Introduction to Regression Analysis

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Introduction to Regression Analysis

• Straight lines, fitted values, residual values,
  sums of squares, relation to the analysis of
  variance

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Population characteristics

• Expected value
• Is a conditional mean it is dependent on
• The conditional mean is called the regression
• When this mean is a straight line, we write

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Alternately
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The Errors

• Are independent
• This assumption is important
• Excluded Longitudinal data repeated measures
  split units cross overs clustering

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The sample gives the estimates of the population
characteristics

• For example
• Some books write
• Your choice (but be clear!)

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A simple example

• X Y
• 1 2
• 2 3
• 3 3
• 4 7
• 5 10

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Residual Sum of Squares

• Without X
• With X
• The difference is the sum of squares
  attributable to X The Regression SS

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Analysis of variance table

• Source SS df MS
• Regression 40 1 40
• Residual 6 3 2
• Total 46 4
• F 40/2 20 cf F(1,3)
• p-value P(F gt 20) 0.0208

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Regression analysis

• . regr y x
• Source SS df MS
  Number of obs 5
• -------------------------------------------
  F( 1, 3) 20.00
• Model 40 1 40
  Prob gt F 0.0208
• Residual 6 3 2
  R-squared 0.8696
• -------------------------------------------
  Adj R-squared 0.8261
• Total 46 4 11.5
  Root MSE 1.4142
• --------------------------------------------------
  ----------------------------
• y Coef. Std. Err. t
  Pgtt 95 Conf. Interval
• -------------------------------------------------
  ----------------------------
• x 2 .4472136 4.47
  0.021 .5767667 3.423233
• _cons -1 1.48324 -0.67
  0.548 -5.720331 3.720331
• --------------------------------------------------
  ----------------------------

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Centering

• . gen xcx-3
• . regr y xc
• Source SS df MS
  Number of obs 5
• -------------------------------------------
  F( 1, 3) 20.00
• Model 40 1 40
  Prob gt F 0.0208
• Residual 6 3 2
  R-squared 0.8696
• -------------------------------------------
  Adj R-squared 0.8261
• Total 46 4 11.5
  Root MSE 1.4142
• --------------------------------------------------
  ----------------------------
• y Coef. Std. Err. t
  Pgtt 95 Conf. Interval
• -------------------------------------------------
  ----------------------------
• xc 2 .4472136 4.47
  0.021 .5767667 3.423233
• _cons 5 .6324555 7.91
  0.004 2.987244 7.012756
• --------------------------------------------------
  ----------------------------