# Unified Equation Representation

    Sys.setenv(OMP_THREAD_LIMIT = 1) # Reducing core use, to avoid accidental use of too many cores
    library(Colossus)
    library(data.table)

The underlying functions in Colossus specify a survival model using a series of vectors, which can be overly complicated at times. Colossus includes functions to interpret formulas, and then call the underlying functions.

## Equation Expression

Generally, a survival model is defined by using a Surv object, initialized with the interval and event columns, and a list of columns. Because Colossus can handle much more complicated models, the definition has to be more complicated. Instead of a Surv object, the model is specified with ‘cox’, ‘poisson’, or ‘finegray’, each also including a strata option. The right side of the equation is listed similarly. Columns are added to subterm types (loglinear, linear, plinear, etc.). Each subterm has a list of columns and a term number. Finally, the method to combine terms is added as a separate item (multiplicative, additive, etc.).

\[ \begin{aligned} \text{Surv(interval, event) } \sim \text{name + ... + factor(name)} \\ \text{'Survival(interval, event) } \sim \text{ subterm(name, factor(name), term_number) + ... + term_model()'} \end{aligned} \]

Based on the number of entries, the format of the time interval is assumed. Cox models use an entry time, an exit time, and an event status. These can be either named ‘start’, ‘tend’, and ‘event’ or included in order. If none are named, a left truncated interval is assumed. A stratified cox model includes the named entry ‘strata’ or the right-most entry is assumed to be the stratification column. A Fine-Gray model adds a weighting column to the standard Cox interval representation, assumed to be an entry named ‘weight’ or the right-most entry. In total, the assumed order is: entry time, exit time, event status, strata column, and weighting column.

\[ \begin{aligned} \text{'Cox(tstart, tend, event) } \sim \text{ ...'} \\ \text{'Cox_Strata(tstart, tend, event, strata) } \sim \text{ ...'} \\ \text{'FineGray(tstart, tend, event, weight) } \sim \text{ ...'} \\ \text{'FineGray_Strata(tstart, tend, event, strata, weight) } \sim \text{ ...'} \end{aligned} \]

Poisson models use a measure of duration (named ‘pyr’), the number of events (named ‘event’), and any strata columns. The assumed order is ‘pyr’, ‘event’, and the strata columns listed sequentially. The duration and event columns can be named, but the naming of the strata columns is currently not supported.

\[ \begin{aligned} \text{'Pois(pyr, event) } \sim \text{ ...'} \\ \text{'Pois_Strata(pyr, event, strata_0) } \sim \text{ ...'} \\ \text{'Pois_Strata(pyr, event, strata_0, strata_1, ...) } \sim \text{ ...'} \end{aligned} \]

The right hand side adds elements to subterms. Each subterm should have the included columns, and optionally the term number. Subterms are assumed to be in term 0 if a term number is not provided, and the model is assumed to be multiplicative excess. The factor option automatically removes the first level as a reference. Note that unused levels can be listed first to use all levels.

The following expressions are equivalent.

    term_n <- c(0, 0, 1)
    tform <- c("loglin", "loglin", "lin")
    names <- c("dose0", "dose1", "dose2")
    modelform <- "M"
    tstart <- "t0"
    tend <- "t1"
    event <- "lung"

    Model_Eq <- Cox(t0, t1, lung) ~ loglinear(dose0, dose1, 0) +
      linear(dose2, 1) + ME()
    Model_Eq <- Cox(t0, t1, lung) ~ loglinear(dose0, dose1) +
      linear(dose2, 1)

    df <- data.table(
      dose0 = 1:4, dose1 = 2:5, dose2 = 3:6,
      t0 = c(0, 0, 1, 0), t1 = 5:8, lung = c(1, 0, 0, 1)
    )
    res <- get_form(Model_Eq, df, nthreads = 1)
    formula <- res$model
    new_data <- res$data

The following tables cover the subterms and term models that are currently implemented. Every option listed has multiple equivalent aliases that all refer to the same subterm or model formula.

Survival Type | Equivalent Aliases
--- | ---
Cox | “cox”, “coxph”
Cox stratified | “cox_strata”, “coxph_strata”
Poisson | “poisson”, “pois”
Poisson stratified | “poisson_strata”, “pois_strata”
Fine-Gray | “finegray”, “fg”
Fine-Gray stratified | “finegray_strata”, “fg_strata”
Logistic | “logit”, “logistic”
Matched Case-Control with one group | “casecon”, “casecontrol”, “case_control”
Matched Case-Control grouped by time at risk | “casecon_time”, “casecontrol_time”, “case_control_time”
Matched Case-Control grouped by strata | “casecon_strata”, “casecontrol_strata”, “case_control_strata”
Matched Case-Control grouped by time at risk and strata | “casecon_strata_time”, “casecontrol_strata_time”, “case_control_strata_time”, “casecon_time_strata”, “casecontrol_time_strata”, “case_control_time_strata”

Subterm Type | Equivalent Aliases
--- | ---
plin | “plin”, “plinear”, “product-linear”
lin | “lin”, “linear”
loglin | “loglin”, “loglinear”, “log-linear” , “exponential”, “exp”
loglin_slope/loglin_top | “loglin-dose”, “loglinear-dose”, “log-linear-dose”
lin_slope/lin_top | “lin-dose”, “linear-dose”, “linear-piecewise”
quad_slope | “quadratic”, “quad”, “quad-dose”, “quadratic-dose”
step_slope/step_int | “step-dose”, “step-piecewise”
lin_quad_slope/lin_quad_int | “lin-quad-dose”, “linear-quadratic-dose”, “linear-quadratic-piecewise”
lin_exp_slope/lin_exp_int/lin_exp_exp_slope | “lin-exp-dose”, “linear-exponential-dose”, “linear-exponential-piecewise”

Term Type | Equivalent Aliases
--- | ---
Multiplicative | “m”, “multiplicative”
Multiplicative Excess | “me”, “multiplicative-excess”
Additive | “a”, “additive”
Product Additive | “pa”, “product-additive”
Product Additive Excess | “pae”, “product-additive-excess”
Geometric Mixture Model | “gmix”,“geometric-mixture”
Geometric Mixture Model, relative error terms | “gmix-r”,“relative-geometric-mixture”
Geometric Mixture Model, excess error terms | “gmix-e”,“excess-geometric-mixture”