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Marginal Structural Model (MSM) for Treatment Switching

Source: R/msm.R
msm.Rd

Excludes data after treatment switching when fitting the switching model to estimate the probabilities of not switching and then switching. The inverse of these probabilities (inverse probability of treatment weights) are then used as weights in a Cox model including data after switching to estimate the adjusted hazard ratio.

Usage

msm(
  data,
  id = "id",
  stratum = "",
  tstart = "tstart",
  tstop = "tstop",
  event = "event",
  treat = "treat",
  swtrt = "swtrt",
  swtrt_time = "swtrt_time",
  base_cov = "",
  numerator = "",
  denominator = "",
  strata_main_effect_only = TRUE,
  ns_df = 3,
  firth = FALSE,
  flic = FALSE,
  stabilized_weights = TRUE,
  trunc = 0,
  trunc_upper_only = TRUE,
  swtrt_control_only = TRUE,
  treat_alt_interaction = TRUE,
  alpha = 0.05,
  ties = "efron",
  boot = FALSE,
  n_boot = 1000,
  seed = 0,
  nthreads = 0
)

Arguments

data

The input data frame that contains the following variables:

  • id: The id to identify observations belonging to the same subject for counting process data with time-dependent covariates.

  • stratum: The stratum.

  • tstart: The starting time of the time interval for counting-process data with time-dependent covariates.

  • tstop: The stopping time of the time interval for counting-process data with time-dependent covariates.

  • event: The event indicator, 1=event, 0=no event.

  • treat: The randomized treatment indicator, 1=treatment, 0=control.

  • swtrt: The treatment switch indicator, 1=switch, 0=no switch.

  • swtrt_time: The time from randomization to treatment switch.

  • base_cov: The baseline covariates (excluding treat) used in the outcome model.

  • numerator: The baseline covariates (excluding treat) used in the numerator switching model for stabilized weights.

  • denominator: The baseline (excluding treat) and time-dependent covariates used in the denominator switching model.

id

The name of the id variable in the input data.

stratum

The name(s) of the stratum variable(s) in the input data.

tstart

The name of the tstart variable in the input data.

tstop

The name of the tstop variable in the input data.

event

The name of the event variable in the input data.

treat

The name of the treatment variable in the input data.

swtrt

The name of the swtrt variable in the input data.

swtrt_time

The name of the swtrt_time variable in the input data.

base_cov

The names of baseline covariates (excluding treat) in the input data for the Cox model.

numerator

The names of baseline covariates (excluding treat) in the input data for the numerator switching model for stabilized weights.

denominator

The names of baseline (excluding treat) and time-dependent covariates in the input data for the denominator switching model.

strata_main_effect_only

Whether to only include the strata main effects in the logistic regression switching model. Defaults to TRUE, otherwise all possible strata combinations will be considered in the switching model.

ns_df

Degrees of freedom for the natural cubic spline for visit-specific intercepts of the pooled logistic regression model. Defaults to 3 for two internal knots at the 33 and 67 percentiles of the treatment switching times.

firth

Whether the Firth's bias reducing penalized likelihood should be used.

flic

Whether to apply intercept correction to obtain more accurate predicted probabilities.

stabilized_weights

Whether to use the stabilized weights. The default is TRUE.

trunc

The truncation fraction of the weight distribution. Defaults to 0 for no truncation in weights.

trunc_upper_only

Whether to truncate the weights from the upper end of the weight distribution only. Defaults to TRUE, otherwise the weights will be truncated from both the lower and upper ends of the distribution.

swtrt_control_only

Whether treatment switching occurred only in the control group. The default is TRUE.

treat_alt_interaction

Whether to include an interaction between randomized and alternative treatments in the outcome model when both randomized arms can switch to alternative treatment.

alpha

The significance level to calculate confidence intervals.

ties

The method for handling ties in the Cox model, either "breslow" or "efron" (default).

boot

Whether to use bootstrap to obtain the confidence interval for hazard ratio. Defaults to FALSE.

n_boot

The number of bootstrap samples.

seed

The seed to reproduce the bootstrap results.

nthreads

The number of threads to use in bootstrapping (0 means the default RcppParallel behavior).

Value

A list with the following components:

  • pvalue: The two-sided p-value.

  • pvalue_type: The type of two-sided p-value for treatment effect, i.e., "Cox model" or "bootstrap".

  • hr: The estimated hazard ratio from the Cox model.

  • hr_CI: The confidence interval for hazard ratio.

  • hr_CI_type: The type of confidence interval for hazard ratio, either "Cox model" or "bootstrap".

  • event_summary: A data frame containing the count and percentage of deaths and switches by treatment arm.

  • data_switch: A list of input data for the switching models by treatment group. The variables include id, stratum, "tstart", "tstop", "cross", denominator, swtrt, and swtrt_time. In addition, stratum variables are converted to dummy variables, and natural cubic spline basis variables are created for the visit-specific intercepts.

  • fit_switch: A list of fitted switching models for the denominator and numerator by treatment group.

  • data_outcome: The input data for the outcome Cox model including the inverse probability of censoring weights. The variables include id, stratum, "tstart", "tstop", "event", "treated", "crossed", "unstablized_weight", "stabilized_weight", base_cov, and treat. If treat_alt_interaction is TRUE, the data set also includes the "treated_crossed" variable.

  • weight_summary: A data frame summarizing the weights by treatment arm.

  • km_outcome: The Kaplan-Meier estimates of the survival functions for the treatment and control groups based on the weighted outcome data truncated at time of treatment switching.

  • lr_outcome: The log-rank test results for the treatment effect based on the weighted outcome data truncated at time of treatment switching.

  • fit_outcome: The fitted outcome Cox model.

  • fail: Whether a model fails to converge.

  • settings: A list containing the input parameter values.

  • fail_boots: The indicators for failed bootstrap samples if boot is TRUE.

  • fail_boots_data: The data for failed bootstrap samples if boot is TRUE.

  • hr_boots: The bootstrap hazard ratio estimates if boot is TRUE.

Details

The hazard ratio and confidence interval under a no-switching scenario are obtained as follows:

  • Exclude observations after treatment switch when fitting the switching model.

  • Define crossover indicators for the last time interval of each subject.

  • Fit the denominator switching model (and numerator model for stabilized weights) using a pooled logistic regression model to estimate the inverse probability of treatment weights (IPTWs).

    • The probability of remaining unswitched is calculated as \(1 - \hat{p}_{\text{switch}}\) and multiplied over time before treatment switch.

    • At the time of switching, this product is multiplied by the predicted probability of switching.

    • After treatment switch, the IPTW remains constant.

    • The inverse of the probability at the start of each interval is used as the interval weight.

  • Fit a weighted Cox model to the outcome survival times, including data after treatment switch, to estimate the hazard ratio.

  • Construct the p-value and confidence interval for the hazard ratio using either robust sandwich variance or bootstrapping. When bootstrapping is used, the confidence interval and p-value are based on a t-distribution with n_boot - 1 degrees of freedom.

References

James M. Robins, Miguel Angel Hernan, and Babette Brumback. Marginal structural models and causal inference in epidemiology. Epidemiology. 2000;11(5):550-560.

Miguel Angel Hernan, Babette Brumback, and James M. Robins. Marginal structural modesl to estimate the causual effect of zidovudine on the survival of HIV-positive men. Epidemiology. 2000;11(5):561-570.

Jing Xu, Guohui Liu, and Bingxia Wang. Bias and Type I error control in correcting treatment effect for treatment switching using marginal structural models in Phase III oncology trials. Journal of Biopharmaceutical Statistics. 2022;32(6):897-914.

Author

Kaifeng Lu, kaifenglu@gmail.com

Examples


sim1 <- tssim(
  tdxo = 1, coxo = 1, allocation1 = 1, allocation2 = 1,
  p_X_1 = 0.3, p_X_0 = 0.3, 
  rate_T = 0.002, beta1 = -0.5, beta2 = 0.3, 
  gamma0 = 0.3, gamma1 = -0.9, gamma2 = 0.7, gamma3 = 1.1, gamma4 = -0.8,
  zeta0 = -3.5, zeta1 = 0.5, zeta2 = 0.2, zeta3 = -0.4, 
  alpha0 = 0.5, alpha1 = 0.5, alpha2 = 0.4, 
  theta1_1 = -0.4, theta1_0 = -0.4, theta2 = 0.2,
  rate_C = 0.0000855, accrualIntensity = 20/30,
  fixedFollowup = FALSE, plannedTime = 1350, days = 30,
  n = 500, NSim = 100, seed = 314159)

fit1 <- msm(
  sim1[[1]], id = "id", tstart = "tstart", 
  tstop = "tstop", event = "event", treat = "trtrand", 
  swtrt = "xo", swtrt_time = "xotime", 
  base_cov = "bprog", numerator = "bprog", 
  denominator = c("bprog", "L"), 
  ns_df = 3, swtrt_control_only = TRUE, boot = FALSE)
  
fit1
#>             n event_n event_pct switch_n switch_pct event_out_n event_out_pct
#> Control   250     208      83.2       77       30.8         208          83.2
#> Treatment 250     185      74.0        0        0.0         185          74.0
#>           trtrand
#> Control         0
#> Treatment       1
#> 
#>                                 Weight summary
#>            N    Min     Q1 Median   Mean     Q3    Max trtrand
#> Control 3148 0.8813 0.9686 0.9976 0.9958 1.0046 1.3166       0
#> 
#>                     Estimate Lower 95% Upper 95%
#> Hazard ratio (HR)   0.555    0.447     0.690    
#> P-value (Cox model) <.0001