distribution of individual covariate

[annieh]

Hi all,

I am comparing two hare populations subject to different management regimes. More specifically I am looking at the affects of hunting on dispersal and survival. I have a number of leverets radio tagged and tag failure and loss is low, what tag loss I have does not appear to be biased and therefore I have opted for Known Fate models in MARK. I have calculated dispersal distance as the distance between the natal site and the breeding site. I want to incorporate this dispersal parameter as an individual covariate in my Known Fate model to assess the costs of dispersal in terms of increased mortality.

However, when assessing the affect of various explanatory variables on dispersal distance, it transpired that I needed to log transform dispersal distance as it was heteroscadastic.

My question is:
Should individual covariates be normally distributed? i.e. should I use the log transformed distance as opposed to dispersal distance. I have tried the analysis with both and they give different results.

Hope this isn't too daft a question!

[sbonner]

Hi Annie,

Regression methods condition on the covariates (i.e., the covariates are treated as fixed not random quantities) and so there are no inherent assumptions regarding the distribution of the covariates.
As a simple example, you can regress a response on a discrete covariate with only 2 or 3 possible values -- and there is no way that the observed distribution of these values will ever be close to normal.

All of the assumptions in regression models (including mark-recapture models with covariates) concern the distribution of the response (a scalar value in simple regression or the capture histories in MR). While it isn't necessary that the covariates follow a specific distribution transforming the covariates can affect the fit of the model (i.e., the distribution of the residuals or the probabilities assigned to the capture histories).

My suggestion is that you think carefully about how you expect the covariate to impact the outcome. E.g., using untransformed dispersal distance would imply that the change in (logit) survival is the same for each unit of distance and using log-transformed distance would imply that the effect per unit increases or decreases at larger distances. If you really aren't sure then you could run different models and compare them to see which explains the data better -- being sure to check the goodness of fit.

Hope that helps,

Simon

[mcmelnychuk]

One other consideration unrelated to the distribution of covariates, and potentially a wrench in the gears, is of what to do with the hares that did not reach breeding sites. What dispersal distance values would you assign to hares that were killed by hunters or predators before they bred? If there actually is a strong relationship between dispersal distance and survival (which seems reasonable), then the animals that were killed were potentially the ones that were in the process of dispersing the furthest, but just didn't make it. If mortality locations are known those could be a surrogate, but they would likely underestimate what the dispersal distances would have been. (I haven't used known fate models, so maybe this point is irrelevant using that structure.)
Cheers, Mike

[annieh]

Thanks Simon for your very helpful comments!

Mike - yes this has been something that has bothered me, and others studying dispersal, for a while - those that died, did they die because they were in the transient phase of dispersal?? Its impossible to know..

I estimated the breeding site as the average of the last three locations in February, the onset of breeding, or the last three prior to death if the individual died before breeding. This has the consequence that older individuals have more time to disperse further, and so dispersal distance is highly correlated with age at which the breeding site was estimated. To try to take into account this flaw, I have also calculated a dispersal rate (distance/age at breeding) and run a new set of models with dispersal rate as an individual covariate.. any comments or suggestions on this would be greatly appreciated!

Many thanks,
Annie

[mcmelnychuk]

It sounds like you've thought a lot about this already and have it under control.
Another possibility to consider could be to measure a distance from natal site to wherever they are at some fixed age, with that age occurring before any individuals have been killed (this couldn't apply if you observed mortalities in young animals, though). This wouldn't be a true dispersal distance because they could still disperse further after the fixed age, but it might be proportional to it, and it would mean that the covariate value represents the same thing for all individuals whether they survived or died.
Good luck with it,
Mike