posbernoulli.b {VGAM} | R Documentation

## Positive Bernoulli Family Function with Behavioural Effects

### Description

Fits a GLM-/GAM-like model to multiple Bernoulli responses where each row in the capture history matrix response has at least one success (capture). Capture history behavioural effects are accommodated.

### Usage

    posbernoulli.b(link = "logit", drop.b = FALSE ~ 1,
                   type.fitted = c("likelihood.cond", "mean.uncond"), I2 = FALSE,
                   ipcapture = NULL, iprecapture = NULL,
                   p.small = 1e-4, no.warning = FALSE)

### Arguments

`link, drop.b, ipcapture, iprecapture` | See `CommonVGAMffArguments` for information about these arguments. By default the parallelism assumption does not apply to the intercept. With an intercept-only model setting `drop.b = TRUE ~ 1` results in the M_0/M_h model.
`I2` | Logical. This argument is used for terms that are not parallel. If `TRUE` then the constraint matrix `diag(2)` (the general default constraint matrix in VGAM) is used, else `cbind(0:1, 1)`. The latter means the first element/column corresponds to the behavioural effect. Consequently it and its standard error etc. can be accessed directly without subtracting two quantities.
`type.fitted` | Details at `posbernoulli.tb`.
`p.small, no.warning` | See `posbernoulli.t`.

### Details

This model (commonly known as M_b/M_{bh} in the capture–recapture literature) operates on a capture history matrix response of 0s and 1s (n x tau). See `posbernoulli.t` for details, e.g., common assumptions with other models. Once an animal is captured for the first time, it is marked/tagged so that its future capture history can be recorded. The effect of the recapture probability is modelled through a second linear/additive predictor. It is well-known that some species of animals are affected by capture, e.g., trap-shy or trap-happy. This VGAM family function does allow the capture history to be modelled via such behavioural effects. So does `posbernoulli.tb` but `posbernoulli.t` cannot.

The number of linear/additive predictors is M = 2, and the default links are (logit p_c, logit p_r)^T where p_c is the probability of capture and p_r is the probability of recapture. The fitted value returned is of the same dimension as the response matrix, and depends on the capture history: prior to being first captured, it is `pcapture`. Afterwards, it is `precapture`.

By default, the constraint matrices for the intercept term and the other covariates are set up so that p_r differs from p_c by a simple binary effect, on a logit scale. However, this difference (the behavioural effect) is more directly estimated by having `I2 = FALSE`. Then it allows an estimate of the trap-happy/trap-shy effect; these are positive/negative values respectively. If `I2 = FALSE` then the (nonstandard) constraint matrix used is `cbind(0:1, 1)`, meaning the first element can be interpreted as the behavioural effect.

### Value

An object of class `"vglmff"` (see `vglmff-class`). The object is used by modelling functions such as `vglm`, and `vgam`.

### Note

The dependent variable is not scaled to row proportions. This is the same as `posbernoulli.t` and `posbernoulli.tb` but different from `posbinomial` and `binomialff`.

### Author(s)

Thomas W. Yee.

### References

See `posbernoulli.t`.

### See Also

`posbernoulli.t` and `posbernoulli.tb` (including estimating N), `deermice`, `dposbern`, `rposbern`, `posbinomial`, `aux.posbernoulli.t`, `prinia`.

### Examples

    # deermice data ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

    # Fit a M_b model
    M.b <- vglm(cbind(y1, y2, y3, y4, y5, y6) ~ 1,
                posbernoulli.b, data = deermice, trace = TRUE)
    coef(M.b)["(Intercept):1"]  # Behavioural effect on the logit scale
    coef(M.b, matrix = TRUE)
    constraints(M.b, matrix = TRUE)
    summary(M.b, presid = FALSE)

    # Fit a M_bh model
    M.bh <- vglm(cbind(y1, y2, y3, y4, y5, y6) ~ sex + weight,
                 posbernoulli.b, data = deermice, trace = TRUE)
    coef(M.bh, matrix = TRUE)
    coef(M.bh)["(Intercept):1"]  # Behavioural effect on the logit scale
    constraints(M.bh)  # (2,1) element of "(Intercept)" is for the behavioural effect
    summary(M.bh, presid = FALSE)  # Significant positive (trap-happy) behavioural effect
    # Approx. 95 percent confidence for the behavioural effect:
    SE.M.bh <- coef(summary(M.bh))["(Intercept):1", "Std. Error"]
    coef(M.bh)["(Intercept):1"] + c(-1, 1) * 1.96 * SE.M.bh

    # Fit a M_h model
    M.h <- vglm(cbind(y1, y2, y3, y4, y5, y6) ~ sex + weight,
                posbernoulli.b(drop.b = TRUE ~ sex + weight),
                data = deermice, trace = TRUE)
    coef(M.h, matrix = TRUE)
    constraints(M.h, matrix = TRUE)
    summary(M.h, presid = FALSE)

    # Fit a M_0 model
    M.0 <- vglm(cbind(    y1 + y2 + y3 + y4 + y5 + y6,
                      6 - y1 - y2 - y3 - y4 - y5 - y6) ~ 1,
                posbinomial, data = deermice, trace = TRUE)
    coef(M.0, matrix = TRUE)
    summary(M.0, presid = FALSE)

    # Simulated data set ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,
    set.seed(123); nTimePts <- 5; N <- 1000  # N is the popn size
    pdata <- rposbern(n = N, nTimePts = nTimePts, pvars = 2, is.popn = TRUE)
    nrow(pdata)  # Less than N (because some animals were never captured)
    # The truth: xcoeffs are c(-2, 1, 2) and cap.effect = +1

    M.bh.2 <- vglm(cbind(y1, y2, y3, y4, y5) ~ x2,
                   posbernoulli.b, data = pdata, trace = TRUE)
    coef(M.bh.2)
    coef(M.bh.2, matrix = TRUE)
    constraints(M.bh.2, matrix = TRUE)
    summary(M.bh.2, presid = FALSE)
    head(depvar(M.bh.2))    # Capture history response matrix
    head(M.bh.2@extra$cap.hist1)  # Info on its capture history
    head(M.bh.2@extra$cap1)  # When it was first captured
    head(fitted(M.bh.2))     # Depends on capture history
    (trap.effect <- coef(M.bh.2)["(Intercept):1"])  # Should be +1
    head(model.matrix(M.bh.2, type = "vlm"), 21)
    head(pdata)
    summary(pdata)
    dim(depvar(M.bh.2))
    vcov(M.bh.2)

    M.bh.2@extra$N.hat     # Estimate of the population size; should be about N
    M.bh.2@extra$SE.N.hat  # SE of the estimate of the population size
    # An approximate 95 percent confidence interval:
    round(M.bh.2@extra$N.hat + c(-1, 1) * 1.96 *  M.bh.2@extra$SE.N.hat, 1)

[Package VGAM version 1.0-3 Index]