make_knots function

Knot locations for M-spline baseline hazard models

Several different algorithms are provided to calculate knot locations for M-spline baseline hazard models. This function is called internally within the nma() function, but may be called directly by the user for more control.

make_knots(
  network,
  n_knots = 7,
  type = c("quantile", "quantile_common", "quantile_lumped", "quantile_longest", "equal",
    "equal_common")
)

Arguments

• network: A network object, containing survival outcomes
• n_knots: Non-negative integer giving the number of internal knots (default 7)
• type: String specifying the knot location algorithm to use (see details). The default used by nma() is "quantile", except when a regression model is specified (using aux_regression) in which case the default is "quantile_common".

Returns

A named list of vectors giving the knot locations in each study.

Details

The type argument can be used to choose between different algorithms for placing the knots:

• "quantile": Creates separate knot locations for each study, internal knots are placed at evenly-spaced quantiles of the observed event times within each study.
• "quantile_lumped": Creates a common set of knots for all studies, calculated as evenly-spaced quantiles of the observed event times from all studies lumped together.
• "quantile_common": Creates a common set of knots for all studies, taking quantiles of the quantiles of the observed event times within each study. This often seems to result in a more even knot spacing than "quantile_lumped", particularly when follow-up is uneven across studies, and may handle differing behaviour in the baseline hazard across studies better than "quantile_longest".
• "quantile_longest": Creates a common set of knots for all studies, using evenly-spaced quantiles of the observed event times in the longest study.
• "equal": Creates separate knot locations for each study, at evenly-spaced times between the boundary knots in each study.
• "equal_common": Creates a common set of knots for all studies, at evenly-spaced times between the earliest entry time and last event/censoring time in the network.

Boundary knots are calculated as follows:

• For separate knot locations in each study, boundary knots are placed at the earliest entry time and last event/censoring time in each study.
• For a common set of knots across all studies, boundary knots are placed at the earliest entry time and last event/censoring time across all studies.

Models with regression on the spline coefficients (i.e. with aux_regression

specified) require a common set of knots across all studies.

Provided that a sufficient number of knots are used, model fit should be largely unaffected by the knot locations. However, sampling difficulties can sometimes occur if knot placement is poor, for example if a knot is placed just before the last follow-up time in a study.

Examples

# Set up newly-diagnosed multiple myeloma network

head(ndmm_ipd)
head(ndmm_agd)

ndmm_net <- combine_network(
  set_ipd(ndmm_ipd,
          study, trt,
          Surv = Surv(eventtime / 12, status)),
  set_agd_surv(ndmm_agd,
               study, trt,
               Surv = Surv(eventtime / 12, status),
               covariates = ndmm_agd_covs))

# The default knot locations
make_knots(ndmm_net, type = "quantile")

# Increasing the number of knots
make_knots(ndmm_net, n_knots = 10)

# Comparing alternative knot positioning algorithms
# Visualise these with a quick function
plot_knots <- function(network, knots) {
  ggplot2::ggplot() +
    geom_km(network) +
    ggplot2::geom_vline(ggplot2::aes(xintercept = .data$knot),
                        data = tidyr::pivot_longer(as.data.frame(knots), cols = dplyr::everything(),
                                                   names_to = "Study", values_to = "knot"),
                        linetype = 2, colour = "grey60") +
    ggplot2::facet_wrap(~Study) +
    theme_multinma()
}

plot_knots(ndmm_net, make_knots(ndmm_net, type = "quantile"))
plot_knots(ndmm_net, make_knots(ndmm_net, type = "quantile_common"))
plot_knots(ndmm_net, make_knots(ndmm_net, type = "quantile_lumped"))
plot_knots(ndmm_net, make_knots(ndmm_net, type = "quantile_longest"))
plot_knots(ndmm_net, make_knots(ndmm_net, type = "equal"))
plot_knots(ndmm_net, make_knots(ndmm_net, type = "equal_common"))