Statistical Test Selection and Practice Problems in Biostatistics

Decision Tree for Choosing a Statistical Test Method

Introduction

Choosing the appropriate statistical test is essential for analyzing data correctly in biostatistics and statistics. The decision depends on the type of variables, the number of groups, and the distribution of the data. This section provides a structured decision tree to guide the selection of statistical tests.

Key Steps in Selecting a Statistical Test

• Identify the research question: Is it about comparing means, proportions, associations, or relationships?

• Determine the type of variables: Are they categorical (nominal, ordinal) or quantitative (interval, ratio)?

• Assess the number of groups: Are you comparing two groups, more than two, or paired samples?

• Check data distribution: Is the data normally distributed or skewed?

• Choose the appropriate test: Use the decision tree to select the test based on the above criteria.

Common Statistical Tests

• t-test: Used to compare means between two groups (independent or paired).

• ANOVA (Analysis of Variance): Used to compare means among more than two groups.

• Chi-square test: Used for categorical data to test associations or differences in proportions.

• Correlation and Regression: Used to assess relationships between quantitative variables.

• Non-parametric tests: Used when data are not normally distributed (e.g., Mann-Whitney U, Wilcoxon signed-rank test).

Comparing Means

Introduction

Comparing means is a fundamental aspect of statistical analysis, especially when evaluating the effect of treatments or interventions. The choice of test depends on the number of groups and whether samples are independent or paired.

• Independent samples t-test: Compares means between two independent groups.

• Pooled t-test: Used when variances are assumed equal between groups.

• Paired samples t-test: Compares means within the same group at two time points.

• ANOVA: Compares means among three or more groups.

• Non-parametric alternatives: Mann-Whitney U test (independent), Wilcoxon signed-rank test (paired), Kruskal-Wallis test (multiple groups).

Example

To compare body weight before and after treatment in the same patients, use a paired t-test if data are normally distributed, or Wilcoxon signed-rank test if not.

Distributions, Variances, and Proportions

Introduction

Statistical tests for proportions and variances are used to analyze categorical data and assess the spread of quantitative data. The choice of test depends on sample size and data type.

• Chi-square test: Tests for association between categorical variables or goodness-of-fit.

• F-test: Compares variances between groups.

• Z-test for proportions: Used for large samples to compare proportions.

• Fisher's exact test: Used for small sample sizes in categorical data.

Example

To test if the proportion of patients losing weight differs between two treatments, use a chi-square test or Fisher's exact test (if sample size is small).

Association Between Two Variables

Introduction

Assessing association between variables helps determine if a relationship exists. The type of test depends on whether variables are categorical or quantitative.

• Pearson correlation: Measures linear association between two quantitative variables.

• Spearman correlation: Non-parametric measure for ordinal or non-normally distributed data.

• Chi-square test of independence: Tests association between two categorical variables.

• Regression analysis: Quantifies the relationship and predicts values.

Example

To assess if weight loss is associated with age, use Pearson correlation if both variables are quantitative and normally distributed.

Practice Problems

Introduction

Practice problems help reinforce understanding of statistical test selection and application. Problems are divided based on data distribution and sample size.

Part 1: Normally Distributed Data

• Compare body mass in patients before and after treatment (paired t-test).

• Compare weight loss between two independent groups (independent t-test).

• Assess effect of treatment in multiple groups (ANOVA).

• Test association between categorical variables (chi-square test).

Part 2: Skewed Data and Small Samples

• Use non-parametric tests (Wilcoxon signed-rank, Mann-Whitney U) for comparisons.

• Fisher's exact test for categorical data with small samples.

Part 3: Proportions and Categorical Data

• Chi-square test for association or difference in proportions.

• Fisher's exact test for small sample sizes.

Summary Table: Statistical Test Selection

Type of Data	Number of Groups	Test (Parametric)	Test (Non-parametric)
Quantitative	2 (independent)	t-test	Mann-Whitney U
Quantitative	2 (paired)	Paired t-test	Wilcoxon signed-rank
Quantitative	>2	ANOVA	Kruskal-Wallis
Categorical	2	Chi-square	Fisher's exact
Categorical	>2	Chi-square	Fisher's exact

Key Formulas

• t-test statistic: $t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$

• ANOVA F-statistic: $F = \frac{\text{Between-group variance}}{\text{Within-group variance}}$

• Chi-square statistic: $\chi^2 = \sum \frac{(O - E)^2}{E}$

• Pearson correlation coefficient: $r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}}$

Additional info:

• Some context and terminology have been inferred to align with standard statistics curriculum.

• Practice problems are designed to reinforce test selection and application for both parametric and non-parametric scenarios.