Bivariate Linear Regression

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Bivariate Linear Regression
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Linear Function

• Y a bX e

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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.

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The Regression Line

• r2 lt 1 ? Predicted Y regresses towards mean Y
• Least Squares Criterion

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Our Beer and Burger Data
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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.

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Error Variance

• What is SSE if r 0?
• If r2 gt 0, we can do better than just predicting
  that Yi is mean Y.

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Standard Error of Estimate

• Get back to the original units of measurement.

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Regression Variance

• Variance in Y due to X
• p is number of predictors.
• p 1 for bivariate regression.

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Coefficient of Determination

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

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Coefficient of Alienation

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

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Testing Hypotheses

• H? b 0
• F t2
• One-tailed p from F two-tailed p from t

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Source Table

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

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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 17 even for a large effect (?
  .5).

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The Regression Line is Similar to a Mean
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Increase n to 10

• Same value of F
• r2 SSregression ? SStotal, 25.6/64.0 .4.

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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 32 even for a large
  effect (? .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.

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Testing Directional Hypotheses

• H? b ?? 0 H1 b gt 0
• For F, one-tailed p .05
• half-tailed p .025.
• P(A?B) P(A)P(B) .5(.05) .025

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Assumptions

• To test H? b 0 or construct a CI
• Homoscedasticity across YX
• Normality of YX
• Normality of Y ignoring X
• No assumptions about X
• No assumptions for descriptive statistics (not
  using t or F)

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Placing Confidence Limits on Predicted Values of
Mean YX

• To predict the mean value of Y for all subjects
  who have some particular score on X

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Placing Confidence Limits on Predicted Values of
Individual YX
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Bowed Confidence Intervals
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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.

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