Chi-Square Tests and Analysis of Variance (ANOVA): Study Notes

Chi-Square Tests and Analysis of Variance (ANOVA)

Chi-Square Goodness of Fit Test

The Chi-Square Goodness of Fit Test is used to determine whether observed categorical frequencies match expected frequencies based on a claimed distribution. This test is commonly applied when you want to check if a die is fair, if survey responses are equally distributed, or if real-world data follows a theoretical law (e.g., Benford's Law).

• Null Hypothesis (H0): The observed frequencies match the claimed distribution.

• Alternative Hypothesis (Ha): The observed frequencies do not match the claimed distribution (at least one probability is different).

• Test Statistic: $\chi^2 = \sum \frac{(O - E)^2}{E}$ Where O = observed frequency, E = expected frequency.

• Expected Frequency (Uniform): $E = \frac{n}{k}$ Where n = total sample size, k = number of categories.

• Degrees of Freedom: $df = k - 1$

• P-value: Area to the right of the calculated $\chi^2$ value in the chi-square distribution with the appropriate degrees of freedom.

• Decision: If p-value < significance level ($\alpha$), reject H0.

Example: Testing if a die is fair by rolling it 60 times and comparing observed frequencies to the expected frequency for each face (10 if fair).

Goodness of Fit Test: Unequal Probabilities

When the claimed probabilities are not equal, expected frequencies are calculated using the given probabilities for each category:

• Expected Frequency (Non-uniform): $E = n \times p$ Where p = claimed probability for the category.

Example: Testing if the first digits of city populations follow Benford's Law, where each digit has a specific probability.

Goodness of Fit Test Using Technology (TI-84)

Statistical calculators like the TI-84 can perform the Goodness of Fit Test efficiently. Data is entered into lists (L1 for observed, L2 for expected), and the test is run using the appropriate function.

• Steps: Enter observed and expected frequencies, select the chi-square GOF test, and interpret the output (test statistic, p-value, conclusion).

Contingency Tables and Expected Frequencies

A contingency table displays the frequency distribution of variables that are categorical. Expected frequencies for each cell (assuming independence) are calculated as:

• $E = \frac{(\text{row total}) \times (\text{column total})}{\text{grand total}}$

Example: Calculating expected frequencies for a drug trial with improvement and placebo/non-placebo groups.

Chi-Square Test of Independence

The Chi-Square Test of Independence determines whether two categorical variables are independent. It uses the same test statistic as the goodness of fit test but is applied to contingency tables.

• Null Hypothesis (H0): The variables are independent.

• Alternative Hypothesis (Ha): The variables are dependent.

• Degrees of Freedom: $df = (r-1)(c-1)$, where r = number of rows, c = number of columns.

Example: Testing if symptom improvement is independent of receiving a placebo.

Chi-Square Test of Homogeneity

The Test of Homogeneity is used to determine if different populations have the same proportion of a characteristic. The procedure is mathematically identical to the test of independence but with different hypotheses.

• Null Hypothesis (H0): All populations have the same proportion of the characteristic.

• Alternative Hypothesis (Ha): At least one population has a different proportion.

Example: Testing if car ownership rates are the same for teens and adults.

One-Way ANOVA (Analysis of Variance)

One-Way ANOVA is used to compare the means of three or more groups to determine if at least one mean is different. It analyzes variance between groups and within groups.

• Null Hypothesis (H0): All group means are equal ($\mu_1 = \mu_2 = \cdots = \mu_k$).

• Alternative Hypothesis (Ha): At least one group mean is different.

• Test Statistic: $F = \frac{\text{Variance Between Groups}}{\text{Variance Within Groups}}$

• Degrees of Freedom:

  • Between: $df_1 = k - 1$

  • Within: $df_2 = N - k$

Example: Comparing average study times across grades 10, 11, and 12.

Post-Hoc Tests: Tukey, Tukey-Kramer, and Bonferroni

If ANOVA indicates at least one mean is different, post-hoc tests identify which means differ. Common tests include:

• Tukey Test: Compares all possible pairs of means. Test statistic: $q_0 = \frac{\bar{x}_2 - \bar{x}_1}{\sqrt{\frac{MSE}{n}}}$

• Tukey-Kramer Test: Used when group sizes are unequal: $q_0 = \frac{\bar{x}_2 - \bar{x}_1}{\sqrt{\frac{MSE}{2} \left(\frac{1}{n_1} + \frac{1}{n_2}\right)}}$

• Bonferroni Test: Adjusts the significance level for multiple comparisons. Multiply the p-value by the number of comparisons.

Example: Comparing mean weight loss across three diet plans after ANOVA shows a difference.

Two-Way ANOVA

Two-Way ANOVA is used to analyze the effect of two independent variables (factors) on a dependent variable, and to test for interaction effects between the factors.

• Interaction Effect: Occurs if the effect of one factor depends on the level of the other factor.

• Test Statistics: $F = \frac{MS(\text{Factor})}{MSE}$ and $F = \frac{MS(\text{Interaction})}{MSE}$

• Interpretation: Test for interaction first. If significant, interpret with caution; if not, test main effects.

Example: Testing how sunlight and fertilizer affect plant growth, and whether their effects interact.

Two-Way ANOVA: Interaction Plots

Interaction plots graphically display means for combinations of two factors. If lines are not parallel, this suggests an interaction effect.

• No Interaction: Lines are roughly parallel.

• Interaction: Lines cross or diverge/converge.

Example: Plotting exam scores by study time and hours of sleep to check for interaction.

Summary Table: Chi-Square and ANOVA Tests

Test	Purpose	Hypotheses	Test Statistic	df
Chi-Square Goodness of Fit	Compare observed to expected frequencies	H0: Observed = Expected Ha: Observed ≠ Expected	$\chi^2 = \sum \frac{(O-E)^2}{E}$	k-1
Chi-Square Independence	Test independence of two variables	H0: Variables independent Ha: Variables dependent	$\chi^2 = \sum \frac{(O-E)^2}{E}$	(r-1)(c-1)
One-Way ANOVA	Compare means of 3+ groups	H0: All means equal Ha: At least one mean differs	$F = \frac{\text{Variance Between}}{\text{Variance Within}}$	k-1, N-k
Two-Way ANOVA	Compare means across 2 factors, test interaction	H0: No effect/interaction Ha: Effect/interaction present	$F = \frac{MS(\text{Factor/Interaction})}{MSE}$	Depends on factors/groups