sim2: The Simulated "sim2" Dataset

In DImodels: Diversity-Interactions (DI) Models

The Simulated "sim2" Dataset

Description

The sim2 dataset was simulated. There are four blocks and four species that vary in proportions (p1 - p4). There are 15 unique sets of proportions identified by the variable community. Each unique community appears once in each block. The response was simulated assuming that there were species identity effects, block effects, an average pairwise interaction effect and a theta value of 0.5.

Usage

data(sim2)

Format

A data frame with 60 observations on the following seven variables:

community

A numeric vector identifying each unique community, i.e., two rows with the same community value also share the same set of p1 to p4 values.

block

A factor taking values 1 to 4 indicating block membership.

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.

response

A numeric vector giving the simulated response variable.

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

DI models take the general form of:

y = Identities + Interactions + Structures + \epsilon

where y is a community-level response, the Identities are the effects of species identities and enter the model as individual species proportions 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 sim2 was simulated with:

• identity effects for the four species with values = 10, 9, 8, 7

• block effects for the four blocks with values = 1, 1.5, 2, 0

• an average pairwise interaction effect = 8

• theta = 0.5 (where \theta is a non-linear parameter included as a power on each pipj product within interaction variables, see Connolly et al 2013 for details)

• \epsilon assumed normally distributed with mean 0 and standard deviation 1.1.

References

Connolly J, T Bell, T Bolger, C Brophy, T Carnus, JA Finn, L Kirwan, F Isbell, J Levine, A Lüscher, V Picasso, C Roscher, MT Sebastia, M Suter and A Weigelt (2013) An improved model to predict the effects of changing biodiversity levels on ecosystem function. Journal of Ecology, 101, 344-355.

Kirwan L, J Connolly, JA Finn, C Brophy, A Lüscher, D Nyfeler and MT Sebastia (2009) Diversity-interaction modelling - estimating contributions of species identities and interactions to ecosystem function. Ecology, 90, 2032-2038.

Examples

####################################
## Code to simulate the sim2 dataset
  


## Simulate dataset sim2 with species identity effects, block effects and 
##  an average pairwise interaction effect with theta=0.5.

## Use the proportions from the first fifteen plots in Switzerland
  data(Switzerland)

## Repeat the 15 plots over four blocks. 
## Give each community type a unique (community) number.
  sim2 <- data.frame(community = rep(1:15, each = 4),
                   block = factor(rep(1:4, times = 15)),
                   p1 = rep(Switzerland$p1[1:15], each = 4),
                   p2 = rep(Switzerland$p2[1:15], each = 4),
                   p3 = rep(Switzerland$p3[1:15], each = 4),
                   p4 = rep(Switzerland$p4[1:15], each = 4))

## Create the average pairwise interaction variable, with theta = 0.5
  AV_variable <- DI_data(prop = c("p1","p2","p3","p4"), data = sim2, 
                         theta = 0.5, what = "AV")
  sim2 <- data.frame(sim2, "AV_theta" = AV_variable)

## To simulate the response, first create a matrix of predictors that includes p1-p4 and  
##  the four block variables and the average pairwise interaction variable with theta=0.5.
  X <- model.matrix(~ p1 + p2 + p3 + p4 + block + AV_theta -1, data = sim2)

## Create a vector of 'known' parameter values for simulating the response.
## The first four are the p1-p4 parameters, the second four are the block effects and 
##  the last one is the interaction parameter.
  sim2_coeff <- c(10,9,8,7,   1,1.5,2,0,    8)

## Create response and add normally distributed error 
  sim2$response <- as.numeric(X %*% sim2_coeff)
  set.seed(328781)
  r <- rnorm(n = 60, mean = 0, sd = 1.1)
  sim2$response <- round(sim2$response + r, digits = 3)
  sim2$AV_theta <- NULL




###################################################################################################
###################################################################################################
## sim2

###########################
## Analyse the sim2 dataset
  
## Load the sim2 data
  data(sim2)
## View the first few entries
  head(sim2)
## Explore the variables in sim2
  str(sim2)
  
## Check that the proportions sum to 1 (required for DI models)
## p1 to p4 are in the 3rd to 6th columns in sim2
  sim2sums <- rowSums(sim2[3:6])
  summary(sim2sums)
  
## Check characteristics of sim2
  hist(sim2$response)
  summary(sim2$response)
  plot(sim2$p1, sim2$response)
  plot(sim2$p2, sim2$response)
  plot(sim2$p3, sim2$response)
  plot(sim2$p4, sim2$response)

## Find the best DI model using autoDI and F-test selection
  auto1 <- autoDI(y = "response", prop = c("p1", "p2", "p3", "p4"), block = "block", data = sim2, 
                  selection = "Ftest")
  summary(auto1)
  
  ## Fit the average pairwise model, including theta, using DI and the AV tag
  m1 <- DI(y = "response", prop = c("p1","p2","p3","p4"), block = "block", DImodel = "AV", 
           estimate_theta = TRUE, data = sim2)
  summary(m1)
  CI_95 <- theta_CI(m1, conf = .95)
  CI_95
  plot(m1)
  library(hnp)


## Check goodness-of-fit using a half-normal plot with a simulated envelope
  library(hnp)
  hnp(m1)


## Graph the profile likelihood
  library(ggplot2)
  ggplot(m1$profile_loglik, aes(x = grid, y = prof)) +
    theme_bw() +
    geom_line() +
    xlim(0,1.5) +
    xlab(expression(theta)) +
    ylab("Log-likelihood") + 
    geom_vline(xintercept = CI_95, lty = 3) + 
    labs(title = "   Log-likelihood versus theta", 
      caption = "dotted vertical lines are upper and lower bounds of 95% CI for theta")
         
## Fit the average pairwise model, including theta, using DI and custom_formula
## A value of theta must be 'chosen'. Take: 0.4533437 from m1. The 'estimate_theta' option is not
##  available with custom_formula.
  AV_variable <- DI_data(prop = c(3:6), data = sim2, theta = 0.4533437, what = "AV")
  sim2a <- data.frame(sim2, "AV_theta" = AV_variable)
  m2 <- DI(y = "response", custom_formula = response ~ 0 + p1 + p2 + p3 + p4 + AV_theta + block, 
           data = sim2a)
## This will adjust the standard errors in m2 for the 'estimation' of theta
  m2$df.residual <- m2$df.residual - 1
## This will adjust the AIC in m2 for the 'estimation' of theta
  m2$aic <- m2$aic + 2
  summary(m2)
  

DImodels documentation built on May 29, 2024, 7:05 a.m.