Regression and Correlation

Jake BlanchardFall 2010

IntroductionWe can use regression to find

relationships between random variables

This does not necessarily imply causation

Correlation can be used to measure predictability

Regression with Constant VarianceLinear Regression: E(Y|

X=x)=+xIn general, variance is function of

xIf we assume the variance is a

constant, then the analysis is simplified

Define total absolute error as the sum of the squares of the errors

Linear Regression

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Variance in Regression AnalysisRelevant variance is conditional:

Var(Y|X=x)

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Confidence IntervalsRegression coefficients are t-

distributed with n-2 dofStatistic below is thus t-

distributed with n-2 dof

And the confidence interval is

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ExampleExample 8.1Data for compressive strength (q)

of stiff clay as a function of “blow counts” (N)

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Plot

Correlation Estimate

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Regression with Non-Constant VarianceNow relax

assumption of constant variance

Assume regions with large conditional variance weighted less

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Example (8.2)Data for maximum settlement (x)

of storage tanks and maximum differential settlement (y)

From looking at data, assume g(x)=x (that is, standard deviation of y increases linearly with x

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Example (8.2) continued

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Multiple Regression

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“Nonlinear” Regression

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Use LINEST in Excel