Hi, I'm Stuart McKerley, nailer from the University of Suffolk, and welcome back to the Sports Biomechanics Lecture Series. As always, we're supported by the International Society of Biomechanics and Sports and sponsored by Viacom. So one of multiple aims or ambitions in organizing this series and probably the most ambitious or one that's the biggest stretch is wouldn't it be amazing if even one person watched one of these lectures and their research improved as a result? So therefore, we've had a direct influence, however small, on the quality of the biomechanics published research literature. So I think with that goal in mind, there have been a few talks aimed towards that. The first one really was Bill Balzopoulos talking about some mechanical misconceptions in inverse dynamics and some of the terminology issues around that. We then had two weeks ago now, an excellent lecture by Kristin Sonani looking at hopefully improving our understanding of statistical analysis that we can then apply to our research. And now following on from Kristin's talk, we've got a talk by Todd Pataki, who is an associate professor at Kyoto University in Japan. And Todd is going to talk about a comparison between discrete and continuous biomechanical data analysis. And this is something really where I'd consider Todd to be the expert on this heavily involved in the packages that people are using to conduct these analysis. And so I'm really grateful to Todd for joining us to tell us a little bit more about this and kind of talk us through it. And as we go along, this talk, or at least Todd's components, is pre-recorded. So if you have any questions, just jot them down in the comment section on YouTube, and we will still make sure that we address each one. So we'll either get back to you and get you a reply individually, or if we think it's something that everyone would benefit from or if there are enough questions, we'll look at maybe recording a separate Q&A video afterwards. Yeah, thank you very much for watching. Thanks, Todd, for joining us, and I'll hand over to you. Thank you very much to Stuart for the invitation and also for the opportunity to talk to you today. I'm going to discuss discrete versus continuous analysis of biomechanical data. And I just finished watching a talk by Kristin Sinani on statistics in sports science. I was part of this lecture series and was released on July 3rd, at least in Japan, that was the date. The topics of Kristin's presentation focused on a variety of fundamental statistics topics, including hypothesis testing, p-values, and power. And in my opinion, the most important topic is computer simulations. I think that computer simulations are the best way to understand fundamental statistical topics. So if you haven't used computer simulations before to understand statistics or to think about statistics, I very much encourage you to watch Kristin's talk and to see how computer simulations can help us understand what statistics is actually doing and its perspectives on experimentation. So this talk will be similar to Kristin's talk in some senses, but it will extend those concepts to more complex dependent variables. So Kristin's talk focused on scalar data, so relatively simple variables like body mass or body height or duration. This talk will extend those ideas to more complex dependent variables that we tend to analyze, especially in sports biomechanics. So our bodies are complex and there are many types of variables that we can measure and analyze, including kinematics, dynamics, and EMG. And under each of these categories, there are many, many variables that we could analyze. And all of these variables change continuously in time. So because they change continuously in time, there are various complexities associated with these variables. And this talk will discuss some of those complexities and how statistics and continuous statistics in particular can deal with those complexities. An overview of the talk. First, I'll give brief overviews of discrete analysis and continuous analysis. And then along the way, we'll ask a few different questions. There are four questions listed here. The first two are what are the main types of continuous analysis and why is continuous analysis valid? And we'll shift to focus on some of the problems with discrete analysis and problems with continuous analysis. And ask a couple more questions. Why does the literature use discrete analysis and is discrete analysis invalid? And then I'll finish this presentation with what I think is the biggest problem underlying this discrete versus continuous issue. Let's consider discrete analysis. But before we do, let's first consider what continuous data are. And from there, it'll be easier to explain discrete analysis. So continuous data can be any dimension. One example is here a one dimensional continuum. The horizontal axis is the continuum domain. This could be something like time. The vertical axis represents the dependent variable value. So this could be something like force. And that dependent variable changes continuously and usually smoothly in time or over that domain. A two dimensional example is something like this. Vertical and horizontal axes represent the domain and the grayscale color represents the dependent variable value.  So I won't bother trying. Here is an example of real data that are one dimensional. So these are ground reaction forces in one dimensional time. We could also have forces that are acting over a 2D service. So this is a 2D continuum, two spatial dimensions plus one dependent variable, the pressure or local force value. And here is an example of three dimensional continuum, something like a strain distribution or stress distribution in bone or some other three dimensional volume. And I think the best way to think about this type of data is as ndmd continua. This is a terminology that appears in a couple of my papers that I developed with some colleagues, but this is not standard terminology. Nevertheless, I think it's the best way to make sense of the wide variety of data that we see in biomechanics. So in this case, the n represents the continuum and the m represents the dependent variable. And this will become clear from some examples. So first example is univariate 0D. Univariate means scalar and 0D is the dimensionality of the continuum. So something like body mass is a 0D, 1D continuum. 0D is the continuum dimensionality, meaning it's not actually a continuum, and m is the dimensionality of the dependent variable. So a scalar, there's just one value. Another example is multivariate 0D, so instantaneous force. Instantaneous force has no time, it's not changing in space or time, so the continuum dimensionality is 0. Force has three degrees of freedom or three orthogonal values, and so the m dimensionality is 3. Example of univariate 1D data. Knee flexion is a single scalar value, but this changes in time, especially when we're analyzing sports maneuvers. And so these data could be regarded as 1D, 1D. Multivariate 0D needs a three-discipline variable pressure. Pressure is a one-dimensional dependent variable, but it can vary continuously over a contact surface, for example, a 2D contact surface. Multivariate ND data example is a bone strain tensor. So strain is a 6 degree of freedom quantity, and that can vary continuously throughout a three-dimensional volume. So I think this ND-MD perspective describes most of the type of data that we analyze in sports and biomechanics. So what is discrete analysis? Well, discrete analysis consists of two main types, point of interest and region of interest. And region of interest analysis is accompanied by region summary metrics, which we'll get to in just a moment. Let's look at some examples. Point of interest example is shown here. These are anterior posterior ground reaction forces during running at different speeds. And point of interest analysis could consist, for example, of looking only at the maximum forces observed in the push-off phase. So around 60 to 80% time in this case, but those gray dots indicate the points of interest. And from these points of interest, we extract values and then analyze those values. An example in 2D, we could have a 2D pressure distribution but only be interested in the local maximum pressure, for example. So we would define points of interest and then extract the pressure from those points and then analyze the data. In pressure, I appreciate that we need an area and not a finite point but usually we analyze it or usually it's measured over an area so that we can quantify that specific point. But that's an aside. The basic process is the same. Region of interest data. Although this rarely appears in the literature and I haven't seen it very often, we can do it for one-dimensional data. It's more common for 2D and 3D data, but the same thing can be done for 1D data. So these are synthetic data. They're not experimental data, but we could define a region of interest. For example, there are three different regions of interest defined here, one between 70 and 80%, one between 65 and 85%, and then one between 60 and 90%. So three regions of interest of progressively greater width. And from those regions of interest, we could extract a single value. For example, the maximum value within that region of interest, the average value within that region of interest, the minimum value within that region of interest, the median value, the integral. There are many different types of summary metrics that can be extracted from a single region of interest. Example in 2D. This is quite common in the literature, so if we have some continuous measurement, like planter pressure or contact pressure, it's typically divided into regions of interest, as indicated on the right. So those are 10 regions of interest that are commonly used in the literature, and so a single value is extracted from those regions, some kind of summary metric, often the maximum pressure from those particular regions, and then analyzed. And this also is applicable to three-dimensional data. So here is an example from the literature where strains and stresses were calculated inside the femoral head, and the femoral head was divided into different regions of interest, and then data from those regions of interest were analyzed as indicated here. Okay, that's it for this section. This is a very brief overview of continuous analysis. Continuous analysis is a set of techniques for analyzing the entire continuum, and most continuous techniques do not discard data. The easiest way to understand continuous techniques is by example. So in the previous section, we considered discrete analysis and continuum is divided into points of interest or regions of interest, and then analyzed, often using bar graphs or tables or something like that. Continuous analysis is usually presented on the continuum itself. So here is an example of continuum analysis. This is an SPM or statistical parametric mapping analysis, and we'll discuss SPM a little bit later, because this is reasonably representative of the types of results that you can get with continuous data analysis. So for this example, we have main and standard deviation trajectories on the left-hand side. These are two different groups. The horizontal axis is time. The vertical axis is the dependent variable. In this case, this is a joint angle, and the data in the literature are often presented in this way with means and standard deviation clouds like this. And we can see here, towards the end of the trajectory, right around probably 95%, we can see a bit of a deviation or a bit of a separation between those two trajectories. The right-hand side is an example of SPM continuous analysis results. So here we have a t-statistic that varies continuously in time, just like the mean and standard deviation, and at the end of time, right around time equals 95%, we see a significant signal. So this is expressing significant differences between those two groups, the black group and the red group on the left-hand side. We're going to talk about what these types of results mean when we consider SPM, but for now, just note that it's possible, in general, through continuous analysis to start with continuous data, like continuous means and standard deviations, and to produce statistical results that remain continuous. So it's not necessary to extract discrete values from continuous data in order to analyze them statistically. That's the only point here. What are the main types of continuous analysis? There are many techniques in the literature, both in the biomechanics literature and outside the biomechanics literature. So what are the main types and how are they related to each other? The main methods are indicated here, functional data analysis, statistical parametric mapping, dimensionality reduction, and machine learning. The orange font indicates a broad family of techniques, and green font indicates a specific technique. So functional data analysis refers to a broad family of techniques. SPM, statistical parametric mapping, refers to a specific technique. PCA, which often appears in the literature, is a specific type of dimensionality reduction, and machine learning techniques specific types include artificial neural networks and support vector machines, for example. This is not a comprehensive list of techniques, and also there are many overlaps between these different techniques, but these are at least a rough categorization of the types of techniques you'll see in the literature. Let's talk about functional data analysis. Functional data analysis, this is one definition. Functional data analysis refers to a collection of methods for analyzing data over a curved surface or continuum. So this sounds like a nice definition, but it has a main problem in that is that this encompasses all analysis techniques, including discrete analysis. So it's basically saying that it's analyzing continuous data. But I don't think that's accurate, at least enough to distinguish it from the other types of techniques that were referenced on the previous slide. So I'll try to explain what FDA is, at least from its origins. So on the left-hand side is a depiction of a basis function or a spline function. This is a third-order spline function. If you're not familiar with spline, it's basically like a third-order polynomial where there are two local extrema. On the left, there's a local maximum, on the right, there's a local minimum. So when it's a third-order spline, it's basically like a third-order polynomial and you can model some smooth changes. So FDA, certainly at the start, used these types of basis functions to model continuous data. So on the right-hand side, there are a number of discrete data points that you can see depicted as circles. And basis functions, like splines, can be used to model the continuous changes. So the idea is that we have noisy measurements but we know that these are physically continuous processes. So something like force, for example, we know continuously changes in time. So if we measured the force and it was very noisy like this, we can model it in a continuous manner as we expect it to occur in the physical world. So this is one key strength is that once you model the data using basis functions, you can calculate the values that we expect to be the most likely values for the real physical phenomenon at absolutely any point in time. Statistical parametric mapping takes a little bit of a different perspective from FDA and I'll show you some results first and then we'll discuss how it differs. So this is an example of an experiment that was experimental data that was analyzed using SPM. This experiment is a simple right-hand squeezing experiment. So right-hand squeezing for 30 seconds and then resting for 30 seconds and squeezing for 30 seconds and resting for 30 seconds. And the red area depicts the part of the brain where brain activity was above and beyond background noise.   It's still active. You're thinking about a variety of different things including a very inspirational talk, you perhaps heard on the internet. But once you start squeezing your hand, then presumably brain function changes and then we use various techniques to try to identify how that activity changes. So in this case, we see that brain activity in the contralateral motor cortex is above and beyond that background noise when we squeeze our hands. So in my own research, I adopted this brain imaging technique called SPM to biomechanical data. And the interpretation here is mathematically identical. This is an experiment involving walking where in one condition the feet were externally rotated and walking so a bit like duck walking versus just normal walking. And this figure depicts the part of the foot where contact pressures were greater when your foot is externally rotated. So as we expect, the pressures are higher along the entire medial side of the foot because that's a part of the foot that you push off with when you externally rotate your feet. So SPM doesn't care if it's a brain or if it's a foot. The basic maths are the same. You look for the underlying noise which in the case of the foot is just normal walking. This is the variance we expect during normal walking and then here is where those pressures are above and beyond that normal walking case. So this is for 2D dimensional data analysis but SPM can also be used for one dimensional data analysis but we'll talk about this in just a second. A brief history of SPM. The theory for SPM was established in the 1970s. It first appeared in the literature at around 1990. This was in the neuroimaging literature and the seminal paper was published in 1994, 1995 in human brain mapping. And in my opinion, the most comprehensive paper was published in 2004 in neuroimage. Then SPM first appeared in the biomechanics literature in 2008 and then also in 2009 by an independent group for bone analysis. I think it was bone density analysis initially but you have bone strain. Any kind of quantity that you can calculate for bone can be analyzed using SPM. To give you an example of SPM's influence in the literature this 1995 seminal paper by Friston et al. has been cited approximately 10,000 times. That is according to today's Google Scholar. I appreciate Google Scholar might not be exactly right but certainly in the neighborhood of 10,000 citations. To give you an idea of Carl Friston's impact on continuous data analysis and specifically in the area of brain science he has an H index of 234 and an I 10 index of 963. That means he has 963 papers that have been cited at least 10 times and he has 234 papers that have each been cited at least 234 times. His work in statistical parametric mapping and its uses in human brain function have been widely influential. This shows the adoption of SPM over a year, over time. SPM in the neuroimaging literature is indicated in dark grey and SPM in the biomechanics literature is indicated in light blue. The original SPM paper was published or the seminal SPM paper was published around 1995 and since then there has been a linear or exponential increase in the number of citations of SPM per year and currently it has well over 3,000 citations per year but probably many more papers use SPM. This is just the citation. Approximately 3,000 papers per year, let's leave it at that. The first SPM paper in biomechanics appeared in 2008 and since then it has seen a linear or if not exponential growth in citation and now there are well over 100 citations per year of SPM in biomechanics. To give you a context of where SPM fits in with classical theory and this will I hope help to show partially how SPM and FDA differ but in the next slide we'll see more specifically how they differ. So if we think of zero-dimensional data, scalar data and vector data each type of data there is has a randomness model and then various applied forms so for zero-d scalar data the randomness model is the Gaussian or the normal distribution and from this theoretical construct of randomness, the Gaussian distribution stem a variety of applied types of tests including t-tests, regression and ANOVA. If instead we start with a random vector then the randomness model is the multivariate Gaussian and the applied form of those, the analogous applied tests include hotelings t-squared tests, canonical correlation analysis and multivariate analysis of variants or MANOVA. If our variable is instead n-dimensional, so not zero-dimensional but instead on a continuum, the randomness model for scalar and vector data comes from random field theory which was first published in the 1970s but the applied form of random field theory is SPM so SPM takes all of these applied hypothesis testing techniques including t-tests and regression and ANOVA and the multivariate forms and allows you to conduct them in four n-dimensional data using random field theory. So I think an FDA statistician's perspective of the relation between FDA and SPM is this. I think an FDA statistician would say that SPM is a small subset of FDA and with this I would agree from a certain perspective but I think it's not very instructive to regard it like this. I think it's more instructive to regard it in terms of classical techniques especially for hypothesis testing. So here is an alternative view for how these techniques relate in hypothesis testing. So let's start at the bottom. There's classical methods that were born from the zero-dimensional Gaussian model of randomness so the Gaussian distribution gave us everything from t-tests to ANOVA and MANCOVA. SPM extends those classical methods directly into continuous data analysis using random field theory and the n-dimensional Gaussian model. FDA also allows you to do this but it corresponds to hypothesis testing mainly in non-parametric inference. So FDA is somewhat removed from classical methods and what removes it is its basis function modeling. So FDA uses basis functions to model data and this in some senses removes it from classical methods which don't use that modeling and SPM also doesn't use that type of modeling so SPM is more closely connected to classical methods. My own opinion is that a better encompassing term for these methods is continuum data analysis but you won't see continuum data analysis in the literature I don't think. I think that's a reasonable term to encompass FDA and SPM but anyway this is one perspective on these methods. Another perspective is as follows. We can regard scientific data analysis as falling into two main categories. Classical hypothesis testing so things like t-test and regression and ANOVA ranging all the way up to multivariate analysis of covariance and we can divide it into a second category which is Bayesian or modern inference techniques. Another type of data analysis is a more engineering type of data analysis where dimensionality reduction becomes very important and machine learning techniques become very important. I'm not suggesting that these can be used for scientific analysis just that these are traditionally engineering techniques. So these scientific techniques are used in their ideal sense to test theoretical predictions and in their ideal engineering sense to find and exploit patterns in data. SPM is much more of a classical technique and FDA is a technique that expands essentially everything. Gets a little bit into classical hypothesis testing but not nearly as much as SPM but many different types of analyses especially PCA principle component analysis are used often within FDA analysis. In my opinion I think a continuous data analysis term or this terminology is appropriate for encompassing everything for continuous data analysis types. Okay that's it for types of continuous data analysis. Why is continuous analysis valid? Well to answer this question the easiest way to do so is with a demo. I'll be back in a moment with a demo. This will be a demo to explain classical hypothesis testing and then how it relates to continuous data analysis and specifically hypothesis testing for continuous data. So let's start with zero dimensional data. So here we have two groups of data. Group A and blue and group B in red. The thick horizontal bars represent the sample means and the individual dots represent individual observations. There are five observations for each group. Now here the data, the true mean value of each group is actually zero but because we've randomly sampled data we've taken five random values from the normal distribution the mean of those five values is not exactly zero. There's some variation and also the mean of the two groups is not exactly the same because they're different random samples. However the true mean for both groups is zero. I can press this new random data button and get a new sample and in this sample the mean difference has reversed. Previously group A had a greater mean than group B and now group B has a greater mean than group A and I can press this button many times and we can generate many random samples. In each case the true population mean is zero but because of random sampling the sample mean is not zero and each time there's a new sample mean and each time there's a new difference between groups. Now I'm going to calculate the t-value. The t-value represents the difference between groups divided by the variance. In this case the variance is one so we can think of the t-value just as the mean difference. Here the group A value is greater than the group B mean value so the t-value is positive.  If I press this button many times we see that the difference between groups is also randomly fluctuating sometimes above zero sometimes below zero. Now I'm going to start saving these t-values that I'm calculating in a histogram and so when I press a new random data this is a value of about minus one and a half so there's one tick for that value. Press it again this time it's around one so there's one tick for that value so I'm going to press this again and again many many times and as I do we can start to see a distribution building up. Now let's compare this distribution that we're manually computing just by pressing the random data and calculating the t-value each time let's compare this to the analytical distribution so I'm going to click this analytical button this represents the student's t-distribution or the distribution that we expect theoretically and we can see already that the distribution is converging to that expected distribution so most of the values the t-values are very close to zero there are quite a few that are between minus one and plus one but there are very few that are greater than two or less than minus two so if I were to press this button an infinite number of times I should converge precisely to that analytical distribution but I won't attempt to do so I think for now we can be reasonably convinced that it will converge to that analytical distribution so how is hypothesis testing conducted? Well we compute the 95th percentile here which means that 95% of the data are less than this value and then we compare our experimental result to that threshold so here the value is less than the threshold so we would conclude no significant difference if I press this a few times we should see it traverse that threshold at some point one in every 20 pushes approximately still not over the threshold ok there we go that one's over the threshold we see a fairly big difference between group A and group B here and this exceeds that threshold so in this case we would conclude a significant difference between group A and group B does that mean that there is an actual difference between group A and group B? of course not all it means is that when there is precisely no difference between these groups we would see this T value relatively rarely we would see a T value this high and only 5% of a large number of experiments so it's important to recognize that the P value and the statistics in general do not represent the behavior of this specific experiment they represent the behavior of this type of experiment over an infinite number of possible random samples so concluding significant difference doesn't mean that there is a difference it just means that when there is no difference we wouldn't expect the result this extreme so many very often ok now let's generalize this to one dimensional continuous data so this is almost exactly the same as the demo above but this time we have one dimensional continuous data the data can be smooth to various values so here are quite smooth values over here and here are quite rough values over here I'll first choose something relatively smooth to start so like before we have two groups one blue and one red and we have five observations for each group and the mean for each group is depicted as a thick line so I can press new random data and we get a new set of random curves so these are Gaussian random fields one dimensional smooth Gaussian random fields and we can generate more of them and each time see a different type of mean and different differences between the two groups and now we can calculate the t value so let me calculate the t value here so the t value represents the differences between groups normalized by the variance when group A is greater than group B when the mean group A is greater than the mean group B we have a positive t value and when the opposite is true we have a negative t value so if we were to generate new random data we see that the t continuum on the right is randomly varying it is also smooth, similarly smooth to the Gaussian random fields and it is randomly varying now the quantity we are interested in here is the maximum t value why the maximum t value? because we were interested in the greatest difference that we can expect by chance so if there is truly no difference between these groups what is the greatest difference that we can expect by chance? so here we see the maximum t value here it is approximately three and as I keep pressing this value we can see that the maximum value randomly changes so let's start to save these t values so press new random data this maximum t value is around two so there is one tick for that t value let's press it again there is one tick a little bit lower and then we can press it a number of times and we start to build up a distribution just like above now let's look at the analytical distribution this is the analytical distribution that we get from random field theory and we can see that the data that we are generating generally converges to this so this distribution is a generalization of students t distribution which is depicted up here in green to the continuum case and this represents the expected values of the maximum value of the t statistic based on this particular smoothness so I won't press this an infinite number of times but again like above the analytical distribution represents the expectation for an infinite number of experiments but let's accept that pushing it one hundred or maybe two hundred times as I've done here demonstrates sufficient convergence to that analytical expectation so how do we conduct hypothesis testing with these continuous data just like before we calculate the 95th percentile for this distribution and we check whether our maximum t value exceeds that threshold so here it kind of exceeds but not really most of the time it won't exceed that threshold when there truly is zero difference between groups but occasionally it will and one in every twenty simulations are so should produce a maximum t value greater than that threshold still don't have one but we should get one any minute now any minute now there we go we have one value that exceeds the threshold so in this case we would conclude that there is a significant difference between the group A and the group B means but like above does this mean that there is an actual difference between the group A and group B means no it means only that when there is precisely no difference we would expect a maximum t value like this relatively infrequently let's repeat one final time this time with rougher data so instead of using smooth data I'm going to use rough data and generate random fields and save their values so each time I'm pressing this we get different maximum t values and we're starting to build up a histogram let's look at the analytical distribution where we can see that that analytical distribution is different than the one above and we can also see that this simulation is converging to that analytical distribution so why is the distribution different because the data are rougher when the data are rougher we expect greater maximum t values with greater probability so random field theory and statistical parametric mapping use this smoothness information to predict the maximum test statistic maximum t value that we would expect by chance and we can see that the threshold is also depends upon the smoothness so this is why continuous data analysis in general is valid for hypothesis testing because to conduct hypothesis testing all we need is a distribution that agrees with some random behavior in the system of interest and a threshold beyond which data traverse relatively infrequently so in the classical hypothesis testing and classical statistics use a simple scalar 0 dimensional data but if our data is instead 1 dimensional and 1 dimensional smooth we can also conduct hypothesis testing we just use a different distribution and this distribution comes from random field theory that was the demo I think this demo helps to clarify an important difference between FDA and SPM so on the left and the right we've got two different data sets a left hand data set is kinematics hip flexion and hip extension and on the right hand side we have a pedal force the dynamic data pedal force during cycling so we can see that first of all that these data are relatively smooth the FDA and SPM approaches are contrasted here that these top two panels FDA would look at these data and model these data using basis functions for example spline basis functions SPM instead looks at the residuals so if we were to subtract the mean trajectory or the mean 1D value from each of these panels we get the residuals on the bottom so the residuals are the differences from the mean and the residuals have a mean of 0 by definition just like in the previous demo and they also have a certain smoothness just like in the demo so SPM models those or it regards those residuals as one dimensional smooth Gaussian random fields so this kind of this slide emphasizes different perspectives on what is being modeled by FDA and SPM okay that's it for this section there are a few problems with discrete analysis and these are the main ones first of all is resolution so if we had a pressure distribution as indicated in the center panel in panel B and we used a region of interest type analysis as indicated in panel A and we extract the maximum pressure in each region we would end up with a pressure distribution like indicated in panel C so we've effectively reduced the resolution of our measurement by applying region of interest analysis and if you know about the Nyquist sampling theorem you'll know that in general it's not a very good idea to reduce sampling resolution it can distort signal a more serious scientific problem is regional conflation and regional conflation can be thought of as follows a single value does not necessarily represent what is truly happening in that region and let's see an example of why that is the case these are pressure distributions during walking at three different speeds slow, normal and fast if we looked at the maximum pressure data across many trials at a single location in this planter pressure distribution so that's indicated by the asterisk in the medial forefoot just under the hallux we would see a general linear correlation between walking speed and maximum pressure as indicated in the panel on the bottom left so slow speeds are blue normal walking speeds are black and fast walking speeds are red and we can see a general increase in pressure with walking speed as we would expect if we were to analyze these data using regional and continuous techniques we would see some conflicting results let's consider the region of interest analysis first which is on the left side of these right hand results so first of all there are two sets of colors red and blue red indicates positive correlation blue indicates negative correlation we can see here that most of the foot has increased pressures with walking speed but the lateral forefoot has decreased pressures that's what the region of interest analyses say the continuous analyses on the right have a slightly shows something slightly different they do show positive correlation between walking speed and pressure in the heel and in the very front of the foot but over the entire midfoot region and even the posterior part of the forefoot there is a negative correlation between walking speed and walking speed and pressure so here we could say that the midfoot is conflated with the anterior heel so in the region of interest analysis on the left we see a bright red red result indicating relatively strong correlation between walking speed and pressure but this is actually coming from the anterior heel the midfoot itself actually exhibits the opposite trend so we would reach the opposite statistical conclusion in this case to what the data actually actually say the regional measurements actually say so the problem of reduced resolution can become scientifically important when there is regional conflation and so this is statistically problematic but I would argue that it's biomechanically very problematic because the biomechanical interpretations of these two sets of results are very different the biomechanical interpretation of the region of interest result might go something like this pressures in general increase across the entire foot but in the lateral forefoot there is decreased pressure this suggests that the people walking move pressures closer to their midfoot and tend to unload their lateral forefoot which is certainly a possibility but this disagrees with the continuous data analysis results on the right which say that the anterior and posterior portions of the foot increase with pressure but the entire central portion of the foot has decreased with walking speed this implies that the foot is being actively stiffened to prevent contact in the midfoot region with walking so these mechanisms and this biomechanical interpretation is quite different and we started with the same data but using a region of interest analysis approach I would argue that we've reached a rather incorrect biomechanical conclusion a final problem with discrete analysis is results interpretation and here is an example from the 3D literature this is an example the 3D finite element analysis literature this is a simple example where forces applied either at one of two locations the hip contact force FH is applied either at pin one location or pin two location and we can see the average strain distributions in the femoral head in the bottom left when the hip force is applied to pin one we see a strain distribution that's very to the left or that's biased to the left and when it's applied to pin right we can see that the strain distribution shifts towards that pin shifts towards the right as we expect data using SPM on the bottom right shows fairly clearly where those distributions vary the warm colors indicate where there are higher strains when the load is on the right and blue areas indicate the opposite where strains are higher when the force is applied to the left hand side so this is an SPM result but results that you typically see in the literature are depicted in the top left region of interest analysis so the femoral head is divided into a number of regions and then bar graphs are plotted and my head hurts trying to think of these data trying to understand what region one is and region two is and region three is where they are anatomically how they these conditions actually what is actually happening in each region due to the different experimental conditions there's much more mental activity required to interpret those results SPM results are much easier to interpret because our brains are very excellent visual pattern recognition machines we can tell at a glance what is happening by looking at these continuous results when we separate them into discrete values it becomes much more of a chore for our brains to make sense of the data if there are these problems with discrete analysis why then does the literature use it and there are a few reasons I think tradition data complexity recently developed theory and software availability first of all tradition biomechanics comes from a tradition of planar analysis so from very early studies in biomechanics we started with separate data recordings and separate analysis so data in the sagittal plane would be examined separate from the data in the frontal plane and that type of data analysis still persists today so even though we clearly have one thing here we have walking and we're interested in analyzing walking kinematics from its earliest times biomechanics separated data into more digestible chunks and there are certainly anatomical and mechanical, physical reasons for doing so but I would argue that this tradition kind of promotes the idea of taking something that's complex and separating it into more easily analyzed bits this is the tradition of our field and the data are very complex and this is another reason we are tempted to analyze them discreetly so here are data, these are ground reaction force data just the vertical component and the posterior component so let me orient you to this figure first of all the panel on the right is vertical ground reaction force the horizontal axis is time and the vertical axis is vertical ground reaction force the panel on the bottom is anterior posterior ground reaction force the horizontal axis is time again and the vertical axis is ground is the anterior posterior force now we're quite accustomed to seeing data like this in the literature usually they are presented separately but because this is a single physical thing, ground reaction force is a single vector and in this case we're just considering two components of it we could also consider it in the anterior posterior vertical plane which is depicted on the left graph and we rarely see data like this in literature but these are the data nonetheless so what we really got here is something considerably more complex that looks something like this, this abstract kind of 3D type trajectory that's varying in time and two vector components are varying simultaneously in time so this and this is just two components usually we measure three components that's not really possible to visualize three components because we would require four dimensional visualization but even in this case with just two components we've got something that's geometrically very complex and it's not easy to think of how we should analyze this subjectively it's tempting to reduce this into simpler values like the maximum value across the maximum force across the entire measurement for example so we have these complex data and we've we're accustomed to extracting relatively simple values from these complex data another reason the literature uses discrete analysis is because the theory necessary for continuous analysis was developed relatively recently it showed this figure before but basically the theory for continuous analysis was developed in the 1970s but didn't appear really in the literature until approximately the year 2000 and sometime in the last 30 years certainly so but biomechanics is much older than that and so this technique came along and biomechanics I think is starting to catch up to the fact that we have continuous data analysis and there are continuous data analysis techniques so let's start to use them a final limitation and why I think the literature does not use or still uses discrete analysis quite often is that not very much software is available the main SPM software package is for MATLAB and it's seen a number of iterations since it was first introduced in the mid 1990s the current version is SPM12 you can download it for free but it's not easy to use it's quite easy to use for neuroimaging experiments for analyzing fMRI data for example but it's very difficult to figure out how it can be used for biomechanical data so in my research I developed a package called SPM1D which basically takes the statistical core and the computational chunks of the SPM package for neuroimaging and adopts it for biomechanical data and for one-dimensional data in particular it was first released in 2010 and at the time I admit it wasn't very good I think it's considerably better now but it's still in version 0.4 and that was released a few years ago and it's only available for Python and MATLAB so if you can't program in Python or can't program in MATLAB they're useful but hopefully in the future SPM software and FDA software as well will become available and much easier to use I have picked on discrete analysis a little bit and so for ballots let's consider problems with continuous analysis and there are many many problems with continuous analysis and here is a list of issues this is just some of them let's just consider one main one which is registration registration is often referred to as a temporal normalization in the biomechanics literature at least for one-dimensional data registration is a more general term that represents all kinds of temporal and spatial resolution across different data dimensionalities so here is one example involving anterior, posterior ground reaction forces during running and sprinting the different sprinting running speeds are indicated by different colors and we can see that different local maximum values occur at different times in the in percent stands so we could register these data so they look something like this so now the maximum and minimum values are much better aligned and so now we could apply a type of analysis to these data so with most studies in biomechanics do not register the data like as they stop at normal temporal normalization which is just considering the start and end of stands as the end points and then interpolating between those two points but registration in general or non-linear registration allows us to warp time and warp the individual observations to better match each other and this can be done manually by specifying specific points like local maximum for example it can also be done algorithmically using a variety of different approaches if we were to analyze these two cases we might expect some different results and we would receive different results in general I personally haven't seen cases where there's a massive problem in this particular case if we look at the results so we have on the top two panels linear and non-linear registration in the bottom we have the SPM results so we can see some differences in the SPM results between these two techniques but not substantial ones and not ones where we would greatly have a greatly different biomechanical interpretation of the data so this is kind of what I've seen with sets that non-linear registration certainly does affect the ultimate results but not necessarily in a biomechanically meaningful way I would also argue that registration is a problem not only for continuous data analysis but also for discrete data analysis because if we were to extract data let's say at time equals 20% for data that were temporarily normalized to just between start and end of stance then we're not necessarily looking at the homologously identical data or homologous data if we don't register the data then we have a danger of not considering homologous events so I think that registration is a problem indeed for continuous analysis but it's also a problem for discrete analysis and perhaps the biggest limitation is that continuous analysis solves one problem and one problem only the only problem that it solves is the zero-dimensional to n-dimensional problem so in the previous demo we showed that classical statistical theory and particularly particularly how Gaussian data form an expected distribution when applied to a certain type of experiment in that case a two sample experiment we see that this basic process in this theory generalizes to continuous data one dimensional data is the example we considered in the demo but it also generalizes to two-dimensional and three-dimensional and n-dimensional data but this is the only problem that SPM solves it does not solve the weaknesses of hypothesis testing and hypothesis testing has a number of weaknesses and in particular it can be greatly misused or abused if it's not understood correctly so continuous analysis is great I think it solves one problem very very well but it certainly doesn't solve all problems and it doesn't solve most problems is discrete analysis invalid? well this is an important question and one that's a little bit difficult to answer so first I'll answer it from a purely statistical perspective and the way I'll do so is by addressing this question how serious is the discrete versus continuous problem well we did a simulation study to investigate what would happen if we were to analyze a single the single maximum effect observed in one-dimensional continuous data when in fact there was no difference so just like the simulation before we've got two different groups there's actually precisely zero difference between the groups but due to random sampling we see some difference and if we were to look at the maximum difference and run our statistical analysis discreetly using that value what are the chances that we would incorrectly conclude that there is actually a difference well here are the results if there is just one scalar trajectory or one variant continuum the probability of incorrectly the probability of incorrectly identifying a difference is 34% so usually the convention is 5% so we're happy if 5% of the experiments that we conduct involve false positives or involve type one error but if we use applied discrete analysis to continuous data with just a single scalar trajectory that probability is much higher up to about 34% if we have one vector trajectory so three components in a single vector something like ground reaction force probability of a falsely rejecting the null hypothesis jumps to 76% and if we go up to two three component vectors or six three component vectors we approach 100% probability that we will incorrectly reject the null hypothesis so given that there are often more than six vector trajectories analyzed in biomechanical studies for example six different joints ten different joints EMG, kinematics, dynamics there is a very high probability of finding an effect a significant effect if you use discrete analysis of those data now that is the statistical answer does that mean that published discrete analyses are invalid I would say no not necessarily because often the effects that we see are much greater than the threshold for significance and in that case often the continuous analysis and the discrete analysis will agree however if we are talking about borderline effects something with a p-value of 0.04 for example 0.03 even 0.01 I would say there is a very good chance that those results would not be considered significant from a continuous analysis perspective so are those analyses invalid or are those results invalid it's impossible to say for sure I would say very strong effects probably are not invalid but borderline effects I would say they may indeed be invalid from a continuous data analysis perspective in my opinion the biggest problem underlying the discrete versus continuous problem is the exploratory perspective of the biomechanics literature so when we think about the human body and what we can measure in a biomechanics experiment there are a tremendous number of variables kinematics there are more than 300 joints in the body each joint has 3 degrees of freedom at least 3 degrees of freedom and we also have a variety of kinematic variables including position and velocity there are 6 degrees of dynamic freedom there are external forces contact pressures we can measure EMG from a number of muscles so if we can measure absolutely every biomechanical variable at every joint in just a normal sports biomechanics way kinematics dynamics in EMG we're talking about well over 10,000 points of data at every single point in time this is a tremendous amount of data and it's not always clear what we should measure and why we should measure it and I think this leads us into the temptation to analyze continuous data discreetly so from the literature here are some example purpose statements and I'm not picking on these statements in particular I think they're representative of the literature more broadly the purpose of our investigation was to evaluate the influence of hamstrings musculotenderness stiffness on lower extremity kinematics and kinetics during landing the objective of this study was to examine the kinetic and kinematic differences between the first and second landings in a vertical jump we hypothesize that the serious elastic tissues of the medial gastronomious muscle have a compliance which allows the muscle to operate with optimal efficiency during normal walking and running so these are I think illustrative of the typical types of purpose statements or hypotheses that we see in the biomechanics literature there are two major problems I believe with these types of statements one is that the variables are unclear they focus on things like kinematic differences or kinetic differences well what variables should we analyze in order to refute the ideas that are being proposed those are often not considered precisely before the experiment is conducted usually we think of which joints and which muscles and things like that we want to measure but not a specific variable that would allow us to refute some preconceived notions of how the biomechanical system ought to work so there is no clear way to refute hypotheses like this because there are no specific variables if we have no specific variables then I would argue that our purpose statements and our hypotheses are scientifically weak and the perspective I think is this one the exploratory perspective that we tend to adopt is this one and this is certainly the perspective that I adopted when I was a graduate student so the perspective on a single experiment is that we use the experiment to identify differences between different populations so on the left we have populations a population of blue people and a population of red people we randomly sample people from those two populations let's say five people each and we put them into an experiment and then that experiment shoots out some group differences and we're tempted to interpret this as evidence of group differences and that's that's okay from one perspective but the problem is that this is just one random sampling and identifying differences for one random sampling is not necessarily evidence for true population differences statistics perspective the problem is like this there are no red people there are no blue people there are only purple people and one experiment that is conducted is just one possible manifestation of group differences that we can expect from that population when there's truly no differences between the two random samples that we extract and if we repeat this experiment many times not once not twice but an infinite number of times each time we will get slightly different group differences and those group differences form a distribution and allow us to compute the likelihood of specific group differences and this is what is meant to be encapsulated in the probability value in the p-value our literature tends to use the p-value as an indication of the strength of an effect which it is to a certain extent but the strength of an effect only pertains to a specific experiment what we are more interested in scientifically is whether or not the differences we are identifying true representations of group differences and in order to reach those conclusions we have to adopt a more subtle and more careful interpretation of what the purpose of an experiment is and what specific experimental results mean in the broader context of experimentation and scientific pursuit so in summary discrete analysis types, the main types are point of interest and region of interest there are a variety of continuous analysis techniques, the main ones that you will probably see in the literature are FDA and SPM functional data analysis and statistical parametric mapping principle component analysis PCA is often used as well there are a number of historical reasons why we use discrete analysis instead of continuous analysis and those include currently I think the biggest obstacles are the literature inertia that is just the way things have been done for a while, but also software availability there is some software but it's not convenient to use yet a big problem is that discrete analysis of continuous data is generally invalid unless one has a specific a priori hypothesis regarding the specific discrete variables that one analyzes this is the statistical perspective and the statistical perspective would say that discrete analysis is invalid in this case, this does not mean that specific results are invalid but it may mean that specific discrete analysis results should be interpreted with a great deal of caution when the effects are not very large the underlying problem in my opinion is that our literature focuses on an exploratory perspective and uses largely exploratory studies which have generally clear purposes and generally clear hypotheses but that these hypotheses and purposes do not pertain to specific variables and instead pertain to broader concepts like kinematics in general or dynamics in general and in my opinion identifying specific variables before we conduct experiments could go a long way to strengthen our literature or we can use continuous analysis to make sure that our conclusions regarding continuous data are more statistically robust I'll leave you with one quote from the great Fisher to call in the statistician after the experiment is done maybe no more than asking him to perform a post-mortem examination he may be able to say what the experiment died of so this is not to say that statisticians are necessary for each experiment the bigger point here is that we should critically think of statistics before conducting experiments and even use a simulation similar to the ones I showed in this presentation similar to the ones Kristen showed in her presentation so we can understand actually the types of effects that we expect to see when there truly are effects or when there truly is a specific effect this will strengthen the statistical quality of our analyses and that's it thank you very much for listening brilliant thanks Todd that was really really good I think I was saying before that someone commented on Kristen's statistics talk saying the time flew by when they were watching it and I think I didn't quite understand what they meant and I would just kind of watch that one back there as almost a viewer of it being pre-recorded and felt exactly the same thing see I really enjoyed that and I think the quote that summed it up for me there was a point where you said we have continuous data and there are continuous data analyses so let's use them I think yeah that in a message that kind of sums it up for me really well yes I think just a quick one before we go on to any questions for people watching do please go back and have a look at some of the previous lectures and we're going to have a week break next week so if you subscribe to the channel on YouTube and click the bell button you should get a notification when things are announced and back up for what's coming after the break but also look out next week over on the isps youtube channel there's going to be I think 10 minute presentations all from what would have been part of the isps annual conference so that's definitely something to look out for as well okay so question time so I think yeah just a few questions I've thought of to maybe expand what you've already covered Todd into some different areas or some follow up questions to help people better use I guess use the techniques appropriately so the first one is around power analysis so people hopefully are familiar with power analysis in the typical discrete analysis if we're planning on using a continuous analysis such as SPM how do we go about kind of calculating how many participants or subjects might be required right so the only real difference between typical discrete power analysis and doing it for continuous data is that things become continuous so for example if you're doing a power analysis with a simple scalar discrete data you would define an effect and that effect would be a number like say 0.5 or 0.6 or something like that and the problem is that when you have one one-dimensional data or two-dimensional data or three-dimensional data you could have a simple scalar effect like that but it also could be a continuous effect so let's say that you are interested in detecting an effect in ground reaction force during walking and that let's say the normal ground reaction force, maximum ground reaction force is around 1.1 to 1.2 body weights and if that goes down let's say to 1.0 body weights for whatever reason that you would that would cause you to reject the hypothesis well you could specify the effect just as a drop of let's say 0.1 body weights or you could specify the effect as continuous so I guess I would look at it as starting with a mean continuous value of ground reaction force or of joint kinematics or something like that and then you should create a different one-dimensional or two-dimensional or three-dimensional continuum that would cause you to reject your null hypothesis in other words you are going to create some fictional continuous signal that has biomechanical meaning that would cause you to reject null hypothesis and once you define that effect then the rest is pretty easy so there is some software for doing it that's not terribly easy to use but that's the basic idea all you have to do is define the effect in that continuous sense I was trying to unmute myself there so yeah brilliant how what tools are available to help people do that if that makes sense yeah so there is a toolbox called power1d if you search the internet for power1d or one word you should come to a tool this is a separate software tool from the SPM1d package that I wrote and it's separate because it's not quite easy to use it is available it's only available in python at the moment but yeah if you search for power1d you should find the software and also a bit of documentation about it okay yeah thanks for that generally as we are sticking on power analysis do you find that generally more participants or less participants would be required the SPM based power analysis compared to a traditional power analysis right that's a great question and I think I'll answer it quickly I think it's about the same actually it's somewhat counterintuitively it's about the same I would have expected that you would need more participants because you're considering more data but the thing about biomechanical data is that there tends to be rather large effects and since you're considering the whole curve in some senses you're more able to detect those effects when they exist so of the power analyses I've done for different types of data like ground reaction forces, kinematics things like that the number of subjects that are required is somewhere around usually between 5 and 10 and these are for not particularly large effects but again it does depend on the effect size and the nature of your variants so it's difficult to give an overall general answer but at least for not terribly small effects and yeah around somewhere around 10 or even less maybe appropriate excellent so I think in that answer you've potentially just removed yet another barrier that people might have perceived to doing some of these analyses so yeah that's a really useful answer thank you yeah the next question I had was around obviously the P in SPM stands for parametric where does this sit around parametric assumptions do we still need to test for them if so how can you comment on that a little bit please right yeah so there is parametric testing there's also non-parametric testing so when assumptions of parametric testing are not met you can do non-parametric analyses that have a very similar interface in the software but yes you can test for normality and this is the primary assumption of the parametric techniques that the residuals are distributed normally or in other words if you subtract the mean from the mean one-dimensional curve from all of the measured curves the remaining the remaining curves are called residuals and they have a mean of zero by definition and they have some smooth variation about that so the question is whether or not those things those one-dimensional squiggles about the zero line are normally distributed so we can test for normality very much like you test for normality for simple scalar data and those tests are available in the SPM1D package but I would kind of tend to recommend a different approach I think it's fine to test for normality that's not a it is possible to do so but I tend to run analyses both using both parametric and then non-parametric methods and 99 times out of 100 I've seen that the results are qualitatively identical so the results you get out of this type of hypothesis testing for continuous data is a threshold where if your data exceeds that threshold you say it's significant statistically significant but the threshold is at a slightly different height if you do non-parametric versus parametric analysis but the practical implications for your interpretation of the results are often negligible so this is kind of an indirect way of testing the assumptions so as long as the parametric approach does not change your qualitative interpretation of the data then the assumption of normality okay, thank you and I think the last question I've got written down is around another kind of maybe issue with the way we often do statistics in biomechanics or sport science is around multiple comparisons so I know that SPM controls for the fact that you're essentially performing multiple comparisons across the time series but what about multiple SPM analysis if that makes sense I guess like you said at the end of the presentation the hypothesis should be specific but if we do have more than one test relating to the same hypothesis to say the hypothesis was I don't know that there would be a kinematic difference and you've assessed four different kinematic time series each using SPM how would you recommend controlling for that right, so you could I suppose the easiest way to do it would be to calculate a Bonferroni correction for these tests so the Bonferroni correction assumes that these tests are independent that's usually not met but this is a conservative approach to it so there is no more conservative approach than a Bonferroni correction meaning it's the most severe correction so if your results still hold after a Bonferroni correction then you can be reasonably confident that you've detected something so all you have to do is adjust the alpha threshold so if you have four tests like you said and calculate the adjusted threshold from 5% it would go down to something like 0.1 0.12 something like that if there are four variables 0.01 or 0.012 and then you plug that into your four separate analyses and that will raise the threshold a little bit to ensure that your overall false positive rate stays at about 5% Excellent that's as simple as I was hoping the answer was going to be as in a lot of the techniques people are already familiar with transfer over so yeah that's a really positive again maybe removing another barrier yeah so I think if people have watched this maybe heard of some of these techniques for the first time or had a bit of a reminder and thought yeah I think I want to go away and learn a bit more about this what would you recommend so are there any resources or yeah where would you recommend people start if they want to learn more I always like starting with the software and just playing around with data sets I don't particularly like reading user manuals or papers I just like playing around with software tools so I would recommend starting with the software and seeing what it does and seeing the kinds of results that it produces trying to incorporate your own data set into it and then starting to think critically about what the underlying mechanisms are what's in that black box that you don't initially understand and what is in that black box is well documented in the scientific literature but yeah I would say first start with the software and then start thinking about the theory and there are a list of references in my opinion are the best places to start for a theory and those are available at spm1d.org spm1d.org there's a section a references section there that has a number of yeah a number of links to papers from the literature thanks yeah I think I think that's good advice for learning any type of coding I think generally get some data get some code and play around with it I think for anyone who hasn't had a go maybe have to be familiar with MATLAB or Python but I know in MATLAB at least I've played around with the SPM code and there are on the website there are examples you download it and you can run the example you can change a few bits and see what happens or you can remove the bit at the start where it reads in the data and reading your own data instead and just see what happens I think you don't have to be able to code SPM in MATLAB you just have to be able to read in your own data and understand what's happening enough that you can tweak it and make some sensible decisions yes I think again I probably mirror that advice I've just pick it up and have a go yeah and I think if anyone watching this has any questions or has any further questions either leave a comment on YouTube or get in touch with either myself or Todd and we'll try and answer them but if if there's enough interest or enough follow up questions to make it worthwhile then rather than replying individually we've said that we'll consider recording a separate video with some answers so that hopefully everyone can benefit yeah so thank you ever so much Todd I really enjoyed it and I think hopefully that's another talk that could really benefit the community and some people could benefit from and it could really hopefully make an impact on some people's analysis but thank you very much you're very welcome and thanks so much for this lecture series it's yeah there's some fantastic talks on it and I'm very happy to be a part of it so thank you


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Not too long ago, we discussed some of the worst topics to bring up when meeting someone for the first time. Now let's talk about some good ones.  Number two. Bucket lists. Talking about your dreams, your aspirations, places in the world that you want to travel. Three. Pets. If you both have pets, you likely have hundreds if not thousands of photos and videos to share. And if they don't have pets, ask them if they could have a pet, what their dream pet would be. Number four. Shows. A lot of us, well, the whole world, have spent a lot of time inside as of late, and we've watched a lot of shows. Number five. Books. Similar to point four. People are reading more than before. Crunch. But it's true. Books can be a great conversation starter, discussing genres that you prefer, what you're currently diving into, what you would recommend. Maybe it's blogs or e-magazines. Comment down below if you'd like more of the worst conversation starters or the best conversation starters for the next one.