Code 
library(tidyverse)
library(patchwork)
library(likelihood)
library(kableExtra)
source("../scripts/aux.R")

annual_pet <- read_csv("../data/spei_climate.csv") |> 
  dplyr::select(sp_elev, year, monthly_pet, monthly_tmed, monthly_prec) |> 
  group_by(sp_elev, year) |>
  summarise(pet = sum(monthly_pet), 
            prec = sum(monthly_prec),
            tmed = mean(monthly_tmed, na.rm = TRUE)) |> 
  rowwise() |> 
  mutate(water_balance = prec - pet)

ratio <- read_csv("../data/ratio_abinpp.csv") |> 
  dplyr::select(year, sp_code, elev_code, sp_elev, ratio_abi_npp = ratio) 

df <- ratio |> 
  inner_join(annual_pet) |> 
  pivot_longer(pet:water_balance, values_to = "mean_climate", 
               names_to = "climate_variable")

Univariate models

Temperature

\[(ABI:NPP)_i=MR \times g(T_{med}) + \epsilon_i\] where:

  • \((ABI:NPP)_i\) is the value of the ABI:NPP ratio for each site and year
  • \(MR\) is a parameter representing the maximum ABI:NPP ratio (dimensionless)
  • \(\epsilon_i\) is the random error for each observation \(i\).
  • \(g(T_{med})\) is a function with values between 0 and 1, which depends on temperature. This function, \(g(T_{med})\), can take different forms.

Proposed Functions for Temperature (Table S6)

  • Gaussian (mtrat1)
    \[g(T_{med})=\exp \left[ -\frac{1}{2} \left(\frac{TMED-T_{OPT}} {T_b}\right)^2 \right]\]

  • Log-normal (mtrat3)
    \[g(T_{med})=\exp \left[ -\frac{1}{2} \left(\frac{\log{(\frac{TMED}{T_{OPT}}}) }{T_b}\right)^2 \right]\]

  • Logistic modified (mtrat6)
    \[g(T_{med})=(1-T_a) \times \left[\frac{1}{1+\exp (-k \times (TMED - T_{OPT})) }\right] + T_a\]

where:

  • \(TMED\) is the value of the annual mean temperature
  • \(T_{OPT}\) = optimum value of the annual mean temperature at which occur the maximum ratio (Lognormal and Gaussian models) or that represent the half-saturation point (logistic part in the Logistic modified model)
  • \(T_b\) = scale parameter
  • \(k\) = rate that determines the steepness of the logistic curve
  • \(T_a\) = scaling factor that adjust the influence of the logistic function.

The error term in the model

\[(ABI:NPP)_i=MR \times g(T_{med}) + \epsilon_i\] is assumed to follows a normal distribution \(\epsilon_i \sim \mathcal{N}(0,1)\)

Code 
dtmed <- df |> 
  filter(climate_variable == "tmed") |> 
  dplyr::select(rat = ratio_abi_npp, tmed = mean_climate)



mtrat1 <- function(MR, Topt, Tb)
{
  tmed_effect <- exp(-0.5*((dtmed$tmed - Topt)/Tb)^2)
  MR * tmed_effect
}

mtrat3 <- function(MR, Topt, Tb)
{
  tmed_effect <- exp(-0.5*(log(dtmed$tmed/Topt)/Tb)^2)
  MR * tmed_effect
}

mtrat6 <- function(MR, k, Topt, Ta){
  tmed_effect <- (1-Ta)*(1 / (1 + exp(- k *( dtmed$tmed - Topt)))) + Ta 
  MR*tmed_effect
}


name_models <- data.frame(
  modelo = paste0("mtrat",c(1,3,6)), 
  name_modelo = c("Gaussian",
                  "Log-normal",
                  "Logistic_Mod")
)

var=list(mean = "predicted", x = "rat", tmed = "tmed", log = TRUE)  

## Guassian model (temp)
set.seed(1234)
result_mtrat1 <- anneal(
  model = mtrat1, 
  var = var, 
  source_data = dtmed,
  par = list(MR = 0.7, Topt = 10.5, Tb = 1.5),
  par_lo = list(MR = 0, Topt = 5, Tb = 0.01),
  par_hi = list(MR = 1, Topt = 20, Tb = 5),
  pdf = dnorm,
  dep_var = "rat",
  initial_temp = 5, 
  temp_red = 0.975, 
  max_iter = 100000, 
  show_display = FALSE
)

result_mtrat1 |> 
  likelihood::write_results("../output/models/mtrat1.txt")

#Log-normal (temp)
set.seed(1234)

result_mtrat3 <- anneal(
  model = mtrat3, 
  var = var, 
  source_data = dtmed,
  par = list(MR = 0.7, Topt = 10.5, Tb = 1.5),
  par_lo = list(MR = 0, Topt = 5, Tb = 0.01),
  par_hi = list(MR = 1, Topt = 20, Tb = 5),
  pdf = dnorm,
  dep_var = "rat",
  initial_temp = 5, 
  temp_red = 0.975, 
  max_iter = 100000, 
  show_display = FALSE
)

result_mtrat3 |> 
  likelihood::write_results("../output/models/mtrat3.txt")

# Logistic mod. (temp)
set.seed(1234)

result_mtrat6 <- anneal(
  model = mtrat6, 
  var = var, 
  source_data = dtmed,
  par = list(MR = 0.7, Ta = 0.09, k = -1, Topt = 10),
  par_lo = list(MR = 0, Ta = 0, k = -2, Topt = 0),
  par_hi = list(MR = 1, Ta = 0.2, k = 2, Topt = 20),
  pdf = dnorm,
  dep_var = "rat",
  initial_temp = 5, 
  temp_red = 0.975, 
  max_iter = 100000, 
  show_display = FALSE
)

result_mtrat6 |> 
  likelihood::write_results("../output/models/mtrat6.txt")

all_results <-
  mget(
    grep("^result_mtrat", ls(), value = TRUE)
  ) |>
  purrr::imap_dfr(~ MLE_results_format(.x, yvar = "rat"), .id = "modelo") |>
  arrange(aic_cor) |>
  mutate(modelo = str_remove(modelo, "result_"))


# compute delta AIC and add computation time
models_results <- all_results |>
  # inner_join(all_time) |>
  inner_join(name_models) |>
  relocate(name_modelo, .after = modelo) |> 
  mutate(deltaAIC = dAIC(aic_cor))

models_results |> 
  write_csv("../data/models_ratio_tmed_summary.csv")

Precipitation

\[(ABI:NPP)_i=MR \times g(PREC) + \epsilon_i\] where:

  • \((ABI:NPP)_i\) is the value of the ABI:NPP ratio for each site and year

  • \(MR\) is a parameter representing the maximum ABI:NPP ratio (dimensionless)

  • \(\epsilon_i\) is the random error for each observation \(i\)

  • \(g(PREC)\) is a function with values between 0 and 1, which depends on precipitation. This function, \(g(PREC)\), can take different forms. #### Proposed Functions for Precipitation (Table S6)

  • Gaussian (mprecrat1)
    \[g(Prec)=\exp \left[ -\frac{1}{2} \left(\frac{PREC-P_{OPT}} {P_b}\right)^2 \right]\]

  • Log-normal (mprecrat3)
    \[g(Prec)=\exp \left[ -\frac{1}{2} \left(\frac{\log{(\frac{PREC}{P_{OPT}}}) }{P_b}\right)^2 \right]\]

  • Gompertz (mprecrat5)
    \[g(Prec)=P_a \times \exp \left[ - P_d \times \exp (-Pc \times PREC)\right]\]
    where:

  • \(PREC\) is the value of the annual precipitation

  • \(P_{OPT}\) = value of the precipitation at which the ABI:NPP ratio is expected to be optimum

  • \(P_b\) = scale parameter that adjust for the variability in the response of the ABI:NPP ratio to changes in precipitation

  • \(P_a\) = scaling factor

  • \(P_d\) = modulates the amplitude of the exponential response

  • \(P_c\) = parameter to control the sensitivity of the response to precipitation

Code 
dprec<- df |> 
  filter(climate_variable == "prec") |> 
  dplyr::select(rat = ratio_abi_npp, prec = mean_climate)

mprecrat1 <- function(MR, Popt, Pb)
{
  prec_effect <- exp(-0.5*((dprec$prec - Popt)/Pb)^2)
  MR * prec_effect 
}

mprecrat3 <- function(MR, Popt, Pb)
{
  prec_effect <- exp(-0.5*(log(dprec$prec/Popt)/Pb)^2)
  MR * prec_effect 
}

mprecrat5 <-  function(MR, Pa, Pd, Pc)
{
  prec_effect <- Pa * exp(-Pd * exp (-Pc * dprec$prec))
  MR * prec_effect
}

name_models <- data.frame(
  modelo = paste0("mprecrat",c(1,3,5)),
  name_modelo = c("Gaussian",
                  "Log-normal",
                  "Gompertz")
)

# Gaussian (prec)
set.seed(1234)

result_mprecrat1 <- anneal(
  model = mprecrat1, 
  var = var, 
  source_data = dprec,
  par = list(MR = 0.7, Popt = 500, Pb = 200),
  par_lo = list(MR = 0, Popt = 300, Pb = 0 ),
  par_hi = list(MR = 1, Popt = 1000, Pb = 600),
  pdf = dnorm,
  dep_var = "rat",
  initial_temp = 5, 
  temp_red = 0.975, 
  max_iter = 100000, 
  show_display = FALSE
)

result_mprecrat1 |> 
  likelihood::write_results("../output/models/mprecrat1.txt")

# Log-normal (prec)

result_mprecrat3 <- anneal(
  model = mprecrat3, 
  var = var, 
  source_data = dprec,
  par = list(MR = 0.7, Popt = 500, Pb = 2),
  par_lo = list(MR = 0, Popt = 300, Pb = 0),
  par_hi = list(MR = 1, Popt = 1000, Pb = 5),
  pdf = dnorm,
  dep_var = "rat",
  initial_temp = 5, 
  temp_red = 0.975, 
  max_iter = 100000, 
  show_display = FALSE
)

result_mprecrat3 |> 
  likelihood::write_results("../output/models/mprecrat3.txt")

# Gompertz (prec)
set.seed(1234)

result_mprecrat5 <- anneal(
  model = mprecrat5, 
  par = list(MR = 0.5, Pa = 0.4, Pd = 0.3, Pc = 0.003),
  par_lo = list(MR = 0, Pa = 0.2, Pd = 0.01, Pc = 0.0001),
  par_hi = list(MR = 1, Pa = 0.8, Pd = 5, Pc = 0.01), 
  var = var, 
  source_data = dprec,
  pdf = dnorm,
  dep_var = "rat",
  initial_temp = 5, 
  temp_red = 0.975, 
  max_iter = 100000, 
  show_display = FALSE
)

result_mprecrat5 |> 
  likelihood::write_results("../output/models/mprecrat5.txt")

all_results <-
  mget(
    grep("^result_mprecrat", ls(), value = TRUE)
  ) |>
  purrr::imap_dfr(~ MLE_results_format(.x, yvar = "rat"), .id = "modelo") |>
  arrange(aic_cor) |>
  mutate(modelo = str_remove(modelo, "result_"))

models_results <- all_results |>
  inner_join(name_models) |>
  relocate(name_modelo, .after = modelo) |> 
  mutate(deltaAIC = dAIC(aic_cor))

models_results |> 
  write_csv("../data/models_ratio_prec_summary.csv")

Multiplicative models (Temperature and Precipitation)

\[(ABI:NPP)_i=MR \times g(T_{med}) \times g(PREC) + \epsilon_i\] where:

  • \((ABI:NPP)_i\) is the value of the ABI:NPP ratio for each site and year
  • \(MR\) is a parameter representing the maximum ABI:NPP ratio (dimensionless)
  • \(\epsilon_i\) is the random error for each observation \(i\)
  • \(g(T_{med})\) is a function with values between 0 and 1, which depends on temperature. This function, \(g(T_{med})\), can take different forms
  • \(g(PREC)\) is a function with values between 0 and 1, which depends on precipitation. This function, \(g(PREC)\), can take different forms
Code 
d <- dtmed |> inner_join(dprec)

var=list(mean = "predicted", x = "rat", prec = "prec", tmed = "tmed", log = TRUE)  

### Temperature (Log-normal); Precipitation (Log-normal) (**mcombrat1**)
mcombrat1 <- function(MR, Topt, Tb, Popt, Pb){
  # log-normal temperature
  # log-normal prec 
  tmed_effect <- exp(-0.5*(log(d$tmed/Topt)/Tb)^2) 
  prec_effect <- exp(-0.5*(log(d$prec/Popt)/Pb)^2)
  MR * tmed_effect * prec_effect
}

set.seed(2023)

result_mcombrat1 <- anneal(
  model = mcombrat1, 
  var = var, 
  source_data = d,
  par = list(MR = 0.7, Topt = 10.5, Tb = 1.5, Popt = 500, Pb = 2), 
  par_lo = list(MR = 0, Topt = 5, Tb = 0.01, Popt = 300, Pb = 0),
  par_hi = list(MR = 1, Topt = 20, Tb = 5, Popt = 1000, Pb = 5),
  pdf = dnorm,
  dep_var = "rat",
  initial_temp = 5, 
  temp_red = 0.975, 
  max_iter = 200000, 
  show_display = FALSE
)

result_mcombrat1 |> 
  likelihood::write_results("../output/models/mcombrat1.txt")

### Temperature (Log-normal); Precipitation (Gauss) (**mcombrat2**)
mcombrat2 <- function(MR, Topt, Tb, Popt, Pb){
  # log-normal temperature
  # gauss prec 
  tmed_effect <- exp(-0.5*(log(d$tmed/Topt)/Tb)^2) 
  prec_effect <- exp(-0.5*((d$prec - Popt)/Pb)^2)
  MR * tmed_effect * prec_effect
}

set.seed(2023)

result_mcombrat2 <- anneal(
  model = mcombrat2, 
  var = var, 
  source_data = d,
  par = list(MR = 0.7, Topt = 10.5, Tb = 1.5, Popt = 500, Pb = 200), 
  par_lo = list(MR = 0, Topt = 5, Tb = 0.01, Popt = 300, Pb = 0),
  par_hi = list(MR = 1, Topt = 20, Tb = 5, Popt = 1000, Pb = 600),
  pdf = dnorm,
  dep_var = "rat",
  initial_temp = 5, 
  temp_red = 0.975, 
  max_iter = 200000, 
  show_display = FALSE
)

result_mcombrat2 |> 
  likelihood::write_results("../output/models/mcombrat2.txt")

### Temperature (Log-normal); Precipitation (Gompertz) (**mcombrat3**)
mcombrat3 <- function(MR, Topt, Tb, Pa, Pd, Pc){
  # log-normal temperature
  # gompertz prec 
  tmed_effect <- exp(-0.5*(log(d$tmed/Topt)/Tb)^2) 
  prec_effect <- Pa * exp(-Pd * exp (-Pc * d$prec))
  MR * tmed_effect * prec_effect
}

set.seed(1234)
result_mcombrat3 <- anneal(
  model = mcombrat3, 
  var = var, 
  source_data = d,
  par = list(MR = 0.7, Topt = 10.5, Tb = 1.5, Pa = 0.4, Pd = 0.3, Pc = 0.003), 
  par_lo = list(MR = 0, Topt = 5, Tb = 0.01, Pa = 0.2, Pd = 0.01, Pc = 0.0001),
  par_hi = list(MR = 1, Topt = 20, Tb = 5, Pa = 0.8, Pd = 5, Pc = 0.01),
  pdf = dnorm,
  dep_var = "rat",
  initial_temp = 5, 
  temp_red = 0.975, 
  max_iter = 200000, 
  show_display = FALSE
)

result_mcombrat3 |> 
  likelihood::write_results("../output/models/mcombrat2.txt")

### Temperature (Gauss); Precipitation (Log-normal) (**mcombrat4**)
mcombrat4 <- function(MR, Topt, Tb, Popt, Pb){
  # gauss temperature
  # log-normal prec 
  tmed_effect <- exp(-0.5*((d$tmed - Topt)/Tb)^2)
  prec_effect <- exp(-0.5*(log(d$prec/Popt)/Pb)^2)
  MR * tmed_effect * prec_effect
}

set.seed(2023)

# change model 
result_mcombrat4 <- anneal(
  model = mcombrat4, 
  var = var, 
  source_data = d,
  par = list(MR = 0.7, Topt = 10.5, Tb = 1.5, Popt = 500, Pb = 2), 
  par_lo = list(MR = 0, Topt = 5, Tb = 0.01, Popt = 300, Pb = 0),
  par_hi = list(MR = 1, Topt = 20, Tb = 5, Popt = 1000, Pb = 5),
  pdf = dnorm,
  dep_var = "rat",
  initial_temp = 5, 
  temp_red = 0.975, 
  max_iter = 200000, 
  show_display = FALSE
)

result_mcombrat4 |> 
  likelihood::write_results("../output/models/mcombrat1.txt")

### Temperature (Gauss); Precipitation (Gauss) (**mcombrat5**)
mcombrat5 <- function(MR, Topt, Tb, Popt, Pb){
  # gauss temperature
  # gauss prec 
  tmed_effect <- exp(-0.5*((d$tmed - Topt)/Tb)^2) 
  prec_effect <- exp(-0.5*((d$prec - Popt)/Pb)^2)
  MR * tmed_effect * prec_effect
}

set.seed(2023)

result_mcombrat5 <- anneal(
  model = mcombrat5, 
  var = var, 
  source_data = d,
  par = list(MR = 0.7, Topt = 10.5, Tb = 1.5, Popt = 500, Pb = 200), 
  par_lo = list(MR = 0, Topt = 5, Tb = 0.01, Popt = 300, Pb = 0),
  par_hi = list(MR = 1, Topt = 20, Tb = 5, Popt = 1000, Pb = 600),
  pdf = dnorm,
  dep_var = "rat",
  initial_temp = 5, 
  temp_red = 0.975, 
  max_iter = 200000, 
  show_display = FALSE
)

result_mcombrat5 |> 
  likelihood::write_results("../output/models/mcombrat5.txt")

### Temperature (Gauss); Precipitation (Gompertz) (**mcombrat6**)
mcombrat6 <- function(MR, Topt, Tb, Pa, Pd, Pc){
  # gauss temperature
  # gompertz prec 
  tmed_effect <- exp(-0.5*((d$tmed - Topt)/Tb)^2)
  prec_effect <- Pa * exp(-Pd* exp (-Pc * d$prec))
  MR * tmed_effect * prec_effect
}

set.seed(1234)

result_mcombrat6  <- anneal(
  model = mcombrat6, 
  var = var, 
  source_data = d,
  par = list(MR = 0.7, Topt = 10.5, Tb = 1.5, Pa = 0.4, Pd = 0.3, Pc = 0.003), 
  par_lo = list(MR = 0, Topt = 5, Tb = 0.01, Pa = 0.2, Pd = 0.01, Pc = 0.0001),
  par_hi = list(MR = 1, Topt = 20, Tb = 5, Pa = 0.8, Pd = 5, Pc = 0.01),
  pdf = dnorm,
  dep_var = "rat",
  initial_temp = 5, 
  temp_red = 0.975, 
  max_iter = 200000, 
  show_display = FALSE
)

result_mcombrat6 |> 
  likelihood::write_results("../output/models/mcombrat6.txt")

### Temperature (Logistic mod.); Precipitation (Log-normal) (**mcombrat7**)
mcombrat7 <- function(MR, Ta, k, Topt, Popt, Pb){
  # log-mod temperature
  # log-normal prec 
  tmed_effect <- (1-Ta)*(1 / (1 + exp(- k *( dtmed$tmed - Topt)))) + Ta 
  prec_effect <- exp(-0.5*(log(d$prec/Popt)/Pb)^2)
  MR * tmed_effect * prec_effect
}

set.seed(123456)

result_mcombrat7 <- anneal(
  model = mcombrat7, 
  var = var, 
  source_data = d,
  par = list(MR = 0.7, Ta = 0.09, k = -1, Topt = 10, Popt = 500, Pb = 2), 
  par_lo = list(MR = 0, Ta = 0, k = -2, Topt = 0, Popt = 300, Pb = 0),
  par_hi = list(MR = 1, Ta = 0.2, k = 2, Topt = 20, Popt = 1000, Pb = 5),
  pdf = dnorm,
  dep_var = "rat",
  initial_temp = 5, 
  temp_red = 0.975, 
  max_iter = 200000, 
  show_display = FALSE
)

result_mcombrat7 |> 
  likelihood::write_results("../output/models/mcombrat7.txt")

### Temperature (Logistic mod.); Precipitation (Gaussian) (**mcombrat8**)
mcombrat8 <- function(MR, Ta, k, Topt, Popt, Pb){
  # log-mod temperature
  # gauss prec 
  tmed_effect <- (1-Ta)*(1 / (1 + exp(- k *( dtmed$tmed - Topt)))) + Ta 
  prec_effect <- exp(-0.5*((d$prec - Popt)/Pb)^2)
  MR * tmed_effect * prec_effect
}

set.seed(2023)

result_mcombrat8 <- anneal(
  model = mcombrat8, 
  var = var, 
  source_data = d,
  par = list(MR = 0.7, Ta = 0.09, k = -1, Topt = 10, Popt = 500, Pb = 200), 
  par_lo = list(MR = 0, Ta = 0, k = -2, Topt = 0, Popt = 300, Pb = 0),
  par_hi = list(MR = 1, Ta = 0.2, k = 2, Topt = 20, Popt = 1000, Pb = 600),
  pdf = dnorm,
  dep_var = "rat",
  initial_temp = 5, 
  temp_red = 0.975, 
  max_iter = 200000, 
  show_display = FALSE
)

result_mcombrat8 |> 
  likelihood::write_results("../output/models/mcombrat8.txt")

### Temperature (Logistic mod.); Precipitation (Gompertz) (**mcombrat9**)
mcombrat9 <- function(MR, Ta, k, Topt, Pa, Pd, Pc){
  # log-mod temperature
  # gompertz prec 
  tmed_effect <- (1-Ta)*(1 / (1 + exp(- k *( dtmed$tmed - Topt)))) + Ta 
  prec_effect <- Pa * exp(-Pd * exp (-Pc * d$prec))
  MR * tmed_effect * prec_effect
}

set.seed(1234)
result_mcombrat9  <- anneal(
  model = mcombrat9, 
  var = var, 
  source_data = d,
  par = list(MR = 0.7, Ta = 0.09, k = -1, Topt = 10, Pa = 0.4, Pd = 0.3, Pc = 0.003), 
  par_lo = list(MR = 0, Ta = 0, k = -2, Topt = 0, Pa = 0.2, Pd = 0.01, Pc = 0.0001),
  par_hi = list(MR = 1, Ta = 0.2, k = 2, Topt = 20, Pa = 0.8, Pd = 5, Pc = 0.01),
  pdf = dnorm,
  dep_var = "rat",
  initial_temp = 5, 
  temp_red = 0.975, 
  max_iter = 200000, 
  show_display = FALSE
)

result_mcombrat9 |> 
  likelihood::write_results("../output/models/mcombrat9.txt")

Comparison multiplicative models

  • It corresponds with table S4 of the manuscript
Code 
all_results <-
  mget(
    grep("^result_mcombrat", ls(), value = TRUE)
  ) |>
  purrr::imap_dfr(~ MLE_results_format(.x, yvar = "rat"), .id = "modelo") |>
  arrange(aic_cor) |>
  mutate(modelo = str_remove(modelo, "result_"))

name_models <- data.frame(
  modelo = paste0("mcombrat",c(1:9)),
  name_modelo = c("Temp (Log-normal) | Prec (Log-normal)", 
                  "Temp (Log-normal) | Prec (Gaussian)", 
                  "Temp (Log-normal) | Prec (Gompertz)", 
                  "Temp (Gaussian) | Prec (Log-normal)", 
                  "Temp (Gaussian) | Prec (Gaussian)", 
                  "Temp (Gaussian) | Prec (Gompertz)", 
                  "Temp (Logistic_Mod) | Prec (Log-normal)", 
                  "Temp (Logistic_Mod) | Prec (Gaussian)", 
                  "Temp (Logistic_Mod) | Prec (Gompertz)")
)


# compute delta AIC and add computation time
models_results <- all_results |>
  inner_join(name_models) |>
  relocate(name_modelo, .after = modelo) |> 
  mutate(deltaAIC = dAIC(aic_cor)) |> 
  mutate(w = wAIC(deltaAIC)) 

models_results |>
  write_csv("../data/models_ratio_combined_summary.csv")
Code 
m <- models_results
colnames(m) <- c(
  "code", "Model name", "Log-Likelihood", "#par",   "AICc", "AIC", "R2", "Slope",   "RMSE", "Delta-AICc", "Weight AIC"
)

m |> 
  kbl(digits = c(0,0,2,0,2,2,3,3,3,3,3), escape = FALSE) |> 
  kable_styling()
code Model name Log-Likelihood #par AICc AIC R2 Slope RMSE Delta-AICc Weight AIC
mcombrat1 Temp (Log-normal) | Prec (Log-normal) -233.77 5 477.79 477.54 0.444 0.999 0.101 0.000 0.184
mcombrat2 Temp (Log-normal) | Prec (Gaussian) -233.79 5 477.82 477.58 0.436 1.002 0.102 0.038 0.180
mcombrat4 Temp (Gaussian) | Prec (Log-normal) -233.84 5 477.91 477.67 0.416 1.004 0.103 0.128 0.172
mcombrat5 Temp (Gaussian) | Prec (Gaussian) -233.86 5 477.95 477.71 0.408 0.998 0.104 0.167 0.169
mcombrat7 Temp (Logistic_Mod) | Prec (Log-normal) -233.64 6 479.62 479.28 0.501 0.989 0.095 1.833 0.073
mcombrat8 Temp (Logistic_Mod) | Prec (Gaussian) -233.66 6 479.67 479.33 0.491 1.001 0.096 1.883 0.072
mcombrat3 Temp (Log-normal) | Prec (Gompertz) -233.78 6 479.91 479.57 0.439 1.019 0.101 2.123 0.064
mcombrat6 Temp (Gaussian) | Prec (Gompertz) -233.84 6 480.02 479.68 0.415 0.998 0.103 2.230 0.060
mcombrat9 Temp (Logistic_Mod) | Prec (Gompertz) -233.64 7 481.74 481.28 0.501 1.005 0.096 3.951 0.025