15: Simple Linear Regression



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15: Linear Regression

  • Expected change in Y per unit X

Introduction (p. 15.1)

  • X = independent (explanatory) variable
  • Y = dependent (response) variable
  • Use instead of correlation
    • when distribution of X is fixed by researcher (i.e., set number at each level of X)
    • studying functional dependency between X and Y

Illustrative data (bicycle.sav) (p. 15.1)

  • Same as prior chapter
  • X = percent receiving reduce or free meal (RFM)
  • Y = percent using helmets (HELM)
  • n = 12 (outlier removed to study linear relation)

Regression Model (Equation) (p. 15.2)

  • “y hat”

How formulas determine best line (p. 15.2)

  • Distance of points from line = residuals (dotted)
  • Minimizes sum of square residuals
  • Least squares regression line

Formulas for Least Squares Coefficients with Illustrative Data (p. 15.2 – 15.3)

  • SPSS output:

Alternative formula for slope

Interpretation of Slope (b) (p. 15.3)

  • b = expected change in Y per unit X
  • Keep track of units!
    • Y = helmet users per 100
    • X = % receiving free lunch
  • e.g., b of –0.54 predicts decrease of 0.54 units of Y for each unit X

Predicting Average Y

  • ŷ = a + bx
    • Predicted Y = intercept + (slope)(x)
    • HELM = 47.49 + (–0.54)(RFM)
  • What is predicted HELM when RFM = 50?
    • ŷ = 47.49 + (–0.54)(50) = 20.5
    • Average HELM predicted to be 20.5 in neighborhood where 50% of children receive reduced or free meal
  • What is average Y when x = 20?
    • ŷ = 47.49 +(–0.54)(20) = 36.7

Confidence Interval for Slope Parameter (p. 15.4)

  • 95% confidence Interval for ß =
  • where
  • b = point estimate for slope
  • tn-2,.975 = 97.5th percentile (from t table or StaTable)
  • seb = standard error of slope estimate (formula 5)
  • standard error of regression

Illustrative Example (bicycle.sav)

  • 95% confidence interval for 
    • = –0.54 ± (t10,.975)(0.1058)
    • = –0.54 ± (2.23)(0.1058)
    • = –0.54 ± 0.24
    • = (–0.78, –0.30)

Interpret 95% confidence interval

    • Model:
    • Point estimate for slope (b) = –0.54
    • Standard error of slope (seb) = 0.24
    • 95% confidence interval for  = (–0.78, –0.30)
    • Interpretation:
    • slope estimate = –0.54 ± 0.24
    • We are 95% confident the slope parameter falls between –0.78 and –0.30

Significance Test (p. 15.5)

  • H0: ß = 0
  • tstat (formula 7) with df = n – 2
  • Convert tstat to p value
  • df =12 – 2 = 10
  • p = 2×area beyond tstat on t10
  • Use t table and StaTable

Regression ANOVA (not in Reader & NR)

  • SPSS also does an analysis of variance on regression model
  • Sum of squares of fitted values around grand mean = (ŷi – ÿ)²
  • Sum of squares of residuals around line = (yi– ŷi
  • Fstat provides same p value as tstat
  • Want to learn more about relation between ANOVA and regression? (Take regression course)

Distributional Assumptions (p. 15.5)

  • Linearity
  • Independence
  • Normality
  • Equal variance

Validity Assumptions (p. 15.6)

  • Data = farr1852.sav
    • X = mean elevation above sea level
    • Y = cholera mortality per 10,000
  • Scatterplot (right) shows negative correlation
  • Correlation and regression computations reveal:
    • r = -0.88
    • ŷ = 129.9 + (-1.33)x
    • p = .009
  • Farr used these results to support miasma theory and refute contagion theory
    • But data not valid (confounded by “polluted water source”)

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