Author: Jarrett Byrnes
Date: 2015-06-25

# Assessing Multifunctionality with the multifunc package

#### Jarrett Byrnes

#### 2015-06-25

## Intro

This is a step-by-step walk-through of the analysis of the German BIODEPTH site in the paper to illustrate the different analyses we conducted to evaluate the relationship between biodiversity and ecosystem multifunctionality.

We begin by loading the multifunc package. If it is not currently installed, it can be found on github, and installed using the devtools library.

    library(devtools)
    install_github("multifunc", username="jebyrnes")

    library(multifunc)

    #for plotting
    library(ggplot2)
    library(patchwork)

    #for data
    library(tidyr)
    library(dplyr)
    library(purrr)
    library(forcats)

    #for analysus
    library(car)

Next, we load in the BIODEPTH data. We use some of the helper functions in the package to identify the column numbers of the relevant functions for later use.

    #Read in data  to run sample analyses on the biodepth data
    data(all_biodepth)

    allVars<-c("biomassY3", "root3", "N.g.m2",  "light3", "N.Soil", "wood3", "cotton3")

    varIdx<-which(names(all_biodepth) %in% allVars)

As we want to work with just the German data, we next subset down the data to just Germany. We then look at the whether species were seeded or not, and determine which species were indeed seeded in Germany. If a column of species is full of zeroes, then it was not seeded in these plots. We will need the column indices of relevant species for later use in the overlap analyses.

    #######
    # Now, specify what data we're working with - Germany
    # and the relevant variables we'll be working with
    #######

    germany<- all_biodepth %>%
      filter(location=="Germany")

    # which variables have > 2/3 of the values not NA?
    german_vars <- whichVars(germany, allVars)

    # What are the names of species in this dataset
    # that have at least some values > 0?
    species <- relevantSp(germany,26:ncol(germany))

    spIDX <- which(names(germany) %in% species) #in case we need these

## Single Function Approach

First, we will demonstrate the qualitative single function approach. As we’re using ggplot2 for plotting, we’ll use tidyr to pivot the data long for a nice faceted plot of each relationship.

    germanyForPlotting <- germany %>%
      select(Diversity, german_vars) %>%
      pivot_longer(cols = german_vars,
                   names_to = "variable",
                   values_to = "value") %>%
      mutate(variable = factor(variable))

    #make the levels of the functions into something nice for plots
    levels(germanyForPlotting$variable) <- c('Aboveground Biomass', 'Root Biomass', 'Cotton Decomposition', 'Soil Nitrogen', 'Plant Nitrogen')

Nest, as we want to display additional information about the fit of diversity to each function, we’ll need to iterate over each function, derive a fit, and then make some labels to be used in the eventual plot.

    germanyLabels <- germanyForPlotting %>%
      group_by(variable) %>%
      nest() %>%
      mutate(fits = map(data, ~lm(value ~ Diversity, data=.x)),
             stats = map(fits, broom::glance)) %>%
      unnest(stats) %>%
      ungroup() %>%
      mutate(labels = paste0("p = ", round(p.value, 3), ", ",
                            expression(R^2), " = ", round(r.squared, 2)),
             labels =  gsub("p = 0 ", "p < 0.001 ", labels),
             Diversity = 7,
             value=c(2000, 1200, 1.5, 6, 20))

Finally, we will put it all together into a single plot.

    ggplot(aes(x=Diversity, y=value),data=germanyForPlotting) +
      geom_point(size=3)+
      facet_wrap(~variable, scales="free") +
      theme_bw(base_size=15)+
      stat_smooth(method="lm", colour="black", size=2) +
      xlab("Species Richness") +
      ylab("Value of Function") +
      geom_text(data=germanyLabels,
                aes(label=labels)) +
     theme(panel.grid = element_blank())

Next, we ask whether the best performing species differs between functions. To do that, we have to pull out all monoculture plots, then reshape the data and run a quick qualitative analysis on it to determine the best performer.

    #pull out only those plots planted in monoculture
    monoGermany<- germany %>%
      filter(rowSums(select(., ACHMIL1:VICTET1))==1) %>%
      #get mono names
      pivot_longer(ACHMIL1:VICTET1,
                   names_to = "mono") %>%
        filter(value !=0) %>%
      select(-value) %>%
    #pivot by function
      pivot_longer(german_vars,
                   names_to = "variable") %>%

    #get the mean for each monoculture
    group_by(variable, mono) %>%
    summarize(value = mean(value))

    #now, who is the best performer?
    monoGermany %>%
      group_by(variable) %>%
      filter(value == max(value))

    # # A tibble: 5 × 3
    # # Groups:   variable [5]
    #   variable  mono       value
    #   <chr>     <chr>      <dbl>
    # 1 N.Soil    TRIREP1    4.81
    # 2 N.g.m2    TRIPRA1   16.9
    # 3 biomassY3 TRIPRA1  818.
    # 4 cotton3   TRIREP1    0.854
    # 5 root3     FESRUB1 1072.

## Overlap Approach

One of the functions we are looking at, soil N, first needs to be reflected before we can apply the overlap approach.

    germany <- germany %>%
      mutate(N.Soil = -1*N.Soil +max(N.Soil, na.rm=T))

We can examine any single species and generate coefficient values and ‘effects’ - i.e., whether a species has a positive (1), negative (-1), or neutral (0) effect on a function. Let’s examine this for biomass production in year 3.

    spList<-sAICfun("biomassY3", species, germany)
    spList

    # $pos.sp
    # [1] "DACGLO1" "FESPRA1" "LATPRA1" "LOTCOR1" "TRIPRA1" "TRIREP1"
    #
    # $neg.sp
    # [1] "CAMPAT1" "PHLPRA1" "RANACR1"
    #
    # $neu.sp
    #  [1] "ACHMIL1" "ALOPRA1" "ANTODO1" "ARRELA1" "BROHOR1" "CENJAC1" "CHRLEU1"
    #  [8] "CREBIE1" "CYNCRI1" "FESRUB1" "GERPRA1" "HOLLAN1" "KNAARV1" "LEOAUT1"
    # [15] "LOLPER1" "LYCFLO1" "PIMMAJ1" "PLALAN1" "RUMACE1" "TAROFF1" "VICCRA1"
    # [22] "VICSEP1"
    #
    # $functions
    # [1] "biomassY3"
    #
    # $coefs
    #     ACHMIL1     ALOPRA1     ANTODO1     ARRELA1     BROHOR1     CAMPAT1
    #      0.0000      0.0000      0.0000      0.0000      0.0000   -391.5405
    #     CENJAC1     CHRLEU1     CREBIE1     CYNCRI1     DACGLO1     FESPRA1
    #      0.0000      0.0000      0.0000      0.0000    104.6404    135.6735
    #     FESRUB1     GERPRA1     HOLLAN1     KNAARV1     LATPRA1     LEOAUT1
    #      0.0000      0.0000      0.0000      0.0000    248.5566      0.0000
    #     LOLPER1     LOTCOR1     LYCFLO1     PHLPRA1     PIMMAJ1     PLALAN1
    #      0.0000    375.6694      0.0000   -297.7452      0.0000      0.0000
    #     RANACR1     RUMACE1     TAROFF1     TRIPRA1     TRIREP1     VICCRA1
    #   -140.2996      0.0000      0.0000    492.6311    325.0173      0.0000
    #     VICSEP1 (Intercept)
    #      0.0000    335.8382
    #
    # $effects
    # ACHMIL1 ALOPRA1 ANTODO1 ARRELA1 BROHOR1 CAMPAT1 CENJAC1 CHRLEU1 CREBIE1 CYNCRI1
    #       0       0       0       0       0      -1       0       0       0       0
    # DACGLO1 FESPRA1 FESRUB1 GERPRA1 HOLLAN1 KNAARV1 LATPRA1 LEOAUT1 LOLPER1 LOTCOR1
    #       1       1       0       0       0       0       1       0       0       1
    # LYCFLO1 PHLPRA1 PIMMAJ1 PLALAN1 RANACR1 RUMACE1 TAROFF1 TRIPRA1 TRIREP1 VICCRA1
    #       0      -1       0       0      -1       0       0       1       1       0
    # VICSEP1
    #       0

Using this as our starting point, we can use the getRedundancy function to generate a) an effect matrix for all functions, b) a coefficient matrix for all functions, and c) a standardized coefficient matrix for all functions.

    redund<-getRedundancy(german_vars, species, germany)
    coefs<-getRedundancy(german_vars, species, germany, output="coef")
    stdCoefs<-stdEffects(coefs, germany, german_vars, species)

    #for example
    redund

    #           ACHMIL1 ALOPRA1 ANTODO1 ARRELA1 BROHOR1 CAMPAT1 CENJAC1 CHRLEU1
    # biomassY3       0       0       0       0       0      -1       0       0
    # root3           1       0      -1       0       1       0       0       0
    # N.g.m2          1       0       0      -1       0       0       0       0
    # N.Soil         -1       0      -1       0       1      -1       0       0
    # cotton3         0       1       1       0      -1       1       0       0
    #           CREBIE1 CYNCRI1 DACGLO1 FESPRA1 FESRUB1 GERPRA1 HOLLAN1 KNAARV1
    # biomassY3       0       0       1       1       0       0       0       0
    # root3           0       0       0       0       1       0      -1       0
    # N.g.m2         -1       0       0       1       0       0      -1       0
    # N.Soil          1       0       1      -1       0       0       0       0
    # cotton3         0       0       0       0       0       0       0       0
    #           LATPRA1 LEOAUT1 LOLPER1 LOTCOR1 LYCFLO1 PHLPRA1 PIMMAJ1 PLALAN1
    # biomassY3       1       0       0       1       0      -1       0       0
    # root3           1       0       1       0       0       0       0      -1
    # N.g.m2          0       0       1       1       0      -1       0      -1
    # N.Soil          1       0       0       1       0       1       0       0
    # cotton3        -1       0      -1       0       0       0       0       1
    #           RANACR1 RUMACE1 TAROFF1 TRIPRA1 TRIREP1 VICCRA1 VICSEP1
    # biomassY3      -1       0       0       1       1       0       0
    # root3          -1       0       0      -1       0       0       0
    # N.g.m2         -1       0       0       1       1       0       0
    # N.Soil          1       0       0      -1      -1       0       0
    # cotton3         0       0       0       1       1       0       0

Using the effect matrix, we then estimate the average number of species affecting all possible combinations of functions - both positive and negative - and then plot the results.

    #plot the num. functions by fraction of the species pool needed
    posCurve<-divNeeded(redund, type="positive")

    posCurve$div<-posCurve$div/ncol(redund)

    pc<-qplot(nfunc, div, data=posCurve, group=nfunc, geom=c("boxplot"))+
      geom_jitter(size=4, position = position_jitter(height=0.001, width = .04))+
      ylab("Fraction of Species Pool\nwith Positive Effect\n")+
      xlab("\nNumber of Functions")+theme_bw(base_size=24)+ylim(c(0,1))

    negCurve<-divNeeded(redund, type="negative")

    negCurve$div<-negCurve$div/ncol(redund)

    nc<-qplot(nfunc, div, data=negCurve, group=nfunc, geom=c("boxplot"))+
      geom_jitter(size=4, position = position_jitter(height=0.001, width = .04))+
      ylab("Fraction of Species Pool\nwith Negative Effect\n")+
      xlab("\nNumber of Functions")+theme_bw(base_size=24)+ylim(c(0,1))

    #combine these into one plot
    pc + nc +
       plot_annotation(tag_levels = 'A')

We could have also looked at overlap indices, such as the Sorenson index.

    allOverlapPos <- map_df(2:nrow(redund),
                            ~getOverlapSummary(redund, m = .x, denom = "set"),
                            .id = "m")

    allOverlapPos

    # # A tibble: 4 × 4
    #   m     meanOverlap sdOverlap     n
    #   <chr>       <dbl>     <dbl> <dbl>
    # 1 1          0.283      0.206    10
    # 2 2          0.0658     0.116    10
    # 3 3          0          0         5
    # 4 4          0         NA         1

One we’ve examined the consequences of functional overlap, we can then move on to compare the number and size of these effects. First, what is the relative balance of positive to negative contributions for each species?

    posNeg<-data.frame(Species = colnames(redund),
                       Pos = colSums(filterOverData(redund)),
                       Neg = colSums(filterOverData(redund, type="negative")))

    #plot it
    ggplot(aes(x=Pos,y= Neg), data=posNeg) +
      geom_jitter(position = position_jitter(width = 0.05, height = 0.05), size=5, shape=1) +
      theme_bw(base_size=18) +
      xlab("\n# of Positive Contributions per species\n") +
      ylab("# of Negative Contributions per species\n") +
      annotate("text", x=0, y=3, label="a)") +
      stat_smooth(method="glm", colour="black", size=2,
                  method.args = list(family=poisson(link="identity"))) +
      geom_abline(slope = 1, intercept = 0, lty = 2, color = "red")

So, a generally positive relationship. We can see from the figure that the slope is less than 1:1, so positive effects accumulate faster. Still, more positive effects = more negative effects. What about the size of those positive versus negative effects?

    #now get the standardized effect sizes
    posNeg<-within(posNeg, {

      stdPosMean <- colSums(filterCoefData(stdCoefs))/Pos
      stdPosMean[which(is.nan(stdPosMean))] <-0

      stdNegMean <- colSums(filterCoefData(stdCoefs, type="negative"))/Neg
      stdNegMean[which(is.nan(stdNegMean))] <-0
    })

    ggplot(aes(x=stdPosMean,y= stdNegMean), data=posNeg) +
      #  geom_point(size=3) +
      geom_jitter(position = position_jitter(width = .02, height = .02), size=5, shape=1) +
      theme_bw(base_size=18) +
      xlab("\nAverage Standardized Size of\nPositive Contributions") +
      ylab("Average Standardized Size of\nNegative Contributions\n") +
      stat_smooth(method="lm", colour="black", size=2)+
      annotate("text", x=0, y=0.25, label="b)") +
      geom_abline(slope=-1, intercept=0, size=1, colour="black", lty=3)

## Averaging Approach

Now we’ll try the averaging approach. The first step is to create a set of columns where we have standardized all of the functions of interest, and then create an average of those standardized functions.

    #add on the new functions along with the averaged multifunctional index
    germany<-cbind(germany, getStdAndMeanFunctions(germany, german_vars))
    #germany<-cbind(germany, getStdAndMeanFunctions(germany, vars, standardizeZScore))

We can then plot the averaged multifunctionality quite simply

    #plot it
    ggplot(aes(x=Diversity, y=meanFunction),data=germany)+geom_point(size=3)+
      theme_bw(base_size=15)+
      stat_smooth(method="lm", colour="black", size=2) +
      xlab("\nSpecies Richness") +
      ylab("Average Value of Standardized Functions\n")

We may want to look at all of the functions on a standardized scale, and add in the averaged line for comparison as well. We can do this by reshaping the data, and plotting it.

    #reshape for plotting everything with ggplot2
    germanyMeanForPlotting <- germany %>%
      select(Diversity, biomassY3.std:meanFunction) %>%
      pivot_longer(cols = c(biomassY3.std:meanFunction),
                   names_to = "variable") %>%
        mutate(variable = fct_inorder(variable))
    #nice names for plotting
    levels(germanyMeanForPlotting$variable) <- c('Aboveground Biomass', 'Root Biomass', 'Cotton Decomposition', 'Soil Nitrogen', 'Plant Nitrogen', 'Mean Multifuncion Index')

    #plot it
    ggplot(aes(x=Diversity, y=value),data=germanyMeanForPlotting)+geom_point(size=3)+
      facet_grid(~variable) +
      theme_bw(base_size=15)+
      stat_smooth(method="lm", colour="black", size=2) +
      xlab("\nSpecies Richness") +
      ylab("Standardized Value of Function\n")

Last, let’s test the statistical fit of the effect of diversity on the averaged multifunctionality index.

    #statistical fit
    aveFit<-lm(meanFunction ~ Diversity, data=germany)
    Anova(aveFit)

    # Anova Table (Type II tests)
    #
    # Response: meanFunction
    #            Sum Sq Df F value    Pr(>F)
    # Diversity 0.29502  1   32.63 4.042e-07 ***
    # Residuals 0.52440 58
    # ---
    # Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

    summary(aveFit)

    #
    # Call:
    # lm(formula = meanFunction ~ Diversity, data = germany)
    #
    # Residuals:
    #       Min        1Q    Median        3Q       Max
    # -0.200533 -0.069321  0.006506  0.062717  0.288942
    #
    # Coefficients:
    #             Estimate Std. Error t value Pr(>|t|)
    # (Intercept) 0.385422   0.017049  22.606  < 2e-16 ***
    # Diversity   0.015360   0.002689   5.712 4.04e-07 ***
    # ---
    # Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    #
    # Residual standard error: 0.09509 on 58 degrees of freedom
    # Multiple R-squared:   0.36,   Adjusted R-squared:  0.349
    # F-statistic: 32.63 on 1 and 58 DF,  p-value: 4.042e-07

## Threshold Approach

First, we need to create a new data set that looks at the number of functions greater than or equal to a threshold across a wide range of thresholds.

    germanyThresh<-getFuncsMaxed(germany, german_vars, threshmin=0.05, threshmax=0.99, prepend=c("plot","Diversity"), maxN=7)

Next, let’s perform an analysis on just the 0.8 threshold data.

    mfuncGermanyLinear08<-glm(funcMaxed ~ Diversity, data=subset(germanyThresh, germanyThresh$thresholds=="0.8"), family=quasipoisson(link="identity"))

    Anova(mfuncGermanyLinear08, test.statistic="F")

    # Analysis of Deviance Table (Type II tests)
    #
    # Response: funcMaxed
    # Error estimate based on Pearson residuals
    #
    #           Sum Sq Df F value Pr(>F)
    # Diversity  6.375  1  6.4928 0.0135 *
    # Residuals 56.944 58
    # ---
    # Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

    summary(mfuncGermanyLinear08)

    #
    # Call:
    # glm(formula = funcMaxed ~ Diversity, family = quasipoisson(link = "identity"),
    #     data = subset(germanyThresh, germanyThresh$thresholds ==
    #         "0.8"))
    #
    # Deviance Residuals:
    #     Min       1Q   Median       3Q      Max
    # -1.4741  -0.9796  -0.8101   0.8364   2.0621
    #
    # Coefficients:
    #             Estimate Std. Error t value Pr(>|t|)
    # (Intercept)  0.27756    0.11302   2.456   0.0171 *
    # Diversity    0.05056    0.02476   2.042   0.0457 *
    # ---
    # Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    #
    # (Dispersion parameter for quasipoisson family taken to be 0.9817929)
    #
    #     Null deviance: 66.909  on 59  degrees of freedom
    # Residual deviance: 60.534  on 58  degrees of freedom
    # AIC: NA
    #
    # Number of Fisher Scoring iterations: 6

To get a better sense of how robust these results are to threshold choice, let’s plot the diversity versus number of functions greater than or equal to a threshold for four different thresholds.

    gcPlot<-subset(germanyThresh, germanyThresh$thresholds %in% qw(0.2, 0.4, 0.6, 0.8)) #note, using qw as %in% is a string comparison operator

    gcPlot$percent<-paste(100*gcPlot$thresholds, "%", sep="")

    qplot(Diversity, funcMaxed, data=gcPlot, facets=~percent) +
      stat_smooth(method="glm",
                  method.args = list(family=quasipoisson(link="identity")),
                  colour="red", lwd=1.2) +
      ylab(expression("Number of Functions" >= Threshold)) +
      xlab("Species Richness") +
      theme_bw(base_size=14) +
      geom_text(data=data.frame(percent = unique(gcPlot$percent),
                           lab = paste(letters[1:4], ")", sep=""),
                           Diversity=2,
                           funcMaxed=6
                           ), mapping=aes(x=Diversity, y=funcMaxed, label=lab))

Given that variation, let’s look at the entire spread of thresholds.

    germanyThresh$percent <- 100*germanyThresh$thresholds
    ggplot(data=germanyThresh, aes(x=Diversity, y=funcMaxed, group=percent)) +
        ylab(expression("Number of Functions" >= Threshold)) +
        xlab("Species Richness") +
        stat_smooth(method="glm",
                    method.args = list(family=quasipoisson(link="identity")),
                    lwd=0.8, fill=NA, aes(color=percent)) +
        theme_bw(base_size=14) +
        scale_color_gradient(name="Percent of \nMaximum", low="blue", high="red")

## Multiple Threshold Approach

To systematically explore the variation in the relationship based on threshold choice, let’s look at the slope of the relationship and its confidence interval over all thresholds. We will then plot this relationship.

    germanyLinearSlopes<-getCoefTab(funcMaxed ~ Diversity,
                                    data = germanyThresh,
                                    coefVar = "Diversity",
                                    family = quasipoisson(link="identity"))

    ######
    # Plot the values of the diversity slope at
    # different levels of the threshold
    ######
    germanSlopes <- ggplot(germanyLinearSlopes, aes(x=thresholds*100,
                                                    y = estimate,
                                                    ymax = estimate + 1.96 * std.error,
                                                    ymin = estimate - 1.96 * std.error)) +
      geom_ribbon(fill="grey50") +
      geom_point() +
      ylab("Change in Number of Functions per Addition of 1 Species\n") +
      xlab("\nThreshold (%)") +
      geom_abline(intercept=0, slope=0, lwd=1, linetype=2) +
      theme_bw(base_size=14)

    germanSlopes

We can easily see a number of indices mentioned in the text - Tmin, Tmax, Tmde, etc. Let’s pull them out with our indices function.

    germanIDX <- getIndices(germanyLinearSlopes, germanyThresh, funcMaxed ~ Diversity)
    germanIDX

    #   Tmin Tmax Tmde Rmde.linear Pmde.linear     Mmin    Mmax    Mmde nFunc
    # 1 0.11 0.81 0.41   0.1734241    0.554957 5.050544 1.06222 4.53135     5

We can now add annotations to our earlier plot to make it more rich, and show us those crucial threshold values.

    germanyLinearSlopes$estimate[which(germanyLinearSlopes$thresholds==germanIDX$Tmde)]

    # [1] 0.1734241

    germanyThresh$IDX <- 0
    germanyThresh$IDX [which(germanyThresh$thresholds %in%
                               c(germanIDX$Tmin, germanIDX$Tmax, germanIDX$Tmde))] <- 1

    ggplot(data=germanyThresh, aes(x=Diversity, y=funcMaxed, group=percent)) +
      ylab(expression("Number of Functions" >= Threshold)) +
      xlab("Species Richness") +
      geom_smooth(method="glm",
                  method.args = list(family=quasipoisson(link="identity")),
                  fill=NA, aes(color=percent, lwd=IDX)) +
      theme_bw(base_size=14) +
      scale_color_gradient(name="Percent of \nMaximum", low="blue", high="red") +
      scale_size(range=c(0.3,5), guide="none") +
      annotate(geom="text", x=0, y=c(0.2,2,4.6), label=c("Tmax", "Tmde", "Tmin")) +
      annotate(geom="text", x=16.7, y=c(germanIDX$Mmin, germanIDX$Mmax, germanIDX$Mmde), label=c("Mmin", "Mmax", "Mmde"))

Or we can see where they are on the curve of slopes at different thresholds.

    germanSlopes + annotate(geom="text", y=c(-0.01, -0.01, -0.01, germanIDX$Rmde.linear+0.02), x=c(germanIDX$Tmin*100, germanIDX$Tmde*100, germanIDX$Tmax*100, germanIDX$Tmde*100),  label=c("Tmin", "Tmde", "Tmax", "Rmde"), color="black")

## Comparison of Sites

Furthermore, we can extend this to the more of the BIODEPTH dataset to compare the strength of multifunctionality across sites. We begin by loading in the data and calculating slopes at different thresholds. We’ll subset down to Sheffield, Portugal, and Sweden for simplicity, but this could be done for all sites quite easily.

    ####
    # Now we will look at the entire BIODEPTH dataset
    #
    # Note the use of dplyr to generate the new data frame using all locations from biodepth.
    # If you're not using dplyr for your data aggregation, you should be!
    # It will save you a lot of time and headaches in the future.
    ####

    #Read in data  to run sample analyses on the biodepth data
    data(all_biodepth)
    sub_biodepth<-subset(all_biodepth, all_biodepth$location %in% c("Sheffield", "Portugal", "Sweden"))
    sub_biodepth$location<-factor(sub_biodepth$location, levels=c("Portugal", "Sweden", "Sheffield"))

    allVars<-qw(biomassY3, root3, N.g.m2,  light3, N.Soil, wood3, cotton3)
    varIdx<-which(names(sub_biodepth) %in% allVars)

    #re-normalize so that everything is on the same
    #sign-scale (e.g. the maximum level of a function is
    #the "best" function)
    sub_biodepth <- sub_biodepth %>%
      group_by(location) %>%
      mutate(
        light3 <- ifelse(sum(is.na(light3)==0),
          -1*light3+max(light3, na.rm=T),
          NA),

        N.Soil <- ifelse(sum(is.na(N.Soil)==0),
          -1*N.Soil+max(N.Soil, na.rm=T),
          NA)
      )  %>%
      ungroup()

    #get thresholds
    bdThreshes<-sub_biodepth %>%
      group_by(location) %>%
      nest() %>%
      mutate(fmaxed = map(data, getFuncsMaxed,
                             vars=allVars,
                             maxN=8)) %>%
      ungroup() %>%
      unnest(fmaxed)

    ####look at slopes

    #note, maxIT argument is due to some fits needing more iterations to converge
    bdLinearSlopes<-getCoefTab(funcMaxed ~ Diversity,
                               data=bdThreshes,
                               groupVar=c("location", "thresholds"),
                               coefVar="Diversity",
                               family=quasipoisson(link="identity"),
                               control=list(maxit=800))

    # Error : no valid set of coefficients has been found: please supply starting values
    # Error : no valid set of coefficients has been found: please supply starting values
    # Error : no valid set of coefficients has been found: please supply starting values
    # Error : no valid set of coefficients has been found: please supply starting values
    # Error : no valid set of coefficients has been found: please supply starting values
    # Error : no valid set of coefficients has been found: please supply starting values
    # Error : no valid set of coefficients has been found: please supply starting values
    # Error : no valid set of coefficients has been found: please supply starting values
    # Error : no valid set of coefficients has been found: please supply starting values
    # Error : no valid set of coefficients has been found: please supply starting values
    # Error : no valid set of coefficients has been found: please supply starting values
    # Error : no valid set of coefficients has been found: please supply starting values
    # Error : no valid set of coefficients has been found: please supply starting values
    # Error : no valid set of coefficients has been found: please supply starting values
    # Error : no valid set of coefficients has been found: please supply starting values
    # Error : no valid set of coefficients has been found: please supply starting values
    # Error : no valid set of coefficients has been found: please supply starting values
    # Error : no valid set of coefficients has been found: please supply starting values
    # Error : no valid set of coefficients has been found: please supply starting values
    # Error : no valid set of coefficients has been found: please supply starting values
    # Error : no valid set of coefficients has been found: please supply starting values
    # Error : no valid set of coefficients has been found: please supply starting values
    # Error : no valid set of coefficients has been found: please supply starting values

Great, now we can look at the indices of these three new sites. We’ll add Germany in so we can compare them against the German values.

    indexTable <- lapply(levels(bdLinearSlopes$location), function(x){
      slopedata <- subset(bdLinearSlopes, bdLinearSlopes$location==x)
      threshdata <- subset(bdThreshes, bdThreshes$location==x)
      ret <- getIndices(slopedata, threshdata, funcMaxed ~ Diversity)
      ret<-cbind(location=x, ret)
      ret
    })
    indexTable <- do.call(rbind, indexTable)

    indexTable <- rbind(data.frame(location="Germany", germanIDX), indexTable)

    indexTable

    #     location Tmin Tmax Tmde Rmde.linear Pmde.linear     Mmin     Mmax     Mmde
    # 1    Germany 0.11 0.81 0.41   0.1734241   0.5549570 5.050544 1.062220 4.531350
    # 13  Portugal   NA 0.29 0.19   0.1715245   0.4002239       NA 4.466414 5.705561
    # 11    Sweden 0.05 0.47 0.11   0.1486650   0.2973299 5.943906 1.998405 5.725218
    # 12 Sheffield 0.06   NA 0.55   0.1198584   0.3595751 3.709067       NA 3.212588
    #    nFunc
    # 1      5
    # 13     6
    # 11     6
    # 12     4

Finally, let’s compare a visualization of the relevant curves at different thresholds to the relationship between threshold and slope of curve to give the table of numbers some visual meaning.

    library(gridExtra)

    bdCurvesAll<-qplot(Diversity, funcMaxed, data=bdThreshes, group=thresholds, alpha=I(0)) +
      facet_wrap(~location)+#, scales="free") +
      scale_color_gradient(low="blue", high="red", name="Proportion of \nMaximum", guide=FALSE) +
      stat_smooth(method="glm", lwd=0.8, fill=NA,
                  method.args = list(family=gaussian(link="identity"),
                                     control=list(maxit=200)), aes(color=thresholds)) +
      ylab("\nNumber of Functions ≥ Threshold\n\n") +
      xlab("Species Richness") +
      theme_bw(base_size=15)  +
      geom_text(data=data.frame(location = levels(bdThreshes$location), lab = paste(letters[1:3], ")", sep=""), thresholds=c(NA, NA, NA)), x=1, y=7, mapping=aes(label=lab))

    #Plot it!
    slopePlot<-ggplot(bdLinearSlopes, aes(x=thresholds*100,
                                          y=estimate,
                                          ymin = estimate - 2*std.error,
                                          ymax = estimate + 2*std.error)) +
      geom_ribbon(fill="grey50") +
      geom_point() +
      ylab("Change in Number of Functions \nper Addition of 1 Species\n") +
      xlab("Threshold (%)") +
      facet_wrap(~location)+#, scale="free") +
      geom_abline(intercept=0, slope=0, lwd=0.6, linetype=2) +
      theme_bw(base_size=15)+
      geom_text(data=data.frame(location = levels(bdThreshes$location),
                                lab = paste(letters[4:6], ")",
                                            sep=""),
                                thresholds=c(NA, NA, NA),
                                std.error=c(NA, NA, NA),
                                estimate = c(NA, NA, NA)
                                ),
                x=0.05, y=0.27, mapping=aes(label=lab))

    ###Plot them in a single figure
    grid.arrange(bdCurvesAll,
                 slopePlot)