Classical inference and design efficiency Zurich SPM Course 2014 Jakob Heinzle heinzle@biomed.ee.ethz.ch Translational Neuromodeling Unit (TNU) Institute for Biomedical Engineering (IBT) University and ETH Zürich Many thanks to K. E. Stephan, G. Flandin and others for material Translational Neuromodeling Unit

Statistical Inference Overview Image time-series Spatial filter Design matrix Statistical Parametric Map Realignment Smoothing General Linear Model Statistical Inference RFT Normalisation p <0.05 Anatomical reference Parameter estimates Classical Inference and Design Efficiency

A mass-univariate approach Time Classical Inference and Design Efficiency

Estimation of the parameters i.i.d. assumptions: 𝜀~𝑁(0, 𝜎 2 𝐼) 𝛽 = ( 𝑋 𝑇 𝑋) −1 𝑋 𝑇 𝑦 OLS estimates: 𝛽 1 =3.9831 𝛽 2−7 ={0.6871, 1.9598, 1.3902, 166.1007, 76.4770, −64.8189} 𝛽 𝛽 8 =131.0040 𝑦 = +𝜀 Ways to reduce the noise. 𝜀 = 𝜎 2 = 𝜀 𝑇 𝜀 𝑁−𝑝 𝛽 ~𝑁 𝛽, 𝜎 2 ( 𝑋 𝑇 𝑋) −1 Classical Inference and Design Efficiency

Contrasts 𝑐 𝑇 𝛽 ~𝑁 𝑐 𝑇 𝛽, 𝜎 2 𝑐 𝑇 ( 𝑋 𝑇 𝑋) −1 𝑐 A contrast selects a specific effect of interest. A contrast 𝑐 is a vector of length 𝑝. 𝑐 𝑇 𝛽 is a linear combination of regression coefficients 𝛽. [1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [1 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] 𝑐= [1 0 0 0 …] 𝑇 𝑐 𝑇 𝛽=𝟏× 𝛽 1 +𝟎× 𝛽 2 +𝟎× 𝛽 3 +𝟎× 𝛽 4 +… = 𝜷 𝟏 𝑐= [1 0 0 −1 0 …] 𝑇 𝑐 𝑇 𝛽=𝟏× 𝛽 1 +𝟎× 𝛽 2 +𝟎× 𝛽 3 +−𝟏× 𝛽 4 +… = 𝜷 𝟏 − 𝜷 𝟒 𝑐 𝑇 𝛽 ~𝑁 𝑐 𝑇 𝛽, 𝜎 2 𝑐 𝑇 ( 𝑋 𝑇 𝑋) −1 𝑐 Classical Inference and Design Efficiency

Hypothesis Testing - Introduction Is the mean of a measurement different from zero? Mean of several measurements Many experiments Ratio of effect vs. noise  t-statistic Null Distribution of T Null distribution What distribution of T would we get for m = 0? m1, s1 Exp A m2, s2 Exp B 𝑇= 𝜇 𝜎 𝑛 𝜎 𝜇 = 𝜎 1,2 𝑛 Classical Inference and Design Efficiency

Hypothesis Testing To test an hypothesis, we construct “test statistics”. Null Hypothesis H0 Typically what we want to disprove (no effect).  The Alternative Hypothesis HA expresses outcome of interest. Test Statistic T The test statistic summarises evidence about H0. Typically, test statistic is small in magnitude when the hypothesis H0 is true and large when false.  We need to know the distribution of T under the null hypothesis. Null Distribution of T Animation how the null distribution is generated. Based on classical statistics^. Classical Inference and Design Efficiency

Hypothesis Testing u t t 𝑝 𝑇>𝑡| 𝐻 0 Significance level α: Acceptable false positive rate α.  threshold uα Threshold uα controls the false positive rate t  Null Distribution of T Conclusion about the hypothesis: We reject the null hypothesis in favour of the alternative hypothesis if t > uα t p-value: A p-value summarises evidence against H0. This is the chance of observing a value more extreme than t under the null hypothesis. p-value Null Distribution of T 𝑝 𝑇>𝑡| 𝐻 0 Classical Inference and Design Efficiency

T-test - one dimensional contrasts – SPM{t} effect of interest > 0 ? = amplitude > 0 ? b1 = cTb > 0 ? cT = 1 0 0 0 0 0 0 0 Question: b1 b2 b3 b4 b5 ... Null hypothesis: H0: cTb=0 T = contrast of estimated parameters variance estimate Test statistic: Classical Inference and Design Efficiency

T-contrast in SPM For a given contrast c: ResMS image beta_???? images # beta images = p = # columns of X con_???? image spmT_???? image SPM{t} Classical Inference and Design Efficiency

T-test: a simple example Passive word listening versus rest Q: activation during listening ? Null hypothesis: b1 = 0 SPMresults: Threshold T = 3.2057 {p<0.001} voxel-level p uncorrected T ( Z º ) Mm mm mm 13.94 Inf 0.000 -63 -27 15 12.04 -48 -33 12 11.82 -66 -21 6 13.72 57 -21 12 12.29 63 -12 -3 9.89 7.83 57 -39 6 7.39 6.36 36 -30 -15 6.84 5.99 51 0 48 5.65 -63 -54 -3 𝑡= 𝑐 𝑇 𝛽 var 𝑐 𝑇 𝛽 Classical Inference and Design Efficiency

T-test: summary T-test is a signal-to-noise measure (ratio of estimate to standard deviation of estimate). H0: vs HA: Alternative hypothesis: T-contrasts are simple combinations of the betas; the T-statistic does not depend on the scaling of the regressors or the scaling of the contrast. Classical Inference and Design Efficiency

Scaling issue 𝛽 = ( 𝑋 𝑇 𝑋) −1 𝑋 𝑇 𝑦 [1 1 1 1 ] / 4 Subject 1 The T-statistic does not depend on the scaling of the regressors. The T-statistic does not depend on the scaling of the contrast. Contrast depends on scaling. [1 1 1 ] Subject 5 / 3 Be careful of the interpretation of the contrasts themselves (eg, for a second level analysis): sum ≠ average 𝛽 = ( 𝑋 𝑇 𝑋) −1 𝑋 𝑇 𝑦 Classical Inference and Design Efficiency

F-test - the extra-sum-of-squares principle Model comparison: Null Hypothesis H0: True model is X0 (reduced model) X0 X1 X0 Test statistic: ratio of explained variability and unexplained variability (error) RSS RSS0 Bessere Designmatrix. 1 = rank(X) – rank(X0) 2 = N – rank(X) Full model ? or Reduced model? Classical Inference and Design Efficiency

F-test - multidimensional contrasts – SPM{F} Test multiple linear hypotheses: Null Hypothesis H0: b3 = b4 = b5 = b6 = b7 = b8 = 0 cTb = 0 X0 X1 (b3-8) 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 cT = Bessere Designmatrix. Full model ? Is any of b3-8 not equal 0? Classical Inference and Design Efficiency

F-contrast in SPM ( RSS0 - RSS ) ResMS image beta_???? images ess_???? images spmF_???? images ( RSS0 - RSS ) SPM{F} Classical Inference and Design Efficiency

F-test example: movement related effects Design matrix 2 4 6 8 10 20 30 40 50 60 70 80 contrast(s) Design matrix 2 4 6 8 10 20 30 40 50 60 70 80 contrast(s) Classical Inference and Design Efficiency

F-test: summary F-tests can be viewed as testing for the additional variance explained by a larger model wrt a simpler (nested) model  model comparison. F tests a weighted sum of squares of one or several combinations of the regression coefficients b. In practice, we don’t have to explicitly separate X into [X1X2] thanks to multidimensional contrasts. Hypotheses: In testing uni-dimensional contrast with an F-test, for example b1 – b2, the result will be the same as testing b2 – b1. It will be exactly the square of the t-test, testing for both positive and negative effects. Classical Inference and Design Efficiency

Orthogonal regressors Variability described by 𝑋 1 Variability described by 𝑋 2 Testing for 𝑋 1 Testing for 𝑋 2 Variability in Y Classical Inference and Design Efficiency

Correlated regressors Shared variance Variability described by 𝑋 1 Variability described by 𝑋 2 Variability in Y Classical Inference and Design Efficiency

Correlated regressors Testing for 𝑋 1 Variability described by 𝑋 1 Variability described by 𝑋 2 Variability in Y Classical Inference and Design Efficiency

Correlated regressors Testing for 𝑋 2 Variability described by 𝑋 1 Variability described by 𝑋 2 Variability in Y Classical Inference and Design Efficiency

Correlated regressors Variability described by 𝑋 1 Variability described by 𝑋 2 Variability in Y Classical Inference and Design Efficiency

Correlated regressors Testing for 𝑋 1 Variability described by 𝑋 1 Variability described by 𝑋 2 Variability in Y Classical Inference and Design Efficiency

Correlated regressors Testing for 𝑋 2 Variability described by 𝑋 1 Variability described by 𝑋 2 Variability in Y Classical Inference and Design Efficiency

Correlated regressors Testing for 𝑋 1 and/or 𝑋 2 Variability described by 𝑋 1 Variability described by 𝑋 2 Variability in Y Classical Inference and Design Efficiency

Design orthogonality For each pair of columns of the design matrix, the orthogonality matrix depicts the magnitude of the cosine of the angle between them, with the range 0 to 1 mapped from white to black. If both vectors have zero mean then the cosine of the angle between the vectors is the same as the correlation between the two variates. Classical Inference and Design Efficiency

Correlated regressors: summary We implicitly test for an additional effect only. When testing for the first regressor, we are effectively removing the part of the signal that can be accounted for by the second regressor:  implicit orthogonalisation. Orthogonalisation = decorrelation. Parameters and test on the non modified regressor change. Rarely solves the problem as it requires assumptions about which regressor to uniquely attribute the common variance.  change regressors (i.e. design) instead, e.g. factorial designs.  use F-tests to assess overall significance. Original regressors may not matter: it’s the contrast you are testing which should be as decorrelated as possible from the rest of the design matrix x2 x2 x2 x^ 2 x^ = x2 – x1.x2 x1 x1 2 x1 x1 x^ 1 Classical Inference and Design Efficiency

Design efficiency The aim is to minimize the standard error of a t-contrast (i.e. the denominator of a t-statistic). This is equivalent to maximizing the efficiency e: Noise variance Design variance If we assume that the noise variance is independent of the specific design: This is a relative measure: all we can really say is that one design is more efficient than another (for a given contrast). Classical Inference and Design Efficiency

Design efficiency A B A+B A-B 𝑐= [1 0] 𝑇 : 𝑒 𝑐,𝑋 =18.1 𝑐= [0.5 0.5] 𝑇 : 𝑒 𝑐,𝑋 =19.0 𝑐= [1 −1] 𝑇 : 𝑒 𝑐,𝑋 =95.2 𝑋 𝑇 𝑋= 1 −0.9 −0.9 1 A+B A-B High correlation between regressors leads to low sensitivity to each regressor alone. We can still estimate efficiently the difference between them. Classical Inference and Design Efficiency

Bibliography: Statistical Parametric Mapping: The Analysis of Functional Brain Images. Elsevier, 2007. Plane Answers to Complex Questions: The Theory of Linear Models. R. Christensen, Springer, 1996. Statistical parametric maps in functional imaging: a general linear approach. K.J. Friston et al, Human Brain Mapping, 1995. Ambiguous results in functional neuroimaging data analysis due to covariate correlation. A. Andrade et al., NeuroImage, 1999. Estimating efficiency a priori: a comparison of blocked and randomized designs. A. Mechelli et al., NeuroImage, 2003. Classical Inference and Design Efficiency