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(v2.1.1.9133) update math formulae

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dr. M.S. (Matthijs) Berends
2025-01-27 22:43:35 +01:00
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man/antibiogram.Rd
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@ -232,48 +232,26 @@ You can also use functions from specific 'table reporting' packages to transform
}
\section{Why Use WISCA?}{
WISCA, as outlined by Barbieri \emph{et al.} (\doi{10.1186/s13756-021-00939-2}), stands for
Weighted-Incidence Syndromic Combination Antibiogram, which estimates the probability
of adequate empirical antimicrobial regimen coverage for specific infection syndromes.
This method leverages a Bayesian hierarchical logistic regression framework with random
effects for pathogens and regimens, enabling robust estimates in the presence of sparse
data.
The Bayesian model assumes conjugate priors for parameter estimation. For example, the
coverage probability \ifelse{latex}{\deqn{$theta$}}{$theta$} for a given antimicrobial regimen
is modeled using a Beta distribution as a prior:
WISCA, as outlined by Barbieri \emph{et al.} (\doi{10.1186/s13756-021-00939-2}), stands for Weighted-Incidence Syndromic Combination Antibiogram, which estimates the probability of adequate empirical antimicrobial regimen coverage for specific infection syndromes. This method leverages a Bayesian hierarchical logistic regression framework with random effects for pathogens and regimens, enabling robust estimates in the presence of sparse data.
\ifelse{latex}{\deqn{$theta$ \sim \text{Beta}($alpha$_0, $beta$_0)}}{
\ifelse{html}{\figure{beta_prior.png}{options: width="300" alt="Beta prior"}}{$theta$ ~ Beta($alpha$_0, $beta$_0)}}
The Bayesian model assumes conjugate priors for parameter estimation. For example, the coverage probability \eqn{\theta} for a given antimicrobial regimen is modelled using a Beta distribution as a prior:
where \eqn{$alpha$_0} and \eqn{$beta$_0} represent prior successes and failures, respectively,
informed by expert knowledge or weakly informative priors (e.g., \eqn{$alpha$_0 = 1, $beta$_0 = 1}).
\deqn{\theta \sim \text{Beta}(\alpha_0, \beta_0)}
The likelihood function is constructed based on observed data, where the number of covered
cases for a regimen follows a binomial distribution:
where \eqn{\alpha_0} and \eqn{\beta_0} represent prior successes and failures, respectively, informed by expert knowledge or weakly informative priors (e.g., \eqn{\alpha_0 = 1, \beta_0 = 1}). The likelihood function is constructed based on observed data, where the number of covered cases for a regimen follows a binomial distribution:
\ifelse{latex}{\deqn{y \sim \text{Binomial}(n, $theta$)}}{
\ifelse{html}{\figure{binomial_likelihood.png}{options: width="300" alt="Binomial likelihood"}}{y ~ Binomial(n, $theta$)}}
\deqn{y \sim \text{Binomial}(n, \theta)}
Posterior parameter estimates are obtained by combining the prior and likelihood using
Bayes' theorem. The posterior distribution of \eqn{$theta$} is also a Beta distribution:
Posterior parameter estimates are obtained by combining the prior and likelihood using Bayes' theorem. The posterior distribution of \eqn{\theta} is also a Beta distribution:
\ifelse{latex}{\deqn{$theta$ | y \sim \text{Beta}($alpha$_0 + y, $beta$_0 + n - y)}}{
\ifelse{html}{\figure{posterior_beta.png}{options: width="300" alt="Beta posterior"}}{$theta$ | y ~ Beta($alpha$_0 + y, $beta$_0 + n - y)}}
\deqn{\theta | y \sim \text{Beta}(\alpha_0 + y, \beta_0 + n - y)}
For hierarchical modeling, pathogen-level effects (e.g., differences in resistance
patterns) and regimen-level effects are modelled using Gaussian priors on log-odds.
This hierarchical structure ensures partial pooling of estimates across groups,
improving stability in strata with small sample sizes. The model is implemented using
Hamiltonian Monte Carlo (HMC) sampling.
For hierarchical modelling, pathogen-level effects (e.g., differences in resistance patterns) and regimen-level effects are modelled using Gaussian priors on log-odds. This hierarchical structure ensures partial pooling of estimates across groups, improving stability in strata with small sample sizes. The model is implemented using Hamiltonian Monte Carlo (HMC) sampling.
Stratified results are provided based on covariates such as age, sex, and clinical
complexity (e.g., prior antimicrobial treatments or renal/urological comorbidities).
For example, posterior odds ratios (ORs) are derived to quantify the effect of these
covariates on coverage probabilities:
Stratified results can be provided based on covariates such as age, sex, and clinical complexity (e.g., prior antimicrobial treatments or renal/urological comorbidities) using \code{dplyr}'s \code{\link[=group_by]{group_by()}} as a pre-processing step before running \code{\link[=wisca]{wisca()}}. In this case, posterior odds ratios (ORs) are derived to quantify the effect of these covariates on coverage probabilities:
\ifelse{latex}{\deqn{\text{OR}_{\text{covariate}} = \frac{\exp($beta$_{\text{covariate}})}{\exp($beta$_0)}}}{
\ifelse{html}{\figure{odds_ratio.png}{options: width="300" alt="Odds ratio formula"}}{OR_covariate = exp(beta_covariate) / exp(beta_0)}}
\deqn{\text{OR}_{\text{covariate}} = \frac{\exp(\beta_{\text{covariate}})}{\exp(\beta_0)}}
By combining empirical data with prior knowledge, WISCA overcomes the limitations
of traditional combination antibiograms, offering disease-specific, patient-stratified
@ -360,7 +338,8 @@ antibiogram(example_isolates,
ureido <- antibiogram(example_isolates,
antibiotics = ureidopenicillins(),
ab_transform = "name"
ab_transform = "name",
wisca = TRUE
)
# in an Rmd file, you would just need to return `ureido` in a chunk,
@ -394,3 +373,6 @@ plot(ab1)
plot(ab2)
}
}
\author{
Implementation: Dr. Larisse Bolton and Dr. Matthijs Berends
}