Hierarchical models with RStan (Part 1)

Real-world data sometime show complex structure that call for the use of special models. When data are organized in more than one level, hierarchical models are the most relevant tool for data analysis. One classic example is when you record student performance from different schools, you might decide to record student-level variables (age, ethnicity, social background) as well as school-level variable (number of student, budget). In this post I will show how to fit such models using RStan. As there is much to say and try on such models I restrict myself in this post to a rather simple example, I will extend this to more complex situations in latter posts.

A few words about RStan:

If you don’t know anything about STAN and RStan make sure to check out this webpage. In a few words RStan is an R interface to the STAN programming language that let’s you fit Bayesian models. A classical workflow looks like this:

  1. Write a STAN model file ending with a .stan
  2. In R fit the model using the RStan package passing the model file and the data to the stan function
  3. Check model fit, a great way to do it is to use the shinystan package

First example with simulated data:

Say that we recorded the response of 10 different plant species to rising temperature and nitrogen concentration. We measured the biomass of 10 individuals per species along a gradient of both temperature and nitrogen concentration and we would like to know how these two variables affect plant biomass. In hierarchical model we let regression parameters vary between the species, this means that, for example, species A might have a more positive slope between temperature and biomass than species B. Note however that we do not fit separate regression to each species, rather the regression parameters for the different species are themselves being fitted to a statistical distribution.

In mathematical terms this example can be written:

\mu_{ij} = \beta_{0j} + \beta_{1j} * Temperature_{ij} + \beta_{2j} * Nitrogen_{ij}

\beta_{0j} \sim N(\gamma_{0},\tau_{0})

… (same for all regression coefficients) …

y_{ij} \sim N(\mu_{ij},\sigma)

For observations i: 1 … N and species j: 1 … J.

This is how such a model looks like in STAN:

/*A simple example of an hierarchical model*/
data {
  int<lower=1> N; //the number of observations
  int<lower=1> J; //the number of groups
  int<lower=1> K; //number of columns in the model matrix
  int<lower=1,upper=J> id[N]; //vector of group indeces
  matrix[N,K] X; //the model matrix
  vector[N] y; //the response variable
parameters {
  vector[K] gamma; //population-level regression coefficients
  vector<lower=0>[K] tau; //the standard deviation of the regression coefficients

  vector[K] beta[J]; //matrix of group-level regression coefficients
  real<lower=0> sigma; //standard deviation of the individual observations
model {
  vector[N] mu; //linear predictor
  gamma ~ normal(0,5); //weakly informative priors on the regression coefficients
  tau ~ cauchy(0,2.5); //weakly informative priors, see section 6.9 in STAN user guide
  sigma ~ gamma(2,0.1); //weakly informative priors, see section 6.9 in STAN user guide
  for(j in 1:J){
   beta[j] ~ normal(gamma,tau); //fill the matrix of group-level regression coefficients 
  for(n in 1:N){
    mu[n] = X[n] * beta[id[n]]; //compute the linear predictor using relevant group-level regression coefficients 

  y ~ normal(mu,sigma);

You can copy/paste the code into an empty text editor and save it under a .stan file.

Now we turn into R:

#load libraries
#where the STAN model is saved
#simulate some data
N<-100 #sample size
J<-10 #number of plant species
id<-rep(1:J,each=10) #index of plant species
K<-3 #number of regression coefficients
#population-level regression coefficient
#standard deviation of the group-level coefficient
#standard deviation of individual observations
#group-level regression coefficients
beta<-mapply(function(g,t) rnorm(J,g,t),g=gamma,t=tau) 
#the model matrix
y<-vector(length = N)
for(n in 1:N){
  #simulate response data
#run the model

The MCMC sampling takes place (took about 90 sec per chain on my computer), and then I got this warning message: “Warning messages:
1: There were 61 divergent transitions after warmup. Increasing adapt_delta above 0.8 may help. See
2: Examine the pairs() plot to diagnose sampling problems”

Here is an explanation for this warning: “For some intuition, imagine walking down a steep mountain. If you take too big of a step you will fall, but if you can take very tiny steps you might be able to make your way down the mountain, albeit very slowly. Similarly, we can tell Stan to take smaller steps around the posterior distribution, which (in some but not all cases) can help avoid these divergences.” from here. This issue occur quite often in hierarchical model with limited sample size, the simplest solution being to re-parameterize the model, in other words to re-write the equations so that the MCMC sampler has an easier time sampling the posterior distribution.

Below is a new STAN model with a non-centered parameterization (See Section 22.6 in STAN user guide):

parameters {
  vector[K] gamma; //population-level regression coefficients
  vector<lower=0>[K] tau; //the standard deviation of the regression coefficients
  //implementing Matt's trick
  vector[K] beta_raw[J];
  real<lower=0> sigma; //standard deviation of the individual observations
transformed parameters {
  vector[K] beta[J]; //matrix of group-level regression coefficients
  //computing the group-level coefficient, based on non-centered parametrization based on section 22.6 STAN (v2.12) user's guide
  for(j in 1:J){
    beta[j] = gamma + tau .* beta_raw[j];
model {
  vector[N] mu; //linear predictor
  gamma ~ normal(0,5); //weakly informative priors on the regression coefficients
  tau ~ cauchy(0,2.5); //weakly informative priors, see section 6.9 in STAN user guide
  sigma ~ gamma(2,0.1); //weakly informative priors, see section 6.9 in STAN user guide
  for(j in 1:J){
   beta_raw[j] ~ normal(0,1); //fill the matrix of group-level regression coefficients 
  for(n in 1:N){
    mu[n] = X[n] * beta[id[n]]; //compute the linear predictor using relevant group-level regression coefficients 
  y ~ normal(mu,sigma);

Note that the data model block is identical in the two cases.

We turn back to R:

#re-parametrize the model
#no more divergent iterations, we can start exploring the model
#a great way to start is to use the shinystan library
#model inference
Inference for Stan model: hierarchical1_reparam.
4 chains, each with iter=2000; warmup=1000; thin=1; 
post-warmup draws per chain=1000, total post-warmup draws=4000.

          mean se_mean   sd  2.5%   25%   50%  75% 97.5% n_eff Rhat
gamma[1]  1.96    0.00 0.17  1.61  1.86  1.96 2.07  2.29  2075    1
gamma[2] -0.03    0.02 0.77 -1.53 -0.49 -0.04 0.43  1.55  1047    1
gamma[3]  2.81    0.02 0.49  1.84  2.52  2.80 3.12  3.79   926    1
tau[1]    0.34    0.01 0.21  0.02  0.19  0.33 0.46  0.79  1135    1
tau[2]    2.39    0.02 0.66  1.47  1.94  2.26 2.69  4.04  1234    1
tau[3]    1.44    0.01 0.41  0.87  1.16  1.37 1.65  2.43  1317    1
sigma     1.04    0.00 0.09  0.89  0.98  1.04 1.10  1.23  2392    1

Samples were drawn using NUTS(diag_e) at Thu Nov 10 14:11:41 2016.
For each parameter, n_eff is a crude measure of effective sample size,
and Rhat is the potential scale reduction factor on split chains (at 
convergence, Rhat=1).

The regression parameters were all decently estimated except for the second slope coefficient, the simulated value was -1.

All MCMC samples for all coefficient can be easily extracted and used to compute whatever your interest is:

#extract the MCMC samples

#plot average response to explanatory variables
#get predicted values for each MCMC sample
pred_x1<-apply(mcmc_hier$gamma,1,function(beta) X_new %*% beta)
#now get median and 95% credible intervals
#same stuff for the second explanatory variables
pred_x2<-apply(mcmc_hier$gamma,1,function(beta) X_new %*% beta)

Here we basically generated new model matrices where only one variable was moving at a time, this allowed us to get the model prediction for the effect of say temperature on plant biomass under average nutrient conditions. These predictions were obtained by multiplying the model matrix with the coefficients for each MCMC sample (the first apply command), and then we can get from these samples the median with 95% credible intervals (the second apply command).

Now we can plot this (code for the plots at the end of the post)


Another important plot is the variation in the regression parameters between the species, again this is easily done using the MCMC samples:

#now we could look at the variation in the regression coefficients between the groups doing caterpillar plots
#we may also add the population-level median estimate
  labs(y="Regression parameters")


The cool thing with using STAN is that we can extend or modify the model in many ways. This will be the topics of future posts which will include: crossed and nested design, multilevel modelling, non-normal distributions and much more, stay tuned!

Code for the first plot:

plot(y~X[,2],pch=16,xlab="Temperature",ylab="Response variable",col=cols[id])
plot(y~X[,3],pch=16,xlab="Nitrogen concentration",ylab="Response variable",col=cols[id])
mtext(text = "Population-level response to the two\nexplanatory variables with 95% CrI",side = 3,line = 0,outer=TRUE)
legend(x=2.1,y=10,legend=paste("Gr",1:10),ncol = 1,col=cols,pch=16,bty="n",xpd=NA,title = "Group\nID")


150 years of ecology: lessons for the future

Last week I was at the annual meeting of the german, austrian and swiss ecologists (GfÖ) which took place in Marburg (DE). The title of the conference referred to the creation of the word Oecologie in 1866 by Ernst Haeckel and the conference aimed at “reflect upon the progress we have made in the past and […] to identify the many unsolved scientific questions that still confront us.” Volkmar Wölters.

How I found the conference:

  • Marburg is a cool town, the “Altstadt” is super-pretty with loads of cosy bars to extend interactions on scientific and non-scientific topics well into the night (and morning for the bravest).
  • As always for a GfÖ conference, the participants were mostly PhDs and PostDocs from Germany, little diversity in terms of disciplines most of the people being hardcore ecologists.
  • Weird repartition of talks within the session, parallel to my talk on the links between plant diversity and herbivory/predation through consumer community shifts was a talk (in another session) on plant diversity increase food quantity and quality with positive effect on social bees. While two talks later in my session we heard about plant richness negative effect on root decomposition …
  • Great social events, the ice breaker was supported by awesome food, the BBQ night revealed the great musical talent of prominent German ecologists (including GfÖ president) and also fantastic dynamism when it comes to dancing on hits from the 80ies.
  • If I got one lesson for the future it is: be a generalist, develop skills and interact with associated disciplines like remote sensing (as Nathalie Pettorelli) or social science to bring new ideas, new concepts and new data into ecology.


Overview of the week:

(All these are based on my notes and my vague, selective memories of the talks, it is certainly not an unbiased vision of the conference)

Day 1:

I held my talk on the first day almost right after the keynote speech of Susanne Fritz so my notes on her talk are rather sparse. She argued that ecologists should embrace the historical dimension of their dataset, for example selective extinction (big species going extinct first) might explain some present-day trait distribution. After the keynote speech I held my talk despite technical failures (spent 15min shouting my slides in a too large lecture hall), I received little feedback from it so I guess that what I am doing is OK. Felix Fornoff presented some nice results on trap-nesting bees, he found that tree diversity effect on these insects are explained by changes in canopy area. As the canopy grows the climatic conditions under the trees become more stable (less wind, less temperature extreme) which positively affect abundance and richness of trap-nesting bees. After the coffee break I had the pleasure to hear an inspiring talk by Leana Zoller on the effect of artificial light on night pollination. In her experiment she put some LED street light into areas not affected by light pollution and recorded its effect on Ciriseum oleacerum fitness, she found strong effect of light pollution in this naive flora and fauna, seed mass of C. oleacerum declined by 20% in light polluted compared to control plot.

Day 2:

A nice talk by Per Schleuss caught my attention on the second day just before lunch, he looked at the degradation Kobresia grasslands in Tibet due to desiccation (climate change) and overgrazing in the area. He found that 70% of the soil organic carbon trapped by the system was either being eroded leading to important river pollution or being mineralized a process which release carbon dioxide in the atmosphere. Christian Wirth was the next keynote speaker with a provocative title: “After the hype: a reality check for trait-based functional biodiversity research”. The talk nicely outlined the different ways to quantify trait variation (mean, dispersion, range) each having different implications for ecosystem functions. He also showed that ecosystem functions are not determined by one unique “super-trait” but that on average five traits are necessary per ecosystem function. He went on talking about intraspecific trait variation saying that “if there are too many individuals varying in their traits to hope to put together a community piece by peice and to predict ecosystem function, then forget about plant traits”. A sobering statement as more and more data accumulate showing large within-species variation in trait values. At the end of the day Sanne Van Den Berge talked biodiversity of hedgerows and tree line in Belgium, in the area of study such structure represent 0.7% of the land cover but host 45% of the floral diversity. Unfortunately these structure are endangered by human pressures especially the older elements which also have the highest diversity. She also reported that local diversity increased by, on average, 4 species between 1974 and 2015, something to add (maybe) to the current debate on biodiversity trends.

Day 3:

I spent the morning sitting in the movement ecology session hearing some nice talks, in particular one by Wiebke Ullmann on hare movement in agricultural landscape and how energy expenditure was affected by home range size which was affected by farm size. As well as one by Lisa Fisler on hoverflies migration through the Swiss Alps showing impressive videos and data that these marvellous tiny creature actively fly against the winds through mountain passes up to 1900m! After lunch I went to a cool session one curious natural history observation or as one of the speaker putted it: “[he is] completely fed up with hypothesis-driven science”. Mark-Olivier Rödel presented great insights on the adaptation of the red-rubber frog which spend the dry season within ant nests completely unarmed by these ferocious insects who hunt and feed on other frog in the region. Then Manfred Türke delighted us with his observations of slug poops. It appears that many mite species survive the passage through the slug intestine and found on average 3 surviving mites per droppings. He thinks that mites dramatically increase their dispersal ability through the use of the “slug-highsped train”.

Day 4:

The last day started with a keynote speech by Shahid Naeem where he reflected on the evolution of conservation paradigms from nature for itself in the 60ies to nature for people in the 2000s to people and nature in current times. He argued that biodiversity is a multidimensional concept (taxonomic diversity, functional diversity …) and one need to look and preserve all dimensions to support ecosystem functions. Going on he argued that human well-being should be the primary focus of contemporary ecological approaches since if higher human well-being is being targeted then we need high provision of services from the ecosystems coming from good functioning of ecosystems which supported by biodiversity (CQFD), therefore society by protecting biodiversity will protect itself. This makes sense even if I have little love for utilitarian argument when it comes to biodiversity conservation. Later on that day I heard an inspiring talk by Jasper Wubs on how one can steer restoration in different directions by using different soil inoculum. In other words if you inoculate some heathland soils into a bare ground plot you get an heathland plant communities, and if 10 meters away you inoculate grassland soils you get grassland communities. How great is that? Finally I listened with great attention to Florian Hartig talk on model selection and its effect on prediction and inference using a simulation study. He showed that unconditional model averaging and LASSO led to the best performance when it comes to prediction while for inference keeping the full model led to the best results.


All in all the GfÖ meeting are always pretty nice, loads of little scientific nuggets here and there and great interactions with cool people. Next meeting will be in Belgium for a very exciting joined meeting with the British Ecological Society, already looking forward to it.



Shiny and Leaflet for Visualization Awards

Next week will be the meeting of the German (and Swiss and Austrians) ecologists in Marburg and the organizing team launched a visualization contest based on spatial data of the stores present in the city. Nadja Simons and I decided to enter the contest, our idea was to link the store data to the city bus network promoting a sustainable and safe way of movement through the town (we are ecologists after all).

We used leaflet to prepare the map and plot the bus lines, bus stops and stores. On top of that because all this information is rather overwhelming to grasp (more than 200 stores and 50 bus stops), we implemented a shiny App to allow the user to restrict its search of the best Bars in Marburg.

All the code is on GitHub and the App can accessed by clicking here.

Go there, enjoy, and if you want to learn a bit about the process of developing the App come back here.


First a few words on what is so cool about leaflet:

  • There is a lot of maps available
  • With the AwesomeMarker plugin, available in the github repo of the package, you can directly tap into three awesome libraries for icons: font awesome, glyphicons and ionicons
  • Leaflet understand HTML code, we used it to provide a clickable link to the bus timetables
  • Using groups makes it easy to interactively add or remove group of objects from the map

Having this was already nice, but putting it into a shiny App was even more exciting. Here are some of the main points:

  • A very important concept in shiny is the concept of reactivity, the way I understood it is that a reactive object is a small function that will get updated every time the user input some elements, see this nice tutorial for more on this.
  • Our idea of the App was that stores should appear when the mouse is passed over their nearest bus stop. The issue there is that the stores must then disappear if the mouse moves out. The trick is to create a set of reactiveValues that are NULL as long as the event (mouse passes over a bus stop) does not occure AND return to NULL when the event is finished (mouse moves out), helped by this post, we were able to implement what we wanted.
  • Shiny gives you a lot of freedom to create and customize the design of the App, what I found very cool was the possibility to have tabs.

Be sure to check the leaflet tutorial page, 90% of my questions were answered there, also check out the many articles on shiny.

See you in Marburg if you’ll be around!


Simulating local community dynamics under ecological drift

In 2001 the book by Stephen Hubbell on the neutral theory of biodiversity was a major shift from classical community ecology. Before this book the niche-assembly framework was dominating the study of community dynamics. Very briefly under this framework local species composition is the result of the resource available at a particular site and species presence or absence depends on species niche (as defined in Chase 2003). As a result community assembly was seen as a deterministic process and a specific set of species should emerge from a given set of resources and species composition should be stable as long as the system is not disturbed. Hubbell theory based on a dispersal-assembly framework (historically developed by MacArthur and Wilson) was a departure from this mainstream view of community dynamics by assuming that local species composition are transient with the absence of any equilibrium. Species abundance follow a random walk under ecological drift, species are continuously going locally extinct while a constant rain of immigrants from the metacommunity bring new recruits. The interested reader is kindly encouraged to read: this, that or that.

Being an R geek and after reading Chapter 4 of Hubbell’s book, I decided to implement his model of local community dynamic under ecological drift. Do not expect any advanced analysis of the model behavior, Hubbell actually derive in this chapter analytical solutions for this model. This post is mostly educational to get a glimpse into the first part of Hubbell’s neutral model.

Before diving into the code it is helpful to verbally outline the main characteristics of the model:

  • The model is individual-based and neutral, assuming that all individuals are equal in their probability to die or to persist.
  • The model assume a zero-sum dynamic, the total number of individual (J) is constant, imagine for example a forest. When a tree dies it allows for a new individual to grow and use the free space
  • At each time step a number of individual (D) dies and are replaced by new recruits coming either from the metacommunity (with probability m) or from the local communities (with probability 1-m)
  • The species identity of the new recruit is proportional to the species relative abundance either in the metacommunity (assumed to be fixed and having a log-normal distribution) or in the local community (which obviously varies)

We are now equipped to immerse ourselves into R code:

#implement zero-sum dynamic with ecological drift from Hubbel theory (Chp4)
  #the metacommunity SAD
  #initialize the local community
  begin<-table(sample(x=1:R,size = J,replace = TRUE,prob = metaSAD))
  #loop through the time and get community composition
  for(i in 2:(Time+1)){
    newC<-helper_untb(vec = mat[(i-1),],R = R,D = D,m = m,metaSAD=metaSAD)
#the function that compute individual death and 
#replacement by new recruits 
  #remove D individuals from the community
  ind_new<-ind[-sample(x = 1:length(ind),size = D,replace=FALSE)]
  #are the new recruit from the Local or Metacommunity?
  #sample species ID according to the relevant SAD
  ind_recruit<-c(ind_new,sapply(recruit,function(isLocal) {

I always like simple theories that can be implemented with few lines of code and do not require fancy computations. Let’s put the function in action and get for a closed community (m=0) of 10 species the species abundance and the rank-abundance distribution dynamics:

library(reshape2) #for data manipulation I
library(plyr) #for data manipulation II
library(ggplot2) #for the nicest plot in the world
#look at population dynamics over time
comm1<-untb(R = 10,J = 100,D = 10,m = 0,Time = 200)
#quick and dirty first graphs
+geom_path()+labs(y="Species abundance")
+scale_color_discrete(name="Species ID")
#let's plot the RAD curve over time
pre_rad<-apply(comm1,1,function(x) sort(x,decreasing = TRUE))
+geom_line()+labs(x="Species rank",y="Species relative abundance")


We see there an important feature of the zero-sum dynamics, with no immigration the model will always (eventually) lead to the dominance of one species. This is because with no immigration there are so-called absorbing states, species abundance are attracted to these states and once these values are reached species abundance stay constant. In this case there are only two absorbing states for species relative abundance: 0 or 1, either you go extinct or you dominate the community.

In the next step we open up the community while varying the death rate and we track species richness dynamic:

#apply this over a range of parameters
 + m=c(0,0.01,0.05,0.1,0.5),Time=200)
comm_list<-alply(pars,1,function(p) do.call(untb,as.list(p)))
#get species abundance for each parameter set
rich<-llply(comm_list,function(x) apply(x,1,function(y) length(y[y!=0])))
rich<-llply(rich,function(x) data.frame(Time=0:200,Rich=x))


As we increase the probability that new recruit come from the metacommunity, the impact of increasing the number of death per time step becomes null. This is because the metacommunity is static in this model, even if species are wiped out by important local disturbance there will always be new immigrants coming in maintaining species richness to rather constant levels.

There is an R package for everything these days and the neutral theory is not an exception, check out the untb package which implement the full neutral model (ie with speciation and dynamic metacommunities) but also allow to fit the model to empirical data and to get model parameter estimate.

Exploring the diversity of Life using Rvest and the Catalog of Life

I am writing the general introduction for my thesis and wanted to have a nice illustration of the diversity of Arthropods compared to other phyla (my work focus on Arthropods so this is a nice motivation). As the literature I have had access so far use pie charts to graphically represent these diversities and knowing that pie chart are bad, I decided to create my own illustration.

Fortunately I came across the Catalogue of Life website which provide (among other things) an overview of the number of species in each phylum. So I decided to try and have a go at directly importing the data from the website into R using the rvest package.

Let’s go:


#load the packages
library(scales)#for comma separator in ggplot2 axis

#read the data
#some minor data re-formatting
#re-format the data frame keeping only animals, plants and
#add a kingdom column
#remove the nasty commas and turn into numeric

Now we are read to make the first plot

 labs(y="Number of Species",x="",title="The diversity of life")



This is a bit overwhelming, half of the phyla have less than 1000 species so let’s make a second graph only with the phyla comprising more than 1000 species. And just to make things nicer I sort the phyla by the number of species:

 labs(y="Number of Species",x="",
 title="The diversity of multicellular organisms from the Catalog of Life")+



That’s it for a first exploration of the powers of rvest, this was actually super easy I expected to have to spend much more time trying to decipher xml code, but rvest seems to know its way around …

This graph is also a nice reminder that most of the described multicellular species out there are small crawling beetles and that we still know alarmingly very little about their ecology and status of threat. As a comparison all the vertebrates (all birds, mammals, reptiles, fishes and some other taxa) are within the Chordata and having a total of 50,000 described species. An even more sobering thought is the fact that the total number of described species is only a fraction of what is left undescribed.

Exploring Spatial Patterns and Coexistance

Today is a rainy day and I had to drop my plans for going out hiking, instead I continued reading “Self-Organization in Complex Ecosystems” from Richard Solé and Jordi Bascompte.

As I will be busy in the coming weeks with spatial models at the iDiv summer school I was closely reading chapter three on spatial self-organization and decided to try and implement in R one of the Coupled Map Lattice Models that is described in the book.

The model is based on a discrete time version of the Lotka-Volterra competition model, for example for two competing species we model the population normalized abundances (the population abundance divided by the carrying capacity) N1 and N2 using the following equations:

N_{1,t+1} = \phi_{N_{1}}(N_{1,t},N_{2,t}) = N_{1,t} * exp[r_{N_{1}} * (1 - N_{1} - \alpha_{12} * N_{2})]

N_{2,t+1} = \phi_{N_{2}}(N_{1,t},N_{2,t}) = N_{2,t} * exp[r_{N_{2}} * (1 - N_{2} - \alpha_{21} * N_{1})]

Where r is the growth rate and the \alpha are the interspecific competition coefficients. Note that since we use normalized abundance the intraspecific competition coefficient is equal to one.

Space can be introduced in this model by considering dispersion between neighboring cells. Say we have a 50 x 50 lattice map and that individuals disperse between neighboring cells with a probability D, as a result the normalized population abundances in cell k at time t+1 will be:

N_{1,t+1,k} = (1 - D) * \phi_{N_{1}}(N_{1,t,k},N_{2,t,k}) + \frac{D}{8} * \sum_{j=1}^{8} \phi_{N_{1}}(N_{1,t,j},N_{2,t,j})

Where j is an index of all 8 neighboring cells around cell k. We have a similar equation for the competing species.

Now let’s implement this in R (you may a nicer version of the code here).

I start by defining two functions: one that will compute the population dynamics based on the first two equations, and the other one that will return for each cell in lattice the row numbers of the neighboring cells:

#Lotka-Voltera discrete time competition function
#@Ns are the population abundance
#@rs are the growth rate
#@as is the competition matrix
  #return population abundance at time t+1

#an helper function returning line numbers of all neighbouring cells
  lnb<-which(dat[,"x"]%in%(vec["x"]+c(-1,-1,-1,0,1,1,1,0)) &
+ dat[,"y"]%in%(vec["y"]+c(-1,0,1,1,1,0,-1,-1)))
  #remove own line

Then I create a lattice, fill it with the two species and some random noise.


The next step is to initialize all the parameters, in the example given in the book the two species have equal growth rate, equal dispersal and equal interspecific competition.

#competition parameters
#growth rate
#dispersal rate
#infos on neighbouring cells
neigh<-apply(dat,1,function(x) find_neighbour(x,dat))

Now we are ready to start the model (the code is rather ugly, sorry for that, saturday coding makes me rather lazy …)

#model the dynamic assuming absorbing boundary (ie ind that go out of the grid are lost to the system)
for(i in generation){  
    #population dynamics within one grid cell
    #grab the neighbouring cells
    #the number of immigrant coming into the focal grid cell
    imm<-(ds/8)*rowSums(apply(neigh_cell,1,function(y) {
    + +imm[2],lnb=x["lnb"])

First let’s reproduce figure 3.8 from the book:

#look at coexistence between the two species within one cell
cell525<-ldply(list_dat,function(x) c(x[525,"N1"],x[525,"N2"]))
+ measure.vars=1:2,variable.name="Species",value.name="Pop")

#look at coexistence across the system
all<-ldply(list_dat,function(x) c(sum(x[,"N1"]),sum(x[,"N2"])))
+ variable.name="Species",value.name="Pop")



And now figure 3.9 plus an extra figure looking at the correlation in population abundance between the two species at the beginning and the end of the simulation:

#compare species-species abundance correlation at the beginning and at the end of the simulation
exmpl$Gen<-rep(c("Gen : 1","Gen : 60"),each=2500)


#look at variation in spatial patterns at the beginning and the end of the simulation
+ measure.vars=c("N1","N2"),variable.name="Species",value.name="Pop")



Neat stuff, I am looking forward to implement other models in this book!


Discussion on Biodiversity trends in the Anthropocene

So today at our group meeting I made a talk inspired by the McGill et al (2015) paper.

The basic messages that I take from this paper are threefold: (i) biodiversity is a complex and vague concept and it is therefore very easy, if one is not careful, to end up comparing apples and oranges when trying to synthesize informations on biodiversity trends, (ii) spatial and temporal scales have a huge impacts on biodiversity patterns, we should therefore be explicit, when talking about biodiversity trends, about the scales under consideration and (iii) we are in a desperate need for community data collected in a standardized way across many sites (mostly in Africa and in the tropics) and repeatedly over time.

In the presentation (which can be found here) I showed studies looking at different biodiversity trends heavily relying on the literature in the McGill paper but also adding results that were published since this paper came out (here, here and here). I deliberately spent more time talking about trends in local species richness because I knew that it would be discussed afterwards as quite a few people in our group (including myself) work on biodiversity experiments.

Then I talked about the controversies that arose from the studies reporting flat trends in local species richness. When the Vellend et al paper came out in 2013 I was actually in Jena sitting with the group of Nico Eisenhauer and since the Vellend paper took quite a hard stance against grassland biodiversity experiment field site, some people in the group publish a short reply. At that time I felt not really concerned about these issues (I was after all in the first year of my PhD) and when I look at the presentation that I held in that year it is funny to note that I presented results from the Murphy and Romanuk study which was easier to include in the classic rhetoric of: “we are losing biodiversity and it will affect ecosystem functioning” rather than presenting the results from Vellend and colleagues. I briefly mentioned it in the presentation so will do the same here: there is a great article with great comments in the dynamic ecology blog on this topic.

More recently a paper led by Gonzalez appeared in Ecology criticizing two specific papers reporting no trends in local species richness: Vellend et al 2013 (again!) and Dornelas et al 2014. They raised three major points: (i) the samples in both studies in not representative of global species richness and anthropogenic impacts, it is therefore not possible to claim that the results in these studies are globally valid, they are rather showing a biased picture of the trend in local species richness. For me this is the major point. In contrast to the more recent study by Elahi et al where the author raised already in the abstract the caveat that their sample was not a random sample of coastal habitat due to the under-representation of impacted sites,Vellend and Dornelas papers can be read as being globally valid. (ii) Time-series duration affect the results with longer time-series being more likely to report loss of species richness than shorter ones. There it is interesting to compare FigS2 in the Vellend paper with Fig3 in the Gonzalez paper, this is exactly the same figure the only difference is that the x-axis (study duration) is logged in one case and not in the other. I leave it to the discretion of the readers to reflect whether study duration impact on species richness trends is more likely to be linear than to be logarithmic. (iii) depending on the baseline used, different results might be reached. For example a piece of forest habitat have been completely logged and richness was measured before and after logging, what should be the baseline for looking at local richness trends over time? The data before logging? Or after logging? Gonzalez et al argue that the undisturbed site should be used while Vellend et al argue that in this case the disturbance will have a major impact on ecosystem functioning and therefore classical BEF studies manipulating species richness within a certain habitat type are of little value.

I personally think that issues (ii) and (iii) are easily corrected by data re-analysis which can be then published and may keep the discussion going, these two points are to my mind not really major ones. The issue of non-representativeness is much bigger, since most of the sites in the Vellend and Dornelas paper are far from being representative of human impacts on natural systems one needs to be careful when interpreting their results (which can be said for many study out there). Anyway the perfect dataset to look at biodiversity trends do not exist so in the meantime it is nice to have such studies to generate some debate, shake some of our dusty, untested ideas that we have (like biodiversity is declining) and then go out and collect more data to maybe one day be able to build this perfect dataset.

In the discussion that issued after the talk it was interesting to hear that we talked much more about big concept like: are scientists responsible for the interpretation that the public make of their results, since there are so many publications out there one is able to cherry-pick studies to prove any points, the scientific success model is driving us towards big project, big groups, big paper with catchy title, complex stats and colourful figures. We spent more time talking about how we do science and how the system is less than optimal rather than talking about biodiversity trends.

Anyhow I could go on for quite some pages on this topic, I’ll stop it here hoping to find motivation to write more posts in the near future.

Plotting regression curves with confidence intervals for LM, GLM and GLMM in R

Once models have been fitted and checked and re-checked comes the time to interpret them. The easiest way to do so is to plot the response variable versus the explanatory variables (I call them predictors) adding to this plot the fitted regression curve together (if you are feeling fancy) with a confidence interval around it. Now this approach is preferred over the partial residual one because it allows the averaging out of any other potentially confounding predictors and so focus only on the effect of one focal predictor on the response. In my work I have been doing this hundreds of time and finally decided to put all this into a function to clean up my code a little bit. As always a cleaner version of this post is available here.
Let’s dive into some code (the function is at the end of the post just copy/paste into your R environment):

#####LM example######
#we measured plant biomass for 120 pots under 3 nutrient treatments and across a gradient of CO2
#due to limited place in our greenhouse chambers we had to use 4 of them, so we established a blocking design
data<-data.frame(C=runif(120,-2,2),N=gl(n = 3,k = 40,labels = c("Few","Medium","A_lot")),Block=rep(rep(paste0("B",1:4),each=10),times=3))

##         Block
## N        B1 B2 B3 B4
##   Few    10 10 10 10
##   Medium 10 10 10 10
##   A_lot  10 10 10 10

#the paramters of the models
#simulate a response vector
#fit the model

## Call:
## lm(formula = Biom ~ Block + C * N, data = data)
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2.11758 -0.68801 -0.01582  0.75057  2.55953 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  10.0768     0.2370  42.521  < 2e-16 ***
## BlockB2      -0.3011     0.2690  -1.119 0.265364    
## BlockB3       2.3322     0.2682   8.695 3.54e-14 ***
## BlockB4      -1.4505     0.2688  -5.396 3.91e-07 ***
## C             0.7585     0.1637   4.633 9.89e-06 ***
## NMedium       0.4842     0.2371   2.042 0.043489 *  
## NA_lot        2.4011     0.2335  10.285  < 2e-16 ***
## C:NMedium     0.7287     0.2123   3.432 0.000844 ***
## C:NA_lot      3.2536     0.2246  14.489  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## Residual standard error: 1.028 on 111 degrees of freedom
## Multiple R-squared:  0.9201, Adjusted R-squared:  0.9144 
## F-statistic: 159.8 on 8 and 111 DF,  p-value: < 2.2e-16

Here we would normally continue and make some model checks. As output from the model we would like to plot the effect of CO2 on plant biomass for each level of N addition. Of course we want to average out the Block effect (otherwise we would have to plot one separate line for each Block). This is how it works:

pred<-plot_fit(m,focal_var = "C",inter_var = "N")

##            C   N       LC     Pred       UC
## 1 -1.9984087 Few 8.004927 8.706142 9.407358
## 2 -1.7895749 Few 8.213104 8.864545 9.515986
## 3 -1.5807411 Few 8.417943 9.022948 9.627952
## 4 -1.3719073 Few 8.618617 9.181350 9.744084
## 5 -1.1630735 Few 8.814119 9.339753 9.865387
## 6 -0.9542397 Few 9.003286 9.498156 9.993026

#the function output a data frame with columns for the varying predictors
#a column for the predicted values (Pred), one lower bound (LC) and an upper one (UC)
#let's plot this
plot(Biom~C,data,col=c("red","green","blue")[data$N],pch=16,xlab="CO2 concentration",ylab="Plant biomass")
legend("topleft",legend = c("Few","Medium","A lot"),col=c("red","green","blue"),pch=16,lwd=3,title = "N addition",bty="n")


The cool thing is that the function will also work for GLM, LMM and GLMM. For mixed effect models the confidence interval is computed from parametric bootstrapping:

######LMM example#######
#now let's say that we took 5 measurements per pots and we don't want to aggregate them
data<-data.frame(Pots=rep(paste0("P",1:120),each=5),Block=rep(rep(paste0("B",1:4),each=5*10),times=3),N=gl(n = 3,k = 40*5,labels=c("Few","Medium","A_lot")),C=rep(runif(120,-2,2),each=5))
#a random intercept term

## Linear mixed model fit by REML ['lmerMod']
## Formula: Biom ~ Block + C * N + (1 | Pots)
##    Data: data
## REML criterion at convergence: 1765.1
## Scaled residuals: 
##     Min      1Q  Median      3Q     Max 
## -2.6608 -0.6446 -0.0340  0.6077  3.2002 
## Random effects:
##  Groups   Name        Variance Std.Dev.
##  Pots     (Intercept) 0.1486   0.3855  
##  Residual             0.9639   0.9818  
## Number of obs: 600, groups:  Pots, 120
## Fixed effects:
##             Estimate Std. Error t value
## (Intercept)  9.93917    0.13225   75.15
## BlockB2     -0.42019    0.15153   -2.77
## BlockB3      2.35993    0.15364   15.36
## BlockB4     -1.36188    0.15111   -9.01
## C            0.97208    0.07099   13.69
## NMedium      0.36236    0.13272    2.73
## NA_lot       2.25624    0.13189   17.11
## C:NMedium    0.70157    0.11815    5.94
## C:NA_lot     2.78150    0.10764   25.84
## Correlation of Fixed Effects:
##           (Intr) BlckB2 BlckB3 BlckB4 C      NMedim NA_lot C:NMdm
## BlockB2   -0.572                                                 
## BlockB3   -0.575  0.493                                          
## BlockB4   -0.576  0.495  0.493                                   
## C          0.140 -0.012 -0.018 -0.045                            
## NMedium   -0.511  0.003  0.027  0.007 -0.118                     
## NA_lot    -0.507  0.008  0.003  0.003 -0.119  0.502              
## C:NMedium -0.134  0.019  0.161  0.038 -0.601  0.175  0.071       
## C:NA_lot  -0.107  0.077  0.020  0.006 -0.657  0.078  0.129  0.394

#again model check should come here
#for LMM and GLMM we also need to pass as character (vector) the names of the random effect
pred<-plot_fit(m,focal_var = "C",inter_var = "N",RE = "Pots")
#let's plot this
plot(Biom~C,data,col=c("red","green","blue")[data$N],pch=16,xlab="CO2 concentration",ylab="Plant biomass")
legend("topleft",legend = c("Few","Medium","A lot"),col=c("red","green","blue"),pch=16,lwd=3,title = "N addition",bty="n")

Please note a few elements:
– so far the function only return 95% confidence intervals
– I have tested it on various types of models that I usually build but there are most certainly still some bugs hanging around so if the function return an error please let me know of the model you fitted and the error returned
– the bootstrap computation can take some time for GLMM so be ready to wait a few minute if you have a big complex model
– the function accept a vector of variable names for the inter_var argument, it should also work for the RE argument even if I did not tried it yet

Happy plotting!

Here is the code for the function:

#for lm, glm, lmer and glmer models
#@m : the fitted lm, glm or merMod object (need to be provided)
#@focal_var: a character, the name of variable of interest that will be plotted on the x axis, ie the varying variable (need to be provided)
#@inter_var: a character or character vector, the name of variable interacting with the focal variable, ie categorical variables from which prediction will be drawn for each level across the focal_var gradient
#@RE: if giving a merMod object give as character or character vector of the name of the random effects variable (so far I only tried with one RE)
#@n: a numeric, the number of data point that will form the gradient of the focal variable
#@n_core: the number of core used to compute the bootstrapped CI for GLMM models

  #turn all character variable to factor
  #make a sequence from the focal variable
  #grab the names and unique values of the interacting variables
  #add the focal variable to this list
  #expand.grid on it
  #remove varying variables and non-predictors
    fixed<-lapply(dat_red,function(x) if(is.numeric(x)) mean(x) else factor(levels(x)[1],levels = levels(x)))
    #the number of rows in the new_dat frame
    #create the new_dat frame starting with the varying focal variable and potential interactions
    #get the name of the variable to average over, debug for conditions where no variables are to be avergaed over
    name_f<-names(dat_red)[sapply(dat_red,function(x) ifelse(is.factor(x),TRUE,FALSE))]
  #get the predicted values
    pred<-predict(m,newdata = new_dat,se.fit=TRUE)
  if(cl=="glm" | cl=="negbin"){
    #predicted values on the link scale
  if(cl=="glmerMod" | cl=="lmerMod"){
    #for bootstrapped CI
  #average over potential categorical variables  
    if(cl=="glmerMod" | cl=="lmerMod"){
      coef_f<-lapply(name_f,function(x) fixef(m)[grep(paste0("^",x,"\\w+$"),names(fixef(m)))])
      coef_f<-lapply(name_f,function(x) coef(m)[grep(paste0("^",x,"\\w+$"),names(coef(m)))])
    pred$fit<-pred$fit+sum(unlist(lapply(coef_f,function(x) mean(c(0,x)))))
  #to get the back-transform values get the inverse link function
  #get the back transformed prediction together with the 95% CI for LM and GLM
  if(cl=="glm" | cl=="lm"){
  #for GLMM need to use bootstrapped CI, see ?predict.merMod
  if(cl=="glmerMod" | cl=="lmerMod"){
    predFun<-function(.) mm%*%fixef(.)
    bb<-bootMer(m,FUN=predFun,nsim=200,parallel="multicore",ncpus=n_core) #do this 200 times
    bb$t<-apply(bb$t,1,function(x) linkinv(x))
    #as we did this 200 times the 95% CI will be bordered by the 5th and 195th value
    bb_se<-apply(bb$t,1,function(x) x[order(x)][c(5,195)])
  #the output

Ecology at the Interface in Rome

Last week I was at the European Ecological Federation conference in Rome, I presented the results of one experiment that we ran last year (THE one big experiment of my PhD). You can find the slides here.

All in all it was a week with ups and downs, some session were highly interesting and generated intense discussion while others were just no-question-next-talk session. Below is a summary of the stuff that got stuck in my head over the week.

Day 1: I arrived pretty late at the conference so could only listen to a few talk in the evening session, Caroline Müller  gave a nice talk on the combined effect of drought and plant chemotypes on insect herbivores (caterpillar). An intriguing results was that plants under shoot herbivory had higher concentration of defensive molecules in the roots than plants with no herbivory. The reason why this happen is left to speculation.

Day 2: The day started at 8.30am sharp by two plenary lectures, the second one given by Prof. Musso (a colleague from the TUM!) explored the ecology of architecture and how to develop new materials and home to ensure sustainable town. Particularly on how to develop incentives to promote low-consuming lifestyles in cities and countries with democracy. I then went to a symposium on scale non-linearity of drivers of environmental changes. There Stefano Larsen gave a nice talk on temporal community shifts of stream invertebrates. His results show that local decline in species richness were due to specialist species extinction that were then not able to re-colonize the area from the regional species pool. In the afternoon I went to the Biodiversity and Ecosystem session which self-organized itself with no chair, leaving a pleading Martin Winter at the end of the session asking: “Somebody should close the session”!

Day 3: Again an early start at 8.30 (not so sharp this time), I went first to the tropical ecology session which started by two great talk from Hannah Tuomisto on fern species distribution in the amazonian forest and Jens-Christian Svenning on historical legacies in palm global species distribution. Jens-Christian did not read my previous post on partial residual plots as he used them to picture most of the relationships he explored, too bad. I then ran to the agricultural ecology session where Emanuelle Porcher gave her talk on the effect of wheat genetic diversity on predation rates, she found weak positive effect of genetic diversity and some seasonal variation that she explained by contrasting climatic conditions. The day continued with two plenaries the second one by Christopher Kennedy on the metabolism of megacities where he presented his approach considering cities as ecosystem and analyzing fluxes of energy entering and exiting the cities and how they moves amongst the compartment in the cities. His results show that despite certain expectation that larger cities are more efficient in using energy, larger cities consume more energy due to larger wealth (GDP) in these big cities. Larger wealth cause larger amount of waste and higher electricity use. This is a particularly challenging issue as more and more are moving towards cities especially in Asia where a new consuming middle class is arising.

Day 4: On thursday morning I went to the high nature value symposium, a concept I was not familiar with but which is basically a framework to identify agrosystems with potentially high diversity and rare/endangered ecosystem type. The talk by James Moran on the implementation of this concept in Ireland was very interesting, he developed a 10-point grading system to assess the ecosystem health (being of course relative to the habitat) and depending on the grade the farmer get more or less money. This system by being pretty close to the 15-point grading system used to assess meat quality (and hence the price paid for a cow) helps farmers grasping the concept of ecosystem health. One quote from this speaker is also worth noting: “If you depend on somebody to translate your results it will be lost in translation”. I then went to a symposium on ecologists’ strategies at science-policy interface with plenty of great talks some given by social scientists other by ecologists involved in this area. One striking talk was by Zoe Nyssa on unexpected negative feedback of conservation action, she did a literature review on this issue in conservation journals and found many instances where conservation programs led to unexpected results. What was particularly interesting was the lively discussion after the end of the session were ecologist and social scientist exchanged on the way to improve communication between ecologists and policy-maker and the society, I have been wanted for a long time to write a post on ecological advocacy, maybe this will motivate me … In the afternoon one quote form a chair asking a question made its way to my notebook: “How did you select your model? The current approach is to fit all possible models and compare them with AIC”, hmmm well depending on your objectives this approach might work, but if you are trying to find mechanisms/test hypothesis this will most certainly not work (see here). Thursday evening I also went out to discover Rome vibrant nightlife with the INGEE people, heavy rain earlier that evening apparently negatively impacted the participation rate, but it was pretty relaxed and nice to chat with some Italian scientists.

So a nice little conference with plenty of things to keep oneself busy (but not too much) and some cool interaction.

Two little annoying stats detail

UPDATED: Thanks to Ben and Florian comments I’ve updated the first part of the post


A very brief post at the end of the field season on two little “details” that are annoying me in paper/analysis that I see being done (sometimes) around me.

The first one concern mixed effect models where the models built in the contain a grouping factor (say month or season) that is fitted as both a fixed effect term and as a random effect term (on the right side of the | in lme4 model synthax). I don’t really understand why anyone would want to do this and instead of spending time writing equations let’s just make a simple simulation example and see what are the consequences of doing this:

#an example of a situation measuring plant biomass on four different month along a gradient of temperature
#the coefficient
#the simulated coefficient for Months are 0.5, 1.2 -0.9
#a simple lm
coef(m_fixed) #not too bad
## (Intercept)        temp      month2      month3      month4 
##   0.9567796   2.0654349   0.4307483   1.2649599  -0.8925088

#a lmm with month ONLY as random term

## (Intercept)        temp 
##    1.157095    2.063714


## $month
##   (Intercept)
## 1  -0.1916665
## 2   0.2197100
## 3   1.0131908
## 4  -1.0412343

VarCorr(m_rand) #the estimated sd for the month coeff

##  Groups   Name        Std.Dev.
##  month    (Intercept) 0.87720 
##  Residual             0.98016

sd(c(0,0.5,1.2,-0.9)) #the simulated one, not too bad!

## [1] 0.8831761

#now a lmm with month as both fixed and random term
m_fixedrand<-lmer(biom~temp+month+(1|month),data) fixef(m_fixedrand) ## (Intercept) temp month2 month3 month4 ## 0.9567796 2.0654349 0.4307483 1.2649599 -0.8925088 ranef(m_fixedrand) #very, VERY small ## $month ## (Intercept) ## 1 0.000000e+00 ## 2 1.118685e-15 ## 3 -9.588729e-16 ## 4 5.193895e-16 VarCorr(m_fixedrand) ## Groups Name Std.Dev. ## month (Intercept) 0.40397 ## Residual 0.98018 #how does it affect the estimation of the fixed effect coefficient? summary(m_fixed)$coefficients ## Estimate Std. Error t value Pr(>|t|)
## (Intercept)  0.9567796  0.2039313  4.691676 9.080522e-06
## temp         2.0654349  0.1048368 19.701440 2.549792e-35
## month2       0.4307483  0.2862849  1.504614 1.357408e-01
## month3       1.2649599  0.2772677  4.562233 1.511379e-05
## month4      -0.8925088  0.2789932 -3.199035 1.874375e-03


##               Estimate Std. Error    t value
## (Intercept)  0.9567796  0.4525224  2.1143256
## temp         2.0654349  0.1048368 19.7014396
## month2       0.4307483  0.6390118  0.6740851
## month3       1.2649599  0.6350232  1.9919901
## month4      -0.8925088  0.6357784 -1.4038048

#the numeric response is not affected but the standard error around the intercept and the month coefficient is doubled, this makes it less likely that a significant p-value will be given for these effect ie higher chance to infer that there is no month effect when there is some

#and what if we simulate data as is supposed by the model, ie a fixed effect of month and on top of it we add a random component
#an lmm model
fixef(m_fixedrand2) #weird coeff values for the fixed effect for month

## (Intercept)        temp      month2      month3      month4 
##   -2.064083    2.141428    1.644968    4.590429    3.064715

c(0,eff[3:5])+rnd.eff #if we look at the intervals between the intercept and the different levels we can realize that the fixed effect part of the model sucked in the added random part

## [1] -2.66714133 -1.26677658  1.47977624  0.02506236


##  Groups   Name        Std.Dev.
##  month    (Intercept) 0.74327 
##  Residual             0.93435

ranef(m_fixedrand2) #again very VERY small

## $month
##     (Intercept)
## 1  1.378195e-15
## 2  7.386264e-15
## 3 -2.118975e-14
## 4 -7.752347e-15

#so this is basically not working it does not make sense to have a grouping factor as both a fixed effect terms and a random term (ie on the right-hand side of the |)

Take-home message don’t put a grouping factor as both a fixed and random term effect in your mixed effect model. lmer is not able to separate between the fixed and random part of the effect (and I don’t know how it could be done) and basically gives everything to the fixed effect leaving very small random effects. The issue is abit pernicious because if you would only look at the standard deviation of the random term from the merMod summary output you could not have guessed that something is wrong. You need to actually look at the random effects to realize that they are incredibely small. So beware when building complex models with many fixed and random terms to always check the estimated random effects.

Now this is only true for factorial variables, if you have a continuous variable (say year) that affect your response through both a long-term trend but also present some between-level (between year) variation, it actually makes sense (provided you have enough data point) to fit a model with this variable as both a fixed and random term. Let’s look into this:

#an example of a situation measuring plant biomass on 10 different year along a gradient of temperature
#the coefficient
#a simple lm

lm(formula = y ~ temp + year, data = data)

Min       1Q   Median       3Q      Max
-2.27455 -0.83566 -0.03557  0.92881  2.74613

Estimate Std. Error t value Pr(>|t|)
(Intercept)  1.41802    0.25085   5.653 1.59e-07 ***
temp         2.11359    0.11230  18.820  < 2e-16 ***
year        -1.54711    0.04036 -38.336  < 2e-16 ***
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.159 on 97 degrees of freedom
Multiple R-squared:  0.9508,    Adjusted R-squared:  0.9498
F-statistic: 937.1 on 2 and 97 DF,  p-value: < 2.2e-16
coef(m_fixed) #not too bad

(Intercept)        temp        year
1.418019    2.113591   -1.547111
#a lmm

Linear mixed model fit by REML ['lmerMod']
Formula: y ~ temp + year + (1 | factor(year))
Data: data

REML criterion at convergence: 304.8

Scaled residuals:
Min      1Q  Median      3Q     Max
-2.0194 -0.7775  0.1780  0.6733  1.9903

Random effects:
Groups       Name        Variance Std.Dev.
factor(year) (Intercept) 0.425    0.6519
Residual                 1.004    1.0019
Number of obs: 100, groups:  factor(year), 10

Fixed effects:
Estimate Std. Error t value
(Intercept)  1.40266    0.49539   2.831
temp         2.01298    0.10191  19.752
year        -1.54832    0.07981 -19.400

Correlation of Fixed Effects:
(Intr) temp
temp  0.031
year -0.885  0.015
fixef(m_rand) #very close to the lm estimation

(Intercept)        temp        year
1.402660    2.012976   -1.548319
abline(0,1) #not too bad
VarCorr(m_rand) #the estimated sd for the within-year variation

Groups       Name        Std.Dev.
factor(year) (Intercept) 0.65191
Residual                 1.00189

Interestingly we see in this case that the standard error (and the related t-value) of the intercept and year slope are twice as big (small for the t-values) in the LMM compared to the LM. This means that not taking into account between-year random variation leads us to over-estimate the precision of the long-term temporal trend (we might conclude that there are significant effect when there are a lot of noise not taken into account). I still don’t fully grasp how this work, but thanks to the commenter for pointing this out.

The second issue is maybe a bit older but I saw it appear in a recent paper (which is a cool one excpet for this stats detail). After fitting a model with several predictors one wants to plot their effects on the response, some people use partial residuals plot to do this (wiki). The issue with these plots is that when two variables have a high covariance the partial residual plot will tend to be over-optimistic concerning the effect of variable X on Y (ie the plot will look much nice than it should be). Again let’s do a little simulation on this:

#say we measure plant biomass in relation with measured temperature and number of sunny hours say per week
#the variance-covariance matrix between temperature and sunny hours
#simulate some data

#partial residual plot of sun
plot(data$sun,sun_res,xlab="Number of sunny hours",ylab="Partial residuals of Sun")


#plot of sun effect while controlling for temp
plot(biom~sun,data,xlab="Number of sunny hours",ylab="Plant biomass")


#same stuff for temp
plot(biom~temp,data,xlab="Temperature",ylab="Plant biomass")


The first graph is a partial residual plot, from this graph alone we would be tempted to say that the number of hour with sun has a large influence on the biomass. This conclusion is biased by the fact that the number of sunny hours covary with temperature and temperature has a large influence on plant biomass. So who is more important temperature or sun? The way to resolve this is to plot the actual observation and to add a fitted regression line from a new dataset (sun_new in the example) where one variable is allowed to vary while all others are fixed to their means. This way we see how an increase in the number of sunny hour at an average temperature affect the biomass (the second figure). The final graph is then showing the effect of temperature while controlling for the effect of the number of sunny hours.

Happy modelling!