The data set comes from one control subject under the event-related design 5.13 with pictures showing every 2 seconds. There are four types of pictures included:
(a) HRF estimates by TFE (b) HRF estimates by sFIR
Figure 5.12: HRF estimates in one voxel. The voxel is selected by using the highest t statistic generated by SPM among the whole brain. TFE gives much better estimation in the tail of HRF than sFIR.
standard, neutral, scary, target circle. The data comes from the paper under review: Hart, Bizzell, Gu, Perkins, Belger (in review) Fronto-Limbic Changes in Children and Adolescents with Familial High Risk for Schizophrenia. The standard pictures, regarded as the background instead of stimulus, are the most frequent presentations during the scan session. Thus, three types of stimulus are considered as Neutral, Scary, Target. The whole scanning session included eight runs, and each of them lasts for 4 minutes. TR is 2 seconds. The region of interest (ROI) here is right Amygdala with 38 voxels.
In order to see whether there is activation in Amygdala, we applied both methods, SPM and TFE, to 38 voxels. SPM has t statistic by using the coefficients in the general linear model to detect activation. Figure 5.14 shows no activation detected by SPM. Even though the coefficients are different, the corresponding t statistic is not significant for detecting the activation. Figure 5.15 shows some activation detected by TFE for Neutral and Target stimulus and no activation for Target. The activation is detected by F statistic in TFE by using the task frequency information. For each
Figure 5.13: Event-related visual data design. The upper graph shows the experimen- tal design with three kinds of events including Neutral (black), Scary (Red), Target (blue). The bottom time series is an example of the fMRI data from one voxel.
single frequency, we used its fundamental frequency and did the hypothesis testing on the frequency to see its activation by voxels. In the 38 voxels, 4 voxels (F values: 7.4, 4.7, 6.3, 4.7) are detected to activate by Neutral and 6 voxels (F values: 5.3, 9.1, 12.5, 7.5, 5.0, 7.5) by Scary. It is confirmed by the researcher exploring the visual fields that Scary evokes more activation than Nautral and also Target has little effect in Amygdala in control subject. Based on the real data, we can see TFE is more sensitive than SPM in detecting the activation based on the frequency information.
The next step is to see the HRF estimates in the activation region. As we already have the activated voxels from TFE, we applied two methods, TFE and sFIR, to estimate HRF in the activated voxels and region. Figure 5.16 shows Neutral HRF estimates by both TFE and sFIR. As they uses the same time series, the HRF esti- mates can be compared side by side. Both of the HRF estimates have similar shape, but the HRF estimates by TFE have higher amplitude. Figure 5.17 shows Scary HRF estimates by both TFE and sFIR.
(a) No neutral activation by SPM
(b) No scary activation by SPM
(c) No target activation by SPM
Figure 5.14: The activation detected by SPM in event-related visual data. The value of coefficients in GLM is displayed in 2×19 = 38 map on the left, and the activation detection is on the right. By thresholding at level 95% of t statistics in SPM, there is no activation detected for either of the three types of stimulus
(a) Neutral (b) Scary (c) Target
Figure 5.15: The activation detected by TFE in event-related visual data (the same data as in Figure 5.14). There are activated voxels detected for Neutral and Scary stimulus.
(a) Neutral HRF by TFE
(b) Neutral HRF by sFIR
Figure 5.16: The HRF estimates for the four activated voxels by Neutral stimulus. The upper row is from TFE and the bottom from sFIR. The left graph is the HRF estimates for each voxels, and the right is the HRF estimates by averaging the time series in the four activated voxels. For Neutral stimulus, F-value=7.4, 4.7, 6.3, 4.7 in the four activated voxels. TFE gives smoother estimate in averaged time series than sFIR.
(a) Scary HRF by TFE
(b) Scary HRF by sFIR
Figure 5.17: The HRF estimates for the six activated voxels by Scary stimulus. The upper row is from TFE and the bottom from sFIR. The left graph is the HRF es- timates for each voxels, and the right is the HRF estimates by averaging the time series in the six activated voxels. For Scary stimulus, F-value =5.3, 9.1, 12.5, 7.5, 5.0, 7.5 in the six activated voxels. TFE gives smoother estimate in averaged time series than sFIR.
Chapter 6
Sampling Properties
The finite sample performance of our proposed methods has been well illustrated via simulations in Chapter 4 and the real applications in Chapter 5. In this chapter, we derive the asymptotic properties of the TFE method. Here is a brief outline. Section 6.1 starts with the asymptotic normality of the estimator including its expectation and covariance. Section 6.2 derives the hypothesis testing procedure associated with TFE. Section 6.3 shows that the weighted least squares estimator has a smaller variance than the ordinary least squares estimator, hence more efficient.