Hypothesis Testing: Parametric and Nonparametric Tests, Odds Ratio, and Relative Risk

Parametric and Nonparametric Tests

Introduction to Hypothesis Testing

Hypothesis testing is a fundamental aspect of inferential statistics, allowing researchers to draw conclusions about populations based on sample data. Tests can be broadly classified as parametric or nonparametric depending on the assumptions made about the data.

• Parametric tests assume underlying statistical distributions (often normality) and are typically used with interval or ratio data.

• Nonparametric tests do not require such assumptions and are suitable for ordinal data or when normality cannot be assumed.

Analysis of Variance (ANOVA)

Comparing Means in More Than Two Groups

ANOVA is a parametric test used to determine whether there are statistically significant differences between the means of three or more independent groups.

• Purpose: To test if at least one group mean is different from the others.

• Example: Comparing the mean blood pressure among three different treatment groups.

Similarity with T-Test

• When the number of groups (k) is two, ANOVA and the independent samples t-test yield equivalent results.

• Test statistics: $t_{obs}^2 = F_{obs}$

Limitations of ANOVA

• Assumes samples are independent, have equal variances (homoscedasticity), and are normally distributed.

• ANOVA only indicates that a difference exists, not which groups differ.

• Post-hoc tests (e.g., Tukey HSD) are required to identify specific group differences.

Nonparametric Tests for Comparing Means

When to Use Nonparametric Tests

Nonparametric tests are appropriate when data are not normally distributed or are measured at the ordinal level (e.g., ranks).

• Common in survey data where responses are ranked.

• Nonparametric tests compare medians rather than means.

Common Nonparametric Tests

Test	Parametric Equivalent
Mann-Whitney U Test	Independent samples t-test
Wilcoxon Signed Rank Test	Pared samples t-test
Kruskal-Wallis Test	One-way ANOVA

Hypothesis Testing Comparing Proportions

Odds and Probability

Odds measure the likelihood of an event occurring compared to it not occurring. Probability measures the likelihood of an event occurring out of all possible outcomes.

• Odds formula: $\text{Odds} = \frac{\text{Number of events with outcome}}{\text{Number of events without outcome}}$

• Probability formula: $\text{Probability} = \frac{\text{Number of events with outcome}}{\text{Total number of events}}$

• Example: Rolling a 6 on a die: Odds = 1/5, Probability = 1/6

Odds Ratio (OR)

The odds ratio quantifies the strength of association between two events, such as exposure and outcome.

• Definition: Ratio of the odds of an event in one group to the odds in another group.

• Formula: $\text{OR} = \frac{A/C}{B/D}$, where:

	Severe	Mild	Total
Diabetes	184 (A)	394 (B)	578
No diabetes	287 (C)	5024 (D)	5311
Total	471	5418

• Calculation: $\text{OR} = \frac{(184/287)}{(394/5024)} = 8.175$

Interpretation of OR

• OR = 1: No association (independence)

• OR > 1: Positive association (risk factor)

• OR < 1: Negative association (protective factor)

• OR shows association, not causation.

• OR > 3.5 is generally considered large.

• Example: OR = 8.175 means a person with diabetes is 8.175 times more likely to have severe COVID than a person without diabetes.

Relative Risk (RR)

Relative risk (risk ratio) compares the probability of an event occurring in an exposed group to the probability in an unexposed group.

• Definition: Ratio of probabilities between two groups.

• Formula: $\text{RR} = \frac{A/(A+B)}{C/(C+D)}$

• Calculation: $\text{RR} = \frac{184/(184+394)}{287/(287+5024)} = 5.89$

Interpretation of RR

• RR = 1: Exposure does not affect outcome

• RR < 1: Exposure reduces risk (protective factor)

• RR > 1: Exposure increases risk (risk factor)

Comparison: Odds Ratio vs Relative Risk

	Odds Ratio (OR)	Relative Risk (RR)
Definition	Ratio of odds of A in the presence of B and the odds of A in the absence of B	Ratio of probability of A in the presence of B and the probability of A in the absence of B
Formula	$(A/C)/(B/D)$	$(A/(A+B))/(C/(C+D))$
Approximation	If A is much smaller than C, OR will be similar to RR. When the disease is rare, OR ≈ RR.	More suitable for cases with medium to high probability.
Usage	Used for case-control studies, logistic regression, and when the outcome is rare.	Used for cohort studies and measuring effect of exposure for common outcomes.

Summary

• ANOVA is used for comparing means across multiple groups when assumptions are met; nonparametric alternatives are available for non-normal data.

• Odds, odds ratio, and relative risk are essential for comparing proportions and associations between categorical variables.

• Understanding when to use OR vs RR is crucial for proper interpretation in epidemiological studies.