Ch 14, Linear Regr'

1
Ch 14, Linear Regr.

• Quantitative variable measured with numbers
• Categorical variable specifies categories, not
  numeric

2
Ch 14, Linear Regr.

• One mean, one prob use quantitative vars
• Chi2 two category vars
• Two means, two prob one (simple) category, one
  quantitative
• ANOVA one quantitative, one category (multiple
  levels)

3
Ch 14, Linear Regr.

• Regression relates two quantitative vars
• House price vs time to sell
• SAT vs graduation QPR
• Fat in diet vs blood pressure

4
Ch 14, Linear Regr.

• Linear regression uses a linear function to
  relate X and Y
• (We will only consider case of a single X, but
  the method can be extended to several X
  variables)
• Y response var
• X predictor or explanatory var

5
Ch 14, Linear Regr.

• Yhat b0 b1X e
• E has normal distn with mean0 and unknown SD
• (Think t distn)
• Need to find b0 and b1

6
Ch 14, Linear Regr.

• Least squares
• Min S (Y Yhat)2
• Use Tools, Data Analysis, Regression

7
Ch 14, Linear Regr.

• See Ch4, p 165
• Sxx ? (x-xavg)(x-xavg)
• Sxy ? (x-xavg)(y-yavg)
• Syy ? (y-yavg)(y-yavg)

8
Ch 14, Linear Regr.

• Slope Sxy/Sxx
• Note units are y per x
• Intercept is based on (xavg, yavg) being on the
  line
• So intercept yavg slopexavg

9
Ch 14, Linear Regr.

• The value of S (Y Yhat)2 is SSE
• SSR (like SSTr) S (Yavg Yhat)2
• dfR 1 (one explanatory var)
• DfeN-2
• Calculate PV using F distn
• H0 Y does not depend on X (slope0)
• Ha slope not 0

10
Ch 14, Linear Regr.

• Since b1 depends on Y and Y is random, b1 is also
  random
• With est SD, use t distn
• SD(slope)SD(E)/vSxx
• And SD(E) RMSE
• DfN-2
• Est SD from Excel
• PV for two-sided t test is same as for F
• Can also find conf intervals

11
Ch 14, Linear Regr.

• If Y depends on X, then does X depend on Y?
• Can reverse roles
• New slope is NOT reciprocal of old slope
• ANOVA is not the same
• But PV is exactly the same

12
Ch 14, Linear Regr.

• Correlation can be thought of as a way of
  measuring relationships that is symmetric in X
  and Y
• -1 lt corr lt 1
• Corr2 product of two slopes
• (Note that the two slopes always have the same
  sign)

13
Ch 14, Linear Regr.

• Index of determination, R2
• R2 SSR/SSTotal
• percent of total variation explained by the
  model
• R2 corr2

14
Ch 14, Linear Regr.

• P 648 for t test for correlation
• TR/v (1-R2)/(N-2)
• (Note N-2 df in denominator)
• Usually only find Pvalue, not CI
• Two-sided PV is exactly same as F in ANOVA

15
Ch 14, Linear Regr.

• Consider 2 sample t test
• Table 10.3 on professors salaries
• Let X0 for public and 1 for private
• Let Y be all the salaries
• Compare to 2 sample t test