DImodels"

In DImodels: Diversity-Interactions (DI) Models

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)

library(DImodels)

Getting Started with DImodels

The DImodels package is designed to make fitting Diversity-Interactions models easier. 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 (from a pool of S species) on community-level responses. Data suitable for DI models will include (at least) for each experimental unit: a response recorded at a point in time, and a set of proportions of S species $p_1$, $p_2$, ..., $p_S$ from a point in time prior to the recording of the response. The proportions sum to 1 for each experimental unit.

Main changes in the package from version 1.3 to version 1.3.1

• A fortify function method has been added to supplement the data fitted to a linear model with model fit statistics.
• A describe_model function is added which can be used to get a short text summary of any DI model.
• Meta-data about a DI model can be accessed via the attributes function.

Main changes in the package from version 1.2 to version 1.3

• The DI and autoDI functions now have an additional parameter called ID which enables the user to group the species identity effects (see examples below).
• The predict function now has flexibility to calculate confidence and prediction intervals for the predicted values.

Main changes in the package from version 1.1 to version 1.2

• There are two new functions added to the package:
  • predict: Make predictions from a fitted DI model without having to worry about theta, and the interaction terms in the data.
  • contrasts_DI: Create contrasts for a DI model.

Main changes in the package from version 1.0 to version 1.1

• DI_data_prepare is now superseded by DI_data (see examples below)

DImodels installation and load

The DImodels package is installed from CRAN and loaded in the typical way.

install.packages("DImodels")
library("DImodels")

Accessing an introduction to Diversity-Introductions models

It is recommended that users unfamiliar with Diversity-Interactions (DI) models read the introduction to DImodels, before using the package. Run the following code to access the documentation.

?DImodels

Datasets included in the DImodels package

There are seven example datasets included in the DImodels package: Bell, sim1, sim2, sim3, sim4, sim5, Switzerland. Details about each of these datasets is available in their associated help files, run this code, for example:

?sim3

In this vignette, we will describe the sim3 dataset and show a worked analysis of it.

The sim3 dataset

The sim3 dataset was simulated from a functional group (FG) Diversity-Interactions model. There were nine species in the pool, and it was assumed that species 1 to 5 come from functional group 1, species 6 and 7 from functional group 2 and species 8 and 9 from functional group 3, where species in the same functional group are assumed to have similar traits. The following equation was used to simulate the data.

$$ y = \sum_{i=1}^{9}\beta_ip_i + \omega_{11}\sum_{\substack{i,j = 1 \ i<j}}^5p_ip_j + \omega_{22}p_6p_7 + \omega_{33}p_8p_9 \ + \omega_{12}\sum_{\substack{i \in {1,2,3,4,5} \ j \in {6,7}}}p_ip_j + \omega_{13}\sum_{\substack{i \in {1,2,3,4,5} \ j \in {8,9}}}p_ip_j + \omega_{23}\sum_{\substack{i \in {6,7} \ j \in {8,9}}}p_ip_j + \gamma_k + \epsilon$$ Where $\gamma_k$ is a treatment effect with two levels (k = 1,2) and $\epsilon$ was assumed IID N(0, $\sigma^2$). The parameter values are in the following table.

| Parameter | Value |        | Parameter | Value | | ----------- | ----------- | ----------- | ----------- | ----------- | | $\beta_1$ | 10 | | $\omega_{11}$ | 2 | | $\beta_2$ | 9 | | $\omega_{22}$ | 3 | | $\beta_3$ | 8 | | $\omega_{33}$ | 1 | | $\beta_4$ | 7 | | $\omega_{12}$ | 4 | | $\beta_5$ | 11 | | $\omega_{13}$ | 9 | | $\beta_6$ | 6 | | $\omega_{23}$ | 3 | | $\beta_7$ | 5 | | $\gamma_1$ | 3 | | $\beta_8$ | 8 | | $\gamma_2$ | 0 | | $\beta_9$ | 9 | | $\sigma$ | 1.2 |

Here, the non-linear parameter $\theta$ that can be included as a power on each $p_ip_j$ component of each interaction variable (Connolly et al 2013) was set equal to one and thus does not appear in the equation above.

The 206 rows of proportions contained in the dataset design_a (supplied in the package) were used to simulate the sim3 dataset. Here is the first few rows from design_a:

library(DImodels)
data("design_a")
knitr::kable(head(design_a))

Where community is an identifier for unique sets of proportions and richness is the number of species in the community.

The proportions in design_a were replicated over two treatment levels, giving a total of 412 rows in the simulated dataset. The sim3 data can be loaded and viewed in the usual way.

data("sim3")
knitr::kable(head(sim3, 10))

Exploring the data

There are several graphical displays that will help to explore the data and it may also be useful to generate summary statistics.

hist(sim3$response, xlab = "Response", main = "")
# Similar graphs can also be generated for the other species proportions.
plot(sim3$p1, sim3$response, xlab = "Proportion of species 1", ylab = "Response")
summary(sim3$response)

Implementing an automated DI model fitting process using autoDI

The function autoDI in DImodels provides a way to do an automated exploratory analysis to compare a range of DI models. It works through a set of automated steps (Steps 1 to 4) and will select the 'best' model from the range of models that have been explored and test for lack of fit in that model. The selection process is not exhaustive, but provides a useful starting point in analysis using DI models.

auto1 <- autoDI(y = "response", prop = 4:12, treat = "treatment", 
                FG = c("FG1","FG1","FG1","FG1","FG1","FG2","FG2","FG3","FG3"), data = sim3, 
                selection = "Ftest")

The output of autoDI, works through the following process:

1. Step 1 fitted the average interactions (AV) model and uses profile likelihood to estimate the non-linear parameter $\theta$ and tests whether or not it differs from one. $\theta$ was estimated to be 0.96814 and was not significantly different from one ($p = 0.4572$). Therefore, subsequent steps assumed $\theta=1$ when fitting the DI models.
2. Step 2 fitted five different DI models, each with a different form of species interactions and treatment was always included. The functional group model (FG) was the selected model. This assumes that pairs of species interact according to functional group membership.
3. Step 3 provided a test for the treatment and indicated that the treatment, included as an additive factor, was significant and needed in the model ($p < 0.0001$).
4. Step 4 provides a lack of fit test, here there was no indication of lack of fit in the model selected in Step 3 ($p = 0.6423$).

Further details on each of these steps are available in the autoDI help file. Run the following code to access the documentation.

?autoDI

All parameter estimates from the selected model can be viewed using summary.

summary(auto1)

If the final model selected by autoDI includes a value of theta other than 1, then a 95% confidence interval for $\theta$ can be generated using the theta_CI function:

theta_CI(auto1, conf = .95)

Here, this code would not run, since the final model selected by autoDI does not include theta estimated.

Fitting individual models using the DI function

For some users, the selection process in autoDI will be sufficient, however, most users will fit additional models using DI. For example, while the treatment is included in autoDI as an additive factor, interactions between treatment and other model terms are not considered. Here, we will first fit the model selected by autoDI using DI and then illustrate the capabilities of DI to fit specialised models.

Fitting the final model selected by autoDI using DI

m1 <- DI(y = "response", prop = 4:12, 
         FG = c("FG1","FG1","FG1","FG1","FG1","FG2","FG2","FG3","FG3"), treat = "treatment", 
         DImodel = "FG", data = sim3)
summary(m1)

Re-fitting the final model selected by autoDI estimating theta using update_DI

m1_theta <- update_DI(object = m1, estimate_theta = TRUE)
coef(m1_theta)

Grouping the species identity effects in the model

The species identity effects in a DI model can be grouped by specifying groups for each species using the ID argument. The ID argument functions similar to the FG argument and accepts a character list of same length as number of species in the model. The identity effects of species belonging in the same group will be grouped together.

Grouping all identity effects into a single term

m1_group <- update_DI(object = m1_theta, 
                      ID = c("ID1", "ID1", "ID1", "ID1", "ID1",
                             "ID1", "ID1", "ID1", "ID1"))
coef(m1_group)

Grouping identity effects of specific species

m1_group2 <- update_DI(object = m1_theta, 
                       ID = c("ID1", "ID1", "ID1", 
                              "ID2", "ID2", "ID2", 
                              "ID3", "ID3", "ID3"))
coef(m1_group2)

Note: Grouping ID effects will not have an effect on the calculation of the interaction effects, they would still be calculated by using all species.

Read the documentation of DI and autoDI for more information and examples using the ID parameter.

?DI
?autoDI

Fitting customised models using the DI function

There are two ways to fit customised models using DI; the first is by using the option DImodel = in the DI function and adding the argument extra_formula = to it, and the second is to use the custom_formula argument in the DI function. If species interaction variables (e.g., the FG interactions or the average pairwise interaction) are included in either extra_formula or custom_formula, they must first be created and included in the dataset. The function DI_data can be used to compute several types of species interaction variables.

Including treatment by species identity term interactions using extra_formula

m2 <- DI(y = "response", prop = 4:12, 
         FG = c("FG1","FG1","FG1","FG1","FG1","FG2","FG2","FG3","FG3"), treat = "treatment", 
         DImodel = "FG", extra_formula = ~ (p1 + p2 + p3 + p4):treatment,
         data = sim3)
summary(m2)

Including treatment by species interaction terms using extra_formula

First, we create the FG pairwise interactions, using the DI_data function with the what argument set to "FG".

FG_matrix <- DI_data(prop = 4:12, FG = c("FG1","FG1","FG1","FG1","FG1","FG2","FG2","FG3","FG3"), 
                     data = sim3, what = "FG")
sim3a <- data.frame(sim3, FG_matrix)

Then we fit the model using extra_formula.

m3 <- DI(y = "response", prop = 4:12, 
         FG = c("FG1","FG1","FG1","FG1","FG1","FG2","FG2","FG3","FG3"),
         treat = "treatment", DImodel = "FG", 
         extra_formula = ~ (bfg_FG1_FG2 + bfg_FG1_FG3 + bfg_FG2_FG3 +
                              wfg_FG1 + wfg_FG2 + wfg_FG3) : treatment, data = sim3a)
summary(m3)

Fitting only a subset of the FG interaction terms using custom_formula

First, we create a dummy variable for level A of the treatment (this is required for the glm engine that is used within DI and because there is no intercept in the model).

sim3a$treatmentA <- as.numeric(sim3a$treatment == "A")

Then we fit the model using custom_formula.

m3 <- DI(y = "response",
         custom_formula = response ~ 0 + p1 + p2 + p3 + p4 + p5 + p6 + p7 + p8 + p9 +
           treatmentA + bfg_FG1_FG2 + bfg_FG1_FG3 + bfg_FG2_FG3, data = sim3a)
summary(m3)

Making predictions and testing contrasts for DI models

Predictions using a DI model

We can make predictions from a DI model just like any other regression model using the predict function. The user does not need to worry about adding any interaction terms or adjusting any columns if theta is not equal to 1. Only the species proportions along with any additional experimental structures is needed and all other terms in the model will be calculated for the user.

# Fit model
m3 <- DI(y = "response", prop = 4:12, 
         treat = "treatment", DImodel = "AV", 
         extra_formula = ~ (AV) : treatment, data = sim3a)

predict_data <- sim3[c(1, 79, 352), 3:12]
# Only species proportions and treatment is needed
print(predict_data)
# Make prediction
predict(m3, newdata = predict_data)

Uncertainity around predictions

# The interval and level parameters can be used to calculate the 
# uncertainty around the predictions

# Get confidence interval around prediction
predict(m3, newdata = predict_data, interval = "confidence")

# Get prediction interval around prediction
predict(m3, newdata = predict_data, interval = "prediction")

# The function returns a 95% interval by default, 
# this can be changed using the level argument
predict(m3, newdata = predict_data, 
        interval = "prediction", level = 0.9)

Contrasts for DI models

The contrasts_DI function can be used to compare and formally test for a difference in performance of communities within the same as well as across different experimental structures

Comparing the performance of the monocultures of different species at treatment A

contr <- list("p1vsp2" = c(1, -1, 0, 0,  0,  0, 0, 0,  0, 0, 0, 0),
              "p3vsp5" = c(0,  0, 1, 0, -1,  0, 0, 0,  0, 0, 0, 0),
              "p4vsp6" = c(0,  0, 0, 1,  0, -1, 0, 0,  0, 0, 0, 0),
              "p7vsp9" = c(0,  0, 0, 0,  0,  0, 1, 0, -1, 0, 0, 0))
the_C <- contrasts_DI(m3, contrast = contr)
summary(the_C)

Comparing across the two treatment levels for monoculture of species 1

contr <- list("treatAvsB" = c(1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0))
the_C <- contrasts_DI(m3, contrast = contr)
summary(the_C)

Comparing between two species mixtures

mixA <- c(0.25, 0,      0.25, 0,      0.25, 0,      0.25, 0, 0, 0, 0, 0)
mixB <- c(0,    0.3333, 0,    0.3333, 0,    0.3333, 0,    0, 0, 0, 0, 0)

# We have the proportions of the individual species in the mixtures, however
# we still need to calculate the interaction effect for these communities
contr_data <- data.frame(rbind(mixA, mixB))
colnames(contr_data) <- names(coef(m3))

# Adding the interaction effect of the two mixtures
contr_data$AV <- DI_data_E_AV(prop = 1:9, data = contr_data)$AV
print(contr_data)

# We can now subtract the respective values in each column of the two 
# mixtures and get our contrast
my_contrast <- as.matrix(contr_data[1, ] - contr_data[2, ])
rownames(my_contrast) <- "mixAvsB"

the_C <- contrasts_DI(m3, contrast = my_contrast)
summary(the_C)

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.

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