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<!DOCTYPE html>
<html>
<head>
<title>Bayesian analysis of Neuroimaging data with R</title>
<meta charset="utf-8">
<meta name="author" content="Chris Hammill Jason Lerch" />
<meta name="date" content="2018-06-14" />
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class: center, middle, inverse, title-slide
# Bayesian analysis of Neuroimaging data with R
### Chris Hammill<br>Jason Lerch
### 2018-06-14
---
<style>
.remark-code {
font-size: 75%;
}
.remark-slide-scaler {
overflow: scroll;
}
.small {
font-size: 65%;
}
.remark-slide-number {
font-size: 10pt;
margin-bottom: -11.6px;
margin-right: 10px;
color: #FFFFFF; /* white */
opacity: 1; /* default: 0.5 */
}
</style>
# Outline
1. Quick overview of algorithms and datasets used today
1. Anatomy and hierarchies
1. Massively univariate classical statistics
1. Meet Reverend Thomas
1. A simple model
1. A more involved model
1. Diagnostics
1. A complex example (if we have time)
1. ...
1. Profit?
---
# Following Along
- To follow along with today's session you will need to either:
- run R with `c("knitr", "tidyverse", "forcats", "ggplot2", "broom", "data.tree", "treemap", "rstanarm", "bayesplot", "pheatmap", "ggrepel")` on your laptop
- use R from our singularity container
---
# Get these slides and code
- From a terminal run:
```bash
git clone https://github.com/Mouse-Imaging-Centre/Scinet-SS-bayesian-neuroimaging-R
cd Scinet-SS-bayesian-neuroimaging-R
## cp in some extra files <use scp if you're working on your own laptop>
cp /bb/scinet/course/ss2018/3_bm/7_mrir/ADNI2_BL_MAGeT_Hippocampus_subfields.csv .
cp /bb/scinet/course/ss2018/3_bm/7_mrir/flathierarchy.Rds .
cp /bb/scinet/course/ss2018/3_bm/7_mrir/twolevelhierarchy.Rds .
```
---
# Singularity set up
- First ssh to niagara and set your port to forward
```bash
ssh -L8787:localhost:<your-port> <you>@niagara.scinet.utoronto.ca
ssh -L<your-port>:localhost:<your-port> tds01
```
- To use the singularity container you will need to set an environment variable with a temporary RStudio password. You will also
need a temporary directory on SCRATCH, this assumes you don't have a directory with this name already
```bash
export RSTUDIO_PASSWORD="dont_use_this_fake_password"
mkdir $SCRATCH/tmp
module load singularity
```
---
# Run the container
- Run the container <this command will remain running>, can be sent to the background
```bash
module load singularity/2.5.1
singularity exec \
--bind /bb/scinet/course/ss2018/3_bm/7_mrir/rstudio_auth.sh:/usr/lib/rstudio-server/bin/rstudio_auth.sh \
--bind $SCRATCH/tmp:/tmp \
--bind $SCRATCH:$SCRATCH \
/bb/scinet/course/ss2018/3_bm/7_mrir/RMINC-develop-2018-06-13-6793fefb8a5a.img \
rserver \
--auth-none 0 \
--auth-pam-helper-path rstudio_auth.sh \
--www-port <your-port>
```
- Then navigate to localhost:8787 in your browser, use your scinet username and `$RSTUDIO_PASSWORD$`
---
# Optional step to compile the slides
- You'll need the package "xaringan" to render the slides
- On singularity this is slightly involved, you'll need a library directory
```r
dir.create("~/tmp_rlibs")
.libPaths(new = "~/tmp_rlibs")
install.packages("xaringan")
```
---
class: middle
# The dataset: ADNI
* baseline scans from ADNI2
* multi-site initiative to study Alzheimer's and Mild Cognitive Impairment
With much gratitude to Nikhil Bhagwat of the CoBrA Lab (PI: Chakravarty) for providing us with processed data.
---
# MAGeT - the algorithm
Chakravarty MM, Steadman P, van Eede MC, Calcott RD, Gu V, Shaw P, Raznahan A, Collins DL, Lerch JP. Performing label-fusion-based segmentation using multiple automatically generated templates. Hum Brain Mapp. 2013 Oct;34(10):2635–54.
---
# MAGeT - the atlas
<img src="hippoatlas.png" width="576" height="400px" />
.small[
Pipitone J, Park MTM, Winterburn J, Lett TA, Lerch JP, Pruessner JC, Lepage M, Voineskos AN, Chakravarty MM, Alzheimer’s Disease Neuroimaging Initiative. Multi-atlas segmentation of the whole hippocampus and subfields using multiple automatically generated templates. Neuroimage. 2014 Nov 1;101:494–512.
Winterburn JL, Pruessner JC, Chavez S, Schira MM, Lobaugh NJ, Voineskos AN, Chakravarty MM. A novel in vivo atlas of human hippocampal subfields using high-resolution 3 T magnetic resonance imaging. Neuroimage. 2013 Jul 1;74:254–65.
]
---
# Load the data
Read in the processed files
```r
suppressMessages(library(tidyverse))
adni <- read.csv("ADNI2_BL_MAGeT_Hippocampus_subfields.csv")
names(adni)
```
```
## [1] "X" "L_CA1" "L_subiculum" "L_CA4DG" "L_CA2CA3"
## [6] "L_stratum" "L_Alv" "L_Fimb" "L_Fornix" "L_Mam"
## [11] "R_CA1" "R_subiculum" "R_CA4DG" "R_CA2CA3" "R_stratum"
## [16] "R_Alv" "R_Fimb" "R_Fornix" "R_Mam" "PTID"
## [21] "VISCODE" "ImageUID" "ORIGPROT" "COLPROT" "AGE"
## [26] "APOE4" "PTGENDER" "DX_bl" "MMSE" "ADAS11"
## [31] "ADAS13"
```
---
#Reorganize the diagnosis label
```r
library(forcats)
adni <- adni %>%
mutate(DX_bl=fct_relevel(DX_bl,
"CN", "SMC", "EMCI", "LMCI", "AD"))
```
---
# Data organization
Make the dataframe long rather than wide
```r
adniLong <- adni %>%
select(-X) %>%
gather(structure, volume, L_CA1:R_Mam)
adniLong %>%
select(PTID, DX_bl, structure, volume) %>% sample_n(5)
```
```
## PTID DX_bl structure volume
## 8863 073_S_4443 EMCI R_CA2CA3 188.39999
## 7152 013_S_4395 LMCI R_subiculum 255.60001
## 3466 137_S_4536 EMCI L_stratum 446.39997
## 6336 002_S_4270 CN R_CA1 642.00003
## 4577 057_S_5295 SMC L_Fimb 86.64124
```
---
# A quick look at the data
```r
library(ggplot2)
ggplot(adniLong) + aes(x=DX_bl, y=AGE) + geom_boxplot()
```
<!-- -->
---
# A quick look at hippocampal volume
```r
adniLong %>% group_by(PTID) %>%
summarize(DX=DX_bl[1], GENDER=PTGENDER[1], HPC=sum(volume)) %>%
ggplot() + aes(DX, HPC, colour=GENDER) + geom_boxplot()
```
<!-- -->
---
# Classic Statistics
1. Decide on model
1. Apply model to every ROI/voxel separately
1. Widen confidence intervals to account for multiple comparisons
---
# map for looping over structures
```r
library(broom)
adniLong %>%
split(.$structure) %>%
map(~lm(volume ~ AGE+PTGENDER+DX_bl, .)) %>%
map_dfr(tidy, .id='roi') %>% head(14)
```
```
## roi term estimate std.error statistic p.value
## 1 L_Alv (Intercept) 196.5742731 29.2852653 6.7123952 3.973773e-11
## 2 L_Alv AGE 2.5174188 0.3956152 6.3633019 3.580089e-10
## 3 L_Alv PTGENDERMale 65.3630633 5.6039202 11.6638106 7.719764e-29
## 4 L_Alv DX_blSMC 16.4673321 9.2922494 1.7721578 7.680579e-02
## 5 L_Alv DX_blEMCI 16.8111358 8.1134240 2.0720149 3.863178e-02
## 6 L_Alv DX_blLMCI 27.8697000 8.3720730 3.3288888 9.178674e-04
## 7 L_Alv DX_blAD 44.5716230 8.5053623 5.2404144 2.126327e-07
## 8 L_CA1 (Intercept) 706.9671959 41.3804715 17.0845612 6.521867e-55
## 9 L_CA1 AGE 0.2790874 0.5590095 0.4992534 6.177587e-01
## 10 L_CA1 PTGENDERMale 87.9815655 7.9184141 11.1110083 1.619634e-26
## 11 L_CA1 DX_blSMC 36.2846228 13.1300726 2.7634747 5.870110e-03
## 12 L_CA1 DX_blEMCI -1.3742345 11.4643766 -0.1198700 9.046207e-01
## 13 L_CA1 DX_blLMCI -29.5212053 11.8298511 -2.4954841 1.280902e-02
## 14 L_CA1 DX_blAD -49.0521356 12.0181907 -4.0814909 4.993610e-05
```
---
# better overview
```r
rTable <- adniLong %>%
split(.$structure) %>%
map(~lm(volume ~ AGE+PTGENDER+DX_bl, .)) %>%
map_dfr(tidy, .id='roi') %>%
filter(startsWith(term, "DX")) %>%
select(roi, term, statistic) %>%
spread(term, statistic)
```
---
# better overview
```r
rTable
```
```
## roi DX_blAD DX_blEMCI DX_blLMCI DX_blSMC
## 1 L_Alv 5.2404144 2.0720149 3.3288888 1.7721578
## 2 L_CA1 -4.0814909 -0.1198700 -2.4954841 2.7634747
## 3 L_CA2CA3 -1.1964159 -0.2493317 -0.9781210 1.2829814
## 4 L_CA4DG -7.0736847 -1.0666621 -4.5515264 1.8258561
## 5 L_Fimb -5.8276864 -1.3677552 -5.1790266 1.0673108
## 6 L_Fornix -0.1788469 1.1224113 -1.1998364 1.6453122
## 7 L_Mam -3.0547427 0.4167804 -0.9110615 1.0460497
## 8 L_stratum -7.5668406 -0.8851018 -4.2477610 2.0285441
## 9 L_subiculum -8.4089152 -1.3472510 -4.6988241 1.4397638
## 10 R_Alv 2.2203457 1.3354607 1.9552464 1.0756636
## 11 R_CA1 -2.3190902 0.2005679 -1.8465551 2.0229622
## 12 R_CA2CA3 -5.0678893 -1.2333109 -2.5235391 0.1698605
## 13 R_CA4DG -4.2709485 -0.4105204 -3.5381430 0.7337568
## 14 R_Fimb -6.0683126 -1.5310185 -4.3249336 -0.1997899
## 15 R_Fornix -1.1565583 0.3071713 -2.3384244 1.1363678
## 16 R_Mam -1.9359947 1.2241687 -0.8748597 1.2207012
## 17 R_stratum -8.1615304 -1.7229391 -4.4048893 0.9723020
## 18 R_subiculum -7.7494464 -1.0529968 -4.5346801 1.7301202
```
---
# better overview
```r
DT::datatable(rTable %>% remove_rownames() %>%
column_to_rownames("roi") %>% round(2), options=list(pageLength=7))
```
<div id="htmlwidget-5e56aa2e737d4c5dc806" style="width:100%;height:auto;" class="datatables html-widget"></div>
<script type="application/json" data-for="htmlwidget-5e56aa2e737d4c5dc806">{"x":{"filter":"none","data":[["L_Alv","L_CA1","L_CA2CA3","L_CA4DG","L_Fimb","L_Fornix","L_Mam","L_stratum","L_subiculum","R_Alv","R_CA1","R_CA2CA3","R_CA4DG","R_Fimb","R_Fornix","R_Mam","R_stratum","R_subiculum"],[5.24,-4.08,-1.2,-7.07,-5.83,-0.18,-3.05,-7.57,-8.41,2.22,-2.32,-5.07,-4.27,-6.07,-1.16,-1.94,-8.16,-7.75],[2.07,-0.12,-0.25,-1.07,-1.37,1.12,0.42,-0.89,-1.35,1.34,0.2,-1.23,-0.41,-1.53,0.31,1.22,-1.72,-1.05],[3.33,-2.5,-0.98,-4.55,-5.18,-1.2,-0.91,-4.25,-4.7,1.96,-1.85,-2.52,-3.54,-4.32,-2.34,-0.87,-4.4,-4.53],[1.77,2.76,1.28,1.83,1.07,1.65,1.05,2.03,1.44,1.08,2.02,0.17,0.73,-0.2,1.14,1.22,0.97,1.73]],"container":"<table class=\"display\">\n <thead>\n <tr>\n <th> <\/th>\n <th>DX_blAD<\/th>\n <th>DX_blEMCI<\/th>\n <th>DX_blLMCI<\/th>\n <th>DX_blSMC<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"pageLength":7,"columnDefs":[{"className":"dt-right","targets":[1,2,3,4]},{"orderable":false,"targets":0}],"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[7,10,25,50,100]}},"evals":[],"jsHooks":[]}</script>
---
# Frequentist Null Hypothesis Testing
- To form conclusions in frequentism we typically lean on null hypothesis testing.
- Null hypotheses are parameter values for your model you'd like to disprove
- If your statistics (and more extreme statistics) would be very unlikely given your null model
you reject the null hypothesis, and conclude that the null hypothesis is not correct.
- Choosing a threshold for this probability (e.g. 0.05) and rejecting when your p-value
is below the threshold gives you a fixed probability of making a "Type I" error,
which conveniently is equal to your threshold.
- So if we reject all p-values when they are below 0.05 we have a 5% chance of
rejecting when the null model is in fact true.
- If this is confusing, you're not alone, this is very hard to wrap your mind around.
---
# Zoom in on a single structure
```r
adniLong %>%
split(.$structure) %>%
map(~lm(volume ~ AGE+PTGENDER+DX_bl, .)) %>%
first %>%
summary
```
```
##
## Call:
## lm(formula = volume ~ AGE + PTGENDER + DX_bl, data = .)
##
## Residuals:
## Min 1Q Median 3Q Max
## -398.90 -49.05 -7.69 45.79 280.80
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 196.5743 29.2853 6.712 3.97e-11 ***
## AGE 2.5174 0.3956 6.363 3.58e-10 ***
## PTGENDERMale 65.3631 5.6039 11.664 < 2e-16 ***
## DX_blSMC 16.4673 9.2922 1.772 0.076806 .
## DX_blEMCI 16.8111 8.1134 2.072 0.038632 *
## DX_blLMCI 27.8697 8.3721 3.329 0.000918 ***
## DX_blAD 44.5716 8.5054 5.240 2.13e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 73.1 on 696 degrees of freedom
## Multiple R-squared: 0.2577, Adjusted R-squared: 0.2513
## F-statistic: 40.28 on 6 and 696 DF, p-value: < 2.2e-16
```
---
# Interpreting these results
- After fitting our models we're left with:
1. coefficient estimates
1. t-statistics
1. p-values
- We know if our model assumptions are satisfied our t-statistics
have a known distribution.
- From this distribution we can figure out the probability of t-statistics
as large or larger than the one we observed (p-value)
---
# Dealing with Many Tests
- If you're testing a lot of hypotheses, a 5% chance of making a mistake adds up
- After 14 tests you have a better than a 50/50 chance of having made at least one mistake
- How do we control for this?
- Two main approaches Family-Wise Error Rate (FWER) control and False-Discovery Rate (FDR) control.
---
# FWER
- In family-wise error rate control, we try to limit the chance we will at least one
type I error.
- Quite conservative, so in neuroimaging we tend to use False Discovery Rate control.
---
# FDR
- Instead of trying to control our chances of making at least one mistake, let's try to control the
fraction of mistakes we make.
- To do this we employ the Benjamini-Hochberg procedure.
- The Benjamini-Hochberg procedure turns our p-values in q-values. Rejecting all q-values below some
threshold controls the expected number of mistakes.
- For example if we reject all hypotheses with q < 0.05, we expect about 5% of our results to be
false discoveries (type I errors).
- If we have 100's or more tests we can accept a few mistakes in the interest of finding the
important results.
---
# multiple comparisons - omnibus FDR
```r
pTable <- adniLong %>%
split(.$structure) %>%
map(~lm(volume ~ AGE+PTGENDER+DX_bl, .)) %>%
map_dfr(tidy, .id='roi') %>%
filter(startsWith(term, "DX")) %>%
select(roi, term, p.value) %>%
spread(term, p.value) %>%
remove_rownames() %>%
column_to_rownames("roi") %>%
as.matrix()
qTable <- pTable
qTable[,] <- p.adjust(pTable, 'fdr')
```
---
# multiple comparisons - omnibus FDR
```r
qTable
```
```
## DX_blAD DX_blEMCI DX_blLMCI DX_blSMC
## L_Alv 1.913694e-06 0.09933886 3.304323e-03 0.15800049
## L_CA1 1.997444e-04 0.90462074 3.842707e-02 0.01921127
## L_CA2CA3 3.408123e-01 0.86311637 4.042228e-01 0.32715109
## L_CA4DG 5.305411e-11 0.37376998 3.761830e-05 0.14463584
## L_Fimb 8.840563e-08 0.30175056 2.338860e-06 0.37376998
## L_Fornix 8.773547e-01 0.36287262 3.408123e-01 0.19015000
## L_Mam 8.017881e-03 0.75494833 4.350941e-01 0.37376998
## L_stratum 2.186873e-12 0.44355718 1.038839e-04 0.10430127
## L_subiculum 1.682195e-14 0.30501574 2.064548e-05 0.27069092
## R_Alv 7.124433e-02 0.30501574 1.183444e-01 0.37376998
## R_CA1 5.726437e-02 0.87735472 1.423331e-01 0.10430127
## R_CA2CA3 3.715510e-06 0.34081228 3.706287e-02 0.87735472
## R_CA4DG 9.976500e-05 0.75494833 1.628422e-03 0.52953604
## R_Fimb 2.547278e-08 0.23302004 8.394683e-05 0.87735472
## R_Fornix 3.569035e-01 0.82778710 5.658467e-02 0.36168535
## R_Mam 1.198641e-01 0.34081228 4.435572e-01 0.34081228
## R_stratum 5.578523e-14 0.16607426 6.300383e-05 0.40422277
## R_subiculum 7.859835e-13 0.37376998 3.761830e-05 0.16607426
```
---
# Hippocampal anatomical hierarchy
Setting up a simple hierarchy, dividing the hippocampus in grey matter and tracts.
1. Hippocampal Formation
1. Grey Matter
1. CA1
1. CA2/CA3
1. CA4/DG
1. subiculum
1. stratum
1. Mammilary bodies
1. White Matter
1. Alveus
1. Fimbria
1. Fornix
---
# Hippocampal anatomical hierarchy
```r
library(data.tree)
hpc <- Node$new("HPC")
gm <- hpc$AddChild("GM")
ca1 <- gm$AddChild("CA1")
lca1 <- ca1$AddChild("left CA1")
lca1$volumes <- adni$L_CA1
rca1 <- ca1$AddChild("right CA1")
rca1$volumes <- adni$R_CA1
ca23 <- gm$AddChild("CA2CA3")
lca23 <- ca23$AddChild("left CA2CA3")
lca23$volumes <- adni$L_CA2CA3
rca23 <- ca23$AddChild("right CA2CA3")
rca23$volumes <- adni$R_CA2CA3
ca4 <- gm$AddChild("CA4DG")
lca4 <- ca4$AddChild("left CA4DG")
lca4$volumes <- adni$L_CA4DG
rca4 <- ca4$AddChild("right CA4DG")
rca4$volumes <- adni$R_CA4DG
subiculum <- gm$AddChild("subiculum")
lsubiculum <- subiculum$AddChild("left subiculum")
lsubiculum$volumes <- adni$L_subiculum
rsubiculum <- subiculum$AddChild("right subiculum")
rsubiculum$volumes <- adni$R_subiculum
stratum <- gm$AddChild("stratum")
lstratum <- stratum$AddChild("left stratum")
lstratum$volumes <- adni$L_stratum
rstratum <- stratum$AddChild("right stratum")
rstratum$volumes <- adni$R_stratum
mam <- gm$AddChild("Mammillary bodies")
lmam <- mam$AddChild("left Mammillary bodies")
lmam$volumes <- adni$L_Mam
rmam <- mam$AddChild("right Mammillary bodies")
rmam$volumes <- adni$R_Mam
wm <- hpc$AddChild("WM")
alveus <- wm$AddChild("Alveus")
lalveus <- alveus$AddChild("left Alveus")
lalveus$volumes <- adni$L_Alv
ralveus <- alveus$AddChild("right Alveus")
ralveus$volumes <- adni$R_Alv
fimbria <- wm$AddChild("Fimbria")
lfimbria <- fimbria$AddChild("left Fimbria")
lfimbria$volumes <- adni$L_Fimb
rfimbria <- fimbria$AddChild("right Fimbria")
rfimbria$volumes <- adni$R_Fimb
fornix <- wm$AddChild("Fornix")
lfornix <- fornix$AddChild("left Fornix")
lfornix$volumes <- adni$L_Fornix
rfornix <- fornix$AddChild("right Fornix")
rfornix$volumes <- adni$R_Fornix
```
---
# Hippocampal anatomical hierarchy
```r
hpc
```
```
## levelName
## 1 HPC
## 2 ¦--GM
## 3 ¦ ¦--CA1
## 4 ¦ ¦ ¦--left CA1
## 5 ¦ ¦ °--right CA1
## 6 ¦ ¦--CA2CA3
## 7 ¦ ¦ ¦--left CA2CA3
## 8 ¦ ¦ °--right CA2CA3
## 9 ¦ ¦--CA4DG
## 10 ¦ ¦ ¦--left CA4DG
## 11 ¦ ¦ °--right CA4DG
## 12 ¦ ¦--subiculum
## 13 ¦ ¦ ¦--left subiculum
## 14 ¦ ¦ °--right subiculum
## 15 ¦ ¦--stratum
## 16 ¦ ¦ ¦--left stratum
## 17 ¦ ¦ °--right stratum
## 18 ¦ °--Mammillary bodies
## 19 ¦ ¦--left Mammillary bodies
## 20 ¦ °--right Mammillary bodies
## 21 °--WM
## 22 ¦--Alveus
## 23 ¦ ¦--left Alveus
## 24 ¦ °--right Alveus
## 25 ¦--Fimbria
## 26 ¦ ¦--left Fimbria
## 27 ¦ °--right Fimbria
## 28 °--Fornix
## 29 ¦--left Fornix
## 30 °--right Fornix
```
---
# Hippocampal anatomical hierarchy
```r
SetGraphStyle(hpc, rankdir="LR")
plot(hpc)
```
<div id="htmlwidget-ad8109ac80e4352e3d65" style="width:720px;height:432px;" class="grViz html-widget"></div>
<script type="application/json" data-for="htmlwidget-ad8109ac80e4352e3d65">{"x":{"diagram":"digraph {\n\ngraph [rankdir = \"LR\"]\n\n\n\n \"1\" [label = \"HPC\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\"] \n \"2\" [label = \"GM\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\"] \n \"3\" [label = \"CA1\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\"] \n \"4\" [label = \"left CA1\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\"] \n \"5\" [label = \"right CA1\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\"] \n \"6\" [label = \"CA2CA3\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\"] \n \"7\" [label = \"left CA2CA3\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\"] \n \"8\" [label = \"right CA2CA3\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\"] \n \"9\" [label = \"CA4DG\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\"] \n \"10\" [label = \"left CA4DG\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\"] \n \"11\" [label = \"right CA4DG\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\"] \n \"12\" [label = \"subiculum\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\"] \n \"13\" [label = \"left subiculum\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\"] \n \"14\" [label = \"right subiculum\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\"] \n \"15\" [label = \"stratum\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\"] \n \"16\" [label = \"left stratum\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\"] \n \"17\" [label = \"right stratum\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\"] \n \"18\" [label = \"Mammillary bodies\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\"] \n \"19\" [label = \"left Mammillary bodies\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\"] \n \"20\" [label = \"right Mammillary bodies\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\"] \n \"21\" [label = \"WM\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\"] \n \"22\" [label = \"Alveus\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\"] \n \"23\" [label = \"left Alveus\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\"] \n \"24\" [label = \"right Alveus\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\"] \n \"25\" [label = \"Fimbria\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\"] \n \"26\" [label = \"left Fimbria\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\"] \n \"27\" [label = \"right Fimbria\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\"] \n \"28\" [label = \"Fornix\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\"] \n \"29\" [label = \"left Fornix\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\"] \n \"30\" [label = \"right Fornix\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\"] \n \"1\"->\"2\" \n \"1\"->\"21\" \n \"2\"->\"3\" \n \"2\"->\"6\" \n \"2\"->\"9\" \n \"2\"->\"12\" \n \"2\"->\"15\" \n \"2\"->\"18\" \n \"21\"->\"22\" \n \"21\"->\"25\" \n \"21\"->\"28\" \n \"3\"->\"4\" \n \"3\"->\"5\" \n \"6\"->\"7\" \n \"6\"->\"8\" \n \"9\"->\"10\" \n \"9\"->\"11\" \n \"12\"->\"13\" \n \"12\"->\"14\" \n \"15\"->\"16\" \n \"15\"->\"17\" \n \"18\"->\"19\" \n \"18\"->\"20\" \n \"22\"->\"23\" \n \"22\"->\"24\" \n \"25\"->\"26\" \n \"25\"->\"27\" \n \"28\"->\"29\" \n \"28\"->\"30\" \n}","config":{"engine":"dot","options":null}},"evals":[],"jsHooks":[]}</script>
---
# Aggregate up the tree
```r
hpc$Do(function(x){
x$volumes <- Aggregate(x, "volumes", rowSums)
}, traversal="post-order", filterFun=isNotLeaf)
hpc$Do(function(x) {
x$meanVolume <- mean(x$volumes)
})
```
---
# Tree graph
```r
library(treemap)
ToDataFrameTable(hpc, "pathString", "meanVolume", "name") %>%
mutate(path=strsplit(pathString, "/"),
struct=map_chr(path, ~ .x[3]),
tissue=map_chr(path, ~ .x[2])) %>%
select(struct, tissue, name, meanVolume) %>%
treemap(index=c("tissue", "struct"), vSize="meanVolume")
```
<!-- -->
---
# Statistics on the tree
```r
hpc$Do(function(x){
adni$volumes <- x$volumes
x$stats <-
lm(volumes ~ AGE+PTGENDER+DX_bl, adni) %>%
tidy() %>%
filter(startsWith(term, "DX")) %>%
select(term, estimate, statistic, p.value)
})
```
---
# Statistics on the tree
```r
print(hpc, AD=function(x)
x$stats %>% filter(term=="DX_blAD") %>% select(statistic) )
```
```
## levelName AD
## 1 HPC -4.5728017
## 2 ¦--GM -6.4256670
## 3 ¦ ¦--CA1 -3.2923199
## 4 ¦ ¦ ¦--left CA1 -4.0814909
## 5 ¦ ¦ °--right CA1 -2.3190902
## 6 ¦ ¦--CA2CA3 -3.5392995
## 7 ¦ ¦ ¦--left CA2CA3 -1.1964159
## 8 ¦ ¦ °--right CA2CA3 -5.0678893
## 9 ¦ ¦--CA4DG -5.9874539
## 10 ¦ ¦ ¦--left CA4DG -7.0736847
## 11 ¦ ¦ °--right CA4DG -4.2709485
## 12 ¦ ¦--subiculum -8.6168787
## 13 ¦ ¦ ¦--left subiculum -8.4089152
## 14 ¦ ¦ °--right subiculum -7.7494464
## 15 ¦ ¦--stratum -8.2297918
## 16 ¦ ¦ ¦--left stratum -7.5668406
## 17 ¦ ¦ °--right stratum -8.1615304
## 18 ¦ °--Mammillary bodies -2.6843408
## 19 ¦ ¦--left Mammillary bodies -3.0547427
## 20 ¦ °--right Mammillary bodies -1.9359947
## 21 °--WM 0.7325428
## 22 ¦--Alveus 4.1207504
## 23 ¦ ¦--left Alveus 5.2404144
## 24 ¦ °--right Alveus 2.2203457
## 25 ¦--Fimbria -6.5869583
## 26 ¦ ¦--left Fimbria -5.8276864
## 27 ¦ °--right Fimbria -6.0683126
## 28 °--Fornix -0.6995234
## 29 ¦--left Fornix -0.1788469
## 30 °--right Fornix -1.1565583
```
---
# Statistics on the tree
```r
print(hpc, AD=function(x)
x$stats %>% filter(term=="DX_blAD") %>%
select(statistic),
filterFun=isNotLeaf )
```
```
## levelName AD
## 1 HPC -4.5728017
## 2 ¦--GM -6.4256670
## 3 ¦ ¦--CA1 -3.2923199
## 4 ¦ ¦--CA2CA3 -3.5392995
## 5 ¦ ¦--CA4DG -5.9874539
## 6 ¦ ¦--subiculum -8.6168787
## 7 ¦ ¦--stratum -8.2297918
## 8 ¦ °--Mammillary bodies -2.6843408
## 9 °--WM 0.7325428
## 10 ¦--Alveus 4.1207504
## 11 ¦--Fimbria -6.5869583
## 12 °--Fornix -0.6995234
```
---
# Statistics on the tree
```r
hpc$Do(function(x){
x$ad <- -log10(x$stats %>% filter(term=="DX_blAD") %>%
select(p.value))
})
ToDataFrameTable(hpc, "pathString", "meanVolume", "name", "ad") %>%
mutate(path=strsplit(pathString, "/"),
struct=map_chr(path, ~ .x[3]),
tissue=map_chr(path, ~ .x[2])) %>%
select(struct, tissue, name, meanVolume, ad) %>%
treemap(index=c("tissue", "struct"), vSize="meanVolume",
vColor="ad", type="value")
```
<!-- -->
---
class: center
# And now for Bayesianism!
---
# Why Bayesian Statistics?
Have you ever...
1. Been confused about what a p-value means?
1. Been frustrated that a difference in significance doesn't mean a significant difference?
1. Known some values for a parameter are impossible but been unable to use that to your advantage?
1. Wanted to ask more interesting questions than whether or not a parameter is or isn't zero?
1. Wanted to use information from the literature to improve your estimates?
---
# Why Bayesian Statistics?
Have you ever...
1. Been confused about what a p-value means?
1. Been frustrated that a difference in significance doesn't mean a significant difference?
1. Known some values for a parameter are impossible but been unable to use that to your advantage?
1. Wanted to ask more interesting questions than whether or not a parameter is or isn't zero?
1. Wanted to use information from the literature to improve your estimates?
Then Bayesian statistics might be right for you!
---
## How you ask?
1. **De-emphasize binary descisions.**
Bayesians avoid null hypothesis tests, instead focusing on estimating their parameters of
interest, and reporting their uncertainty.
1. **Posterior Distributions**
Bayesian analyses produce a distribution of possible parameter values (the posterior), that
can be used to ask many interesting questions about values. E.g. what is the probability the
effect in the hippocampus is larger than the effect in the anterior cingulate cortex.
1. **Prior Information**
Bayesian analyses can use prior information. Bayesian analysis requires an *a priori* assessment
of how likely certain parameters are. This can be vague (uninformative) or can precise (informative)
and steer your analysis away from nonsensical results.
---
class: center
# Meet The Reverend
Reverend Thomas Bayes
---
class: middle
## Bayes' Theorem
- Bayes noticed this useful property for the probabilities for two events "A" and "B"
$$ \color{red}{P(A | B)} = \frac{{\color{blue}{P(B | A)}\color{orange}{P(A)}}}{\color{magenta}{P(B)}} $$
- `\(\color{red}{P(A|B)}\)`: The probability of A given that B happened
- `\(\color{blue}{P(B|A)}\)`: The probability of B given that A happened
- `\(\color{orange}{P(A)}\)`: The probability of A
- `\(\color{magenta}{P(B)}\)`: the probability of B
- Bayes did this in the context of the binomial distribution
---
class: middle
# But who's that behind him!
---
class: center
# It's Pierre-Simon Laplace
---
## Bayesian Statistics
- Laplace generalized Bayes Theorem into it's modern form. While working on sex-ratios in French births.
- For light reading on the history of bayesianism consider reading [the theory that would not die](https://yalebooks.yale.edu/book/9780300188226/theory-would-not-die)
## Bayes in brief
- Start with some parameters `\(\theta\)`
- Collect some data `\(D\)`
- And deduce the probability of different values of `\(\theta\)` given that you observed `\(D\)`
- Key difference between Bayesianism and Frequentism is that view that `\(\theta\)` has an associated
probability distribution. In frequentism `\(\theta\)` is an unknown constant.
---
# Different Probabilities
- Frequentists believe that probabilities represent the long-run proportion of events
- Under this model `\(P(\theta)\)` doesn't make much sense.
- Ramsey and DeFinetti showed that probability can also represent degree of belief.
- Under this model `\(P(\theta)\)` is an assesment of what you think the the parameter will be.
- For some of the philosophy underpinning bayesian reasoning consider reading
[Bayesian philosophy of science](http://www.laeuferpaar.de/Papers/BookFrame_v1.pdf)
---
class: middle
# Bayes' Theorem Redux
$$ \color{red}{P(\theta | D)} = \frac{{\color{blue}{P(D | \theta)}\color{orange}{P(\theta)}}}{\color{magenta}{\int P(D | \theta)P(\theta)d\theta}} $$
**Posterior**: `\(\color{red}{P(\theta|D)}\)`:
the probability of our parameters given our data
**Likelihood**: `\(\color{blue}{P(D|\theta)}\)`
The probability of our data given our parameters
**Prior**: `\(\color{orange}{P(\theta)}\)`
The probability of our parameters before we saw the data
**Normalizing Constant**: `\(\color{magenta}{\int P(D | \theta)P(\theta)d\theta}\)`
The probability of the data averaged over all possible parameter sets
---
class: middle
# Bayes' Theorem Redux
$$ \color{red}{P(\theta | D)} \propto \color{blue}{P(D | \theta)}\color{orange}{P(\theta)}$$
**Posterior**: `\(\color{red}{P(\theta|D)}\)`:
the probability of our parameters given our data
**Likelihood**: `\(\color{blue}{P(D|\theta)}\)`
The probability of our data given our parameters
**Prior**: `\(\color{orange}{P(\theta)}\)`
The probability of our parameters before we saw the data
---
class: middle
# Bayes' Theorem Redux
$$ \color{red}{P(\theta | D)} \propto \color{orange}{P(\theta)}\color{blue}{P(D | \theta)}$$
**Posterior**: `\(\color{red}{P(\theta|D)}\)`:
the probability of our parameters given our data
**Prior**: `\(\color{orange}{P(\theta)}\)`
The probability of our parameters before we saw the data
**Likelihood**: `\(\color{blue}{P(D|\theta)}\)`
The probability of our data given our parameters
**Pardon the re-ordering**
---
class: middle
# Posterior
`\(\color{red}{P(\theta|D)}\)`
- The goal of bayesian statistics
- The posterior is probability distribution over parameters.
- Depends on the data we observed.
- Can be used to answer interesting questions. For
example how likely is an effect between two biologically meaninful boundaries.
---
class: middle
# Prior
`\(\color{orange}{P(\theta)}\)`
- This is what we knew before the experiment.
- The prior is also a probability distribution over parameters.
- Doesn't depend on the data we saw.
- Gives a probability for any value the parameters could take.
---
class: middle
# Likelihood
`\(\color{blue}{P(D | \theta)}\)`
- This is how probable our data is given a hypothetical parameter set
- The likelihood is a probability distribution over data (not parameters)
- Is still a function of parameters.
---