Evaluate covariates

Water deer covariates

Our case-study results have so far been quite clear-cut:

Chinese water deer are easier to detect in open grasslands, but distance from the coast probably doesn’t influence deer density

It’s unlikely that your own field data will be as easy to interpret!

You will probably have a larger and more complex set of hypotheses to test
You will have more covariates to interpret and compare
Your covariates may be correlated with each other

Complex covariate models

If you only compare simple hypotheses, each involving a single covariate, the model selection process automatically determines which covariates matter

However, building more complex models complicates how we interpret our results:

We have multiple covariates, e.g. 🌿 🐦 🌺
Some models contain several covariates, e.g. 🦋 = 🌿 + 🐦
Each covariate appears in multiple models, in different combinations with other covariates

When your hypotheses overlap (i.e. covariates appear in more than one model), the success of an individual model is insufficient to tell you how useful each predictor is

Correlated covariates

Information Theory attempts to break correlations amongst covariates through each variable appearing alone and alongside all the others

Model comparison can show us which covariates are related to the response, and which appear to be related simply because they are correlated with another predictor

However, it will be easier to interpret your results both statistically and ecologically if you minimise correlated covariates

Exclude highly-correlated covariates

We recommend excluding highly-correlated covariates (correlation > 0.7) from your models

Of your highly-correlated variables, retain the one most likely to be the driver of variation in density or detectability

Importance of a covariate

We have three approaches to determine the importance of a particular covariate:

Sum the model weights (probabilities) of all models in which that covariate appears
Calculate evidence ratios between pairs of models that either include or omit a particular covariate
Determine whether the confidence interval for the effect of a covariate includes zero

These approaches are useful both to help you understand your results, and to communicate them clearly

Calculate summed model weights

To determine the importance of a covariate, sum the model weights for each model in which a particular covariate appears

Model weights reminder

A model’s weight indicates its probability of being the best model
The summed weight of all models equals 1

So the closer to 1 a summed model weight for given parameter is, the more important that parameter is in determining density or detectability

Summed weights: DistToCoast

We’ll use AICcmodavg’s importance() function, specifying which parameter (density or detectability) we’re interested in, and the covariate we want to evaluate

1SumWt_DTC <- importance(models,
2    c.hat = cHat,
3    parm = "DistToCoast",
4    parm.type = "lambda",
5    second.order = TRUE)

1: Model list
2: Estimate of \(\hat{c}\)
3: Covariate of interest
4: Effect on density (lambda)
5: Use AICc

Importance of DistToCoast

SumWt_DTC


Importance values of 'lam(DistToCoast)':

w+ (models including parameter): 0.05 
w- (models excluding parameter): 0.95

Models that include DistToCoast as a predictor of density only have a summed weight of 5%, compared to 95% for models that omit DistToCoast as a covariate

This is very strong evidence that distance to the coast is not a useful predictor of water deer density in this study

Summed weights: Landcover

Sum detectability model weights by hand

Although the help pages for AICcmodavg says it can calculate summed model weights for covariates predicting detectability, we haven’t succeeded with this. Instead, we’ll extract and sum the model weights for detectability covariates by hand

For detectability, we will only include models built with the best supported detection function - half-normal

Check which models in your QAICctable to include:


Model selection based on QAICc:
(c-hat estimate = 1.240156)

          K  QAICc Delta_QAICc QAICcWt Cum.Wt Quasi.LL
hnLC_.    4 227.71        0.00    0.94   0.94  -107.00
hnLC_DTC  5 233.43        5.72    0.05   1.00  -106.71
hazLC_.   5 238.65       10.94    0.00   1.00  -109.33
hazLC_DTC 6 247.36       19.65    0.00   1.00  -109.28
haz._DTC  5 280.27       52.56    0.00   1.00  -130.14
hn._DTC   4 283.08       55.37    0.00   1.00  -134.68
haz._.    4 292.71       65.00    0.00   1.00  -139.50
hn._.     3 296.97       69.26    0.00   1.00  -143.99

The half-normal models containing Landcover are models 1 and 2 in our table, so we’ll use those numbers to index our QAICc table in the next step

Importance of Landcover

1SumWt_LC <- sum(QAICctable$QAICcWt[
2    c(1,2)])
SumWt_LC

1: Sum the model weights for
2: our half-normal models containing Landcover to predict detectability

[1] 0.9959845

Models that include Landcover as a predictor of density have a summed weight of 100%, compared to 0% for models that omit Landcover as a covariate

The evidence from summed model weights is even stronger here, with models including landcover securing all of the total model weight

This adds to our conviction that landcover strongly influences detectability of water deer in this study

Evidence ratios

Pairs of models can also be compared using evidence ratios:

\[ Weight_{Model B} / Weight_{Model A} \]

Evidence ratios are useful to:

Indicate the relative likelihood of one model compared with another
Help you determine the effect of including a parameter, or excluding it

Calculate an evidence ratio

If we’re still undecided about the importance of a covariate, we can contrast pairs of models which are identical apart from whether they include or exclude that covariate

For example, check the importance of distance from the coast by contrasting the half-normal model containing both DistToCoast and Landcover (hnLC_DTC) with the model containing only Landcover (hnLC_.):

Calculate an evidence ratio between hnLC_DTC and hnLC_.:

1evidenceDTC <- evidence(QAICctable,
2    model.high = "hnLC_.",
3    model.low = "hnLC_DTC")
evidenceDTC

1: Specify the AIC comparison table
2: More highly ranked model (smaller dAICc)
3: Lower ranked model (larger dAICc)


Evidence ratio between models 'hnLC_.' and 'hnLC_DTC':
17.43

The evidence ratio between these two models is 17.43, indicating that the model omitting DistToCoast is over 17 times more likely than the model containing DistToCoast

Further evidence ratios

For more complex model sets, we might want to investigate a covariate more thoroughly

We could calculate the evidence ratio for more pairs of models further down our AIC table

For example, in our water deer case-study we could compare the models that exclude landcover as a predictor: hn._DTC and hn._.. This would tell us the importance of DistToCoast without taking the influence of landcover into account

Every pairwise comparison of models, one of which includes and one of which omits the covariate, gives us information about the relative importance of that covariate

The evidence ratio will vary in size depending on:

The pair of models we are comparing
The importance of the other covariates in that pair