# Example Complex Analysis Function: Modelling Cox Regression

#### Emily de la Rua and Gabriel Becker

#### 2025-01-09

## Introduction

In this vignette we will demonstrate how a complex analysis function can be constructed in order to build highly-customized tables with `rtables`. This example will detail the steps in creating an analysis function to calculate a basic univariable Cox regression summary table to analyze the treatment effect of the `ARM` variable and any covariate/interaction effects for a survival analysis. For a Cox regression analysis function with more customization options and the capability of fitting multivariable Cox regression models, see the `summarize_coxreg()` function from the `tern` package, which builds upon the concepts used in the construction of this example.

The packages used in this vignette are:

    library(rtables)
    library(dplyr)

## Data Pre-Processing

First, we prepare the data that will be used to generate a table in this example. We will use the example `ADTTE` (Time-To-Event Analysis) dataset `ex_adtte` from the `formatters` package, which contains our treatment variable `ARM`, several variables that can be chosen as covariates, and censor variable `CNSR` from which we will derive the event variable `EVENT` required for our model. For the purpose of this example, we will use age (`AGE`) and race (`RACE`) as our covariates.

We prepare the data as needed to observe the desired effects in our summary table. `PARAMCD` is filtered so that only records of overall survival (OS) are included, and we filter and mutate to include only the levels of interest in our covariates. The `ARM` variable is mutated to indicate that `"B: Placebo"` should be used as the reference level of our treatment variable, and the `EVENT` variable is derived from `CNSR`.

    adtte <- ex_adtte

    anl <- adtte %>%
      dplyr::filter(PARAMCD == "OS") %>%
      dplyr::filter(ARM %in% c("A: Drug X", "B: Placebo")) %>%
      dplyr::filter(RACE %in% c("ASIAN", "BLACK OR AFRICAN AMERICAN", "WHITE")) %>%
      dplyr::mutate(RACE = droplevels(RACE)) %>%
      dplyr::mutate(ARM = droplevels(stats::relevel(ARM, "B: Placebo"))) %>%
      dplyr::mutate(EVENT = 1 - CNSR)

## Creating Helper Functions: Cox Regression Model Calculations

### `tidy` Method for `summary.coxph` Objects: `tidy.summary.coxph`

This method allows the `tidy` function from the `broom` package to operate on `summary.coxph` output, extracting the values of interest to this analysis and returning a tidied `tibble::tibble()` object.

    tidy.summary.coxph <- function(x, ...) {
      is(x, "summary.coxph")
      pval <- x$coefficients
      confint <- x$conf.int
      levels <- rownames(pval)
      pval <- tibble::as_tibble(pval)
      confint <- tibble::as_tibble(confint)

      ret <- cbind(pval[, grepl("Pr", names(pval))], confint)
      ret$level <- levels
      ret$n <- x[["n"]]
      ret
    }

### Function to Estimate Interaction Effects: `h_coxreg_inter_effect`

The `h_coxreg_inter_effect` helper function is used within the following helper function, `h_coxreg_extract_interaction`, to estimate interaction effects from a given model for a given covariate. The function calculates the desired statistics from the given model and returns a `data.frame` with label information for each row as well as the statistics `n`, `hr` (hazard ratio), `lcl` (CI lower bound), `ucl` (CI upper bound), `pval` (effect p-value), and `pval_inter` (interaction p-value). If a numeric covariate is selected, the median value is used as the sole “level” for which an interaction effect is calculated. For non-numeric covariates, an interaction effect is calculated for each level of the covariate, with each result returned on a separate row.

    h_coxreg_inter_effect <- function(x,
                                      effect,
                                      covar,
                                      mod,
                                      label,
                                      control,
                                      data) {
      if (is.numeric(x)) {
        betas <- stats::coef(mod)
        attrs <- attr(stats::terms(mod), "term.labels")
        term_indices <- grep(pattern = effect, x = attrs[!grepl("strata\\(", attrs)])
        betas <- betas[term_indices]
        betas_var <- diag(stats::vcov(mod))[term_indices]
        betas_cov <- stats::vcov(mod)[term_indices[1], term_indices[2]]
        xval <- stats::median(x)
        effect_index <- !grepl(covar, names(betas))
        coef_hat <- betas[effect_index] + xval * betas[!effect_index]
        coef_se <- sqrt(betas_var[effect_index] + xval^2 * betas_var[!effect_index] + 2 * xval * betas_cov)
        q_norm <- stats::qnorm((1 + control$conf_level) / 2)
      } else {
        var_lvl <- paste0(effect, levels(data[[effect]])[-1]) # [-1]: reference level
        giv_lvl <- paste0(covar, levels(data[[covar]]))
        design_mat <- expand.grid(effect = var_lvl, covar = giv_lvl)
        design_mat <- design_mat[order(design_mat$effect, design_mat$covar), ]
        design_mat <- within(data = design_mat, expr = {
          inter <- paste0(effect, ":", covar)
          rev_inter <- paste0(covar, ":", effect)
        })
        split_by_variable <- design_mat$effect
        interaction_names <- paste(design_mat$effect, design_mat$covar, sep = "/")
        mmat <- stats::model.matrix(mod)[1, ]
        mmat[!mmat == 0] <- 0
        design_mat <- apply(X = design_mat, MARGIN = 1, FUN = function(x) {
          mmat[names(mmat) %in% x[-which(names(x) == "covar")]] <- 1
          mmat
        })
        colnames(design_mat) <- interaction_names
        coef <- stats::coef(mod)
        vcov <- stats::vcov(mod)
        betas <- as.matrix(coef)
        coef_hat <- t(design_mat) %*% betas
        dimnames(coef_hat)[2] <- "coef"
        coef_se <- apply(design_mat, 2, function(x) {
          vcov_el <- as.logical(x)
          y <- vcov[vcov_el, vcov_el]
          y <- sum(y)
          y <- sqrt(y)
          y
        })
        q_norm <- stats::qnorm((1 + control$conf_level) / 2)
        y <- cbind(coef_hat, `se(coef)` = coef_se)
        y <- apply(y, 1, function(x) {
          x["hr"] <- exp(x["coef"])
          x["lcl"] <- exp(x["coef"] - q_norm * x["se(coef)"])
          x["ucl"] <- exp(x["coef"] + q_norm * x["se(coef)"])
          x
        })
        y <- t(y)
        y <- by(y, split_by_variable, identity)
        y <- lapply(y, as.matrix)
        attr(y, "details") <- paste0(
          "Estimations of ", effect, " hazard ratio given the level of ", covar, " compared to ",
          effect, " level ", levels(data[[effect]])[1], "."
        )
        xval <- levels(data[[covar]])
      }
      data.frame(
        effect = "Covariate:",
        term = rep(covar, length(xval)),
        term_label = as.character(paste0("  ", xval)),
        level = as.character(xval),
        n = NA,
        hr = if (is.numeric(x)) exp(coef_hat) else y[[1]][, "hr"],
        lcl = if (is.numeric(x)) exp(coef_hat - q_norm * coef_se) else y[[1]][, "lcl"],
        ucl = if (is.numeric(x)) exp(coef_hat + q_norm * coef_se) else y[[1]][, "ucl"],
        pval = NA,
        pval_inter = NA,
        stringsAsFactors = FALSE
      )
    }

### Function to Extract Effect Information: `h_coxreg_extract_interaction`

Using the previous two helper functions, `h_coxreg_extract_interaction` uses ANOVA to extract information from the given model about the given covariate. This function will extract different information depending on whether the effect of interest is a treatment/main effect or an interaction effect, and returns a `data.frame` with label information for each row (corresponding to each effect) as well as the statistics `n`, `hr`, `lcl`, `ucl`, `pval`, and `pval_inter` (for interaction effects only). This helper function is used directly within our analysis function to analyze the Cox regression model and extract relevant information to be processed and displayed within our output table.

    h_coxreg_extract_interaction <- function(effect, covar, mod, data) {
      control <- list(pval_method = "wald", ties = "exact", conf_level = 0.95, interaction = FALSE)
      test_statistic <- c(wald = "Wald", likelihood = "LR")[control$pval_method]
      mod_aov <- withCallingHandlers(
        expr = car::Anova(mod, test.statistic = test_statistic, type = "III"),
        message = function(m) invokeRestart("muffleMessage")
      )
      msum <- if (!any(attr(stats::terms(mod), "order") == 2)) summary(mod, conf.int = control$conf_level) else mod_aov
      sum_anova <- broom::tidy(msum)
      if (!any(attr(stats::terms(mod), "order") == 2)) {
        effect_aov <- mod_aov[effect, , drop = TRUE]
        pval <- effect_aov[[grep(pattern = "Pr", x = names(effect_aov)), drop = TRUE]]
        sum_main <- sum_anova[grepl(effect, sum_anova$level), ]
        term_label <- if (effect == covar) {
          paste0(levels(data[[covar]])[2], " vs control (", levels(data[[covar]])[1], ")")
        } else {
          unname(formatters::var_labels(data, fill = TRUE)[[covar]])
        }
        y <- data.frame(
          effect = ifelse(covar == effect, "Treatment:", "Covariate:"),
          term = covar, term_label = term_label,
          level = levels(data[[effect]])[2],
          n = mod[["n"]], hr = unname(sum_main["exp(coef)"]), lcl = unname(sum_main[grep("lower", names(sum_main))]),
          ucl = unname(sum_main[grep("upper", names(sum_main))]), pval = pval,
          stringsAsFactors = FALSE
        )
        y$pval_inter <- NA
        y
      } else {
        pval <- sum_anova[sum_anova$term == effect, ][["p.value"]]

        ## Test the interaction effect
        pval_inter <- sum_anova[grep(":", sum_anova$term), ][["p.value"]]
        covar_test <- data.frame(
          effect = "Covariate:",
          term = covar, term_label = unname(formatters::var_labels(data, fill = TRUE)[[covar]]),
          level = "",
          n = mod$n, hr = NA, lcl = NA, ucl = NA, pval = pval,
          pval_inter = pval_inter,
          stringsAsFactors = FALSE
        )
        ## Estimate the interaction
        y <- h_coxreg_inter_effect(
          data[[covar]],
          covar = covar,
          effect = effect,
          mod = mod,
          label = unname(formatters::var_labels(data, fill = TRUE)[[covar]]),
          control = control,
          data = data
        )
        rbind(covar_test, y)
      }
    }

## Creating a Helper Function: `cached_model`

Next, we will create a helper function, `cached_model`, which will be used within our analysis function to cache and return the fitted Cox regression model for the current covariate. The `df` argument will be directly inherited from the `df` argument passed to the analysis function, which contains the full dataset being analyzed. The `cov` argument will be the covariate that is being analyzed depending on the current row context. If the treatment effect is currently being analyzed, this value will be an empty string. The `cache_env` parameter will be an environment object which is used to store the model for the current covariate, also passed down from the analysis function. Of course, this function can also be run outside of the analysis function and will still cache and return a Cox regression model.

Using these arguments, the `cached_model` function first checks if a model for the given covariate `cov` is already stored in the caching environment `cache_env`. If so, then this model is retrieved and returned by `cached_model`. If not, the model must be constructed. This is done by first constructing the model formula, `model_form`, starting with only the treatment effect (`ARM`) and adding a covariate effect if one is currently being analyzed. Then a Cox regression model is fit using `df` and the model formula, and this model is both returned and stored in the caching environment object as `cache_env[[cov]]`.

    cached_model <- function(df, cov, cache_env) {
      ## Check if a model already exists for
      ## `cov` in the caching environment
      if (!is.null(cache_env[[cov]])) {
        ## If model already exists, retrieve it from cache_env
        model <- cache_env[[cov]]
      } else {
        ## Build model formula
        model_form <- paste0("survival::Surv(AVAL, EVENT) ~ ARM")
        if (length(cov) > 0) {
          model_form <- paste(c(model_form, cov), collapse = " * ")
        } else {
          cov <- "ARM"
        }
        ## Calculate Cox regression model
        model <- survival::coxph(
          formula = stats::as.formula(model_form),
          data = df,
          ties = "exact"
        )
        ## Store model in the caching environment
        cache_env[[cov]] <- model
      }
      model
    }

## Creating the Analysis Function: `a_cox_summary`

With our data prepared and helper function created, we can proceed to construct our analysis function `a_cox_summary`, which will be used to populate all of the rows in our table. In order to be used to generate both data rows (for interaction effects) and content rows (for main effects), we must create a function that can be used as both `afun` in `analyze` and `cfun` in `summarize_row_groups`. Therefore, our function must accept the `labelstr` parameter.

The arguments of our analysis function will be as follows:

- `df` - a `data.frame` of the full dataset required to fit the Cox regression model.
- `labelstr` - the `string` label for the variable being analyzed in the current row/column split context.
- `.spl_context` - a `data.frame` containing the `value` column which is used by this analysis function to determine the name of the variable/covariate in the current split. For more details on the information stored by `.spl_context` see `?analyze`.
- `stat` and `format` - `string`s that indicate which statistic column we are currently in and what format should be applied to print the statistic.
- `cache_env` - an `environment` object that can be used to store cached models so that we can prevent repeatedly fitting the same model. Instead, each model will be generated once per covariate and then reused. This argument will be passed directly to the `cached_model` helper function we defined previously.
- `cov_main` - a `logical` value indicating whether or not the current row is summarizing covariate main effects.

The analysis function works within a given row/column split context by using the current covariate (`cov`) and the `cached_model` function to obtain the desired Cox regression model. From this model, the `h_coxreg_extract_interaction` function is able to extract information/statistics relevant to the analysis and store it in a `data.frame`. The rows in this `data.frame` that are of interest in the current row/column split context are then extracted and the statistic to be printed in the current column is retrieved from these rows. Finally, the formatted cells with this statistic are returned as a `VerticalRowsSection` object. For more detail see the commented function code below, where the purpose of each line within `a_cox_summary` is described.

    a_cox_summary <- function(df,
                              labelstr = "",
                              .spl_context,
                              stat,
                              format,
                              cache_env,
                              cov_main = FALSE) {
      ## Get current covariate (variable used in latest row split)
      cov <- tail(.spl_context$value, 1)

      ## If currently analyzing treatment effect (ARM) replace empty
      ## value of cov with "ARM" so the correct model row is analyzed
      if (length(cov) == 0) cov <- "ARM"

      ## Use cached_model to get the fitted Cox regression
      ## model for the current covariate
      model <- cached_model(df = df, cov = cov, cache_env = cache_env)

      ## Extract levels of cov to be used as row labels for interaction effects.
      ## If cov is numeric, the median value of cov is used as a row label instead
      cov_lvls <- if (is.factor(df[[cov]])) levels(df[[cov]]) else as.character(median(df[[cov]]))

      ## Use function to calculate and extract information relevant to cov from the model
      cov_rows <- h_coxreg_extract_interaction(effect = "ARM", covar = cov, mod = model, data = df)
      ## Effect p-value is only printed for treatment effect row
      if (!cov == "ARM") cov_rows[, "pval"] <- NA_real_
      ## Extract rows containing statistics for cov from model information
      if (!cov_main) {
        ## Extract rows for main effect
        cov_rows <- cov_rows[cov_rows$level %in% cov_lvls, ]
      } else {
        ## Extract all non-main effect rows
        cov_rows <- cov_rows[nchar(cov_rows$level) == 0, ]
      }
      ## Extract value(s) of statistic for current column and variable/levels
      stat_vals <- as.list(apply(cov_rows[stat], 1, function(x) x, simplify = FALSE))
      ## Assign labels: covariate name for main effect (content) rows, ARM comparison description
      ## for treatment effect (content) row, cov_lvls for interaction effect (data) rows
      nms <- if (cov_main) labelstr else if (cov == "ARM") cov_rows$term_label else cov_lvls
      ## Return formatted/labelled row
      in_rows(
        .list = stat_vals,
        .names = nms,
        .labels = nms,
        .formats = setNames(rep(format, length(nms)), nms),
        .format_na_strs = setNames(rep("", length(nms)), nms)
      )
    }

## Selecting Parameters

We are able to customize our Cox regression summary using this analysis function by selecting covariates (and their labels), statistics (and their labels), and statistic formats to use when generating the output table. We also initialize a new environment object to be used by the analysis function as the caching environment to store our models in. For the purpose of this example, we will choose all 5 of the possible statistics to include in the table: n, hazard ratio, confidence interval, effect p-value, and interaction p-value.

    my_covs <- c("AGE", "RACE") ## Covariates
    my_cov_labs <- c("Age", "Race") ## Covariate labels
    my_stats <- list("n", "hr", c("lcl", "ucl"), "pval", "pval_inter") ## Statistics
    my_stat_labs <- c("n", "Hazard Ratio", "95% CI", "p-value\n(effect)", "p-value\n(interaction)") ## Statistic labels
    my_formats <- c(
      n = "xx", hr = "xx.xx", lcl = "(xx.xx, xx.xx)", pval = "xx.xxxx", pval_inter = "xx.xxxx" ## Statistic formats
    )
    my_env <- new.env()
    ny_cache_env <- replicate(length(my_stats), list(my_env)) ## Caching environment

## Constructing the Table

Finally, the table layout can be constructed and used to build the desired table.

We first split our `basic_table` using `split_cols_by_multivar` to ensure that each statistic exists in its own column. To do so, we choose a variable (in this case `STUDYID`) which shares the same value in every row, and use it as the split variable for every column so that the full dataset is used to compute the model for every column. We use the `extra_args` argument for which each list element’s element positions correspond to the children of (columns generated by) this split. These arguments are inherited by all following layout elements operating within this split, which use these elements as argument inputs. To elaborate on this, we have three elements in `extra_args`: `stat`, `format`, and `cache_env` - each of which are arguments of `a_cox_summary` and have length equal to the number of columns (as defined above). For each use of our analysis function following this column split, depending on the current column context, the corresponding element of each of these three list elements will be inherited from `extra_args` and used as input. For example, if `analyze_colvars` is called with `a_cox_summary` as `afun` and is performing calculations for column 1, `my_stats[1]` (`"n"`) will be given as argument `stat`, `my_formats[1]` (`"xx"`) as argument `format`, and `my_cache_env[1]` (`my_env`) as `cache_env`. This is useful for our table since we want each column to print out values for a different statistic and apply its corresponding format.

Next, we can use `summarize_row_groups` to generate the content row for treatment effect. This is the first instance where `extra_args` from the column split will be inherited and used as argument input in `cfun`.

After generating the treatment effect row, we want to add rows for covariates. We use `split_rows_by_multivar` to split rows by covariate and apply appropriate labels.

Following this row split, we use `summarize_row_groups` with `a_cox_summary` as `cfun` to generate one content row for each covariate main effect. Once again the contents of `extra_args` from the column split are inherited as input. Here we specify `cov_main = TRUE` in the `extra_args` argument so that main effects rather than interactions are considered. Since this is not a split, this instance of `extra_args` is not inherited by any following layout elements. As `cov_main` is a singular value, `cov_main = TRUE` will be used within every column context.

The last part of our table is the covariate interaction effects. We use `analyze_colvars` with `a_cox_summary` as `afun`, and again inherit `extra_args` from the column split. Using an `rtables` “analyze” function generates data rows, with one row corresponding to each covariate level (or median value, for numeric covariates), nested under the content row (main effect) for that same covariate.

    lyt <- basic_table() %>%
      ## Column split: one column for each statistic
      split_cols_by_multivar(
        vars = rep("STUDYID", length(my_stats)),
        varlabels = my_stat_labs,
        extra_args = list(
          stat = my_stats,
          format = my_formats,
          cache_env = ny_cache_env
        )
      ) %>%
      ## Create content row for treatment effect
      summarize_row_groups(cfun = a_cox_summary) %>%
      ## Row split: one content row for each covariate
      split_rows_by_multivar(
        vars = my_covs,
        varlabels = my_cov_labs,
        split_label = "Covariate:",
        indent_mod = -1 ## Align split label left
      ) %>%
      ## Create content rows for covariate main effects
      summarize_row_groups(
        cfun = a_cox_summary,
        extra_args = list(cov_main = TRUE)
      ) %>%
      ## Create data rows for covariate interaction effects
      analyze_colvars(afun = a_cox_summary)

Using our pre-processed `anl` dataset, we can now build and output our final Cox regression summary table.

    cox_tbl <- build_table(lyt, anl)
    cox_tbl
    #>                                                                         p-value       p-value
    #>                                      n    Hazard Ratio      95% CI      (effect)   (interaction)
    #> ————————————————————————————————————————————————————————————————————————————————————————————————
    #> A: Drug X vs control (B: Placebo)   247       0.97       (0.71, 1.32)    0.8243
    #> Covariate:
    #>   Age                               247                                               0.7832
    #>     34                                        0.92       (0.68, 1.26)
    #>   Race                              247                                               0.7441
    #>     ASIAN                                     1.03       (0.68, 1.57)
    #>     BLACK OR AFRICAN AMERICAN                 0.78       (0.41, 1.49)
    #>     WHITE                                     1.06       (0.55, 2.04)