Hybrid Mixture Model

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Description

The argument hcov in secr.fit is used to fit a hybrid mixture model. ‘Hybrid’ refers to a flexible combination of latent classes (as in a finite mixture) and known classes (cf groups or sessions). A hybrid mixture model includes a parameter ‘pmix’ for the mixing proportion and optionally allows detection parameters to be modelled as class-specific ( ~ h2). This is particularly useful for modelling sex ratio and sex differences in detection, and matches the Bayesian sex-specific model of Gardner et al. (2010).

For observed animals all of unknown class the model is identical to a finite mixture (i.e. latent-class) model. For observed animals all of known class, the classes are no longer ‘latent’ and the model is equivalent to a grouped model with an additional binomial factor for class membership.

Assumptions

hcov identifies a single individual covariate (the class covariate) that should be a factor with two levels, or contain character values that will be coerced to a factor (e.g., ‘f’, ‘m’). Missing values (NA) are used for individuals of unknown class. If hcov has more than two levels, all but the first two levels are converted to NA (but see exception for h3 models below).

It is assumed that the probability of recording a missing value for the class covariate is independent of the true class membership (e.g., sex equally likely to be recorded for males and females).

Operational details

A hybrid mixture model is fitted whenever hcov is not NULL. Mixture models include a parameter ‘pmix’, the mixing proportion. If the covariate identified by hcov is missing ('' or NA) for all individuals and a mixture term (h2 or h3) appears in the detection model (e.g., g0 ~ h2) then a conventional finite mixture model is fitted (cf Pledger 2000, Borchers & Efford 2008).

As with finite mixture models, any detection parameter (g0, sigma etc.) may be modelled as depending on mixture class by model specifications such as (g0 ~ h2, sigma ~ h2). See Examples.

In general hcov has been designed for two classes and two classes are assumed if neither ‘h2’ nor ‘h3’ appears in the model formulae. However, there is a small exception: hcov may have three non-missing levels if ‘h3’ appears in a model formula. Note that h2 cannot be combined with h3; h3 is for advanced use only and has not been fully tested.

The number of fitted parameters is the same as the corresponding finite mixture model if mixture terms (‘h2’, ‘h3’) appear in the model formulae. Otherwise (no mixture terms) estimating pmix requires a single extra parameter. The estimate of pmix then depends solely on the observed class proportions in the covariate, and the beta variance-covariance matrix will show zero covariance of pmix with other detection parameters.

Models for pmix

Variation in the parameter pmix may be modelled across sessions i.e., models such as pmix ~ session or pmix ~ Session are valid, as are formulae involving session covariates defined in the sessioncov argument of secr.fit.

If no mixture term appears in the formula for pmix then one is added automatically (usually ‘h2’). This serves mostly to keep track of values in the output.

Attempting to model pmix as a function of individual covariates or other within-session terms (t, b etc.) will cause an error.

Interpreting output

When you display a fitted secr model the parameter estimates are in a final section headed 'Fitted (real) parameters evaluated at base levels of covariates'. The same output may be obtained by calling the predict method directly. Calling predict has the advantage that you can obtain estimates for levels of the covariates other than the base levels, by specifying newdata. An example below shows how to specify h2 in newdata. [Note: predict is generic, and you must consult ?predict.secr to see the help for the specific implementation of this method for fitted secr objects].

The output from predict.secr for a mixture model is a list with one component for each (possibly latent) class. Each row corresponds to a fitted real parameter: ordinarily these include the detection parameters (e.g., g0, sigma) and the mixing proportion (pmix).

In the case of a model fitted by maximizing the full likelihood (CL = FALSE), density D will also appear in the output. Note that only one parameter for density is estimated, the total density across classes. This total density figure appears twice in the output, once for each class.

The standard error (SE.estimate) is shown for each parameter. These are asymptotic estimates back-transformed from the link scale. The confidence limits are also back-transformed from the link scale (95% CI by default; vary alpha in predict.secr if you want e.g. 90% CI).

The mixing proportion pmix depends on the composition of the sample with respect to hcov and the detection model. For a null detection model the mixing proportion is exactly the proportion in the sample, with appropriate binomial confidence limits. Otherwise, the mixing proportion adjusts for class differences in the probability and scale of detection (see Examples).

The preceding refers to the default behaviour when pmix ~ h2. It is possible also to fix the mixing proportion at any arbitrary value (e.g., fixed = list(pmix = 0.5) for 1:1 sex ratio).

On output the classes are tagged with the factor levels of hcov, regardless of how few or how many individuals were actually of known class. If only a small fraction were of known class, and there is cryptic variation unrelated to hcov, then the association between the fitted classes and the nominal classes (i.e. levels of hcov) may be weak, and should not be trusted.

Limitations

Hybrid mixture models are incompatible with groups as presently implemented.

The hcov likelihood conditions on the number of known-class individuals. A model fitted with hcov = NULL or with a different hcov covariate has in effect a different data set, and likelihoods, deviances or AICs cannot be compared. AIC can be used to compare models provided they all have the same hcov covariate in the call to secr.fit, whether or not h2 appears in the model definition.

Likelihood

The likelihood of the hybrid mixture model is detailed in an appendix of the vignette secr-finitemixtures.pdf.

References

Borchers, D.L. and Efford, M.G. (2008) Spatially explicit maximum likelihood methods for capture–recapture studies. Biometrics 64, 377–385.

Gardner, B., Royle, J.A., Wegan, M.T., Rainbolt, R. and Curtis, P. (2010) Estimating black bear density using DNA data from hair snares. Journal of Wildlife Management 74, 318–325.

Pledger, S. (2000) Unified maximum likelihood estimates for closed capture–recapture models using mixtures. Biometrics 56, 434–442.

See Also

secr.fit

Examples

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## Not run: 

## house mouse dataset, morning trap clearances
## 81 female, 78 male, 1 unknown
morning <- subset(housemouse, occ = c(1,3,5,7,9))
summary(covariates(morning))

## speedy model fitting with coarse mask
mmask <- make.mask(traps(morning), buffer = 20, nx = 32)

## assuming equal detection of males and females
## fitted sex ratio p(female) = 0.509434 = 81 / (81 + 78)
fit.0 <- secr.fit(morning, hcov = "sex", mask = mmask, trace = FALSE)
predict(fit.0)

## allowing sex-specific detection parameters
## this leads to new estimate of sex ratio 
fit.h2 <- secr.fit(morning, hcov = "sex", mask = mmask, trace = FALSE,
    model = list(g0 ~ h2, sigma ~ h2))
predict(fit.h2)

## specifying newdata for h2 - equivalent to predict(fit.h2)
predict(fit.h2, newdata = data.frame(h2 = factor(c('f','m'))))

## conditional likelihood fit of preceding model
## estimate of sex ratio does not change 
fit.CL.h2 <- secr.fit(morning, hcov = "sex", mask = mmask, trace = FALSE,
    CL = TRUE, model = list(g0 ~ h2, sigma ~ h2))
predict(fit.CL.h2)

## did sexes differ in detection parameters?
fit.CL.0 <- secr.fit(morning, hcov = "sex", mask = mmask, trace = FALSE,
    CL = TRUE, model = list(g0 ~ 1, sigma ~ 1))
LR.test(fit.CL.h2, fit.CL.0)

## did sex ratio deviate from 1:1?
fit.CL.h2.50 <- secr.fit(morning, hcov = "sex", mask = mmask, trace = FALSE,
    CL = TRUE, model = list(g0 ~ h2, sigma ~ h2), fixed = list(pmix = 0.5))
LR.test(fit.CL.h2, fit.CL.h2.50)

## did sexes show extra-compensatory variation in lambda0?
## (Efford and Mowat 2014)
fit.CL.a0 <- secr.fit(morning, hcov = "sex", mask = mmask, trace = FALSE,
    CL = TRUE, model = list(a0 ~ 1, sigma ~ h2))
LR.test(fit.CL.h2, fit.CL.a0)

## trend in ovenbird sex ratio, assuming sex-specific detection
omask <- make.mask(traps(ovenCH), buffer = 300, nx = 32)
fit.sextrend <- secr.fit(ovenCH, model = list(g0~h2, sigma~h2, pmix~Session),
    hcov = "Sex", CL = TRUE, mask = omask, trace = FALSE)
predict(fit.sextrend)[1:5]


## End(Not run)

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