nll_frailty function

Negative log-likelihood function: frailty

Function calculating negative log-likelihood (nll) for the observed patterns of mortality in infected and uninfected treatments when it assumed there is unobserved variation in virulence.

nll_frailty(
  a1 = a1,
  b1 = b1,
  a2 = a2,
  b2 = b2,
  theta = theta,
  data = data,
  time = time,
  censor = censor,
  infected_treatment = infected_treatment,
  d1 = "Weibull",
  d2 = "Weibull",
  d3 = ""
)

Arguments

• a1, b1: location and scale parameters describing background mortality
• a2, b2: location and scale parameters describing mortality due to infection
• theta: parameter describing variance of unobserved variation in virulence
• data: name of data frame containing survival data
• time: name of data frame column identifying time of event; time > 0
• censor: name of data frame column idenifying if event was death (0) or right-censoring (1)
• infected_treatment: name of data frame column identifying if data are from an infected (1) or uninfected (0) treatment
• d1, d2: names of probability distributions chosen to describe background mortality and mortality due to infection, respectively; both default to the Weibull distribution
• d3: name of probability distribution chosen to describe unobserved frailty; choice of 'gamma' or 'inverse Gaussian'

Returns

numeric

Details

The function assumes the unobserved variation in the rate of mortality due to infection is continuously distributed and follows either the gamma distribution or the inverse Gaussian distribution, with mean = 1 and variance = theta.

The nll is based on five parameter functions for the location and scale parameters for background mortality and mortality due to infection, respectively, plus a parameter estimating the variance of the unobserved variation in virulence.

Examples

### Example 1: unobserved variation in virulence described by gamma distribution

# step #1: parameterise nll function to be passed to 'mle2'
    m01_prep_function <- function(a1 = a1, b1 = b1, a2 = a2, b2 = b2, theta = theta){
      nll_frailty(
        a1 = a1, b1 = b1, a2 = a2, b2 = b2, theta = theta,
        data = data_lorenz,
        time = t,
        censor = censored,
        infected_treatment = g,
        d1 = "Gumbel",
        d2 = "Weibull",
        d3 = "Gamma"
        )}

# step #2: send 'prep_function' to 'mle2' for maximum likelihood estimation
    m01 <- mle2(
      m01_prep_function,
      start = list(a1 = 20, b1 = 5, a2 = 3, b2 = 0.1, theta = 2)
      )

    summary(m01)

### Example 2: unobserved variation in virulence described by inverse Gaussian distribution

    m02_prep_function <- function(a1 = a1, b1 = b1, a2 = a2, b2 = b2, theta = theta){
      nll_frailty(
        a1 = a1, b1 = b1, a2 = a2, b2 = b2, theta = theta,
        data = data_lorenz,
        time = t,
        censor = censored,
        infected_treatment = g,
        d1 = "Gumbel",
        d2 = "Weibull",
        d3 = "Inverse Gaussian"
        )}

    m02 <- mle2(
      m02_prep_function,
        start = list(a1 = 20, b1 = 5, a2 = 3, b2 = 0.1, theta = 2)
        )

    summary(m02)

# compare model estimates by AICc
    AICc(m01, m02, nobs = 256)