Chapter 9

The aim of these exercises is to extend the analyses from Chapter 8, focusing on identifying the relative importance of predictors in linear models that include at least one continuous predictor.

A. Floral traits and fitness components in milkweed

Recall the example of La Rosa and Conner (2017) from the Chapter 8 exercises. They examined effects of up to six floral traits on fitness components of milkweeds, Asclepias spp. The fitness components were male and female pollination success and female reproductive success.

The data are available from Dryad here. Fitness component estimates were relativized by dividing by the mean, and the traits were standardized to a mean of zero and standard deviation of one. You can also get the data from larosa.csv.

df <- read.csv("../data/larosa.csv")
knitr::kable(head(df,10), booktabs=TRUE) %>%
     kableExtra::kable_styling(latex_options = c("HOLD_position","scale_down","striped"))
species plant.id gyn.width hood.length hood.height horn.reach slit.length gap.width display.flowers.1day removals.per.flower insertions.per.flower fruits geo_mean relz.gyn.w relz.hood.l relz.hood.h relz.horn.r relz.slit.l relz.gap.w remins.std.gyn.w remins.std.hood.l remins.std.hood.h remins.std.horn.r remins.std.slit.l remins.std.gap.w std.floral.display.1day rel.removal.per.flower rel.insertion.per.flower fruit.std.gyn.w fruit.std.hood.l fruit.std.hood.h fruit.std.horn.r fruit.std.slit.l fruit.std.gap.w fruit.std.display.size rel.fruits notes total.flower.quantity poll.duration.per10min.no.flies. poll.visits.per10min.no.flies.
Asyr AS01 0.2190 0.3557 0.5095 0.2038 0.1918 0.0464 168 3.0308 0.3231 7 0.2954 0.7414 1.2042 1.7249 0.6900 0.6493 0.1571 0.0126 0.7453 0.0231 -0.0052 0.1851 -1.2587 0.8513 1.5032 0.7706 -0.0338 0.6940 -0.0218 -0.0925 0.1087 -1.2607 0.7907 1.176 325.0 0.512 0.480
Asyr AS02 0.2042 0.2776 0.5293 0.1820 0.1843 0.0597 120 3.0732 1.0244 3 0.2727 0.7488 1.0180 1.9410 0.6674 0.6758 0.2189 -1.2374 -1.1126 0.4765 -0.6511 -0.4590 -0.1277 0.1527 1.5242 2.4433 -1.2902 -1.1224 0.4261 -0.7732 -0.5484 -0.1345 0.1036 0.504 369.0 0.375 2.413
Asyr AS03 0.2233 0.2771 0.5083 0.1809 0.1687 0.0664 117 0.3226 0.1935 10 0.2699 0.8274 1.0267 1.8833 0.6703 0.6251 0.2460 0.3758 -1.1245 -0.0044 -0.6837 -1.7987 0.4420 0.1090 0.1600 0.4616 0.3313 -1.1340 -0.0489 -0.8075 -1.9150 0.4329 0.0607 1.680 372.0 0.000 0.000
Asyr AS04 0.2081 0.2403 0.4241 0.1520 0.1796 0.0515 76 1.2475 0.2970 3 0.2484 0.8377 0.9673 1.7071 0.6118 0.7229 0.2073 -0.9080 -1.9999 -1.9328 -1.5400 -0.8626 -0.8251 -0.4878 0.6187 0.7084 -0.9591 -1.9899 -1.9533 -1.7100 -0.9601 -0.8288 -0.5262 0.504 303.0 0.188 0.665
Asyr AS05 0.2136 0.3544 0.5368 0.1940 0.1842 0.0459 148 3.5573 0.9008 3 0.2941 0.7262 1.2049 1.8250 0.6596 0.6262 0.1560 -0.4435 0.7144 0.6483 -0.2956 -0.4676 -1.3013 0.5602 1.7643 2.1484 -0.4922 0.6638 0.5957 -0.3985 -0.5571 -1.3030 0.5044 0.504 262.0 4.736 11.667
Asyr AS07 0.2276 0.2984 0.4867 0.2161 0.1953 0.0840 61 1.5000 0.2400 1 0.2835 0.8030 1.0527 1.7170 0.7624 0.6890 0.2963 0.7389 -0.6178 -0.4991 0.3592 0.4857 1.9387 -0.7061 0.7439 0.5724 0.6964 -0.6386 -0.5375 0.2916 0.4153 1.9232 -0.7409 0.168 150.0 2.138 0.668
Asyr AS08 0.2103 0.3155 0.4390 0.1806 0.1967 0.0581 69 0.1833 0.3333 0 0.2751 0.7644 1.1468 1.5957 0.6564 0.7150 0.2112 -0.7222 -0.2110 -1.5915 -0.6926 0.6059 -0.2638 -0.5896 0.0909 0.7950 -0.7724 -0.2409 -1.6163 -0.8169 0.5379 -0.2700 -0.6264 0.000 90.0 0.000 0.000
Asyr AS09 0.2261 0.3385 0.5744 0.2101 0.1898 0.0597 388 2.5053 0.3915 1 0.3022 0.7481 1.1200 1.9005 0.6952 0.6280 0.1975 0.6122 0.3361 1.5094 0.1815 0.0134 -0.1277 4.0535 1.2426 0.9337 0.5690 0.2940 1.4462 0.1042 -0.0665 -0.1345 3.9399 0.168 421.5 8.181 5.167
Asyr AS10 0.2214 0.3508 0.5406 0.2295 0.1997 0.0525 47 2.0200 0.5600 0 0.3026 0.7317 1.1593 1.7865 0.7584 0.6599 0.1735 0.2153 0.6287 0.7353 0.7563 0.8636 -0.7400 -0.9099 1.0018 1.3357 0.1700 0.5801 0.6817 0.7100 0.8008 -0.7442 -0.9413 0.000 50.0 0.000 0.000
Asyr AS11 0.2175 0.3166 0.4320 0.1501 0.1746 0.0535 35 0.1389 0.0278 0 0.2685 0.8102 1.1793 1.6092 0.5591 0.6504 0.1993 -0.1141 -0.1849 -1.7518 -1.5963 -1.2920 -0.6550 -1.0845 0.0689 0.0663 -0.1611 -0.2154 -1.7747 -1.7693 -1.3981 -0.6595 -1.1131 0.000 36.0 3.410 0.628 
df_syr<-subset(df,species=="Asyr")
df_vir<-subset(df, species=='Avir')
df_tub<-subset(df, species=='Atub')

For each fitness component and for each species separately (i.e. 6 models in total):

Refit the linear models from the Chapter 8 exercises but now use the recommended methods from this chapter (hierarchical partitioning or LMD, PMVD) to assess each predictor’s relative importance in each of the models.

If you’re really keen, there’s a third species in the dataframe (Atub).

Now use AIC and Aikake weights to find the most parsimonious model (best fit with fewest predictors) for each combination of fitness component and species. If there are multiple models with similar AICs, then use full (zero) model averaging to produce a final model.

B. Diet and land-use drives mercury accumulation in wolverines

Peraza et al. (2023) studied what factors drove mercury accumulation in muscle tissues of a high-altitude carnivore, the wolverine (Gulo gulo). Wolverine muscle (for Hg) and hair (for N and C stable isotopes) samples were obtained from carcasses submitted by trappers and hair snags across four Canadian provinces. We will focus on total Hg concentration (µg.gdw) in muscle as the response and 14 predictor variables measured at the point of collection:

  • delta15N and delta13C from hair samples,
  • longitude,
  • latitude,
  • precipitation,
  • mean maximum and mean minimum temperatures,
  • elevation,
  • soil organic C,
  • distance from nearest coast,
  • mean soil pH at 10cm and 60cm depths, and
  • Hg net and Hg wet deposition rates.

Start by reading in the data.

peraza <- read.csv("../data/peraza clean.csv")
knitr::kable(head(peraza,10), booktabs=TRUE) %>%
     kableExtra::kable_styling(latex_options = c("HOLD_position","scale_down","striped"))
ageclass sex long lat thg delta15n delta13c hgdep hgwet prec tempmax tempmin elev dist soc sph10 sph60 X
Adult Male -139.2 64.4 0.095 5.775 -24.480 8.644 5.731 30.459 0.242 -10.445 1205 496304.4 19.000 56 58 NA
Yearling Female -136.5 62.2 1.184 9.570 -25.438 10.364 3.209 22.764 3.282 -8.797 954 721306.5 144.104 56 60 NA
Adult Female -137.4 62.8 0.498 6.692 -25.216 12.601 2.918 25.705 2.845 -8.578 687 653101.6 114.319 54 60 NA
Yearling Female -133.1 60.7 0.308 5.747 -25.773 9.165 3.253 28.955 4.186 -7.525 926 883503.6 69.691 56 58 NA
Adult Male -135.4 63.3 0.058 6.330 -25.684 9.067 2.700 31.001 3.142 -9.092 598 590826.8 106.367 66 70 NA
Yearling Male -139.2 64.4 0.084 5.594 -23.896 8.644 5.731 30.459 0.242 -10.445 1205 496304.4 19.000 56 58 NA
Adult Male -140.6 64.6 0.120 4.000 -33.459 11.238 2.964 25.016 3.363 -8.605 442 501614.8 93.346 57 61 NA
Yearling Male -137.4 62.8 0.595 7.223 -26.486 12.601 2.918 25.705 2.845 -8.578 687 653101.6 114.319 54 60 NA
Yearling Male -137.4 62.8 0.309 7.280 -25.037 12.601 2.918 25.705 2.845 -8.578 687 653101.6 114.319 54 60 NA
Adult Male -132.0 61.3 0.052 4.728 -25.333 6.673 3.062 31.631 3.376 -9.474 1048 830036.0 81.455 55 58 NA

We will fit a multiple linear regression model relating total Hg to the predictors.

Do the usual pre-analysis checks of assumptions using boxplots, a scatterplot matrix and VIFs.

Note the strong collinearity between min and max temperature, between soil pH at 10 and 60cm and between distance from coast and latitude. The authors removed min temperature, latitude and pH at 60cm from their model.

Fit a multiple regression model relating log total Hg to the remaining 11 predictors and check the residual plot.

Any indication of outliers of concern?

We recommend you proceed with the model with all data, but note that Perazo et al. omitted 17 observations as outliers so your results will differ somewhat from theirs.

What conclusions would you draw from your linear model?

Now use the methods from the first question to evaluate the relative importance of the different predictors and find the most parsimonious model.

C. Predictors of insect richness in freshwater streams

Tonkin et al. (2015) Tonkin et al (2022) surveyed 80 freshwater stream sites in mid-latitude China to determine how different climate and catchment (watershed) variables predicted the richness of three different insect groups (Ephemeroptera, Plecoptera, Trichoptera; collectively abbreviated as EPT). They recorded 32 predictor variables in total but to avoid collinearity (r > 0.7), only 17 variables were included in the analyses (see their Table 1).

The full dataset is available at http://dx.doi.org/10.6084/m9.figshare.1305679 but a tidied-up version including only the non-collinear predictors is available here. We will also not include region (a categorical variable) as a predictor, resulting in 16 predictors. Tonkin et al used Poisson regression models to link richness to these predictors but for the purposes of this chapter, we will treat richness as normally distributed (its distribution wasn’t very skewed - you can use boxplots to see if you agree).

tonkin <- read.csv("../data/tonkin.csv")
head(tonkin, 10)
##    sitecode region ept ephem plec trich trees_bl  trees_nl    shrub herbaceous
## 1      BS01   East   7     6    0     1  0.00000 100.00000  0.00000          0
## 2      BS02   East  14    10    1     3  0.00000 100.00000  0.00000          0
## 3      BS03   East  16    13    0     3  0.00000  84.61538 15.38462          0
## 4      BS04   East  17    14    0     3  0.00000 100.00000  0.00000          0
## 5      BS05   East  11     6    2     3  0.00000 100.00000  0.00000          0
## 6      BS06   East  12    10    1     1  0.00000 100.00000  0.00000          0
## 7      BS07   East  17     7    6     4  0.00000 100.00000  0.00000          0
## 8      BS08   East  18    12    4     2 42.85714  57.14286  0.00000          0
## 9      BS09   East  27    13    6     8 66.66667  33.33333  0.00000          0
## 10     BS10   East   9     7    1     1  0.00000 100.00000  0.00000          0
##    cultivated water bio1 bio4 bio8 bio15 bio18    ai  pet elevation    slope
## 1           0     0  170 8420  211    55   573 14284 1153        99 2.631222
## 2           0     0  157 8406  198    53   588 15367 1062       292 2.318342
## 3           0     0  164 8476  205    54   578 14841 1103       186 2.871562
## 4           0     0  165 8503  206    54   556 14424 1094       145 1.416681
## 5           0     0  164 8354  205    54   585 14670 1124       211 3.630091
## 6           0     0  157 8264  197    54   615 15977 1084       338 5.192832
## 7           0     0  166 8407  206    55   580 14747 1127       157 2.908305
## 8           0     0  165 8400  206    54   582 15317 1104       167 6.563747
## 9           0     0  159 8327  199    54   603 15790 1081       285 6.270956
## 10          0     0  166 8487  207    54   572 14708 1115       126 3.333718
##    catch_size
## 1          76
## 2          11
## 3          13
## 4           5
## 5           1
## 6           8
## 7           1
## 8           7
## 9           6
## 10          4

Fit a multiple linear regression model relating total EPT richness to the 16 predictors. Note that the ratio of observations to predictors is very marginal!

Which predictors had the strongest influence on EPT richness?

Now use a standard regression tree to relate richness to the predictors.

How do the results compare?

Tonkin et al used boosted regression trees (BRTs) - Fit a BRT

Use the same settings as they did (with the default bag fraction of 0.5):

  • a slow learning rate of 0.0005 (to ensure 1000 trees were reached) and
  • tree complexity at 5.
  • model validation was based on 10-fold cross-validation.

References

La Rosa, Raffica J., and Jeffrey K. Conner. 2017. “Floral Function: Effects of Traits on Pollinators, Male and Female Pollination Success, and Female Fitness Across Three Species of Milkweeds ( Asclepias ).” American Journal of Botany 104 (1): 150–60. https://doi.org/gr2r9p.
Peraza, Inés, John Chételat, Murray Richardson, Thomas S. Jung, Malik Awan, Steve Baryluk, Ashu Dastoor, et al. 2023. “Diet and Landscape Characteristics Drive Spatial Patterns of Mercury Accumulation in a High-Latitude Terrestrial Carnivore.” Edited by Asif Qureshi. PLOS ONE 18 (5): e0285826. https://doi.org/gsn8rr.
Tonkin, Jonathan D., Deep Narayan Shah, Mathias Kuemmerlen, Fengqing Li, Qinghua Cai, Peter Haase, and Sonja C. Jähnig. 2015. “Climatic and Catchment-Scale Predictors of Chinese Stream Insect Richness Differ Between Taxonomic Groups.” Edited by Kyle A. Young. PLOS ONE 10 (4): e0123250. https://doi.org/gsn58k.