# binomialff: Binomial Family Function In VGAM: Vector Generalized Linear and Additive Models

## Description

Family function for fitting generalized linear models to binomial responses

## Usage

 ```1 2``` ```binomialff(link = "logitlink", multiple.responses = FALSE, parallel = FALSE, zero = NULL, bred = FALSE, earg.link = FALSE) ```

## Arguments

 `link` Link function; see `Links` and `CommonVGAMffArguments` for more information.
 `multiple.responses` Multivariate response? If `TRUE`, then the response is interpreted as M independent binary responses, where M is the number of columns of the response matrix. In this case, the response matrix should have Q columns consisting of counts (successes), and the `weights` argument should have Q columns consisting of the number of trials (successes plus failures). If `FALSE` and the response is a (2-column) matrix, then the number of successes is given in the first column, and the second column is the number of failures.
 `parallel` A logical or formula. Used only if `multiple.responses` is `TRUE`. This argument allows for the parallelism assumption whereby the regression coefficients for a variable is constrained to be equal over the M linear/additive predictors. If `parallel = TRUE` then the constraint is not applied to the intercepts. `zero` An integer-valued vector specifying which linear/additive predictors are modelled as intercepts only. The values must be from the set {1,2,...,M}, where M is the number of columns of the matrix response. See `CommonVGAMffArguments` for more information. `earg.link` Details at `CommonVGAMffArguments`. `bred` Details at `CommonVGAMffArguments`. Setting `bred = TRUE` should work for multiple responses (`multiple.responses = TRUE`) and all VGAM link functions; it has been tested for `logitlink` only (and it gives similar results to brglm but not identical), and further testing is required. One result from fitting bias reduced binary regression is that finite regression coefficients occur when the data is separable (see example below). Currently `hdeff.vglm` does not work when `bred = TRUE`.

## Details

This function is largely to mimic `binomial`, however there are some differences.

When used with `cqo` and `cao`, it may be preferable to use the `clogloglink` link.

## Value

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

## Warning

See the above note regarding `bred`.

The maximum likelihood estimate will not exist if the data is completely separable or quasi-completely separable. See Chapter 10 of Altman et al. (2004) for more details, and safeBinaryRegression and `hdeff.vglm`. Yet to do: add a `sepcheck = TRUE`, say, argument to further detect this problem and give an appropriate warning.

## Note

If `multiple.responses` is `FALSE` (default) then the response can be of one of two formats: a factor (first level taken as failure), or a 2-column matrix (first column = successes) of counts. The argument `weights` in the modelling function can also be specified as any vector of positive values. In general, 1 means success and 0 means failure (to check, see the `y` slot of the fitted object). Note that a general vector of proportions of success is no longer accepted.

The notation M is used to denote the number of linear/additive predictors.

If `multiple.responses` is `TRUE`, then the matrix response can only be of one format: a matrix of 1's and 0's (1 = success).

Fisher scoring is used. This can sometimes fail to converge by oscillating between successive iterations (Ridout, 1990). See the example below.

Thomas W. Yee

## References

McCullagh, P. and Nelder, J. A. (1989). Generalized Linear Models, 2nd ed. London: Chapman & Hall.

Altman, M. and Gill, J. and McDonald, M. P. (2004). Numerical Issues in Statistical Computing for the Social Scientist, Hoboken, NJ, USA: Wiley-Interscience.

Ridout, M. S. (1990). Non-convergence of Fisher's method of scoring—a simple example. GLIM Newsletter, 20(6).

`hdeff.vglm`, `Links`, `rrvglm`, `cqo`, `cao`, `betabinomial`, `posbinomial`, `zibinomial`, `double.expbinomial`, `seq2binomial`, `amlbinomial`, `simplex`, `binomial`, `simulate.vlm`, safeBinaryRegression, `residualsvglm`.

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42``` ```shunua <- hunua[sort.list(with(hunua, altitude)), ] # Sort by altitude fit <- vglm(agaaus ~ poly(altitude, 2), binomialff(link = clogloglink), data = shunua) ## Not run: plot(agaaus ~ jitter(altitude), shunua, ylab = "Pr(Agaaus = 1)", main = "Presence/absence of Agathis australis", col = 4, las = 1) with(shunua, lines(altitude, fitted(fit), col = "orange", lwd = 2)) ## End(Not run) # Fit two species simultaneously fit2 <- vgam(cbind(agaaus, kniexc) ~ s(altitude), binomialff(multiple.responses = TRUE), data = shunua) ## Not run: with(shunua, matplot(altitude, fitted(fit2), type = "l", main = "Two species response curves", las = 1)) ## End(Not run) # Shows that Fisher scoring can sometime fail. See Ridout (1990). ridout <- data.frame(v = c(1000, 100, 10), r = c(4, 3, 3), n = rep(5, 3)) (ridout <- transform(ridout, logv = log(v))) # The iterations oscillates between two local solutions: glm.fail <- glm(r / n ~ offset(logv) + 1, weight = n, binomial(link = 'cloglog'), ridout, trace = TRUE) coef(glm.fail) # vglm()'s half-stepping ensures the MLE of -5.4007 is obtained: vglm.ok <- vglm(cbind(r, n-r) ~ offset(logv) + 1, binomialff(link = clogloglink), ridout, trace = TRUE) coef(vglm.ok) # Separable data set.seed(123) threshold <- 0 bdata <- data.frame(x2 = sort(rnorm(nn <- 100))) bdata <- transform(bdata, y1 = ifelse(x2 < threshold, 0, 1)) fit <- vglm(y1 ~ x2, binomialff(bred = TRUE), data = bdata, criter = "coef", trace = TRUE) coef(fit, matrix = TRUE) # Finite!! summary(fit) ## Not run: plot(depvar(fit) ~ x2, data = bdata, col = "blue", las = 1) lines(fitted(fit) ~ x2, data = bdata, col = "orange") abline(v = threshold, col = "gray", lty = "dashed") ## End(Not run) ```

### Example output  ```Loading required package: stats4
v r n     logv
1 1000 4 5 6.907755
2  100 3 5 4.605170
3   10 3 5 2.302585
Deviance = 22.72791 Iterations - 1
Deviance = 28.29819 Iterations - 2
Deviance = 19.01799 Iterations - 3
Deviance = 21.07849 Iterations - 4
Deviance = 17.97632 Iterations - 5
Deviance = 18.59905 Iterations - 6
Deviance = 17.90326 Iterations - 7
Deviance = 18.42591 Iterations - 8
Deviance = 17.88036 Iterations - 9
Deviance = 18.37207 Iterations - 10
Deviance = 17.87175 Iterations - 11
Deviance = 18.3519 Iterations - 12
Deviance = 17.86832 Iterations - 13
Deviance = 18.34386 Iterations - 14
Deviance = 17.86692 Iterations - 15
Deviance = 18.34059 Iterations - 16
Deviance = 17.86634 Iterations - 17
Deviance = 18.33924 Iterations - 18
Deviance = 17.8661 Iterations - 19
Deviance = 18.33868 Iterations - 20
Deviance = 17.866 Iterations - 21
Deviance = 18.33845 Iterations - 22
Deviance = 17.86596 Iterations - 23
Deviance = 18.33835 Iterations - 24
Deviance = 17.86595 Iterations - 25
Deviance = 22.72791 Iterations - 1
Deviance = 28.29819 Iterations - 2
Deviance = 19.01799 Iterations - 3
Deviance = 21.07849 Iterations - 4
Deviance = 17.97632 Iterations - 5
Deviance = 18.59905 Iterations - 6
Deviance = 17.90326 Iterations - 7
Deviance = 18.42591 Iterations - 8
Deviance = 17.88036 Iterations - 9
Deviance = 18.37207 Iterations - 10
Deviance = 17.87175 Iterations - 11
Deviance = 18.3519 Iterations - 12
Deviance = 17.86832 Iterations - 13
Deviance = 18.34386 Iterations - 14
Deviance = 17.86692 Iterations - 15
Deviance = 18.34059 Iterations - 16
Deviance = 17.86634 Iterations - 17
Deviance = 18.33924 Iterations - 18
Deviance = 17.8661 Iterations - 19
Deviance = 18.33868 Iterations - 20
Deviance = 17.866 Iterations - 21
Deviance = 18.33845 Iterations - 22
Deviance = 17.86596 Iterations - 23
Deviance = 18.33835 Iterations - 24
Deviance = 17.86595 Iterations - 25
Warning messages:
1: glm.fit: algorithm did not converge
2: glm.fit: algorithm did not converge
3: In glm(r/n ~ offset(logv) + 1, weight = n, binomial(link = "cloglog"),  :
fitting to calculate the null deviance did not converge -- increase 'maxit'?
(Intercept)
-5.157362
VGLM    linear loop  1 :  deviance = 22.72791
VGLM    linear loop  2 :  deviance = 28.29819
Taking a modified step.
VGLM    linear loop  2 :  deviance = 18.08526
VGLM    linear loop  3 :  deviance = 17.80972
VGLM    linear loop  4 :  deviance = 18.20771
Taking a modified step.
VGLM    linear loop  4 :  deviance = 17.46081
VGLM    linear loop  5 :  deviance = 17.46624
Taking a modified step.
VGLM    linear loop  5 :  deviance = 17.43666
VGLM    linear loop  6 :  deviance = 17.43668
Taking a modified step.
VGLM    linear loop  6 :  deviance = 17.43661
VGLM    linear loop  7 :  deviance = 17.43661
Taking a modified step.
VGLM    linear loop  7 :  deviance = 17.43661
Warning message:
In vglm.fitter(x = x, y = y, w = w, offset = offset, Xm2 = Xm2,  :
some quantities such as z, residuals, SEs may be inaccurate due to convergence at a half-step
(Intercept)
-5.400638
VGLM    linear loop  1 :  coefficients = -0.10015601,  2.10784287
VGLM    linear loop  2 :  coefficients = -0.11063311,  3.70413974
VGLM    linear loop  3 :  coefficients = -0.10881416,  5.58548660
VGLM    linear loop  4 :  coefficients = -0.11736263,  7.34812525
VGLM    linear loop  5 :  coefficients = -0.13796225,  8.32167765
VGLM    linear loop  6 :  coefficients = -0.14943194,  8.49745083
VGLM    linear loop  7 :  coefficients = -0.15084587,  8.49588971
VGLM    linear loop  8 :  coefficients = -0.15083084,  8.49572183
VGLM    linear loop  9 :  coefficients = -0.15082982,  8.49572911
VGLM    linear loop  10 :  coefficients = -0.15082987,  8.49572905
VGLM    linear loop  11 :  coefficients = -0.15082987,  8.49572904
(Intercept)      -0.1508299
x2                8.4957290

Call:
vglm(formula = y1 ~ x2, family = binomialff(bred = TRUE), data = bdata,
criter = "coef", trace = TRUE)

Pearson residuals:
Min      1Q   Median         3Q   Max
logitlink(prob) -642.8 -0.3786 -0.09315 -0.0004856 1.054

Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept)  -0.1508     0.4626  -0.326    0.744
x2            8.4957     2.0883   4.068 4.74e-05 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1