7/12/2016. GROUP ANALYSIS Martin M. Monti UCLA Psychology AGGREGATING MULTIPLE SUBJECTS VARIANCE AT THE GROUP LEVEL

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1 GROUP ANALYSIS Martn M. Mont UCLA Psychology NITP AGGREGATING MULTIPLE SUBJECTS When we conduct mult-subject analyss we are tryng to understand whether an effect s sgnfcant across a group of people. Whether somethng s sgnfcant depends on the varance we assess t aganst: Classcal statstcal hypothess testng proceeds by comparng the dfference between the expected and hypotheszed effect aganst the yardstck of varance. [Holmes & Frston, 1998] VARIANCE AT THE GROUP LEVEL Fxed Effects (FFX): s about the ntra-subject varablty. An effect s compared aganst the yardstck of the precson wth whch t can be measured (for each subject). The dfferent subjects are consdered to be fxed. Random Effects (RFX): s about the nter-subject varablty. An effect s compared aganst the yardstck of how much varablty there s across dfferent subjects. The dfferent subjects are consdered to be a random sample from a greater populaton. Mxed Effects (MFX): s about ntra-subject & ntersubject varablty. 1

2 FIXED EFFECTS: INTRA-SUBJECT VARIABILITY Only varaton (over sessons) s measurement error True Response magntude s fxed Adapted from T Nchols RANDOM EFFECTS: INTER-SUBJECT VARIABILITY Source of varaton Response magntude Response magntude s random Each subject/sesson has random magntude But note, the populaton mean s fxed Adapted from T Nchols MIXED EFFECTS Two sources of varaton Measurement error Response magntude Response magntude s random Each subject/sesson has random magntude But note, the populaton mean s fxed Adapted from 7 T Nchols

3 IN OTHER WORDS FFX Model: y ~ (0, w ) d Subj. effect Meas. error IN OTHER WORDS FFX Model: y ~ (0, w ) d But d s a random varable! d d pop z z ~ (0, ) b Populaton effect Subj. varablty (around dpop) IN OTHER WORDS FFX Model: y ~ (0, w ) d But d s a random varable! d d z z ~ (0, ) pop MFX Model: y d pop Populaton effect z Subj. varablty (around d pop ) Meas. error b 3

4 IN OTHER WORDS FFX Model: y ~ (0, w ) d But d s a random varable! d d z z ~ (0, ) pop MFX Model: y d z ) pop b ( A HAIRY EXAMPLE Queston: Do M & F har dffer n length? erment: Take 5 hars from each of 8 Ss (4F, 4M) [ w=1, b=49] σ FFX : σ w Nn = = 0.01 σ RFX : σ b N = 49 4 = 1.5 σ MFX : σ w Nn + σ b N = 49 4 = 1.6 By Jeanette Mumford 4

5 GROUP ANALYSIS STRATEGIES: FFX FIXED V RANDOM Fxed sn t wrong, just usually sn t of nterest Fxed Effects Inference I can see an effect n ths sample Random Effects Inference I can extend my nference to the populaton: I expect to see the effect across the populaton 15 GROUP ANALYSIS STRATEGIES (I): ALL-IN-ONE Complete sngle-level GLM that relates varous parameters of nterest at the group level to the full set of (tme seres) data avalable Y XX g g Concatenated tmeseres Sngle subject desgn matrx Second level desgn matrx (e.g., pat v vol) Group level parameter error term 5

6 GROUP ANALYSIS STRATEGIES (I): ALL-IN-ONE Complete sngle-level GLM that relates varous parameters of nterest at the group level to the full set of (tme seres) data avalable Y XX g X g g Concatenated tmeseres Sngle subject desgn matrx Second level desgn matrx (e.g., pat v vol) Group level parameter GROUP ANALYSIS STRATEGIES (I): ALL-IN-ONE Complete sngle-level GLM that relates varous parameters of nterest at the group level to the full set of (tme seres) data avalable Computatonally ntense approach What f you acqure 1 more dataset? GROUP ANALYSIS STRATEGIES (II): THE SUMMARY STATISTIC APPROACH 6

7 GROUP ANALYSIS STRATEGIES (II): THE SUMMARY STATISTIC APPROACH GROUP ANALYSIS STRATEGIES (II): ND LEVEL 1. Perform an OLS [the SPM way] Assume that 1 st level varances ( w ) are the same for all subjects (.e., homoschedastcty)* Assume that 1 st level desgn matrces are the same for all subjects (.e., are balanced)* Estmate b from the (c) β carred forward from the 1 st level analyses, use t to assess the average group effect. Essentally, ths s a t-test! + Rapd & smple Are w truly the same (dstracted subjects, learnng, )? Are 1 st level matrces truly the same (forgotten v recalled)? GROUP ANALYSIS STRATEGIES (II): ND LEVEL. Perform a GLS (WLS) [the FSL way] Carry forward (c) β as well as 1 st level varance ( w ) Estmate b, defne (for each subject j) the overall varance s: σ wj + σ b Perform a GLS where each subject s ( nd level) data s weghted by her overall varance. + Bad subjects wth a large w wll be down-weghted + Statstcally more correct (presumably better for more usng desgns beyond smple t-test) Computatonally more ntensve (teratve calculaton of varance) 7

8 GROUP ANALYSIS STRATEGIES (II): THE SUMMARY STATISTIC APPROACH The debate: Frston (SPM): Assume homoscedastc 1 st level varances and do an OLS. Beckmann 03 (FSL): must use lower level varance n group estmaton, else no longer equvalent to the all-n-one approach Frston 05 (SPM): OLS s robust to unequal varances (but can estmate the covarance structure [usng ReML] from frst level [only sgnfcant voxels] and carry that forward). Smth 05 (FSL): Wthn subject varablty can actually be farly large Mumford 09: OLS s robust even n the presence of outlers and volatons of homoschedastcty, but only for 1 sample t-test. GLS always more optmal strategy. RECAP. FFX nferences are vald, but only wth respect to the sample. May be of nterest for sngle case studes, or small rare populatons you can fully sample.. MFX nferences are vald over the populaton you sample from because you are accountng for samplng varablty. Ths s what you want to do for a typcal group study.. The Summary statstc approach s effcent. Run 1 st levels ndependently, then combne the results. If you run 1 more subject, then you only have to re-run the group. MASSIVE UNIVARIATE APPROACH 8

9 Source: Jonathan Peelle HOW THESE DATA WERE GENERATED Source: Jonathan Peelle MULTIPLE COMPARISONS PROBLEM When you make 1 test, what s the probablty that a postve result s, n fact, not true (.e., false postve) a (say, 5%) If we make tests, what s the overall probablty (.e., famly-wse probablty) of false postves? 1 (1 a) (at a nomnal 5%: 9.75%) If we make n tests, what s the overall probablty (.e., famly-wse probablty) of false postves? 1 (1 a) n 9

10 MULTIPLE COMPARISONS PROBLEM How many tests do we perform n fmri analyss? Over (say) 100,000 null voxels, how many tmes wll we ncorrectly reject H 0? ~5,000 voxels (on average!) P r o b F P # Comparsons FISHY STATISTICS Stmul: pctures of faces (w/emotonal expressons). Task: determne what emotons depcted faces were experencng. Desgn: blocks of 1 sec actvaton/rest Analyss: standard data processng wth SPM Subject: 1 dead Atlantc Salmon. FALSE ACTIVATIONS UNDER H0 P < 0.05 (168 voxels) P < 0.01 (364 voxels) P < (3 voxels) 10

11 HOW MUCH CORRECTION? A B C t =.10, p < 0.05 (uncorrected) t = 3.60, p < (uncorrected) t = 7.15, p < 0.05, Bonferron Corrected Poor Specfcty (rsk of false postves) Good Power Good Specfcty Poor Power (rsk of false negatves) CORRECTION FOR MULTIPLE COMPARISONS man strateges: 1. Famly Wse Error (FWE): Control for the probablty of any false postves (e.g., Bonferron, Random Feld Theory, Permutaton). False Dscovery Rate (FDR): Control proporton of false postves among rejected tests FWE (I): BONFERRONI Man dea: make each ndvdual test more strngent, so overall you end up wth your total (.e., famly-wse) desred false postve rate. Bonf a FW n a P T a H a FW n ( 0) 1 For example: Desred famlywse false postve rate: a FW = 0.05 Total number of (ndependent) tests: 100,000 Then the Bonferron-corrected false postve level for each ndvdual test s: a Bonf a n FW ,000 11

12 FWE (I): BONFERRONI Assumes ndependent tests FMRI data spatally correlated (vasculature, spatal smoothng), so the number of ndependent tests s less than the number of voxels Overly strngent Increases Type II error Dffcult to fnd what s n n order to calculate the correct a bonf FWE (II): RANDOM FIELD THEORY Allows to fnd a threshold n a set of data where t s not easy (or even mpossble) to fnd the number of ndependent varables 3 step approach:. Estmate how smooth the data s ( resels ).Compute how many peaks would be above the threshold by chance ( Euler Characterstc ).Calculate the threshold that yelds desred FWER 1. SMOOTHNESS PARAMETRIZATION We can't compute the # of ndependent voxels, but we can compute the number of resoluton elements (.e. resels ). 1

13 . EULER CHARACTERISTIC Topologcal measure [c] Threshold an mage at u EC = # of blobs - # holes At hgh u: EC = # of blobs P(blob) = E[EC] Under H0, a FWE = E[EC] 3. THRESHOLD a FW E[ c] R(4log )( ) e 3 Z 1 Z t te Gven the smoothness of my data (R), what threshold (Z) do I need to set so that I have a FW chance (~E[EC]) of havng peak above threshold? FALSE DISCOVERY RATE (FDR) FDR controls the expected proporton of false postve values among supra-threshold values (.e., false clams v false tests): p < 0.05 FWE means: There s only a 5% chance any result s a false postve. p < 0.05 FDR means: No more than 5% of actve voxels are false postves. 13

14 FALSE DISCOVERY RATE (FDR) COMPARING CORRECTION METHODS Sgnal Nose Sgnal+Nose NO CORRECTION (a = 0.1) On average, 10% of the 'false' voxels are ncorrectly declared actve. In each experment we have about 10% false alarms 14

15 FWE (a = 0.1) FDR (a = 0.1) RESOURCES Mont M.M. (011) Statstcal analyss of fmri tme-seres: A crtcal evaluaton of the GLM approach. Fronters n Human Neuroscence, 5(8). Mumford, J. A., and Nchols, T. (009). Smple group fmri modelng and nference. Neuromage 47, Mumford, J. A., and Poldrack, R. A. (007). Modelng group fmri data. Soc. Cogn. Affect. Neurosc., Beckmann, C. F., Jenknson, M., and Smth, S. M. (003). General multlevel lnear modelng for group analyss n fmri. Neuromage 0, Poldrack R.A., Mumford J.A., Nchols T.E. (011) Handbook of Functonal MRI Analyss, Cambrdge Unversty Press. Lazar, N. (008). The statstcal analyss of functonal MRI data. Sprnger. Frston K.J., et al Statstcal Parametrc Mappng: The Analyss of Functonal Bran Images, chapter 8. 15

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