Linear Models in Medical Imaging. John Kornak MI square February 19, 2013

Transcription

1 Linear Models in Medical Imaging John Kornak MI square February 19, 2013

2 Acknowledgement / Disclaimer Many of the slides in this lecture have been adapted from slides available in talks available on the SPM web site.

3 Overview Motivation Linear model formulation Region of interest analyses Pixel/voxel based analyses Multiple comparisons for images Bayesian image analysis methods

4 Motivation Linear Models = cornerstone of statistical methods - linear regression to complex models Imaging data statistical methods to look for regional effects or at the voxel level Across subjects: Tissue differences between groups (e.g., voxel/tensor-based morphometry) Within subjects: PET (positron emission tomography), fmri (functional MRI) activation in brain due to thought, stimulus or task Non-brain: Diffusion (DWI, DTI, tractography), Bone mineral density etc., etc.

5 FMRI Data Serial Snapshots of Volunteers brain Time Active Passive Baseline

6 Software SPM PET, fmri, VBM and TBM, EEG/MEG ( needs Matlab) FSL fmri primarily + DTI ( R AnalyzeFMRI package + see medical imaging view + linear models in general ( Also, many R books and free ones (see R web site) + online tutorials

7 Challenges Generating suitable statistical imaging models Dealing with highly multivariate responses (curse of dimensionality) Defining imaging hypotheses Creating computationally efficient analysis procedures

8 Aims of Statistical Modeling Summarize data Estimation: point and interval estimates Inference: hypotheses / relationships Prediction

9 Aims of Statistical Modeling Summarize data Estimation: point and interval estimates Inference: hypotheses / relationships Prediction

10 Statistical Modeling Strategy Propose a model for the data Fit the model Assess the model s adequacy Fit other plausible models Compare all fitted models Interpret the best model

11 Statistical Models: Definitions Univariate response variable y i (for exp. unit i) Covariates (x i1, x i2,..., x ik ) = (predictors of interest and nuisance variables) x i T Data is: { y i,x T i ;i = 1,...,n}, n experimental units Continuous covariates: e.g. blood pressure, druglevel, age (random or controlled) Factors: e.g. diagnosis, gender, drinking level (low, medium, high)

12 The (General) Linear Model A simple linear model might take the form: e.g. y i = β i + x i2 β 2 + x i 3 β x im β m + ε i y i = β mean + x i,age β age + x i,gender β gender + x i,diagnosis β diagnosis + ε i ε i ~ N(0,σ 2 ), i.i.d., i =1,.,n i.i.d. = independently and identically distributed

13 The (General) Linear Model For univariate data: y i = x i T β + ε i, β = (β 1,...,β m ) T i = 1,...,n is a set of unknown parameters or in matrix notation y = X T β + ε This can be extended to a multivariate response Y = X T B + E

14 Ex. Hippocampal Volume HCV ~ Age + Diagnosis (Wilkinson notation) Diagnosis can be normal control (NC) or Alzheimer s disease (AD)

15 Ex. Hippocampal Volume HCV ~ Age + Diagnosis + Age*Diagnosis (Wilkinson notation) Diagnosis can be normal control (NC) or Alzheimer s disease (AD)

16 Structural T1 weighted MRI s Hippocampal volumes manually traced Volume measure = response for each subject Disease status encoded 1 for AD and 0 for NC (the term) x diagnosis = x diag.

17 y i = β 1 + x i,age β age + x i,diag. β diag. + x i.age x i,diag. β inter + ε i Case 1 HCV age

18 y i = β 1 + x i,age β age + x i,diag. β diag. + x i.age x i,diag. β inter + ε i Case 2 HCV age

19 Case 3 HCV NC AD age

20 Case 4 HCV NC AD age

21 y i = β 1 + x i,age β age + x i,diag. β diag. + x i.age x i,diag. β inter + ε i Case 4 HCV NC AD age

22 Linear models can be more general - only needs to be linear in the parameters: We can have: y = x β + x 2 β + exp(x )β + x π x β + ε i age 1 age 2 height 3 age height 4 i But not i = 1,...,n y i = x age β 1 + exp( x height β 2 )

23 Estimation Minimize squared error (Least Squares Error) = Maximum Likelihood Estimation for linear model ˆβ = (X T X) 1 X T y E( ˆβ) = β V ( ˆβ) = σ 2 (X T X) 1 Estimate by sum of squares error ˆσ 2 = n or divide by n-1 for unbiased estimate

24 Inference Model Comparison Take linear model y = X T β + ε And add constraint this defines a new model that is a simplification of the previous one

25 Inference Model Comparison E.g., cf. model y i = β 1 + β 2 x i1 + β 3 x i2 + ε i to simplification with β 3 = 0 (0,0,1) β 1 β 2 β 3 i.e. y i = β 1 + β 2 x i + ε i = 0 i.e. Aβ = c

26 What about &? β 2 = 0 β 3 = 0 Aβ = c β 1 β 2 β 3 = 0 0

27 And what about? β 2 = β 3 β 1 ( ) Aβ = c β 2 β 3 = 0 Is there a difference between 2 conditions? E.g. is the activation effect of reading a word vs. imagining the object different?

28 Definition: Linear model nested in another if 1 st model can be obtained by linear constraint on the 2 nd Nesting tree: age + gender age gender null

29 F-test for General Linear Hypothesis ( ) y = X T β + ε ε N n 0,σ 2 I n Consider This is the General Linear Hypothesis

30 Under, i.e., Aβ = c F = (SSE nested SSE larger ) / ( p larger p nested ) (SSE larger ) / (n p larger ) F plarger p nested,n p larger p denotes the number of model parameters n denotes the number of data points SSE = Deviance = sum of squares error /residuals Tests whether ratio of variances = 1

31 FMRI Data: Set of Volumes (over time) or Set of Time-Series (over space) Serial Snapshots of Volunteers brain Time Active Passive Baseline

32 image data kernel design matrix parameter estimates realignment & motion correction smoothing Linear Model model fitting statistic image Thresholding & Random Field Theory normalisation anatomical reference Statistical Parametric Map (test statistics) Corrected thresholds & p-values

33 Estimation The estimation entails finding the parameter values such that the linear combination best fits the data at each voxel Active Passive Baseline β 1 + β 2 + β 3

34 Parameter Estimates Same model for all voxels Different parameters for each voxel 0.83 ˆβ = ˆβ = beta_0001.img beta_0002.img ˆβ = beta_0003.img

35 y X T β SPM View β 1 β 2 β 1 +β 2 +β 3 β 1 +β 2 +β 3 β 3 Note: We trust: Long series with large effects and small error

36 Spatial Modeling

37 Spatial Hypotheses Question - how do we extend from standard univariate hypotheses to answering spatially motivated questions? Not easy - curse of dimensionality (millions of voxels) A B A 2D 1D B e.g. in 1D it makes sense to infer A is less than B, but what is the equivalent in 2D?

38 Spatial Testing Solutions Summarize the image into one dimensional quantities for testing (e.g. region of interest analysis) Consider the overall test as a combination of individual voxel tests (voxel based analysis) Perform shape/object analysis on objects defined via landmarks Build Bayesian image analysis models

39 Spatial Testing Solutions Summarize the image into one dimensional quantities for testing (e.g. region of interest analysis) Consider the overall test as a combination of individual voxel tests (voxel based analysis) Perform shape/object analysis on objects defined via landmarks Build Bayesian image analysis models

40 Voxel based analysis Each voxel obtains a test statistic from the linear model, e.g. t or F Forms statistical maps of the statistics

41 image data kernel design matrix parameter estimates realignment & motion correction smoothing Linear Model model fitting statistic image Thresholding & Random Field Theory normalisation anatomical reference Statistical Parametric Map (test statistics) Corrected thresholds & p-values

42 Hypothesis Testing Null Hypothesis H 0 Test statistic T t observed realization of T α-level Acceptable false positive risk Level α = Pr( T>u α H 0 ) Threshold u α controls false positive risk at level α u α Null Distribution of T α

43 Multiple Comparisons Problem Which of 100,000 voxels are statistically significant? α =0.05 5,000 false positive voxels

44 Assessing Statistic Images Where s the signal or change? High Threshold Med. Threshold Low Threshold t > 5.5 t > 3.5 t > 0.5 Good Specificity Good Sensitivity Poor Sensitivity (risk of false negatives) Poor Specificity (risk of false positives) How can we determine a sensible threshold level?

45 Multiple Comparison Solutions : Measuring False Positives Familywise Error Rate (FWER) Familywise Error Existence of one or more false positives False Discovery Rate (FDR) FDR = E[FP/(TP+FP)] TP+FP voxels declared active, FP falsely so Realized false discovery rate: FP/(TP+FP)

46 Bonferroni Correction FWE, α, for N independent voxels is approximately α = Nv (v = voxel-wise error rate) To control FWE set v = α / N Independent Voxels Spatially Correlated Voxels Bonferroni is too conservative for brain images

47 FWER MCP Solutions: Random Field Theory Euler Characteristic χ u Topological Measure No holes Never more than 1 blob #blobs - #holes At high thresholds, just counts blobs Random Field FWER = Pr(Max voxel u H o ) = Pr(One or more blobs H o ) Pr(χ u 1 H o ) E(χ u H o ) Threshold See description at Suprathreshold Sets

48 Random Field Theory Limitations Multivariate normality (Gaussianity) Virtually impossible to check Sufficient smoothness FWHM smoothness 3-4 voxel size Smoothness estimation Estimate is biased when images not sufficiently smooth Several layers of approximations Lattice Image Data Continuous Random Field

49 Multiple Comparison Solutions: Measuring False Positives Familywise Error Rate (FWER) Familywise Error Existence of one or more false positives False Discovery Rate (FDR) FDR = E[FP/(TP+FP)] TP+FP voxels declared active, FP falsely so Realized false discovery rate: FP/(TP+FP)

50 False Discovery Rate For any threshold, all voxels can be crossclassified: Accept Null Reject Null Null True TN FP Null False FN TP N A N R Realized FDR rfdr = FP /(TP+FP) = FP /N R Special case: if N R = 0, rfdr = 0 But only can observe N R, don t know TP & FP We therefore control the expected rfdr FDR = E(rFDR)

51 False Discovery Rate Illustration: Noise Signal Signal+Noise

52 Control of Per Comparison Rate at 10% 11.3% 11.3% 12.5% 10.8% 11.5% 10.0% 10.7% 11.2% 10.2% 9.5% Percentage of Null Pixels that are False Positives Control of Familywise Error Rate at 10% Occurrence of Familywise Error FWE Control of False Discovery Rate at 10% 6.7% 10.4% 14.9% 9.3% 16.2% 13.8% 14.0% 10.5% 12.2% 8.7% Percentage of Observed Above Threshold Pixels that are False Positives

53 Benjamini & Hochberg Procedure Select desired limit q on FDR Order p-values, p (1) p (2)... p (V) Let r be largest i such that p (i) i/v q/c(v) Reject all hypotheses corresponding to p (1),..., p (r) p-value NB, no spatial consideration 0 1 Journal of the Royal Statistical Society Series B (1995) 57: i/v p (i) i/v q/c(v) 0 1

54 Also, Non-Parametric Testing If H 0 is true then time order irrelevant (if noise really white) Therefore permute the timepoints and obtain test statistics If true test statistic is extreme compared to others then reject H 0

55 Types of Spatial Inference Individual voxel level Cluster level Set level Bayesian model based

56 Voxel-level Inference Retain voxels above α-level threshold u α Gives best spatial specificity H 0 at a single voxel can be rejected u α space Significant Voxels No significant Voxels

57 Cluster-level Inference Two step-process Define clusters by arbitrary threshold u clus Retain clusters larger than α-level threshold k α u clus space Cluster not significant k α k α Cluster significant

58 Cluster-level Inference Typically better sensitivity Worse spatial specificity The null hyp. of entire cluster is rejected Only means that one or more of voxels in cluster active u clus space Cluster not significant k α k α Cluster significant

59 Set-level Inference Count number of blobs c uses minimum blob size k to count significant activity if number of blobs > n(k,u ) Worst spatial specificity Only can reject global null hypothesis clus u clus k k space Here c = 1; only 1 cluster larger than k

60 A flexible Bayesian Approach Model the form of activity Provides an adaptive thresholding approach space Active voxels

61 Bayesian Model y = data, parameter estimates of statistics z = binary activation map modeled as a MRF x = activation level field modeled as a MRF = residual error MRF = Markov Random Field (similar random field but defined on a lattice)

62 Model Illustration x zx + z = 0 z = 1 z = 0

63 Model Illustration x zx fit z = 0 z = 1 z = 0

64 y x z zx + ε

65 Other Topics and Omissions Hemodynamic response function Multiple subjects (random and mixed effects models) PCA, ICA Multivariate analysis with variogram modeling Space-time modeling