When can the cause of a population decline be determined?
Dr. Trevor Hefley
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10/06/2016
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Inferring the factors responsible for declines in abundance is a prerequisite to preventing the extinction of wild populations. Many of the policies and programs intended to prevent extinctions operate on the assumption that the factors driving the decline of a population can be determined. Exogenous factors that cause declines in abundance can be statistically confounded with endogenous factors such as density dependence. To demonstrate the potential for confounding, we used an experiment where replicated populations were driven to extinction by gradually manipulating habitat quality. In many of the replicated populations, habitat quality and density dependence were confounded, which obscured causal inference. Our results show that confounding is likely to occur when the exogenous factors that are driving the decline change gradually over time. Our study has direct implications for wild populations, because many factors that could drive a population to extinction change gradually through time.
Hefley, T.J., M.B. Hooten, J.M. Drake, R.E. Russell, D.P. Walsh (In press) When can the cause of a population decline be determined? Ecology Letters
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- [00:00:00.000](applause)
- [00:00:01.296]Well hello, thank you Jessica for the introduction
- [00:00:04.310]and putting this all together, and it's great to be back
- [00:00:07.437]to Harden Hall, I think this is the first time since
- [00:00:10.611]I graduated, so, with that, I just want to present
- [00:00:15.789]some recent work.
- [00:00:18.402]Originally actually, I started it at the beginning
- [00:00:20.479]of my PhD, but as many things go, it's kind of drawn out
- [00:00:25.602]for many years now.
- [00:00:27.045]And it's a question I've been really interested in
- [00:00:31.075]for wild populations, but I ran into some difficulties,
- [00:00:35.017]and this really is a story of those difficulties.
- [00:00:39.682]And so I want to start off with some motivation for
- [00:00:41.541]this question in the context of ecological research.
- [00:00:46.050]So in 1999, Robert May presented this paper in the Royal
- [00:00:52.085]Society of Biological Sciences on unanswered questions
- [00:00:54.942]in ecology, and really the idea behind it was his ideas
- [00:01:00.326]of some of the biggest questions that needed to be answered
- [00:01:03.602]in the next century.
- [00:01:06.011]And so I just want to go through that list.
- [00:01:08.301]The first one on that list is actually what determines
- [00:01:10.775]population density, right?
- [00:01:13.048]And I remember reading this probably six or seven years ago
- [00:01:16.856]and this really stuck with me because this seemed
- [00:01:19.574]like a very simple question, right?
- [00:01:21.764]You know, what determines the levels of certain populations
- [00:01:24.994]of wildlife or plants or humans, if you will.
- [00:01:28.740]And he made a pretty good argument that, you know,
- [00:01:31.682]for the most part we could give a verbal description
- [00:01:35.485]describing why a population has a certain level, but,
- [00:01:38.465]for the most part we really lacked any firm understanding
- [00:01:42.520]of that, and of course to understand why a population
- [00:01:45.632]would be changing in density, we have to understand
- [00:01:48.152]what determines the level.
- [00:01:49.925]Now if we look at some of the other questions on the list,
- [00:01:52.494]we could say spatial structure populations,
- [00:01:55.059]or scale of ecological studies in space and time,
- [00:01:58.480]those seem a little more difficult to me to answer.
- [00:02:02.571]Stability and complexity in ecological communities,
- [00:02:04.996]that's a pretty difficult question, when we start thinking
- [00:02:07.876]of interacting organisms.
- [00:02:11.192]And just to throw the whole list up there, when I saw
- [00:02:13.236]the first question I was fairly convinced that that was
- [00:02:15.392]the easiest of them to answer.
- [00:02:18.907]And you know, beginning at the first few years of my PhD
- [00:02:22.122]working on that with Bobwhite Quail.
- [00:02:27.876]So that's really the context of this question is,
- [00:02:29.569]it's still one of those questions out there that for the
- [00:02:32.247]most part we still need to work on to get good answers for.
- [00:02:39.603]So when we think about what determines population density,
- [00:02:43.345]there's really two general factors out there,
- [00:02:46.725]so there's endogenous factors that are influenced by
- [00:02:49.112]the density in the population, so those could be things
- [00:02:51.792]like density dependent, and then there's these exogenous
- [00:02:54.757]factors that are not influenced by density, so these are
- [00:02:57.538]things like weather, you know, possibly predators out there.
- [00:03:03.408]But there are factors that the actual number of individuals
- [00:03:06.145]in the population typically don't influence, right?
- [00:03:08.908]So for example the number of, you know, piping plumbers
- [00:03:12.065]out there do not influence the weather.
- [00:03:14.412]So those are really the two factors out there and when
- [00:03:17.233]we build models we wanna have both of those factors in it.
- [00:03:21.822]And when you think of declining populations for example,
- [00:03:25.673]many of those may be declining due to exogenous factors,
- [00:03:28.539]right, so, exogenous factors such as land-use change
- [00:03:31.347]or environmental contamination like DDT concentration.
- [00:03:35.091]And then of course climate change, right?
- [00:03:38.236]So they're being driven to extinction due to some external
- [00:03:41.699]forcing that more or less they can't control.
- [00:03:45.004]This might be different than for example in the Lee effect,
- [00:03:48.588]where we see the endogenous factor driving
- [00:03:53.009]the population extinct.
- [00:03:55.690]And then finally we want to ask the question,
- [00:03:59.111]which exogenous factors are causing decline, right?
- [00:04:02.825]For many populations of concern, that's really the question
- [00:04:05.844]we want to answer, and typically we have a single time
- [00:04:08.817]series from a single population, and that's the important
- [00:04:12.281]part today here, is we have one time series, we don't
- [00:04:15.932]typically have replication, we'll talk about the case
- [00:04:19.085]of replication, but we typically don't also have any
- [00:04:22.107]reference populations out there, right?
- [00:04:24.490]So we have a single population, and I just wanna go through
- [00:04:26.922]a few examples of wild populations that I'm working on
- [00:04:30.429]right now, so what we have here in this figure is the
- [00:04:33.730]number of pronghorn antelope, or at least the index
- [00:04:38.228]of abundance, for an area in Northwest North Dakota.
- [00:04:43.505]And we can see through time, you know, the Pronghorn
- [00:04:45.895]abundance has increased drastically, then collapsed,
- [00:04:49.244]increased again and now it most recently collapsed.
- [00:04:52.749]And we have some exogenous drivers and those are shown
- [00:04:56.685]in the plots in the subset plots with the red.
- [00:05:01.949]So what we have there is the number of Pronghorn Antelope
- [00:05:04.788]harvested each year, so typically when there are more
- [00:05:07.716]antelope there's a larger harvest, but we also have the
- [00:05:10.446]number of oil wells per square kilometer.
- [00:05:13.215]And we'd like to know, you know for example, which of these
- [00:05:15.937]factors may be driving the population, if any.
- [00:05:21.211]And there's many other factors that could be in here,
- [00:05:23.014]and so Katie Christy and I are working on that right now.
- [00:05:28.430]And you know there's factors such as the weather,
- [00:05:30.449]or some of the land-use characteristics in the area,
- [00:05:33.624]but that just sets the context of we'd like to understand
- [00:05:36.055]what's driving the changes in abundance.
- [00:05:41.143]Personally I'm interested in the Bobwhite Quail population,
- [00:05:44.894]so this is from Nemaha County, Nebraska, and we see
- [00:05:47.712]long term trends of decline in Bobwhite Quail populations,
- [00:05:52.473]and ideally we'd like to take some co-variates related
- [00:05:55.835]to land-use cover and try to understand why that population
- [00:05:59.858]is declining.
- [00:06:01.310]I mean that's really where I started to run into some
- [00:06:03.664]issues related to actually answering this question.
- [00:06:07.966]So we're gonna talk about some experimental data,
- [00:06:11.683]so I think one of the things, well, actually that I learned
- [00:06:15.108]as an undergrad, so how many people know what
- [00:06:17.549]this is a photo of?
- [00:06:20.144]Anyone?
- [00:06:25.185](chuckles)
- [00:06:26.317]So this is actually fire beetle populations.
- [00:06:30.944]So Drew Tire teaches 450, that's a natural resources
- [00:06:35.836]population dynamics class, and one of the exercises in there
- [00:06:39.052]is to manage beetle populations, right?
- [00:06:43.172]And I think one of the takeaways of that exercise that
- [00:06:46.097]stuck with me is that if we can simplify the situation
- [00:06:48.497]that we have, we might be able to learn something about
- [00:06:53.284]the tools that we use in wildlife ecology and management.
- [00:06:56.593]And so that was really my main motivation for the data set
- [00:06:59.735]that I'm gonna use for this study.
- [00:07:02.737]So the data is on Daphnia, so the situation is microcosm
- [00:07:07.742]experiments of daphnia populations, so these daphnia
- [00:07:11.581]populations were raised in these acryllic containers,
- [00:07:15.111]and basically the populations were started off
- [00:07:17.969]at 10 individuals each, and they were fed a fixed amount
- [00:07:21.817]of food for about 10, or about 150 days, and they were
- [00:07:25.402]allowed to reach equilibrium.
- [00:07:27.543]And there were 60 of these populations, and half of them
- [00:07:30.425]were assigned to a treatment group and half were assigned
- [00:07:32.865]to a control group.
- [00:07:34.512]The control group received constant food throughout time,
- [00:07:37.325]and the treatment group received a reduction in the amount
- [00:07:40.971]of food, so every month the food was reduced by 25%
- [00:07:43.157]until they were driven to extinction.
- [00:07:46.379]I mean this seems like a very simplified setting,
- [00:07:48.730]because for the most part we consensus the population,
- [00:07:52.160]there's no detection error in there, you know?
- [00:07:54.593]We can count them almost perfectly, we don't really need
- [00:07:58.147]to even do any kind of statistical modeling to know
- [00:08:00.498]that the reduction in food drove the population
- [00:08:03.225]to extinction, right?
- [00:08:04.955]And in this specific case we also have replication, right?
- [00:08:08.647]And so we should start testing our tools that we have
- [00:08:12.781]on simple data sets like this where we kind of know
- [00:08:16.891]the answer from the get go, right?
- [00:08:18.841]So we know food should have a significant effect on
- [00:08:20.896]population abundance and hopefully we'd be able to
- [00:08:24.400]determine that from our modeling.
- [00:08:28.898]So this is just an example of a single population,
- [00:08:32.117]again there's 60 of them, and this the deteriorating
- [00:08:34.803]environment, so the population was started off at 10
- [00:08:37.485]individuals on day 1, and then we saw some transient
- [00:08:41.156]fluctuations, until about day 105, and then a day 150
- [00:08:47.936]the food reductions began, and the inset plot there in red
- [00:08:52.145]is the amount of food, so it's this geometric decline
- [00:08:55.232]in food levels, 25% each month until the population
- [00:08:58.885]went extinct.
- [00:09:00.497]So again we have 30 of these that were driven to extinction
- [00:09:02.933]and then 30 that were in the control environment.
- [00:09:05.570]All the populations were extant within the experiment
- [00:09:08.691]and the control environment, so, they basically showed
- [00:09:12.268]just random fluctuations over time.
- [00:09:18.141]So when I first attacked this problem, I kind of went at it
- [00:09:22.482]from a different perspective than what I'm going to
- [00:09:24.588]present it, but I wanna first start off with just standard
- [00:09:27.340]regression models, right?
- [00:09:29.197]So can we determine the effect of food using standard
- [00:09:31.258]regression models, and then we'll move into more mechanistic
- [00:09:34.230]models related to population growth.
- [00:09:37.944]So the most simple analysis that we might do is just a
- [00:09:41.458]simple linear regression model, this is the kind of analysis
- [00:09:43.794]we could do in Excel if we wanted.
- [00:09:46.441]An idea is that we could use the log transform abundance,
- [00:09:51.094]so that might be something we commonly do for ecological
- [00:09:55.178]datas, log transformed accounts, I mean we just have
- [00:09:58.610]an intercept in a slope and X is our food level at each
- [00:10:01.337]time T, and then we have some error term epsilon
- [00:10:04.647]that is normally distributed.
- [00:10:06.747]So those are some of the assumptions of the model,
- [00:10:09.098]and when we do that we get a regression coefficient
- [00:10:11.160]that is 2.33, right, so it's positive and it's statistically
- [00:10:16.519]significant.
- [00:10:18.239]I should mention that I did scale the food amount so that
- [00:10:20.682]a value of 1 indicated a full food level, so that was about
- [00:10:24.486]800 microliters of this freeze dried green algae each day,
- [00:10:29.208]and a value of zero means no food.
- [00:10:31.807]But this is exactly the kind of inference we'd expect
- [00:10:34.156]from a single time series from a single population,
- [00:10:36.636]we know food level is related to abundance, and we're
- [00:10:39.407]able to determine that from this model, right?
- [00:10:43.620]But you know, as with any statistical analysis, we would
- [00:10:47.555]check the assumptions of our model and that's what we're
- [00:10:49.655]going to do next.
- [00:10:50.977]So what I have on the two figures here, the blue points are
- [00:10:54.232]just the log transform abundance, and the line through it
- [00:10:59.397]is the regression line that best fit.
- [00:11:01.380]So the model possibly doesn't fit that bad, but when we
- [00:11:04.651]look at the residual plots, we definitely see some kind of
- [00:11:07.587]auto correlation in the residuals.
- [00:11:09.939]This is to be expected, because, you know, a population
- [00:11:12.587]definitely depends on the population size at the time
- [00:11:16.055]before it, so you know there's no reason we'd probably
- [00:11:19.422]start out with this model, but it is the simplest one
- [00:11:22.638]we could use.
- [00:11:24.869]So we look at these auto correlated residuals and maybe our
- [00:11:26.855]next guess is to use a model for auto correlation.
- [00:11:32.892]So what we have here is a linear regression model,
- [00:11:35.499]and it's called a linear model with correlated errors.
- [00:11:39.248]So this term epsilon here, we said it was normally
- [00:11:41.825]distributed before, but it had an independence assumption
- [00:11:45.256]associated with it and all we're doing is replacing that
- [00:11:47.926]with a correlation matrix here denoted by C, and that
- [00:11:51.763]correlation matrix might depend on some parameters fi here.
- [00:11:56.475]And that accounts for the fact that the residuals have
- [00:11:59.279]some inherent correlation among them.
- [00:12:02.844]When we do that we get a coefficient estimate of 0.36
- [00:12:07.300]and the 95% confidence interval no longer suggests that
- [00:12:13.034]it's statistically significant, right?
- [00:12:15.307]So this compares to the linear regression model without
- [00:12:18.153]correlated errors, you know the coefficient was much larger
- [00:12:21.669]and we got basically different inference here.
- [00:12:25.012]Alright, so, that really struck me as a bit odd,
- [00:12:30.999]because it's not really something I would expect to happen
- [00:12:32.813]in this type of analysis, I might expect the confidence
- [00:12:35.537]interval to be wider but I sure wouldn't expect the
- [00:12:37.683]coefficient to actually change.
- [00:12:41.291]If we think about using some kind of model scoring
- [00:12:46.494]technique, we can use like AIC, and when we fit the
- [00:12:50.625]correlated errors model, we get an AIC of 30.8,
- [00:12:54.800]and that compares to the linear regression model
- [00:12:56.775]of about 54.8.
- [00:12:58.379]So again if we were to judge which model we should use,
- [00:13:03.214]and we should be probably very careful here using AIC
- [00:13:06.186]for correlated error models.
- [00:13:08.621]But if we were to do that, we would choose the linear
- [00:13:12.594]regression model with correlated errors and our conclusions
- [00:13:15.765]would be that the food levels don't have a statistically
- [00:13:18.942]significant effect on population abundance which we know
- [00:13:22.581]is a priority not true.
- [00:13:26.068]So this really begs the question, what happened?
- [00:13:28.571]What went wrong here?
- [00:13:30.103]This is really the simplest analysis we could have done
- [00:13:32.003]with the data.
- [00:13:34.911]So let's take the linear regression model with the
- [00:13:37.688]correlated errors, same model as before, and we can
- [00:13:40.258]reparameterize it, motivated by the Karhunen-Loeve expansion
- [00:13:45.119]so basically what we do here is we observe that the epsilon
- [00:13:50.646]term is a realization of a Gaussian Process, right?
- [00:13:54.039]And we can decompose a finite realization of that Gaussian
- [00:13:57.754]process into an orthogonal basis factor, so, using ighem
- [00:14:01.472]basis factors and spectral basis coefficients,
- [00:14:05.805]and that's all that this expansion does here, it's just a
- [00:14:08.614]reparameterization of the correlated error model.
- [00:14:12.924]And the interesting thing to note is this matrix Z here,
- [00:14:17.139]so it's a matrix of Igen basis factors.
- [00:14:19.735]It kind of acts like co-variates in a sense.
- [00:14:23.525]Very similar to co-variates X.
- [00:14:28.909]And one thing we can do here is we can plot the coefficient
- [00:14:34.087]of determination, so an R squared value, versus the columns
- [00:14:39.247]of that matrix Z, so the igen vectors in the matrix Z,
- [00:14:42.261]and we see that the second igen vector has a correlation
- [00:14:46.974]with the covariate food of about .95.
- [00:14:51.560]So that's a very high level, and on the opposite of that
- [00:14:56.070]is the food level covariate, so again, ranging between
- [00:14:59.361]zero and one, it's showing a geometric decline,
- [00:15:02.140]and the second igen vector, so the second column
- [00:15:04.285]of that matrix Z.
- [00:15:06.685]So they have a negative relationship, but it's very highly
- [00:15:09.898]correlated, right?
- [00:15:11.279]So we know in regression models when you have a high level
- [00:15:14.091]of multi-collinearity that we could see coeffients change
- [00:15:18.268]sign, we could get different imprints, and that holds true
- [00:15:21.322]for the correlated errors model, in the sense that the
- [00:15:24.456]correlation between igen vectors and the covariates
- [00:15:27.584]is kind of hidden in that second order correlation matrix.
- [00:15:31.591]So that's a technical explanation of really what went wrong
- [00:15:34.735]in this situation.
- [00:15:39.373]So this is well known, I mean this is nothing new.
- [00:15:41.368]And this has been shown over the last 10 or probably 15
- [00:15:45.749]years in the statistics literature, particularly related to
- [00:15:50.308]disease mapping, so spatial statistics for diseases.
- [00:15:54.604]So Brian Rich first really gave a good technical explanation
- [00:15:58.161]of coefficients changing when you add spatially structured
- [00:16:03.237]covariates in a regression model.
- [00:16:08.618]In ecological literature, the effect has obviously been
- [00:16:11.910]noted, so, the title of this, you know, it says
- [00:16:14.799]incorporating spatial autocorrelation may invert
- [00:16:17.280]observed patterns, so when you account for this
- [00:16:20.175]autocorrelation you may see opposite sign coefficients
- [00:16:25.006]than you would in the standard model.
- [00:16:27.156]And that's exactly what Brian Rich was saying in his
- [00:16:30.591]analysis and giving the technical explanation of it.
- [00:16:35.049]In the journal Ecography, they basically went through
- [00:16:37.705]a large number of data sets, you know, all the way from
- [00:16:40.881]modeling the distribution of plants to rare species
- [00:16:44.317]and stuff like that and they showed that in a large number
- [00:16:47.002]of those data sets they actually observed this coefficient
- [00:16:49.728]shift, so the actual magnitude in some cases the sign
- [00:16:55.258]of the coefficients changing.
- [00:17:00.602]I think probably the breakthrough paper came from Jim Hodges
- [00:17:04.480]an American Statistician, and he basically lays it out
- [00:17:09.641]really well here in this case, looking at the same data
- [00:17:13.014]that Brian Rich did before, but he talks about adding
- [00:17:15.294]spatially correlated errors can essentially mess up the
- [00:17:17.898]fixed effects you love, so the fixed effects we love are
- [00:17:21.040]things like the food level covariate, right?
- [00:17:23.137]We wanna estimate the coefficient associated with that,
- [00:17:25.779]and when we start talking about incorporating the temporal
- [00:17:29.032]correlation in there or the spatial correlation or,
- [00:17:31.954]in the case of spatial temporal data, the spatial temporal
- [00:17:34.978]correlation, they may compete with the co-variates
- [00:17:39.377]that are in our model.
- [00:17:41.378]Again, this is kind of hidden in the assumptions we've made
- [00:17:44.352]within the data set.
- [00:17:49.238]And then finally John Fieberg has a really interesting paper
- [00:17:52.370]talking about biotylemmetry data, so linking, for example,
- [00:17:57.669]movement rates with physiological states of animals,
- [00:18:01.921]and notes a very similar thing occurring in there, so,
- [00:18:05.462]again this has been widely documented in different types
- [00:18:09.391]of data other than population time series, so again,
- [00:18:12.667]what I've shown here is really nothing new in the sense
- [00:18:16.751]that we knew about it 10 or 15 years ago, it just wasn't
- [00:18:19.897]clear why it was showing up in time series of populations
- [00:18:24.033]and what we might be able to do about it.
- [00:18:28.418]So this really brings me kind of the end of the regression
- [00:18:30.617]model section, and, in the paper I more or less left off
- [00:18:33.914]at this point because, I don't really think regression
- [00:18:37.074]models, they're not really based on any theory related
- [00:18:39.197]to population growth or decline, right?
- [00:18:41.773]And we have a lot of theory about that in ecology.
- [00:18:46.167]And maybe if we incorporate some of that theory into our
- [00:18:49.408]models that we might be able to alleviate
- [00:18:52.171]this issue compounding.
- [00:18:55.453]So let's start talking about models that explicitly
- [00:18:57.979]incorporate population dynamics and let's see what problems
- [00:19:00.459]or successes we run into in this area.
- [00:19:05.444]So population growth models, what we have here is the
- [00:19:08.784]population dynamics in a steady environment, so we have
- [00:19:11.383]the Ricker population growth model, but you could put
- [00:19:13.778]whatever model you want in here, it could be, you know,
- [00:19:18.676]ordinary differential equation, it could be Everton Holt,
- [00:19:21.957]whatever you like, we have some intrinsic growth rate R
- [00:19:24.601]which is thought to be the maximum achievable growth rate,
- [00:19:27.491]and some carrying capacity K or equilibrium population size,
- [00:19:31.619]so the long term average of the population would fluctuate
- [00:19:35.126]around, and then of course we have some stochastic
- [00:19:37.639]error term there that might be related to environmental
- [00:19:40.727]stochastity or demographic stochasticity.
- [00:19:48.049]So when we start talking about populations in
- [00:19:50.208]a deteriorating environment, we have to think of the
- [00:19:52.771]parameters R and K, so the intrinsic growth rate and the
- [00:19:56.072]carrying capacity being functions of the environment, right?
- [00:20:00.532]So the maximum achievable growth rate in some certain
- [00:20:04.086]environment, or the maximum achieve, well, the stable
- [00:20:08.503]equilibrium population size in some sort of environment.
- [00:20:11.764]So we're gonna make R and K functions of some covariate axis
- [00:20:14.126]covariate in our case would be the food levels.
- [00:20:17.185]And this was really, this work was motivated from
- [00:20:21.582]Peter Abrams who really started to explore this idea
- [00:20:24.927]of small population sizes will warn us
- [00:20:27.707]of impending extinctions.
- [00:20:29.772]And just to give you an idea of why we want to do this is
- [00:20:33.576]sometimes we'll see the intrinsic growth rate change faster
- [00:20:37.041]than, say, the carrying capacity, and we'll see very
- [00:20:39.473]different dynamics of decline.
- [00:20:41.590]So this is actually the breeding bird survey data from Ohio
- [00:20:44.945]for Bobwhite Quail, and you can see that the population
- [00:20:49.907]was rather steady until I think 1978, and then it just
- [00:20:53.870]absolutely collapsed.
- [00:20:56.593]So to explain those types of dynamics which are best
- [00:21:01.476]described by bifurcations or those types of mathematics,
- [00:21:05.807]we want models that can mimic that type of behavior,
- [00:21:09.681]and allowing first order difference equation to have
- [00:21:12.570]parameters that vary due to covariates that are changing
- [00:21:17.284]over time will allow for that type of dynamic.
- [00:21:21.712]So getting back to this idea of population dynamics
- [00:21:25.412]in deteriorating environment, this is really our main tool
- [00:21:29.509]for determining the cause of a population decline,
- [00:21:32.898]this is really the one that aligns with the theory out there
- [00:21:35.783]and we're gonna put some covariates in it and see how
- [00:21:39.203]it performs.
- [00:21:41.364]So we're gonna first start off by letting our intrinsic
- [00:21:44.550]growth rate in a certain environment depend on alpha
- [00:21:47.638]and the covariate, so alpha 1 is just a regression
- [00:21:51.229]coefficient, it's the influence of food levels on the
- [00:21:56.028]on the growth rate, and gamma 1 is very similar, it's
- [00:21:59.769]the influence of food levels on the equilibrium population
- [00:22:03.150]size, and we'll fit that model to the singularization
- [00:22:08.326]of the data, but first I want to reparameterize it,
- [00:22:11.989]and I'm gonna define this parameter theta as just the ratio
- [00:22:15.902]of alpha 1 divided by gamma 1, or at least the negative
- [00:22:19.834]of that, and when we do that we can rewrite the Ricker model
- [00:22:23.143]in a way that exposes the co-linearity, alright?
- [00:22:27.909]So the co-linearity between the X sub T, so that's the food
- [00:22:31.581]level at time T, and the log population, so N sub T.
- [00:22:36.019]And this is really the main insight I had about this problem
- [00:22:41.755]is that as a population declines, right?
- [00:22:45.077]When it's driven by a covariate that's also declining,
- [00:22:47.348]those two are going to be very highly correlated, right?
- [00:22:50.147]And being able tease those apart is going to be a challenge.
- [00:22:59.531]So we'll take this reparameterized Ricker model and this
- [00:23:03.019]is very familiar methods in non-linear regression,
- [00:23:07.700]is when we parameterize a model in such a way that it's
- [00:23:11.004]more computationally stable to fit, and when we do that
- [00:23:14.976]we get regression coefficient alpha, again this influence
- [00:23:18.395]is the intrinsic growth rate, it's positive, but the 95%
- [00:23:22.271]confidence interval includes zero.
- [00:23:25.243]So again, a full food level of 1 corresponds to a growth
- [00:23:28.467]rate of 0.4, but the confidence interval includes zero,
- [00:23:32.636]and we know that these populations that least to persist
- [00:23:35.371]have to have a positive growth rate.
- [00:23:38.517]On the parameter theta, this could be thought of as the
- [00:23:40.562]strength of the density dependence, so it's the reduction
- [00:23:42.939]of the growth rate as population size increases.
- [00:23:45.825]It should be negative which it's estimated as negative,
- [00:23:48.720]but again the 95% confidence interval includes zero.
- [00:23:52.240]We can reparameterize it and get confidence intervals
- [00:23:56.125]using the delta method, and we get an estimated carrying
- [00:23:59.480]capacity so that's that gamma 1 of about 37, so,
- [00:24:03.030]in one of those microcosm experiments, we'd expect there
- [00:24:06.974]to be on average about 37 daphnia through time.
- [00:24:10.976]Okay, and luckily in this case, the confidence interval
- [00:24:13.990]does exclude zero because it really wouldn't biologically
- [00:24:18.812]make sense to have a negative carrying capacity.
- [00:24:21.623]And this goes back to work of Blaine Griffon and John Drake
- [00:24:24.931]who showed within microcosm experiments that when we
- [00:24:28.396]manipulate habitat quality, so through the amount of food,
- [00:24:31.835]and habitat size, through the size of the microcosm,
- [00:24:34.398]that we'd expect changes in both R and K, right?
- [00:24:37.041]This might be a bit contrary to a lot of modeling approaches
- [00:24:41.425]that would only allow the carrying capacity the change
- [00:24:44.024]over time, we do expect to see changes in the intrinsic
- [00:24:46.421]growth rate.
- [00:24:49.731]And finally this model has an AIC C square of 24,
- [00:24:53.688]so far it's the best model we found for the data,
- [00:24:58.692]best fitting.
- [00:25:02.663]The thing that really was interesting here is the
- [00:25:04.597]correlation between X sub T and N sub T is 0.87,
- [00:25:10.239]so that's the coefficient of determination there,
- [00:25:12.213]so if we're familiar with standard reduction models,
- [00:25:14.446]this is a very high level of colinearity.
- [00:25:18.288]So much so that we really wouldn't expect to get reliable
- [00:25:21.882]results, and that's some of the issues we're seeing
- [00:25:25.204]with the confidence intervals here.
- [00:25:29.098]So when you think of more realistic population dynamics,
- [00:25:31.482]so, at a value of X equals zero we might see negative
- [00:25:35.270]growth rates or we might see negative carrying capacity
- [00:25:38.251]in some sense, so we might see a population that is
- [00:25:40.314]essentially doomed to extinction before the food levels
- [00:25:42.960]are completely depleted.
- [00:25:44.936]And what we can do here is just add in an intercept term
- [00:25:47.364]to our growth rate and our carrying capacities, so an alpha
- [00:25:50.372]not and a gamma not.
- [00:25:52.152]And this was the work by John Drake and Blaine Griffon
- [00:25:56.201]in their nature paper where they basically showed that
- [00:25:58.933]you could expect intrinsic growth rate probably to go to
- [00:26:02.107]zero before the carrying capacity does, so, again we'll
- [00:26:05.253]see this type of collapsing dynamics in certain populations.
- [00:26:09.246]So we can fit that model and what we get here is that
- [00:26:12.306]alpha 1, so that's the effect of food levels on intrinsic
- [00:26:16.107]growth rate, the estimate's 1.77, and the 95% confidence
- [00:26:20.150]interval excludes zero so we get a statistically significant
- [00:26:25.410]effect of food, which is exactly what we'd expect.
- [00:26:30.852]For the gamma parameter we get 34 as a point estimate
- [00:26:34.394]but we get a confidence interval that's suspiciously wide
- [00:26:36.751]from negative 190 all the way up to 65, and this is,
- [00:26:42.443]this is again probably due to, well, it is due to the
- [00:26:45.405]co-linearity, except it's difficult to express that in
- [00:26:48.055]this non-linear model.
- [00:26:50.728]And as far as how this model fits, it does have two more
- [00:26:53.906]parameters but it does have the lowest AICc score, so,
- [00:26:59.343]these type of dynamics typically are best supported
- [00:27:02.781]by the data.
- [00:27:07.360]So one question I think everyone probably has at this point
- [00:27:11.104]is when you've limited yourself to a single time series
- [00:27:14.207]from an experiment that has 60 populations, right?
- [00:27:17.390]And so how well does this generalize across populations?
- [00:27:20.584]So what I have here is plots of the 95% confidence intervals
- [00:27:25.390]that's the vertical lines, and the point estimates,
- [00:27:29.467]that's the black dots here, for alpha 1 which is related
- [00:27:32.863]to the effect of food level and intrinsic growth rate,
- [00:27:35.269]and gamma 1 which is related to the effect of food level
- [00:27:38.493]on the equilibrium population size.
- [00:27:42.502]And I think the figure showing the estimates of alpha
- [00:27:45.801]is pretty telling in that, again, given the prior experiment
- [00:27:53.763]by Blake and Griffon, we would expect that the growth rate
- [00:27:59.142]would be positive, right?
- [00:28:01.093]And presumably if we had enough data, statistically
- [00:28:03.607]significant.
- [00:28:05.407]That only happens for two of the 30 populations
- [00:28:07.881]in the treatment, right?
- [00:28:10.648]There's in fact more growth rates that are estimated
- [00:28:12.972]to be negative here.
- [00:28:16.977]And so I think there's four of those.
- [00:28:20.156]For the most part most of the 95% confidence intervals
- [00:28:22.434]include zero, so if we were to conclude the effect of
- [00:28:24.818]food levels on the intrinsic growth rate, we really lack
- [00:28:28.747]the statistical power to be able to determine that.
- [00:28:32.735]Again, that's an interesting thought for declining
- [00:28:35.659]populations because we're trying to collect more and more
- [00:28:38.476]data as they're racing toward extinction, so,
- [00:28:41.248]in many cases we can't just simply go out and collect
- [00:28:44.723]more data, and that co-linearity really adds to the problem
- [00:28:47.951]because given a statistical model, we're gonna need more
- [00:28:51.580]and more data to make the same conclusions.
- [00:28:55.141]As far as the carrying capacity goes, all the estimates
- [00:28:57.457]are positive, most of them have confidence intervals
- [00:29:00.063]that exclude zero, so we're able to estimate the influence
- [00:29:04.278]of environmental deterioration or decreasing food levels
- [00:29:09.346]on the equilibrium population size, but the intrinsic growth
- [00:29:13.939]rate is really difficult.
- [00:29:15.632]And again the intrinsic growth rate is what would lead
- [00:29:17.405]to population collapses.
- [00:29:19.097]So it's an important quantity to measure.
- [00:29:24.094]So I want to talk about a few solutions, because with a
- [00:29:28.064]single population with a single time series, it doesn't
- [00:29:31.852]really look very good, right?
- [00:29:33.965]For the most part we're not gonna be able to determine
- [00:29:36.241]the cause of a population decline.
- [00:29:38.269]So what are some solutions out there?
- [00:29:41.560]For the experimental data we're talking about here,
- [00:29:43.262]we have replication, and I'll talk about replication
- [00:29:46.409]in the real world as it relates to wild populations,
- [00:29:48.979]but I want to talk about replication in this designed
- [00:29:51.037]experiment, we also have the experimental design,
- [00:29:53.440]things we could have changed before we designed the
- [00:29:56.250]experiment and collected the data that might have allowed us
- [00:29:58.775]to be able to make that determination.
- [00:30:01.835]So when I combine, through a combined analysis of about
- [00:30:05.148]60 populations, so this is 30 in the treatment,
- [00:30:07.326]30 in the control, I fit the same model and I fit many
- [00:30:10.813]different types of models, this one actually came out as
- [00:30:15.068]describing the data the best, but this is just the Ricker
- [00:30:17.760]model with the intrinsic growth rate and the carrying
- [00:30:21.557]capacity changing as a function of food level.
- [00:30:24.707]We get estimates of alpha 1 that are positive and the 95%
- [00:30:28.163]confidence interval excludes zero, so exactly what we
- [00:30:31.009]would expect given previous experimental results,
- [00:30:34.596]and we get an estimated gamma parameter of 64, again,
- [00:30:39.048]that's statistically significant, so, we conclude here
- [00:30:42.623]at the end of the day that food levels do in fact influence
- [00:30:45.397]intrinsic growth rate and equilibrium population size,
- [00:30:49.610]but this is given a large amount of data and replication.
- [00:30:54.406]Furthermore, you can take every model I've talked about,
- [00:30:57.414]and we get reliable results, regardless of what model
- [00:31:00.951]is used, statistical inference linking through to
- [00:31:04.967]population decline is invariant to the method that we choose
- [00:31:09.184]and this makes sense, I mean, if we have strong data,
- [00:31:12.029]and we have lots of it, and it's collected in a responsible
- [00:31:15.541]manner, we should be able to have conclusions that
- [00:31:19.265]are invariant to model selection.
- [00:31:23.282]Talking about the experimental design, now let's go back
- [00:31:25.554]to a single time series from a single population,
- [00:31:28.071]what could we have done differently to be able to detect
- [00:31:30.421]the change?
- [00:31:31.862]So what I have here is four simulated data sets.
- [00:31:34.878]The simulated data set in panel A is very similar to the
- [00:31:38.370]data set that was collected, we see this geometric decline
- [00:31:41.713]in food level which is represented by the inset plot
- [00:31:44.680]that is red.
- [00:31:46.459]And then I added various levels of random noise to that
- [00:31:48.734]covariate so typically in the natural world, covariates
- [00:31:52.696]don't change in a geometric fashion, they have some
- [00:31:55.915]random variation in them, and it increased the random
- [00:31:58.193]variation through plot C and D.
- [00:32:01.830]And what I'll show on the next plot is the coefficient
- [00:32:06.408]estimates that we obtained for the intrinsic growth rate
- [00:32:08.964]because that was the most troublesome parameter.
- [00:32:11.236]And there will be functions of the correlations between
- [00:32:13.456]X sub T, the food levels, and N sub T.
- [00:32:17.099]So this shows our coefficient estimates, and it appears,
- [00:32:21.227]you know, we get positive estimates when we have low levels
- [00:32:23.540]of correlation between a covariate and population size
- [00:32:28.960]and low levels are associated with the large amount of
- [00:32:31.271]random variation, so effective we had each time step
- [00:32:34.509]we need to apply a random treatment, the food levels
- [00:32:37.878]to our population to be able to tease apart the effect
- [00:32:41.920]of food level on these parameters.
- [00:32:45.475]When we start seeing levels of about .6 or higher,
- [00:32:48.566]then we really run into this issue of compounding.
- [00:32:51.657]Again, this holds against the correlated errors regression
- [00:32:54.345]model when we start seeing high levels of correlation
- [00:32:58.147]in X sub T and N sub T we start seeing the igem vectors
- [00:33:00.701]being correlated with the covariate, so, it's this type
- [00:33:03.515]of endogenous process where we have density dependence
- [00:33:05.697]acting on the population which depends on food levels,
- [00:33:09.124]but we also have food levels in our population model
- [00:33:12.349]and are competing to explain what's going on.
- [00:33:17.018]So I wanna end with several slides on future direction.
- [00:33:22.731]I don't really think that this problem has a great solution
- [00:33:26.549]that I've presented up to this point, you know,
- [00:33:29.856]we don't really have replicated populations in the real
- [00:33:31.928]world and we certainly don't have controls, but for the
- [00:33:34.712]most part we're talking about observational data so we
- [00:33:37.116]can't go back and say, you know, would you please change
- [00:33:39.936]the land cover in Nemaha County in this year, right?
- [00:33:44.436]So we gotta deal with this in an observational
- [00:33:48.935]study setting.
- [00:33:50.426]So with future direction, I wanna talk about, you know,
- [00:33:52.604]this idea about wild populations, right?
- [00:33:54.461]So, one of the cool things John Drake did in his study
- [00:33:57.364]is he had triplicate counts of those populations,
- [00:33:59.879]and they're basically a census, so if the triplicate counts
- [00:34:04.889]didn't agree he sent the undergrads back to recount them
- [00:34:07.291]until they agreed.
- [00:34:08.935]And so it's about as good as you can get and when we're
- [00:34:12.489]talking about wild populations we certainly can't go out
- [00:34:15.858]there and count every individual.
- [00:34:18.966]And then replication in the real world, I have some ideas
- [00:34:21.000]here, they're not really anything I've started to work on,
- [00:34:25.067]they're more ideas of what I think would be fruit forward.
- [00:34:28.089]And then I want to talk about regularization and relate that
- [00:34:31.519]to priors for Bayesian analyses.
- [00:34:37.069]So let's go back to the Pronghorn Antelope in the northern
- [00:34:42.541]part of North Dakota, Northwest part.
- [00:34:45.870]So when they actually go out and count these antelope,
- [00:34:48.550]this is a very big area and they fly aerial surveys
- [00:34:50.779]and there's certainly detection issues, so that's basically
- [00:34:54.292]to say what you see on this graph is the observed number
- [00:34:56.719]of pronghorn antelope and probably there's more out there,
- [00:35:00.151]so we have this issue with detection error.
- [00:35:04.296]And you know wildlife biologists have numerous tools
- [00:35:07.591]that they can use to account for detection error,
- [00:35:11.189]so one idea here is that we need to formulate models
- [00:35:16.720]that actually estimate the true population size so that
- [00:35:20.151]we can in fact even calculate the correlation to detect
- [00:35:22.752]this type of confounding, because if we go back and use
- [00:35:25.773]the observed population sizes, then you'd be actually
- [00:35:30.982]less correlated with the covariate due to the random
- [00:35:33.740]error associated with detection issues.
- [00:35:38.607]So when we formulate state space models or hierarchical
- [00:35:42.281]models, we might think of using, for example, the inmixture
- [00:35:45.218]approach that Andy Royal developed.
- [00:35:48.021]So we'll have N sub T is from a Poisson distribution
- [00:35:51.400]and we'll put our Ricker population growth model within
- [00:35:53.473]there, but then Y sub T is actually our observed
- [00:35:57.235]population size, so that's our population index.
- [00:36:00.242]And so we're left with estimating N sub T.
- [00:36:03.040]So in a typical Bayesian analysis we could use N sub T
- [00:36:07.007]and calculate the correlation between our covariates
- [00:36:10.544]as a direct quantity, and I think that's a useful diagnostic
- [00:36:13.648]so we at least can detect and know if it's an issue.
- [00:36:17.870]It also turns out when it's an issue you'll typically
- [00:36:20.379]have a very difficult time getting your NCMC chains to mix
- [00:36:24.887]and many diagnostics will show that something is wrong
- [00:36:29.103]or you have an identifiability issue for parameters.
- [00:36:33.244]So that's one idea at least for detecting it, now actually
- [00:36:36.688]talking about alleviating the effect, we can think about
- [00:36:41.531]replication, right?
- [00:36:43.055]And so what I have here is the 30 populations from day 105
- [00:36:48.100]to the end of the experiment which is on day 416,
- [00:36:52.021]and those are in the control environment, so these are
- [00:36:54.573]the ones that received the full amount of food throughout
- [00:36:56.978]the entire period.
- [00:36:59.121]And the interesting thing here is if you calculate
- [00:37:01.609]all paralaise correlations between this time series, right?
- [00:37:05.116]So I think there's a, I wanna say there's 432, I think
- [00:37:09.697]in this case, what we see here is we expect to see
- [00:37:13.709]a histogram centered around zero, with some sort of spread,
- [00:37:20.308]but the thing we see is the histogram actually shifted to
- [00:37:23.206]the positive direction, so what we're saying is that in fact
- [00:37:26.800]in this experiment where we have truly replicated
- [00:37:30.709]populations that are not interacting because they're
- [00:37:33.024]in different containers, they're still correlated,
- [00:37:35.963]so they're not really true replicates, right?
- [00:37:41.188]So the reason they're not true replicates is because
- [00:37:43.478]that error term, the epsilon sub T, for example in the
- [00:37:46.995]Ricker model describes environmental stochasticity
- [00:37:50.798]and demographic stochasticity, and environmental
- [00:37:54.809]stochasticity may be correlated, so when it happens to be
- [00:37:58.284]just slightly warm in the lab that day or in the microcosm
- [00:38:00.588]chambers, it's slightly warm for everyone, so there's
- [00:38:03.938]still some correlation going on.
- [00:38:06.418]So I would say to get truly replicate populations
- [00:38:09.556]you'd probably have to have a population in different labs
- [00:38:12.531]all around the world.
- [00:38:14.023]So the idea of true replication doesn't even exist
- [00:38:16.700]even in this model system.
- [00:38:20.654]And so that should be encouraging from the sense of
- [00:38:22.468]well we don't really need true replicate populations
- [00:38:24.733]to determine the decline or the cause of a decline.
- [00:38:29.447]And so that leaves us with modeling the correlation
- [00:38:32.052]among populations and achieving replication in that manner.
- [00:38:35.612]And so one idea that I've had is to start thinking of
- [00:38:39.045]population dynamics spatially explicit in continuous space.
- [00:38:43.142]So typically we have a difficult time defining populations
- [00:38:46.399]in wildlife ecology because, you know, for example
- [00:38:50.093]I go Nemeha County and I go count the number of
- [00:38:52.406]Bobwhite Quail in one spot, and I move 100 yards down
- [00:38:55.700]the road and I count them again, is that a replicated
- [00:38:58.883]population?
- [00:39:00.612]What if I move 10 miles, is that a replication?
- [00:39:04.034]Well, we really have a difficult time thinking about
- [00:39:06.554]replication in that case, and we ought to start thinking
- [00:39:09.163]about continuous space, or spatially explicit models
- [00:39:12.382]that deal with continuous space.
- [00:39:15.271]And so some of the ideas I've had were to start using
- [00:39:21.352]partial differential equations, so what I have here is
- [00:39:23.723]a reaction diffusion equation, and one portion of that
- [00:39:27.161]equation describes ecological diffusion, so it's a very
- [00:39:31.135]rough diffusion process, so that's the spread or the
- [00:39:34.763]movement of organisms across space and that's the continuous
- [00:39:38.385]spatial portion of it, and then on the other side we have a
- [00:39:42.853]reaction part of the equation which deals with the
- [00:39:46.489]population growth, so we see R and K showing up over there,
- [00:39:49.953]those can be thought of as intrinsic growth rate
- [00:39:52.356]and equilibrium population sizes, although those do not
- [00:39:56.405]necessarily have good analogues in continuous space.
- [00:39:59.927]And we can link this to count data just like we have
- [00:40:03.566]with discrete space models, right?
- [00:40:06.903]So this allows us to model the dependence in a non-separable
- [00:40:09.620]way between time series at different locations.
- [00:40:14.454]And we can link that using a point process model.
- [00:40:18.656]So that's one idea of replication there.
- [00:40:21.431]Some of the other ideas I've had is just simply modeling
- [00:40:23.622]the correlation of the epsilon terms in the Ricker models,
- [00:40:27.385]but I think this type of spatio-temporal model would be
- [00:40:31.719]useful, and again, for things like the breeding birds
- [00:40:33.769]survey data we have replication essentially at multiple
- [00:40:38.891]routes across large areas, so combining those together
- [00:40:42.153]I think will be very useful, but again, this isn't anything
- [00:40:45.658]I've been working on.
- [00:40:49.644]And then I also wanna talk here about regularization.
- [00:40:55.255]So regularization is really a popular tool in statistics
- [00:40:57.606]and machine learning that's really used, I would say
- [00:41:00.444]for model selection, so, if you've heard of the Lasso
- [00:41:05.192]or ridge regression, those are two common methods,
- [00:41:08.731]and it's also used to constrain highly co-linear covariates
- [00:41:13.176]so ridge regression invented back in the 70s basically
- [00:41:17.026]to deal with matrices that you couldn't invert due to
- [00:41:20.694]high levels of co-linearity.
- [00:41:22.886]So maybe those ideas can be applied in this specific
- [00:41:26.317]situation, and that's something I think will be very useful.
- [00:41:31.205]But we should also start thinking about a connection between
- [00:41:35.334]these frequentist tools with Bayesian methods, so Bayesian
- [00:41:39.336]regularization is actually a result of using an informative
- [00:41:43.419]prior, so when we use an informative prior, we get
- [00:41:46.999]regularization, but regularization is something we would
- [00:41:50.454]want to use in many situations when we have high amounts
- [00:41:54.135]of co-linearity.
- [00:41:56.947]So I think this makes a fairly good case for thinking about
- [00:42:02.573]using priors when we're talking about population decline,
- [00:42:05.794]and linking it to, you know, standard methods that we've
- [00:42:09.645]used and Mevanhouten and Tom Hobbs have a really good
- [00:42:12.742]monograph on that, linking those two types of approaches,
- [00:42:17.774]regularization with Bayesian priors and giving good
- [00:42:21.408]interpretations of that.
- [00:42:25.374]But I think it also brings up the case that maybe we
- [00:42:27.567]ought to just start using informative priors in the case
- [00:42:29.957]that we have a single time series from a single population.
- [00:42:33.844]So I keep going back and thinking about like whooping cranes
- [00:42:36.613]for example, there is no way we're ever going to get
- [00:42:39.013]replication there, there's one single population,
- [00:42:42.083]that's all we have.
- [00:42:44.517]So in those cases, can we use prior knowledge about the
- [00:42:47.620]parameters of interest, for example, R and K.
- [00:42:50.799]And if we do use those, we'll get automatic regularization.
- [00:42:55.421]But more guidance is needed, for example, R is the
- [00:42:58.594]intrinsic growth rate in a certain environment but it
- [00:43:00.774]certianly has an upper bound, right?
- [00:43:02.748]There can't be an infinite number of whooping cranes
- [00:43:04.763]produced by a pair each year.
- [00:43:07.234]So there are some upper bounds and this was explored in
- [00:43:09.679]this paper on ecology and evolution, talking about
- [00:43:13.112]intelligent ways to come up with informative priors
- [00:43:17.357]for our population growth models.
- [00:43:19.583]And I really think that's probably the way you're going
- [00:43:22.546]to solve this for a single time series for a single
- [00:43:25.434]population, when we have multiple time series
- [00:43:28.203]in replication I think we should harness those and
- [00:43:31.561]in a type of spacio-temporal analysis, and that's really
- [00:43:38.077]all I have for today, so I'll take any questions,
- [00:43:42.064]and thanks everyone.
- [00:43:43.966](audience applauds)
- [00:43:52.169]Well?
- [00:43:53.002](laughs)
- [00:43:54.843]Okay, this may be a dumb question.
- [00:44:00.663]Back in the earlier models that you were looking at,
- [00:44:05.097]the stochastic term that you had was basically an error
- [00:44:10.120]term at the end of the nonlinear model.
- [00:44:13.368]Have you tried anything like where you've done random
- [00:44:15.550]coefficient rather than putting that random variability
- [00:44:17.652]at the end of the model as a plus E term, putting it
- [00:44:22.752]on the coefficients and modeling the variation,
- [00:44:25.654]you know when you do have replication modeling it there
- [00:44:28.422]rather than at the end of the day?
- [00:44:31.198]Yeah, so the question actually goes back to, if I
- [00:44:36.196]understand it right, let's talk about this model right here,
- [00:44:39.464]so we have R and we have K and they're functions
- [00:44:42.635]of X, right? Right.
- [00:44:44.155]And we could also make those random coeficients
- [00:44:46.962]because each year R may be a little different or X
- [00:44:49.810]may be a little different, and my initial thought on that
- [00:44:53.455]was to use a Gaussian process to model those semi
- [00:44:56.755]parametrically, but what we see is we see that essentially
- [00:45:00.862]we have two semi-parametric terms that are competing
- [00:45:04.233]to explain very similar components.
- [00:45:08.356]And so that led me to, in generalized additive models
- [00:45:11.903]they call it concurvity, but it's basically you have
- [00:45:14.840]two smooth trends, or smooth effects that are competing
- [00:45:19.883]and that's where I wasn't able to make any progress
- [00:45:22.732]at that point, but you're right, thinking of R and K
- [00:45:26.394]as random effects is probably the way to go because
- [00:45:28.914]we don't really know what relationship they should have
- [00:45:31.593]with the covariate.
- [00:45:36.500]Anyone else?
- [00:45:45.650]I just have a methodological question
- [00:45:47.787]about when you were feeding your populations of daphnia,
- [00:45:50.711]did you give them like the same percentage of body weight
- [00:45:56.587]per day, per individual, so did you take into account both
- [00:45:59.428]growth, like somatic growth of individuals as well as
- [00:46:03.393]population growth?
- [00:46:05.172]So that's a great question, I'll say that I never
- [00:46:08.345]collect my own data, so this was actually done in a study,
- [00:46:13.349]the Drake and Griffon study in the Nature paper in 2010,
- [00:46:18.821]and they did not, there's no relationship, but,
- [00:46:21.847]I do see what you're saying there, and that would be a good
- [00:46:25.170]avenue to explore, but they did not.
- [00:46:33.480]Anything else?
- [00:46:37.070]Any last comments, Trevor? Nope.
- [00:46:39.788]Alright thank you everyone for coming.
- [00:46:42.868](applause)
- [00:46:46.133]And just to remind you this was recorded so if you know
- [00:46:47.952]anyone who would benefit from this seminar, please feel
- [00:46:50.217]free to share, it's on the SNR website.
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