- 1 dynamicLM
- 2 Introduction
- 3 Tutorial: basic example
- 4 References
The goal of dynamicLM is to provide a simple framework to make dynamic w-year risk predictions, allowing for penalization, competing risks, time-dependent covariates, and censored data.
If you use our library, please reference it:
Anya H Fries*, Eunji Choi*, Julie T Wu, Justin H Lee, Victoria Y Ding, Robert J Huang, Su-Ying Liang, Heather A Wakelee, Lynne R Wilkens, Iona Cheng, Summer S Han, Software Application Profile: dynamicLM—a tool for performing dynamic risk prediction using a landmark supermodel for survival data under competing risks, International Journal of Epidemiology, Volume 52, Issue 6, December 2023, Pages 1984–1989, https://doi.org/10.1093/ije/dyad122
Anya H Fries*, Eunji Choi*, Summer S Han, Penalized landmark supermodels (penLM) for dynamic prediction for time-to-event outcomes in high-dimensional data. BMC Med Res Methodol, Volume 25, Article Number 22, January 2025, https://doi.org/10.1186/s12874-024-02418-9
“Dynamic prediction” involves obtaining prediction probabilities at baseline and later points in time; it is essential for better-individualized treatment. Personalized risk is updated with new information and/or as time passes.
An example is cancer treatment: we may want to predict a 5-year risk of recurrence whenever a patient’s health information changes. For example, we can predict w-year risk of recurrence at baseline (time = 0) given their initial covariates Z(0) (e.g.,30 years old, on treatment), and we can then predict w-year risk at a later point s given their current covariates Z(s) (e.g., 30+s years old, off treatment). Note that here the predictions make use of the most recent covariate value of the patient.
The landmark model for survival data is is a simple and powerful approach to dynamic prediction for many reasons:
- Time-varying effects are captured by considering interaction terms between the prediction (“landmark”) time and covariates
- Time-dependent covariates can be used, in which case, for prediction at landmark time s, the most updated value Z(s) will be used. Note that covariates do not have to be time-dependent because time-varying effects will be captured regardless.
- Both standard survival data and competing risks can be handled. In standard survival analysis only one event (e.g., recurrence) is considered, with possible censoring. Competing risks consider the time-to-first-event (‘time’) and the event type (‘cause’), for example analyzing recurrence with the competing risk of death.
- Penalization enables effective handling of high-dimensional data by penalizing model coefficients to either select covariates or shrink coefficients.
It is built on hazards, like a (cause-specific) Cox model. From a
landmark time
We offer a short introduction to the theory behind the landmark supermodel.
In short, the creation of the landmark model for survival data is built on the concept of risk assessment times (i.e., landmarks) that span risk prediction times of interest. In this approach, a training dataset of the study cohort is transformed into multiple censored datasets based on a prediction window of interest and the predefined landmarks. A model is fit on these stacked datasets (i.e., supermodel), and dynamic risk prediction is then performed by using the most up-to-date value of a patient’s covariate values.
Further references: Putter and Houwelingen describe landmarking extensively here and here.
You can download dynamicLM from GitHub
with:
# install.packages("devtools")
devtools::install_github("thehanlab/dynamicLM")Package documentation can be found in this
pdf
and more information about any function can be obtained by running
?function_name.
This is a basic example which shows you how to use dynamicLM to make
dynamic 5-year predictions and check calibration and discrimination
metrics.
We have an alternative tutorial which uses cancer relapse data. This is the example first used at the time of publishing the Software Application Profile in IJE, and only includes an unpenalized landmark supermodel.
First, load the library:
library(dynamicLM)Data can come in various forms, with or without time-dependent covariates:
- Static data, with one entry per patient. Here, landmark time-varying effects are still considered for dynamic risk prediction.
- Longitudinal (long-form) data, with multiple entries for each patient with updated covariate information.
- Wide-form data, with a column containing the time at which the covariate changes from 0 to 1.
We illustrate the package using the long-form PBC data sets from the
survival package, which gives the time-to-event of a transplant under
the competing event of death. Using get_pbc_long(), we combine pbc
(containing baseline information and follow-up status) with pbcseq
(repeated laboratory values). In the merged data, the follow-up time has
column name tstart. Running ?pbc or ?pbcseq gives more information
on the data.
pbc_df <- get_pbc_long()
head(pbc_df)
#> id time status trt age stage tstart tstop endpt albumin alk.phos ascites
#> 1 1 1.1 2 1 58.76523 4 0.0 0.5 0 2.60 1718 1
#> 2 1 1.1 2 1 58.76523 4 0.5 1.1 2 2.94 1612 1
#> 3 2 12.3 0 1 56.44627 3 0.0 0.5 0 4.14 7395 0
#> 4 2 12.3 0 1 56.44627 3 0.5 1.0 0 3.60 2107 0
#> 5 2 12.3 0 1 56.44627 3 1.0 2.1 0 3.55 1711 0
#> 6 2 12.3 0 1 56.44627 3 2.1 4.9 0 3.92 1365 0
#> ast bili chol edema hepato platelet protime spiders male
#> 1 138.0 14.5 261 1 1 190 12.2 1 0
#> 2 6.2 21.3 261 1 1 183 11.2 1 0
#> 3 113.5 1.1 302 0 1 221 10.6 1 0
#> 4 139.5 0.8 302 0 1 188 11.0 1 0
#> 5 144.2 1.0 302 0 1 161 11.6 1 0
#> 6 144.2 1.9 302 0 1 122 10.6 1 0We first specify which variables are fixed or longitudinal (time-varying).
outcome <- list(time = "time", status = "status")
fixed_variables <- c("male", "stage", "trt", "age")
varying_variables <- c("albumin", "alk.phos", "ascites", "ast", "bili",
"edema", "hepato", "platelet", "protime", "spiders")
covars <- list(fixed = fixed_variables, varying = varying_variables)We will produce 5-year dynamic predictions of transplant (w). Landmark
time points (lms) are set as every year between 0 and 4 years to train
the model. This means we are only interested in predicting the 5-year
risk of transplant for patients both at baseline and at later points -
up to 4 years after diagnosis.
w <- 5 # Predict the 5-year outcome of transplant
lms <- seq(0, 4, by = 1) # Risk assessment time points (every year for 4 years)With this, we are ready to build the super data set that will train the model. We print intermediate steps for illustration.
There are three steps:
stack_data(): stacks the landmark data sets- An optional additional update for more complex columns that vary with landmark-times: For example, here we update the value of age.
add_interactions(): Landmark time interactions are added, note the additional columns created. We will consider linear and quadratic landmark interactions with the covariates (given byfunc_covars = c("linear", "quadratic")) and the landmarks (func_lms = c("linear", "quadratic")).
Note that these return an object of class LMdataframe. This has a
component data which contains the dataset itself.
We illustrate the process in detail by printing the entries at each step for one individual and some example columns.
example_columns <- c("id", "time", "status", "trt", "age", "albumin", "ascites")
pbc_df[pbc_df$id == 1, c("tstart", example_columns)]
#> tstart id time status trt age albumin ascites
#> 1 0.0 1 1.1 2 1 58.76523 2.60 1
#> 2 0.5 1 1.1 2 1 58.76523 2.94 1We first stack the datasets over the landmarks (see the new column ‘LM’) and update the treatment covariate. One row is created for each landmark that the individual is still alive at. In this row, if time is greater time than the landmark time plus the window, it is censored at this value (this occurs in the first row, for example, censored at 0+5), and the most recent value all covariates is used (in our case, only treatment varies).
# Stack landmark datasets
covars <- list(fixed = fixed_variables, varying = varying_variables)
# covars <- list(fixed = NULL, varying = NULL)
lmdata <- stack_data(pbc_df, outcome, lms, w, covars, format = "long",
id = "id", rtime = "tstart")
data <- lmdata$data
print(data[data$id == 4, c("LM", example_columns)])
#> LM id time status trt age albumin ascites
#> 16 0 4 5.0 0 1 54.74059 2.54 0
#> 18 1 4 5.3 2 1 54.74059 2.80 0
#> 19 2 4 5.3 2 1 54.74059 2.92 0
#> 191 3 4 5.3 2 1 54.74059 2.92 0
#> 21 4 4 5.3 2 1 54.74059 2.59 0We then (optionally) update more complex LM-varying covariates. Here we create update the age covariate, based on age at time 0.
lmdata$data$age <- lmdata$data$age + lmdata$data$LMLastly, we add landmark time-interactions. We use the following naming
convention: _LM1 refers to the first interaction in func_covars,
_LM2 refers to the second interaction in func_covars, etc. For
example, with linear and quadratic terms, albumin_LM1 refers to
albumin * LM, albumin_LM2 is albumin * LM^2.
Similarly, LM1 and LM2 are created from func_lm. Here, LM1 =
LM and LM2 = LM^2.
An optional additional argument is pred_covars which can limit the
covariates that will have landmark time interactions.
lmdata <- add_interactions(lmdata,
func_covars = c("linear", "quadratic"),
func_lms = c("linear", "quadratic"))
data <- lmdata$data
print(data[data$id == 1,
c("LM", example_columns,
paste0(example_columns[4:7], "_LM1"),
paste0(example_columns[4:7], "_LM2"))])
#> LM id time status trt age albumin ascites trt_LM1 age_LM1 albumin_LM1
#> 1 0 1 1.1 2 1 58.76523 2.60 1 0 0.00000 0.00
#> 2 1 1 1.1 2 1 59.76523 2.94 1 1 59.76523 2.94
#> ascites_LM1 trt_LM2 age_LM2 albumin_LM2 ascites_LM2
#> 1 0 0 0.00000 0.00 0
#> 2 1 1 59.76523 2.94 1One can print lmdata. The argument verbose allows for additional
stored objects to be printed (default is FALSE).
print(lmdata, verbose = TRUE)Note that lmdata$all_covs returns a vector with all the covariates
that have landmark interactions.
print(lmdata$all_covs)A traditional (unpenalized) or penalized landmark supermodel can be fit to the data.
Please note that convergence warnings in cox.fit are common throughout
but can be disregarded if the coefficients obtained from the model are
reasonable (Therry Therneau, the author of the survival package, has
stated this in previous discussions
here
and here).
To fit a supermodel, a stacked data set, formula, and method need to be
provided. The input to dynamic_lm varies slightly depending on if
standard survival data or competing events are being considered.
In the case of standard survival data:
formula <- "Surv(LM, time, status) ~ stage +
bili + bili_LM1 + bili_LM2 +
albumin + albumin_LM1 + albumin_LM2 +
LM1 + LM2 + cluster(id)"
supermodel <- dynamic_lm(lmdata, as.formula(formula), "coxph", x = TRUE) In the case of competing risks (as for this example data):
formula <- "Hist(time, status, LM) ~ stage +
bili + bili_LM1 + bili_LM2 +
albumin + albumin_LM1 + albumin_LM2 +
LM1 + LM2 + cluster(id)"
supermodel <- dynamic_lm(lmdata, as.formula(formula), "CSC", x = TRUE) print(supermodel)
#>
#> Landmark cause-specific cox super model fit for dynamic prediction of window size 5:
#>
#> $model
#> ----------> Cause: 1
#> coef exp(coef) se(coef) robust se z p
#> stage 0.726673 2.068189 0.201920 0.352789 2.060 0.0394
#> bili 0.056299 1.057914 0.059149 0.037374 1.506 0.1320
#> bili_LM1 0.050673 1.051979 0.056364 0.038767 1.307 0.1912
#> bili_LM2 0.002579 1.002583 0.012455 0.009569 0.270 0.7875
#> albumin -0.273729 0.760539 0.710437 0.608541 -0.450 0.6528
#> albumin_LM1 -0.235161 0.790443 0.764169 0.617457 -0.381 0.7033
#> albumin_LM2 0.022656 1.022915 0.180097 0.142674 0.159 0.8738
#> LM1 0.635287 1.887563 2.683794 2.191284 0.290 0.7719
#> LM2 -0.179084 0.836036 0.626355 0.496641 -0.361 0.7184
#>
#> Likelihood ratio test=NA on 9 df, p=NA
#> n= 1175, number of events= 56
#>
#>
#> ----------> Cause: 2
#> coef exp(coef) se(coef) robust se z p
#> stage 0.609239 1.839032 0.087574 0.140112 4.348 1.37e-05
#> bili 0.135984 1.145664 0.014712 0.015599 8.718 < 2e-16
#> bili_LM1 0.003599 1.003605 0.018316 0.019694 0.183 0.855012
#> bili_LM2 0.001171 1.001172 0.005023 0.005295 0.221 0.824928
#> albumin -0.977390 0.376292 0.262343 0.296724 -3.294 0.000988
#> albumin_LM1 0.006638 1.006660 0.310891 0.303998 0.022 0.982578
#> albumin_LM2 -0.035341 0.965276 0.075791 0.071283 -0.496 0.620050
#> LM1 -0.160122 0.852039 1.052334 1.054453 -0.152 0.879302
#> LM2 0.094909 1.099559 0.252121 0.246167 0.386 0.699833
#>
#> Likelihood ratio test=NA on 9 df, p=NA
#> n= 1175, number of events= 281There are additional ways of printing/accessing the model.
# E.g., of additional arguments to print
# * cause: only print this cause-specific model
# * verbose: show additional stored objects
print(supermodel, cause = 1, verbose = TRUE)
# Coefficients can easily be accessed via
coef(supermodel)Dynamic log hazard ratios can be plotted:
par(mfrow = c(1,3))
plot(supermodel)
The hazard ratio (not log HR) can also be plotted setting
logHR = FALSE. Specifying the covars arguments allows for a subset
of dynamic hazard ratios to be plotted and conf_int = FALSE removes
the confidence intervals.
plot(supermodel, logHR = FALSE, covars = c("bili", "albumin"), conf_int = FALSE)If the super dataset is not created via the functions stack_data() and
add_interactions and is simply a dataframe, additional parameters must
be specified to fit a model. (In this case, see ?dynamic_lm.data.frame
for the additional parameters and the details section of
?add_interactions for how the interaction terms must be named).
With a large dataset and time-dependent effects, the supermodel has
numerous parameters. To handle this high-dimensionality and ensure
generalizability, one can alternatively fit a penalized model, where
coefficients are shrunk or selected by penalizing the pseudo-partial
likelihood
Penalization balances model complexity and goodness-of-fit, where the
optimal weight
To fit a penalized landmark supermodel, the lmdata is the only required
input. First, either a coefficient path (using pen_lm) or a
cross-validated model (using cv.pen_lm) is created for multiple
penalties (lambdas dynamic_lm.
The code largely makes calls to the glmnet library.
A coefficient path can be fit as follows. By default the argument
alpha is 1, which represents the LASSO (L1) penalty. By using
alpha = 0, we are using a Ridge (L2) penalty.
path <- pen_lm(lmdata, alpha = 0)Each line on the plot represents one variable and shows how its coefficient changes against the L1 norm of the coefficient vector (i.e., for different penalties). The top axis shows how many variables are selected.
par(mfrow = c(1, 2))
plot(path, all_causes = TRUE)
Alternatively, the path can be printed.
print(path, all_causes = TRUE)If you want to specify only a subset of covariates to fit the path to,
this is done with the y argument:
path1 <- pen_lm(lmdata, y = c("male", "male_LM1", "male_LM2",
"trt", "trt_LM1", "trt_LM2"))A cross-validated can be fit as follows. By default the argument alpha
is 1, which representies the LASSO (L1) penalty. By using alpha = 0,
we are using a Ridge (L2) penalty.
cv_model <- cv.pen_lm(lmdata, alpha = 0) To print the outcome for all causes:
print(cv_model, all_causes = TRUE)To plot the cross-validation curve:
par(mfrow = c(1, 2))
plot(cv_model, all_causes = TRUE)
To specify only a subset of covariates to fit to
cv_model1 <- cv.pen_lm(lmdata, y = c("male", "male_LM1", "male_LM2",
"trt", "trt_LM1", "trt_LM2"))To fit a landmark supermodel, either a coefficient path or
cross-validated model is passed to dynamic_lm() with a value for
lambda. This value can be numeric (as many lambdas as causes) or, for
cross-validated models, can be the minimum from cross-validation
(“lambda.min”) or within 1-standard error from the minimum
(“lambda.1se”).
supermodel_pen <- dynamic_lm(cv_model, lambda = "lambda.1se")One can print the covariates:
print(supermodel_pen, all_causes = TRUE)The largest coefficients can also be plot. Here, we plot the 15 largest
coefficients. For further arguments, see ?plot.penLMCSC or
?plot.penLMcoxph.
# Add more space on the sides
par(mar = c(5, 10, 1, 7)) # default is c(5.1, 4.1, 4.1, 2.1)
plot(supermodel_pen, max_coefs=15)
The hazard ratios can also be plot, as for an unpenalized model, by
setting HR = TRUE.
par(mfrow=c(1,3))
plot(supermodel_pen, covars = c("bili", "stage", "edema"),
HR = TRUE, ylim = c(0, 0.25))
Once dynamic_lm has been run, the same prediction procedures and model
evaluation, etc., can be performed regardless of how the model has been
fit.
Predictions for the training data can easily be obtained. This provides w-year risk estimates for each individual at each of the training landmarks they are still alive.
p1 <- predict(supermodel)
p2 <- predict(supermodel_pen)
print(p1)
#> $preds
#> LM risk
#> 1 0 0.02705205
#> 2 0 0.03687079
#> 3 0 0.07795531
#> 4 0 0.07450562
#> 5 0 0.04540469
#> 6 0 0.03708919
#> [ omitted 1170 rows ]One can print the predictions. The argument verbose allows for
additional stored objects to be printed (default is FALSE).
print(p1, verbose = TRUE)Test data can be a stacked dataset (lmdata):
p_test <- predict(supermodel, lmdata)Alternatively, test data can be a data frame. As a prediction is made for an individual at a specific prediction time, both a prediction (“landmark”) time (e.g., at baseline, at 2 years, etc) and an individual (i.e., covariate values set at the landmark time-point) must be given. For example, we can prediction w-year risk from baseline using an entry from the very original data frame.
example_test <- pbc_df[1:5, ]
example_test$age <- example_test$age + example_test$tstart
print(example_test[, c("tstart", example_columns)])
#> tstart id time status trt age albumin ascites
#> 1 0.0 1 1.1 2 1 58.76523 2.60 1
#> 2 0.5 1 1.1 2 1 59.26523 2.94 1
#> 3 0.0 2 12.3 0 1 56.44627 4.14 0
#> 4 0.5 2 12.3 0 1 56.94627 3.60 0
#> 5 1.0 2 12.3 0 1 57.44627 3.55 0
p_test <- predict(supermodel, example_test, lms = "tstart", cause = 1)Calibration plots, which assess the agreement between predictions and
observations in different percentiles of the predicted values, can be
plotted for each of the landmarks used for prediction. Entering a named
list of prediction objects in the first argument allows for comparison
between models. This list can be of supermodels or prediction objects
(created by calling predict()).
par(mfrow = c(1, 3), pty = "s") # square axes
outlist <- calplot(list("LM" = p1, "penLM" = p2),
times = c(0,2,4), # landmarks to plot at
method = "quantile", q=10, # method for calibration plot
# Optional plotting parameters to alter
ylim = c(0, 0.52), xlim = c(0, 0.52), lwd = 1,
xlab = "Predicted Risk", ylab = "Observed Risk",
legend = TRUE, legend.x = "bottomright")
Predictive performance can also be assessed using time-dependent dynamic AUC (AUCt) or time-dependent dynamic Brier score (BSt).
- AUCt is defined as the percentage of correctly ordered markers when comparing a case and a control – i.e., those who incur the pr imary event within the window w after prediction and those who do not.
- BSt provides the average squared difference between the primary event markers at time w after prediction and the absolute risk estimates by that time point.
Predictive performance can also be assessed using the summary (average) time-dependent dynamic area under the receiving operator curve or time-dependent dynamic Brier score. This enables one score per model or one comparison for a pair of models.
scores <- score(list("LM" = p1, "penLM" = p2),
times = c(0, 2, 4)) # landmarks at which to assessThese results can be printed:
print(scores) # print everything
print(scores, summary = FALSE) # only print AUCt and BSt
print(scores, landmarks = FALSE) # only print summary metricsThese results can also be plot with point wise confidence intervals.
par(mfrow = c(1, 4))
plot(scores, summary = TRUE)
Additional parameters control which plots to include and additional information. The first plot below shows how one can remove confidence intervals and plot the time-dependent contrasts. One can also plot if model summary metrics are significantly different or not. The second plot adds the p-values of the significant comparisons to a plot. The third plots the contrasts directly (third example).
See ?plot.LMScore for more information.
# Three plots and make extra space below for the x-labels
par(mfrow = c(1, 3), mar = c(9, 4, 4, 3))
# E.g., plot only the time-dependent AUC contrasts without CIs
plot(scores, brier = FALSE, landmarks = TRUE,
summary = FALSE, contrasts = TRUE, se = FALSE)
# E.g., plot only summary BS and add p-value comparisons
plot(scores, auc = FALSE, landmarks = FALSE, summary = TRUE,
ylim = c(0, 0.15),
las = 2, # Rotate x-axis labels
add_pairwise_contrasts = TRUE, # Include the contrasts
cutoff_contrasts = 0.05, # Significance cutoff, default 0.05
pairwise_heights = c(0.1, 0.13), # Height of the contrast labels
# (only 2/3 are significant so
# only need to specify 2 heights)
width = 0.01) # Width of ends of bars
# E.g., plot summary BS contrasts
plot(scores, auc = FALSE, landmarks = FALSE, summary = TRUE,
contrasts = TRUE, las = 2)
Bootstrapping can be performed by calling score() and setting the
arguments split.method = "bootcv" and B (the number of bootstrap
replications). Note that the argument x = TRUE must be specified when
fitting the model (i.e., when calling dynamic_lm()).
scores <- score(list("LM" = supermodel, "penLM" = supermodel_pen),
times = c(0, 2, 4), metrics = "auc",
split.method = "bootcv", B = 10) # 10 bootstrapsExternal validation can be performed by passing in test predictions
(created by calling predict()) or by specifying the supermodel as the
object argument and passing new data through the data argument. This
data can be a LMdataframe or a dataframe (in which case lms must be
specified).
newdata <- pbc_df[pbc_df$tstart == 0, ] # Use the data from baseline as "new" data
par(mfrow = c(1,1))
outlist <- calplot(list("LM" = supermodel, "penLM" = supermodel_pen),
cause = 1,
data = newdata,
lms = 0, # landmark time of the newdata
method = "quantile", q = 10,
ylim = c(0, 0.25), xlim = c(0, 0.25))
score(list("LM" = supermodel, "penLM" = supermodel_pen),
cause = 1, data = newdata, lms = 0)Individual risk score trajectories can be plotted. As with predict(),
the data input is in the form of the original data. For example, we can
consider two individuals with the same stage and similar albumin levels,
but with different bilirunbin levels.
We turn our data into long-form data to plot.
# Select individuals
idx <- pbc_df$id %in% c(231, 29)
# Prediction time points
x <- seq(0, 4, by = 0.5)
# Stack landmark datasets
dat <- stack_data(pbc_df[idx, ], outcome, x, w, covars, format = "long",
id = "id", rtime = "tstart")$data
dat$age <- dat$age + dat$LM
dat <- dat[order(dat$id, dat$LM), ]print(dat[dat$LM %in% 0:3, c("id", "time", "status", "LM", supermodel$lm_covs)])
#> id time status LM stage bili albumin
#> 182 29 5.0 0 0 2 0.7 3.78
#> 184 29 6.0 0 1 2 0.6 3.90
#> 1842 29 7.0 0 2 2 0.6 3.90
#> 1851 29 8.0 0 3 2 0.5 3.74
#> 1517 231 3.2 2 0 4 3.4 1.96
#> 1518 231 3.2 2 1 4 4.4 3.25
#> 15191 231 3.2 2 2 4 8.1 3.06
#> 15201 231 3.2 2 3 4 14.8 2.89plotrisk(supermodel, dat, format = "long", ylim = c(0, 0.35),
x.legend = "topleft")
We can see that the individual with higher bilirunbin levels (id=231) has a higher and increasing 5-year risk of transplant. This can be explained by the dynamic hazard rate of bilirunbin (seen above). Further, the risk of transplant rapidly increases when the bilirunbin levels rise.
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