Sociology 601 Martin Lecture 17: November 25

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Sociology 601 (Martin)Lecture 17 November 25

• Correlation (Agresti and Finlay 9.4)
• The problem we need to fix 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 the murder rate to murders per 100
  persons per year, then Yhat -.00086 .00058X
• (does this mean the association is now weaker?)

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the correlation a standardized slope

• An accepted solution for the problem of scale is
  to standardize both axes, then calculate the
  slope.
• b ?Y / ?X
• r (?Y /sY)/(?X /sX) (?Y /?X )(sX/sY)
  b(sX/sY)
• where

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More on the correlation.

• 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 ?

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Measuring linear association the correlation.

• 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
  

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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 larger
  the degree of linear association.
• r 0 when there is no linear trend in the
  relationship between X and Y.

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More 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 ß)
• this means that a correlation says nothing about
  causal direction!
• The correlation is valid only when a straight
  line is a reasonable model for the relationship
  between X and Y.

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Examples of the correlation coefficient r

• b 1, r 1
• b 5, r 1
• b .2, r 1
• b -1, r -1
• b .5, r .8
• b .5, r .3
• b 0, linear assumption holds
• b 0, linear assumption does not hold

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Finding 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
• (obs18)
• attend regul
• -------------------------------
• attend 1.0000
• regul -0.6133 1.0000

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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.

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Proportional reduction in error

• Predicting Y without using X
• Y Ybar e1
• E1 ? e12 ? (observed Y predicted Y)2
• Predicting Y using X
• Y Yhat e2 a bX e2
• E2 ? e22 ? (observed Y predicted Y)2
• Proportional reduction in error
• r2 PRE (E1 E2 ) / E1 (TSS SSE) / TSS

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Proportional reduction in error.

• calculating r 2 for 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.

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Proportional reduction in error.

• r 2 is 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 2 is, 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.