Statistical Methods Study Sheet

Template Overview

Mastering statistical methods requires understanding the exact mathematical assumptions, formula derivations, effect size calculations, and programming syntax associated with each test. The Statistical Methods Study Sheet serves as a rigorous personal reference for graduate students, data scientists, and quantitative researchers. It pairs theoretical mechanics with practical R and Python code blocks for rapid consultation.

Markdown Template Structure

Copy the block below or click Download .md File to add this template directly to your personal vault:

---
title: "Stats Reference: [Topic / Test Family]"
date: [YYYY-MM-DD]
course_or_project: "[Quantitative Methodology / Biostatistics]"
target_test_family: "[Parametric Tests / Non-Parametric / GLM / Bayesian]"
tags: [statistics, methodology, quantitative, study-sheet, data-science, r-stats]
---

# Statistical Methods Study Sheet — [Topic / Test Family]

## 1. Test Selection Decision Flowchart
- **Data Type:** Continuous dependent variable with categorical independent variable (2 groups).
- **Normality Check:** Shapiro-Wilk test `p > 0.05` → Parametric path (`Independent Samples t-test`).
- **If Normality Fails:** Mann-Whitney U test (Wilcoxon rank-sum non-parametric test).
- **Homogeneity of Variance Check:** Levene's test `p > 0.05` → Student's t-test; if `p < 0.05` → Welch's t-test.

## 2. Theoretical Assumptions & Diagnostics Matrix
| Assumption Name | Description & Requirement | Diagnostic Check Tool | Violation Remedy / Contingency |
|---|---|---|---|
| Normality of Residuals | Residuals follow a Gaussian distribution | Q-Q Plot, Shapiro-Wilk test | Log/Box-Cox transformation or non-parametric test |
| Homoscedasticity | Equal variance across all factor levels | Levene's test, Residuals vs Fitted plot | Apply Welch's correction or heteroscedastic robust errors |
| Independence of Observations | Data points collected without serial correlation | Durbin-Watson statistic | Use mixed-effects models with random intercepts |

## 3. Mathematical Equations & Formula Derivation
- **Test Statistic Formula:**
  $$t = \frac{\bar{X}_1 - \bar{X}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$$
- **Effect Size Metric (Cohen's d):**
  $$d = \frac{\bar{X}_1 - \bar{X}_2}{s_{pooled}}$$
  *Where:* $d = 0.2$ (Small), $d = 0.5$ (Medium), $d = 0.8$ (Large).

## 4. Code Implementation (R & Python)
### Python (SciPy & Statsmodels)
```python
import scipy.stats as stats

# Welch's two-sample t-test (robust against unequal variance)
stat, p_val = stats.ttest_ind(group_a, group_b, equal_var=False)
print(f"t-statistic: {stat:.4f}, p-value: {p_val:.4e}")

R (Base & Tidyverse)

# Welch Two Sample t-test
t_result <- t.test(latency ~ hardware_group, data = benchmark_data, var.equal = FALSE)
print(t_result)

5. Common Traps & Interpretation Hazards

• P-Value Fallacy: A statistically significant p-value (p < 0.001) does not imply a practically meaningful effect size if sample size $N$ is massive.
• Multiple Comparison Hazard: Running pairwise tests without Holm-Bonferroni or Benjamini-Hochberg FDR correction multiplies false positives.

6. Self-Testing Practice Exercises

• Derive the degrees of freedom calculation for Welch-Satterthwaite equation.
• Explain why non-parametric tests lose statistical power relative to parametric alternatives when normality holds.

## Filled Real-World Example

Here is a realistic example demonstrating how this template captures actual deliberations:

```markdown
---
title: "Stats Reference: Two-Way ANOVA & Interaction Effects"
date: 2026-10-09
course_or_project: "PSYCH 610: Advanced Quantitative Methods"
target_test_family: "Parametric GLM Family"
tags: [statistics, methodology, quantitative, study-sheet, data-science, r-stats]
---

# Statistical Methods Study Sheet — Two-Way ANOVA & Interaction Effects

## 1. When to Use
Use when investigating the effect of two categorical independent variables (factors) on a single continuous dependent variable, and examining whether an interaction effect exists between the factors.

## 2. Core Mathematical Model
$$Y_{ijk} = \mu + \alpha_i + \beta_j + (\alpha\beta)_{ij} + \epsilon_{ijk}$$

## 3. Diagnostic Code Snippet
```r
model <- aov(transcription_speed ~ model_size * hardware_type, data = df)
summary(model)
TukeyHSD(model)

4. Interpretation Rule

If the interaction term (lphaeta)_{ij} is statistically significant (p < 0.05), do not interpret main effects in isolation; plot the interaction curves and report simple main effects.

## When to Use and When to Avoid

### When to Use
- Use when preparing for doctoral comprehensive exams in quantitative disciplines.
- Use as a practical workbench reference while analyzing empirical research datasets.
- Use to master statistical modeling assumptions and mathematical formula derivations.

### When to Avoid
- Avoid using for qualitative thematic studies with zero numerical data.
- Avoid using as a substitute for consulting an expert biostatistician on high-stakes clinical trials.

## On-Device AI Integration

Summarize your recording in Oyma with this local instruction:

```markdown
Review the statistical methods study sheet:
1. Verify the mathematical test formulas and assumption diagnostics for accuracy.
2. Ensure both R and Python code snippets follow idiomatic, modern data science practices.
3. Formulate three challenging mock examination questions based on test assumption violations.
Output in structured Markdown.

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