Covariance components II autocorrelation & nonsphericity Alexa Morcom Oct. 2003

Methods by blondes vs. mullets?

Nonsphericity - what is it and why do we care? Need to know expected behaviour of parameters under H0 - less intrinsic variability means fewer df, so liberal inference Null distribution assumed normal Further assumed to be ‘iid’ - errors are identical and independently distributed “Estimates of variance components are used to compute statistics and variability in these estimates determine the statistic’s d.f.”

An illustration... y = X* b + e y1 = X* b1 + e1 y2 b2 e2 A GLM with just 2 observations y = X* b + e y1 = X* b1 + e1 y2 b2 e2 e ~ N(0, s) iid iid assumptions e ~ N(0, C e) error covariance matrix C e

Spherical e2 e1 C e = 1 0 0 1

Non-identical e2 e1 C e = 4 0 0 1

Non-independent e2 e1 C e = 1 3 0.5 5

Varieties of nonsphericity in fMRI Temporal autocorrelation - 1st level Correlated repeated measures - 2nd level Unequal variances between groups - 2nd level Unequal within-subject variances - 1st level* Unbalanced designs at 1st level* (Spatial ‘nonsphericity’ or smoothness)

A traditional psychology example Repeated measures of RT across subjects RTs to levels 2 & 3 may be more highly correlated than those to levels 1 & 2

Sphericity Compound symmetry s11 s12 … s1k s21 s22 … s2k … … … ... sk1 sk2 … skk s2 rs2 … rs2 rs2 s2 … rs2 … … … ... rs2 rs2 … s2 n subjects k treatments sij = sample var/ cov Variance of difference between pair of levels constant Not easy to see! By inspection: Treatment variances equal, treatment covariances equal

The traditional psychology solution Sphericity - most liberal condition for SS to be distributed as F ratio A measure of departure from sphericity: e SS but approx. by F with Greenhouse-Geisser corrected d.f. (based on Satterthwaite approx): A fudge in SPSS because e must be estimated, and this is imprecise (later…) so correction slightly liberal F [(k-1)e, (n-1)(k-1)e]

A more general GLM y = X*b + e OLS Wy = WX*b + We W/GLS Weighting by W to render Cov(We) iid or known

A more general GLM y = X*b + e OLS Wy = WX*b + We W/GLS Weighting by W to render Cov(We) iid or known bw = (WX)-y Cb = (WX)- WCe W T(WX)-T i.e. covariance of parameter estimates depends on both the design and the error structure ... ^ ^

A more general GLM y = X*b + e OLS Wy = WX*b + We W/GLS Weighting by W to render Cov(We) iid or known bw = (WX)-y Cb = (WX)- WCe W T(WX)-T i.e. covariance of parameter estimates depends on both the design and the error structure ... If Ce is iid with var = s 2, then W = I; Cb Ce = s 2I ^ ^ ^

A more general GLM y = X*b + e OLS Wy = WX*b + We W/GLS Weighting by W to render Cov(We) iid or known bw = (WX)-y Cb = (WX)- WCe W T(WX)-T i.e. covariance of parameter estimates depends on both the design and the error structure ... If Ce is iid with var = s 2, then W = I; Cb Ce = s 2I If single covariance component, direct estimation Otherwise iterative, or determine Ce first ... ^ ^ ^

Colouring & whitening... Imposed ‘ temporal smoothing ’ W=S (SPM99) Sy = SX*b + Se Cb = (SX)- SCe S T(SX)-T S is known and Ce assumed ‘swamped’ Resulting d.f. adjustment = Satterthwaite (but better than Greenhouse-Geisser) ^

Colouring & whitening... Imposed ‘ temporal smoothing ’ W=S (SPM99) Sy = SX*b + Se Cb = (SX)- SCe S T(SX)-T S is known and Ce assumed ‘swamped’ Resulting d.f. adjustment = Satterthwaite (but better than Greenhouse-Geisser) Prewhitening: if Ce is assumed known, premultiply by W = Ce½ (SPM2) b by OLS then is best estimator & Cb = (XT Ce-1X)-1 ^ ^ ^

Effects on statistics t = cT b (cTCbc )½ Estimation is better - increased precision of b Minimum covariance of estimator maximises t as Cb is in denominator (& depends on X & Ce: compare S, ‘bigger’ denominator) Precise determination of d.f. as function of W (i.e. Ce) & design matrix X (if S, fewer) t = cT b (cTCbc )½

Estimating multiple covariance components Doing this at every voxel would require ReML at every voxel (my contract is too short…) As in SPSS, such estimation of Ce would be imprecise, and inference ultimately too liberal: Ce = rrT + X Cb XT (critical ‘circularity’… ) To avoid this, SPM2 uses spatial (cross-voxel) pooling of covariance estimation This way, Ce estimate is precise & (prewhitened OLS) estimation proceeds noniteratively

1st level nonsphericity Model Ce as linear combination of bfs: C(l)e = Si (l1Q1 + l 2Q2) Timeseries autocorrelations in fMRI (Low freq. 1/f removed by high-pass filter) White noise is Q1 Lag 1 autoregressive AR(1) is Q2

Estimated Ce Q1 Q2

2nd level nonsphericity Here model unequal variance across measures, &/or unequal covariance between measures C(l)e = Si (l1Q1 + l 2Q2 … + … l iQi) No. of bfs depends on no. of measures & options selected Nonsphericity? Correlated repeated measures?

Variance for each measure for all subjects Covariance of each pair of measures for all subjects 3 measures: 3 diagonals Q1- Q3 3 off-diag Q4- Q6

What difference does it make? SPM99 OLS method (applied incorrectly) & assuming iid - big t, lots of df, liberal Worsley & Friston’s SPM99 method with Satterthwaite df correction - smaller t, fewer df, valid but not ideal (cons) SPM2 Gauss-Markov (ideal) estimator with prewhitening - full no. of df along with correct t value

Limitations of 2 level approach y = X(1)b(1) + e(1) b(1) = X(2) b(2) + e(2) y = X(1)X(2)b(2) + X(1)e(2) + e(1) Cov(y) = X(1)Ce(2)X(1)T + Ce(1) (into ReML) 2-stage ‘summary statistic’ approach assumes ‘mixed effects’ covariance components are separable at the 2 levels Specifically, assumes design X & variance same for all subjects/ sessions, even if nonsphericity modelled at each level