Lecture%20Video%20W11D3%20-%20Practice%20problem%20session

Study Overview

• Participants: Males and females followed over a year.

• Parameters: Blood pressure measured initially and post-treatment after administering two different drugs.

• Objective: Analyze reduction in blood pressure as the response variable.

Key Variables

• Response Variable (y): Reduction in blood pressure.

  • Defined as the difference in blood pressure measurements before and after treatment.

• Predictor Variables:

  • x1: Drug type (binary predictor)

    • Drug 1: coded as 1

    • Drug 2: coded as 0

  • x2: Gender (binary predictor)

    • Male: coded as 1

    • Female: coded as 0

Model Specification without Interaction

• Model: In cases where drug effectiveness is assumed to be similar for both genders, no interaction term is needed:

  • [ y = \beta_0 + \beta_1x_1 + \beta_2x_2 + \epsilon ]

Model Specification with Interaction

• If there is a suspicion that drug effectiveness may differ by gender, include an interaction term:

  • Model:

  • [ y = \beta_0 + \beta_1x_1 + \beta_2x_2 + \beta_3 (x_1 \times x_2) + \epsilon ]

Hypothesis Testing

• To assess interaction:

  • Null Hypothesis (H0): ( \beta_3 = 0 ) (no interaction)

  • Alternative Hypothesis (H1): ( \beta_3
    eq 0 ) (interaction present)

• If interaction is confirmed, to compare drug effectiveness:

  • Test: ( \beta_1 = 0 ) and ( \beta_3 = 0 ) vs at least one non-zero

Conceptual Insights

• Interaction Definition: Occurs when the effect of one variable depends on another.

• Determining Interaction:

  • If drug effectiveness is consistent across genders, no interaction is needed.

  • If potential differences exist, interaction terms should be included in the model.

Continuous vs. Binary Interaction Example

• Model incorporating both types:

  • Predictors: Continuous predictor (x1) and binary predictor (x2 - gender).

• Determine signs for coefficients based on modeled relationships:

  • Beta 1: Slope for females, sign derived from visual model analysis (i.e., upward or downward trend).

  • Beta 3: Represents the difference between male and female slopes,

    • Determine by assessing the slopes visually.

  • Beta 2: The intercept difference between males and females.

Evaluating Interactions in Practical Context

• Consider real-life scenarios:

  • Analyze slope for specific predictors in two-way interaction context (e.g., elevation and rainfall).

  • Assess increases in response under various configurations of predictors with interactions included.

Key Takeaways

• Recognize when interactions are necessary based on assumptions of differences in treatment effects across groups.

• Evaluate null and alternative hypotheses relevant to model components, especially interaction terms.

• Interpretation of coefficients becomes complex with interactions; careful distinction between main effects and slopes required.