Interpretation of Beta-values

[Chanelle Potgieter]

Good day,

I just want to confirm... a lot of articles interpret beta-values and their standard errors in their results (for occupancy modeling), rather than the predicted occupancy or detection probability estimate.

What is the beta-value exactly? Is it just the covariate coefficient one gets from the summary table of a model? And why do one interpret the beta-value instead of the  predicted occupancy or detection probability estimate?

Kind regards,

Chanelle Potgieter

Honours Student

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[Marc Kery]

Dear Chanelle,

to understand what a statistical model is, it is very helpful to write it down in algebra. In fact, I think that one cannot understand what a 'beta value' is without referring to the algebra of the model.

An occupancy model can be described as two linked logistic regression models, one for the true (and partially latent) presence/absence, denoted z, and the other for the observed (and false-negative-error-prone) detection/nondetection data, denoted y.

Let's assume you fit a model where a covariate X (say, vegetation density) affects both the latent state z and the observations y, through its effects on occupancy probability and detection probability (PS1). Indexing sites by i and repeated measurements (a.k.a., repeated visits, occasions, ...) by j, we can write this simple static occupancy model as follows:

# Submodel for latent state z, governed by occupancy prob. psi

z_i ~ Bernoulli(psi_i)

logit(psi_i) = beta0 + beta1 * x_i                                       # Covariate model linking psi and X linearly on the link scale

# Submodel for observed data y, governed by detection prob. p

y_i,j ~ Bernoulli(z_i * p_i)

logit(p_i) = alpha0 + alpha1 * x_i                                      # Covariate model linking p and X linearly on the link scale

When you fit this model, you get estimates of the four parameters beta0, beta1, alpha0 and alpha1. Two of them are an intercept and the other two are slope parameters that describe how a probability (on the logit = log(odds) = log(p / (1-p)) scale) depends linearly on the covariate X.

If you only need to know whether X affects (in a correlative way !) occupancy or detection, then you can simply check whether it is significantly different from zero and look at its sign: if the sign is positive, then (say) occupancy probability is greater for larger values of X than for smaller values, vice versa. This is often the main thing that we want to know.

However, due to the non-linear nature of the link function (i.e., the logit), it is near impossible to directly see what a change in x really does to psi without plotting predictions from the model.

Therefore, most of the times we will fill into the estimated regression equation (i.e., beta0 + beta1 * x_i) a large range of values for X and see what this produces in terms of occupancy probability psi. In unmarked we have predict() functions to do this. These yield not only point estimates for these predictions, but also their standard errors and 95% confidence intervals. Almost always we will present the results from such a prediction exercise in a plot.

Hope that helps. But do learn to understand these models in algebra. Without that I can't see how one can use them in a responsible manner, to be honest.

Best regards  --- Marc

(PS1) Note that it is one of the great assets of these kinds of hierarchical models, that we can estimate the effects of the same covariate on both parts of the model, i.e., here, on both occupancy and detection probability.

(PS2) Sometime we need a website for unmarked and/or HMecology and then put there a FAQ section. (A possibly improved version of) This post here could then go under the heading "what is a beta value ?"

---

From: unma...@googlegroups.com <unma...@googlegroups.com> on behalf of Chanelle Potgieter <u215...@tuks.co.za>

Sent: Tuesday, July 30, 2024 17:16
To: unmarked <unma...@googlegroups.com>
Subject: [unmarked] Interpretation of Beta-values

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[Chanelle Potgieter]

Dear Marc,

Thanks a lot. It makes much more sense now !

I would like to show  my results from one of my best models (according to AIC model selection), and just ask for assistance  regarding results interpretation, if I may.

Kindly refer to the attached images with regards to my queries.

RAI_Jackal refers to the relative abundance of Jackal and I am looking at civet occupancy.

Will I interpret the results in the first picture as: "Civet occupancy increases with larger values of relative jackal abundance (Beta-value = 8.85 +- SE)?

And then how will I interpret a covariate's influence on detection or occupancy if it is categorical (a factor), like vegetation type? Do I need to interpret all three estimates or does it depend on what my hypothesis is?

I also noticed that my standard error values are very large, and there may be several reasons for that... In my case I believe it is due to the small sample size... Can I however still present these results, and just discuss my limitations in the end?

# Fit the model
  occ27 <- occu(formula = ~ Veg_Type ~ RAI_Jackal, data = sample.unmarkedFrame_cov)
  occ27

# Vegetation type influence on detection#

preds <- data.frame(Veg_Type = c("Central Sandy Bushveld", "Waterberg Mountain Bushveld", "Western Sandy Bushveld"))
  fits.occ27.psi <- predict(occ27, type = "det", newdata=preds, appendData = TRUE)
  fits.occ27.psi$type <- "det"
  fits.occ27.psi
 
 

# Relative Jackal abundance influence on occupancy #

# Create a sequence for RAI_Jackal#
  RAI_Jackal_seq <- seq(min(site_cov$RAI_Jackal, na.rm = TRUE),
                        max(site_cov$RAI_Jackal, na.rm = TRUE),
                        by = 0.01)
 
  # Predict occupancy probabilities for the sequence
  predict_occ27_Jackal <- predict(occ27,
                                  newdata = data.frame(RAI_Jackal = RAI_Jackal_seq),
                                  type = "state")
 
  # Combine the predictions with the RAI_Jackal sequence
  results <- cbind(predict_occ27_Jackal, RAI_Jackal = RAI_Jackal_seq)
  print(results)
 
  preds <- data.frame(RAI_Jackal_seq)
  fits.occ27.psi <- predict(occ27, type = "state", newdata=preds, appendData = TRUE)
  fits.occ27.psi$type <- "psi"
  fits.occ27.psi

Lastly, the code above (bold) is what I used to obtain predicted occupancy and detection estimates for the model. I am just unsure where and when we should interpret these results (results obtained from the predict function)? 

The second and third image is the results I obtained from the code in bold.

Kind regards,

--

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[Achaz von Hardenberg]

Hi Chanelle, 

I suspect that  those large SE actually indicate that there is no effect of Jackal on occupancy. In your AIC model selection, have you tried an intercept only model (~1 ) for the occupancy part? I suspect that it will result in being the best fitting model.

Hope this helps! 

Achaz von Hardenberg

Associate Professor

Department of Earth and Environmental Sciences, University of Pavia, Italy 

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[Chanelle Potgieter]

Dear Achaz,

Thank you for your response. 

Attached is a screenshot of my best models, I am only looking at the AIC models with Delta_AICc values  between 2 and 7, so my null model unfortunately does not fall in this range.

Any suggestion/s how I could interpret the estimates for my best model, despite the standard errors being large?

I am not sure if the estimates may be incorrect as a result of how I structured my data in my covariate dataset? 

Kind regards,

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[Marc Kery]

Dear Chanelle,

can you show us a summary of the unmarked data frame ? I agree with you that some of these unusually large numbers may be due to small sample size.

Thanks and best regards  ---- Marc

---

From: unma...@googlegroups.com <unma...@googlegroups.com> on behalf of Chanelle Potgieter <u215...@tuks.co.za>

Sent: Wednesday, July 31, 2024 11:23
To: unma...@googlegroups.com <unma...@googlegroups.com>
Subject: Re: [unmarked] Interpretation of Beta-values

 

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[Chanelle Potgieter]

Dear Marc,

Kindly find attached the summary of my unmarked data frame.

Kind regards,

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