The  Fight  Against  Alzheimer’s  Disease  

Rules  for  Construc.ng  and    Repor.ng  Second-­‐level  Sta.s.cs    

 

Donald  G.  McLaren,  Ph.D.  GRECC,  ERNM  Veteran’s  Hospital  

Department  of  Neurology,  MGH/HMS      

hGp://www.mar.nos.org/~mclaren      

02/16/2012  

Financial  Disclosures  

•  None.  

MATLAB Basics

•  Column Major (rows then columns) •  * versus .* (. Can be used with many functions) •  (),[],{} •  Strings versus numbers •  Variable Names are: a, a1, a3 not a(1) a(2) a{2} •  = versus == •  4D data versus 2D processing •  SPM •  Nifti files

General Linear Model (GLM)

Y=aX+b

Brain Imaging

•  The data: http://labs.jam3.ca/category/voxel-engine-

jam3-labs/

Brain Imaging

•  From the beginning (almost)….

[ 5 6 7 5 ]

Analyses steps in the GLM

(1) Does an analysis of variance separately at each voxel

(2) Makes statistic from the results of this analysis, for each voxel

(3) Inferences

Converging  Evidence  of  Func.onal  Imaging  in  Alzheimer’s  

(Buckner  et  al.  2005)  

First-­‐Level  Analysis  

BOLD  Imaging  

A

0Time'(TR)

40 80 120 160

0Time'(TR)

40 80 120 1600Time'(TR)

40 80 120 160

0Time'(TR)

40 80 120 160 0Time'(TR)

40 80 120 160

B

C D

F G

Signal'(a.u.)

Signal'(a.u.)

Signal'(a.u.)

Signal'(a.u.)

Signal'(a.u.)

0Time'(TR)

2 4 6 8

0Time'(TR)

40 80 120 160

E

Signal'(a.u.)

N

PV(self)

PV(sem)

fMRI  Ac.vity  

HueGel  at  al.  

Ac.vity  in  the  Brain  

S.mulus  

Hemodynam

ic  Inverse  Problem  

(Adapted  from  Bucker  2003)  

What Model Should I Use?

•  Block Designs: Assume a Model •  Event-Related Designs:

– FIR / No Assumed Model – Assume a Model + Derivatives – Assume a Model

Other  Elements  of  the  GLM  

•  AR(1)  •  Highpass  filter  •  Mo.on  correc.on  factors  •  Excluding  .mepoints  

Quality  Control  

•  Normaliza.on  –  Did  it  work?  •  Ar.facts  –  Check  ResMS  image  or  raw  data?  •  Ac.vity  –  Do  you  get  the  expected  ac.vity  at  the  first  level  (visual  cortex  with  visual  s.muli)?  

 •  Movement  –  TR  to  TR,    overall  movement  

•  Signal  Spikes…  

Second-­‐Level  Modeling  

•  These  are  all  random  effects  (because  of  variance  correc.ons  and  using  beta’s  from  the  first  level)  

•  Mean  across  subjects  divided  by  variance  across  subjects.  – Low  subjects  with  very  low  variance  between  them  can  lead  to  a  significant  finding,  even  if  no  subject  was  significant  at  the  single  subject  level  

–  Implica.ons  for  analysis  (e.g.  SLBT??)  

Types of Data – Dependent Variable

•  Functional Connectivity – Correlation •  Functional Connectivity -- ICA •  Task Data

– Single Condition – Multiple Conditions – Multiple Predictors Per Condition

•  Context-Dependent Connectivity •  Other Types

Types of Data – Independent Variable

•  Group •  Behavioral Data

•  Performance •  Neuropsychological Scores •  Emotional Scales

•  Age •  Total Brain Volume, Hippocampal Volume,

etc. •  Images

The General Linear Model

npnpnnn

pp

pp

pp

xxxY

xxxYxxxYxxxY

εββββ

εββββ

εββββ

εββββ

+++++=

+++++=

+++++=

+++++=

,2,21,10

3,32,321,3103

2,22,221,2102

1,12,121,1101

εββββ +++++= ppXXXY 22110

=Y Y ε+observed = predicted + random error

Y = Xβ +εIn Matrix Form

Summary of the GLM

Y = X . β + ε

Observed data: SPM uses a mass univariate approach – that is each voxel is treated as a separate column vector of data. Y is the BOLD signal at various time points at a single voxel

Design matrix: Several components which explain the observed data, i.e. the BOLD time series for the voxel Timing info: onset vectors, Om

j, and duration vectors, Dmj

HRF, hm, describes shape of the expected BOLD response over time Other regressors, e.g. realignment parameters

Parameters: Define the contribution of each component of the design matrix to the value of Y Estimated so as to minimise the error, ε, i.e. least sums of squares

Error: Difference between the observed data, Y, and that predicted by the model, Xβ . Not assumed to be spherical in fMRI

Example • 

Scan no Voxel 1 Taskdifficulty

1 57.84 52 57.58 43 57.14 44 55.15 25 55.90 36 55.67 17 58.14 68 55.82 39 55.10 110 58.65 611 56.89 512 55.69 2

X can contain values quantifying experimental variable

Y X

Parameters & error

this line is a 'model' of the data

slope β = 0.23

Intercept c = 54.5

•  β: slope of line relating x to y –  ‘how much of x is

needed to approximate y?’

•  ε = residual error –  the best estimate

of β minimises ε: deviations from line

–  Assumed to be independently, identically and normally distributed

Y = βx + c + ε

GLM •  Important to model all known variables,

even if not experimentally interesting: – e.g. head movement,

block and subject effects – minimise residual error

variance for better stats – effects-of-interest are the

regressors you’re actually interested in

subjects

covariates

conditions: effects of interest

GLM – One Sample T-test

•  load SPM.mat; figure; imagesc(SPM.xX.X)

GLM – One Sample T-test

GLM – Paired T-Test

GLM – Two Sample T-Test

•  In SPM, the 1s will change if the variance is not equal and is set to unequal

GLM – Multiple Regression

x1 x2 c

GLM – Repeated Measures

GLM – Biological Parametric Mapping

Implementation MATLAB/SPM

Statistical Tests

•  T-Test

•  F-Test

Implementing the T-test

Variance Estimate

Sqrt(Var*cT(XTX)-1c)

c = +1 0 0 0 0 0 0 0

T =

contrast of estimated

parameters

t-test H0: cTβ = 0

variance estimate

Fixed Effects Magnitudes, Model-Free Example

•  Using the numerator from the T-test, we get c*b as the magnitude of the effect. = c

= b

•  c has the unique properties of summing to zero, and in certain cases c*cT is equal to 1. This is the basis for the problem, that will be shown in a few slides.

[ ] - 0.46159 -0.0949 .30717 .4317 .4578 .09565 -0.30571 -0.43014

- 0.13263 - 0.11653 .069775 .042275 .119375 .072575

- 0.02633 - 0.02853

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥Graphical Illustration of the contrast matrix

Fixed Effects Magnitudes, Model-Free Example

•  Using the numerator from the T-test, we get c*b as the magnitude of the effect. = c

= b

[ ] - 0.46159 -0.0949 .30717 .4317 .4578 .09565 -0.30571 -0.43014

- 0.13263 - 0.11653 .069775 .042275 .119375 .072575

- 0.02633 - 0.02853

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥

Por.ons  of  the  contrast  matrix  and    the  β  coefficient  matrix.                                        represents  a  mul.plica.on  of  two  matrix  elements    Add  the  products  of  the  above  mul.plica.on  to  obtain  the  magnitude.  

Graphical Illustration of the contrast matrix

Fixed Effects Magnitudes, Model-Free Example

•  Using the numerator from the T-test, we get c*b as the magnitude of the effect. = c

= b

[ ] - 0.46159 -0.0949 .30717 .4317 .4578 .09565 -0.30571 -0.43014

- 0.13263 - 0.11653 .069775 .042275 .119375 .072575

- 0.02633 - 0.02853

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥

Por.ons  of  the  contrast  matrix  and    the  β  coefficient  matrix.                                        represents  a  mul.plica.on  of  two  matrix  elements    Add  the  products  of  the  above  mul.plica.on  to  obtain  the  magnitude.  

Graphical Illustration of the contrast matrix

Fixed Effects Magnitudes, Model-Free Example

•  Using the numerator from the T-test, we get c*b as the magnitude of the effect. = c

= b

[ ] - 0.46159 -0.0949 .30717 .4317 .4578 .09565 -0.30571 -0.43014

- 0.13263 - 0.11653 .069775 .042275 .119375 .072575

- 0.02633 - 0.02853

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥

Por.ons  of  the  contrast  matrix  and    the  β  coefficient  matrix.                                        represents  a  mul.plica.on  of  two  matrix  elements    Add  the  products  of  the  above  mul.plica.on  to  obtain  the  magnitude.  

Graphical Illustration of the contrast matrix

Fixed Effects Magnitudes, Model-Free Example

•  Using the numerator from the T-test, we get c*b as the magnitude of the effect. = c

= b

[ ] - 0.46159 -0.0949 .30717 .4317 .4578 .09565 -0.30571 -0.43014

- 0.13263 - 0.11653 .069775 .042275 .119375 .072575

- 0.02633 - 0.02853

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥

Por.ons  of  the  contrast  matrix  and    the  β  coefficient  matrix.                                        represents  a  mul.plica.on  of  two  matrix  elements    Add  the  products  of  the  above  mul.plica.on  to  obtain  the  magnitude.  

Graphical Illustration of the contrast matrix

Fixed Effects Magnitudes, Model-Free Example

•  Using the numerator from the T-test, we get c*b as the magnitude of the effect. = c

= b

[ ] - 0.46159 -0.0949 .30717 .4317 .4578 .09565 -0.30571 -0.43014

- 0.13263 - 0.11653 .069775 .042275 .119375 .072575

- 0.02633 - 0.02853

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥

Por.ons  of  the  contrast  matrix  and    the  β  coefficient  matrix.                                        represents  a  mul.plica.on  of  two  matrix  elements    Add  the  products  of  the  above  mul.plica.on  to  obtain  the  magnitude.  

Graphical Illustration of the contrast matrix

Fixed Effects Magnitudes, Model-Free Example

•  Using the numerator from the T-test, we get c*b as the magnitude of the effect. = c

= b

[ ] - 0.46159 -0.0949 .30717 .4317 .4578 .09565 -0.30571 -0.43014

- 0.13263 - 0.11653 .069775 .042275 .119375 .072575

- 0.02633 - 0.02853

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥

Por.ons  of  the  contrast  matrix  and    the  β  coefficient  matrix.                                        represents  a  mul.plica.on  of  two  matrix  elements    Add  the  products  of  the  above  mul.plica.on  to  obtain  the  magnitude.  

Graphical Illustration of the contrast matrix

Fixed Effects Magnitudes, Model-Free Example

•  Using the numerator from the T-test, we get c*b as the magnitude of the effect. = c

= b

[ ] - 0.46159 -0.0949 .30717 .4317 .4578 .09565 -0.30571 -0.43014

- 0.13263 - 0.11653 .069775 .042275 .119375 .072575

- 0.02633 - 0.02853

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥

Por.ons  of  the  contrast  matrix  and    the  β  coefficient  matrix.                                        represents  a  mul.plica.on  of  two  matrix  elements    Add  the  products  of  the  above  mul.plica.on  to  obtain  the  magnitude.  

Graphical Illustration of the contrast matrix

Fixed Effects Magnitudes, Model-Free Example

•  Using the numerator from the T-test, we get c*b as the magnitude of the effect. = c

= b

[ ] - 0.46159 -0.0949 .30717 .4317 .4578 .09565 -0.30571 -0.43014

- 0.13263 - 0.11653 .069775 .042275 .119375 .072575

- 0.02633 - 0.02853

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥

Por.ons  of  the  contrast  matrix  and    the  β  coefficient  matrix.                                        represents  a  mul.plica.on  of  two  matrix  elements    Add  the  products  of  the  above  mul.plica.on  to  obtain  the  magnitude.  

Graphical Illustration of the contrast matrix

Fixed Effects Magnitudes, Model-Free Example

•  Using the numerator from the T-test, we get c*b as the magnitude of the effect. = c

= b

[ ] - 0.46159 -0.0949 .30717 .4317 .4578 .09565 -0.30571 -0.43014

- 0.13263 - 0.11653 .069775 .042275 .119375 .072575

- 0.02633 - 0.02853

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥

Por.ons  of  the  contrast  matrix  and    the  β  coefficient  matrix.                                        represents  a  mul.plica.on  of  two  matrix  elements    Add  the  products  of  the  above  mul.plica.on  to  obtain  the  magnitude.  

Graphical Illustration of the contrast matrix

Fixed Effects Magnitudes, Model-Free Example

•  Using the numerator from the T-test, we get c*b as the magnitude of the effect. = c

= b

•  c has the unique properties of summing to zero, and in certain cases c*cT is equal to 1.

[ ] - 0.46159 -0.0949 .30717 .4317 .4578 .09565 -0.30571 -0.43014

- 0.13263 - 0.11653 .069775 .042275 .119375 .072575

- 0.02633 - 0.02853

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥

Por.ons  of  the  contrast  matrix  and    the  β  coefficient  matrix.                                        represents  a  mul.plica.on  of  two  matrix  elements    Add  the  products  of  the  above  mul.plica.on  to  obtain  the  magnitude.  

Graphical Illustration of the contrast matrix

Implementing the F-test

F = error

variance estimate

additional variance

accounted for by effects of

interest

0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1

H0: cTβ = 0 c =

T/r/F  Notes  

•  If  F  is  a  single  row  contrast,  then  F=T^2  •  There  are  formulas  to  convert  between  T/r  and  other  sta.s.cs  (e.g.  cohen’s  d)  

Constructing Contrasts

Construc.ng  Contrasts  

•  What  is  the  null  hypothesis?  

•  Make  the  null  hypothesis  equal  to  0  

•  Label  the  columns  based  on  the  weigh.ng  of  the  components  of  the  null  hypothesis  – For  repeated  measures,  form  the  sub-­‐elements  of  the  contrast,  then  apply  the  weights  

Constructing Contrasts •  S1G1C1: [1 zeros(1,10) 1 0 1 0 0 1 0 0 0 0 0] •  S1G1C2: [1 zeros(1,10) 1 0 0 1 0 0 1 0 0 0 0] •  S2G1C1: [0 1 zeros(1,9) 1 0 1 0 0 1 0 0 0 0 0] •  G1: [ones(1,6)/6 zeros(1,5) 1 0 1/3 1/3 1/3 1/3 1/3 1/3 0 0 0] •  G1vsG2: [ones(1,6)/6 ones(1,5)/5 1 -1 0 0 0 1/3 1/3 1/3 -1/3 -1/3

-1/3]

Implementation MATLAB/SPM

Caveat 1: What is analyzed…

•  Missing Data – NaN – Zeros

Caveat 2: Designs

•  Between-subject Designs

•  Within-subject Designs

•  Mixed Designs

Variance  Correc.ons  

•  The  issue  of  non-­‐sphericity  

Repeated  Measures  in  FSL  

•  Limited  to  designs  that  have  no  viola.ons  of  sphericity.  

Mixed  Designs  

•  Only  look  at  within-­‐subject  effects  

Mixed  Designs  in  FSL  

Along  with  the  general  caveats,  you  can  not  use  randomise  with  mixed  designs.  

Covariates  

•  If  you  have  a  single  group:  – Demeaning  covariate  will  not  change  the  slope  – Demeaning  makes  the  group  term  the  mean  of  the  group;  whereas  not  demeaning  makes  the  group  term  the  intercept.  

Covariates  

•  If  you  have  a  mul.ple  groups:  – Demeaning  covariate  will  not  change  the  slope,  no  maGer  how  you  demean  it  

– Demeaning  within  each  group  à  controlling  for  the  covariate,  but  group  means  are  uneffected  

– Demeaning  across  everyone  à  controlling  for  the  covariate,  but  group  means  are  effected.  If  you  do  this,  you  should  refer  to  group  tests  as  a  comparison  of  covariate-­‐adjusted  means  

Inferences

Inferences

•  Cluster – AlphaSim – Extent Threshold – SPM8b – False Discovery Rate

•  Voxel – False Discovery Rate (FDR) – Family-wise Error Correction (FWE)

Inferences – AlphaSim/Extent

•  Generate X random sets of p-values, smooth them, count the number of voxels in the largest cluster in each set.

•  Determine how many voxels are in the largest clusters X% of the time to find the threshold

•  Use the result for the extent threshold

Inferences – AlphaSim/Extent Data  set  dimensions:    nx  =      100      ny  =      100      nz  =        25      (voxels)  dx  =    4.00      dy  =    4.00      dz  =    4.00      (mm)    Gaussian  filter  widths:    sigmax  =    0.00      FWHMx  =    0.00    sigmay  =    0.00      FWHMy  =    0.00    sigmaz  =    0.00      FWHMz  =    0.00      Cluster  connec.on  =  Nearest  Neighbor  Threshold  probability:  pthr  =  5.000000e-­‐03      Number  of  Monte  Carlo  itera.ons  =    1000      Cl  Size      Frequency          CumuProp          p/Voxel      Max  Freq              Alpha              1              1213114          0.985345    0.00499870                    0          1.000000              2            17579            0.999623    0.00014624                626          1.000000              3            454            0.999992    0.00000561                364          0.374000              4            10            1.000000    0.00000016                  10          0.010000  

Inferences -- FDR

•  q% of significant voxels are false positives

•  Rank all p-values •  Find max(i) such that pi ≤ i/V*q

– V is search space

Inferences -- FDR

p(i)

i/V i/V × q

0 1

0 1

Topological  FDR  

•  Correc.on  for  the  expected  number  of  peaks  or  clusters  using  the  false  discovery  rate.  

The  Search  Space  

•  How  many  voxels  (resels)  are  you  tes.ng  •  resels  –  resolu.on  elements  

Finding  Clusters  and  Peaks  [voxels  voxelstats  clusterstats  regions  mapparameters  UID]  =peak_nii(‘myimage.nii’,mapparameters)    UID  -­‐-­‐  This  was  added  to  allow  the  user  to  add  a  unique  ID  to  the  analysis.  If  this  field  is  not  specified,  then  the  default  will  be  a  .mestamp  (e.g.  _20111108T150137).  To  avoid  the  using  a  .mestamp,  set  this  field  to  ‘’.  sign  -­‐-­‐  This  can  either  be  ‘pos’  or  ‘neg.  This  specifies  the  direc.on  of  the  contrast  to  test.  If  not  specified,  the  program  will  default  to  ‘pos’.  NOTE:  F-­‐contrasts  can  only  be  posi.ve.  thresh  -­‐-­‐  This  specifies  the  voxel  threshold  for  finding  clusters.  This  is  only  op.onal  because  the  program  will  default  to  a  value  of  0.  The  threshold  has  to  be  a  number  and  can  either  be  the  T/F  sta.s.c,  a  p-­‐value,  or  any  other  number.  This  should  almost  always  be  specified.  type  -­‐-­‐  This  specifies  the  sta.s.c  type,  'T',  'F',  ‘Z’,  or  'none'.  cluster  -­‐-­‐    This  specifies  the  minimum  cluster  size  required  to  keep  a  cluster  in  the  results.  The  default  is  0.  This  is  BAD!!!.  df1  -­‐-­‐The  numerator  degrees  of  freedom  for  T/F-­‐test  df2  -­‐-­‐The  denominator  degrees  of  freedom  for  F-­‐test.    label-­‐-­‐  This  specifies  which  labeling  scheme  to  use.  

OrthoView  

What  to  report  in  papers  

•  Be  explicit  about  the  model  –  What  are  the  factors  –  What  are  the  covariates  –  What  did  you  set  as  the  variance  and  dependence  for  each  factor  

•  Be  explicit  about  the  contrast  you  are  using  •  Be  explicit  about  how  to  interpret  the  contrast  

–  Group  means,  group  intercepts,  covariate  adjusted  group  means  

•  Be  explicit  about  the  thresholds  used  –  Correc.ons  for  mul.ple  comparisons  –  Small  Volume  Correc.on  (corrected  in  SPM8  in  late  Feb.  2012)  

Acknowledgements  •  Harvard  Aging  Brain  Project  

–  Dr.  Reisa  Sperling  –  Dr.  Alireza  Atri  –  Dr.  Aaron  Schultz  –  Aishwarya  Sreenivasan  –  Andrew  Ward  –  Dr.  Koene  Van  Dijk  –  Dr.  Willem  Huijbers  –  Dr.  Dorene  Rentz  –  Dr.  Trey  Hedden    

•  Dr.  Darren  Gitelman  •  Jus.n  Vincent  •  Dr.  Bob  Spunt  •  Wisconsin  Alzheimer’s  

Disease  Research  Center  

–  Imaging  Core  (Dr.  Sterling  Johnson,  Dr.  Michele  Ries,  Dr.  Guofan  Xu,  Elisa  Canu,  Erik  Kastman)  

–  Dr.  Mark  Sager  –  Dr.  Sanjay  Asthana  

•  Mark McAvoy •  Tim Hess •  Ron Serlin  

Useful  Mailing  Lists  •  SPM  –  hGp://www.jiscmail.ac.uk/list/spm.html  

•  FSL  -­‐-­‐  hGp://www.jiscmail.ac.uk/list/fsl.html  

•  Freesurfer  -­‐-­‐  hGp://surfer.nmr.mgh.harvard.edu/fswiki/FreeSurferSupport  

•  CARET  -­‐-­‐  hGp://brainvis.wustl.edu/wiki/index.php/Caret:Mailing_List  

•  I  highly  recommend  reading  the  posts  on  these  lists  as  they  will  save  you  .me  in  the  future.  

Open  Neuroscience