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Assess the calibration of a multistate model

calib_msm.Rd

Calculates the underlying data for calibration plots of the predicted transition probabilities from a multistate model using three methods.

  1. BLR-IPCW: Binary logistic regression with inverse probability of censoring weights.

  2. MLR-IPCW: Multinomial logistic regression with inverse probability of censoring weights, based on the nominal calibration framework of van Hoorde et al. (2014, 2015)

  3. Pseudo-values: Pseudo-values estimated using the Aalen-Johansen estimator (Aalen OO, Johansen S, 1978).

Usage

calib_msm(
  data_ms,
  data_raw,
  j,
  s,
  t,
  tp_pred,
  tp_pred_plot = NULL,
  calib_type = "blr",
  curve_type = "rcs",
  rcs_nk = 3,
  loess_span = 0.75,
  loess_degree = 2,
  loess_surface = c("interpolate", "direct"),
  loess_statistics = c("approximate", "exact", "none"),
  loess_trace_hat = c("exact", "approximate"),
  loess_cell = 0.2,
  loess_iterations = 4,
  loess_iterTrace = FALSE,
  mlr_smoother_type = c("sm.ps", "sm.os", "s"),
  mlr_ps_int = 4,
  mlr_degree = 3,
  mlr_s_df = 4,
  mlr_niknots = 4,
  weights = NULL,
  w_function = NULL,
  w_covs = NULL,
  w_landmark_type = "state",
  w_max = 10,
  w_stabilised = FALSE,
  w_max_follow = NULL,
  pv_group_vars = NULL,
  pv_n_pctls = NULL,
  pv_precalc = NULL,
  pv_ids = NULL,
  CI = FALSE,
  CI_type = "bootstrap",
  CI_R_boot = NULL,
  CI_seed = NULL,
  transitions_out = NULL,
  assess_moderate = TRUE,
  assess_mean = TRUE,
  ...
)

Arguments

data_ms

Validation data in msdata format

data_raw

Validation data in data.frame (one row per individual)

j

Landmark state at which predictions were made

s

Landmark time at which predictions were made

t

Follow up time at which calibration is to be assessed

tp_pred

Data frame or matrix of predicted transition probabilities at time t, if in state j at time s. There must be a separate column for the predicted transition probabilities into every state, even if these predicted transition probabilities are 0.

tp_pred_plot

Data frame or matrix of predicted risks for each possible transition over which to plot the calibration curves. Argument provided to enable user to apply bootstrapping manually.

calib_type

Whether calibration plots are estimated using BLR-IPCW ('blr'), MLR-IPCW ('mlr') or pseudo-values ('pv')

curve_type

Whether calibration curves are estimated using restricted cubic splines ('rcs') or loess smoothers ('loess')

rcs_nk

Number of knots when curves are estimated using restricted cubic splines

loess_span

Span when curves are estimated using loess smoothers

loess_degree

Degree when curves are estimated_ using loess smoothers

loess_surface

see loess.control

loess_statistics

see loess.control

loess_trace_hat

see loess.control

loess_cell

see loess.control

loess_iterations

see loess.control

loess_iterTrace

see loess.control

mlr_smoother_type

Type of smoothing applied. Takes values s (see s), sm.ps (see sm.ps) or sm.os (see sm.os).

mlr_ps_int

the number of equally-spaced B spline intervals in the vector spline smoother (see sm.ps)

mlr_degree

the degree of B-spline basis in the vector spline smoother (see sm.ps)

mlr_s_df

degrees of freedom of vector spline (see s)

mlr_niknots

number of interior knots (see sm.os)

weights

Vector of inverse probability of censoring weights

w_function

Custom function for estimating the inverse probability of censoring weights

w_covs

Character vector of variable names to adjust for when calculating inverse probability of censoring weights

w_landmark_type

Whether weights are estimated in all individuals uncensored at time s ('all') or only in individuals uncensored and in state j at time s ('state')

w_max

Maximum bound for inverse probability of censoring weights

w_stabilised

Indicates whether inverse probability of censoring weights should be stabilised or not

w_max_follow

Maximum follow up for model calculating inverse probability of censoring weights. Reducing this to t + 1 may aid in the proportional hazards assumption being met in this model.

pv_group_vars

Variables to group by before calculating pseudo-values

pv_n_pctls

Number of percentiles of predicted risk to group by before calculating pseudo-values

pv_precalc

Pre-calculated pseudo-values

pv_ids

Id's of individuals to calculate pseudo-values for

CI

Size of confidence intervals as a %

CI_type

Method for estimating confidence interval (currently restricted to bootstrap)

CI_R_boot

Number of bootstrap replicates when estimating the confidence interval for the calibration curve

CI_seed

Seed for bootstrapping procedure

transitions_out

Transitions for which to calculate calibration curves. Will do all possible transitions if left as NULL.

assess_moderate

TRUE/FALSE whether to estimate data for calibration plots

assess_mean

TRUE/FALSE whether to estimate mean calibration

...

Extra arguments to be passed to w_function (custom function for estimating weights)

Value

calib_msm returns a list containing two elements: plotdata and metadata. The plotdata element contains the data for the calibration plots. This will itself be a list with each element containing calibration plot data for the transition probabilities into each of the possible states. Each list element contains patient ids (id) from data_raw, the predicted transition probabilities (pred) and the estimated observed event probabilities (obs). If a confidence interval is requested, upper (obs_upper) and lower (obs_lower) bounds for the observed event probabilities are also returned. If tp_pred_plot is specified, column (id) is not returned. The metadata element contains metadata including: a vector of the possible transitions, a vector of which transitions calibration curves have been estimated for, the size of the confidence interval, the method for estimating the calibration curve and other user specified information.

Details

Observed event probabilities at time t are estimated for predicted transition probabilities tp_pred out of state j at time s.

calib_type = 'blr' estimates calibration curves using techniques for assessing the calibration of a binary logistic regression model (Van Calster et al., 2016). A choice between restricted cubic splines and loess smoothers for estimating the calibration curve can be made using curve_type. Landmarking (van Houwelingen HC, 2007) is applied to only assess calibration in individuals who are uncensored and in state j at time s. Calibration can only be assessed in individuals who are also uncensored at time t, which is accounted for using inverse probability of censoring weights (Hernan M, Robins J, 2020). See method BLR-IPCW from Pate et al., (2024) for a full explanation of the approach.

calib_type = 'mlr' estimates calibration scatter plots using a technique for assessing the calibration of multinomial logistic regression models, namely the nominal calibration framework of van Hoorde et al. (2014, 2015). Landmarking (van Houwelingen HC, 2007) is applied to only assess calibration in individuals who are uncensored and in state j at time s. Calibration can only be assessed in individuals who are also uncensored at time t, which is accounted for using inverse probability of censoring weights (Hernan M, Robins J, 2020). See method BLR-IPCW from Pate et al., (2024) for a full explanation of the approach.

calib_type = 'pv' estimates calibration curves using using pseudo-values (Andersen PK, Pohar Perme M, 2010) calculated using the Aalen-Johansen estimator (Aalen OO, Johansen S, 1978). Calibration curves are generated by regressing the pseudo-values on the predicted transition probabilities. A choice between restricted cubic splines and loess smoothers for estimating the calibration curve can be made using curve_type. Landmarking (van Houwelingen HC, 2007) is applied to only assess calibration in individuals who are uncensored and in state j at time s. The nature of pseudo-values means calibration can be assessed in all landmarked individuals, regardless of their censoring time. See method Pseudo-value approach from Pate et al., (2024) for a full explanation of the approach.

Two datasets for the same cohort of inidividuals must be provided. Firstly, data_raw must be a data.frame with one row per individual containing the variables for the time until censoring (dtcens), and an indicator for censoring dtcens_s, where (dtcens_s = 1) if an individual is censored at time dtcens, and dtcens_s = 0 otherwise. When an individual enters an absorbing state, this prevents censoring from happening (i.e. dtcens_s = 0). data_raw must also contain the desired variables for estimating the weights. Secondly, data_ms must be a dataset of class msdata, generated using the [mstate] package. This dataset is used to apply the landmarking and identify which state individuals are in at time t. While data_ms can be derived from data_raw, it would be inefficient to do this within calibmsm::calib_msm due to the bootstrapping procedure, and therefore they must be inputted seperately.

Unless the user specifies the weights using weights, the weights are estimated using a cox-proportional hazard model, assuming a linear functional form of the variables defined in w_covs. We urge users to specify their own model for estimating the weights. The weights argument must be a vector with length equal to the number of rows of data_raw.

Confidence intervals cannot be produced for the calibration scatter plots (calib_type = 'mlr'). For calibration curves estimated using calib_type = 'blr', confidence intervals can only be estimated using bootstrapping (CI_type = 'bootstrap). This procedure uses the internal method for estimating weights, we therefore encourage users to specify their own bootstrapping procedure, which incorporates their own model for estimating the weights. Details on how to do this are provided in the vignette BLR-IPCW-manual-bootstrap. For calibration curves estimated using calib_type = 'pv', confidence intervals can be estimated using bootstrapping (CI_type = 'bootstrap) or parametric formulae (CI_type = 'parametric). For computational reasons we recommend using the parametric approach.

The calibration plots can be plotted using plot.calib_msm and plot.calib_mlr.

References

Aalen OO, Johansen S. An Empirical Transition Matrix for Non-Homogeneous Markov Chains Based on Censored Observations. Scand J Stat. 1978;5(3):141-150.

Andersen PK, Pohar Perme M. Pseudo-observations in survival analysis. Stat Methods Med Res. 2010;19(1):71-99. doi:10.1177/0962280209105020

Hernan M, Robins J (2020). “12.2 Estimating IP weights via modeling.” In Causal Inference: What If, chapter 12.2. Chapman Hall/CRC, Boca Raton.

Pate, A., Sperrin, M., Riley, R. D., Peek, N., Van Staa, T., Sergeant, J. C., Mamas, M. A., Lip, G. Y. H., Flaherty, M. O., Barrowman, M., Buchan, I., & Martin, G. P. Calibration plots for multistate risk predictions models. Statistics in Medicine. 2024;April:1–23. doi: 10.1002/sim.10094.

Van Calster B, Nieboer D, Vergouwe Y, De Cock B, Pencina MJ, Steyerberg EW (2016). “A calibration hierarchy for risk models was defined: From utopia to empirical data.” Journal of Clinical Epidemiology, 74, 167–176. ISSN 18785921. doi:10.1016/j.jclinepi.2015. 12.005. URL http://dx.doi.org/10.1016/j.jclinepi.2015.12.005

Van Hoorde K, Vergouwe Y, Timmerman D, Van Huffel S, Steyerberg W, Van Calster B (2014). “Assessing calibration of multinomial risk prediction models.” Statistics in Medicine, 33(15), 2585–2596. doi:10.1002/sim.6114.

Van Hoorde K, Van Huffel S, Timmerman D, Bourne T, Van Calster B (2015). “A spline-based tool to assess and visualize the calibration of multiclass risk predictions.” Journal of Biomedical Informatics, 54, 283–293. ISSN 15320464. doi:10.1016/j.jbi.2014.12.016. URL http://dx.doi.org/10.1016/j.jbi.2014.12.016.

van Houwelingen HC (2007). “Dynamic Prediction by Landmarking in Event History Analysis.” Scandinavian Journal of Statistics, 34(1), 70–85.

Yee TW (2015). Vector Generalized Linear and Additive Models. 1 edition. Springer New, NY. ISBN 978-1-4939-4198-8. doi:10.1007/978-1-4939-2818-7. URL https://link.springer.com/book/10.1007/978-1-4939-2818-7.

Examples

# Estimate BLR-IPCW calibration curves for the predicted transition
# probabilities at time t = 1826, when predictions were made at time
# s = 0 in state j = 1. These predicted transition probabilities are stored in tps0.

# Extract the predicted transition probabilities out of state j = 1
tp_pred <- dplyr::select(dplyr::filter(tps0, j == 1), any_of(paste("pstate", 1:6, sep = "")))

# Now estimate the observed event probabilities for each possible transition.
dat_calib <-
calib_msm(data_ms = msebmtcal,
 data_raw = ebmtcal,
 j=1,
 s=0,
 t = 1826,
 tp_pred = tp_pred,
 w_covs = c("year", "agecl", "proph", "match"))

# Summarise the output
summary(dat_calib)
#> The method used to assess calibration was BLR-IPCW
#> 
#> There were non-zero predicted transition probabilities into states  1,2,3,4,5,6
#> 
#> Calibration curves have been estimated for transitions into states  1,2,3,4,5,6
#> 
#> Calibration was assessed at time 1826 and calibration was assessed in a landmarked cohort of individuals in state j = 1 at time s = 0
#> 
#> A confidence interval was not estimated
#> 
#> The estimated data for calibration plots are stored in list element `plotdata`:
#> 
#> $state1
#>   id      pred       obs
#> 2  2 0.1140189 0.1095897
#> 4  4 0.1383878 0.1036308
#> 
#> $state2
#>   id      pred       obs
#> 2  2 0.2316569 0.1698031
#> 4  4 0.1836189 0.1855591
#> 
#> $state3
#>   id       pred       obs
#> 2  2 0.08442692 0.1248583
#> 4  4 0.07579429 0.1166606
#> 
#> $state4
#>   id      pred       obs
#> 2  2 0.2328398 0.2427580
#> 4  4 0.2179331 0.2243106
#> 
#> $state5
#>   id      pred       obs
#> 2  2 0.1481977 0.1909795
#> 4  4 0.1538475 0.1654523
#> 
#> $state6
#>   id      pred       obs
#> 2  2 0.1888598 0.2069354
#> 4  4 0.2304185 0.2542212
#> 
#> 
#> 
#> The estimated mean calibration are stored in list element `mean`:
#> 
#>        state1        state2        state3        state4        state5 
#> -0.0216273416 -0.0152282576  0.0254839288  0.0097158314 -0.0003011927 
#>        state6 
#>  0.0032309988