Statistics and Chi-Square Tests
Nonparametric Statistics
- Definition: Nonparametric statistics do not assume a specific distribution for the data.
- Commonly used when parametric assumptions cannot be met.
Parametric vs Nonparametric Tests
Parametric Tests:
- Examples: t-tests, ANOVA
- Assumed underlying data distribution: normal
- Defined by parameters like means, standard deviations
Nonparametric Tests:
- Examples: chi-square tests, sign test
- Used for nominal data or when parametric assumptions are violated
- Generally less power than parametric tests
Chi-Square Goodness-of-Fit Test
- Function: Analyzes frequency data to see if observed counts differ from expected counts.
- Common use case: Determine preferences (e.g., preferred drinks like Coke vs Pepsi).
- Measures how well observed frequencies match expected frequencies.
Assumptions of Chi-Square Test:
- 1 categorical independent variable with two or more levels.
- Dependent variable is a frequency or count.
- If continuous, prefer parametric test.
- Independence of scores (no score dependency across cells).
- Groups are mutually exclusive.
- Minimum of 5 subjects/events expected per group; otherwise, power is reduced too much.
Choosing the Correct Analysis
Dependent Variable is Numerical:
- Independent Variable Numerical: Linear regression
- Independent Variable Categorical: t-test or ANOVA (based on levels in categorical variable)
Dependent Variable is Categorical:
- Independent Variable Numerical: Logistic Regression
- Independent Variable Categorical: Chi-square tests, sign test, binomial test.
Example 1: Chi-Square Test with Divers
Question: Do male deep-sea divers have different numbers of male vs. female first-born?
Hypotheses:
- Null (H0): Pboy = Pgirl
- Alternative (HA): Pboy ≠ Pgirl
Observed Data for 25 Divers:
- Boys: 10, Girls: 15, Total: 25
Expected Values Calculation:
- Calculate expected counts by multiplying probability by total participants:
- Boys: $12.5$, Girls: $12.5$ (based on ratio of 0.5)
Step 2: Calculate Chi-Square ($\chi^2$)
- Formula: \chi^2 = \sum \frac{(Observed - Expected)^2}{Expected}
- For the divers:
- \chi^2 = \frac{(10-12.5)^2}{12.5} + \frac{(15-12.5)^2}{12.5} = 1.00
Step 3: Degrees of Freedom
- df = number \, of \, categories - 1
- For this test, df = 2 - 1 = 1
Step 4: Assess Hypothesis
Critical Value: Compare with chi-square distribution.
If \chi^2{calculated} < \chi^2{critical}, fail to reject H0.
Calculate and compare: (1.00 < 3.841), therefore, fail to reject H0.
Example 2: Ratios in Epilepsy Patients
Question: Ratio of right-handed (RH) to left-handed (LH) patients in 100 patients with epilepsy.
Hypotheses:
- Null (H0): P(RH) = 0.95, P(LH) = 0.05
- Alternative (HA): H0 is not true.
Observed Counts:
- RH: 85; LH: 15
- Total: 100
Step 2: Calculate Expected Values:
- Expected counts (theoretical): RH = 95, LH = 5 based on ratios provided.
Step 3: Calculate Chi-Square ($\chi^2$)
- \chi^2 = \frac{(85-95)^2}{95} + \frac{(15-5)^2}{5} = 21.05
Step 4: Degrees of Freedom
df = 2 - 1 = 1
Rejection: Calculate critical value.
Find that 21.05 > 3.841, thus reject H0.
APA Style Results
- Report: The ratio of right-handed to left-handed people was significantly less than the rate observed in the general population: \chi^2(1, N=100) = 21.05, p < .01.
Example 3: Rats in a Maze
- Question: Do rats prefer one door over the others?
- Ideal hypotheses and approach similar to previous examples with observed counts in a grid format.
Chi-Square Test of Independence
- Used to determine if two categorical variables are independent (e.g., handedness and gender).
Example Setup
- Observed counts provided, calculate expected counts based on marginal totals.
- Follow through steps outlined in previous examples for hypothesis testing.
Step 1 - 3 Processed Similarly
- Compute \chi^2 and associated degrees of freedom, assess through critical values table to determine if results support H0 or HA.