Statistical Analysis Cheat Sheet: Formulas and Methods

Sample Size Calculation

To find what n must be:

n ≥ (Z_1-α/2 / MOE)² * p(1-p)

Wald Confidence Interval

SE^ = sqrt((p^ * (1 – p)) / n)

Odds Ratio Analysis

Calculated by criss-cross of 2x2 table.

Example: The odds for someone to smoke ≥ 5 cigarettes and have lung cancer is 3.40 times that of someone who smokes < 5 cigarettes daily to have lung cancer.

Fisher's Exact Test

Used when cell frequencies are < 5 in a 2x2 table. Tests for independence.

• H₀: Odds ratio = 1
• H_A: Odds ratio ≠ 1

Chi-Square Test of Independence

• H₀: The two variables are independent.
• H_A: The two variables are not independent.

Logit Model

To find the number of odds used in the model, find all combinations of levels (e.g., a 2x3 model has 6 combinations).

Exp(estimate) = odds for that factor.

This means that for a specific FACTOR, you have ODDS times higher odds of being in the CONTEXT than that of the REFERENCE LEVEL, while adjusting for OTHER FACTORS.

Model Formulas

• Predicted probability of success: P(x) = exp(b₀ + B₁X) / (1 + exp(b₀ + B₁X))
• Odds of success: p(x) / (1 - p(x)) = exp(B₀ + B₁X)
• Odds ratio: odds A / odds B = exp(B₁(a - b))

Simple Linear Regression

Degrees of freedom in the complete model:

DF_c = n − number of estimated parameters

DF_c = n − 2 (For the model Y = b₀ + b₁x + E)

Experimental Design

One-Way ANOVA

Research Question: Is the mean for the variable equal among the different groups observed?

• H₀: μ₁ = μ₂ = μ₃
• H_A: At least one μ is different

Complex Model: Y_ij = μ_i + E_ij

Reduced Model: Y_ij = μ + E_ij

Assumptions

• Independence of observations: Cannot be proved through data; must be ensured during experimental design.
• Normality of residuals: Errors are normally distributed N(0, σ²). Check via Q-Q plot (data should follow the 45-degree line). Use Shapiro-Wilk test to confirm.
• Equal Variances: Variance among groups must be equal. Check via box plots (similar IQRs/medians) or Levene's test.

Violated Assumptions

1. Kruskal-Wallis Test: H₀: All medians are equal. H_A: At least one median is not equal.
2. Randomization/Permutation: Shuffle labels to see if grouping has a real effect.

Effects Model

Assumes groups have a non-random effect on the response variable. All variability is attributed to factors.

• Complex: Y_ij = μ + a_i + E_ij
• Reduced: Y_ij = μ + E_ij
• H₀: a_i = 0 for all i
• H_A: At least one a_i ≠ 0

Random Effects Model

Used to find differences in populations rather than specific treatment effects.

• Complex: Y_ij = μ + A_i + E_ij
• Reduced: Y_ij = μ + E_ij

ANCOVA