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

Recommended

Recommended

More Related Content

What's hot

What's hot (19)

Viewers also liked

Viewers also liked (11)

Similar to Corr And Regress

Similar to Corr And Regress (20)

Recently uploaded

Recently uploaded (20)

Corr And Regress