Friedman Test and Categorical Data Analysis

Friedman Test

• Non-parametric equivalent of the One-way repeated measures ANOVA and an extension of the Wilcoxon signed-rank test.

• Used for comparing 3+ dependent groups.

• Requires 3+ observations/measurements for each "participant."

Main Assumptions of Repeated Measures ANOVA

• Normality of residuals

• Sphericity

• If one or more of these assumptions fail, consider the Fredman test.

• Used for continuous or ordinal data.

Hypotheses

• $H_0$: All groups belong to the same distribution.

• $H_a$: At least one group belongs to a different distribution.

• If the distributions of the groups are of the same shape, the hypotheses are:

  • $H_0$: The medians of all groups are equal.

  • $H_a$: At least one group has a median not equal to the others.

Method

1. State $H_0$

2. By construction, each group will have the same number of entries, denoted as $n$, and let $k$ be the number of groups.

3. Rank the data for each participant across groups from 1 to $k$ (Potential ranks: $1, 2, …, k$).

4. If we have repeated values across groups, assign a tied rank, which is the mean of the potential ranks of these data (Actual ranks).

5. Sum the ranks for each group, denoted by $R_i$ for the $i^{th}$ group.

Test Statistic

• The test statistic is given by:
  $F = [\frac{12}{nk(k+1)} \sum<em>{i=1}^{k} R</em>i^2] - 3n(k+1)$

• If we have tied ranks, we use the corrected test statistic:
  $F<em>{corrected} = \frac{(k-1) [\sum</em>{i=1}^{k} R<em>i^2 - C</em>F]}{\sum<em>{i=1}^{n} \sum</em>{j=1}^{k} r<em>{ij}^2 - C</em>F}$
  where $C<em>F = \frac{n k (k+1)^2}{4}$ and $r</em>{ij}$ is the rank corresponding to group $i$ in participant $j$.

• For $k = 3$ and $k = 4$, and $n = 2, 3, 4$, we evaluate the test statistic by using tables.

• If $F$/F{corrected} > F{crit}, we reject $H_0$. Otherwise, we use $\chi^2$ tables with $k-1$ degrees of freedom.

• If $F$/F{corrected} > \chi^2{crit}, we reject $H_0$.

Post Hoc Tests

• Wilcoxon signed-rank tests on all pairs. However, this tends to inflate the Type I error rate.

• To control this, we use the following Bonferroni correction.

• Nemenyi post hoc tests.

Categorical Data Analysis

• Categorical data take nominal values, i.e., no intrinsic ordering (e.g., eye color).

• We address cases where both predictor and outcome variables are categorical (e.g., association between personality and color preference).

• We cannot compare means with categorical data.

• Consider frequencies instead.

Assumptions for Categorical Variables

• Independence: Each person, item, etc., only contributes to one cell in the frequency table.

• Expected Frequencies:

  • For $\chi^2$ goodness-of-fit test and 2 x 2 tables, no expected frequencies should be < 5.

  • In longer tables, all expected frequencies > 1 and no more than 20% of expected frequencies < 5.

The $\chi^2$ Tests

• These can be used for goodness-of-fit and to test for association between categorical variables.

The Chi-Square ($\chi^2$) Distribution

• Let $Z<em>1, …, Z</em>k \sim N(0, 1)$, then
  $S = \sum<em>{i=1}^{k} Z</em>i^2 \sim \chi^2_v$

• $S$ has a probability density function (pdf):
  $P_v(x) = \begin{cases} \frac{1}{2^{v/2} \Gamma(v/2)} x^{v/2 - 1} e^{-x/2}, &amp; x &gt; 0 \ 0, &amp; \text{otherwise} \end{cases}$
  where $\Gamma$ denotes the Gamma function.

• For $n \in \mathbb{N}$, $\Gamma(n) = (n-1)!$

• For $z \in \mathbb{C}$, \Re(z) > 0,
  $\Gamma(z) := \int_{0}^{\infty} x^{z-1} e^{-x} dx$

Properties

• $\chi^2$ distributions have a positive skew.

• As $v \rightarrow \infty$, $\chi^2$ becomes symmetrical.

• Let $X \sim \chi^2_v$, then
  $E(X) = v$
  $Var(X) = 2v$
  $\frac{X - v}{\sqrt{2v}} \sim N(0, 1)$

Calculations

• If $X \sim \chi^2<em>v$, then $E(X) = \int x P</em>v(x) dx = v$
  $Var(X) = E(X^2) - [E(X)]^2 = 2v$
  $E(X^2) = \int x^2 P_v(x) dx = (v+2)v$

• If $\lbrace X<em>i \rbrace</em>{i=1}^{N}$ is a sequence of independent $\chi^2$ random variables with d.f. $v<em>1, v</em>2, …, v<em>N$ respectively, then $\sum</em>{i=1}^{N} X<em>i \sim \chi^2</em>{\sum<em>{i=1}^{N} v</em>i}$

$\chi^2$ Goodness-of-Fit Test

• Used to test if categorical data are as expected, or if it fits a certain distribution.

Example

• Exam Mark classification: A, B, C, D, E

• Exam Board: expectation of frequencies in each class.

• Exam Results: test to compare with expected frequencies to determine any potential issues.

Test Statistic

• The test statistic is:
  $X^2 = \sum<em>{i=1}^{K} \frac{(O</em>i - E<em>i)^2}{E</em>i}$
  where K is the number of categories.

• $O_i$ are the observed frequencies.

• $E_i$ are the expected frequencies.

Evaluation

• Evaluate $X^2$ using $\chi^2$ tables with $K-1$ degrees of freedom.

• Hypotheses:

  • $H_0: X^2 = 0$ (no difference between observed and expected frequencies).

  • $H_1: X^2 \neq 0$ (a significant difference between the observed and expected frequencies).

• If X^2 > \chi^2{crit}, we reject $H</em>0$.