Chapter 18 Cross-cutting Questions

18.1 Exercise 1

This exercise will build on Exercise 4.5, which uses data from a paper published by Catherine Polik who took FW8051 in the Spring of 2023 (Polik et al., 2018).

We are going to model the relationship between sea-surface temperature (SST) anomalies and a nitrogen isotropic signature ( $δ^{15} N$ ) measured in two different types of sediment:

Sapropels, which are dark-colored sediments, rich in organic matter.
Marl sediment, which is carbon-poor and next to the Sapropels.

The temperature anomalies (our response variable) represent discrepancies between two different methods for reconstructing sea-surface temperature from molecular signatures in marine sediment cores ( $T E X_{86}$ and $U_{37}^{K^{'}}$ ). The $δ^{15} N$ measurements are lower in Sapropel sediments and likely capture N-cycling dynamics during periods of high productivity. The accelerated growth of ammonia-oxidizing organisms (in the Archaea domain) is thought to impact SST measurements reconstructed using the $T E X_{86}$ method, and thus, likely impact the temperature anomalies.

The data set can be accessed from the Data4Ecologists package as follows:

library(Data4Ecologists)
library(ggplot2)
data(PolikSST)

The data set contains the following variables:

Site = Location where the core was taken during Leg 160 of the Ocean Drilling Program
Sapropel = Three sapropels sampled: S3 (81 ka), S5 (124 ka), and i-282 (2.943 Ma)
mbsf = Meters below seafloor, i.e. the depth of the sample
d15N.TN = The nitrogen isotopic value of the bulk sediment, in permil
TEX86.SST = SST reconsctructed using the TEX86 method, in $^{\circ}$ C
Uk37.SST = SST reconsctructed using the Uk37 method, in $^{\circ}$ C
Temp.Anomaly The difference between TEX86 and Uk37 SST reconstructions
Sediment = Whether the sample comes from inside the sapropel or from the surrounding, carbon-poor marl sediment

Before starting the exercise, filter out any observations that are missing values for either Temp.Anomaly or d15N.TN using the following code:

library(dplyr)
PolikSST <- PolikSST %>% 
  filter(!is.na(Temp.Anomaly)) %>%
  filter(!is.na(d15N.TN))

Fit a model using lm where Temp.Anomaly depends only on d15N.TN, but use a quadratic polynomial to model this relationship. Make sure to use raw polynomials (poly(x, 2, raw = TRUE)) or create the squared term and append it to the data set prior to fitting (e.g., PolikSST$d15.TNsquared <- PolikSST$d15.TN*PolikSST$d15.TN). We will assume that all assumptions of the regression model are met. We can then use derivatives and calculus to estimate the value of d15N.TN associated with the maximum temperature anomaly. Our fitted model implies:

$E [Y | X] = β_{0} + β_{1} X + β_{2} X^{2}$

Let’s take a derivative of this experession with respect to $X$ , set it equal to 0, and solve for $X$ :

$\frac{d E [Y | X]}{d X} = β_{1} + 2 β_{2} X = 0$ .

This implies that the maximum occurs at $X_{m a x} = \frac{- β_{1}}{2 β_{2}}$ .

Plug in the estimated coefficients to estimate the value of d15N.TN associated with the maximum expected temperature anomaly.

Determine a confidence interval for $X_{m a x}$ using bootstrapping or the delta method.
Fit the same model in a Bayesian framework using JAGS. Make sure to check that your model has converged. Then, calculate a 95% credible interval using the posterior distributions of $β_{1}$ and $β_{2}$ .
Explain how uncertainty is measured in the frequentist approach and contrast that to how uncertainty is measured in the Bayesian approach. This needs to be in your own words and should be constructed without the help of large language models/AI. Further, you should be prepared to explain your answer in person if I request that you do so.

References

Polik, C. A., Elling, F. J., & Pearson, A. (2018). Impacts of paleoecology on the TEX86 sea surface temperature proxy in the pliocene-pleistocene mediterranean sea. Paleoceanography and Paleoclimatology, 33(12), 1472–1489.