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Bivariate Linear Regression

Bivariate Linear Regression. Linear Function. Y = a + bX +e. Sources of Error. Error in the measurement of X and or Y or in the manipulation of X. The influence upon Y of variables other than X (extraneous variables), including variables that interact with X.

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Bivariate Linear Regression

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  1. Bivariate Linear Regression

  2. Linear Function Y = a + bX +e

  3. Sources of Error • Error in the measurement of X and or Y or in the manipulation of X. • The influence upon Y of variables other than X (extraneous variables), including variables that interact with X. • Any nonlinear influence of X upon Y.

  4. The Regression Line • r2 < 1  Predicted Y regresses towards mean Y • Least Squares Criterion:

  5. Our Beer and Burger Data

  6. Pearson r is a Standardized Slope • Pearson r is the number of standard deviations that predicted Y changes for each one standard deviation change in X.

  7. Error Variance • What is SSE if r = 0? • If r2 > 0, we can do better than just predicting that Yi is mean Y.

  8. Standard Error of Estimate • Get back to the original units of measurement.

  9. Regression Variance • Variance in Y “due to” X • p is number of predictors. • p = 1 for bivariate regression.

  10. Coefficient of Determination • The proportion of variance in Y explained by the linear model.

  11. Coefficient of Alienation • The proportion of variance in Y that is not explained by the linear model.

  12. Testing Hypotheses • H: b = 0 • F = t2 • One-tailed p from F = two-tailed p from t

  13. Source Table • MStotal is nothing more than the sample variance of the dependent variable Y. It is usually omitted from the table.

  14. Power Analysis • Using Steiger & Fouladi’s R2.exe

  15. Power = 13%

  16. G*Power = 15%

  17. Summary Statement The linear regression between my friends’ burger consumption and their beer consumption fell short of statistical significance, r = .8,beers = 1.2 + 1.6 burgers, F(1, 3) = 5.33, p = .10.The most likely explanation of the nonsiginficant result is that it represents a Type II error. Given our small sample size (N = 5), power was only 13% even for a large effect (ρ = .5).

  18. The Regression Line is Similar to a Mean

  19. Increase n to 10 • Same value of F • r2 = SSregression SStotal, = 25.6/64.0 = .4 (down from .64).

  20. Power Analysis

  21. Power = 33%

  22. G*Power = 36%

  23. Increase n to 10 A .05 criterion of statistical significant was employed for all tests. An a priori power analysis indicated that my sample size (N = 10) would yield power of only 33% even for a large effect (ρ = .5). Despite this low power, the analysis yielded a statistically significant result. Among my friends, beer consumption increased significantly with burger consumption, r = .632,beers = 1.2 + 1.6 burgers, F(1, 8) = 5.33,p = .05.

  24. Testing Directional Hypotheses • H: b  0 H1: b > 0 • For F, one-tailed p = .05 • half-tailed p = .025. • P(AB) = P(A)P(B) = .5(.05) = .025

  25. Assumptions • To test H: b = 0 or construct a CI • Homoscedasticity across Y|X • Normality of Y|X • Normality of Y ignoring X • No assumptions about X • No assumptions for descriptive statistics (not using t or F)

  26. Placing Confidence Limits on Predicted Values of Mean Y|X • To predict the mean value of Y for all subjects who have some particular score on X:

  27. Placing Confidence Limits on Predicted Values of Individual Y|X

  28. Bowed Confidence Intervals

  29. Testing Other Hypotheses • Is the correlation between X and Y the same in one population as in another? • Is the slope for predicting Y from X the same in one population as in another? • Is the intercept for predicting Y from X the same in one population as in another.

  30. Can differ on r but not slope or slope but not r.

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