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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)

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:

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 )²
- 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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