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. Any nonlinear influence of X upon Y.

The Regression Line r 2 < 1  Predicted Y regresses towards mean Y Least Squares Criterion:

Our Beer and Burger Data

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.

Error Variance What is SSE if r = 0? If r 2 > 0, we can do better than just predicting that Y i is mean Y.

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

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

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

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

Testing Hypotheses H  : b = 0 F = t 2 One-tailed p from F = two-tailed p from t

Source Table MS total is nothing more than the sample variance of the dependent variable Y. It is usually omitted from the table.

Power Analysis Using Steiger & Fouladi’s R2.exe

Power = 13%

G*Power = 15%

Summary Statement The linear regression between my friends’ burger consumption and their beer consumption fell short of statistical significance, r =.8, beers = 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).

The Regression Line is Similar to a Mean

Increase n to 10 Same value of F r 2 = SS regression  SS total, = 25.6/64.0 =.4 (down from.64).

Power Analysis

Power = 33%

G*Power = 36%

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 = burgers, F(1, 8) = 5.33, p =.05.

Testing Directional Hypotheses H  : b  0H 1 : b > 0 For F, one-tailed p =.05 half-tailed p =.025. P(A  B) = P(A)P(B) =.5(.05) =.025

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)

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:

Placing Confidence Limits on Predicted Values of Individual Y|X

Bowed Confidence Intervals

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.

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