R: The simulated repeated measures "simRM" dataset

simRM {DImodelsMulti}

R Documentation

The simulated repeated measures "simRM" dataset

Description

The simRM dataset was simulated from a repeated measures DI model. It contains 116 plots comprising of four species that vary in proportions (p1 - p4). There is one simulated response (Y), taken at three differing time points, recorded in a stacked data format (one row per response value). The data was simulated assuming that there were existing covariances between the responses, an additive treatment effect, and both species identity and species interaction effects were present.

Usage

data("simRM")

Format

A data frame with 384 observations on the following seven variables.

plot: A numeric vector identifying each unique experimental unit
p1: A numeric vector indicating the initial proportion of species 1
p2: A numeric vector indicating the initial proportion of species 2
p3: A numeric vector indicating the initial proportion of species 3
p4: A numeric vector indicating the initial proportion of species 4
Y: A numeric vector giving the simulated response for an ecosystem function
time: A three-level factor indicating the time point at which the response Y was recorded

Details

What are Diversity-Interactions (DI) models?

Diversity-Interactions (DI) models (Kirwan et al., 2009) are a set of tools for analysing and interpreting data from experiments that explore the effects of species diversity on community-level responses. We strongly recommend that users read the short introduction to Diversity-Interactions models (available at: DImodels). Further information on Diversity-Interactions models is also available in Kirwan et al., 2009 and Connolly et al., 2013.

Parameter values for the simulation

Repeated measures DI models take the general form of:

{y}_{mt} = {Identities}_{mt} + {Interactions}_{mt} + {Structures}_{t} + {\epsilon}_{mt}

where y are the community-level responses, the Identities are the effects of species identities for each response and enter the model as individual species proportions measured at the beginning of the time period, the Interactions are the interactions among the species proportions, while Structures include other experimental structures such as blocks, treatments, or density.

The dataset simRM was simulated with:

identity effects for the four species for each time:
- time1 = 4.1, 2.1, 3.6, 4.8
- time2 = 2.3, 2.4, 5.1, 5.0
- time3 = 0.7, 2.3, 6.5, 5.9
additive interaction effects for each time:
- time1 = 3.3, 4.0, 1.3, 5.2
- time2 = 3.6, 4.5, 0.5, 6.5
- time3 = 4.5, 5.2, 0.6, 7.8
\epsilon assumed to have a multivaraite normal distribution with mean 0 and Sigma:

[1,] 2.01 -0.11 -0.08

[2,] -0.11 2.96 0.21

[3,] -0.08 0.21 1.03

References

Dooley, A., Isbell, F., Kirwan, L., Connolly, J., Finn, J.A. and Brophy, C., 2015.
Testing the effects of diversity on ecosystem multifunctionality using a multivariate model.
Ecology Letters, 18(11), pp.1242-1251.

Finn, J.A., Kirwan, L., Connolly, J., Sebastia, M.T., Helgadottir, A., Baadshaug, O.H., Belanger, G., Black, A., Brophy, C., Collins, R.P. and Cop, J., 2013.
Ecosystem function enhanced by combining four functional types of plant species in intensively managed grassland mixtures: a 3-year continental-scale field experiment.
Journal of Applied Ecology, 50(2), pp.365-375 .

Kirwan, L., Connolly, J., Finn, J.A., Brophy, C., Luscher, A., Nyfeler, D. and Sebastia, M.T., 2009.
Diversity-interaction modeling: estimating contributions of species identities and interactions to ecosystem function.
Ecology, 90(8), pp.2032-2038.

Examples

###################################################################################################
###################################################################################################

## Modelling Example

# For a more thorough example of the workflow of this package, please see vignette
# DImulti_workflow using the following code:

# vignette("DImulti_workflow")


head(simRM)

# We call DImulti() to fit a series of models, with increasing complexity, and test whether the
# additional terms are worth keeping.

# We begin with an ID DI model, ensuring to use method = "ML" as we will be comparing fixed effects

RMmodel <- DImulti(y = "Y", time = c("time", "AR1"), unit_IDs = "plot",
                   prop = paste0("p", 1:4), data = simRM, DImodel = "ID",
                   method = "ML")
print(RMmodel)

# We can simplify the above model by grouping the ID effect terms into two categories, G1 and G2

RMmodel_ID <- DImulti(y = "Y", time = c("time", "AR1"), unit_IDs = "plot",
                   prop = paste0("p", 1:4), data = simRM, DImodel = "ID",
                   ID = c("G1", "G1", "G2", "G2"),
                   method = "ML")

# We can compare the two models by calling the anova function to perform a likelihood ratio test

anova(RMmodel, RMmodel_ID)

# As the p-value < an alpha value of 0.05, we reject the null hypthesis that the extra terms are
# equal to zero, therefore we continue with the larger model.
# We can confirm this result by selecting the model with the lower AIC value.

AIC(RMmodel); AIC(RMmodel_ID)

# Next, we include the simplest interaction structure available in this package, "AV", which adds
# a single extra term per ecosystem function

RMmodel_AV <- DImulti(y = "Y", time = c("time", "AR1"), unit_IDs = "plot",
                   prop = paste0("p", 1:4), data = simRM, DImodel = "AV",
                   method = "ML")

anova(RMmodel, RMmodel_AV)

# Once again, we select the more complex model.
#
# We can continue increasing the complexity of the interaction structure in the same fashion, this
# time we elect to use the additive interaction structure

RMmodel_ADD <- DImulti(y = "Y", time = c("time", "AR1"), unit_IDs = "plot",
                   prop = paste0("p", 1:4), data = simRM, DImodel = "ADD",
                   method = "ML")

anova(RMmodel_AV, RMmodel_ADD)

# We continue with the additive interactions structure, the next interaction structure to test
# functional group interactions, is not nested with our additive model, so we compare th two using
# information criterion. In this case we choose to use AICc, to better penalise extra terms, as AIC
# becomes biased towards the more complex model as parameters numbers increase.
# The functional group strcuture requires an additional parameter, FG, to specify groups.

RMmodel_FG <- DImulti(y = "Y", time = c("time", "AR1"), unit_IDs = "plot",
                   prop = paste0("p", 1:4), data = simRM, DImodel = "FG",
                   FG = c("G1", "G1", "G2", "G2"),
                   method = "ML")

AICc(RMmodel_ADD); AICc(RMmodel_FG)

# In this case, we choose the lower valued model, which is the additive structure.
#
# The last interaction structure to test is the full pairwise model. Which we can see is not worth
# the extra terms. So our final interaction structure chosen is additive.

RMmodel_FULL <- DImulti(y = "Y", time = c("time", "AR1"), unit_IDs = "plot",
                   prop = paste0("p", 1:4), data = simRM, DImodel = "FULL",
                   method = "ML")

anova(RMmodel_ADD, RMmodel_FULL)

# Finally, we can also increase the model complexity via the inclusion of the non-linear parameter
# theta, which we can estimate, or select a value for. We also choose to estimate using the "REML"
# method as we will do no further fixed effect model comparisons

RMmodel_theta <- DImulti(y = "Y", time = c("time", "AR1"), unit_IDs = "plot",
                      prop = paste0("p", 1:4), data = simRM, DImodel = "ADD",
                      estimate_theta = TRUE, method = "REML")

print(RMmodel_theta)

# We can however test changes to the correlation structure, testing the AR(1) structure we have been
# using against the simpler CS strcuture.

RMmodel_CS <- DImulti(y = "Y", time = c("time", "CS"), unit_IDs = "plot",
                      prop = paste0("p", 1:4), data = simRM, DImodel = "ADD",
                      theta = 0.9991, method = "REML")

AICc(RMmodel_theta); AICc(RMmodel_CS)

# We see that they are practically identical, but the AR(1) strcuture is preffered.

##################################################################################################
#
##################################################################################################

## Code to simulate data

set.seed(523)

props <- data.frame(plot = integer(),
                    p1 = integer(),
                    p2 = integer(),
                    p3 = integer(),
                    p4 = integer())

index <- 1 #row number

#Monocultures
for(i in 1:4) #4 species
{
  for(j in 1:2) #2 technical reps
  {
    props[index, i+1] <- 1
    index <- index + 1
  }
}


#Equal Mixtures
for(rich in sort(rep(2:4, 3))) #3 reps at each richness level
{
  sp <- sample(1:4, rich) #randomly pick species from pool

  for(j in 1:2) #2 technical reps
  {
    for(i in sp)
    {
      props[index, i+1] <- 1/rich #equal proportions
    }
    index <- index + 1
  }
}


#Unequal Mixtures
for(rich in sort(rep(c(2, 3, 4), 15))) #15 reps at each richness level
{
  sp <- sample(1:4, rich, replace = TRUE) #randomly pick species from pool

  for(j in 1:2) #2 technical reps
  {
    for(i in 1:4)
    {
      props[index, i+1] <- sum(sp==i)/rich #equal proportions
    }
    index <- index + 1
  }
}


props[is.na(props)] <- 0

mySimData <- props
mySimData$Y <- NA

ADDs <- DImodels::DI_data(prop=2:5, what=c("ADD"), data=mySimData)
mySimData <- cbind(mySimData, ADDs)

E_AV <- DImodels::DI_data(prop=2:5, what=c("E", "AV"), data=mySimData)
mySimData <- cbind(mySimData, E_AV)

mySimData$plot <- 1:nrow(mySimData)

mySimData$time <- 1
mySimDataT1 <- mySimData
mySimDataT2 <- mySimData
mySimDataT3 <- mySimData
mySimDataT2$time <- 2
mySimDataT3$time <- 3


n <- 3 #Number of Ys
p <- qr.Q(qr(matrix(stats::rnorm(n^2), n))) #Principal Components (make sure it's positive definite)
S <- crossprod(p, p*(n:1)) #Sigma
m <- stats::runif(n, -0.25, 1.5)

#runif(11, -1, 7) #decide on betas randomly

for(i in 1:nrow(mySimData))
{
  #Within subject error
  error <- MASS::mvrnorm(n=1, mu=m, Sigma=S)
  mySimDataT1$Y[i] <- 4.1*mySimDataT1$p1[i] + 2.1*mySimDataT1$p2[i] + 3.6*mySimDataT1$p3[i] +
  4.8*mySimDataT1$p4[i] + 3.3*mySimDataT1$p1_add[i] + 4.0*mySimDataT1$p2_add[i] +
  1.3*mySimDataT1$p3_add[i] + 5.2*mySimDataT1$p4_add[i] + error[1]
  mySimDataT2$Y[i] <- 2.3*mySimDataT2$p1[i] + 2.4*mySimDataT2$p2[i] + 5.1*mySimDataT2$p3[i] +
  5.0*mySimDataT2$p4[i] + 3.6*mySimDataT2$p1_add[i] + 4.5*mySimDataT2$p2_add[i] +
  0.5*mySimDataT2$p3_add[i] + 6.5*mySimDataT2$p4_add[i] + error[2]
  mySimDataT3$Y[i] <- 0.7*mySimDataT3$p1[i] + 2.3*mySimDataT3$p2[i] + 6.5*mySimDataT3$p3[i] +
  5.9*mySimDataT3$p4[i] + 4.5*mySimDataT3$p1_add[i] + 5.2*mySimDataT3$p2_add[i] +
  0.6*mySimDataT3$p3_add[i] + 7.8*mySimDataT3$p4_add[i] + error[3]
}

mySimData <- rbind(mySimDataT1, mySimDataT2)
mySimData <- rbind(mySimData, mySimDataT3)

mySimData$time <- as.factor(mySimData$time)
mySimData$plot <- as.factor(mySimData$plot)

[Package DImodelsMulti version 1.1.1 Index]