Modeling Zero-Inflated eDNA qPCR Data with artemis 

library(artemis)

Introduction to Zero-inflated models

A zero-inflated model is a mixture model between two processes that both can result in zero values in the response variable. We use these models when we have more zero’s in the response variable than would be expected under a single model. Thus, the “zero-inflated” tries to capture the mechanisms that could results in these extra zeros.

For work in eDNA, there are several potential processes that could fall under a zero-inflated process, e.g. filter failures, reagent failures, etc. These are distinct from the zero values caused by the measured value falling under the limit of detection.

Note: The above applies to cases when the threshold of detection, i.e. the sensitivity of the assay, is relatively high. In cases when the assay is sensitive to single copies of DNA, the expected distribution is not Normal, and hence different modeling decisions are warranted.

In a zero inflated model, there are two components:

  • the non-zero model, which is functionally similar to the non-zero inflated case
  • the zero-model, which determines the rate at which zero values can occur independent of the non-zero model.

these two models are combined into a mixture where the likelihood is defined as

p(y) = p(zero|X_1) + (1 - p(zero|X_1)) * p(y|X_2)

where p(zero|X_1) is the probability of zero, given data X_1 and p(y|X_2) is the likelihood of y given data X_2. Note that the zero and non-zero components can be predicated on different data (i.e. the zero and non-zero models do not require matching terms).

When to Use Zero-inflated models

There are two criteria that make zero-inflated models appropriate:

  1. The limit of detection is higher than single-digit copies, thus the distribution of non-zeros is approximately normal. When the limit of detection is in the single-digits, the Poisson (count) models in artemis are more appropriate. These models have the additional benefit of capturing zero-inflated process inherit in low-copy number situations.

  2. There are more zero values in the observed data than explained by the predictors. This can be assessed by inspecting the residual values (difference between predicted values and observed values). This presents as data values that are predicted to be high concentrations, but are observed to be low or zero concentrations. Note: this assumes the model fits well to non-zero values. If the model does not fit well across all values, model refinement should be attempted.

Fitting Zero-inflated Models in artemis

Zero-inflated models can be fit in artemis via the eDNA_zinf_lm*() functions. For these functions, the model is specified as in the glm package, where the model formula for the non-zero model and the zero-inflated model are separated by a | in the model formula, e.g. 

eDNA_zinf_lm(Cq ~ Distance_m + Volume_mL | Distance_m, eDNA_data, ...)

In this formula, the term ~ Distance_m + Volume_ml specifies the predictors for the non-zero model, whereas the | Distance_m term specifies the predictors for the zero-model.

The non-zero model follows the same form as the eDNA_lm*() models, namely a censored-data model using normally distributed errors.

The zero model is a Binomial Logit model, where the chance of an individual observation being a zero give the predictors is estimated via a GLM using a binomial distribution and a logit link function.

The zero-inflated models can be summarized, printed, and plotted as any other model from the artemis package.

Evaluating a zero-inflated model

The zero-inflated models can be compared using the loo() function, allowing model comparison.