calculating odds ratios in spatial correlation model

[shannonbarbermeyer]

Are odds ratios calculated the same way in spatial correlation models even if you have more than one covariate and theta's can just be ignored?

For example is the following correct?
estimate std.error
A1 ccupancy psi -1.978540 (0.855428)
B2 :detection thta0 -1.942184 (0.224220)
B3 :detection thta1 1.790065 (0.293362)
A4 :detection psiPreyMedHigh 4.149201 (1.202894)
A5 :detection psiHumanLow 0.958883 (0.894612)
B1 :detection p1 -1.255232 (0.403439)
B2 :detection p1ObsGood 2.240959 (0.436597)

A site with a prey index of medium/high has 63 times greater probability of being occupied than one with a prey index of low. But because I have another covariate - human impacts - don't I need to incorporate that in as well in the above statement because A1 psi represents the case where human impacts are high and prey index is low?

And an observer with good expertise has a 9 times greater probability of detecting the animal given the animal's presence at a site.

Also, I'm not sure why A4 and A5 say "detection" in the output because they were on the occupancy part of the design matrix along with psi and the theta's. Maybe just a bug in the way the output reads?

Thank you,
Shannon

[darryl]

Shannon
Because you have no interactions between the covariates, you can just interpret them individually. Strictly speaking you should also be talking about the odds of occupancy, not probability of occupancy, and the odds ratio relates to a 1-unit increase in the covariate of interest.

If anyone is a bit in the dark on what Shannon is talking about, here's a brief run down.

psi = Pr(site occupied)
psi/(1-psi) = odds of a site being occupied, eg odds of 2:1, site is twice as likely to be occupied than unoccupied, ie psi = 2/3
odds ratio (OR) = odds[2]/odds[1], ie ratio of odds at 2 different types of places.

If you have a covariate in a model for psi, and are using the logit link (ie logistic regression), and the estimated effect size for the covariate is beta, then the odds ratio for a 1 unit increase in that covariate will be exp(beta)

Cheers
Darryl

[cooch]

Because you have no interactions between the covariates, you can just interpret them individually. Strictly speaking you should also be talking about the odds of occupancy, not probability of occupancy, and the odds ratio relates to a 1-unit increase in the covariate of interest.

Probably time to write this up generally - same issue applies in MARK as well (obviously).

[shannonbarbermeyer]

Strictly speaking you should also be talking about the odds of occupancy, not probability of occupancy, and the odds ratio relates to a 1-unit increase in the covariate of interest.

Thank you for your help, Darryl.

Two quick clarifying questions:

1) does the 1-unit of increase ever apply to categorical covariates? Say if the levels are only 1,0? Or does it only ever apply to continuous?

2) can you explain what you mean by strictly referring to odds and not probability? I'm confused because I think of a fair coin where one has 50/50 odds of getting a head and also a 50% probability of getting a head.

Thank you very much for your help,
Shannon

[cooch]

Strictly speaking you should also be talking about the odds of occupancy, not probability of occupancy, and the odds ratio relates to a 1-unit increase in the covariate of interest.

Thank you for your help, Darryl.

Two quick clarifying questions:

1) does the 1-unit of increase ever apply to categorical covariates? Say if the levels are only 1,0? Or does it only ever apply to continuous?

2) can you explain what you mean by strictly referring to odds and not probability? I'm confused because I think of a fair coin where one has 50/50 odds of getting a head and also a 50% probability of getting a head.

Thank you very much for your help,
Shannon

Easiest way to think of it is as follows:

The coefficients in the logit scale (i.e., β) are interpretable as the natural log of the odds ratios. The odds of success (say, survival) is the ratio of the probability of success to the probability of failure.

Logistic regression (which is analogous to what we're doing here) is one way to generalize the odds ratio beyond two binary variables. Suppose you have a binary response variable Y (say, live or die) and a binary predictor variable X (say, male or female), and in addition we have other predictor variables Z1, ..., Zp that may or may not be binary (a bunch of covariates). If we use multiple logistic regression to regress Y on X, and the set of covariates Z1, ..., Zp, then the estimated coefficient beta(x) for X is related to a conditional odds ratio.

[darryl]

Strictly speaking you should also be talking about the odds of occupancy, not probability of occupancy, and the odds ratio relates to a 1-unit increase in the covariate of interest.

Thank you for your help, Darryl.

Two quick clarifying questions:

1) does the 1-unit of increase ever apply to categorical covariates? Say if the levels are only 1,0? Or does it only ever apply to continuous?

Applies to both, the mathematics doesn't care. It's a 1 unit increase for that covaraite whether it be an indicator variable, continuous covariate etc. Could be a change of 0 -> 1, or 5->6 or -99 -> -98, doesn't matter.

2) can you explain what you mean by strictly referring to odds and not probability? I'm confused because I think of a fair coin where one has 50/50 odds of getting a head and also a 50% probability of getting a head.

Well, you'd started off ok in your post talking about an odds ratio, but then with the interpretation you said "... 63 times greater probability of being occupied than one with a prey index of low." It should be "...63 times greater odds of being occupied than one with a prey index of low."

When we have 'even odds' or 50:50, or 1:1, that says the probability of success is equal to the probability of failure, ie 0.5. Now consider the roll of a dice, what's the probability of rolling a 6? 1/6. What's the odds of rolling a 6? (1/6)/(5/6), so 0.2:1 or 1:5. See chapter 3 of MacKenzie et al 2006 for more.

[shannonbarbermeyer]

Great - thank you!

Shannon