The Effects of Autocorrelated Noise and Biased HRF in fmri Analysis Error Rates

Transcription

1 The Effects of Autocorrelated Noise and Biased HRF in fmri Analysis Error Rates Ariana Anderson University of California, Los Angeles Departments of Psychiatry and Behavioral Sciences David Geffen School of Medicine July 29, / 23 Ariana Anderson Autocorrelation Induced Bias in fmri GLM Analysis

2 1 Background of fmri Imaging 2 Model-Driven Analysis Methods 3 Simulation 4 Conclusion 2 / 23 Ariana Anderson Autocorrelation Induced Bias in fmri GLM Analysis

3 Mechanics of fmri The fmri Blood Oxygen Level Dependent (BOLD) signal is an indirect measure of neuronal activity captured during an fmri scan. Neuronal Activation Oxygenated Blood Influx Oxygen Concentration Increases Regional MR Signal Increases 3 / 23 Ariana Anderson Autocorrelation Induced Bias in fmri GLM Analysis

4 Mechanics of fmri The fmri Blood Oxygen Level Dependent (BOLD) signal is an indirect measure of neuronal activity captured during an fmri scan. Analysis of the BOLD signal is performed under the assumption that neuronal activity coincides with increased blood flow. Neuronal Activation Oxygenated Blood Influx Oxygen Concentration Increases Regional MR Signal Increases 3 / 23 Ariana Anderson Autocorrelation Induced Bias in fmri GLM Analysis

5 Mechanics of fmri Some regions with high CSF content will have a high-intensity signal under a T2-weighted scan. 4 / 23 Ariana Anderson Autocorrelation Induced Bias in fmri GLM Analysis

6 Terminology Voxel: A 3-dimensional analogue of a pixel. A voxel is a region usually between 1 5mm 3 5 / 23 Ariana Anderson Autocorrelation Induced Bias in fmri GLM Analysis

7 Terminology Voxel: A 3-dimensional analogue of a pixel. A voxel is a region usually between 1 5mm 3 Hemodynamic Response Function: A function modeling the change in blood flow due to some trigger. 5 / 23 Ariana Anderson Autocorrelation Induced Bias in fmri GLM Analysis

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9 General Linear Model Analysis Y = XB + ɛ, where Y = time course of voxels, X = design matrix and regressors, B = parameters, ɛ = error. Use parametric tests to infer how well a given voxel follows the time-course of a theoretical active voxel. 7 / 23 Ariana Anderson Autocorrelation Induced Bias in fmri GLM Analysis

10 General Linear Model Analysis Y = XB + ɛ, where Y = time course of voxels, X = design matrix and regressors, B = parameters, ɛ = error. Use parametric tests to infer how well a given voxel follows the time-course of a theoretical active voxel. To get the design matrix for the task function, convolve the hemodynamic response function with task paradigm g f = b a f (τ)g(t τ)dτ 7 / 23 Ariana Anderson Autocorrelation Induced Bias in fmri GLM Analysis

11 General Linear Model Analysis Y = XB + ɛ, where Y = time course of voxels, X = design matrix and regressors, B = parameters, ɛ = error. Use parametric tests to infer how well a given voxel follows the time-course of a theoretical active voxel. To get the design matrix for the task function, convolve the hemodynamic response function with task paradigm g f = b a f (τ)g(t τ)dτ This method allows one to account for confounding effects such as drift or respiration, by incorporating them into the design matrix. 7 / 23 Ariana Anderson Autocorrelation Induced Bias in fmri GLM Analysis

12 8 / 23 Ariana Anderson Autocorrelation Induced Bias in fmri GLM Analysis

13 HRF Variations Aging 9 / 23 Ariana Anderson Autocorrelation Induced Bias in fmri GLM Analysis

14 HRF Variations Aging Disease (Alzheimers), Injury (Stroke) 9 / 23 Ariana Anderson Autocorrelation Induced Bias in fmri GLM Analysis

15 HRF Variations Aging Disease (Alzheimers), Injury (Stroke) Pharmocological treatments 9 / 23 Ariana Anderson Autocorrelation Induced Bias in fmri GLM Analysis

16 HRF Variations Aging Disease (Alzheimers), Injury (Stroke) Pharmocological treatments CO 2 Concentrations 9 / 23 Ariana Anderson Autocorrelation Induced Bias in fmri GLM Analysis

17 10 / 23 Ariana Anderson Autocorrelation Induced Bias in fmri GLM Analysis

18 Y = XB + ɛ, where Y = time course of voxels, X = design matrix and regressors, B = parameters, ɛ = error. The Gauss-Markov Theorem guarantees that this estimate is the Best Linear Unbiased Estimate (BLUE) if E(ɛ i ) = 0, V (ɛ i ) = σ 2 <, and cov(ɛ i, ɛ j ) = 0, for i j. 11 / 23 Ariana Anderson Autocorrelation Induced Bias in fmri GLM Analysis

19 Y = XB + ɛ, where Y = time course of voxels, X = design matrix and regressors, B = parameters, ɛ = error. The Gauss-Markov Theorem guarantees that this estimate is the Best Linear Unbiased Estimate (BLUE) if E(ɛ i ) = 0, V (ɛ i ) = σ 2 <, and cov(ɛ i, ɛ j ) = 0, for i j. The final assumption is violated as the data are serially correlated over time both during rest and task periods (HRF). 11 / 23 Ariana Anderson Autocorrelation Induced Bias in fmri GLM Analysis

20 Generalized Least Squares: Estimate ˆβ GLS s.t. ɛ N(0, σ 2 Σ) 12 / 23 Ariana Anderson Autocorrelation Induced Bias in fmri GLM Analysis

21 Generalized Least Squares: Estimate ˆβ GLS s.t. ɛ N(0, σ 2 Σ) The GLS introduces a matrix P frequently used to decorrelate the residuals, and models the data as P 1 Y = P 1 X β + P 1 ɛ and minimizes the Mahalanobis distance to find β, such that ˆβ = argmin β (Y X β) Σ 1 (Y X β) 12 / 23 Ariana Anderson Autocorrelation Induced Bias in fmri GLM Analysis

22 Generalized Least Squares: Estimate ˆβ GLS s.t. ɛ N(0, σ 2 Σ) The GLS introduces a matrix P frequently used to decorrelate the residuals, and models the data as P 1 Y = P 1 X β + P 1 ɛ and minimizes the Mahalanobis distance to find β, such that ˆβ = argmin β (Y X β) Σ 1 (Y X β) If P P = Σ, then Var(P 1 ɛ) = P 1 Var(ɛ)P 1 = σ 2 I 12 / 23 Ariana Anderson Autocorrelation Induced Bias in fmri GLM Analysis

23 Generalized Least Squares: Estimate ˆβ GLS s.t. ɛ N(0, σ 2 Σ) The GLS introduces a matrix P frequently used to decorrelate the residuals, and models the data as P 1 Y = P 1 X β + P 1 ɛ and minimizes the Mahalanobis distance to find β, such that ˆβ = argmin β (Y X β) Σ 1 (Y X β) If P P = Σ, then Var(P 1 ɛ) = P 1 Var(ɛ)P 1 = σ 2 I This also guarantees that cov(ɛ i, ɛ j ) = 0 for i j and the conditions are met for this estimate of ˆβ GLS to be optimal as BLUE. In practice, P either imposed (coloring) or estimated (whitening), but the efficiency and accuracy of these approaches aren t guaranteed. 12 / 23 Ariana Anderson Autocorrelation Induced Bias in fmri GLM Analysis

24 Autocorrelations in data result in serially correlated residuals, which in turn do not guarantee BLUE for ˆβ 13 / 23 Ariana Anderson Autocorrelation Induced Bias in fmri GLM Analysis

25 Autocorrelations in data result in serially correlated residuals, which in turn do not guarantee BLUE for ˆβ HRF is assumed to be independent of covariates and location, which is not always true. 13 / 23 Ariana Anderson Autocorrelation Induced Bias in fmri GLM Analysis

26 Simulating and measuring bias Simulate two time series of length 100 that differ by the addition of autocorrelated noise that is unrelated to the task being performed, and compare the parameter estimates ˆβ with those estimated from data that has uncorrelated (independent) additive white noise, 14 / 23 Ariana Anderson Autocorrelation Induced Bias in fmri GLM Analysis

27 Simulating and measuring bias Simulate two time series of length 100 that differ by the addition of autocorrelated noise that is unrelated to the task being performed, and compare the parameter estimates ˆβ with those estimated from data that has uncorrelated (independent) additive white noise, y autocorrelated = β(h X ) + g E + ɛ 14 / 23 Ariana Anderson Autocorrelation Induced Bias in fmri GLM Analysis

28 Simulating and measuring bias Simulate two time series of length 100 that differ by the addition of autocorrelated noise that is unrelated to the task being performed, and compare the parameter estimates ˆβ with those estimated from data that has uncorrelated (independent) additive white noise, y autocorrelated = β(h X ) + g E + ɛ y independent = β(h X ) + ɛ 14 / 23 Ariana Anderson Autocorrelation Induced Bias in fmri GLM Analysis

29 Simulating and measuring bias Simulate two time series of length 100 that differ by the addition of autocorrelated noise that is unrelated to the task being performed, and compare the parameter estimates ˆβ with those estimated from data that has uncorrelated (independent) additive white noise, y autocorrelated = β(h X ) + g E + ɛ y independent = β(h X ) + ɛ 200,000 total realizations with randomization of β, X, g, E, h and ɛ 14 / 23 Ariana Anderson Autocorrelation Induced Bias in fmri GLM Analysis

30 Parameter Settings X = 10 activations of length TR Unif [1, 5]. 15 / 23 Ariana Anderson Autocorrelation Induced Bias in fmri GLM Analysis

31 Parameter Settings X = 10 activations of length TR Unif [1, 5]. β Unif [1, 5]. 15 / 23 Ariana Anderson Autocorrelation Induced Bias in fmri GLM Analysis

32 Parameter Settings X = 10 activations of length TR Unif [1, 5]. β Unif [1, 5]. h is the HRF modeled by a double-gamma function dilated by δ Unif [0, 1]. 15 / 23 Ariana Anderson Autocorrelation Induced Bias in fmri GLM Analysis

33 Parameter Settings X = 10 activations of length TR Unif [1, 5]. β Unif [1, 5]. h is the HRF modeled by a double-gamma function dilated by δ Unif [0, 1]. ɛ N(0, 1). 15 / 23 Ariana Anderson Autocorrelation Induced Bias in fmri GLM Analysis

34 Parameter Settings X = 10 activations of length TR Unif [1, 5]. β Unif [1, 5]. h is the HRF modeled by a double-gamma function dilated by δ Unif [0, 1]. ɛ N(0, 1). Autocorrelation is introduced into y autocorrelated by convolving a random noise time series E = (η 1, η 2,..., η T ) with η N(0, 1), with an exponential decay function g(t) = r t where r Unif [0, 1]. 15 / 23 Ariana Anderson Autocorrelation Induced Bias in fmri GLM Analysis

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36 Bias Model Parameter HRF Known HRF Biased HRF Ignored y autocorrelated Mean y independent Mean y autocorrelated Variance y independent Variance Table: Mean and Variance of Bias, demonstrating that in the presence of autocorrelated noise, a biased HRF leads to drastic bias in the estimates of ˆβ. Autocorrelated noise is a known occurrence that is usually ignored in most theoretical models and analysis techniques. 17 / 23 Ariana Anderson Autocorrelation Induced Bias in fmri GLM Analysis

37 Data HRF Task Length β ˆβ decay rate r δ HRF Bias y auto Known y auto Biased y auto Ignored y ind Known y ind Biased y ind Ignored Table: Correlations of Parameters with Bias = β ˆβ for yautocorrelated and y independent data. A higher correlation indicates that biased parameter assumptions lead to biased estimates of ˆβ, which in turn leads to even higher false positive and negatives in detecting active regions. 18 / 23 Ariana Anderson Autocorrelation Induced Bias in fmri GLM Analysis

38 Typical Solution Include temporal derivative of the task function in design matrix: y autocorrelated = β(h X ) + g E + ɛ, where X = (x, x, confounders). 19 / 23 Ariana Anderson Autocorrelation Induced Bias in fmri GLM Analysis

39 Model Sensitivity Specificity First Order (no derivatives) Second Order (with derivatives) Table: Derivative Adjustment within GLM Sensitivity: Probability of detection given activation occurred. 20 / 23 Ariana Anderson Autocorrelation Induced Bias in fmri GLM Analysis

40 Model Sensitivity Specificity First Order (no derivatives) Second Order (with derivatives) Table: Derivative Adjustment within GLM Sensitivity: Probability of detection given activation occurred. Specificity: Probability of no detection given no activation occurred. 20 / 23 Ariana Anderson Autocorrelation Induced Bias in fmri GLM Analysis

41 Positive correlation of the estimate ˆβ with bias shows that estimates that are large are more likely to be biased, implying a greater Type 1 error. 21 / 23 Ariana Anderson Autocorrelation Induced Bias in fmri GLM Analysis

42 Positive correlation of the estimate ˆβ with bias shows that estimates that are large are more likely to be biased, implying a greater Type 1 error. The decay rate parameter r was positively correlated with bias, indicating that the longer the temporal autocorrelation present in the white noise, the greater the expected variance. 21 / 23 Ariana Anderson Autocorrelation Induced Bias in fmri GLM Analysis

43 Conclusions Estimates become more biased and noisy both because of the addition of autocorrelated noise in the data and also because of mis-specification of the HRF. 22 / 23 Ariana Anderson Autocorrelation Induced Bias in fmri GLM Analysis

44 Conclusions Estimates become more biased and noisy both because of the addition of autocorrelated noise in the data and also because of mis-specification of the HRF. When the HRF is ignored completely, the model doesn t exhibit changes in the estimates during the fitting of y autocorrelated data, but its bias is greater than for all other models and data. 22 / 23 Ariana Anderson Autocorrelation Induced Bias in fmri GLM Analysis

45 Conclusions Estimates become more biased and noisy both because of the addition of autocorrelated noise in the data and also because of mis-specification of the HRF. When the HRF is ignored completely, the model doesn t exhibit changes in the estimates during the fitting of y autocorrelated data, but its bias is greater than for all other models and data. Models that include temporal derivatives perform worse than those without, in terms of sensitivity and specificity. 22 / 23 Ariana Anderson Autocorrelation Induced Bias in fmri GLM Analysis

46 Conclusions Estimates become more biased and noisy both because of the addition of autocorrelated noise in the data and also because of mis-specification of the HRF. When the HRF is ignored completely, the model doesn t exhibit changes in the estimates during the fitting of y autocorrelated data, but its bias is greater than for all other models and data. Models that include temporal derivatives perform worse than those without, in terms of sensitivity and specificity. Suggestion: When temporal derivatives are modeled for, significance tests need to assess for both first order and derivative effects jointly. 22 / 23 Ariana Anderson Autocorrelation Induced Bias in fmri GLM Analysis

47 Thank you! 23 / 23 Ariana Anderson Autocorrelation Induced Bias in fmri GLM Analysis