Package 'trps'

Calculate and add $\alpha$

Description

Calculate $\alpha$ for a two source trophic position model using equations from Post 2002.

Usage

add_alpha(data, abs = FALSE)

Arguments

data	data.frame of stable isotope samples with mean values for two baselines. For aquatic ecosystems, baseline one needs to come from a benthic source and baseline two needs to come from a pelagic source. Baseline $\delta^{13}$C columns need to be named c1 and c2, with the consumer's $\delta^{13}$C column named d13c.
abs	logical that controls whether the absolute value is taken for the numerator and denominator. Default is FALSE meaning that the absolute value is not taken.

Details

$\alpha = (\delta^{13}C_c - \delta ^{13}C_2) / (\delta ^{13}C_1 - \delta ^{13}C_2)$

where $\delta^{13}C_c$ is the isotopic value for consumer, $\delta^{13}C_1$ is the mean isotopic value for baseline 1 and $\delta^{13}C_2$ is the mean isotopic value for baseline 2.

Value

a data.frame that has alpha, min_alpha, and max_alpha added.

Examples

combined_iso |>
  add_alpha()

---

Stable isotope data for amphipods (baseline 1)

Description

Stable isotope data ($\delta^{13}$C and $\delta^{15}$N) for amphipods collected from an ecoregion in Lake Ontario.

Usage

baseline_1_iso

Format

data.frame containing 14 rows and 5 variables

common_name

name of the spcies (i.e., Amphipoda)

ecoregion

ecoregion where samples were collected

d13c_b1

observed values for $\delta^{13}$C

d15n_b1

observed values for $\delta^{15}$N

---

Stable isotope data for dreissenids (baseline 2)

Description

Stable isotope data ($\delta^{13}$C and $\delta^{15}$N) for dreissenid collected from an ecoregion in Lake Ontario.

Usage

baseline_2_iso

Format

data.frame containing 12 rows and 5 variables

common_name

name of the spcies (i.e., Dreissenids)

ecoregion

ecoregion where samples were collected

d13c_b2

observed values for $\delta^{13}$C

d15n_b2

observed values for $\delta^{15}$N

---

Stable isotope data for lake trout, amphipods (benthic baseline; baseline 1) and dreissenids (pelagic baseline; baseline 2),

Description

Stable isotope data ($\delta^{13}$C and $\delta ^{15}$N) for lake trout collected from two ecoregions in Lake Ontario. Values of $\delta ^{13}$C and $\delta ^{15}$N for a benthic baseline (amphipods; baseline 1; d13c_b1 and d15n_b1) and pelagic baseline (dreissenids; baseline 2; d13c_b2 and d15n_b2) with the means for each baseline calculated (c1, n1, c2, and n2).

Usage

combined_iso

Format

data.frame containing 117 rows and 13 variables

id

row id number

common_name

name of the spcies (i.e., Lake Trout)

ecoregion

ecoregion where samples were collected

d13c

observed values for $\delta^{13}$C of consumer

d15n

observed values for $\delta^{15}$N of consumer

d13c_b1

observed values for $\delta^{13}$C of baseline 1

d15n_b1

observed values for $\delta^{15}$N of baseline 1

d13c_b2

observed values for $\delta^{13}$C of baseline 2

d15n_b2

observed values for $\delta^{15}$N of baseline 2

c1

mean values for $\delta^{13}$C of baseline 1

n1

mean values for $\delta^{15}$N of baseline 1

c2

mean values for $\delta^{13}$C of baseline 2

n2

mean values for $\delta^{15}$N of baseline 2

---

Stable isotope data for lake trout (consumer)

Description

Stable isotope data ($\delta^{13}$C and $\delta^ {15}$N) for lake trout collected from an ecoregion in Lake Ontario.

Usage

consumer_iso

Format

data.frame containing 30 rows and 6 variables

common_name

name of the spcies (i.e., Lake Trout)

ecoregion

ecoregion where samples were collected

d13c

observed values for $\delta^{13}$C

d15n

observed values for $\delta^{15}$N

---

Bayesian model - One Source Trophic Position

Description

Estimate trophic position using a one source model derived from Post 2002 using a Bayesian framework.

Usage

one_source_model(bp = FALSE)

Arguments

bp	logical value that controls whether informed priors are supplied to the model for $\delta^{15}$N baseline. Default is FALSE meaning the model will use uninformed priors, however, the supplied data.frame needs values for $\delta^{15}$N baseline (n1).

Details

$\delta^{15}N = \delta^{15} N_1 + \Delta N \times (tp - \lambda_1)$

$\delta^{15}$N are values from the consumer, $\delta^{15} N_1$ is mean $\delta^{15}$N values of baseline 1, $\Delta$N is the trophic discrimination factor for N (i.e., dn mean of 3.4), $tp$ is trophic position, and $\lambda_1$ is the trophic level of baselines which are often a primary consumer (e.g., 2).

The data supplied to brms() needs to have the following variables d15n, n1, and l1 ($\lambda$) which is usually 2.

Value

returns model structure for one source model to be used in a brms() call.

See Also

brms::brms()

Examples

one_source_model()

---

Bayesian priors - One Source Trophic Position

Description

Create priors for one source trophic position model derived from Post 2002.

Usage

one_source_priors(bp = FALSE)

Arguments

bp	logical value that controls whether informed priors are supplied to the model for $\delta^{15}$N baseline. Default is FALSE meaning the model will use uninformed priors, however, the supplied data.frame needs values for $\delta^{15}$N baseline (n1).

Value

returns priors for one source model to be used in a brms() call.

See Also

brms::brms()

Examples

one_source_priors()

---

Adjust Bayesian priors - One Source Trophic Position

Description

Adjust priors for one source trophic position model derived from Post 2002.

Usage

one_source_priors_params(
  n1 = NULL,
  n1_sigma = NULL,
  dn = NULL,
  dn_sigma = NULL,
  tp_lb = NULL,
  tp_ub = NULL,
  sigma_lb = NULL,
  sigma_ub = NULL,
  bp = FALSE
)

Arguments

n1	mean ($\mu$) prior for the mean $\delta^{15}$N baseline. Defaults to 9.
n1_sigma	variance ($\sigma$) for the mean $\delta^{15}$N baseline. Defaults to 1.
dn	mean ($\mu$) prior value for $\Delta$N. Defaults to 3.4.
dn_sigma	variance ($\sigma$) for $\delta^{15}$N. Defaults to 0.25.
tp_lb	lower bound prior for trophic position. Defaults to 2.
tp_ub	upper bound prior for trophic position. Defaults to 10.
sigma_lb	lower bound prior for $\sigma$. Defaults to 0.
sigma_ub	upper bound prior for $\sigma$. Defaults to 10.
bp	logical value that controls whether informed priors are supplied to the model for $\delta^{15}$N baseline. Default is FALSE meaning the model will use uninformed priors, however, the supplied data.frame needs values for $\delta^{15}$N baseline (n1).

Details

$\delta^{15}N = \delta^{15} N_1 + \delta N \times (tp - \lambda_1)$

This function allows the user to adjust the priors for the following variables in the equation above:

• The mean (n1; $\mu$) and variance (n1_sigma; $\sigma$) for the mean $\delta^{15}$N for a given baseline ($\delta^{15}N_1$). This prior assumes a normal distribution.

• The mean (dn; $\mu$) and variance (dn_sigma; $\sigma$) of $\Delta$N (i.e, trophic enrichment factor). This prior assumes a normal distribution.

• The lower (tp_lb) and upper (tp_ub) bounds for trophic position. This prior assumes a uniform distribution.

• The lower (sigma_lb) and upper (sigma_ub) bounds for variance ($\sigma$). This prior assumes a uniform distribution.

Value

stanvars object to be used with brms() call.

See Also

one_source_priors(), one_source_model(), and brms::brms()

Examples

one_source_priors_params()

---

Bayesian model - Two Source Trophic Position

Description

Trophic position using a two source model derived from Post 2002 using a Bayesian framework.

Usage

two_source_model(bp = FALSE, lambda = NULL)

Arguments

bp	logical value that controls whether informed priors are supplied to the model for both $\delta^{15}$N and $\delta^{15}$C baselines. Default is FALSE meaning the model will use uninformed priors, however, the supplied data.frame needs values for both $\delta^{15}$N and $\delta^{15}$C baseline (c1, c2, n1, and n2).
lambda	numerical value, 1 or 2, that controls whether one or two $\lambda$s are used. See details for equations and when to use 1 or 2. Defaults to 1.

Details

We will use the following equations from Post 2002:

1. $\delta^{13}C_c = \alpha \times (\delta ^{13}C_1 - \delta ^{13}C_2) + \delta ^{13}C_2$

2. $\delta^{15}N = \Delta N \times (tp - \lambda_1) + n_1 \times \alpha + n_2 \times (1 - \alpha)$

3. $\delta^{15}N = \Delta N \times (tp - (\lambda_1 \times \alpha + \lambda_2 \times (1 - \alpha))) + n_1 \times \alpha + n_2 \times (1 - \alpha)$

For equation 1)

where $\delta^{13}C_c$ is the isotopic value for consumer, $\alpha$ is the ratio between baselines and consumer $\delta^{13}C$, $\delta^{13}C_1$ is the mean isotopic value for baseline 1, and $\delta^{13}C_2$ is the mean isotopic value for baseline 2

For equation 2) and 3)

$\delta^{15}$N are values from the consumer, $n_1$ is $\delta^{15}$N values of baseline 1, $n_2$ is $\delta^{15}$N values of baseline 2, $\Delta$N is the trophic discrimination factor for N (i.e., mean of 3.4), tp is trophic position, and $\lambda_1$ and/or $\lambda_2$ are the trophic levels of baselines which are often a primary consumer (e.g., 2 or 2.5).

The data supplied to brms() when using baselines at the same trophic level (lambda argument set to 1) needs to have the following variables, d15n, c1, c2, n1, n2, l1 ($\lambda_1$) which is usually 2. If using baselines at different trophic levels (lambda argument set to 2) the data frame needs to have l1 and l2 with a numerical value for each trophic level (e.g.,2 and 2.5; $\lambda_1$ and $\lambda_2$).

Value

returns model structure for two source model to be used in a brms() call.

See Also

brms::brms()

Examples

two_source_model()

---

Bayesian model - Two Source Trophic Position with $\alpha_r$

Description

Estimate trophic position using a two source model with $\alpha_r$ derived from Post 2002 and Heuvel et al. 2024 using a Bayesian framework.

Usage

two_source_model_ar(bp = FALSE, lambda = NULL)

Arguments

bp	logical value that controls whether informed priors are supplied to the model for both $\delta^{15}$N baselines. Default is FALSE meaning the model will use uninformed priors, however, the supplied data.frame needs values for both $\delta^{15}$N baseline (n1 and n2).
lambda	numerical value, 1 or 2, that controls whether one or two lambdas are used. See details for equations and when to use 1 or 2. Defaults to 1.

Details

We will use the following equations derived from Post 2002 and Heuvel et al. 2024:

1. $\alpha = (\delta^{13} C_c - \delta ^{13}C_2) / (\delta ^{13}C_1 - \delta ^{13}C_2)$

2. $\alpha = \alpha_r \times (\alpha_{max} - \alpha_{min}) + \alpha_{min}$

3. $\delta^{15}N = \Delta N \times (tp - \lambda_1) + n_1 \times \alpha_r + n_2 \times (1 - \alpha_r)$

4. $\delta^{15}N = \Delta N \times (tp - (\lambda_1 \times \alpha_r + \lambda_2 \times (1 - \alpha_r))) + n_1 \times \alpha_r + n_2 \times (1 - \alpha_r)$

For equation 1)

This equation is a carbon source mixing model with $\delta^{13}C_c$ is the isotopic value for consumer, $\delta^{13}C_1$ is the mean isotopic value for baseline 1 and $\delta^{13}C_2$ is the mean isotopic value for baseline 2. This equation is added to the data frame using add_alpha().

For equation 2)

$\alpha$ is being corrected using equations in Heuvel et al. 2024 with $\alpha_r$ being the corrected value (bound by 0 and 1), $\alpha_{min}$ is the minimum $\alpha$ value calculated using add_alpha() and $\alpha_{max}$ being the maximum $\alpha$ value calculated using add_alpha().

For equation 3) and 4)

$\delta^{15}$N are values from the consumer, $n_1$ is $\delta^{15}$N values of baseline 1, $n_2$ is $\delta^{15}$N values of baseline 2, $\Delta$N is the trophic discrimination factor for N (i.e., mean of 3.4), tp is trophic position, and $\lambda_1$ and/or $\lambda_2$ are the trophic levels of baselines which are often a primary consumer (e.g., 2 or 2.5).

The data supplied to brms() when using baselines at the same trophic level (lambda argument set to 1) needs to have the following variables, d15n, n1, n2, l1 ($\lambda_1$) which is usually 2. If using baselines at different trophic levels (lambda argument set to 2) the data frame needs to have l1 and l2 with a numerical value for each trophic level (e.g., 2 and 2.5; $\lambda_1$ and $\lambda_2$).

Value

returns model structure for two source model to be used in a brms() call.

See Also

brms::brms()

Examples

two_source_model_ar()

---

Bayesian model - Two Source Trophic Position with $\alpha_r$ and carbon mixing model

Description

Estimate trophic position using a two source model with $\alpha_r$ derived from Post 2002 and Heuvel et al. 2024 using a Bayesian framework.

Usage

two_source_model_arc(bp = FALSE, lambda = NULL)

Arguments

bp	logical value that controls whether informed priors are supplied to the model for both $\delta^{15}$N and $\delta^{15}$C baselines. Default is FALSE meaning the model will use uninformed priors, however, the supplied data.frame needs values for both $\delta^{15}$N and $\delta^{15}$C baseline (c1, c2, n1, and n2).
lambda	numerical value, 1 or 2, that controls whether one or two lambdas are used. See details for equations and when to use 1 or 2. Defaults to 1.

Details

We will use the following equations derived from Post 2002 and Heuvel et al. 2024:

1. $\alpha = (\delta^{13} C_c - \delta ^{13}C_2) / (\delta ^{13}C_1 - \delta ^{13}C_2)$

2. $\alpha = \alpha_r \times (\alpha_{max} - \alpha_{min}) + \alpha_{min}$

3. $\delta^{13}C = c_1 \times \alpha_c + c_2 \times (1 - \alpha_c)$

4. $\delta^{15}N = \Delta N \times (tp - \lambda_1) + n_1 \times \alpha_c + n_2 \times (1 - \alpha_c)$

5. $\delta^{15}N = \Delta N \times (tp - (\lambda_1 \times \alpha_c + \lambda_2 \times (1 - \alpha_c))) + n_1 \times \alpha_c + n_2 \times (1 - \alpha_c)$

For equation 1)

This equation is a carbon source mixing model with $\delta^{13}C_c$ is the isotopic value for consumer, $\delta^{13}C_1$ is the mean isotopic value for baseline 1 and $\delta^{13}C_2$ is the mean isotopic value for baseline 2.

For equation 2)

$\alpha$ is being corrected using equations in Heuvel et al. 2024. with $\alpha_r$ being the corrected value (bound by 0 and 1), $\alpha_{min}$ is the minimum $\alpha$ value calculated using add_alpha() and $\alpha_{max}$ being the maximum $\alpha$ value calculated using add_alpha().

For equation 3)

This equation is a carbon source mixing model with $\delta^{13}$C being estimated using c_1, c_2 and $\alpha_c$ calculated in equation 1.

For equation 4) and 5)

$\delta^{15}$N are values from the consumer, $n_1$ is $\delta^{15}$N values of baseline 1, $n_2$ is $\delta^{15}$N values of baseline 2, $\Delta$N is the trophic discrimination factor for N (i.e., mean of 3.4), tp is trophic position, and $\lambda_1$ and/or $\lambda_2$ are the trophic levels of baselines which are often a primary consumer (e.g., 2 or 2.5).

The data supplied to brms() when using baselines at the same trophic level (lambda argument set to 1) needs to have the following variables, d15n, n1, n2, l1 ($\lambda_1$) which is usually 2. If using baselines at different trophic levels (lambda argument set to 2) the data frame needs to have l1 and l2 with a numerical value for each trophic level (e.g., 2 and 2.5; $\lambda_1$ and $\lambda_2$).

Value

returns model structure for two source model to be used in a brms() call.

See Also

brms::brms()

Examples

two_source_model_arc()

---

Bayesian priors - Two Source Trophic Position

Description

Create priors for two source trophic position model derived from Post 2002.

Usage

two_source_priors(bp = FALSE)

Arguments

bp

logical value that controls whether informed priors are supplied to the model for both $\delta^{15}$N and $\delta^{15}$C baselines. Default is FALSE meaning the model will use uninformed priors, however, the supplied data.frame needs values for both $\delta^{15}$N and $\delta^{15}$C baseline (c1, c2, n1, and n2).

Value

returns priors for two source model to be used in a brms() call.

See Also

two_source_model() and brms::brms()

Examples

two_source_priors()

---

Bayesian priors - Two Source Trophic Position with $\alpha_r$

Description

Create priors for trophic position using a two source model with $\alpha_r$ derived from Post 2002 and Heuvel et al. 2024.

Usage

two_source_priors_ar(bp = FALSE)

Arguments

bp	logical value that controls whether informed priors are supplied to the model for $\delta^{15}$N baselines. Default is FALSE meaning the model will use uninformed priors, however, the supplied data.frame needs values for both $\delta^{15}$N baseline (n1, and n2).

Value

returns priors for two source model to be used in a brms() call.

See Also

brms::brms()

Examples

two_source_priors_ar()

---

Bayesian priors - Two Source Trophic Position with $\alpha_r$ and carbon mixing model

Description

Create priors for trophic position using a two source model with $\alpha_r$ derived from Post 2002 and Heuvel et al. 2024.

Usage

two_source_priors_arc(bp = FALSE)

Arguments

bp

logical value that controls whether informed priors are supplied to the model for both $\delta^{15}$N and $\delta^{15}$C baselines. Default is FALSE meaning the model will use uninformed priors, however, the supplied data.frame needs values for both $\delta^{15}$N and $\delta^{15}$C baseline (c1, c2, n1, and n2).

Value

returns priors for two source model to be used in a brms() call.

See Also

brms::brms()

Examples

two_source_priors_arc()

---

Adjust Bayesian priors - Two Source Trophic Position

Description

Adjust priors for two source trophic position model derived from Post 2002.

Usage

two_source_priors_params(
  a = NULL,
  b = NULL,
  c1 = NULL,
  c1_sigma = NULL,
  c2 = NULL,
  c2_sigma = NULL,
  n1 = NULL,
  n1_sigma = NULL,
  n2 = NULL,
  n2_sigma = NULL,
  dn = NULL,
  dn_sigma = NULL,
  tp_lb = NULL,
  tp_ub = NULL,
  sigma_lb = NULL,
  sigma_ub = NULL,
  bp = FALSE
)

Arguments

a	($\alpha$) exponent of the random variable for beta distribution. Defaults to 1. See beta distribution for more information.
b	($\beta$) shape parameter for beta distribution. Defaults to 1. See beta distribution for more information.
c1	mean ($\mu$) prior for the mean of the first $\delta^{13}$C baseline. Defaults to -21.
c1_sigma	variance ($\sigma$)for the mean of the first $\delta^{13}$C baseline. Defaults to 1.
c2	mean ($\mu$) prior for or the mean of the second $\delta^{13}$C baseline. Defaults to -26.
c2_sigma	variance ($\sigma$)for the mean of the first $\delta^{13}$C baseline. Defaults to 1.
n1	mean ($\mu$) prior for the mean of the first $\delta^{15}$N baseline. Defaults to 8.
n1_sigma	variance ($\sigma$)for the mean of the first $\delta^{15}$N baseline. Defaults to 1.
n2	mean ($\mu$) prior for or the mean of the second $\delta^{15}$N baseline. Defaults to 9.5.
n2_sigma	variance ($\sigma$) for the mean of the second $\delta^{15}$N baseline. Defaults to 1.
dn	mean ($\mu$) prior value for $\Delta$N. Defaults to 3.4.
dn_sigma	variance ($\sigma$) for $\delta^{15}$N. Defaults to 0.5.
tp_lb	lower bound for priors for trophic position. Defaults to 2.
tp_ub	upper bound for priors for trophic position. Defaults to 10.
sigma_lb	lower bound for priors for $\sigma$. Defaults to 0.
sigma_ub	upper bound for priors for $\sigma$. Defaults to 10.
bp	logical value that controls whether informed priors are supplied to the model for both $\delta^{15}$N and $\delta^{15}$C baselines. Default is FALSE meaning the model will use uninformed priors, however, the supplied data.frame needs values for both $\delta^{15}$N and $\delta^{15}$C baseline (c1, c2, n1, and n2).

Details

We will use the following equations from Post 2002:

1. $\delta^{13}C_c = \alpha * (\delta ^{13}C_1 - \delta ^{13}C_2) + \delta ^{13}C_2$

2. $\delta^{15}N = \Delta N \times (tp - \lambda_1) + n_1 \times \alpha + n_2 \times (1 - \alpha)$

3. $\delta^{15}N = \Delta N \times (tp - (\lambda_1 \times \alpha + \lambda_2 \times (1 - \alpha))) + n_1 \times \alpha + n_2 \times (1 - \alpha)$

• The random exponent ($\alpha$; a) and shape parameters ($\beta$; b) for $\alpha$. This prior assumes a beta distribution.

• The mean (c1; $\mu$) and variance (c1_sigma; $\sigma$) of the mean for the first $\delta^{13}C$ for a given baseline. This prior assumes a normal distributions.

• The mean (c2;$\mu$) and variance (c2_sigma; $\sigma$) of the mean for the second $\delta^{13}C$ for a given baseline. This prior assumes a normal distributions.

• The mean (n1; $\mu$) and variance (n1_sigma; $\sigma$) of the mean for the first $\delta^{15}N$ for a given baseline. This prior assumes a normal distributions.

• The mean (n2;$\mu$) and variance (n2_sigma; $\sigma$) of the mean for the second $\delta^{15}$N for a given baseline. This prior assumes a normal distributions.

• The mean (dn; $\mu$) and variance (dn_sigma; $\sigma$) of $\Delta$N (i.e, trophic enrichment factor). This prior assumes a normal distributions.

• The lower (tp_lb) and upper (tp_ub) bounds for priors for trophic position. This prior assumes a uniform distributions.

• The lower (sigma_lb) and upper (sigma_ub) bounds for variance ($\sigma$). This prior assumes a uniform distributions.

Value

stanvars object to be used with brms() call.

See Also

two_source_priors(), two_source_model(), and brms::brms()

Examples

two_source_priors_params()

---

Adjust Bayesian priors - Two Source Trophic Position with $\alpha_r$

Description

Create priors for trophic position using a two source model with $\alpha_r$ derived from Post 2002 and Heuvel et al. 2024.

Usage

two_source_priors_params_ar(
  a = NULL,
  b = NULL,
  n1 = NULL,
  n1_sigma = NULL,
  n2 = NULL,
  n2_sigma = NULL,
  dn = NULL,
  dn_sigma = NULL,
  tp_lb = NULL,
  tp_ub = NULL,
  sigma_lb = NULL,
  sigma_ub = NULL,
  bp = FALSE
)

Arguments

a	($\alpha$) exponent of the random variable for beta distribution. Defaults to 1. See beta distribution for more information.
b	($\beta$) shape parameter for beta distribution. Defaults to 1. See beta distribution for more information.
n1	mean ($\mu$) prior for first $\delta^{15}$N baseline. Defaults to 8.0.
n1_sigma	variance ($\sigma$)for first $\delta^{15}$N baseline. Defaults to 1.
n2	mean ($\mu$) prior for second $\delta^{15}$N baseline. Defaults to 9.5.
n2_sigma	variance ($\sigma$) for second $\delta^{15}$N baseline. Defaults to 1.
dn	mean ($\mu$) prior value for $\Delta$N. Defaults to 3.4.
dn_sigma	variance ($\sigma$) for $\delta^{15}$N. Defaults to 0.5.
tp_lb	lower bound for priors for trophic position. Defaults to 2.
tp_ub	upper bound for priors for trophic position. Defaults to 10.
sigma_lb	lower bound for priors for $\sigma$. Defaults to 0.
sigma_ub	upper bound for priors for $\sigma$. Defaults to 10.
bp	logical value that controls whether informed baseline priors are supplied to the model for $\delta^{15}$N baselines. Default is FALSE meaning the model will use uninformed priors, however, the supplied data.frame needs values for both $\delta^{15}$N baseline (n1 and n2)

Details

We will use the following equations derived from Post 2002 and Heuvel et al. 2024:

1. $\alpha = (\delta^{13} C_c - \delta ^{13}C_2) / (\delta ^{13}C_1 - \delta ^{13}C_2)$

2. $\alpha = \alpha_r \times (\alpha_{max} - \alpha_{min}) + \alpha_{min}$

3. $\delta^{15}N = \Delta N \times (tp - \lambda_1) + n_1 \times \alpha_r + n_2 \times (1 - \alpha_r)$

4. $\delta^{15}N = \Delta N \times (tp - (\lambda_1 \times \alpha_r + \lambda_2 \times (1 - \alpha_r))) + n_1 \times \alpha_r + n_2 \times (1 - \alpha_r)$

For equation 1)

This equation is a carbon source mixing model with $\delta^{13}C_c$ is the isotopic value for consumer, $\delta^{13}C_1$ is the mean isotopic value for baseline 1 and $\delta^{13}C_2$ is the mean isotopic value for baseline 2. This equation is added to the data frame using add_alpha().

For equation 2)

$\alpha$ is being corrected using equations in Heuvel et al. 2024 with $\alpha_r$ being the corrected value (bound by 0 and 1), $\alpha_{min}$ is the minimum $\alpha$ value calculated using add_alpha() and $\alpha_{max}$ being the maximum $\alpha$ value calculated using add_alpha().

For equation 3) and 4)

$\delta^{15}$N are values from the consumer, $n_1$ is $\delta^{15}$N values of baseline 1, $n_2$ is $\delta^{15}$N values of baseline 2, $\Delta$N is the trophic discrimination factor for N (i.e., mean of 3.4), tp is trophic position, and $\lambda_1$ and/or $\lambda_2$ are the trophic levels of baselines which are often a primary consumer (e.g., 2 or 2.5).

This function allows the user to adjust the priors for the following variables in the equation above:

• The random exponent ($\alpha$; a) and shape parameters ($\beta$; b) for $\alpha_r$. This prior assumes a beta distribution.

• The mean (n2;$\mu$) and variance (n2_sigma; $\sigma$) of the second $\delta^{15}$N for a given baseline. This prior assumes a normal distributions.

• The mean (c1;$\mu$) and variance (c1_sigma; $\sigma$) of the second $\delta^{13}$C for a given baseline. This prior assumes a normal distributions.

• The mean (c2;$\mu$) and variance (c2_sigma; $\sigma$) of the second $\delta^{13}$C for a given baseline. This prior assumes a normal distributions.

• The mean (dn; $\mu$) and variance (dn_sigma; $\sigma$) of $\Delta$N (i.e, trophic enrichment factor). This prior assumes a normal distributions.

• The lower (tp_lb) and upper (tp_ub) bounds for priors for trophic position. This prior assumes a uniform distributions.

• The lower (sigma_lb) and upper (sigma_ub) bounds for variance ($\sigma$). This prior assumes a uniform distributions.

Value

stanvars object to be used with brms() call.

See Also

two_source_priors_ar(), two_source_model_ar(), and brms::brms()

Examples

two_source_priors_params_ar()

---

Adjust Bayesian priors - Two Source Trophic Position with $\alpha_r$ and carbon mixing model

Description

Adjust priors for trophic position using a two source model with $\alpha_r$ derived from Post 2002 and Heuvel et al. 2024

Usage

two_source_priors_params_arc(
  a = NULL,
  b = NULL,
  n1 = NULL,
  n1_sigma = NULL,
  n2 = NULL,
  n2_sigma = NULL,
  c1 = NULL,
  c1_sigma = NULL,
  c2 = NULL,
  c2_sigma = NULL,
  dn = NULL,
  dn_sigma = NULL,
  tp_lb = NULL,
  tp_ub = NULL,
  sigma_lb = NULL,
  sigma_ub = NULL,
  bp = FALSE
)

Arguments

a	($\alpha$) exponent of the random variable for beta distribution. Defaults to 1. See beta distribution for more information.
b	($\beta$) shape parameter for beta distribution. Defaults to 1. See beta distribution for more information.
n1	mean ($\mu$) prior for first $\delta^{15}$N baseline. Defaults to 8.0.
n1_sigma	variance ($\sigma$)for first $\delta^{15}$N baseline. Defaults to 1.
n2	mean ($\mu$) prior for second $\delta^{15}$N baseline. Defaults to 9.5.
n2_sigma	variance ($\sigma$) for second $\delta^{15}$N baseline. Defaults to 1.
c1	mean ($\mu$) prior for first $\delta^{13}$C baseline. Defaults to -21.
c1_sigma	variance ($\sigma$)for first $\delta^{13}$C baseline. Defaults to 1.
c2	mean ($\mu$) prior for second $\delta^{13}$C baseline. Defaults to -26.
c2_sigma	variance ($\sigma$) for second $\delta^{13}$C baseline. Defaults to 1.
dn	mean ($\mu$) prior value for $\Delta$N. Defaults to 3.4.
dn_sigma	variance ($\sigma$) for $\delta^{15}$N. Defaults to 0.25.
tp_lb	lower bound for priors for trophic position. Defaults to 2.
tp_ub	upper bound for priors for trophic position. Defaults to 10.
sigma_lb	lower bound for priors for $\sigma$. Defaults to 0.
sigma_ub	upper bound for priors for $\sigma$. Defaults to 10.
bp	logical value that controls whether informed baseline priors are supplied to the model for $\delta^{15}$N baselines. Default is FALSE meaning the model will use uninformed priors, however, the supplied data.frame needs values for both $\delta^{15}$N baseline (n1 and n2)

Details

We will use the following equations derived from Post 2002 and Heuvel et al. 2024:

1. $\alpha = (\delta^{13} C_c - \delta ^{13}C_2) / (\delta ^{13}C_1 - \delta ^{13}C_2)$

2. $\alpha = \alpha_r \times (\alpha_{max} - \alpha_{min}) + \alpha_{min}$

3. $\delta^{13}C = c_1 \times \alpha_r + c_2 \times (1 - \alpha_r)$

4. $\delta^{15}N = \Delta N \times (tp - \lambda_1) + n_1 \times \alpha_r + n_2 \times (1 - \alpha_r)$

5. $\delta^{15}N = \Delta N \times (tp - (\lambda_1 \times \alpha_r + \lambda_2 \times (1 - \alpha_r))) + n_1 \times \alpha_r + n_2 \times (1 - \alpha_r)$

For equation 1)

This equation is a carbon source mixing model with $\delta^{13}C_c$ is the isotopic value for consumer, $\delta^{13}C_1$ is the mean isotopic value for baseline 1 and $\delta^{13}C_2$ is the mean isotopic value for baseline 2.

For equation 2)

$\alpha$ is being corrected using equations in Heuvel et al. 2024. with $\alpha_r$ being the corrected value (bound by 0 and 1), $\alpha_{min}$ is the minimum $\alpha$ value calculated using add_alpha() and $\alpha_{max}$ being the maximum $\alpha$ value calculated using add_alpha().

For equation 3)

This equation is a carbon source mixing model with $\delta^{13}$C being estimated using c_1, c_2 and $\alpha_r$ calculated in equation 1.

For equation 4) and 5)

$\delta^{15}$N are values from the consumer, $n_1$ is $\delta^{15}$N values of baseline 1, $n_2$ is $\delta^{15}$N values of baseline 2, $\Delta$N is the trophic discrimination factor for N (i.e., mean of 3.4), tp is trophic position, and $\lambda_1$ and/or $\lambda_2$ are the trophic levels of baselines which are often a primary consumer (e.g., 2 or 2.5).

This function allows the user to adjust the priors for the following variables in the equation above:

• The random exponent ($\alpha$; a) and shape parameters ($\beta$; b) for $\alpha_r$. This prior assumes a beta distribution.

• The mean (n2;$\mu$) and variance (n2_sigma; $\sigma$) of the second $\delta^{15}$N for a given baseline. This prior assumes a normal distributions.

• The mean (c1;$\mu$) and variance (c1_sigma; $\sigma$) of the second $\delta^{13}$C for a given baseline. This prior assumes a normal distributions.

• The mean (c2;$\mu$) and variance (c2_sigma; $\sigma$) of the second $\delta^{13}$C for a given baseline. This prior assumes a normal distributions.

• The mean (dn; $\mu$) and variance (dn_sigma; $\sigma$) of $\Delta$N (i.e, trophic enrichment factor). This prior assumes a normal distributions.

• The lower (tp_lb) and upper (tp_ub) bounds for priors for trophic position. This prior assumes a uniform distributions.

• The lower (sigma_lb) and upper (sigma_ub) bounds for variance ($\sigma$). This prior assumes a uniform distributions.

Value

stanvars object to be used with brms() call.

See Also

two_source_priors_arc(), two_source_model_arc(), and brms::brms()

Examples

two_source_priors_params_ar()

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