Corr And Regress

Topics Covered: ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

The relationship between x and y ,[object Object],[object Object],[object Object],[object Object]

Scattergrams Y X Y X Y X Y Y Y Positive correlation Negative correlation No correlation

Variance vs Covariance ,[object Object],[object Object],[object Object]

Variance vs Covariance ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Covariance ,[object Object],[object Object],[object Object]

Example Covariance x y ( )( ) 0 3 - 3 0 0 2 2 - 1 - 1 1 3 4 0 1 0 4 0 1 - 3 - 3 6 6 3 3 9 What does this number tell us? x x i  y y i  x i x  y i y  3  x 3  y   7

Problem with Covariance: ,[object Object]

Example of how covariance value relies on variance 4.67 Covariance: 1166.67 Covariance: 28 Sum of x error * y error : 7000 Sum of x error * y error : 50 51 50 51 Mean 9 47 48 2500 0 1 7 4 48 49 900 20 21 6 1 49 50 100 40 41 5 0 50 51 0 50 51 4 1 51 52 100 60 61 3 4 52 53 900 80 81 2 9 53 54 2500 100 101 1 X error * y error y x x error * y error y x Subject Low variance data High variance data

Solution: Pearson’s r ,[object Object],[object Object],[object Object],[object Object]

Limitations of r ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Regression ,[object Object],[object Object]

Best-fit Line ,[object Object],[object Object],[object Object],[object Object],[object Object],= ŷ , predicted value intercept ε ŷ = a x + b ε = residual error = y i , true value slope

Least Squares Regression ,[object Object],Residual ( ε ) = y - ŷ Sum of squares of residuals = Σ (y – ŷ ) 2 Model line: ŷ = ax + b ,[object Object],[object Object],a = slope, b = intercept

Finding b ,[object Object],ε ε b b ,[object Object],b

Finding a ,[object Object],b ,[object Object],b b

Minimising sums of squares ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Values of a and b sums of squares (S) Gradient = 0 min S

The maths bit ,[object Object],[object Object],[object Object]

The solution ,[object Object],[object Object],[object Object],[object Object],[object Object],a = r s y s x r = correlation coefficient of x and y s y = standard deviation of y s x = standard deviation of x

The solution cont. ,[object Object],[object Object],[object Object],[object Object],y = ax + b b = y – ax b = y – ax b = y - r s y s x r = correlation coefficient of x and y s y = standard deviation of y s x = standard deviation of x x

Back to the model ,[object Object],[object Object],[object Object],Rearranges to: a b a a ŷ = ax + b = r s y s x r s y s x x + y - x r s y s x ŷ = (x – x) + y

How good is our model? ,[object Object],[object Object],[object Object],This is the variance explained by our regression model This is the variance of the error between our predicted y values and the actual y values, and thus is the variance in y that is NOT explained by the regression model s y 2 = ∑ (y – y) 2 n - 1 SS y df y = s ŷ 2 = ∑ (ŷ – y) 2 n - 1 SS pred df ŷ = s error 2 = ∑ (y – ŷ) 2 n - 2 SS er df er =

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],How good is our model cont.

How good is our model cont. ,[object Object],[object Object],[object Object],[object Object]

Is the model significant? ,[object Object],[object Object],=......= complicated rearranging ,[object Object],t (n-2) = r (n - 2) √ 1 – r 2 (because F = t 2) So all we need to know are r and n F (df ŷ ,df er ) = s ŷ 2 s er 2 r 2 (n - 2) 2 1 – r 2

General Linear Model ,[object Object],[object Object],[object Object]

Multiple regression ,[object Object],[object Object],[object Object],[object Object],[object Object]

SPM ,[object Object],[object Object],[object Object],[object Object]

Corr And Regress

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Corr And Regress