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Sociology 601 Class 17: October 28, 2009

Sociology 601 Class 17: October 28, 2009. Review (linear regression) new terms and concepts assumptions reading regression computer outputs Correlation (Agresti and Finlay 9.4) the correlation coefficient r relationship to regression coefficient b r-squared: the reduction in error.

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Sociology 601 Class 17: October 28, 2009

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  1. Sociology 601 Class 17: October 28, 2009 • Review (linear regression) • new terms and concepts • assumptions • reading regression computer outputs • Correlation (Agresti and Finlay 9.4) • the correlation coefficient r • relationship to regression coefficient b • r-squared: the reduction in error

  2. Review: Linear Regression • New terms and concepts • slope • intercept •  •  • negative and positive slopes • zero slope • least squares regression • predicted value • residuals • sums of squares error

  3. Review: Linear Regression • Assumptions • random sample (errors are independent) • linear • no heteroscedasticity • no outliers • Linearity, heteroscedasticity, and outliers can be checked with scattergrams and crosstabs • before computing regressions • on residuals

  4. A Problem with Regression Coefficients Regression coefficients don’t measure the strength of an association in a way that is easily compared across different models with different variables or different scales. • Rescaling one or both axes changes the slope b. • Example: murder rate and poverty rate for 50 US States. Yhat = -.86 + .58X , • where Y = murder rate per 100,000 per year • and X = poverty rate per 100 • If we rescale y, the murder rate, to murders per 100 persons per year, then Yhat = -.00086 + .00058X • (does this mean the association is now weaker?) • If we rescale x, the poverty rate, to proportion in poverty (0.00 -> 1.00), then Yhat = -.86 + 58X • (does this mean the association is now stronger?)

  5. the correlation – a standardized slope • An accepted solution for the problem of scale is to standardize both axes (e.g., change them into z-scores with mean zero and a standard deviation of 1), then calculate the slope. b = Y / X r = (Y /sY)/(X /sX) = (Y /X )*(sX/sY)= b*(sX/sY) where

  6. The Correlation Coefficient, r • r is called … • the Pearson correlation (or simply the correlation) • the standardized regression coefficient (or the standardized slope) • r = b*(sX/sY) • r is a sample statistic we use to estimate a population parameter 

  7. Calculating r: an example Calculating r for the murder and poverty example • b = .58, sX = 4.29, sY = 3.98 • r = b*(sX/sY) = .58*(4.29/3.98) = .629= .63 • alternatively (if the murder rate is per 100 persons), • b = .00058, sX = 4.29, sY = .00398 • r = b*(sX/sY) = .00058*(4.29/.00398) = .629 = .63

  8. Properties of the correlation coefficient r: • –1  r  1 • r can be positive or negative, and has the same sign as b. • r = ± 1 when all the points fall exactly on the prediction line. • The larger the absolute value of r, the stronger the linear association. • r = 0 when there is no linear trend in the relationship between X and Y.

  9. Properties of the Correlation Coefficient r: • The value of r does not depend on the units of X and Y. • The correlation treats X and Y symmetrically • (unlike the slope β) • this means that a correlation implies nothing about causal direction! • The correlation is valid only when a straight line is a reasonable model for the relationship between X and Y.

  10. Calculating a correlation coefficient using STATA • Recall the religion and state control study, where high levels of state regulation were associated with low levels of weekly church attendance. • . correlate attend regul • (obs=18) • | attend regul • -------------+------------------ • attend | 1.0000 • regul | -0.6133 1.0000

  11. An alternative interpretation of r: proportional reduction in error • Old interpretation for murder and poverty example: r = .63, the murder rate for a state is expected to be higher by 0.63 standard deviations for each 1.0 standard deviation increase in the poverty rate. • New interpretation: by using poverty rates to predict murder rates, we explain ?? percent of the variation in states’ murder rates.

  12. Proportional reduction in error: • Predicting Y without using X: Y = Ybar + e1; E1 =  e12 =  (observed Y – predicted Y)2 = Total Sums of Squares = TSS • Predicting Y using X: Y = Yhat + e2 = a + bX + e2; E2 =  e22 =  (observed Y – predicted Y)2 = Sum of Squared Error = SSE • Proportional reduction in error: r2 = PRE = (E1 – E2 ) / E1 = (TSS – SSE) / TSS

  13. Proportional reduction in error. • calculating r 2for the murder and poverty example: • r 2 = .629 2 = .395 • alternatively (using computer output), • r 2 = (TSS – SSE) / TSS = (777.7 – 470.4)/777.7 = .395 • interpretation: 39.5% of the variation in states’ murder rates is explained by its linear relationship with states’ poverty rates.

  14. R-square • r 2is also called the coefficient of determination. • Properties of r 2: • 0  r 2  1 • r 2 = 1 (its maximum value) when SSE = 0. • r 2 = 0 when SSE = TSS. (furthermore, b = 0) • the higher r 2is, the stronger the linear association between X and Y. • r 2 does not depend on the units of measurement. • r 2 takes the same value when X predicts Y as when Y predicts X.

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