R: Maximum Likelihood Estimation

mle2 {bbmle} | R Documentation

## Maximum Likelihood Estimation

### Description

Estimate parameters by the method of maximum likelihood.

### Usage

    mle2(minuslogl, start, method, optimizer,
        fixed = NULL, data=NULL,
        subset=NULL,
    default.start=TRUE, eval.only = FALSE, vecpar=FALSE,
    parameters=NULL,
    parnames=NULL,
    skip.hessian=FALSE,
    hessian.opts=NULL,
    use.ginv=TRUE,
    trace=FALSE,
    browse_obj=FALSE,
    gr=NULL,
    optimfun,...)
    calc_mle2_function(formula,parameters, links, start,
       parnames, use.deriv=FALSE, data=NULL,trace=FALSE)

### Arguments

`minuslogl` | Function to calculate negative log-likelihood, or a formula
`start` | Named list. Initial values for optimizer
`method` | Optimization method to use. See `optim`.
`optimizer` | Optimization function to use. Currently available choices are "optim" (the default), "nlm", "nlminb", "constrOptim", "optimx", and "optimize". If "optimx" is used, (1) the `optimx` package must be explicitly loaded with `load` or `require`(Warning: Options other than the default may be poorly tested, use with caution.)
`fixed` | Named list. Parameter values to keep fixed during optimization.
`data` | list of data to pass to negative log-likelihood function: must be specified if `minuslogl` is specified as a formula
`subset` | logical vector for subsetting data (STUB)
`default.start` | Logical: allow default values of `minuslogl` as starting values?
`eval.only` | Logical: return value of `minuslogl(start)` rather than optimizing
`vecpar` | Logical: is first argument a vector of all parameters? (For compatibility with `optim`.) If `vecpar` is `TRUE`, then you should use `parnames` to define the parameter names for the negative log-likelihood function.
`parameters` | List of linear models for parameters. MUST BE SPECIFIED IN THE SAME ORDER as the start vector (this is a bug/restriction that I hope to fix soon, but in the meantime beware)
`links` | (unimplemented) specify transformations of parameters
`parnames` | List (or vector?) of parameter names
`gr` | gradient function
`...` | Further arguments to pass to optimizer
`formula` | a formula for the likelihood (see Details)
`trace` | Logical: print parameter values tested?
`browse_obj` | Logical: drop into browser() within the objective function?
`skip.hessian` | Bypass Hessian calculation?
`hessian.opts` | Options for Hessian calculation, passed through to the `hessian` function
`use.ginv` | Use generalized inverse (`ginv`) to compute approximate variance-covariance
`optimfun` | user-supplied optimization function. Must take exactly the same arguments and return exactly the same structure as `optim`.
`use.deriv` | (experimental, not yet implemented): construct symbolic derivatives based on formula?

### Details

The `optim` optimizer is used to find the minimum of the negative log-likelihood. An approximate covariance matrix for the parameters is obtained by inverting the Hessian matrix at the optimum.

The `minuslogl` argument can also specify a formula, rather than an objective function, of the form `x~ddistn(param1,...,paramn)`. In this case `ddistn` is taken to be a probability or density function, which must have (literally) `x` as its first argument (although this argument may be interpreted as a matrix of multivariate responses) and which must have a `log` argument that can be used to specify the log-probability or log-probability-density is required. If a formula is specified, then `parameters` can contain a list of linear models for the parameters.

If a formula is given and non-trivial linear models are given in `parameters` for some of the variables, then model matrices will be generated using `model.matrix`. `start` can be given:

- as a list containing lists, with each list corresponding to the starting values for a particular parameter;

- just for the higher-level parameters, in which case all of the additional parameters generated by `model.matrix` will be given starting values of zero (unless a no-intercept formula with `-1` is specified, in which case all the starting values for that parameter will be set equal)

- [to be implemented!] as an exhaustive (flat) list of starting values (in the order given by `model.matrix`)

The `trace` argument applies only when a formula is specified. If you specify a function, you can build in your own `print()` or `cat()` statement to trace its progress. (You can also specify a value for `trace` as part of a `control` list for `optim()`: see `optim`.)

The `skip.hessian` argument is useful if the function is crashing with a "non-finite finite difference value" error when trying to evaluate the Hessian, but will preclude many subsequent confidence interval calculations. (You will know the Hessian is failing if you use `method="Nelder-Mead"` and still get a finite-difference error.)

If convergence fails, see the manual page of the relevant optimizer (`optim` by default, but possibly `nlm`, `nlminb`, `optimx`, or `constrOptim` if you have set the value of `optimizer`) for the meanings of the error codes/messages.

### Value

An object of class `"mle2"`.

### Warning

Do not use a higher-level variable named `.i` in `parameters` – this is reserved for internal use.

### Note

Note that the `minuslogl` function should return the negative log-likelihood, -log L (not the log-likelihood, log L, nor the deviance, -2 log L). It is the user's responsibility to ensure that the likelihood is correct, and that asymptotic likelihood inference is valid (e.g. that there are "enough" data and that the estimated parameter values do not lie on the boundary of the feasible parameter space).

If `lower`, `upper`, `control$parscale`, or `control$ndeps` are specified for `optim` fits, they must be named vectors.

The requirement that `data` be specified when using the formula interface is relatively new: it saves many headaches on the programming side when evaluating the likelihood function later on (e.g. for profiling or constructing predictions). Since `data.frame` uses the names of its arguments as column names by default, it is probably the easiest way to package objects that are lying around in the global workspace for use in `mle2` (provided they are all of the same length).

When `optimizer` is set to "optimx" and multiple optimization methods are used (i.e. the `methods` argument has more than one element, or `all.methods=TRUE` is set in the control options), the best (minimum negative log-likelihood) solution will be saved, regardless of reported convergence status (and future operations such as profiling on the fit will only use the method that found the best result).

### See Also

`mle2-class`

### Examples

    x <- 0:10
    y <- c(26, 17, 13, 12, 20, 5, 9, 8, 5, 4, 8)
    d <- data.frame(x,y)

    ## in general it is best practice to use the `data' argument,
    ##  but variables can also be drawn from the global environment
    LL <- function(ymax=15, xhalf=6)
        -sum(stats::dpois(y, lambda=ymax/(1+x/xhalf), log=TRUE))
    ## uses default parameters of LL
    (fit <- mle2(LL))
    fit1F <- mle2(LL, fixed=list(xhalf=6))
    coef(fit1F)
    coef(fit1F,exclude.fixed=TRUE)

    (fit0 <- mle2(y~dpois(lambda=ymean),start=list(ymean=mean(y)),data=d))
    anova(fit0,fit)
    summary(fit)
    logLik(fit)
    vcov(fit)
    p1 <- profile(fit)
    plot(p1, absVal=FALSE)
    confint(fit)

    ## use bounded optimization
    ## the lower bounds are really > 0, but we use >=0 to stress-test
    ## profiling; note lower must be named
    (fit1 <- mle2(LL, method="L-BFGS-B", lower=c(ymax=0, xhalf=0)))
    p1 <- profile(fit1)

    plot(p1, absVal=FALSE)
    ## a better parameterization:
    LL2 <- function(lymax=log(15), lxhalf=log(6))
        -sum(stats::dpois(y, lambda=exp(lymax)/(1+x/exp(lxhalf)), log=TRUE))
    (fit2 <- mle2(LL2))
    plot(profile(fit2), absVal=FALSE)
    exp(confint(fit2))
    vcov(fit2)
    cov2cor(vcov(fit2))

    mle2(y~dpois(lambda=exp(lymax)/(1+x/exp(lhalf))),
       start=list(lymax=0,lhalf=0),
       data=d,
       parameters=list(lymax~1,lhalf~1))

    ## Not run:
    ## try bounded optimization with nlminb and constrOptim
    (fit1B <- mle2(LL, optimizer="nlminb", lower=c(lymax=1e-7, lhalf=1e-7)))
    p1B <- profile(fit1B)
    confint(p1B)
    (fit1C <- mle2(LL, optimizer="constrOptim", ui = c(lymax=1,lhalf=1), ci=2,
       method="Nelder-Mead"))

    set.seed(1001)
    lymax <- c(0,2)
    lhalf <- 0
    x <- sort(runif(200))
    g <- factor(sample(c("a","b"),200,replace=TRUE))
    y <- rnbinom(200,mu=exp(lymax[g])/(1+x/exp(lhalf)),size=2)
    d2 <- data.frame(x,g,y)

    fit3 <- mle2(y~dnbinom(mu=exp(lymax)/(1+x/exp(lhalf)),size=exp(logk)),
        parameters=list(lymax~g),data=d2,
        start=list(lymax=0,lhalf=0,logk=0))

    ## End(Not run)

[Package bbmle version 1.0.19 Index]