Okay, in this video, we're going to briefly discuss the assumptions of two factor general linear models. We're going to begin with a very brief review of the assumptions of general linear models in general. And then I want to highlight one particular assumption which is Independence. And highlight how we need to be more careful when looking for independence. And a two-factor general linear model than we do for a one factor general linear model. So the assumptions of general linear models overall are listed here and we've seen them before. And we, when we discussed one factor, general linear models. And you'll recall that they are listed here in order of decreasing importance. So our most important assumption is that our data were sampled randomly. And that's because you want to avoid any bias in the estimates that we make. We also need the data to be independent. And in previous videos we discussed the concept of independence. And that's what we're going to expand on a little bit in this video. Third, we had the assumption of equal variance. And fourth, we have normally distributed residuals. And we've discussed before how general linear models can be quite robust to deviations from normality. Okay? I want you to reach back in your minds now and hopefully remember a slide that looks like this when we talked about independence with respect to one factor general linear models. In previous videos, I discussed the concept of independence. And I'd said that if a dataset includes data that are not independent, then that can potentially lead to results that are unreliable if your analysis does not account for that lack of independence. And I highlighted a number different ways in which non-independence can creep into an experiment. Things like chin, similar relatedness. So maybe family structure in a, in a dataset may be similar environments. And I also highlighted the fact that making multiple measurements from the same individual can be a source of non-independence. And in these next few slides that I'm gonna be showing you, I'm just going to use this example of measuring individuals more than once as our focal example of non-independence. So just to be absolutely clear, The example that I'm walking through the next few slides does not only apply to cases where we've measured individuals more than once, but it applies more generally for other sources of non-independence as well. Now there's a lot of preamble for one slide. So what's, what's the focus or what's the point of this slide? The point of this slide, as we've discussed previously, is that the way in which non-independence can arise in, in an experiment and cause or not cause issues with pseudo replication can be fairly subtle. So I want you to start with looking at the left here. So on the left, this is an example of a one, of an experimental design for one factor general linear model, which is really straightforward. Each letter represents a data point from an independent individual. And so you can see here that all the letters are different, which means we have all of our data points are in, are independent of one another. So in this left-hand case here, based on this information alone. So based just on the information about which individuals had been measured and whether or not we've measured more than one individual. Based on that information alone, we would not worry about C to replication. It's always possible there could be other details. An experiment that could cause pseudo replication we introduced here, but we're ignoring those situations for now. We're just focusing on this simplified scenario. What we pointed out previously is that the scenario that we have on the right also will not cause us worry for pseudo replication. And you can see that we have multiple data points that come from the same individual. But each data point, each of those observations from a particular individual or all are all associated with different treatments. So we have only one measurement from individual a in treatment one versus treatment 11, measurement and individual a and treatment 21 individual from treatment a in three. So because we do not have more than one measurement from a particular individual with, in a treatment. We don't have to worry about whether or not this, this situation will not introduce problem to pseudo replication for our analysis. Instead, we've talked previously about how this new situation on the left, this would be a source of concern because in this case you can see within each treatment we have more than one measurement per individual. And so this could be a source history to replication if we did not analyze these data appropriately. For example, using the mean value of each individual. Within our treatments or using some other form of analysis, something like a mixed effects model, which is something we'll talk about in future videos. Where those types of analyses can account for pseudo replication or can account for this kind of experimental design. And thereby help us avoid problems of CDO replication. Okay? So this is what we've talked about previously is one factor, general linear models. How does this situation translate into a case where we have a two factor general linear model? Well, let's start by focusing on the situation on the right pace. We're just focusing here. So again, just to reorient, just to reorient everyone to this figure. We have three treatments. Add each treatment, we have six data points. Where each data point came from a separate individual. So this means we have six individuals, a, B, C, D, F in this experiment are the same individuals that were measured and each of our three treatments. Okay? If we're to be analyzing these data with a one factor general linear model than we do not need to worry about pseudo replication. Imagine however, that we wanted to take this experimental design and make it slightly more complicated by saying that we're going to take half of the individuals and treatment one and allocate them to one treatment. Say we're going to say that their females and say that the other half of the individuals and treatment one are going to be males. And we're gonna do the same thing for each of those other treatments. So this is what I've done here. Just partaken align and slipping down to divide our datasets for each of our different treatments in half. And I've changed the description of the experiment from just same treatment one, treatment two, treatment three to now be an explicit and saying, let's imagine or experiment, test three different drugs. Drug one, drug to drug three. So in this experiment, we have two factors. We have sex, where we have both females and males, and we have three different drug treatments, drug one, drug-to-drug, three. In this case. Now with this experimental design, we now do have the potential for pseudo replication. I want you to consider two perspectives on this experiment. And in each case I want you to stop the video and think about them, okay? The first perspective that I want you to think about is from the perspective of the drug treatment. So if we're interested, so if we're going to do a two factor general linear model for these data. I want you to ask yourself with respect to the drug treatment. So trying to determine the differences between drug one drug to drug 3x3. Do we have to worry about pseudo replication? Okay, just stop and think about that for a moment. Okay? Now that you've given that a good think, you may see that with respect to the drug treatment. You can see that no, from that perspective, we do not need to worry about CDO replication because that the, the perspective I'm just going to go back one slide. That's the perspective that we had here when we had our one factor general linear model will return to our two-factor case now. Okay? So from the perspective of drug, we do not need to worry about pseudo replication. I want you to now stop and consider this experiment from the perspective of sex, okay, or the factor Sx. Stop the video, think about that for a moment. And then started again. Okay, now that you've given that a good think, what we'll have likely seen is that from the perspective of the factor sx, you will see that in that case we do have C2 replication. And that's because if you look at females, look within the female treatment, we now have three data points that all came from the same individual. So we all have three measurements from the same individual. Likewise for individual B, likewise for individual F. Okay? So from the perspective of the factor Sx, this experiment, if we were to analyze it with a two-factor general linear model and not do anything further to account for this non-independence, then our results would be really suspect force due to replication. So what would our ideal experiment look like for two-factor general linear model if you want to avoid pseudo replication. So for a two factor general linear model, what we want to have in this case is we want to have all of our data points for our two factors to be fully independent. And that's why every single letter here that I've indicated different, which indicates to us that all of our data points came from separate individuals. Again, just to really emphasize this point, the fact that I'm focusing on measuring multiple individuals. I've, I focused on VAT source of non-independence. Purely as one example, I could have created this whole video using another source of non-independence. For example, if we had individuals that were housed in the same cages, or if we were working with flies that were fed on different batches of medium, and some flies all experienced one medium and other flies experienced another. That point is, just to really emphasize the fact I'm focusing on multiple measurements from individuals has only one kind of case in which this non-independence can arise. Okay? So these ideas we had been discussing this video are far more general then measurements from single individuals. So we'll stop this video there. I hope this, this has been helpful. The take-home message here is that when you are assessing a dataset to determine whether or not it is reasonable to analyze with a two factor general linear model. Please be careful when looking for sources of non-independence. And I'll stop video there and say, thank you very much.