16 - Group Analysis in fMRI: Part 1 of 2
October 16, 2018
May 30, 2018
Gang Chen, NIMH
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Gang Chen, NIMH
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GANG CHEN: So the layout of this talk, basically, it's about group analysis. It's a big topic. In the early days of fMRI data analysis, people-- you could do away with just simple t-test, but nowadays, as the experiment designs become more and more sophisticated, we may need complicated models. So that's why we need to spend quite a bit of time covering the group analysis part.
So first, we're going to talk about some basic concepts, terminologies, to clear. Then we switch to those specific modeling approaches at the group level. Then there some miscellaneous issues, like centering, inter-subject correlation, inter-class correlation.
Then lastly, we'll talk about something new, which is a little bit dramatically different from the typical group analysis. Because the typical group analysis, we usually do lots of ways through the whole brain. That approach is nowadays-- it is a little bit troubling because of the correction approach. Usually, I mean, now the rigors requires you to perform so-called vigorous cluster correction.
It's-- my personal opinion is overly penalizing, too conservative. So as a user, you probably feel like sometimes, it's very difficult to get your clusters to pass such a threshold approach. So we're developing a new approach which, right now, we don't do it. Hopefully, and they will instead-- will focus on some RIs. Those RIs can be based on previous studies, based on beta analysis, or based on some atlas available.
Suppose you have 200 something, 250 RIs based on atlas. Then you can focus on their-- do the group analysis on those RIs. That will give some legroom to get around the multiple comparison issue. We can talk about-- spend probably one hour about that. But I probably will get to that tomorrow morning, the first session. So that's the structure of this talk.
So first of all, why do we need group analysis? The reasoning is simple, we do science. We need to say something about the population. We don't-- we will code some subjects, right, like 20 subjects.
In the end, we don't say something specifically about those individual subjects. Instead, our interest, our focus, is on some population or the difference between the populations, like patients versus controls. So that's why we need to make some general statement about the whole population at the population level. So that's basically the reason we perform this group analysis step.
So this simplest case, when we do group analysis, suppose we have-- here we have seven subjects. So suppose we just focus on one voxel. At the group level, basically we want to say something about the centrality, the average of those seven subjects. I mean, when we have seven numbers, there are a couple of ways to say something about those seven subjects.
First, we have the average, which is the mean. So that's some statistic that says the centrality of those seven numbers. That's one way to describe it. Another way is how spread out those seven numbers are, which is-- that is the standard error, the concept of standard deviation. So that's how concentrated-- that number shows how concentrated or how spread out those seven numbers are.
So those are the two quantities. The first is basically shows the average. The second one shows the reliability of that mean, that average. So we combine those two numbers. That's the t-statistic.
So the mean divided by the standard error of the mean, that's the t-statistic. So the t is a dimensionless number, right? So the mean is something interpretable in our context. It's the percentage signal change. But the t does not have a unit. It's just a dimensionless number.
So in the end, we have two values. One is the mean, other is the t-statistic, right? So those-- unfortunately, in neural imaging, people pretty much focus on the t-statistic. That doesn't mean the average is not important. Instead, it is important, but it is just that the current trend in the field is a little bit unhealthy.
So then we-- based on the t-statistic, we can say something about the mean, right? If the t is relatively big, then based on the traditional statistic, so we have the null hypothesis, then we say it's reasonable to reject the null hypothesis. And we can see, oh, this voxel, we can reasonably say-- make a statement that it's activated.
That's the concept of statistical significance. So we set the threshold, for example, [INAUDIBLE]. So that's one group of subjects.
Now suppose we have two groups, right? So suppose, still one voxel, seven numbers-- seven subjects in one group, another seven subjects in other group. Same thing, but we just compare the two groups.
So first of all, give the mean of the group difference, which is the two means, then get the difference. So that's the mean difference. Then standard deviation, same thing, we get the standard deviation of the group means-- group difference. So then we calculate the t-statistic.
Another way is called paired t-test. That's, we have one group of subjects but we have two conditions, right? So like house versus phase. But this scenario can't be reduced to one sample t-test, because we only have seven subjects. Each subject, we can get-- reduce the two conditions. Basically, get the difference between the two conditions.
So it's-- essentially, it's just the equivalent to a one sample t-test. So in the end, you get the mean of the condition difference, then the standard deviation of the group mean. So that's the cartoonish description of the three different types of t-tests.
So then from modeling perspective, we can say something. I mean, it's pretty much the same. We have seven numbers, so we'll have the mean. So right, divide by seven, we get the average. For example, the average is, in this case, 0.92% single change.
So if we're writing a model, that down below, the equation, it's essentially regression model or it's a beta model, but it's the regression, we don't have any explanatory viable. It's just the intercept. So it is a regression model, but it's a special case. So that's--
On the left hand side, notice that I put the beta values. That's because the betas are found-- each individual object, right? So that's the estimated beta. On the right hand side, just the intercept, positive residuals.
You may also notice, on the left hand side, I put a hat above the beta value. What does that hat mean? In mathematical notation, the hat means it's estimated values. They are not direct observations from--
So we will code the subjects. The direct observations are the EPI time series. Those betas are-- we usually went through many steps, pre-processing, then individual subject regression analysis. So the betas are large measurements, right? So they are-- basically, it's our estimations.
So since they are estimations, they are not-- they have some reliability issue. That means we're not 100% sure those values are accurate or not.
So each beta, in fact, each beta has some-- has a standard error associated with it. But the typical fMRI group analysis, we ignore that reliability information. We ignore the standard error. We just focus on the beta. That's just some lazy approach.
That approach works reasonably well if we ignore that reliability information, but we could improve that, but we'll fix-- come back to that issue later. So let's just stick to the traditional approach. Just take the beta variables from the individual subjects, then do a typical group analysis.
So in this case, we just have one sample t-test, right? So that's the beta model. So in the end, the last line shows that we-- keep in mind, we have two values at each voxel. One is the group average. Then we'll have the t-statistic.
So the group analysis, the traditional approach, the conventional statistic, is always will have the concept of the null hypothesis, where it pretend nothing going on in the brain. That's our starting point.
Then we calculate the t-statistics, calculate the average, then if the t-statistic is large enough, so then we can reject the null hypothesis. Then we make a decision, every voxel or every region in the brain, whether we claim that region is activated, or the other regions are not-- I mean, it's not statistically significant.
So that, in the end, it's-- the conclusion is a binarized decision. So that's very unfortunate, because in reality, we know the brain is not-- either they are activated or not activated. Probably, most likely, most of the regions are activated. That's just because most regions, the effect size is too small.
I mean, for example, one region, the signal is 0.21% signal change. It's too weak. If we really want to properly detect such a weak effect, we may probably recruit hundreds of subjects.
So just because of that one region fails to reach this statistical significance, does not necessarily mean that region is not active. We cannot make such a statement. So that, we always need to keep in mind. We are making-- remember, what conclusion you make, it's about statistical significance. It's not about the effect size significance, right?
So it could be, one region fail-- the failure of a region reaching this statistical significance simply means it's just the failure to reach that statistic threshold. Doesn't mean there is low activation in that region. So you always need to keep that in mind.
Of course, there are many caveats, other than that's just a binarized decision. And it's, every subject's brain is unique in the sense the shape is different. Each region's location is probably slightly different, right? So even the region size, different.
So when we do group analysis, we have to warp individual subject brains to some standard space. That warping, that alignment, of course, is not perfect, right? So that may create some problem about the-- I mean, would have some impact on the statistical significance, right, if their alignment is not perfectly aligned.
So that's also a reason to motivate us to do the RI-based approach, which we'll talk about tomorrow morning. So that's-- we'll see how that achieve maybe-- I mean, as a alternative, maybe achieve better power, detection power.
Another way-- perspective I want to point out is that nowadays, the group model is all so big. It's gradually becoming more and more complicated. In the early days, students' t-test would be good enough, right? t-test, you can do one sample, two sample, or paired t-test.
So later, I mean, gradually, people start to use ANOVA approach. And then ANOVA is-- that means you have one or more than one factor, right, as a categorical variable, like groups, multiple groups, or you have tasks, multiple tasks, or the combination of the two. So that's ANOVA.
Then you may have a general linear model. By general linear model, that means you may have some quantitative variables, right? So like age, want to control the age variability across subjects, or IQ, or reaction time. So once you throw in one or more quantitative variables, then you have ANCOVA or general linear model.
Then if you have a longitudinal study, then you may have missing data, for example. That's another complication, or you have some quantitative variable. That quantitative variable is a within-subject quantitative variable. What does it mean?
Like reaction time. Suppose you have two conditions, house and phase. Both conditions, you have reaction time. So that's called a within-subject quantitative variable.
Unlike age, each subject, you only have one number associated with that subject. But your reaction time, suppose you have two conditions, house and phase, so that's a within-subject covariate. How to deal with that? That requires a linear mixed-effects model approach.
So those are the possible scenarios we will cover at the group level. So that's why we-- sometimes we need big models. That means we put all the possible variables in-- throw in the model. Then that's the big approach.
Before I talk about the specific modeling approaches, we need to clarify some of the terminologies. So for example, at the group level, remember, it's-- regardless what models-- what model you want to adapt, either t-test, ANOVA, ANCOVA, general linear model, or even linear mixed-effects modeling, we have y. I mean, on the left-- the variable on the left hand side, the y.
That y is, we call that response variable or outcome variable. In the old days, people may call that dependent variable, but nowadays, between these, probably we-- that terminology is fading away.
So on the right hand side, that depends on the nature of each variable. You may have like for example, factor. Factor is a categorical variable, right, to categorize their-- so it's not a quantitative variable. So it's a qualitative variable.
So that, basically, there are two big categories. One is, we categorize the tasks, right? So that's called within-subject-- I mean, in psychology, people call that within-subject factor. Sometimes, they may be called repeated measures factor, right? So like house versus phase.
Or if it's a emotion study, you may have positive emotion, negative, or neutral emotions, right? So you have three, in that case, three levels for that factor. So that's one kind of a factor.
Another one is called the between subjects factor. So that, you would classify each subject based on something like, if it's male versus female, so that's gender. Or it's-- whether it's a patient or a control, so that's another way. Or you may have genotypes, or handedness, left handed versus right handed. So that's between subject factor.
Lastly, I want to mention, it's not popular, but under some context, people describe the subject itself as a factor. It's called the random effects factor. So people usually don't mention it, but for modeling perspective, the subjects are-- I mean, 10 subjects, you basically have a factor with 10 levels.
The reason we call it random effects factor, I'll come back to this point later, but the reason is that we don't care about-- I mean, in the end, we don't want to say something about the individual subject effect at all. But the reason we code them, we use them as representatives for the population, so they are, in fact, under some scenarios-- actually, most scenarios, the subjects are a variable in the model. Just-- usually, we don't explicitly say them, but they're-- from modeling perspective, there are some terms associated with each subject.
So the second half of the slide, basically, we describe the second big category, the quantitative variable. Sometimes people call it covariate, but the word covariate is a little bit messy in neural imaging, because there are two different ways, two different usages, people-- when they use the word covariate.
So sometimes, people use the word covariate to mean a variable of low interest. So in that use, with that usage, people can mean, can be a factor or can be a quantitative variable, right? But in that case, people simply say, I don't care about-- I'm not going to describe the effect of that variable.
So that it can be a factor. For example, gender, right? So that's-- in other software packages, people say, we model a gender as a covariate. When they say that, simply that mean they just throw in that variable into the model as an additive effect.
Whether it's an interaction or not, they don't even mention. They don't-- usually don't do it. You just see, somehow, there is a magic, modern approach, sort of to control that, the viability of that variable. So that can be a factor or can be a quantity, like age.
So that usage a little bit vague, and also for modeling perspective, it's problematic. I will come back to that later too. So to avoid that confusion, I usually simply-- I don't avoid the word covariate, or when I use it, I usually say it's a quantitative covariate, to clarify, to make it explicit.
So I usually just describe each variable based on the nature of the variable, whether it's categorical, which is a factor, or it's a quantitative variable. So for me, covariate, usually I mean it's a quantitative variable.
So now, let's switch to another way-- descriptive terminology for a variable, experimental variable. It's fixed effects versus random effects. Fixed effects, that's pretty much usually like typical variables, like tasks, right? Like positive, or negative, or neutral. So that factor, emotion factor is three levels.
So we care about each level of that factor, so that's why called fixed effects. So in a sense, that's pretty much like in physics, something we want to measure, right? In the end, we need to discuss it. It is a constant in the model, under the traditional statistical modern approach.
So we will see later exactly what it means from the modeling perspective. So that's typical. Like factors, even quantitative variables, they are fixed, in a sense, they are parameters. We consider them as a constant.
So in contrast-- oh, before we talk about it, let's use the example I showed yesterday to-- for the-- we used the example to describe the modeling approach, which we don't make an assumption about the hemodynamic response function. Instead, we modelled the hemodynamic response function with tens, multiple tens, [INAUDIBLE] function.
So we used this experiment design, which we have four conditions, right? So 2 by 2, so either it's a human versus tool-- that's-- first column is human. The second column is a tool. So that's one factor.
Another factor is the image type, whether it's a real image versus a thought image. So the first well is the real image. The second well is the thought image. So it's a 2 by 2 design, two factors. First factor is human versus tool. The second factor is real image versus a thought image.
So with those four conditions, we had two factors. So that's-- people sometimes call it factorial design. So 2 by 2-- each factor has two levels.
So then-- and both factors are fixed effects, that's because we-- in the end, we care about the specific effect for each level under each combination of that 2 by 2 design. So the other, in contrast, the other concept is called the random effect factor. For fMRI, that's-- the only-- usually the only one is the subject, because as I mentioned a couple of times before, the subjects in the model actually are-- there are parameters associated.
For example, in this model, this is a linear mixed effects model that-- better to describe the concept of random effects factor. So on the left hand side, here I use y sub i. So that's the-- usually the effects estimates from each individual subject.
On the right hand side, we have the design matrix X. That's the fixed effects design matrix, plus the beta. Here the beta is-- I mean, the design matrix can be those factors, like task factors, positive, negative, neutral, or can be a gender, right, male versus females. So the model is through these fixed effects.
So in the fixed, in a sense, the betas are constants. We don't know them. We don't know their values, but the assumption is they are fixed. They are constants. So we need to estimate them.
However, in this the second part, this part, so there is a design matrix. There is-- also, there is a parameter. But the design-- this is the random effects part. So the B sub i, that's correspond to each subject.
So this is the fixed effects. That means we assume it's like pretty much like the physics, the gravitational constant, the same thing. But here, we assume each individual subject has some unique personality, a unique response in each region. So this here, that's the deviation part of each subject from the population effect. So that's the random effects part.
The last term is the residuals. Residual, it's just fluctuation. So this part, that's why we call the effects. That shows that each individual is different from the population effect.
So in the end, we don't talk about this part. And also, the reason we call it random effect is that because those-- the B sub i, they're not fixed. And we make assumption, that assumption usually it's the typical one of the Gaussian distribution. So it is a random variable, unlike the fixed effect here.
This is, they're constant, right? We don't know it but we need to estimate them. But here, that's just, we make assumption, pretty much like the residuals. We assume they follow Gaussian distribution.
We already talked about the difference between the fixed versus random effects. So the next thing I want to talk about is the concept of a main effect versus the interaction. Main effect, well, that's when we have two, at least two factors.
Main effect, that's, I mean, everybody already learned this in basic statistic classes, right? So main effect-- basically, suppose you have two or more levels for a factor. So like emotion, we have positive, negative, neutral, three levels.
So the main effect basically is omnibus test. So three conditions, that means the null hypothesis or the three conditions are equal. So then we need to do F-test. That F-test, if turns out it's significant, then we can say at least one of the conditions is different from the other two.
But with three conditions, we are not so sure where exactly the difference is, because with three, that can be the first one is different from the other two, or can be the second or the third. So we don't know. There's a vagueness associated with such a F-test. So usually, we have to go through-- people call it a post hoc test. So part-- that omnibus test.
So even with two conditions, like a positive versus neutral, suppose just two conditions, if you do such a F-test, that's usually, it's-- usually, it's, a t-test is good enough. But if you really want to go an F-test, do a main effect, then you have to get a F. In that case, the F is-- it's always positive. It doesn't tell you the directionality, even though the F-test in theory is equivalent to a paired t-test.
So that's the main effect. Interaction, that's when we have two factors, right? So to two factors.
Suppose we have gender, male versus female, then we have two conditions, positive versus negative. So whenever we say there is an interaction, basically it's a way-- geometrically describe those two lines are not parallel over each other [INAUDIBLE]. So that means they have interaction.
So the two parts basically describe exactly the same thing. They just-- we use two different ways to describe it. On the left hand side, the x-axis shows the gender, right? So the two conditions is described with two colors, with two different lines.
On the right hand side, the x-axis is the two conditions, negative and neutral. The gender, male versus female, are described with the two lines. So it's just two different ways to visualize the interaction effect. So on the right hand side, immediately we see they intersect, those two lines. So that's-- they intuitively shows the interaction effect. On the left hand side, we have to extend the two lines to show the interaction effect.
So that's just two factors. If we've more than two factors, three way or even four way interaction-- it depends on how sophisticated your experiment design is. So then let's just talk about the factors.
If you have factor, one factor and one quantitative variable, you may also have the interaction between those two different types of variables, right? Suppose you have gender, male versus female. You have age. We can also talk about interaction. In that case, it's just that age effect is different between the two groups.
So that's why when I say in the field, the other software packages usually say, a covariate of age. But when they say that, usually they mean they make a strong assumption, the age effect is the same across the two groups. They didn't model the interaction between the two groups and the age effect.
So from a modeling perspective, that, whenever they say that, I already know that they have some-- they already make a strong assumption without modeling the interaction effect. So that's why I usually don't focus on the word. When people say covariate, it simply mean that it's a variable of low interest. So that usually indicates they have some problem with their modeling approach.
So I mean, lastly, here I mention, even the two quantitative variables, two covariates, you can also talk about the interaction. And it's, for example, even just one variable, like age, we can talk about interaction. Age, if we just put the age as a additive effect in the model, you already assume it's a linear-- linearity effect, right?
So potentially, you could have self interaction. That means you could have higher order-- I mean, quadratic or even cubic. We don't know unless you try it out. All right.
So the complication of interactions, different types of-- different types or different order of interactions, right? So for example, this slide basically shows you have two groups or two conditions. Then if age is a covariate-- so if we make-- on the left hand side, we make an assumption there's no interaction, basically you just put the age as a additive effect. Then you just-- the model just-- if you don't tell it to have an interaction, they just-- you get what you are looking for, right?
On the right hand side, then potentially maybe there is an interaction effect. So when you analyze the data, don't ask for it. Of course, you will not get the effect.
So that's basically, I think, is the terminology perspective, so some background information. So now let's look at a particular example. So it's reasonably simple scenario for nowadays, for a neural imaging study.
So suppose we have a 2 by 3 mixed ANCOVA. What do I mean? So first of all, it's a 2 by 3. That means we have two factors. The first factor will have two levels. Second factor has three levels. That's the part, 2 times 3, right? So it's two factors.
Mixed, that word is, the two factors are different type. One is between. The other one is within.
So ANCOVA, that means we have a third experimental variable. It's some quantitative variable. So that's basically those three words, what they mean.
So specifically, the first factor is a group, the two groups. We have patients versus control. The second factor, Factor B, is a condition. It's a emotion experiment. We have three levels, positive, negative, and neutral.
So then we have-- if we consider subjects, it's a factor which is a random effect factor. So in this case, we have totally 30 subjects. 15 is patients. I mean, this autistic spectrum disorder kids. Then the other 15 are healthy controls.
So then if we want to control the variability of age, so we want to model the age as a quantitative variable-- so that's basically the situation we have. How do we do with this, right? It's a 2 by 3 plus we throw in age as a covariate.
So even for such a simple data structure, it's for-- your imaging software package, this is a challenge, actually. It's not a simple data structure to handle, actually. So how do we do it?
Well, let's ignore the covariate, the age first, right? So let's just focus on the ANOVA part. If we do ANOVA, again, this is a 2 by 3 mixed ANOVA, right? So not ANCOVA. Let's put the age aside.
So for that, if you look at a textbook, look at the F-statistic 2 by 3 ANOVA, so those three are the two factors or two main effects plus the interaction. If you look at the formula, it's like this. I mean, we don't pay attention to those wiggles under this.
But the F-statistic is basically the ratio of two, so I highlight the denominators. Those two main effects have different denominator. So the first factor-- first, that between subject factor, that denominator is different order to F-test. So that, it makes our life miserable, because if we do this, use the traditional approach, for example, using so-called the univariate, general linear model approach to handle this, then you have to be very careful about the denominator.
Unfortunately, in fMRI, this part is not done properly, simply because the denominator is different. If we use the univariate general linear model approach, people would-- they have to-- I mean, if not careful, they usually use the same denominator.
So that cause-- they failed. During the past 25 years, so many studies, they are done improperly in the literature, because this approach is adopted in some software packages. So that's just the simple case.
A 2 by 3 or even 2 by 2, if you have to narrow with the two factors, one is between, the other one is within, the between subject factor F-test usually is not done correctly. So if you use the linear mixed effect model approach--
So that's because they use this so-called general linear model. And they only have this residual part. That's why. So that's why they use the same denominator for all the three F-tests.
So that approach, even though it's popular in neural imaging, but it's-- I mean, it's not-- only [INAUDIBLE] the F-statistic. If we were to post our test for the t-test, the problem continues, continues to extend to the t-test. So there are multiple issues. So I don't want to go to details, but it's described here, the problems, even for those t-tests.
So that's one scenario. So here, actually this slide shows the F-statistic is not done correctly when you use the general linear model approach. They don't use the term general linear model, but they use something else, but the problem is the same. So also the t-tests, so those are the scenarios. So it can be problematic.
Another scenario is that-- not just-- they mix the type, even for the situation when you have two factors, both factors are within-subject, that problem is even worse, because if we still use the general linear model approach, both main effects for those two factors would be wrong. So the general linear model approach-- and that problem also would be extended to the post hoc tests as well.
So because of those problems, with the general linear model approach, and also, you cannot handle covariate when you do ANOVA. So there is a better way to do it. This is a not something in theory is complicated. I mean, people have adopted a better approach for many years.
Like in SAS, in SPSS, nowadays, in R, people have been doing this correctly for many years already. So it's not so complicated, simply because the neural imaging field has not switched and has not yet [INAUDIBLE], or they are not willing to correct the problem.
So this approach is widely available outside of neural imaging. It's, instead of using univariate general linear model, we use multivariate general linear model. So that simply solve the problem. So I don't want to go to the technical details.
Basically, on the left hand side, instead of one column, based on the within-subject factor, we can use the matrix instead of a vector. So that solve all the problems. First, the F-statistc would be properly formulated. In addition to that, we can put the quantitative variables into the model easily, as long it's between-subject quantitative variable, like age, like IQ, like-- whatever other-- brain volume, for example.
So of course, this will not be able to handle within-subject covariate. Like reaction time, you have multiple conditions. Then this would not be good enough. So except for that, and also accept for missing data scenario, this multivariate general linear model would be very adaptive.
So that's an extension. You have any-- which the program 3dMEMA-- basically it's-- the underlying mechanism is about-- it's this a multivariate general linear model approach.
Another point about-- I'm switching to a topic-- is that, as I briefly mentioned before, when we go to group level, usually we only take the betas from the individual subject analysis, for group analysis, right? So whenever we do that, we make a assumption. We assume those betas are equally reliable.
What I mean when I say that, equally reliable? That means that we ignore the standard errors of those betas. Ignoring, that means they are-- suppose there are 10 subjects. The betas are equally reliable. Basically, the standard errors are the same across those subjects. So that's usually the case.
Then you may ask, can we do it better, or what's the consequence if we ignore it? So the alternative is, we don't ignore the standard error. Instead, we take-- we consider the standard error. Put it in the model. So that's the alternative approach.
That approach is-- I mean, in other fields like meta analysis, not in fMRI-- the meta analysis in fMRI is messy. I won't touch it. But the other field, when people do meta analysis, basically you have multiple studies, right? You either synthesize or summarize across studies. Usually, you need to take both betas and the standard errors into consideration.
So how do we do it? That's the question. Well, one way to look at this is the old approach. You only take the betas. Basically, you treat those betas as equally reliable. That means their weights, the weights is the same.
But here, we don't treat them equally. Suppose there are 10 subjects. We have to discriminate those 10 subjects based on how reliable each subject is. So one subject, if the beta-- and also, there is a standard error. If that standard error is small, that means it's more reliable. I'm going to trust this subject more, versus another subject with a less reliable error. So that's what I call discrimination.
So I'm going to weight it based on each subject's-- their reliability of the effect estimate. So then you may ask, where do I get that reliability information? That information is available. It's just that we don't realize it. It's embedded in the t-statistic.
Why? Remember, what is t? t is beta divide by standard error. So we know the t. We know the beta. Then now it's just beta divided by t. Then you get the denominator for the t, which is the standard error.
When do we individual subject analysis, we have beta and the t. We just take both pieces of information to group analysis. The program will automatically calculate standard error. I mean, just beta divided by t. So then we just weight it. We just differentiate those subjects.
So that's-- the approach is-- I'm sure it's pretty intuitive. It just-- that is done in-- in fMRI, the program is called the 3dMEMA, Mixed Effect Multilevel Analysis. So that, basically, as a user, you just take both beta and the t as input. Feed it into this program.
Well, sounds nice, right? But there are a couple of downsides as well, of course. First of all, the computation cost is much higher, because there is more input. Also, because this-- the algorithm is more complicated as well, not just the input amount of data. So that takes much longer. That's one thing.
Another thing is, from a modeling perspective, like only do simple scenarios. Like you do-- you have two groups or you have two conditions. Simply, basically, it's like a typical t-test. You-- simple t-test, paired t-test, or two sample, two groups. Or multiple groups, that's fine.
But if you have combinations or you have multiple conditions, then it's getting complicated. Usually, we have to reduce sophisticated scenario into a-- I call it a piecemeal approach, to break into pieces. Then we can use this-- I call it mixed effects approach to handle it.