Linear Hierarchical Models
This presentation is the property of its rightful owner.
Sponsored Links
1 / 15

Linear Hierarchical Models PowerPoint PPT Presentation


  • 78 Views
  • Uploaded on
  • Presentation posted in: General

Linear Hierarchical Models . Corinne Iola Giorgia Silani. SPM for Dummies. Outline. Fixed Effects versus Random Effects Analysis: how linear hierarchical models work Single-subject Multi-subjects Population studies. RFX: an example of hierarchical model.

Download Presentation

Linear Hierarchical Models

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


Linear hierarchical models

Linear Hierarchical Models

Corinne Iola

Giorgia Silani

SPM for Dummies


Outline

Outline

  • Fixed Effects versus Random Effects Analysis: how linear hierarchical models work

    • Single-subject

    • Multi-subjects

    • Population studies


Rfx an example of hierarchical model

RFX: an example of hierarchical model

Y = X(1)(1) + e(1)(1st level) – within subject

:

(1) = X(2)(2) + e(2) (2nd level) – between subject

Y = scans from all subjects

X(n) = design matrix at nth level

(n)= parameters - basically the s of the GLM

e(n)= N(m,2) error we assume there is a Gaussian distribution with a mean (m)

and variation (2)


Linear hierarchical models

Hierarchical form

1st level y = X(1) (1) +(1)

2nd level (1) = X(2) (2) +(2)


Random effects analysis why

Random Effects Analysis: why?

  • Interested in individual differences, but also

  • …interested in what is common

    As experimentalists we know…

  • each subjects’ response varies from trial to trial (with-in subject variability)

  • Also, responses vary from subject to subject (between subject variability)

  • Both these are important when we make inference about the population


Random effects analysis why1

Random Effects Analysis : why?

  • with-in subject variability – Fixed effects analysis (FFX) or 1st level analysis

    • Used to report case studies

    • Not possible to make formal inferences at population level

  • with-in and between subject variability – Random Effect analysis (RFX) or 2nd level analysis

    • possible to make formal inferences at population level


How do we perform a rfx

How do we perform a RFX?

  • RFX (Parameter and Hyperparameters (Variance components)) can be estimated using summary statistics or EM (ReML) algorithm

  • The gold standard approach to parameter and hyperparameter is the EM (expectation maximization)….(but takes more time…)

  • EM

    • estimates population mean effect as MEANEM

    • the variance of this estimate as VAREM

    • For N subjects, n scans per subject and equal within-subject variance we have

      VAREM = Var-between/N + Var-within/Nn

  • Summary statistics

    • Avg[a]

    • Avg[Var(a)]

  • However, for balanced designs (N~12 and same n scans per subject).

    • Avg[a] = MEANEM

    • Avg[Var(a)] = VAREM


Random effects analysis

Random Effects Analysis

  • Multi - subject PET study

  • Assumption - that the subjects are drawn at random from the normal distributed population

  • If we only take into account the within subject variability we get the fixed effect analysis (i.e. 1st level - multisubject analysis)

  • If we take both within and between subjects we get random effects analysis (2nd level analysis)


Linear hierarchical models

1

^

s21

with -in

Single-subject FFX

t = ___

Subj1= -1 1 0 0 0 0 0 0 0 0


Linear hierarchical models

<i>

^

<s2i> with -in

Multi-subject FFX

t = ___

Group= -1 1 -1 1 -1 1 -1 1 -1 1


Linear hierarchical models

<i>

^

^

<s2i> + with -in

<s2i> between

RFX analysis

t = ________

}

Subj1= -1 1 0 0 0 0 0 0 0 0

Subj2= 0 0 -1 1 0 0 0 0 0 0

@2nd level

Subj5= 0 0 0 0 0 0 0 0 -1 1


Linear hierarchical models

Differences between RFX and FFX


Linear hierarchical models

^

^

^

^

^

11

12

1

2

^

^

^

^

2

12

1

11

Random Effects Analysis : an fMRI study

1st Level 2nd Level

DataDesign MatrixContrast Images

SPM(t)

One-sample

t-test @2nd level


Two populations

Two populations

Estimated

population

means

Contrast images

Two-sample

t-test @2nd level


Example multi session study of auditory processing

Example: Multi-session study of auditory processing

SS results

EM results

Friston et al. (2003) Mixed effects and fMRI studies, Submitted.


  • Login