7.15 Coursera Statistical Inference Self Study

I have to start the work today but remind me to do these:

• Update Homework 2 Question 9's answer

Consider a normal distribution. If we were to sample 100 draws from a uniform distribution (which has mean 0.5, and variance 1/12) and take their mean, X^bar. What is the approximate probability of getting as large as 0.51 or larger expressed to 3 decimal places?

answer: 0.365

R code: pnorm(0.51, mean = 0.5, sd = sqrt(1/(12*100)), lower.tail = FALSE)

• Update Homework 2 Question 11's answer

If you roll ten standard dice, take their average, then repeat this process over and over and construct a histogram, what would be its variance expressed to 3 decimal places?

answer: 0.292

original answer: 2.917, I did not realize that there are ten standard dice.

• Do swirl assignments for 9, 10, and 11
• Learn ggplot2 for the courser project (2nd part)

7.14 Coursera Statistical Inference Quiz 3 Question 6

Question 6

To further test the hospital triage system, administrators selected 200 nights and randomly assigned a new triage system to be used on 100 nights and a standard system on the remaining 100 nights. They calculated the nightly median waiting time (MWT) to see a physician. The average MWT for the new system was 4 hours with a standard deviation of 0.5 hours while the average MWT for the old system was 6 hours with a standard deviation of 2 hours. Consider the hypothesis of a decrease in the mean MWT associated with the new treatment. What does the 95% independent group confidence interval with unequal variances suggest vis a vis this hypothesis? (Because there's so many observations per group, just use the Z quantile instead of the T.)

Options:

• When subtracting (old - new) the interval is entirely above zero. The new system appears to be effective. (correct)
• When subtracting (old - new) the interval is entirely above zero. The new system does not appear to be effective.
• When subtracting (old - new) the interval contains 0. The new system appears to be effective. (previously chosen)
• When subtracting (old - new) the interval contains 0. There is not evidence suggesting that the new system is effective. (previously chosen)

R code:

• X <- 6; Sx <- 2
  Y <- 4; Sy <- 0.5
  nx <- 100; ny <- 100
  df <- (Sx^2/nx + Sy^2/ny)^2/((Sx^2/nx)^2/(nx-1) + (Sy^2/ny)^2/(ny-1))
  X - Y + c(-1,1) * qt(0.975, df) * sqrt(Sx^2/nx + Sy^2/ny)
• return: [1] 1.591503 2.408497
• Z quantile: (4 - 6)/(2/sqrt(100))

I replaced qt(0.975, df) with Z quantile at the very beginning. The question is a little confusing me as it ask using the Z quantile instead of the T. Anyway. I attempted this question correct in the last time. 7 out of 7. Perfect!

Side note: remember that the calculation of confidence intervals is different for the situation of equal and unequal variance. For unequal variance, the degree of freedom (df) has to be calculated in a more complicated way.

7.9 Coursera Self Study - Statistical Inference

By today I finished watching Week 1 and Week 2's video lectures and quizzes. I finished the Homework 1, the swirl assignment of the first lecture in the second week, and in the middle of the second lecture of the second week. I started doing the first part of the course project which is due two weeks later. I downloaded and installed the BasicTex and pandoc so that I can convert my R markdown file to a LaTex pdf. This is just something that I learned when trying to do the conversion, that converting an html file to a pdf will not show up the title, but either a md or Rmd file will make that happen.

7.7 VA Reading - Hess 2007 NCN

Hess C, Mukherjee P (2007). Visualizing White matter Pathways in the Living Human Brain: Diffusion Tensor Imaging and Beyond. Neuroimaging Clin N Am; 17(4): 407-26, vii.

Part of the article has been read and recorded already.

Diffusion magnetic resonance imaging limitations and future development
Although several white matter atlases have now been made available, a systematic validation of DTI tractography has yet to be undertaken for the large majority of neural pathways that have been cataloged using the technique.
The tracts delineated using DTI tractography lack polarity - there is at present no method that allows the differentiation of afferent and efferent connections. In the future, the orientation of tracts may be inferred by using combined functional MR imaging-diffusion MR imaging paradigms.
The mathematic technique such as HARDI for reconstruction of multi-modal diffusion have had little impact on the underlying physics that govern the measurement of diffusion-weighted MR imaging data, in that diffusion remains an SNR-limited modality. Need to increase spatial resolution, but signal will lose at the same time and increase the uncertainty with the estimated tensor or other diffusion models.
Translate diffusion MR imaging approaches to 7T or higher field strengths and to phased array head coils with increasing numbers of receiver elements can promise the increase of the available SNR, such that higher spatial and angular resolution can be more reliably achieved.
Example of recent approach: tradeoff between angular and spatial resolution for improving SNR by fusing DTI and HARDI information over concentric shells in q-space.

7.7 VA Reading - Smith 2006 NeuroImage (cont.)

Smith SM, Jenkinson M, Johansen-Berg H, Rueckert D, Nichols TE, Mackay CE, Watkins KE, Ciccarelli O, Cader MZ, Matthews PM, Behrens TEJ (2006). Tract-based spatial statistics: Voxelwise analysis of multi-subject diffusion data. NeuroImage; 31: 1487-505. 

Future directions
Use all available diffusion tensor information, rather than just FA, both in the alignment in preprocessing stage as well as feed into the final statistics. The improvement in accuracy may be modest, though.Can also include other imaging (such as T1-weighted structural images) to help drive the alignment. DTI data may suffer from low SNR, so instead of using FA map to drive the nonlinear registration, using T1-weighted images can qualitatively improve registration robustness. Can also be the segmented white matter image (from T1-weighted image) for registration, which may reduce noise and improve resolution, but have less rich contrast information.
Can extend to other diffusion measurements (mean diffusivity, tensor eigenvalues, principal tract direction, etc.). Also, selecting maximum FA is not necessary, e.g. measure tract thickness.
Useful and fairly straightforward to define a standard-space skeleton. If one was not concerned about inter-group biases resulting from such a predefined space, a standard-space mean FA image and derived skeleton could simplify TBSS analyses.
There is no reason why one has to carry out the cross-subject statistics separately for each voxel.

7.6 VA Reading - Smith 2006 NeuroImage (cont.)

Smith SM, Jenkinson M, Johansen-Berg H, Rueckert D, Nichols TE, Mackay CE, Watkins KE, Ciccarelli O, Cader MZ, Matthews PM, Behrens TEJ (2006). Tract-based spatial statistics: Voxelwise analysis of multi-subject diffusion data. NeuroImage; 31: 1487-505.

Limitations and dangers
A serious limitation of VBM-style approaches is the need for spatial smoothing, and the problem of arbitrarily choosing the spatial smoothing extent. When tract width is smaller than original voxel size, it is very difficult to determine whether a reduction in FA is really due to within-tract FA change or a change in tract thickness. Hard to solve this problem; partly for that reason, the mean FA skeleton is thresholded (typically 0.2), rather than being allowed to fall all the way to zero.
A similar possible confounding effect is within-scan head motion. One could potentially estimate head motion using image entropy measures and/or motion estimates from the eddy-current/head motion preprocessing and feed this into final statistical analyses as a confound regressor, but this would not be guaranteed to remove all related problems, and could remove the effect of interest.
Another area requiring careful interpretation is crossing tracts or tract junctions, which is difficult to be estimated by voxelwise statistics. Practically, a more sophisticated data projection approach is needed. An apparent reduction in FA at junctions can in fact be due to an increase in one of the tracts feeding into the junction, if it is a "weaker" tract than others feeding into the junction (what is a weaker tract?).

• (7.7) http://www.pnas.org/content/102/34/12212.long
  In both WM and thalamus, physiological factors other than myelin that could influence the correlation include intravoxel fiber crossing or axon diameter. In anatomic regions containing intravoxel fiber crossing, increased FA of an individual fiber population can result in a decrease in the overall FA. e.g. between optic radiation and posterior forceps.

Pathology could reduce FA too strongly, leading to exclusion of potential areas from analysis (e.g. due to the thresholding of the mean FA values). This may be a subtle effects due to the nature of most of the pathologies. But in case it happens, an appropriate approach would be to use a target FA image for registration, and mean FA skeleton, derived from a relevant control group.

7.6 Back from Weekend Mode

It's a long weekend without any study. California seashore is definitely beautiful and worth exploring again and again.
But now, I need to better integrate what I have reached so far because resources seem to be so much accumulated.
Over the past month, I have been working on:

• Process B0 masks (which is less important as pointed out by Dieter);
• Read in TBSS basics;
• Learn how to use FSL;
• Get familiar with white matter (and I must, appointed by Pratik);
• Read a bunch of other papers;

I should still spend most of the time at VA in doing the routine job, which is processing B0 masks for the cases.
For TBSS reading, the Smith 2006 has not yet finished, which needs to be done. Then, I need to focus on the paper whose TBSS algorithm is more likely to be used by me. I need to get ready for the TBSS stage and get familiar with the WM atlas I will use.
Then, I should keep getting familiar with FSL toolbox and try playing it with some image samples.
Keep reading and memorizing white matter using the reference.
Read other papers if still have time to do that.