i)^2}{E_i} $$ where:

• $O_i$ = Observed frequency for category $i$
• $E_i$ = Expected frequency for category $i$

Degrees of Freedom

The degrees of freedom (df) for a Chi-Square Goodness of Fit test are determined by the number of categories minus one, and minus the number of parameters estimated from the data: $$ df = k - 1 - p $$ where:

• $k$ = Number of categories
• $p$ = Number of parameters estimated

Calculating Expected Frequencies

Expected frequencies ($E_i$) are calculated based on the null hypothesis. For example, if testing whether a die is fair, the expected frequency for each face is: $$ E_i = \frac{\text{Total Observations}}{\text{Number of Faces}} $$

Assumptions of the Chi-Square Test

• Data should be in the form of frequencies or counts of cases.
• Observations should be independent of each other.
• Expected frequency for each category should be at least 5 to ensure the validity of the test.

Step-by-Step Procedure

1. State the Hypotheses: Define $H_0$ and $H_a$ based on the research question.
2. Choose the Significance Level: Commonly, $\alpha = 0.05$.
3. Calculate Expected Frequencies: Based on the null hypothesis.
4. Compute the Chi-Square Statistic: Using the formula $\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}$.
5. Determine Degrees of Freedom: $df = k - 1 - p$.
6. Find the Critical Value or P-Value: Using Chi-Square distribution tables or software.
7. Make a Decision: Compare the test statistic to the critical value or assess the p-value against $\alpha$ to accept or reject $H_0$.

Interpreting Results

If the calculated Chi-Square statistic exceeds the critical value or if the p-value is less than the chosen significance level, the null hypothesis is rejected. This indicates that there is a significant difference between the observed and expected frequencies.

Example Problem

*Suppose a six-sided die is rolled 60 times, and the observed frequencies for each face are as follows: 1: 8, 2: 12, 3: 10, 4: 10, 5: 10, 6: 10. Test at $\alpha = 0.05$ whether the die is fair.*

1. State the Hypotheses: <
   • $H_0$: The die is fair; each face has an expected frequency of 10.
   • $H_a$: The die is not fair; at least one face has a different expected frequency.
2. Calculate Expected Frequencies: $E_i = 10$ for each face.
3. Compute Chi-Square Statistic: $$ \chi^2 = \frac{(8-10)^2}{10} + \frac{(12-10)^2}{10} + \frac{(10-10)^2}{10} + \frac{(10-10)^2}{10} + \frac{(10-10)^2}{10} + \frac{(10-10)^2}{10} = \frac{4}{10} + \frac{4}{10} = 0.8 $$
4. Degrees of Freedom: $df = 6 - 1 = 5$.
5. Find Critical Value: For $df=5$ and $\alpha=0.05$, $\chi^2_{critical} \approx 11.070$.
6. Decision: $0.8 < 11.070$, so we fail to reject $H_0$.
7. Conclusion: There is no significant evidence to suggest the die is unfair.

Applications of Goodness of Fit Tests

• Testing the fairness of dice or games of chance.
• Evaluating the distribution of categorical survey responses.
• Assessing the fit of observed genetic trait distributions in biology.

Advantages of Goodness of Fit Tests

• Simple to perform and interpret.
• Applicable to various types of categorical data.
• Does not require large sample sizes if expected frequencies are adequate.

Limitations of Goodness of Fit Tests

• Requires a sufficiently large sample size to ensure expected frequencies are reliable.
• Not suitable for small sample sizes or when expected frequencies are less than 5.
• Sensitive to violations of the test assumptions, such as independence of observations.

Comparison Table

Summary and Key Takeaways