Modeling eDNA qPCR Data with `artemis`

library(artemis)

(Adapted from manuscript Espe et al, in review)

Introduction

A primary purpose of the artemis package is to facilitate modeling of qPCR data from eDNA samples. It does this via two modeling functions: eDNA_lm() for fixed effects models and eDNA_lmer() for mixed effects models. These functions mirror the semantics of R’s built in lm() and lme4’s lmer().

Model Inputs

Both modeling functions require the following inputs from the data:

A vector of numeric Cq values (quantification cycles), one for each qPCR replicate. Cq values corresponding to non-detections for your assay should be recorded as the threshold value (the default is 40.0 cycles).
The intercept value \(\alpha\) and the slope value \(\beta\) from a standard curve equation associated with the qPCR analysis. This is used to convert the observed Cq values to the corresponding log concentration of eDNA. This conversion occurs internally.
A threshold value of the most cycles which are attempted in qPCR (defaults to 40 cycles).

An example of qPCR data in the correct format for modeling with artemis can be viewed by calling eDNA_data, which is a data.frame with Cq values from live car experiments completed in the California Sacramento-San Joaquin Delta with Delta Smelt:

head(eDNA_data)
#>         Date FilterID TechRep    Cq Distance_m Volume_mL Biomass_N
#> 1 2017-08-02  cvp-1-1       1 40.00         50        50       100
#> 2 2017-08-02  cvp-1-1       2 38.13         50        50       100
#> 3 2017-08-02  cvp-1-1       3 37.38         50        50       100
#> 4 2017-08-02 cvp-1-10       1 36.24         40       200       100
#> 5 2017-08-02 cvp-1-10       2 40.00         40       200       100
#> 6 2017-08-02 cvp-1-10       3 40.00         40       200       100
#>   StdCrvAlpha_lnForm StdCrvBeta_lnForm
#> 1             21.168            -1.529
#> 2             21.168            -1.529
#> 3             21.168            -1.529
#> 4             21.168            -1.529
#> 5             21.168            -1.529
#> 6             21.168            -1.529
str(eDNA_data)
#> 'data.frame':    180 obs. of  9 variables:
#>  $ Date              : Date, format: "2017-08-02" "2017-08-02" ...
#>  $ FilterID          : chr  "cvp-1-1" "cvp-1-1" "cvp-1-1" "cvp-1-10" ...
#>  $ TechRep           : num  1 2 3 1 2 3 1 2 3 1 ...
#>  $ Cq                : num  40 38.1 37.4 36.2 40 ...
#>  $ Distance_m        : num  50 50 50 40 40 40 40 40 40 40 ...
#>  $ Volume_mL         : num  50 50 50 200 200 200 200 200 200 200 ...
#>  $ Biomass_N         : num  100 100 100 100 100 100 100 100 100 100 ...
#>  $ StdCrvAlpha_lnForm: num  21.2 21.2 21.2 21.2 21.2 ...
#>  $ StdCrvBeta_lnForm : num  -1.53 -1.53 -1.53 -1.53 -1.53 ...

Note that there are no variable levels with missing or NA values in these example data. However if there were NA values in the input data set, any rows with NAs in the data will be dropped when the data is prepped for modeling. This is because Stan models cannot not take NA values. Although NA values will be automatically dropped from the data prior to modeling, we recommend removing NA values as a separate step prior to modeling. This allows inspection and potentially correction of the rows with NA values. For example,

na_vals = !complete.cases(eDNA_data)
eDNA_data[na_vals,] # visual inspection

Fixed effects models with `eDNA_lm()`

Fixed effects models are primarily used with completely randomized experiments without blocking variables. For most observational data or blocked experimental data, mixed effects models are likely more appropriate.

To fit a fixed effects model to the sample eDNA_data where Distance_m is the only predictor, we give the function a model formula and the input data listed above:


model_fit = eDNA_lm(Cq ~ Distance_m, 
                    data = eDNA_data,
                    std_curve_alpha = 21.2, std_curve_beta = -1.5)

Notice that we provide the standard curve parameters (std_curve_alpha and std_curve_beta as separate arguments to the function. In cases where there are multiple standard curve parameters in use in the same dataset (e.g. using data from multiple labs or experiments), the standard curve parameters can each be given as vectors. These vectors must be the same length as the number of rows in the data.

The model functions, similar to lm() in base R, will automatically add an intercept term. You can explicitly omit the intercept if you have a good reason for doing so. Please see ?lm for a more full description of how to specify linear models in R.

Full control of the MCMC algorithm can be accomplished by adding these control arguments to the end of the eDNA_lm*() call, which then passes them on to rstan::sampling(). Available arguments for MCMC control can be found in the help for rstan::sampling.

For example,

model_fit = eDNA_lm(Cq ~ Distance_m, 
                    data = eDNA_data,
                    std_curve_alpha = 21.2, std_curve_beta = -1.5,
                    seed = 1234, 
                    chains = 1) # we don't recommend sampling just 1 chain; the default is 4

Mixed effects models with `eDNA_lmer()`

Random or mixed effects models are typically used when there are grouping factors which need to be accounted for in the model (e.g. blocking variables, subsamplings from a single filter, etc.).

To fit a model with one or more random effect(s), use the eDNA_lmer() function. Random effects are specified using the same syntax as the lme4 package, e.g. (1|random effect).

For example, to specify a random effect for “Year”,

d = eDNA_data # create a copy to modify 
d$Year = factor(sample(2018:2020, size = nrow(d), replace = TRUE)) # create a random variable

model_fit2 = eDNA_lmer(Cq ~ Distance_m + Volume_mL + (1|Year),
                       data = d,
                       std_curve_alpha = 21.2, std_curve_beta = -1.5,
                       seed = 1234)

Summarizing and plotting model output

As with the simulation objects, the model results can be summarized or plotted with default methods using summary() and plot(), or converted to a data.frame object for further manipulation.

summary(model_fit)

plot(model_fit, pars = c("intercept", "betas"))

Additional arguments can be provided to the plot method, which are passed to rstan::plot methods for stanfit objects. More details are available via ?rstan::plot.

Matching lme4 convention, random effects are not included in the default summary() output. You can view a summary of the random effects with ranef(),

ranef(model_fit2)

or by subsetting the stanfit slot of the model object with @, and specifying the random_betas parameters with the pars argument:

rstan::summary(model_fit2@stanfit, pars = "rand_betas", probs = c(0.50, 0.025, 0.975))$summary
plot(model_fit2, pars = "rand_betas")

Further notes on modeling

Because the models implemented in artemis are Bayesian, you will get the most out of their results when you can work with and summarize posterior probabilities. Some helpful resources for this are the Stan User’s Guide, and the stanfit objects vignette from the rstan package.

Useful modeling advice

This is a collection of advice for modeling eDNA data with the artemis package.

Center and scale your predictor values: artemis uses MCMC to estimate values, and this will be more efficient if the predictor values are not on vastly different scales. In general, the MCMC will be the most efficient when the predictors are roughly centered at 0, and have stdev of 1.
Use priors: The default priors in artemis follow the conventions of the rstanarm package, and are weakly informative. When the data do not strongly inform the parameter estimates, the model fit can be improved by specifying stronger priors.

Other information

The underlying Stan models are compiled on install. Thereafter, the models will not need to be re-compiled. The model’s Stan code can be found in the artemis source code here.

Modeling eDNA qPCR Data with artemis