1. Contrasts and Basis Functions Hugo Spiers Adam Liston

2. Overview Contrasts - Hugo –What are they for? –What do I need for a contrast? –What types of model can I use? –What is the best model to use? Basis Functions – Adam

3. What is a contrast used for? The GLM characterises postulated relationships between our experimental manipulations and the data Contrasts allow us to statistically test a set of possible hypothesis about this modelled data

4. What do I need for a contrast? Some data (Y) A design matrix (X) Parameters estimated with GLM (ß) A set of specific hypothesis about the data

5. Simple Example Human Brain Function 2: Chapter 8 Investigation of motor cortex Subject presses a device then rests 4 times Increasing the amount of force exerted with each press

6. Constructing the model How do we model the “press” condition? Hypothesis: We will see a linear increase in activation in motor cortex as the force increases Model this with a regressor with a value for each time point a press occurs These values increase linearly with each press Since the signal is not on average zero (even without stimuli or task) a constant offset needs to be included

7. Constructing the model Do we model the rest periods? The information contained in the data corresponds effectively to the difference between conditions and the rest period Therefore in this case NO

8. Non-parametric Model Time (scans) Regressors 1 2 3 4 5

9. Non-parametric Model Time (scans) A contrast = A linear combination parameters: C’ x ß Example c’ = 1 1 1 1 0 Regressors 1 2 3 4 5

10. Statistical Tests T-test –Tells you whether there is a significant increase or decrease in the contrast specified F-test –Tells you whether there is a significant difference between the conditions in the contrast

11. Non-parametric Model Time (scans) Example c’ = 1 0 0 0 0 Regressors 1 2 3 4 5

12. Non-parametric Model Time (scans) Example c’ = -1 1 0 0 0 Regressors 1 2 3 4 5

13. T-tests in Contrasts A one dimensional contrast T = contrast of estimated parameters variance estimate T = s 2 c’(X’X) + c c’b So, for a contrast in our model of 1 0 0 0 0: T = (ß1x1 + ß2x0 + ß3x 0 + ß4x0 + ß5x0) Estimated variance

14. Non-parametric Model Time (scans) Search for a linear increase Example c’ = 1 2 3 4 0 Regressors 1 2 3 4 5

15. Non-parametric Model Time (scans) Better to 0 centre the contrast Example c’ = -3 -1 1 3 0 Regressors 1 2 3 4 5

16. F-test To test a hypothesis about general effects, independent of the direction of the contrast F = error variance estimate additional variance accounted for by tested effects

17. Non-parametric Model Time (scans) Example Ftest c’ = 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 Regressors 1 2 3 4 5

18. Parametric Models If you have too many regressors you reduce your degrees of freedom and your chance of finding false positives rises Solution: Include regressors that explicitly takes into account prior hypotheses

19. Linear Parametric Model LINEAR PARAMETRIC ALL PRESS MEAN Time

20. 1: 2: 3: Regressors NEW REGRESSORS Main effect of pressing Removed 0 0 Linear Parametric Model

21. Non-linear models 1.Linear 2.Log 3.All press 4.mean Regressors

22. T-test Contrasts with our model 1.Linear 2.Log 3.Press 4.Mean Contrasts Regressors 1 0 0 0 - T1 0 1 0 0 - T2 0 0 1 0 - T3 0 0 0 1 - T4 -1 0 0 0 - T5 0 -1 0 0 - T6 0 0 -1 0 - T7 0 0 0 -1 - T8

23. F- contrast with this model 1.Linear 2.Log 3.Press 4.Mean 1 specified Contrast Regressors 1 2 3 4 (regressor) 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1

24. Practical Example

30. Summary Contrasts are statistical (F or T) tests of specific hypotheses Non-modelled information is taken into account implicitly in contrasts F-Contrasts look for the effects of a group of regressors T-contrasts look for increases or decreases Non-parametric models can give fine grained information about the variables in the contrast But, parametric regressors help reduce the number of regressors and test specific hypotheses directly Parametric increases should be zero centred to specifically test for their effect rather than general increases or decreases relative to the baseline Using linear and non-linear regressors can help to model parametric data more effectively

31. Switching gears… basis functions Once we have the design, how do we relate it to our data?

32. Switching gears… basis functions Once we have the design, how do we relate it to our data? Time series of haemodynamic responses

33. Switching gears… basis functions Once we have the design, how do we relate it to our data? Time series of haemodynamic responses Fit these using some shape…

34. A bad model...

35. A « better » model...

36. Basis functions Can be used in combination to describe any point in space. For instance, the (x, y, z) axes of a graph are basis functions which combine to describe points on the graph Orthogonality?? (x, y, z, ?)

37. Temporal basis functions

38. Fourier Series Any shape can be described by a sum of sines and cosines – violin string Any shape can be described by a sum of sines and cosines – violin string

39. Temporal basis functions

40. Basis functions used in SPM are curves used to ‘describe’ or fit the haemodynamic response.

41. Temporal basis functions

42. Summary The same question can be modelled in multiple ways, but these are not always equally good, and there are many trade-offs. T tests examine specific one-way questions F tests can look significance within any of several questions (like an ANOVA) Basis functions combine to describe the haemodynamic response

43. spanner???! For a “set” of basis functions, how do we For a “set” of basis functions, how do we use the T-test to test for an increase or a decrease?