 So, again, here, this part will address some topics that will be particularly useful for you as soon as we go to multivariate analysis, because this is the topic of this week, but they are still very useful. You can, without problems, using them in univariate regression. The first is a notion of partial regression. And here, actually, there is also another urban myth, like the one with the normality, about partial regression. So our situation when, for instance, we have two or more sets of explanatory variables that would be invoked called for to explain a distribution of any species. For instance, I have worked with people in the forest tree, and they have blocks of variable for their distribution of trees. They had explanatory variables about the climate, about the soil, about natural catastrophic events like fires, for instance, or insect arbors, and another force block with human influence. And we had to build, actually, to build models and try to find out how much of the variance of the tree abundances or the tree communities, we had the uni and multivariate situations, those blocks explained separately or in common. So this is today's topic in general terms. You can use regression modeling to address these situations. So I'll begin, of course, with a simpler case where instead of four groups of explanatory variables, we have only two because it's easier, of course, to begin with this situation. So you can definitely use regression to study the influence of one set or the other separately or the two together. And the natural things are, and go, the result of using both together is not the sum of the result of using separately one and the other. And this is because in natural situations, you always have some correlations between the blocks of variables. So in my forest tree situations, for instance, there were correlations between climatic variables and soil variables because gradients overlapped in some way, not, those may be correlation, those may be causal or non-causal correlations, who cares? I mean, the result is that, numerically speaking, you have correlation between your groups of variables. The only situation where you don't have such correlations are the ones where you build experimental designs such as ANOVA cross designs, balanced cross designs with equal numbers of replicates in each cell. And then the two categorical variables for this cross and ANOVA, those, by definition, are independent, are not related to one another in terms of correlation. So in this case, yes, you would add up the influence of the one and the other. But in general cases where you go out and sample your sites and take environmental variables of various groups, you are certain to find correlation among those explanatory variables. And I insist it's there, that is, that the problem may be. So first, the partial regression is a way of estimating how much of the variation of the response variable can be attributed exclusively to one set of explanatory variables, meaning you have first to remove the effect of the other set. So I'll briefly go through this procedure. But then, when we will switch to what we call variation partitioning, meaning assessing only the magnitude of the influence of the different groups, we don't have to go as far as to compute partial regressions for every possible situation. If we simply have to assess the R squared or the adjusted R squared as they are, this won't be necessary. But anyway, let's first go through the principle of partial regression. And also, if you have a model of multiple regression, a simple LM model in R, and you simply want to assess the influence of each of the explanatory variables separately, you have to remember that actually the regression coefficients affected to those variables are partial coefficients by definition. Each of those coefficients is dependent on the presence of the other explanatory variables. Simply verify, run your multiple regression with all variables and look at the coefficients and then remove one variable among those X variables and recompute the regression and all the regression coefficients will have changed because each of them takes into account the fact that the other are already there, including the fact that they are inter-correlated. And if you want to get the weight, the importance of each of your variables in your regression model, then you'll have to compute what are called the partial regression, the standard partial regression coefficients. Standard meaning that for this purpose you'd have to standardize your explanatory variables before running the regression. In many other packets, not in R, unfortunately, but in many standard statistical packages, the output of the regression gives you the full regression coefficients that take into account the scale, the units and the range of your variables. And also the standardized, so they are called coefficient regression for standardized variables or something like that, or you can do this manually by just standardizing the scale, the whole matrix of explanatory variables, run your LM procedure on this, and then the regression coefficients, the absolute values of those regression coefficients immediately tell you which variable is the most important, which is less important, and so on and so on. This is between brackets, but it may be useful to you. Now here we are in another situation. You have your response variable. You have an X matrix of one group of explanatory variables, which may be, for instance, environmental variables. And you have another one that I call W following Pierre's symbolism. That one may be, well, the climate, for instance, if the other was a soil, in my forest, for example, or it could be environmental in one case and spatial structure in the other case. That was the frame in which we worked from and we still work now. So you want, for instance, to know the influence of your X variables, but taking into account the influence of the W variables. So globally, you want to build a model explaining Y with that part of X that is completely independent of your Y variables. That means that, let's take this Venn diagram that we will use very often from now on, the total variation in variable Y, which will be for matrix Y from tomorrow on, is represented by this rectangle here. And here you have this first circle here representing the whole variation explained by the X matrix, environmental, for instance. And the other circle here is the variation explained by matrix W, which may be the spatial structure. As you see, and it occurs in natural situations, there is overlap between those two circles because, and I insist upon it, because of the correlation between X and W. And of course, everything else around this in the rectangle is that fourth portion called fraction D in our 1992 paper. So it would be useful for you if you retain now those designations for fraction R in the case of two explanatory matrices, we shall always use this fraction A for what is uniquely explained by matrix X when you have taken the influence of W into account. C is the opposite and B is the common fraction. D being the unexplained part of variation, fraction of variation. But now in regression, how do you obtain a vector if you want the Y hat, the fitted values, the partial fitted values, values of Y explained by X holding W constant, meaning you take into account the influence, the A fraction actually, but the corresponding fitted values. First, you compute the residuals of X on, not on Y, on W. This is the most important one. And there resides the second urban myth about partial regression. Many people still think that to run a partial regression, you have to take the influence of W out of Y before computing the regression of those residuals with X. No, you don't do this. This is not correct. And if you do this with W and X and swapping them, you won't obtain the correct partitioning. It simply won't work. No, you have to take the effect of W out of X because the problem, as I have now repeated many times and will probably repeat many more times, is that X and W are correct. And then you compute the residuals on Y on W. Those ones, I put them in second because they are not necessary in all cases, not even. But in the full case of partial regression, you simply remove the effect of W from the X matrix and form the raw response variables. So you have a variable singular for now, for today. So you have here the residuals, so you compute a multiple regression of all the X values explained by W. And then you take the residuals and same now you take the residuals for Y. And from now on, you can apply two methods. The full partial regression, well, no, actually I could call it the semi-partial regression would be to regress Y, so the raw response variable on those residuals. So using these residuals as explanatory variable, but keeping the response variable as it is. It's perfectly doable. It's perfectly valid. And you obtain an R square that is called the semi-partial R square. We will use this notion when we will run variation partitioning among multivariate matrices from tomorrow on. But if you want really to be completely partial, and this will lead to, for instance, the partial correlation coefficient, then you use also the residuals of Y against W, and you run your regression this way. And this, the R square obtained this way is called the partial R square. And actually though the vectors of fitted values, well, the R square are different, but the vector of fitted values only differ by the value of the intercept, okay? So in the symbolism of the Venn diagram that I have used before, the partial correlation coefficient, so the one obtained here with this operation here, the partial correlation coefficient is the square root of fraction A divided by A and D. If you remember, A is the part uniquely explained by X, D is the unexplained part. So the effect of W is completely removed here from this expression. Can be tested with an F statistic with the corresponding degrees of freedom, as usual, so the number of explanatory variables here, and the number of explanatory variables, and the ones in the W matrix in the denominator here. This corresponds to this. In our partitioning diagram, it's as if I had cut a hole in this rectangle. Everything originating from W has completely disappeared, and my partial R square, removing the square root, the partial R square, is the ratio between A and A plus D, and the partial correlation coefficient is simply the square root of this value of this figure. So this is partial regression, full partial regression with the influence of W removed from both sides of our model. The semi-partial correlation coefficient is the square root of A, but this time divided by all fraction, so by the total variance of the data. I did not remove the part of variance of Y explained by W, so it's still here. And A is always the same, and the corresponding F being the same here, as you see. So here, my partial R square, or semi-partial R square, is A divided by all the rest, and divided by the whole surface here, by the total variance of the data, A divided by the total. And if you take the square root, you obtain the semi-partial correlation coefficient. We'll use this in multivariate variation partition. Now these operations that I mentioned, regress, and so on and so forth, you need them actually only if you want to obtain the fitted values, the models themselves. If your only aim is to obtain the magnitude of the individual, the size, the values corresponding to those individual fractions, then you have to go another way, strangely enough. And the reason is in the adjusted R square, because until now I did not make any difference between the two, and actually what I have shown you correspond to, if you don't make any corrections, any adjustments, it corresponds to that biased R square that everybody knows and actually uses in many circumstances, sometimes for good reasons and sometimes not. But what we want, of course, is to use the adjusted R square. The problem is that the adjusted R square, as Pierre defined it, we're using Ezekiel's formula, cannot be computed on a partial regression result. This adjustment does not exist. So you have to resort to a trick, because actually if you have two explanatory variables, you need three computations to obtain all the fractions. You have four fractions, you have two blocks, four fractions, you need three computations. So either, yes, if you don't need the R square, then just want the fitted value as well. Then, of course, you can compute one, one partial, another partial, and any one of the others involving the B fraction, and then by subtraction you will get everything. So then you don't have your corrected, your adjusted R square. To obtain the adjusted R square, you don't use partial regression. So how do you do? You just take, run a first regression of your response data in function of both explanatory matrices. So this will give you the whole envelope of the explained variations. So A plus B plus C. And then a simple, well, a multiple regression if you have many X variables here. So another multiple regression, but with one of your matrices, X for instance, in this context it would be an environment for instance. And yet another one with only space. So above there you have fraction A plus B plus C, here you have A plus B, and in the last one B plus C. So of course by subtraction here, if you subtract this adjusted R square from this one, you obtain A, and from that on you can obtain B and C. So you have a, all fractions can be obtained then, using only these three. Of course, if you have more than two blocks of explanatory variables, then you have more of those computations to make. Be reassured, Pierre took care of this up to the situation of four explanatory variables, or groups of variables through a function called Warpart in Wegan, which make all this automatically. You'll still have to test for the fractions that are testable separately though. So as I mentioned, those R squared above aren't adjusted. So you have to adjust those using Ezekiel's formula. Well, Pierre's function of course takes care of this. So actually, when you have computed this, you obtain an R square for this one. You adjust it. You use the adjusted R square. Here again, adjusted R square, here again the adjusted R square, and then you can proceed. And this method, as I mentioned, has been extended to multivariate response data. By a paper that we wrote together, actually it was published in 1992, but the thinking was made in 1989 during my marvelous year as a postdoc fellow in Pierre's lab. That was when we began our collaboration. And then there were many thinking and writing and so on. And finally it was published in 1992. And this is a paper that has gained some appreciation. It has been cited more than 2,000 times since then. So it still contributes heavily to my citation index in that context. And then it has been much improved. And in particular, for that, all that part about adjusted R square has been introduced in Perez Neto and Corsair's paper in 2006. This is the paper that Pierre showed or an extract of it with the problem of the R square and the adjusted R square with the figures he showed you before this morning. It comes from this paper here. So to summarize your variation partitioning, you do it this way. So once you have adjusted your R square, you obtain A by subtracting B plus C adjusted from the one that has all the fractions. And then, of course, since you have A, you can obtain B by subtracting A from A plus B. This is elementary arithmetic. I will not bore you with this. OK, so you obtain all fractions. Fraction A, which is a unique contribution of matrix X in the symbolism we used before. And fraction C, which is a unique contribution of the other explanatory matrix. Those two fractions can be tested for significance. As in any partial regression, I've shown you the F statistic before. But fraction B cannot. And it took us years to understand actually what was, you know, we are empiricists at the basis, we are not mathematicians. It's not to say that we do anything funny and then, no. We simply try to find solutions. But when we find them, it may work. But in some cases, we have to think over the mathematical consequences and explanations and so on often. And this was always Pierre's attitude. And this is how he progressed and he made so tremendous contributions. He always asks for help by people who may be more knowledgeable than him and me in mathematics. So it took us years to understand what was going on with that damned B fraction. Because as it is, of course people have come and say, yes, but I also want fitted values for my B fraction. Because you may have noticed that I mentioned how to obtain fitted values using partial regression for A. And of course the reverse with C, which is always possible. I did not mention anything about B. And the reason is that B is actually kind of a ghost fraction. It is only there because of that correlation between X and W. Have I told you this already? If anybody says no, we're sleeping all the time. Now, actually, that means that because of those correlations, matrix X makes part of the job of matrix W. And matrix W makes part of the job of matrix X in explaining the variance in Y. Variance in Y is related, some part of the variance in the response variable or variables is related to both X and W because of those correlations between X and W. So this is no real part of variance that you can fit. We tried. We desperately tried for some time, OK? Trying to, well, you can figure out using this, subtracting the A fraction from the fitted values of A from A plus B and then taking the rest of it and figure that this is a B fraction. It seems logical. Yes, until you try the reverse, we should give the same result and it does not give the same result. There is a paper by Meo et al. and N-Corthes in 1998 explaining this problem because this was the result of a postdoc fellowship another postdoc fellowship with Pierre. So definitely you cannot or there are rumors now that some people have found the contorted ways to test for fraction B. I'm not so sure it's so straightforward If it was straightforward, we would have found it and I'm still completely unsure if it really can be done. But in any case, for now, let's consider that it's simply not possible. So this is, for now, not possible. So you can test the unique fractions. You can, when you have more than two explanatory matrices there are some parts of them that are testable or not. To make things simple for you, in Function Varpart, Pierre added a column at the end, testable yes or no. It does not go as far as to test everything because in most cases you don't have to test all those fractions. You simply want the values corresponding to the adjusted R-squares and so on. But if you want to test them, you have to target those that can be tested and those are mentioned in the results of Varpart. And in case of multivariate response data, we strongly advocate the use of adjusted coefficients of determination, not the usual R-square, the unadjusted one, together with RDA. Why do I say this? This is because you could be tempted. And actually the first, well, our 1992 paper was based on canonical correspondence analysis where we didn't have any notion of using an adjusted R-square. We used the R-square produced by CCA, which is actually a proportion of inertia. And maybe it was fortunate because we may not have published that paper altogether if we were conscious about that problem. But in any case, adjustment of the pseudo R-square produced by a CCA is much more difficult than the simple, you cannot use Ezekiel's formula in the case of CCA, canonical correspondence analysis, which is the constrained form of correspondence analysis. You cannot use it. And the only way to obtain something like an adjustment of an R-square is by bootstrap. And if you bootstrap this, you do it twice and you obtain two different values. So this is a little bit, well, it's certainly not straightforward. And well, unless being strictly forced to use it for whatever reason, we simply don't, since we have a simple solution with RDA. So as I mentioned to you before, you can expand this reasoning up to at least four explanatory matrices. Four is the maximum I had to deal with. In some cases, people come up with more than that. But then it splits the variance into so many terms as to be, you end up with very small values that are generally non-significant and are probably not explainable either. So in those cases, you're probably one, at least one of those groups of variable could be construed as a way of separating the data into different subsets and to be analyzed separately with a manageable number of groups of explanatory variables. So be careful about that, because of course, here again, if you have too many of those fractions, then you explain the R-square become very small and often non-significant and the explanation may go out of hand altogether. Well, last comments before we end here, I have some last comments about that famous B fraction. That B fraction has nothing to do with an interaction in ANOVA, because it's easy to think that, well, you have influence of A, influence of X of W, so this must be the interaction. No way it is not. Another sentence you have to write on your ceiling, so you see it when you wake up. And everywhere you need, you deem appropriate, fraction B has nothing to do with an interaction term. In ANOVA, an interaction term measures the effect. Yeah, I always have to read a little bit, because it's always that tricky to express it, of factor on the dependent variable conditioned on the level of another factor, in the simple two case. So you have two factors. The interaction measures how the effect of one factor depending on the other, or on the levels of the other. This is an interaction. I have a question. I would be happy if people here who are not sure about their conception of interaction would raise hands, because I have a set of slides that are especially devoted to that. It's a funny example borrowed from Socalerolf and Rolf with male and female rats consuming fresh and rancid lard. And all the possible situations are represented in those slides. I may not have the time enough to present all those slides, but I could at least do a little piece out of it and provide this in our web page. So are somebody interested in this group of slides? Yes? Yes. Pierre and Vico raised hands. OK, now I can do this, certainly. It was a very funny exercise to build those slides. OK, so just to make sure. So that'd be, and try. And you can analyze ANOVA results, and we'll speak about this. It will be one of my topics. You can analyze ANOVA results using multiple regression thinking. We do this in RDA now. When you have a balanced cross design in ANOVA, and I insist it must be balanced, same number of replicates in all the cells with two factors, what is the value of the B fraction? It is? OK. Did I forget to tell you that the B fraction arises because of a correlation between the two matrices? I think I have told you. So now what is the consequence of this in terms of ANOVA, the two factor ANOVA with the balanced design? What is the value of the B fraction? Zero. Thank you. One out of four, well, in any case, it's better than. I suspect many people thought about a zero, but it's not there. Yes, it's zero. You don't have an overlapping fraction because the two factors are independent from one another. They are not correlated. So you may have an ANOVA with an extremely significant interaction in a balanced cross design, but the B fraction will still be zero. It's always zero. And to go even further in that direction, ANOVA, this is the reason why an ANOVA, it's much easier to compute when your design is balanced. This is exactly because of that damned fraction B. If your design is balanced, you don't have a B fraction. If you have, if you miss a replica somewhere and your design becomes unbalanced in some way, then you have an interaction, but you have a B fraction that contaminates the B fraction, the interaction. So the estimation of the magnitude of interaction becomes more difficult because of that interference of the B fraction. And a final word of caution. In some cases, you end up with such a partitioning. Quantitatively speaking, I mean, a very important B fraction with respect to A and C. This is a dangerous situation in terms of interpretation of ecological interpretation because you cannot tell. In that part here, you can absolutely not tell in terms of your, you know, when you run a regression or any other type of modeling, you have hypotheses that are causal. In no way, you can, using a regression, prove or demonstrate a causality. This is in your hypothesis. A significant model will give you fuel to your hypothesis and reinforce, maybe, the idea that you're right about your causal hypothesis. But in no way, you have proven anything. This also must be clear. But in terms of your hypothesis, when you have such a large B fraction, you cannot readily attribute it. Decide that this is due, essentially, to the causal effect of this part, w, or on that x part. You cannot. So here, you are stuck with the problem because you have real multicollinearity among the variables between the two sets. There are ways to try and improve that by selecting variables. But again, this is a dangerous play. We'll see how, usually, we proceed by selecting variables separately from the two parts and then running the variation partitioning, which is the honest way to do it normally. Because in that case, you don't overly promote the separation of the two groups, which would be dangerous as well on the other side. But in such limited cases, maybe, you would have to think about the reason why in one variables in your two groups are so massively correlated, because this is the origin. So you may run a computer correlation matrix with the two sets of variables together and check whether this one and this one or this one explained by a series. It's not always obvious because sometimes it occurs because one of those here is correlated with together a couple of others in the other matrix. So it's not always easy to track down. But in some situations, you may discover that actually, you have something quite trivial in ecological terms because one, I don't know, maybe climatic gradient here explains several of your variables in this part. So again, here in that case, you would have to resort to ecological thinking to decide whether it is the climate itself or the other environmental variables that explain really the distribution of your species or something like that. So again, it's all a matter of sound thinking. Do all this automatically and, well, blindly run this and take, yeah, well, OK, this is marginally significant. This one is not. I'll explain this. And we have a huge part that is common. And I'll decide that it's due to this one because that part was it. No, no, no, no. Don't play that kind of games. It will end up with nothing. And you will be unable, actually, to justify your choices in such a case. So there are critical cases when the bee fraction is high. OK, with that, I think I am over with my presentation for today. Is there any question? Hand shaking? No? OK, so bon appétit to Monde.