nll_basic function

Negative log-likelihood function: basic model

Function returning the negative log-likelihood (nll) for the 'basic' relative survival model, given the functions' parameters and the observed data.

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

Arguments

• a1, b1: location and scale parameters for background mortality
• a2, b2: location and scale parameters for mortality due to infection
• 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 identifying 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 describing background mortality and mortality due to infection, respectively; both default to the Weibull distribution

Details

By deafult, this function takes arguments for location and scale parameters named; a1, b1, a2, b2. These parameters are components of survival functions describing patterns of background mortality and mortality due to infection. The particular form of these survival functions depends on the probability distributions chosen to describe each source of mortality; d1, d2. The function also takes arguments directing it to the data to be analysed and how they are labelled; data, time, censor, etc.

The nll returned by the function depends on the numerical values of the location and scale parameters, which determine how well the likelihood model describes the observed data. Maximum likelihood estimation functions, e.g., mle2 of the package bbmle, can be used find values of the location and scale parameters minimising the model's nll. The resulting maximum likelihood estimates can be used to describe host mortality due to background mortality and mortality due to infection, including the pathogen's virulence.

The model assumes all the individuals in the infected population are infected. It is also assumes infections are homogeneous, i.e., each infection has the same influence on host survival. Consequently a single hazard function, with a single pair of values for its location and scale parameters, can be used to describe the pattern of mortality due to infection for the infected population as a whole.

Examples

# prepare subset of 'data_blanford'; treatments 'cont' and 'Bb06' of Block 3

  data01 <- subset(data_blanford,
    (data_blanford$block == 3) & (
      (data_blanford$treatment == 'cont') |
         (data_blanford$treatment == 'Bb06')) &
           (data_blanford$day > 0))

 head(data01, 4)

# step #1: 'prep function' linking 'nll_basic' to data
  # and identifying parameters to estimate
    m01_prep_function <- function(a1 = a1, b1 = b1, a2 = a2, b2 = b2){
      nll_basic(
        a1 = a1, b1 = b1, a2 = a2, b2 = b2,
        data = data01,
        time = t,
        censor = censor,
        infected_treatment = inf,
        d1 = 'Weibull', d2 = 'Weibull')
        }

# step #2: send 'prep_function' to 'mle2' for maximum likelihood estimation
  #  starting values specified as
    m01 <- mle2(m01_prep_function,
             start = list(a1 = 2, b1 = 0.5, a2 = 2.5, b2 = 0.25)
             )

    summary(m01)