Lecture 9 | Quizlet

Why is running multiple pairwise t-tests problematic?

It inflates the Type I error rate (family-wise error rate) across comparisons.

Family-Wise Error Rate Formula (3 tests at α = 0.05)

1 − (0.95)³ = 0.143 or 14.3% chance of at least one Type I error.

Number of Pairwise Comparisons Formula

k(k − 1) / 2, where k is the number of groups.

Core Concept of One-Way ANOVA

Compares variance between group means to variance within groups in one test.

Mean Squares Between (MSB) Formula

MSB = SSB / dfB, where dfB = k − 1.

Mean Squares Within / Error (MSW) Formula

MSW = SSW / dfW, where dfW = N − k.

F-Ratio Formula

F = MSB / MSW.

Expected F-Ratio under the Null Hypothesis

Approximately 1.

Purpose of Post-Hoc Tests

Determine specific group differences after a significant overall ANOVA without inflating α.

Examples of Post-Hoc Tests

Tukey's HSD, Bonferroni, Scheffé, Fisher's LSD, Newman-Keuls.

Definition of Correlation

A statistical technique estimating and describing the relationship between two variables.

Two Primary Properties of Correlation Coefficient (r)

Direction (positive/negative) and strength (magnitude bounded between -1 and +1).

Covariance Formula

Cov_x,y = Σ(x − x̄)(y − ȳ) / (n − 1).

Limitation of Covariance

It depends on the units of measurement and is unstandardized.

Pearson's Correlation Coefficient (r) Formula

r = Cov_x,y / (s_x · s_y) or SP / √(SS_x · SS_y).

Degrees of Freedom for Pearson's r Significance Test

df = n − 2.

Null and Alternative Hypotheses for r

H₀: ρ = 0; H₁: ρ ≠ 0.

Third-Variable Problem in Correlation

An unmeasured variable may drive the observed relationship, preventing causal inference.

Impact of Restriction of Range on Correlation

Truncating variable range can artificially decrease (or alter) the observed correlation.

Impact of Outliers on Correlation

A single extreme data point can dramatically inflate or deflate r.

Coefficient of Determination (r²)

The proportion of variance in one variable explained/shared by the other variable.

Effect Size Benchmarks for r

Small: ±0.1; Medium: ±0.3; Large: ±0.5.

Spearman's Rho (r_s)

A non-parametric correlation measuring monotonic consistency using ranked data.

When to Use Spearman's Rho

With ordinal data or non-linear monotonic relationships.

Handling Tied Ranks in Spearman's Rho

Assign each tied score the average of the ranks they occupy.

Point-Biserial Correlation (r_pb)

Measures relationship between one continuous variable and one dichotomous variable.

Phi Coefficient (ϕ)

Measures relationship between two dichotomous variables.