Biostats Final Exam Review

Module 6: ANOVA

General Overview

• ANOVA (Analysis of Variance): Used for comparing three or more groups.

• T-test is suitable for comparing only two groups.

• Key assumptions:

  • Normal distribution of data.

  • Equal variances across groups.

• Levene’s Test: To test for homogeneity of variance.

• If significant, indicates variances differ across groups.

Variance Components

• Between Group Variance: Spread of group scores around the grand mean, indicating the separation between groups.

• Within Group Variance: Spread of scores within each group around their respective means.

F Ratio

• Definition: Ratio of between-group variance to within-group variance.

• F Ratio = (Explained Variance) / (Unexplained Variance).

• Significance of F Ratio:

  • Larger F ratio suggests ANOVA is more likely to be significant.

  • Indicates a difference exists among group means.

• Interpretation:

  • Large F ratio: Small p-value → reject null hypothesis (H0).

  • Small F ratio: Large p-value → fail to reject H0.

Power & Effect Size

• Power: Probability of correctly rejecting a false null hypothesis (H0).

  • High Power: High probability of detecting a true difference.

  • Low Power: Increased risk of failing to detect significant differences (Type II error).

• Effect Size: Quantifies how much group means differ.

  • Small Effect Size: Indicates a small difference.

  • Large Effect Size: Indicates a significant difference among group means.

One Way ANOVA

• Involves one independent variable with three or more levels.

• Components:

  • Comparison of between-group variance (explained) vs. within-group variance (unexplained).

  • Sum of Squares (SS): Reflects variance. Larger SS indicates higher variance.

  • Degrees of Freedom (df): Always one less than total observations: df = k - 1 (where k = total number of groups).

Two Way ANOVA

• Involves two independent variables, each with two or more levels.

• Analyzes:

  • Main effects of each independent variable.

  • Interaction effects between the variables.

• No interaction is indicated by parallel lines; crossing or non-parallel lines indicate interaction.

Repeated Measures ANOVA

• Each subject is tested under all experimental conditions (similar to a paired t-test).

• Controls for differences between subjects.

• Focus is on variance within subjects.

Mixed Design ANOVA

• Involves one independent variable and one repeated factor.

• Example: Symptom treatment over time, with and without exercise.

Multiple Comparison Tests

• ANOVA indicates at least one significant difference but does not specify which group means are different.

• Purpose of Multiple Comparison Tests: Contrast pairs of means against critical value to identify significant differences.

Liberal vs Conservative Tests

• Liberal Tests: More likely to find significant differences with means that are closer together. Higher power, increased chance of Type I error (e.g., Fisher’s LSD).

• Conservative Tests: Require means to be further apart for significant differences. Lower power, fewer Type I errors (e.g., Scheffe’s comparison).

• Balance Test: Tukey test (more conservative than SNK).

Post Hoc Multiple Comparisons

• Post Hoc: All pairwise contrasts explored after significant ANOVA (unplanned).

• Planned Comparisons: Set beforehand to examine specific pairs of means, even when ANOVA isn't significant.

ANOVA with Multifactorial Variables

• Simple Effects: Separate analyses of each row or column within a factorial design.

• Examines the effect of one independent variable at specific levels of another.

Nonparametric Statistics

• Suitable for nominal or ordinal data, generally have lower power than parametric tests.

• Key Features:

  • Do not assume normality of variance.

  • Often used with small samples.

• Parametric vs Nonparametric Tests:

  • Unpaired T-test: Mann-Whitney U test (two independent groups).

  • Paired T-test: Sign test or Wilcoxon signed-ranks test (two related groups).

  • One Way ANOVA: Kruskal-Wallis ANOVA (three or more independent groups).

  • One Way Repeated Measures ANOVA: Friedman two way ANOVA (three or more related groups).

• Useful when data is not normally distributed, has outliers, or fails homogeneity of variance assumptions.

• Ranking Scores: Ranks data from smallest to largest (negative values considered smallest). Ties are assigned average ranks.

Module 7: Chi Square

Chi Square Basics

• Purpose: Tests significance of proportions and determines if differences between observed and expected data are due to chance or relationships.

• Assesses associations between categorical variables but does not imply causation.

• Assumptions:

  • Data represent individual counts.

  • Categories are mutually exclusive.

  • No subject represented twice.

• Hypothesis Testing:

  • If Chi-square > critical value → reject H0.

  • If Chi-square < critical value → fail to reject H0.

Goodness of Fit

• Evaluates if observed data aligns with a specific distribution or expected proportions.

• Hypothesis Testing: H0 states observed proportion does not differ from expected proportion.

• Degrees of Freedom: df = k - 1 (number of categories).

Standardized Residuals

• Identify which categories contribute most to the Chi-square value.

• Formula: Residual = Observed - Expected, (Standardized Residual = Residual / √Expected).

• Larger residuals = greater variation from expected

Independence Test

• Examines association between two categorical variables.

• Hypothesis Testing: H0 assumes no association.

• Data is organized into contingency tables.

• Degrees of Freedom: df = (rows - 1) × (columns - 1).

Correlation

• Measures association between two variables.

• Types of Relationships:

  • Positive: As X increases, Y increases.

  • Negative: As X increases, Y decreases.

• Note: Does not imply causation.

Scatter Plots

• Visual representation of data to clarify patterns.

• Closer points to a straight line indicate stronger associations.

• Outliers: Data points that lie outside the cluster.

Correlation Coefficient

• Used to assess strength and direction of relationship between two variables.

• Range: From -1.0 to +1.0.

• The sign indicates direction, while 0 indicates no relationship.

Pearson Coefficient

• Pearson r: Used for sample data and population parameters (p).

• Applicable when X and Y are continuous variables, normally distributed, on interval/ratio scales.

• Hypothesis Testing:

  • H0 states no relationship (p = 0).

  • HA states a relationship exists (p ≠ 0).

• Spearman Rank Correlation Coefficient (rs): Nonparametric alternative to Pearson's coefficient, based on ranked data or ordinal data.

Module 8: Regression

Linear Regression

• The correlation coefficient describes strength but not prediction.

• Regression models predict outcomes based on shared variance.

• Coefficient of Determination (r²): Represents proportion of variance explained by the independent variable. Ranges from 0 to 1.

Simple Linear Regression

• Predicts how well one variable predicts another.

• Variables:

  • X: Independent variable.

  • Y: Dependent variable.

• Regression Equation: Ŷ = a + bX.

• H0: b = 0 (no relationship).

• Least Squares Method: Fits regression line to minimize sum of squared residuals.

Multiple Regression

• Utilizes multiple independent variables to predict one dependent variable.

• Equation: Ŷ = a + b1X1 + b2X2 + …

• R² indicates percentage of total variance explained by predictors.

• H0: b = 0 for each independent variable.

Standardized Regression Coefficients

• Allows comparison across different units of independent variables.

• Coefficients converted to z-scores or standardized beta weights to measure contribution to prediction.

Collinearity

• Occurs when independent variables are correlated, making some appear less important.

• Higher Variance Inflation Factor (VIF) indicates greater collinearity.

Logistic Regression

• Predicts probability of event occurrence with a dichotomous dependent variable.

• Independent variables can be continuous or categorical.

• Dummy Variables: Used for categorical variables by assigning numerical values.

• Outcome Coding: Target group is coded as 1, reference group as 0.

Odds Ratio

• Measures likelihood of belonging to a target group compared to a reference group.

• OR > 1: Increased risk.

• OR < 1: Decreased risk.