Exercise 12. Single-Species, Single-Season Occupancy Models with Errors_2

Single-SPECIES, SINGLE-Season Occupancy Models with IDENTIFICATION ERRORSOBJECTIVES:To learn and understand the single-season occupancy model that assesses false positive detections, and

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TABLE OF CONTENTS# TOC \o "1-3" \h \z \u ## HYPERLINK \l "_Toc162887063"

##Single-SPECIES, SINGLE-Season Occupancy Models with IDENTIFICATION ERRORS # PAGEREF _Toc162887063 \h ##3### HYPERLINK \l "_Toc162887064" ##OBJECTIVES: # PAGEREF _Toc162887064 \h ##3### HYPERLINK \l "_Toc162887065" ##BASIC INFORMATION

# PAGEREF _Toc162887065 \h ##3### HYPERLINK \l "_Toc162887066"

##BACKGROUND # PAGEREF _Toc162887066 \h ##4### HYPERLINK \l "_Toc162887067"

##EXTENDING THE MODEL TO INCLUDE FALSE-POSITIVES # PAGEREF _Toc162887067 \h

##9### HYPERLINK \l "_Toc162887068" ##ERROR MODEL SPREADSHEET INPUTS # PAGEREF_Toc162887068 \h ##11### HYPERLINK \l "_Toc162887069" ##SPREADSHEET HISTORY PROBABILITIES # PAGEREF _Toc162887069 \h ##13### HYPERLINK \l

"_Toc162887070" ##THE ERROR MODEL MULTINOMIAL LOG LIKELIHOOD # PAGEREF _Toc162887070 \h ##14### HYPERLINK \l "_Toc162887071" ##MAXIMIZING THE LOG LIKELIHOOD # PAGEREF _Toc162887071 \h ##15### HYPERLINK \l "_Toc162887072"

##ERROR MODEL OUTPUT # PAGEREF _Toc162887072 \h ##16### HYPERLINK \l

"_Toc162887073" ##MODEL PSI, P(.), ALPHẶ) # PAGEREF _Toc162887073 \h ##19##

# HYPERLINK \l "_Toc162887074" ##MODEL PSI,P(t); ALPHA=0 # PAGEREF

_Toc162887074 \h ##21### HYPERLINK \l "_Toc162887075" ##MODEL PSI,P(.); ALPHA=0

# PAGEREF _Toc162887075 \h ##24### HYPERLINK \l "_Toc162887076"

##SIMULATING ERROR DATA # PAGEREF _Toc162887076 \h ##25###

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Single-SPECIES, SINGLE-Season Occupancy Models with IDENTIFICATION ERRORS

OBJECTIVES:To learn and understand the single-season occupancy model that

assesses false positive detections, and how it fits into a multinomial maximum likelihood analysis.To use Solver to find the maximum likelihood estimates for the probability of false positive detection, and the probability of detection and site occupancy for each group To assess the -2LogeL of the saturated model

To introduce concepts of model fit.To learn how to simulate single-season

occupancy data with false positive identifications.BASIC INFORMATIONIf youï¿½ve been completing the exercises in this book in order, youï¿½ve learned a great deal about the single-season occupancy modeling, and some interesting variations

of the basic model In this exercise, we describe occupancy models in which false, positive identifications are included in the dataset This model was developed by Andy Royle and Bill Link, and is described in the paper: Royle, J.A., and W A Link 2006 Generalized site occupancy models allowing for falsepositive and false negative errors Ecology 87:835-841 Click on the

worksheet labeled ï¿½Errorsï¿½ and weï¿½ll get started.BACKGROUNDHopefully by now you have a solid understanding that the general occupancy model handles the fact that an encounter history full of zeroes (e.g., ï¿½0000ï¿½) can indicate either that the species of interest was absent from a site, or that species was present but simply undetected You might recall that the encounter history probability for such records handles this by adding the two possible outcomes together:

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however, which might be made by field observers, and those are false positive errors; i.e., observations recorded as 1 but should have been recorded as 0 These kinds of errors almost certainly occur Andy Royle and Bill Link point out that ï¿½even low false positive error rates can induce extreme bias in estimates of site occupancy when they are not accounted for.ï¿½ Hereï¿½s a quick example (based on personal experience) When avian ecologists conduct surveys of breeding birds, the vast majority of detections are obtained when theobserver hears the song or calls issued by individuals in the early morning hours Some species are notoriously hard to differentiate For instance, red-eyed vireos and blue-headed vireos are very similar in song except for

differences in the singing rate and ï¿½sweetness.ï¿½ In such cases, it is fairly easy to misidentify one species as the other You might recall that the single-season occupancy model assumes that these kinds of errors are not made, but false positives can be more common than anyone would like to admit

Nonetheless, they do occur and need to be handled So, how can the season occupancy model be extended to account for false positive detections? Well, believe it or not, the model that Andy Royle and Bill Link used as a starting point was the mixture model So, letï¿½s quickly review the mixture model, and then we will extend the concepts from the basic mixture model to include false errors REVIEW OF SINGLE SEASON MIXTURE MODELSThe idea behind mixture models, also called heterogeneity models, is that sites in the study area are unique in some way, such that there is heterogeneity among sites in terms of detection and occupancy probability Mixture models are used when investigators either donï¿½t record or donï¿½t have access to site- or survey-level covariates Thus, in mixture models, all of the differences among sites are unobservable or unknown Unobservable heterogeneity refers to situations when the factors causing differences in either occupancy probability or

single-detection probability cannot be readily identified This could simply mean we have absolutely no clue what might cause differences, but are willing to accept that there might be differences that we cannot measure For instance, if food resources are a critical predictor of occupancy but cannot be measured readily

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across sites, it might impose heterogeneity among the study sites, where some sites are rich in food resources and others are poor, even though food was not measured directly So, how does one model unobservable heterogeneity? Well, the basic idea is that the study sites can be divided into multiple groups, and each group (not each site) has unique detection probabilities and a unique probability of occupancy The number of groups can be either a discrete number (e.g., 2 groups, 3 groups, etc.) or an infinite number In exercise 6, we focused on a heterogeneity model in which the group number is discrete (n = 2), and heterogeneity was modeled for detection probability only So, for the two-point# mixture model we divide the study sites into two groups The population

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#i#d#e#a# #t#o# #r#e#c#a#l#l# #t#h#a#t# #m#i#x#t#u#r#e models are data hungry and care must be taken to not overparameterize the model Additionally, there

is the additional, sticky issue that the likelihood surface may have local optimạ¿½we discussed both of the issues in detail in exercise 6.EXTENDING THE MODEL TO INCLUDE FALSE-POSITIVES#As we mentioned previously, the two-point mixture model is the basis for the occupancy model in which false-positives and false negatives can be determined Again, the sites are divided into two

groups But this time, the dividing factor is whether the sites are truly

occupied or truly vacant In the diagram to the right, this means that N1 sites are truly occupied, and N2 sites are truly empty Now we write out

encounter history probabilities for each group, starting with group 1 (sites aretruly occupied) Wẹ¿½ll stick with our study in which sites were surveyed fourtimes The probability of realizing a 1111 history is:

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0001, a false positive error was made on survey 4 For history 0101, a false positive error was made on surveys 2 and 4 For history 1111, four false

positives were recorded Holy crow! To write out the encounter history

probabilities for group 2, we need a new parameter, called alpha (or

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spreadsheet In this example, the investigator surveys 250 study sites, with each site being surveyed 4 times The encounter histories are recorded in cellsB4:B19, and the frequency of each history is recorded in cells C4:C19 The total number of sites is given in cell C20, and the number of unique histories

is given in cell C21 (which you might remember indicates the number of terms in our multinomial likelihood function) To avoid over-parameterization, you can only run models with 15 or fewer parameters The nạ¿½ve estimate for occupancy(occupancy unadjusted for detection probability) is computed in cell C22 as the total number of sites which had one or more detections divided by the total number of sites In this case the estimate is around 60% Keep that in mind as

we move through the exercise.#OK, now letï¿½s look at the parameters Notice the spreadsheet is divided into two sections In the first s#e#c#t#i#o#n#

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unoccupied As with other spreadsheet exercises, you enter a 1 when a

parameter is being uniquely estimated, or enter a 0 if the parameter is being forced to be equal to some other parameter Note that we wonï¿½t estimate (1-

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#a#r#e#& #.#w#e# #a#r#e going to let Solver find the betas that maximize the multinomial log likelihood function (see below) SPREADSHEET HISTORY

PROBABILITIESOK! Now we are ready to compute the probability of realizing each history Letï¿½s start with the first history listed, 1111, in cell B4 The probability of realizing a 1111 history is estimated for each group separately

If sites are truly occupied, the probability of realizing a 1111 history is

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#=#G#1#1#*#G#1#2#*#G#1#3#*#G#1#4#*#G#1#5#.# # #A#c#r#o#s#s# #both groups, the probability of realizing a 1111 history is the sum of the two mixing

probabilities, given in cell J4 (=H4+I4) The natural log of the combined history probabilities is computed in cell K4 And so it goes for the remaining histories.Make sense? Spend time now clicking on the formula for each history and for group In our experience, if students understand how the encounter histories are calculated, the rest is a piece of cake.Notice that the sum of cells J4:J19 must equal 1 (cell J20): there are 16 possible histories, and eachhistory has a probability of being realized, but the sum of the probabilities must be 1.00 THE ERROR MODEL MULTINOMIAL LOG LIKELIHOODThe goal of the

analysis, as you might have guessed, is to find the combination of betas that maximizes the multinomial log likelihood function Remember, by changing the betas, we change the parameter estimates linked to each beta, which changes the probability of each encounter history, which changes the LogeL Betas ( MLEs ( Encounter Histories ( LogeLAll thatï¿½s left is to compute the log likelihood,given the frequencies of each history and the historyï¿½s probability The multinomial log likelihood formula that wẹ¿½ve been using is in the blue box below # SHAPE \* MERGEFORMAT ####There are 16 terms in this function, one foreach of the encounter histories The yi in the blue box are the frequencies of each kind of history and the pi in the blue box equation above are the history probabilities The LogeL is computed in cell B26 with the equation

=SUMPRODUCT(C4:C19,K4:K19), which corresponds to the general formula in the bluebox Now all we have to do is maximize this value to find the MLEï¿½s for our dataset MAXIMIZING THE LOG LIKELIHOODBefore we run our first model, we need tomake sure that (1-

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#E#R#R#O#R# #M#O#D#E#L# #O#U#T#P#U#TBefore we study the output, itï¿½s important

to note that the sometimes the parameters for the two groups get switched, meaning that group 1 is really group 2 and vica versa If your results donï¿½t match the results shown on the next few pages, try ï¿½seedingï¿½ the betas with random numbers (e.g., enter =RAND() in a beta cell), and then try running Solveragainï¿½ with random starting betas, your groups might switch back again First, letï¿½s take a look at the parameter estimates found by Solver:#The proportion of sites that were truly occupied (group 1) is 0.35659 (cell G4) Bysubtraction, the proportion of sites that were truly unoccupied (group 2) is (1-

0 35659) = 0.64341 These estimates would almost certainly be different if we assumed no false-positive errors For occupied sites, p1 = 0.73239 (cell G6), p2 = 0.66204 (cell G7), p3 = 0.67353 (cell G8), and p4 = 0.66602 For sites that were truly unoccupied, the probability of making a false positive

observation is fairly high across all four survey#s#:# # #ạ¿½1# #=#

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#B#2#5#:#L#2#6#.#####T#h#e# #L#o#g#e#L# #i#s# #g#i#ven in cell B26 Cell C26 is-2 times cell B26, and is the -2LogeL K is the number of parameters in any given model, and the underlying equation is =SUM(E5:E9,E12:E15) AIC is

computed as the -2LogeL + 2*K AICc is the second order correction of AIC, and uses the number of study sites in the calculation Deviance is computed as the difference between the saturated modelï¿½s -2LogeL and the current modelï¿½s -2LogeL; the lower the number the better Remember that by definition the saturated model is a model in which the data ï¿½fitï¿½ the model perfectly Thesaturated modelï¿½s -2LogeL is computed in the usual way (as in previous

exercises) in cells N4:O21 The model we just ran had a deviance of 6.8799; itï¿½s hard to tell if this is good or not without a goodness of fit test The Model Degrees of Freedom is the number of unique histories minus K In a modelwithout covariates, as long as the Model Degrees of Freedom is positive, you havenï¿½t overparameterized your model C-hat is computed in cells J26 as Deviance divided by DF The C-hat in this case is close to 1 C-hats larger than 1 might indicate some kind of lack of fit The Chi-Square statistic and associated p-value are given in cells K26:L26 The Chi-square computations are provided in the orange cells L4:M19 Click on the button labeled Model 1 to addyour results to the Results Table.#OK, now that youï¿½ve run one model, weï¿½ll run three more: a second error model where p

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Tiêu đề	Single-Species, Single-Season Occupancy Models with Identification Errors
Tác giả	Donovan, T. M., J. Hines
Trường học	University of Vermont
Chuyên ngành	Occupancy Modeling
Thể loại	Exercises
Năm xuất bản	2007
Thành phố	Burlington

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Số trang	2.052
Dung lượng	1,3 MB