Lorelei Howard and Nick Wright MfD 2008 t-tests, anova and regression

Summary

• Correlation
• Regression
• Relevance to SPM

Regression

• Regression: Prediction of one variable from knowledge of one or more other variables.
• Regression v. correlation: Regression allows you to predict one variable from the other (not just say if there is an association).
• Linear regression aims to fit a straight line to data that for any value of x gives the best prediction of y.

Best fit line, minimising sum of squared errors

• Describing the line as in GCSE maths: y = m x + c
• Here, ŷ = bx + a
• ŷ : predicted value of y
• b: slope of regression line
• a: intercept

• Residual error (ε): Difference between obtained and predicted values of y (i.e. y- ŷ).
• Best fit line (values of b and a) is the one that minimises the sum of squared errors (SSerror) (y- ŷ)2

• ε

• ε = residual

• = y i , observed

• = ŷ, predicted

• ŷ = bx + a

How to minimise SSerror

• Minimise (y- ŷ)2 , which is (y-bx+a)2
• Plotting SSerror for each possible regression line gives a parabola.
• Minimum SSerror is at the bottom of the curve where the gradient is zero – and this can found with calculus.
• Take partial derivatives of (y-bx-a)2 and solve for 0 as simultaneous equations, giving:

• Values of a and b

• Sums of squared error (SSerror)

• Gradient = 0
• min SSerror

How good is the model?

• We can calculate the regression line for any data, but how well does it fit the data?
• Total variance = predicted variance + error variance
• sy2 = sŷ2 + ser2
• Also, it can be shown that r2 is the proportion of the variance in y that is explained by our regression model
• r2 = sŷ2 / sy2
• Insert r2 sy2 into sy2 = sŷ2 + ser2 and rearrange to get:
• ser2 = sy2 (1 – r2)
• From this we can see that the greater the correlation the smaller the error variance, so the better our prediction

Is the model significant?

• i.e. do we get a significantly better prediction of y from our regression equation than by just predicting the mean?
• F-statistic:

• F

• (dfŷ,dfer)

• =

• sŷ2

• ser2

• =......=

• r2 (n - 2)2

• 1 – r2

• complicated
• rearranging

• And it follows that:

• t(n-2) =

• r (n - 2)

• √1 – r2

• So all we need to
• know are r and n !

Summary

• Correlation
• Regression
• Relevance to SPM

General Linear Model

• Linear regression is actually a form of the General Linear Model where the parameters are b, the slope of the line, and a, the intercept.
• y = bx + a +ε
• A General Linear Model is just any model that describes the data in terms of a straight line

• 
• 
• 
• 
• 
• 
• 
• 
• 

• +

• 

• =

• 

• +

• Y

• X

• data vector (Voxel)

• design matrix

• parameters

• error vector

• ×

• =

• One voxel: The GLM
• Our aim: Solve equation for β – tells us how much BOLD signal is explained by X

Multiple regression

• Multiple regression is used to determine the effect of a number of independent variables, x1, x2, x3 etc., on a single dependent variable, y
• The different x variables are combined in a linear way and each has its own regression coefficient:
• y = b0 + b1x1+ b2x2 +…..+ bnxn + ε
• The a parameters reflect the independent contribution of each independent variable, x, to the value of the dependent variable, y.
• i.e. the amount of variance in y that is accounted for by each x variable after all the other x variables have been accounted for

SPM

• Linear regression is a GLM that models the effect of one independent variable, x, on one dependent variable, y
• Multiple Regression models the effect of several independent variables, x1, x2 etc, on one dependent variable, y
• Both are types of General Linear Model
• This is what SPM does and will be explained soon…

Summary

• Correlation
• Regression
• Relevance to SPM
• Thanks!

Directory: mfd archive -> 2008
mfd archive -> Bayes for Beginners
2008 -> Estimating the Transfer Function from Neuronal Activity to bold

Download 13.89 Kb.

Share with your friends: