Last updated: 2019-11-10
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Knit directory: HHVtransmission/
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Set up model data. Only pre-processing is removing left censored family starting at week 8.
library(tidyverse)── Attaching packages ─────────────────────────────────────────────────────────────────── tidyverse 1.2.1 ──✔ ggplot2 3.2.1 ✔ purrr 0.3.3
✔ tibble 2.1.3 ✔ dplyr 0.8.3
✔ tidyr 1.0.0 ✔ stringr 1.4.0
✔ readr 1.3.1 ✔ forcats 0.4.0── Conflicts ────────────────────────────────────────────────────────────────────── tidyverse_conflicts() ──
✖ dplyr::filter() masks stats::filter()
✖ dplyr::lag() masks stats::lag()library(conflicted)exposure_data = read_csv("data/exposure_data.csv") Parsed with column specification:
cols(
FamilyID = col_character(),
virus = col_character(),
infant_wks = col_double(),
infectious_1wk = col_double(),
final_infant_wk = col_double(),
infected = col_double(),
momhiv = col_character(),
final_exposure = col_double(),
interpolate_idpar = col_character(),
M = col_double(),
S = col_double(),
HH = col_double(),
obs_infected = col_double(),
final_wk = col_double(),
outcome_time = col_double(),
enrollment_age = col_double()
)exposure_data_long = read_csv("data/exposure_data_long.csv") %>%
mutate(exposure = if_else(count == 1, 0, 10^(count)))Parsed with column specification:
cols(
FamilyID = col_character(),
virus = col_character(),
infant_wks = col_double(),
infectious_1wk = col_double(),
final_infant_wk = col_double(),
infected = col_double(),
momhiv = col_character(),
final_exposure = col_double(),
interpolate_idpar = col_character(),
obs_infected = col_double(),
final_wk = col_double(),
outcome_time = col_double(),
enrollment_age = col_double(),
idpar = col_character(),
count = col_double(),
interpolated = col_logical()
)exposure_data_long %>%
subset(idpar != "HH") %>%
group_by(virus, FamilyID, idpar) %>%
mutate(any_interp = any(interpolated)) %>%
subset(any_interp) %>%
summarize(
first_obs = min(which(is.na(interpolate_idpar)))
) %>%
subset(first_obs > 2)# A tibble: 2 x 4
# Groups: virus, FamilyID [2]
virus FamilyID idpar first_obs
<chr> <chr> <chr> <int>
1 CMV AD S 8
2 HHV-6 AD S 8model_data = exposure_data %>% mutate(cohort = "All") %>% subset(FamilyID != "AD")
model_data_long = exposure_data_long %>% mutate(cohort = "All") %>% subset(FamilyID != "AD")
save(model_data, model_data_long, file = "output/model_data.RData")The probability of being uninfected after a single, weekly exposure can written as following:
\[ s(i) = exp(-\beta_0 - \beta_{E}E_i) \] or \[ s(i) = exp(-\lambda(i)) \]
Instead of continuous time, we consider discretized time denoted by \(i \in \{1, ..., n+1\}\) for n+1 weeks of exposures (with n survived exposures). The likelihood follows for a single, infected participant (note that in discrete time we don’t apply an instantaneous hazard but use 1 - S(n) for the infectious week):
\[ L_j(n_j+1) = \prod_{i = 1}^{n_j} s_j(i) * (1-s_j(n_j+1))\]
for the j\(^{th}\) participant with a unique set of total exposures. We next setup the log-likelihood for the population (m participants) and use \(\Delta\) to denote an observed infection.
\[ \sum_{j = 1}^{m} log L_j(n_j) = \sum_{j = 1}^{m}\sum_{i = 1}^{n_j} log(s_j(i)) + \sum_{j = 1}^{m} \Delta_j * log(1-s_j(n_j+1))\]
The following assumptions are used: at-risk individuals are independent and the risk associated with a weekly exposures is unique from other exposures (i.e., non-infectious exposure weeks are exchangeable). From this formulation, we can find the maximum likelihood estimators for the parameters by minimizing the negative log likelihood. Both parameters are solved numerically.
The null model has the following form for weekly risk:
\[ s_0(i) = exp(-\beta_0) \]
I.e., risk is constant and not affected by household exposure. The null model has a simplified log-likelihood
\[ \sum_{j = 1}^{m} log L_j(n_j) = -\beta_0 \sum_{j = 1}^{m}n_j + log(1-exp(-\beta_0)) \sum_{j = 1}^{m} \Delta_j \]
with closed-form solution for the null risk
\[\hat{\beta_0} = -log(1 - \frac{\sum_{j = 1}^{m} \Delta_j}{\sum_{j = 1}^{m}n_j})\]
sessionInfo()R version 3.6.1 (2019-07-05)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS Catalina 10.15.1
Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/3.6/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.6/Resources/lib/libRlapack.dylib
locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] conflicted_1.0.4 forcats_0.4.0 stringr_1.4.0 dplyr_0.8.3
[5] purrr_0.3.3 readr_1.3.1 tidyr_1.0.0 tibble_2.1.3
[9] ggplot2_3.2.1 tidyverse_1.2.1
loaded via a namespace (and not attached):
[1] tidyselect_0.2.5 xfun_0.10 haven_2.1.1 lattice_0.20-38
[5] colorspace_1.4-1 vctrs_0.2.0 generics_0.0.2 htmltools_0.4.0
[9] yaml_2.2.0 utf8_1.1.4 rlang_0.4.1 pillar_1.4.2
[13] glue_1.3.1 withr_2.1.2 modelr_0.1.5 readxl_1.3.1
[17] lifecycle_0.1.0 munsell_0.5.0 gtable_0.3.0 workflowr_1.4.0
[21] cellranger_1.1.0 rvest_0.3.4 evaluate_0.14 memoise_1.1.0
[25] knitr_1.25 fansi_0.4.0 broom_0.5.2 Rcpp_1.0.2
[29] scales_1.0.0 backports_1.1.5 jsonlite_1.6 fs_1.3.1
[33] hms_0.5.1 digest_0.6.22 stringi_1.4.3 grid_3.6.1
[37] rprojroot_1.3-2 cli_1.1.0 tools_3.6.1 magrittr_1.5
[41] lazyeval_0.2.2 crayon_1.3.4 pkgconfig_2.0.3 zeallot_0.1.0
[45] xml2_1.2.2 lubridate_1.7.4 assertthat_0.2.1 rmarkdown_1.16
[49] httr_1.4.1 rstudioapi_0.10 R6_2.4.0 nlme_3.1-140
[53] git2r_0.26.1 compiler_3.6.1