Untitled Flashcards Set

t-test

• Usage: Compare means of two independent samples.

• Model Form: Yij=μ+αi+ϵijY_{ij} = \mu + \alpha_i + \epsilon_{ij}Yij​=μ+αi​+ϵij​

• EDA: Box plot

• Example(s): Teaching evaluation data (Module 01)

• Assumptions: Independence, equal variance (unless using unequal variances test)

• Null Hypothesis: μ1=μ2\mu_1 = \mu_2μ1​=μ2​

• Follow-up Procedures: None

• Phrasing Conclusions: There is a significant difference in the mean [response] between [group 1] and [group 2].

Paired t-test

• Usage: Compare means of paired samples (same subjects measured at different times).

• Model Form: Same as t-test (slightly more complicated error structure)

• EDA: Profile of how each observation changes over time

• Example(s): Teaching evaluation data (Module 02)

• Assumptions: Independence between pairs

• Null Hypothesis: μ1=μ2\mu_1 = \mu_2μ1​=μ2​ or the true mean difference D=0D = 0D=0

• Follow-up Procedures: None

• Phrasing Conclusions: There is a significant difference in the mean [response] between [group 1] and [group 2].

One-way ANOVA

• Usage: Compare means of a response at two or more factor levels of one factor.

• Model Form: Yi=μ+αi+ϵiY_i = \mu + \alpha_i + \epsilon_iYi​=μ+αi​+ϵi​

• EDA: Box plot

• Example(s): Tire data (Module 02)

• Assumptions: Independence, constant error variance, normality

• Null Hypothesis: α1=α2=⋯=αk=0\alpha_1 = \alpha_2 = \dots = \alpha_k = 0α1​=α2​=⋯=αk​=0 or μ1=μ2=⋯=μk\mu_1 = \mu_2 = \dots = \mu_kμ1​=μ2​=⋯=μk​

• Follow-up Procedures: If F-test significant, use Tukey or Dunnett multiple comparisons.

• Phrasing Conclusions: There is a significant difference in the mean [response] between [factor level] and [other factor level].

Two-way ANOVA

• Usage: When the model has two factors of interest.

• Model Form: Yij=μ+αi+βj+αβij+ϵijY_{ij} = \mu + \alpha_i + \beta_j + \alpha\beta_{ij} + \epsilon_{ij}Yij​=μ+αi​+βj​+αβij​+ϵij​

• EDA: Interaction plot

• Example(s): Advertising data (Module 03)

• Assumptions: Independence, constant error variance, normality

• Null Hypothesis: α1=α2=⋯=αk=0\alpha_1 = \alpha_2 = \dots = \alpha_k = 0α1​=α2​=⋯=αk​=0 and β1=β2=⋯=βk=0\beta_1 = \beta_2 = \dots = \beta_k = 0β1​=β2​=⋯=βk​=0

• Follow-up Procedures: If significant interaction, use Tukey/Dunnett for each factor; otherwise, treat as one-way ANOVA.

• Phrasing Conclusions: Similar to one-way ANOVA but may be more complicated if there are significant interactions.

Blocked ANOVA

• Usage: When there is a confounding factor with a known effect or one we aren't interested in.

• Model Form: Yij=μ+αi+βj+ϵijY_{ij} = \mu + \alpha_i + \beta_j + \epsilon_{ij}Yij​=μ+αi​+βj​+ϵij​

• EDA: Use block factor in plot (e.g., facet over blocks)

• Example(s): Dehumidifier power consumption (Module 03)

• Assumptions: Independence, constant error variance, normality

• Null Hypothesis: α1=α2=⋯=αk=0\alpha_1 = \alpha_2 = \dots = \alpha_k = 0α1​=α2​=⋯=αk​=0 or μ1=μ2=⋯=μk\mu_1 = \mu_2 = \dots = \mu_kμ1​=μ2​=⋯=μk​

• Follow-up Procedures: If F-test significant, use Tukey/Dunnett comparisons (no comparisons for blocking factor).

• Phrasing Conclusions: There is a significant difference in the mean [response] between [factor level] and [other factor level], adjusting for [blocking factor].

Repeated Measures ANOVA

• Usage: For within-subjects factors or multiple measurements per experimental unit.

• Model Form: Same as one-way/two-way ANOVA with more complex error structure.

• EDA: Profile plots

• Example(s): Pulse rate data (Module 04)

• Assumptions: Independence (between subjects), constant error variance, normality

• Null Hypothesis: α1=α2=⋯=αk=0\alpha_1 = \alpha_2 = \dots = \alpha_k = 0α1​=α2​=⋯=αk​=0 or μ1=μ2=⋯=μk\mu_1 = \mu_2 = \dots = \mu_kμ1​=μ2​=⋯=μk​

• Follow-up Procedures: Similar to one-way/two-way ANOVA but may not require multiple comparisons for within-subjects factors.

• Phrasing Conclusions: Same as interpreting other ANOVA models.

Simple Linear Regression

• Usage: Estimating the relationship between one predictor and a response.

• Model Form: Y=β0+β1X1+ϵY = \beta_0 + \beta_1 X_1 + \epsilonY=β0​+β1​X1​+ϵ

• EDA: Scatterplot

• Example(s): Manatee data, supervisors (Module 05)

• Assumptions: Independence, constant error variance, normality, and linearity

• Null Hypothesis: β1=0\beta_1 = 0β1​=0

• Follow-up Procedures: Box-Cox plot for potential power transformations if model is non-linear or has non-constant variance.

• Phrasing Conclusions: [Response] increases by [b1] units for each one-unit increase in X1.

Multiple Linear Regression

• Usage: Similar to simple linear regression, but for multiple predictors.

• Model Form: Y=β0+β1X1+β2X2+⋯+βkXk+ϵY = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \dots + \beta_k X_k + \epsilonY=β0​+β1​X1​+β2​X2​+⋯+βk​Xk​+ϵ

• EDA: Scatterplot Matrix

• Example(s): State Burglary Rate, Homework 4, Patient Satisfaction (Modules 05, 06)

• Assumptions: Independence, constant error variance, normality, and linearity

• Null Hypothesis: β1=β2=⋯=βk=0\beta_1 = \beta_2 = \dots = \beta_k = 0β1​=β2​=⋯=βk​=0

• Follow-up Procedures: Try different predictor sets or transformations, checking adjusted R squared.

• Phrasing Conclusions: [Response] increases by [b1] units for every one-unit increase in X1, holding other variables constant.