Welcome back. This is Section B of the Randomization Lecture where we're going to go over the types of randomization schemes. I'm going to talk about Simple randomization schemes, Restricted randomization schemes and Adaptive randomization schemes. These are, admittedly, somewhat abbreviated overviews of the various types of randomization schemes that can be employed in a clinical trial. Simple randomization is probably what most of us are familiar with, it's analogous to the coin toss again. It's just a complete randomization. Each time you toss the coin is independent of the last time you tossed the coin, so it doesn't matter if you got a heads or tails the last time, this next flip is totally independent and has a 50-50 chance of either heads or tails. The advantage of having a Simple randomization scheme where the last assignment doesn't tell you anything about the next assignment, and the actual probability of the treatment assignments stays fixed from assignment to assignment, from coin toss to coin toss, is that each assignment is completely unpredictable. The fact that a patient was assigned to treatment A yesterday will tell you nothing about what this new patient is going to be assigned to. And in the long run, if you have enough patients, it will equal out if you have a truly, scientifically designed randomization process, and you use a Simple Randomization scheme, the number of patients assigned to each group should be about equal, assuming you have large enough numbers. So the risks associated with a Simple Randomization procedure, is that you may indeed imbalances. These imbalances could be related to the number of patients assigned to each treatment group, for example, you could end up in a trial of 100 patients with 40 assigned to one group and 60 assigned to the other group, when your ideal was a one to one allocation with 50 patients in each group, and that in itself, will lower the statistical power or could lower the statistical power, associated with your clinical trial. Also, you can have an imbalance related to important confounding factors, if there are prognostic factors associated with the outcome and in a Simple Randomization scheme the chips are going to fall where they may. So you could just by chance have more people with severe disease assigned to one group than than the other, or more women assigned to one group than the other. So, there's no control on the characteristics of the patients by treatment group. And if you have large numbers of patients in a trial, that's not such a problem because your large numbers favor the probability that you really will get comparable distribution of potential confounding factors across the treatment groups. But with small numbers, you're more likely to run into imbalances. And again, that can lower the statistical power of your trial. And even if these imbalance, either in the number of patients, or the types of patients, or both, don't really affect the results of your trials, they really are not too severe in terms of numbers, and the factors that they're, the groups are imbalanced on, really don't have an association with the outcome. It still can diminish the credibility of your, results. It can make people questions things. Well, why, why didn't they have equal numbers in both groups? Or, more commonly, oh, well, of course they had the sickest patients were in treatment group A, so that's why treatment group B looked better. And sometimes you can never really address those credibility issues fully, and they can really undermine the results of your trial and give people reasons to not accept the results. As I referred to before, these risks are inversely associated with the number of participants. The more participants you have in a trial, the more that the properties of probability are going to be working in your favor. So, how have clinical trialists and biostatisticians addressed this issue? Well, they've imposed restrictions to the randomization scheme to ensure balance across important factors in the design of, of experiments. So, when somehow there's a constraint added to produce the expected assignment ratio, in the example we've been using, one to one. According to time that the study's been going on, or on specified covariates, such as severity of disease, or gender, or clinic. And the two primary maneuvers that are used in this restricted randomization are blocking and stratification. And I'll go over each and one of them separately. First we'll start with blocking. So let me just tell you what blocking is before I tell you why we do it. A block, if you want to define the block, is a list of treatments that achieves the treatment assignment ratio. Again, we're sticking with that example of one to one. So, a block of two would be an A and a B that we achieve their ratio of one to one. If we use the block size of four, that means that a block would have two As and two Bs in it. And we talk about permuted blockticides, which means, every possible way you could list those two As and those two Bs and you see listed out on the slide is the six possible ways you could order two As and two Bs. You can have AABB, ABAB, and I'll let you read the rest for yourselves. But that's the list of permutations of the order of the treatment assignments inside a block of four, for a one to one treatment assignment. Another important point is that the size of the smallest possible block to use, is the sum of the integers defined in the treatment allocation ratio. So, you can't meet the treatment allocation ratio, if you have a block that is smaller than the sum of the integers. So, I think an example illustrates this best. If the allocation ratio is one to one, the smallest block size is 1 plus 1 is 2. That's the smallest block you can have, to get a one to one ratio. So, if you have a different allocation ratio, say two to one, the smallest possible block size is three because you'd have to have two As and one B. And, if you wanted to go and use larger block sizes, and I'll discuss reasons for using different block sizes in a few minutes, the larger block sizes need to be multiples of the smallest one, so in order to meet the treatment allocation ratio of two to one, the smallest block size you can have is three. The next one is six because that is a multiple of the smallest block size and, it goes up accordingly. So, because within a block size of six, you could meet that allocation ratio of two to one. You could have four As and two Bs. Now if you have a block size of seven that wouldn't work, or even a block size of eight, you couldn't meet that treatment allocation ratio within that block size. So the larger block size are multiples of the, of the smallest one. And then, when we apply this blocking principle and use blocks of treatment assignments, what we really do is produce all possible permutations of the block, similar to how I have listed as the second sub-bullet under the first bullet. We list all the possible ways a block of four could be ordered. And then, when we go to develop a randomization scheme or a ran, a list of the treatment assignments, what we do is randomly choose those block sizes. Randomly choose, from those permutations of a block size of four. So, I think the example we'll come upon in a few slides will help illustrate this point. So why do we do this? It seems sort of confusing, and if it does seem confusing when you first look at this, this is common, that blocking, is one of the more difficult, simple concepts that we deal with. Once you get it, it's very clear. But just that first time understanding exactly what blocking is may take some quiet moments alone looking at it. But the reason we bother to do this is that it ensures balance in the treatment assignment ratio over time. And this makes sense, right? Because if we're using that block size of four, that means after every four patients, even if we have a sample size of 400, after every four patients, we're ensured that we've met the allocation ratio, that two have been assigned to A and two to B. As we go along in the trial, we can't have long runs of As or long runs of Bs that you might have in a simple randomization design, that would be associated with time because it takes time to accrue patients to a trial. You can't have that problem if you use the block size. You'll only have runs of one particular treatment assignment, the longest run possible within a particular block size, and you can see with a block size of four and a one to one allocation ratio, the longest possible run of a treatment is two, for two patients. So, how do you figure out how many permuted blocks do you have with a particular block size? Well, when you start with small block sizes, it's fairly easy, so, if you have the allocation ratio is one to one, right? If the size of the block is two, the number of possible blocks you can have is two, it's AB and BA. And the way you can figure that out is to write them down, or you can use this multinomial coefficient you see illustrated at the bottom of the slide in blue with the factorial signs. Now, don't be scared of, of this formula, it's quite simple. So again, if you are back to a block size of four with the one to one allocation ratio, I showed you on the last page and have listed out all the possible blocks and that is six. But, we can come up with that number six by using the multinomial coefficient. So, the numerator is the factorial of four and you see that 4!, which translates to 4 times 3 times 2 times 1. And then the denominator is the number of each kind of thing, so in this example, in a block size of four, we're going to have two As and two Bs. So the R, we have an R1 and an R2, and both of them equal 2 in this case. So you can see that the 4 factorial is divided by 2 factorial times 2 factorial, and you get the number of possible permuted blocks for a block size of four is six blocks. And I'll let you go through that on your own for block size of six, where it starts to get more complicated and you don't want to have to sit down and write out all possible blocks, in this case 20, and you'd rather be able to figure that out more easily using this shortcut of the multinomial coefficient. So, once we've decided that we're going to have a blocked randomization design, commonly referred to as permuted blocks, and this is probably one of the most common features of a randomized clinical trial design, that most trials do use blocking, but there are some considerations when implementing a block design. First you have to use the same allocation ratio, throughout the trial. You can't modify your allocation ratio, as you go along, and we'll see an example of that when we talk about adaptive randomization. So if you're going to use blocking, you've bought into the fact that your allocation ratio is going to be, the same throughout the trial, which again, is very common, not to change the allocation ratio. Second, it's important to understand that block sizes, and information about the block sizes used in a clinical trial are really on a need to know basis. They in general shouldn't be written down in the protocol that the investigators are going to be looking at, because it gives you a hint about how the randomization scheme is going to work. And if you know the block size, then you'll start to be able to predict what the next assignment is. And that's one of the whole points of randomization, is to make the next assignment unpredictable, so investigators or people won't be influenced by that knowledge about who to enroll in the trial, or when to enroll, roll them. So I've heard people say, well, they see the block size written down in the protocol. You really shouldn't write it down there. Now, there should be separate documentation of the treatment assignment scheme and how it was generated, but generally, the details in the protocol should be limited to those that the clinician needs to know in order to execute the protocol, and be knowledgeable about the general experimental design. Now, one of the tricks that we use to maintain that lack of knowledge about upcoming treatment assignments, is to use more than one block size. Because then, the overall sequence of how treatment assignments are allocated is going to appear to be more random. If you use the standard block size of two in the one to one allocation ratio, people are going to be able to start figuring that out that if, if the first assignment was an A, the next one's going to be a B. And if they know the block size, then they'll know that block's complete and then they're back in the next, when the next block opens up, they're back to not knowing whether it's going to be an A or a B. But once that third assignment is issued, they are going to know the fourth assignment. And it gets a little bit more complicated with larger block sizes. But nonetheless, it takes away from the unpredictability aspects of randomization. And so the way that we try to deal with that, is to use more than one block size so that the treatment assignments appear to be more like a simple randomization scheme, and are less likely to be predicted accurately. And this is especially important in unmasked trials. Because, if you know the treatment in a masked trial, and you know which treatment's going to come up, A or B, and it's masked, well, if it's effectively masked, you still don't know that much. You know that it will be a different treatment, but you don't know which treatment it is. If it's a unmasked trial, and you start to figure the block sizes, then you are going to know the actual treatment that's going to be administered to the next patient, or have a good idea. And certainly that opens the door to more selection bias creeping into the trial or again back to that confounding by indication. Okay, so what are some of the benefits and risks associated with blocking? Well some of the advantages associated with blocking are, that it helps to guarantee overall balance, especially in smaller trials. So, you're more likely at the end to get equal numbers in both groups. It also protects against time related changes that may influence your clinical trial that could be, those changes could be in the composition of the study population. So if you start with more severe patients and get to less severe patients enrolled in the trial as the trial goes on, if you have blocking, you're more likely to have equal representation of those different types of patients in both groups. And those kind of changes, you know, a lot of things can change over the course of a trial. The study population, you may have some new data collection procedures or new instrumentation that is used, implemented in the course of the trial that you'd like to have used equally in both groups. And there could be other forces, outside of the trial, that may influence outcomes that you would like those external forces to be equally distributed across the treatment groups. And by insuring that the allocation ratio is met as you go along in the trial, blocking protects against some of these time related changes. Also, if you have a case where a trial is stopped early, either because of efficacy measures, you've found that a good treatment and you don't believe it's ethical to continue. Or safety reasons that you're more likely to have balance groups because you have this blocking that institutes balance every so often as you go along, and that leads to a more powerful analyses. The disadvantages of blocking, as I mentioned before, is that they can facilitate the prediction of future assignments. If you figure out the block size, you can start to figure out what is likely to be the next assignment. And that disadvantage is more problematic in unmasked trials, or trials that are poorly masked, that people can figure out the treatment assignments. So now I'm going to go on to the second common maneuver that's used to restrict randomization, and that is stratification. And stratification is used to ensure balance in the treatment groups across groups that can be specified before randomization. So by imposing stratification, we can ensure that the treatment assignment is met within a subgroup of the population and those subgroups are commonly clinic, gender, or some measure of disease severity. So, if we stratify by clinic, then we can ensure that at each clinic we are going to meet the allocation ratio. And you can imagine that could seem like intuitively an important thing to do. That if you have clinics scattered across the country or indeed the world, that there may be plenty of population infrastructure issues that influence the outcome for those patients just as much as a treatment may. So you want to ensure that you're balanced in your treatment assignment within each clinic, so that you have a fair comparison of the treatment assignments. And those external factors associate with the clinic don't bias comparison. Generally, if you're going to use stratification, it should be reserved for subgrouping variables that are considered to be strongly related to the outcome, the primary outcome for the trial. They can either be a strong confounder, sort of, as we've talked about before, that some prognostic factor that strongly predicts whether you're going to have a good or a bad outcome, or some factor that is actually even an effect modifier that the treatment effect works differently in different subgroups of the population. So practically, what stratification requires is a separate treatment assignment schedule for each stratum in a Stratified Randomization scheme. So if you have three clinics, that really means, if you're going to institute a randomization scheme that's stratified by clinic, that you'll need three different treatment assignment lists. So here's an example of both stratification and blocking. So the example we're going to use here is for some type of treatment for breast cancer. So we have treatment A or B. Again, back to the one to one treatment allocation ratio, that's what our ideal is. And we're going to stratify it by two different factors. One is center, is the patient coming to Center X or Center Y? We have two different centers. And then we're also going to stratify the treatment assignment based on the patient's postmenopausal status. Is this someone who has gone through menopause or premonopausal person? Into this stratified design, we're going to institute a block size of four. So, if you look at the table, you can see that the first row indicates the center and then, that sort of divides the actual table in half. Within each center, you can have postmenopausal or premenopausal women. So, within each center, you have two types of women. So, what's the treatment assignment gong to be? Well, if you look under clinic X for postmenopausal women, if we are using a block size four, potential first block could be the one listed there with the patient gets assignment B, the second patient assignment A, the third patient assignment B, and then the fourth patient assignment A. And as I talked about before when we were discussing blocking, walah, you end up with two As and two Bs after four patients. But notice that, that's specifically within Center X within postmenopausal women. So you actually have a separate list based on the stratification variables. So, again, if you go to the right one column, we've got the list for premenopausal women who enrolled at Center X, and when a premenopausal woman arrives at Center X to be randomized, she is going to receive an assignment from that second string of As and Bs. So the first woman who fits that characteristic is going to get assignment B, the second woman will get assignment B and the third and fourth women will get assignment A, because that is the, the first block in that particular treatment assignment list for that combination of stratification variables. So, I hope that's clear. Now one point that may be a little hard to absorb at first, is that if you stratify without blocking, there's really no point in stratifying. Because, what you end up doing is segmenting your population into small groups. And then, imposing a Simple Randomization scheme within those segmented groups. So, you could even be more at risk for imbalances within those groups because you have smaller numbers of patients. So it's important to recognize that stratification alone is not a useful maneuver unless you're going to add blocking to ensure that there is balance over time and over assignments within each list that's defined by the stratification variables. So, I'm going to go on to discuss some of the practical aspects of stratification. One thing that you should recognize is that you have to limit it to a few variables. Once people are kind of aware of the benefits of stratification, they tend to think, wouldn't it be nice to have very homogeneous groups? But that can get you into problems, to have too many stratification variables. So, you want to pick ones that are highly related to outcomes. That are really important to control for and sort of a sense of confounding, and also ones that are logistically possible. You have to know the stratification variable status before you randomize. So, you probably don't want to have as a stratification variable some interpretation of a blood test that takes a few days or a few weeks to get the results back, because you can't randomize someone until you know which list to use. So there's a logistical consideration that you need to have those data available at randomization. The typical ones that are used in multicenter clinical trials or clinic, that almost universally considered an, an important stratification variable, because populations and treatments do tend to vary quite widely by center. Sometimes in a surgical trial, because there's concern about the effect of different skill levels, the stratification variable will actually be on the surgeon, involved in the trial, so that you can ensure, that there is an equal allocation of what procedures were done by surgeon. You may also stratify by stage of disease if that is an important prognostic factor, if you have some measure of disease severity and what's the likely prognosis that you think is strongly related to the outcome, then you may consider that as a stratification variable. Probably there's more emphasis than there should be on demographic characteristics as stratification variables. But if you thought that it was important to ensure that you had the same numbers of people in treatment A and treatment B within subgroups based by gender or race or age categories. You could also stratify based on that type of characteristic. The problem that you can come across with having too many stratification variables, as you saw on the last slide, each combination relates in a separate list, is that you get too many strata. And you can't fill up the blocks, and you actually are working against yourself because if you have lots of strata. And each one has a separate list associated with it, and people are coming into the trial and filling up those blocks what you'll, you may end up with is a lot of open blocks, blocks that didn't get completely filled. So a block size of four would be open if only three assignments were filled in that block. And if you have to merge together data from a lot of unopened blocks, you're likely to end up with an imbalance overall in the treatment group allocation. So, you've sort of shot yourself in the foot. So, stratification and blocking are really the two most common restrictive randomization features. And so, the take home points here are that blocking is very important in terms of ensuring that you maintain the design allocation ratio as you go throughout the trail and helps control for a number of things that can change over time. Whereas stratification, is related to baseline characteristics of the patient, or where the patient is at, that you can control the balance of the treatment assignment, that it meets the design assignment within those subgroups of patients. And indeed, for stratification to be effective, you should also apply blocking. So the final type of scheme I'm going to talk about is an Adaptive Randomization scheme. And I'm only going to briefly review this. An Adaptive scheme is a process in which the probability of assignment to the treatment i.e the allocation ratio, does not remain constant over the course of the trial, but is somehow determined by the current balance of participants in the trial, or even the outcomes from patients enrolled in the trial. There are two common types of Adaptive Randomization schemes. One is based on minimization, after the first patient the treatment assignment that yields the smallest in balance is chosen. So, instead of having stratum to ensure balance across important characteristics, in a minimization scheme you can be balancing on a number of characteristics or prognostic factors as you go on in the trial, and ensure that your, have balance as the trial goes on. So then, if you have a patient with a certain set of characteristics, that you have a relative lack of in a certain treatment group, their probability to be assigned to that treatment group would be greater, and you can combine several patient characteristics to do this. But the allocation scheme can't be determined in advance if you're using a minimization design, so you have to have some, you know, real time usually computer resources to be able to effectively institute a minimization scheme. In effect, you're sort of writing or generating the treatment assignment list as you go along to ensure balance, across the treatment groups. Again, you have to decide what factors you want to balance things on, and have data on those factors. Another type of Adaptive Randomization that relies more on outcome assessment, is a play the winner design, where you change the treatment allocation to favor the better treatment based on the primary outcome. So, if one treatment appears more favorable, you preferentially assign patients to the betterment treatment. You give them a greater probability of assigned to the better treatment. To do this, you have to be able to evaluate outcomes relatively quickly. So for the prior patient's outcomes to influence this next patient's treatment assignment, you have to have those data again to implement this kind of design, and therefore, need to be able to evaluate the outcomes relatively, quickly, and you can implement these designs in stages, so you might start with a, fixed allocation ratio and after you get to a certain number of patients, impliment an adaptive ratio. Okay, I know this has been a long section. We've gone over Simple Randomization, Restrictive Randomization, and Adaptive Randomization schemes.