21 - Group Analysis: Part 2 of 2
January 28, 2019
May 30, 2018
Gang Chen, NIMH
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Gang Chen, NIMH
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PRESENTER: Previously, I talk about the two different ways to threshold-- I mean, to come up with the-- control the false positive rate at the whole brain level. So one approach is purely based on the cluster size but based on the spatial extent. So sometimes they'll call it like the mountain approach. You have different mountains, so you compare the mountain.
So the one approach is like the cross-section the area. So that's the primary threshold approach. So the mountain is you cut it at some level, then you compare how much area that is section cut off. So that's both-- one of my favorite theory SPM and Monte Carlo simulation approaching [INAUDIBLE] both projects adopt that.
So with that approach, firstly, you have the voxel-wide threshold. That's the primary threshold. Suppose you choose the p-value. At voxel level, it's 0.01. So you find all the clusters, then you do something else to come up with the cluster threshold.
Suppose it's 55 voxels. Then you come back to your real data. For those clusters, with that primary threshold, if you are greater than 55, OK, then you say, we can talk about those clusters. Otherwise, forget about it.
So even though this cluster is 54, just one voxel less, then that is-- whatever threshold, then people can't do anything about it. Just one voxel different. Is that-- I mean, was it that important? So that's-- immediately, we face that problem.
So just one voxel difference, we cannot talk about it, because nowadays the journal reviewers, they're very picky about this so-called p-value thing. That's a false positive rate. So that's one problem.
Another problem is that they focus on the spatial part then let it cut off. It's just-- I mean, artificially, your reality, the whole-- the brain, even though it's-- I mean, let me put it this way. The law hypothesis itself, it's a strawman. I mean, in reality, nobody cares about that so-called zero activity, low activation. In reality, the real situations, we already know the whole brain pretty much. Almost all regions are some extent activated. You don't get exact zeroes. In the brain.
All the voxels except, I mean, for white matters, CSF, maybe you can say low activation. For other brain matter, nobody can make a claim a reading is really zero. So that's the starting point of law hypothesis. Beta is zero, that's just not reasonable to start with.
So the more reasonable approach is the law hypothesis itself. It's something we can-- with more confidence to make such a claim, instead of-- to have based on the p-value. That-- I mean, I'm going to skip, actually. But that criticism about this-- sorry-- p-value thing. I mean, there are many ways people have debated about this, have been debating about this for many years.
So I'm going to skip another criticism about this, I mean, the p-value thing. So let's see what can we do better. I'm going to skip this as well. Let's use the real data to demonstrate the alternative approach.
So this is a typical fMRI data, but we're focusing on the ROI level. So in this case, we have 124 subjects. All of them are kids. So the data is resting state data. So for each subject, they performed a correlation analysis. Resting state, there are many ways to analyze.
But this is the traditional approach. You pick up a seed region. You want to calculate the correlation between this seed region and the rest of the brain. So you have this correlation map. Then you convert it to z score, facial z score. So you have that z map per subject.
So with 124 subjects, when you go to group level, you have 124 z score three-dimensional volume data. So group level, suppose the model is-- we have this-- it's regression model. It's a simple regression, because we have one experimental variable, which is behavior data. I believe it's a theory of mind measurement.
So the model is-- at each voxel you have the y. On the left-hand side, y is the z score. That's the seed-based correlation. On the right-hand side, you have the intercept. You that the x is the-- that's a theory of mind measurement for each child. So that's the-- if you took the traditional approach, you have this simple regression model, generally linear model. Then you supposed we care about the effect is the b, the theory of mind effect, at each voxel.
So if you do that at the whole brain level, this is the scenario we have. So depends on which primary threshold. In this table, the first row, the voxel-wise threshold is the 0.001. That's pretty much-- nowadays, the journal-- or the journals or reviewers demand use 0.001.
So in other case, if you happen to use a Monte Carlo simulations-- we haven't talked about that yet, s you run simulations with noise. But also you have the spatial structure. Then you decide the threshold. In this case, the number of voxels is 28.
So whenever you have the cluster which is 28 or above, then you can see you can make a confident claim talk about the theory of mind, that behavior effect. So there are only two clusters that managed to survive that threshold. That's on the last column, those are the two ROIs.
So if you adopt a different threshold, 0.005-- that's the second row-- then the threshold, of course, it will need to be more lenient. So the cluster has to be bigger. That's the concept. And it's very intuitive.
That threshold is a 66 voxels. So then you have four clusters. Terms you keep going, move to 0.01. That's probably a little bit difficult to pass those reviewers. But anyway, if you do that, the clusters size threshold is 106. Then you still have four clusters.
You keep doing 0.05-- I mean, that's definitely-- nobody will-- I mean, journals will not even accept it. But if you do it anyway, it would require 467 voxels. In that case, there still are six clusters manage to survive.
So there are two things I want to mention here. First of all, it's a little bit arbitrary. So it depends on what kind of primary threshold you may end up with-- slightly different without. Notice the first row, you only have-- you can claim two clusters. The latter three rows, you get four clusters.
So it's nearly arbitrary. That's troubling. I mean, which one do you trust? Right? I mean-- and it's something-- I mean, should be little bit more rigorous and more-- I mean, but this arbitrary is definitely putting us in a dicey situation. So that's one problem.
Another problem is when is this spatial extent is a nice approach, is a reasonable approach? I mean, let me put another hypothetical scenario. That scenario is, suppose you have two clusters. One cluster-- suppose just focus on the first row-- is pretty heavy, because that cluster is above the threshold 28. Suppose you have 30 voxels-- OK, nice. Everybody is happy. We can say something about that.
The second cluster is only 20 voxels. But that's just the cluster size. How about if we ignore the signal strength? We're not talking about the beta value. We're just based on the t-statistic. So is that a reasonable approach-- just based on the t-statistic, ignore the signal strength? Even though the second cluster may be even stronger, the signal itself is stronger.
But simply because the cluster is smaller-- the smaller can be many ways. It can be, of course, the sample size, or can be just anatomically, intrinsically, that region is smaller. Why do we need to penalize those regions that are just anatomically smaller? Is that reasonable? Is that a nice approach?
So that's-- I mean, there are many ways we can criticize the current approaches. So we can do better to avoid those kinds of dilemmas so how do we do better? That's why I call the current approach is inefficient.
Why is it inefficient? There's some information we throw away, we [INAUDIBLE] we ignore-- we ignore, with only you take advantage, what kind of information. So that's the point I want to keep focusing on here.
So remember, this slide here at the bottom, that's the voxel-wise approach. So we'll have that equation at the same-- suppose that if I do the ROI level, suppose we have some number of ROIs. Basically, that's the same model for each of the ROIs.
So that's right at the top. So those are the equations. Suppose we have some number of ROIs. So 20 ROIs, we would have 20 equations like that. So each equation is just a group analysis. So we'll have 20 models. So that's pretty much the voxel-wise approach into the whole brain.
But now I'm switching to the ROI-based methodology. So in the end, if we do that, like we do the whole brain, 20 ROIs, we'll 20 equations, 20 models. We solve it. Then immediately, we face the problem, the multiple testing problem.
So if we-- but for only, of course-- that's just a larger variable solution. The penalty is just too much for us to handle. So that's why in the field, nobody-- pretty much nobody-- does ROI-based analysis. One-- that's just main reason. Occasionally, you may see that in the paper. That's probably focused on only a few ROIs. They can afford such a penalty.
Otherwise, you don't really see people do ROI-based analysis for whole brain. For example, you have 200 ROIs, you don't see that. That's because that penalty. So that's our starting point.
Suppose we just have 20 ROIs. In this case, actually, we have 20-- I think I believe it's 21 ROIs-- in this real example. So what can we do better? Well, let's look at the equation I put-- I mean, I condensed them. It's the same 21 ROIs.
The reason the traditional approach is that-- look the a and b, the intercept and the slope. Even though there are 21 ROIs, those intercept, we don't make assumption. They just behave independently among those 21 ROIs.
So we have 21 bottles. Those 21 bottles are totally independent with each other, just like the whole brain. Suppose we have 200 voxels. We solved 200,000 equations. But we assume they are-- those 200,000 equations totally independent of each other. They don't share any information, even though they do have some kind of a similarity.
What kind of a similarity? That's my starting point. So we can take advantage. They have to share some commonality. What is that commonality?
Well, we like talking about random things. They're not totally independent with each other, because-- well, here we'll talk about the brain response. In this case, it's some behavior data effect. To Even talk about brain response, they are are not totally random number. They don't go wild from minus infinity to positive infinity. They are not like that.
When I talk about brain response, I mean, the percent it would change it would be-- I mean, it's at the most, it's 3%. That's at the most. So they're probably roughly in some narrow range. So we can use that information.
That's called partial pruning. Why it's called partial pruning? So instead of 21 separate models, we're going to put them together with one model. So that's the pivotal point. But when we put it together, that means we have to somehow code the ROIs as a variable in the model. So that's why you have that extra variable that psi-- psi, I mean, Greek letter here, this one. That codes the ROI variable there.
So once we do that, then-- oh, sorry. Actually, that's the subject, not the Greek letter. It's subject. I misspoke. So we put it into one model. We need either that Greek letter to code the [INAUDIBLE] variability. That's the extra term.
In addition to that, we also do something different from the traditional approach. a and b-- a sub [INAUDIBLE] and b sub [INAUDIBLE]. Previously, those are treated as constant. We don't know them, but they are constant. They're parameters. But now, here, we don't assume they are constant. They are random variables.
So that's why in the model, look at the second row, and make assumption that a sub j, b sub j, they follow some Gaussian distribution. So that's another thing. I mean, that is the crucial point. So I don't assume they are constant. I assume they follow some Gaussian distribution.
So that's the thing I do it, it's dramatically different from the traditional approach. So we treat them as a random variables. So that's why I bound them-- loosely bound them-- with that Gaussian distribution. That distribution-- of course, we just assume the distribution, but we don't know the mean, we don't know the variance. We're going to use the data to find out what's the mean, what's the variance, for each two-- each of the two parameters-- the intercept and slope.
So that's the crucial part. So traditional approach, a sub j and b sub j, they are parameters, but they are fixed. We just [INAUDIBLE] make them. But here I'm doing it differently. I assume they follow some Gaussian distribution. So that's the crucial part.
Then you may argue-- I mean, it this is the so-called Bayesian approach. That's why this is a little bit different. You may Challenge me, say, why what's your rationale to make such an assumption, a Gaussian distribution?
Well, yes, that is a little bit arbitrary. But Gaussian distribution-- I'm not saying exactly what kind of a distribution. The parameter, the mean, the variance, I don't fix them. I just-- I will let the data to find out. The only assumption is that shape.
So people criticize that Bayesian approach is subjective. Yes, it is subjective. But how subjective? I mean, in the traditional approach, you also make assumption about the residuals for the Gaussian distribution. I'm not doing something dramatically different from that. If you can reasonably assume the residuals follow Gaussian distribution, I can do the same thing.
So I'm not-- just the distribution. I'm not saying how low or how spread out the distribution is. I'm just saying it's just some Gaussian distribution. So that's the only thing I'm doing a little bit different from the traditional approach.
So with that, then the reason we call that partial pruning, we don't let those a's and b's, the intercepts and the slopes, to go wild. They're [INAUDIBLE] not bounded, sort of constrained-- constrained by those two Gaussian distributions. That's why it's called a partial pruning or shrinkage. That's another term to describe this.
So that's practically, I just apply those two assumptions, one for the intercept another for the slope. So that's the only thing I do. Well, then you may ask, why is this better? Why the multiple comparison correction issue is gone?
Well, that's because I only have one model, unlike the traditional approach. If your model voxels, you have [INAUDIBLE] 200,000 voxels, you have 200,000 models. If you do the ROI, 21 ROIs, you have 21 models. I only have one model. So there is no such a thing called a multiple comparison issue.
So that's-- now the issue is gone. Now also-- I mean, in place of multiple testing, now I'll just have this so-called partial pruning. I constrain them-- those 21 ROIs-- through this assumption, the coast Gaussianality assumption. So that's only thing I do little bit different. And it's not something unreasonable. That's just a distribution. Right?
So when I do that, I end up with 21 ROIs-- seven rows, three columns. 21 ROIs. For this, we'll apply this approach to this specific data set. So in the end, I don't have p-value anymore. Instead, I have this-- in Bayesian modeling, people talk about a posterior distribution. So we have prior assumption. The prior assumption, basically, they are-- I mean, I just showed you the a's and b's, they are the Gaussian distribution. That's the priors.
Then in the end, I have the distributions. Distributions depend on-- I mean, the vertical line is the 0 value, so the 0. So it depends on where that-- your distribution relative to the 0 value, this is the slope part. In this particular study, We'll focus on the [INAUDIBLE] b, the slopes. I mean, 21 ROIs, we have 21 slopes.
So depending on where each ROIs, the posterior distribution relative to the 0 value, then you can talk about the evidence, whether you have strong or weak evidence. So that's-- then you still have to make a decision, of course. Decision-- well, what do you want to talk about?
Yeah, you can still apply the 0.05 criteria. Well, fine. You can do that. But you don't have to. If it's 0.07,
I mean, the tails, if it's 0.7%, you can still talk about it. We don't have to be that strict. As long as you show the posterior distribution, and it's transparent, I don't want to hide anything.
Unlike the current approach with whole brain-- you only show the 5 clusters. The rest of brain, you just hide it. Or not just hide-- I mean, even if you want to show, there's no way to show it.
But here, [INAUDIBLE] publish result, all the 21 ROIs, I'm going to show it. So I have nothing to hide. So it's more transparent. And you can focus on some ROIs. That doesn't mean when a person reads the paper, you can also say there are the other ROIs.
I mean, this is also healthy for-- in the field, when you do meta-analysis, for example. You don't have to-- based on what you people reveal in the paper-- that's because, I mean, we have nothing to hide here. I mean, even for those ROIs, the evidence is weak. For the people who perform meta-analysis, you can still use the information. So that's another advantage.
So that's one way to summarize the data using the posterior distribution. Of course, you can also condense them to use the error bars, of course, the standard deviation. You can do that, too. I mean, even if we use the 0.05 approach, we end up-- I believe there are 1, 2, 3-- six ROIs.
Even if you use the same criteria 0.05, two tails 0.05, unlike the whole brain analysis, we only manage to have two ROIs to survive our approach. Using the same criteria 0.05, we have six ROIs, the green bars on the right-hand side. If we loosen the criteria to 0.01 tailed, for example, then you can talk about two extra ROIs as well. So any questions about this? Yes.
AUDIENCE: [INAUDIBLE] understand this right, but if your original data you're applying blur, [INAUDIBLE] that would have been biased--
PRESENTER: Blur? Smoothing?
AUDIENCE: If you're blurring the original data, [INAUDIBLE] Gaussian on your area and expand your [INAUDIBLE] or if you're [INAUDIBLE] if, for instance, [INAUDIBLE] Gaussian spread [INAUDIBLE]
PRESENTER: Yeah. This Gaussian is-- I mean, I forgot to mention the ROIs-- you average within the ROI. It's like over each voxel. So 21 ROIs, just you have-- each time you have 21 numbers. Of course, there you have an impact but [INAUDIBLE] the ROI.
How do we define the ROI? I mean, you can base it on atlas, for example. That's pretty fixed. Or if people use the so-called peak voxel, you draw a ball around that voxel, which is what this-- the people in this study, they did, based on using, I believe, 6 millimeter radius, to draw a ball around the peak voxel.
So the smoothing part, that basically, you wanted a bowl with a bigger radius or smaller. So that's relatively-- that's [INAUDIBLE] the impact of a smoothing. Yeah.
So that's a basic approach for this. So that's second way. A third way, basically you can further reduce the-- I mean, summarize the data. There is a 21 rows. Each row shows the mean, the standard deviation of the quantile intervals. So there are different ways to summarize it based on the situation. So this is the most condensed one.
There are more information you can pull out of it. For example, one thing is that you can go back to talk about each subject-- say something about each subject as well. You get to the mean, get the standard deviation, get a quantile interval.
For example, you can talk about some outliers, for example. There are some more stuff which you can explore. I haven't do something about this. But that's a potential area we can [AUDIO OUT] in the future to see if we can do something about it.
Well, then you may ask, how good your model is. Right? Well, we can do something to assess the model or compare to the traditional approach. One way is to-- called a cross [AUDIO OUT] you can calculate the information criteria, leave one out, for example, then we compare the traditional approach. You can immediately see the information criteria, the difference between the two models.
The first one, that's the traditional model, general linear model. That's the information criteria. Second row is the Bayesian approach. So immediately, we see the Bayesian model does a better job. The performance is better. So that's one way to judge the performance of the model.
A second way is these two plots. What are they? The first one is the traditional approach, the general linear model. So that's 21 ROIs, you do 21 separate generally linear models, in this case just simple regression. On the right-hand side is the Bayesian model.
But exactly what do the curves describe? What do they tell us? Well, the black [AUDIO OUT] that's the original data. Same thing. So the two black curves is the same thing. It's the same data. So that's the original data.
What are the blue clouded curves? That's based on the model you want to draw, randomly generate the simulations. So, I mean, when we build a model, we can keep generating new data. So that's one way to-- that's why it's called a cross-validation.
So to see the model performance, so using the traditional model we can randomly generate new data. So you can see the original data is the black one. The blue one is based on the model. So the difference, the discrepancy between two tells us about how accurate your model is.
On the right-hand side is the Bayesian approach. Of course, the Bayesian approach is not perfect either. But I [AUDIO OUT] compare to the traditional approach. That's ROI-- ROI-- I mean, it's like the whole brain analyzes. Each voxel, each ROI, is modeled separately.
So the Bayesian model does do a better job for the peak area. And also, even the tail area, it does a better job as well. So that's a second way to judge the model. The third way, which is a little bit esoteric, I'm not going to talk about-- but that's-- from that criteria, the Bayesian approach also performs better.
So that's-- there are multiple ways to judge the model performance. So we're not just saying, I have this [INAUDIBLE] I have some prior. I mean, there's-- even this, there's room to improve this model as well. But that's a separate issue. But at least currently, compared to the traditional approach, we do show the model is doing a better job. So that's one [INAUDIBLE] we can adopt this approach.
The second potential application is, I call it a cross-regional analysis. That's basically what's in currently in the field graph theory. People use the graph theory to handle, talk about [INAUDIBLE] so-called connectivity. I want to avoid the concept of connectivity, because the word is pretty vague.
I mean, people interpret that as-- I mean, it's-- talk about-- talk about, what it's called, ages-- I mean, when people say [INAUDIBLE], does that mean it's a physical existence they talk about? But anyway, let's see if we have applied the same approach to that scenario.
So graph theory, I mean, there are many methods. But I'll talk about the basic one. Suppose we have this real data-- 17 subjects, each subject we [AUDIO OUT] 17 ROIs. So that it means we'll end up-- each subject will have that correlation matrix. Suppose it's resting state data, or even, I mean, task-based data, you can calculate some sort of a correlation.
But anyway, we end up with a 17 by 17 correlation matrix. But that [INAUDIBLE] of course the diagonals we don't care, because it's all 1's. The half diagonal, we can only put-- I mean, focus on half of it, just like the intersubject correlation. So that's end up-- 17 ROIs, it end up 136 values. Focus on, for example, lower triangular part.
So what are we going to do with those 136 numbers? The facial z score. The current approach-- I mean, the famous one-- is called-- I think some people you probably-- here you probably know that. It's called NBS-- Network-Based Statistics. That's implemented in several packages, including, I believe, con, C-O-N. The package also probably adopts that same approach.
So, I mean, the idea is, with 100-- in this case, 136, z score is just 136 t-tests. Right? Now the question immediately-- how do you correct for multiple testing? That's pretty much like voxels. In this case, instead of 100 and some number of voxels, it's 136 pairs.
But how do we correct that? The NBS approach is, basically, it's-- they borrow the idea from the permutation test at the group label. Randomize, for example-- or R-reversion, not randomize. R-reversion.
So in that case, so the two permutations then use the permutation as a law distribution. Then in the end, the [INAUDIBLE] number to form some sort of a cross [INAUDIBLE] clusters, they form a sort of label in-- they call the edges.
So the idea is very much similar to the permutation. That's the whole brain analysis. So that's what they do. But the problem with that is they don't differentiate. When they show so-called clusters, different formulation of the clusters, they don't differentiate them, because those so-called edges, they have some correlations.
Not all edges share the exact same correlation. So they can't accurately quantify the correlations. So that's one trouble. But also, bigger trouble is that they have to start with a primary threshold, pretty much like the voxel-wise p-value. When they do that thresholding, the result is-- and can be dramatically different. Depends on what kind of primary threshold.
There is law criteria, law standard, to [AUDIO OUT] user what t-value, what t-statistic you want in threshold. If you read the manual, it will just say whatever, anything goes. You can pick up any t-value, any p- or t-value to threshold. So I mean, people have done that to show-- you change different p-value, different threshold, you could-- you end up with dramatically different results. So that actually is just troubling.
Well, the reason why that is-- you do have [INAUDIBLE] arbitraries is just simply because you can't accurately quantify the relationships among those peers. That's why. I mean, it's because the model is so crude that it only characterized the relationships. So that's basically the trouble.
So we can do something better, I believe. And I'm trying right now. So, so far it's promising. Because it's pretty much like what [INAUDIBLE] intersubject correlation, similar thing. We can accurately quantify to the relationships among those ROIs.
So here is a simple result to show you. On the left-hand side, you just do the individual t-test. On the right-hand side, you're using the Bayesian approach. You can see that the right-hand side approach basically shows the more powerful detection. I mean, on the left-hand side, we use the network-based approach to correct for the multiple testing issue.
So that's one thing advantage is, there's no arbitraries involved with [INAUDIBLE] approach, because in the end, you have the posterior distribution, because the model is based on exactly how you characterize the relationships. So there's no room for the idea of arbitraries. Also, the detection power is higher. So that's second thing.
And [INAUDIBLE] we can do something even better. For each ROIs, we can say something about this. You can estimate the distribution-- I forget the term. In graph theory, they talk about the hops. Hops.
Let's talk about the hops. When they find the hops, the hops is defined how many so-called surviving connection-- edges. If we have more edges, that's called a hop. That's called a concentration center In the brain. But that's just discrete, because it's based on how many surviving edges associated with each ROI.
Here we can do something better. I can exactly quantify the contribution from each ROI. So each ROI I can say something about the contribution, the standard error, and the quantile. So we can make, based on the posterior distribution, we can make a more accurate, quantitative judgment about the so-called hops. So that's something-- I mean, I think that this approach would do something better.
Similarly, we can go back to ROI to-- I mean, similar to the situation with different ROIs. So there's [INAUDIBLE] more room to explore. So that's where I keep trying.
Also, we can also do the model comparison, as I showed you, for that seed-based correlation as well. So this-- the model does perform better. The left-hand side is the traditional approach, the t-test approach. The right-hand side [INAUDIBLE] the Bayesian-based. So that's based on the discrepancy between the original data versus the model-based simulations.
So that's second application. There are two other applications, as I mentioned before. The third one would be [INAUDIBLE] based white matter connectivity analysis. That is real connectivity, because it-- the connectivity is based on the bundles, the white matter bundles.
So in that case, you don't have a correlation matrix, but you have a matrix suffer a white matter bundles. But you can do something that would be similar to the correlation matrix, except that you don't have all the elements available. Some of the elements, without this, you can't say there's no-- white matter connectivity.
But from modern perspective, it's pretty much similar, because the matrix is symmetric. So that's the third application. The fourth application is the intersubject correlation I was talking about, naturalistic scanning. You can apply the approach to that at the ROI level as well.
But, I mean, I'm also doing some testing. So far the result is also promising. So that's basically the [INAUDIBLE] Bayesian model based on the linear mixed effects model I showed you previously. So that's another potential [INAUDIBLE]. So far, there's-- I'm working on those four applications.
So for regular speaking, this approach can be applied to the whole brain level to the-- at the voxel. But it's not really practical for two reasons. The first reason is that Bayesian approach, the computation cost is just unbearable. For ROI, for the previous analysis I showed you, that's about five to seven minutes. That's fine.
But if you do whole brain, that's-- [INAUDIBLE] don't want to try it. That's just-- maybe take-- it may take years just for one analysis. So that's the major hurdle to apply the approach to whole brain.
Slight second problem, it's less-- it's more about the assumption [INAUDIBLE] because the voxels in the brain-- the [INAUDIBLE] voxels are closely, highly correlated with each other. [INAUDIBLE] 90% is a little bit challenging. But that can be done, but that requires some-- some work about that.
But the first, the computation cost is the major one. So that's-- basically, this is-- I mean, there are some other few limitations. I mean, the first one, we haven't defined those ROIs. That can be done based on previous studies, based on meta-analysis, or can be based on atlas. Nowadays, there are many atlas available. So we can use that to define ROI.
If you don't have that, those sources, how can I do? One possibility is you recruit more subject-- double your subjects. Then use half of your subjects to define [INAUDIBLE] use the second half to perform the analysis. Or if you don't double the number of subjects, you can double your scanning time. Use half of data to define ROIs then you use second one to perform the analysis. So those are the alternatives.
So I think that's pretty much I want to say. And it's-- I mean, there are many ways we can criticize different models. Even Bayesian approach, of course, you can criticize. But at least, the approach provides a platform for us to criticize it, even if it doesn't do a perfect job. I mean, no model does a great job. We always start just with something then go from there.
So at least the better approach, we do have our platform for us to criticize to compare to alternatives. Unlike the current approach-- nobody is doing any model comparison. Nobody is checking their model performance. We just have an assembly line. We use the software, just goes through all steps, then we have the results. If we are happy, we just the write it, publish the results.
Nobody questions that model, whether it's valid or not. Nobody is doing anything about the model, challenging the model. So here, we can directly show the model performance, check the model's performance, from different ways. So I think that's pretty much I wanted to say. This is just a summary of this presentation.