Chi-Square and Distribution-Free Tests Classrooms

94.1 Chi-Square Values

The Chi-Square ($\chi^2$) distribution is a non-parametric distribution used primarily for hypothesis tests related to categorical data. It's often used to determine if there is a significant difference between observed and expected frequencies.

Chi-Square Statistic ($\chi^2$): Measures the discrepancy between observed and expected frequencies. $$\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}$$ where $O_i$ are observed frequencies and $E_i$ are expected frequencies.
Degrees of Freedom (df): For a goodness-of-fit test, $df = k - 1 - m$, where $k$ is the number of categories and $m$ is the number of parameters estimated from the sample. For a test of independence, $df = (rows - 1)(columns - 1)$.

📊 Analyze Categorical Data!

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94.2 Fitting Data to Distributions

The Chi-Square goodness-of-fit test is used to determine whether observed frequency data fits a particular distribution (e.g., uniform, normal, Poisson). It compares the observed counts in categories to the counts expected under the hypothesized distribution.

Null Hypothesis ($H_0$): The observed data fits the specified distribution.
Alternative Hypothesis ($H_1$): The observed data does not fit the specified distribution.
Steps:
1. State hypotheses.
2. Calculate expected frequencies.
3. Calculate the $\chi^2$ test statistic.
4. Determine degrees of freedom.
5. Compare $\chi^2$ to critical value or calculate p-value.
6. Make a decision.

📈 Test Your Data Fit!

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94.3 Distribution-Free Tests Intro

Distribution-free tests, also known as non-parametric tests, are statistical methods that do not rely on assumptions about the distribution of the population data (e.g., normality). They are particularly useful when data is not normally distributed, sample sizes are small, or data is ordinal.

Advantages:
- No distributional assumptions (e.g., normality).
- Can be used with ordinal or ranked data.
- Less sensitive to outliers.
Disadvantages:
- Less powerful than parametric tests if assumptions are met.
- May require more complex calculations for large datasets.

🤔 When to Use Non-Parametric Tests?

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94.4 Sign Test

The Sign Test is a simple non-parametric test used for paired data (e.g., before-after measurements) to determine if there is a consistent difference between two conditions. It focuses only on the direction of the difference (positive or negative), ignoring the magnitude.

Hypotheses:
- $H_0$: The median difference is zero (no consistent difference).
- $H_1$: The median difference is not zero (or is positive/negative).
Procedure:
1. Calculate the difference for each pair.
2. Assign a '+' for positive differences, '-' for negative differences, and ignore '0' differences.
3. Count the number of '+' signs ($n_+$) and '-' signs ($n_-$).
4. The test statistic is typically the smaller of $n_+$ or $n_-$.
5. Compare to critical values from a binomial distribution or use a normal approximation for large samples.

➕➖ Test Paired Data!

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94.5 Wilcoxon Signed-Rank Test

The Wilcoxon Signed-Rank Test is a non-parametric alternative to the paired t-test. It is used for paired data and considers both the direction and the magnitude of the differences between pairs, making it more powerful than the Sign Test.

Hypotheses:
- $H_0$: The median difference between pairs is zero.
- $H_1$: The median difference is not zero (or is positive/negative).
Procedure:
1. Calculate the absolute differences for each pair.
2. Rank the absolute differences (assigning average ranks for ties).
3. Reassign the original signs to the ranks.
4. Sum the positive ranks ($T_+$) and negative ranks ($T_-$).
5. The test statistic ($T$) is the smaller of $T_+$ and $|T_-|$.
6. Compare $T$ to critical values from a Wilcoxon Signed-Rank table.

📈 Rank Your Differences!

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94.6 Mann–Whitney Test

The Mann-Whitney U Test (also known as the Wilcoxon Rank-Sum Test) is a non-parametric test used to compare two independent samples. It determines if two independent samples are drawn from the same population or from populations with the same median, without assuming normality.

Hypotheses:
- $H_0$: The two populations have the same distribution (or median).
- $H_1$: The two populations have different distributions (or medians).
Procedure:
1. Combine all data from both samples and rank them from smallest to largest.
2. Sum the ranks for each sample ($R_1$, $R_2$).
3. Calculate the U statistics ($U_1$, $U_2$).
4. The test statistic ($U$) is the smaller of $U_1$ and $U_2$.
5. Compare $U$ to critical values from a Mann-Whitney U table or use a normal approximation for large samples.

🤝 Compare Independent Samples!

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