sim2 {DImodels} | R Documentation |

## 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*version 1.3.2 Index]