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Simple Regression Mary M. Whiteside, PhD.

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Presentation on theme: "Simple Regression Mary M. Whiteside, PhD."— Presentation transcript:

1 Simple Regression Mary M. Whiteside, PhD

2 Overview Model Data Least Squares Criterion
Interpreting b0, b1, and R2 Partitioning of the Sum of Squares LINE Assumptions Inferences Diagnostics Caveats

3 Model Y = b0 + b1X + e m(Y|x) = b0 + b1X
where Y and e are random variables X is a variable with fixed values b0, b1 are unknown parameters

4 Data n independent observations of Y for fixed values of X
bi-variate pairs (xi,yi) i = 1, 2, …, n. Data comprise a random sample from a finite population or independent observations of a random variable Example: (female literacy, infant mortality rate)

5 Least Squares Criterion
Minimize sum of squared error from the point (y) to the line (yhat) Unique analytical solution obtained by differentiating the sum of squared error with respect to b0 and b1

6 Interpreting b0, b1 and R2 bo is only interpretable if zero, (0) is in the range of the X data. Then, it is the estimated expected value of Y when x = 0. b1 is the estimated change in the expected value of Y when X increases by 1 (change scale for context) R2 is the reduction in the squared error of Y associated with the linear relationship with X

7 Partitioning of the total sum of squares TSS of Y
(mean adjusted)TSS = MSS + SSE Total (mean adjusted) sum of squares = model sum of squares + sum of squared error S[y - ȳ]2 = S[ŷ – ȳ]2 + S[y – ŷ]2 R2 = 1 – (SSE/TSS)

8 LINE Assumptions Linearity - check scatterplot of data or residuals vs. X Independence - check scatterplots of residuals vs. Yhat, X Normality – histograms of residuals Equal Variance - scatterplots of residuals by Yhat, X

9 Confidence interval b0, b1 (2,3) Prediction interval Ynext (4)
Inferences Tests of H0: b1=0 (r=0) (1) Confidence interval b0, b1 (2,3) Prediction interval Ynext (4) Confidence interval mY|x (5)

10 Caveats Outliers Extrapolations Outliers


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