Inferences for Regression Notes

Chapter 27: Inferences for Regression

Correlation & Regression Review

• Group work for 10-15 minutes to review correlation and regression concepts.

Example: Body Fat and Waist Size

• Examines relationship between % body fat and waist size (inches).

• Scatterplot represents this data.

Understanding Regression

• Regression models the relationship between two quantitative variables:

  • Predictor variable (independent)

  • Response variable (dependent)

• Imagines an idealized regression line where means fall along this line.

Inference in Regression

• Aim to go beyond individual data points by creating confidence intervals and testing hypotheses regarding the slope of the regression line.

Population vs. Sample

• In finding confidence intervals for means, we assumed a true underlying mean.

• For regression, a corresponding collection of % body fat for given waist sizes exists; must be aware distributions change.

Idealized Regression Model

• Parameters are described using Greek letters:

  • Intercept: ( \beta_0 )

  • Slope: ( \beta_1 )

• Represents the fitted line as:

  • ( \muy = \beta0 + \beta_1 x)

• Errors must be included: ( y = \beta0 + \beta1 x + \epsilon )

Assumptions & Conditions

• In regression inference, several assumptions must be checked:

  1. Linearity Assumption:

  • Straight Enough Condition: The scatterplot must show a linear relationship.

  1. Independence Assumption:

  • Individuals in the sample must be representative with independent observations.

  1. Equal Variance Assumption:

  • Spread of residuals should remain uniform (Checked via residual plot).

  1. Normal Population Assumption:

  • Residuals should form a nearly normal distribution – unimodal and symmetric.

Condition Check Order

• Scatterplot: Check for linearity

• Fit Regression Model: Calculate residuals ( \epsilon )

• Residuals vs. Plot: Check scatter for patterns

• Histogram and Normal Probability Plot: To confirm normality of residuals

• Proceed with inference if assumptions hold.

Intuition of Regression Inference

• Sample impacts variability of estimated regression slope and intercept.

• Three factors that influence standard error of slope (( SE(b) )):

  • Scatter around regression line (measured by residual standard deviation ( S_e ))

  • Spread of x values (standard deviation ( S_X ))

  • Sample size ( n )

• Standard error formula:

  • ( SE(b) = \frac{S_e}{\sqrt{n-1}} )

Sampling Distribution

• When conditions are satisfied, the estimated regression slope follows a Student's t-distribution:

  • ( \frac{b1 - \beta1}{SE(b_1)} \sim t(n - 2) )

Regression Inference

• Null Hypothesis: ( H0: \beta1 = 0 ) (No linear relationship)

• Test Statistic:

  • ( t = \frac{b1 - 0}{SE(b1)} )

• Confidence Interval Formula:

  • ( b1 \pm t{(df)} * SE(b_1) )

Example Problem: Math and Anxiety Test Data

• Null Hypotheses: ( H0: \beta1 = 0 )

• Alternative Hypothesis: ( HA: \beta1 \neq 0 )

• Conditions to Verify:

  • Straight Enough Condition: Check scatterplot.

  • Independence: No influence from one student to another.

  • Constancy of Variance: Residuals plot consistent.

  • Normality: Histogram of residuals is unimodal and symmetric.

• Conclusion: Low P-value (0.0084) indicates strong evidence against null hypothesis.

• Confidence Interval Example:

  • ( b_1 = -4.486 \pm 2.074(1.551) = (-7.70, -1.27) )

  • Meaning: 95% confidence that an increase in anxiety correlates with a decline in math scores of 1.27 to 7.70 points.

Conclusion

• Understanding of regression concepts and rigorous checking of assumptions is crucial to valid statistical inference.

• The process allows for practical interpretations relating data attributes and trends through regression analysis.