2 Curve-Fitting
Fitting parametric distributions to a Survival curve is useful for
both event prediction and for gathering trial planning assumptions. The
gestate package supports curve fitting for Weibull, exponential and
Log-normal distributions for both life-table data (fit_KM) patient-level
data (fit_tte_data).
2.1 Example Data Set
In this section, we will use the following randomly-generated data
set, created by simulate_trials, which assumes an underlying Weibull
distribution with parameters alpha = 100, beta = 0.8. For more details
on this step, see section 4.1 in the vignette on trial planning
(“trial_planning”). The second section of this code generates a 7-day
interval lifetable; this is to create a data set that can be used to
illustrate the use of fit_km, which operates on life-table data.
Typically, if you have patient-level data available it is recommended
that you should use that directly instead.
library(gestate)
library(survival)
# Simulate a single data set based upon a Weibull curve with shape 0.8 and scale 100:
# Distributions
example_distribution <- Weibull(alpha=100,beta=0.8)
example_recruitment <- LinearR(rlength=24,Nactive=600,Ncontrol=0)
# Create the full trajectory
maxtime <- 100
example_data_long <- simulate_trials(active_ecurve=example_distribution,control_ecurve=example_distribution,rcurve=example_recruitment,assess=maxtime,iterations=1,seed=1234567,detailed_output=TRUE)
# Shrink the trajectory to an early time point at which event prediction will occur
example_data_short <- set_assess_time(example_data_long,18)
######################
# Create life-table with 7-day 'sampling'
# Note that this is to create an example lifetable and would not normally be done as it is typically better to use patient-level data if available
temp1 <- summary(survfit(Surv(example_data_short[,"Time"],1-example_data_short[,"Censored"])~ 1,error="greenwood"))
out1 <- cbind(temp1$time,temp1$n.risk,temp1$surv,temp1$std.err)
out1 <- rbind(c(0,out1[1,2],1,0),out1)
colnames(out1) <- c("Time","NAR","Survival","Std.Err")
interval <- 1
x1 <- ceiling(max(out1[,"Time"]/interval))*interval
times1 <- seq(from = 0, to = x1,by=interval)
keys1 <- findInterval(times1,out1[,"Time"])
example_lifetable <- out1[keys1,]
example_lifetable[,"Time"] <- times1
2.2 Curve-Fitting Life-Tables: fit_KM
The fit_KM function performs a (weighted) non-linear regression on
the Survival function from a life-table. It is meant for use in
situations where patient-level data is unavailable, for instance where
the only source of information is a Kaplan Meier plot from a
publication, but a parametric fit is required. The function acts as a
wrapper for the ‘survival’ package function survreg, but produces
outputs in keeping with the rest of the ‘gestate’ package, including the
parametric forms for the fitted curves.
In general it has inferior performance to fit_tte_data in all cases
where both methods are able to be applied. With the use of suitable
weighting it is approximately unbiased in its parameter estimation, but
it is not possible to analytically obtain accurate estimates of
parameter estimate variability by this method. It can also be seen to
have higher variability than patient-level MLE-based methods such as
fit_tte_data.
The ‘KMcurve’ argument specifies the name of the life-table, while
the ‘Survival’ and ‘Time’ arguments can be used to change the names of
the relevant columns. Due to the inevitable positive correlation between
Survival points and the means of fitting, it works best if the input
life-table has a constant interval regardless of timing of event
occurrence (e.g. one point every 7 days). With an event-driven interval,
it will typically be biased towards fitting parts of the dataset with
the most events.
The ‘type’ argument specifies which parametric curve type to fit;
‘Weibull’,‘Lognormal’,‘Exponential’ or ‘automatic’. In general, fitting
with an exponential distribution is discouraged as it is a special case
of the Weibull distribution and it does not have the flexibility to
reasonably fit data that deviates from the constant-hazards assumption;
this can lead to poor fits or over-precision unless you are sure of the
underlying distribution. If automatic (the default) is chosen, both
Weibull and Log-normal are fitted and the one with the lowest (weighted)
residual sum of squares is selected. If there are any convergence
issues, the ‘startbeta’ and ‘startsigma’ parameters may be used to tweak
the starting parameter values for Weibull and Log-normal curves
respectively.
To reflect that variability of the Survival function is strongly
associated with the number of patients at risk, it is strongly
recommended to weight the non-linear regression by the number at risk.
Weighting is required by default, but may be turned off with the
‘weighting’ argument. The name of the weights column is specified by the
‘weights’ argument. By default, the weights are the values from the
specified column, but the ‘weight_power’ argument optionally provides a
power to raise them to.
Although the fitting process creates a covariance matrix for the
parameter estimates, these are heavily underestimated since the
regression is performed upon data points that are highly correlated
(since each KM estimate can be described as the previous one multiplied
by the subsequent probability of an event). The matrix can therefore be
considered to be highly misleading. The output therefore does not return
the covariance matrix.
# Curve-fit the example lifetable in automatic mode
fit1 <- fit_KM(KMcurve=example_lifetable,Survival="Survival",Time="Time",Weights="NAR",type="automatic")
fit1
#> $Curvetype
#> [1] "Weibull"
#>
#> $Parameters
#> Alpha Beta
#> 73.4954086 0.9179521
#>
#> $VCov
#> Alpha_Var Beta_Var Covariance
#> NA NA NA
#>
#> $Fit
#> Distribution Metric Value
#> [1,] "Weibull" "Weighted Sum of Squares" "0.391506961862774"
#> [2,] "Lognormal" "Weighted Sum of Squares" "0.590485691670524"
2.3 Curve-Fitting Patient-Level Data: fit_tte_data
Maximum Likelihood Estimation (MLE) is a more efficient curve-fitting
method that also allows for reasonable quantification of errors.
fit_tte_data uses the Survfit R function to MLE fit Weibull, Log normal
or Exponential parametric curves to the Survival function. Where
patient-level data is available, it is therefore recommended to use
fit_tte_data over fit_KM.
The ‘data’ argument specifies the data set, while the ‘Time’ and
‘Event’ arguments provide the column names for the times and event
indicators respectively. The ‘censoringOne’ argument tells the function
whether a 1 represents a censoring (TRUE) or an event (FALSE, default).
As with fit_KM, the ‘type’ argument gives a choice of Weibull, Log
normal, Exponential or automatic fitting. As noted for fit_KM, it is
generally recommended to use Weibull in preference to Exponential
fitting, to avoid unnecessarily fixing the shape parameter. In automatic
mode, the selection is between Weibull and Log-Normal distributions
based on the higher log-likelihood (since both distributions have 2
parameters, this corresponds to the lower AIC). The ‘init’ argument may
be used to provide starting values to Survfit; this will not work in
automatic mode however.
# Curve-fit the example lifetable in automatic mode
fit2 <- fit_tte_data(data=example_data_short,Time="Time",Event="Censored",censoringOne=TRUE,type="automatic")
fit2
#> $Curvetype
#> [1] "Weibull"
#>
#> $Parameters
#> Alpha Beta
#> 74.9044606 0.8971022
#>
#> $VCov
#> Alpha_Var Beta_Var Covariance
#> 427.253348 0.010806 -1.827576
#>
#> $Fit
#> Distribution Metric Value
#> [1,] "Weibull" "Log-Likelihood" "-300.998710542156"
#> [2,] "Lognormal" "Log-Likelihood" "-303.185871359364"
Both fitting functions return a list with the type of Curve that has
been fit, the parameters, and finally the components of the covariance
matrix of the fit.
3 Event Prediction: event_prediction
3.1 Introduction
The event_prediction function uses a combination of fit_tte_data and
the numerical integration approaches found in nph_traj to produce
estimates for event numbers at any given time point. There are four main
inputs required for event prediction: Firstly a patient-level data set
from which an event Curve can be estimated (‘data’). Secondly, an RCurve
reflecting the combined already-observed recruitment and predicted
future recruitment (‘rcurve’). Thirdly, a Curve reflecting the dropout
censoring in the trial (‘dcurve’). Finally, optionally, a set of
parameters describing what has actually been observed at an observed
time point, to allow for conditional predictions; these are described in
section 3.5.
In general, it is not recommended to use an exponential curve fit to
base event predictions on since this imposes a strong constraint of
constant event rates. This can result in poor curve fitting,
over-precision of prediction intervals, and unnecessary bias in the
estimates.
Particular care should be paid to time units since measured event
times are typically in days, while recruitment and timeline planning are
typically in months (see section 3.2 for more on time units).
event_prediction assumes that all times are in units of months, except
for the event times, which may be in units of days (default) or months;
this choice is controlled by the optional ‘units’ argument. gestate
assumes a conversion factor between months and days of 365.25/12. As a
workaround for cases where other units are desirable, if the ‘units’
argument is set to months and all times for all variables are provided
in the same unit, then no conversions are done and all output units will
remain in the unit of the originally provided data.
3.2 Event Curve Fitting
Section 2.3 describes the MLE fitting process and the arguments
required for this. All of these arguments carry over into
event_prediction.
3.3 Recruitment Specification
The RCurve for an event prediction should have units of months and
typically be of type PieceR since the exact numbers of patients that
have so far entered the trial in each month will be known. The program
does not consider variability in future recruitment as it is expected
that by the point that event prediction is performed a sizeable amount
of recruitment will have already occurred, and the variability in future
recruitment will be both low and subject to negative feedback (i.e. if
it drops, efforts are made to increase it and vice versa). As event
prediction is performed in a blinded manner, the randomisation ratio (or
even the number of arms) is irrelevant, and therefore any between-arm
distribution of patients can be specified.
# Create an RCurve incorporating both observed and predicted recruitment with arbitrary randomisation ratio of 1
recruit <- PieceR(matrix(c(rep(1,12),10,15,25,30,45,60,55,50,65,60,55,30),ncol=2),1)
3.4 Censoring Specification
The amount of dropout censoring varies wildly between trials. The
endpoint Overall Survival (OS) ought to have minimal dropout that does
not tangibly affect event prediction, whereas other endpoints are likely
to have enough that it warrants consideration. A dropout distribution
can be supplied to event_prediction via the ‘dcurve’ argument.
It is suggested to use fit_KM or fit_tte_data to curve-fit a version
of the data where dropouts are handled as events, while administrative
censoring and events are handled as censored. This should produce a
reasonable estimate for censoring unconditional on event occurrence
(assuming independent censoring).
3.5 Conditioning
To calculate conditional predictions, it is necessary to specify the
number of observed events at a given time, as well as preferably the
number of patients remaining at risk (total number intended to be
recruited minus events occurred, minus dropouts). The conditional number
at risk is important because if it is known that a larger-than-expected
number of patients have dropped out, then it is clear this will lower
the future expected rate of events (as fewer patients than expected are
at risk). Conditional predictions can be enabled by specifying a
conditional time through the ‘cond_Time’argument. The number of
conditional events is then given by ’cond_Events’ and the conditional
number at risk by ‘cond_NatRisk’. If the latter is not known, gestate
calculates an estimate based upon the expected number at risk at the
conditioning time adjusted by the difference between the conditional and
expected numbers of events at that time.
3.6 Accounting for Effects of Adjudication
For trials with adjudication it is common for event prediction to be
performed upon investigator-reported data due to delays in adjudication.
However, typically a proportion of adjudicated events will be
‘down-adjudicated’, resulting in events used in the prediction not being
‘valid’. To account for this net down-adjudication (and the rarer case
of net up-adjudication), the ‘discountHR’ argument is available. If it
is assumed that patients remain at risk after an observed event,
‘discountHR’ corresponds to the probability of an event being
adjudicated as true. It applies a discount hazard ratio to the fitted
event curve to reflect this. The effect of specifying a value below 1
for this parameter is to ensure there are fewer events predicted
initially, but since the patients remain at risk, the predictions after
the discount is applied will slowly catch up with those without the
discount. This approach can also be used in cases of up-adjudication
where patients may be found to retrospectively have had an event at a
later stage - here the value should be above 1.
The argument’discountHR’ may only be used in conjunction with a
Weibull curve since Lognormal curves are not subject to proportional
hazards. If patients are no longer at risk following an event (e.g. if
trial participation is ended) but they may still be down-adjudicated,
then this may instead be handled by simply multiplying all predicted
event numbers by the probability of adjudication as an event.
If the ‘discountHR’ argument is used, care should be taken with
conditional predictions; in this situation, the number of events to be
conditioned on is the unadjusted (i.e. ‘observed’) number of events,
rather than the adjusted number. Likewise, the conditioned number at
risk should be for the unadjusted events rather than the adjusted. The
program will automatically adjust the conditioning values to reflect the
adjusted scale before performing the calculations.
3.7 Calculations
A Curve object is created from the fitted parameters. This is
combined with the recruitment RCurve and censoring Curve to produce
estimates of the expected event numbers at each integer time point using
the same approach as nph_traj. Conditioned trajectories are created by
scaling future events by the ratio of the conditioned number at risk to
the expected number at risk at the conditioning time. The scaled number
of additional events is then added to the conditioned number of events.
Where no conditioned number at risk is provided, gestate assumes by
default that it is the expected number minus the difference between
conditioned and expected event numbers at the conditioning time;
i.e. that each additional observed event than expected reduces the
expected number at risk by one.
The estimation errors for the parameters are also carried over into
the event prediction calculation and propagated through by the delta
method to enable the estimation of standard errors on both the
conditional and unconditional expected trajectories. The conditional
standard error is always lower at any given time, particularly close to
the time of conditioning (and it is 0 at that time). The standard errors
of the trajectories are then combined with the number at risk and the
event numbers via a beta binomial distribution to create prediction
intervals for the conditional and unconditional predicted
trajectories.
The ‘PI’ argument specifies the width of prediction interval to
produce, by default 0.95 (95%) intervals.
A paper on the general approach to analytic prediction intervals and
the formulae for the calculations is to be published.
3.8 Plotting Event Prediction
For convenience, there is an inbuilt plotting function for event
prediction output; plot_ep. This is designed to automatically read
output from event_prediction, so only requires the ‘data’ argument,
specifying the name of the event_prediction output. By default, it makes
use of the available information, plotting the unconditional trajectory
with prediction CI, as well as the conditional ones if available. The
arguments ‘trajectory’ and ‘which_PI’ can be used to eliminate some or
all of these lines in the plot. The prediction_fitting argument switches
the prediction intervals for confidence intervals of fitting for the
mean trajectory estimate, which can be used for technical/diagnostic
purposes; these parameter fitting CIs are not representative of the
range of events that could occur.
To plot the observed events as well, an observed event trajectory may
also be supplied through the ‘observed’ argument. This accepts either a
vector of events, corresponding to the values at times 1,2,3…, or a
2-column matrix / data-frame with the first column the time points in
ascending order, and the second the event numbers.
A target number of events may also be specified using the ‘target’
argument. The optional ‘max_time’ and ‘max_E’ options override the
automatic plotting area dimensions, while the ‘legend_position’ and
‘no_legend’ arguments control the legend positioning and whether it is
displayed at all. Standard graphical arguments are also accepted.
3.9 Diagnostics: Plotting the Fitted Kaplan Meier Curve
To assess the goodness of fit for the event fitting, it is often
helpful to compare the fitted event distribution to the observed Kaplan
Meier plot. The function plot_km_fit will automatically create such a
plot from the relevant input. It takes two key arguments: ‘fit’ and
‘data’. ‘fit’ should be the name of the output from fit_KM,
fit_tte_data, or event_prediction.’data’ should be the name of the
original patient data. The arguments ‘Time’, ‘Events’ and ‘CensoringOne’
should be specified appropriately, similarly to fit_tte_data or
event_prediction. A series of arguments governing the appearance of the
plot are then available; see the individual help file for plot_km_fit
for more details.
The plot displays the Kaplan Meier curve, the fitted event curve and
the specified confidence interval. If a fit_KM input is provided, no
confidence interval will be displayed as it is not available.
The variance function for the fitted trajectory is calculated from
the covariance matrix and parameter estimates using the delta method.
The interval is then estimated using moment-fitted beta distributions
(mean and variance). These intervals are asymmetric and have better
coverage and properties than using a normal approximation, particularly
where the variance is large or for estimates close to 0 or 1.
In general, for event prediction to be effective, it is necessary for
a close fit between the observed data and the fitted curve. This is
firstly due to the obvious need to model closely the behaviour of
patients for times that have already been observed, but secondly also
because fitting the observed data well is necessary in order to have a
plausible chance of extrapolating to future times. The key exception to
this principle is where overfitting occurs, but this ought not to be
relevant for distributions with only one or two parameters.
3.10 Example
We first create a new example data set with piecewise recruitment
using the simulate_trials function in gestate:
# Create the full length trial showing what 'actually' happens
trial_long <- simulate_trials(active_ecurve=Weibull(50,0.8),control_ecurve=Weibull(50,0.8),rcurve=recruit,fix_events=200,iterations=1,seed=12345,detailed_output=TRUE)
# Create a shortened version with data for use in the event prediction
trial_short <- set_assess_time(data=trial_long,time=10,detailed_output = FALSE)
# Create the trajectory of events
maxtime <- max(ceiling(trial_long[,"Assess"]))
events <- rep(NA,maxtime)
for (i in 1:maxtime){
events[i] <- sum(1-set_assess_time(trial_long,i)[,"Censored"])
}
The ‘trial_short’ object contains the patient data observable at the
time of event prediction (10 months). Note that a ‘1’ in the Censored
column denotes a censoring, so the censoringOne argument needs to be set
to TRUE, and that the unit of time is in months, so the units argument
must be “Months”. For real patient data it is more usual to have units
of days, which is the default.
‘recruit’ (see section 2.3) contains the observed and predicted
patient recruitment as an RCurve object. ‘events’ contains the numbers
of events observed at each time point (both before and after prediction)
and can be used to assess performance. At the time of event prediction,
10, there are 47 observed events and the number at risk is 500-47= 453;
these will be the conditioning parameters. We assume that there is no
dropout censoring.
We can now perform a simple event prediction using Weibull curve type
selection:
predictions <- event_prediction(data=trial_short, Event="Censored", censoringOne=TRUE, type="Weibull", rcurve=recruit,max_time=60, cond_Events=49, cond_NatRisk=451, cond_Time=10, units="Months")
We can then plot the conditional trajectory, with prediction
intervals, alongside the events that were (or will be) observed at each
time point:
# Plot observed events and conditional predictions
plot_ep(predictions,trajectory="conditional",which_PI="conditional",max_time=40,observed=events,target=200,max_E=200)
The plot shows a reasonable agreement between the observed trajectory
and the estimated conditional trajectory, which predicts 200 events
would be reached by month 29. However, the 95% prediction interval is
wide (particularly on the lower bound), and the prediction interval for
the time to reach 200 events is 22-55 months. In practice therefore,
event prediction at this time point is not that useful for estimating a
trial end date.
We can now revisit the event prediction at two later time points (14
and 18 months), and compare the prediction accuracies:
trial_less_short <- set_assess_time(data=trial_long,time=14,detailed_output = FALSE)
predictions2 <- event_prediction(data=trial_less_short, Event="Censored",censoringOne=TRUE, type="Weibull", rcurve=recruit,max_time=60, cond_Events=101,cond_NatRisk=399,cond_Time=14, units="Months")
trial_mature <- set_assess_time(data=trial_long,time=18,detailed_output = FALSE)
predictions3 <- event_prediction(data=trial_mature, Event="Censored",censoringOne=TRUE, type="Weibull", rcurve=recruit,max_time=60, cond_Events=148,cond_NatRisk=352,cond_Time=18, units="Months")
Only 4 months later, the 95% prediction interval is down to 23-34
months, with an estimate of 27 months, while by 18 months, a tight 95%
prediction interval of 23-27 months is possible around an estimate of 25
months. The tightening of the prediction interval over time is partly
due to a reduction in the number of events remaining to be predicted,
but also the increasing precision with which the parameters of the
parametric Kaplan Meier curve fit. As a rule of thumb, 100 observed
events appear to be required to give reasonable precision of estimation
of Weibull parameters, but the greater the length of extrapolation
required, the more precise the estimates have to be to give useful
accuracy.
The conditional prediction interval can therefore demonstrate that in
this case where a target of 200 events is desired, event prediction is
probably too inaccurate to be of use at 10 months, somewhat indicative
at 14 months, and fairly accurate by 18 months. As the trial has
‘progressed’, the estimate of the time of finishing has dropped from 29
months to 27 months and then again to 25 months. Depending on tolerance
for risk, it may be more useful to use a smaller prediction interval
e.g. 80 or 90%.
It should be noted that all of these prediction intervals are
dependent upon the underlying distribution being Weibull, and that the
same distribution applies to all patients, irrespective of e.g. the time
of recruitment. They also depend on the recruitment distribution being
fixed, i.e. containing no variability - this may be unreasonable for
predictions at a very early time point. These limitations will tend to
lead to the interval being narrower than it ought. Conversely, the
prediction intervals do not account for knowledge obtained from prior
trials or beliefs from the trial design stage; if there are strong prior
expectations aligned with the observed data then the prediction
intervals may be wider than they need be.
Finally, we can check the quality of the curve fitting and the degree
of uncertainty around it.
From visual inspection, in all three cases there is reasonable agreement
between the KM and fitted curves (albeit there is not much data at 10
months). This ensures their predictions are likely to at least be
plausible. The width of the confidence interval using 10 months data is
roughly double that for 18 months data. It is clear that the 10-month
time point gives very little confidence about the Survival function when
extrapolating. However, the 14 and 18 month time points provide some
information about likely future behaviour, so do appear to be
potentially useful for event prediction as long as the assumption of a
Weibull distribution is reasonable.
4 Prior Data Integration Approach
4.1 Introduction
When planning trials, we generally assume particular event,
recruitment and dropout rates. In the early stages of a trial, when
there are few events, such assumptions are typically the best source of
information about the future trajectory in a trial, and it is desirable
to integrate these assumptions with the data that is available so far.
Without such assumptions, sufficiently accurate regression to give
useful predictions is commonly not possible until approximately 100
events have been observed. For many trials, this may not be feasible or
useful as either they may have low target event numbers, or they may
need to make early predictions.
To smooth the transition from using planning assumptions to observed
data, while allowing meaningful predictions to be made at early stages
in the trial, an approach to Weibull regression that integrates prior
data has been added in gestate v1.6.0. This allows prior assumptions to
be included into the Weibull regression and subsequent event prediction
via a weighted prior patient data set.
Where an existing patient-level data set is available, this may be
used directly. Where it is not, and only a summary distribution is
known, the first step is to create an appropriate data set with the
required characteristics.
4.1 Creation of Prior Patient-Level Data: create_tte_prior
The create_tte_prior data set takes as inputs a distribution
(specified as a Curve object via ‘curve’), the length of time over which
prior events were observed (‘duration’), and the number of these prior
events (‘events’). From these, it creates patient-level data with the
requisite properties.
Below is an example creating such a data set the peforming Weibull
regression to show the data meets the desired distribution
example_data_prior <- create_tte_prior(curve=example_distribution,duration=100,events=100)
head(example_data_prior)
#> Time Event
#> [1,] 0.08890685 1
#> [2,] 0.30120362 1
#> [3,] 0.56612077 1
#> [4,] 0.86280292 1
#> [5,] 1.18441735 1
#> [6,] 1.52720433 1
tail(example_data_prior)
#> Time Event
#> [154,] 98 0
#> [155,] 98 0
#> [156,] 98 0
#> [157,] 98 0
#> [158,] 98 0
#> [159,] 98 0
fit3 <- fit_tte_data(data=example_data_prior,Time="Time",type="Weibull")
fit3$Parameters
#> Alpha Beta
#> 99.1679412 0.8070656
One thing to be careful about when creating such a data set is that
the created data fits the required distribution without any noise.
Consequently,if the prior is used in conjunction with an observed set of
data that is very similar, it could lead to over-precision in the
fitting. To avoid this, it is recommended to either not use a Weibull
(or exponential) distribution when creating the prior (so the prior data
will not perfectly fit the shape of a Weibull curve), or to use
considerable down-weighting of the prior when performing the fit.
A few output customisation options are available within the function:
The ‘Event’ and ‘Time’ arguments allow the default column names to be
changed, and the ‘censoringOne’ parameter allows a change from the
default of censoring=0 (i.e. events = 1).
4.2 Weibull Curve Fitting with Weighted Prior Data:
fit_tte_data_prior
The fit_tte_data_prior function performs Weibull curve fitting using
prior patient-level data. Currently, it does not support exponential or
log-normal curve fitting.
This function performs maximum likelihood estimation on the pooled
observed and prior data, essentially implementing Weibull regression
with optional weights applied to the prior data.
The function is analogous to the fit_tte_data function with many of
the same arguments, and similar outputs. In addition to the arguments of
fit_tte_data, ‘priordata’, ‘priorTime’, ‘priorEvent’,
‘priorcensoringOne’ and ‘priorweight’ are also added. ‘priordata’ is a
required argument that specifies the data set containing the prior
patient level data. ‘priorTime’, ‘priorEvent’ are analogous to ‘Time’
and ‘Event’ for specifying column names, but for the prior data set. By
default they assume whatever values have been supplied (or used by
default) in the ‘Time’ and ‘Event’ arguments, so only need to be
specified if column names differ between data sets. Similarly,
‘priorcensoringOne’ also defaults to the value of ‘censoringOne’, and
only needs to be set if there is a difference between data sets in the
definition of Events / Censorings.
Finally, ‘priorweight’ specifies the weighting to be applied to the
prior data set. This typically assumes a value between 0 and 1, with 0
meaning no weight at all is given to the prior data (i.e. only observed
data is used), and 1 meaning equal weight is applied (i.e. essentially
the prior and observed data is exchangeable). Values between 0 and 1
imply a down weighting of the prior data, so that each prior patient
only counts as a fraction of a patient in the observed data. By default,
‘priorweight’ assumes the value 1, i.e. exchangeability with observed
data.
In general it is recommended to down weight historical data so that
the number of events represents both a low percentage of the final
target number of events and a relatively low absolute number (< 100).
Doing so ensures that when there are few observed events, reasonable
event prediction is possible. Conversely, as the data maturity
increases, the observed data will outweigh the prior data, ensuring that
differences are correctly accounted for, and over-precision of estimates
is avoided.
4.3 Weibull Event Prediction with Weighted Prior Data:
event_prediction_prior
The event_prediction_prior function performs Weibull event prediction
using optionally-weighted prior patient-level data. Currently, it does
not support exponential or log-normal curve fitting. More details on the
curve fitting can be found above in section 4.2.
The event_prediction_prior function is analogous, and very similar,
to the event_prediction function (see section 3 for full details), but
with the additional ability to integrate prior data. As with the
fit_tte_data_prior curve fitting function, there are several new
arguments ‘priordata’, ‘priorTime’, ‘priorEvent’, ‘priorcensoringOne’
and ‘priorweight’.
‘priordata’ contains the name of the prior data set. ‘priorTime’,
‘priorEvent’, ‘priorcensoringOne’ default to ‘Time’, ‘Event’ and
‘censoringOne’ values respectively, but may be independently set if
there are discrepancies in the naming conventions between the ‘data’ and
‘priordata’ sets.
‘priorweight’ specifies the weighting to be applied to the prior data
set. This typically assumes a value between 0 and 1, with 0 meaning no
weight at all is given to the prior data (i.e. only observed data is
used), and 1 meaning equal weight is applied (i.e. essentially the prior
and observed data is exchangeable). Values between 0 and 1 imply a down
weighting of the prior data, so that each prior patient only counts as a
fraction of a patient in the observed data. By default, ‘priorweight’
assumes the value 1, i.e. exchangeability with observed data.
In general it is recommended to down weight historical data so that
the number of events represents both a low percentage of the final
target number of events and a relatively low absolute number (< 100).
Doing so ensures that when there are few observed events, reasonable
event prediction is possible. Conversely, as the data maturity
increases, the observed data will outweigh the prior data, ensuring that
differences are correctly accounted for, and over-precision of estimates
is avoided.
4.4 Weibull Event Prediction with Weighted Prior Data Example
We start with the same example as in 3.10, where events are drawn
from a Weibull(50,0.8) distribution. However, we now also assume that we
have information from a previous trial, which had 200 events over 36
months with a fitted Weibull(60,0.7) distribution. This historical
information is first turned into a patient-level set of data:
prior <- create_tte_prior(curve=Weibull(50,0.8),duration=36,events=200, Event="Censored", censoringOne=TRUE)
Event prediction can then be performed using the same inputs as for
the example in 3.10, but also including the prior data. Since the
previous trial data may not be exactly comparable, and since we don’t
want historical data to overpower that from the current trial, we
downweight the historical data using a factor of 0.25, to give an
effective number of events of 50:
predictions_prior <- event_prediction_prior(data=trial_short, priordata=prior, priorweight=0.25, Event="Censored", censoringOne=TRUE, type="Weibull", rcurve=recruit, max_time=60, cond_Events=49, cond_NatRisk=451, cond_Time=10, units="Months")
The trajectory predictions for the prior data approach can then be
plotted, along with as a comparison the original one that uses observed
data only:
# Plot observed events and conditional predictions
plot_ep(predictions,trajectory="conditional",which_PI="conditional",max_time=40,observed=events,target=200,max_E=200,main="Kaplan Meier Curve Fit Plot: 10 Months, No Prior Information")
plot_ep(predictions_prior,trajectory="conditional",which_PI="conditional",max_time=40,observed=events,target=200,max_E=200,main="Kaplan Meier Curve Fit Plot: 10 Months, Prior Information")
Including the prior data, even with relatively low weight, greatly
increases the precision of the predictions, although still leaving an
interval width of approximately a year. This highlights an inherent
issue with making event predictions with immature data, even with prior
information about the underlying distribution of events, namely that
event occurrence is a random process and the further you are from trial
completion, the greater the wider the distribution of times to
completion.
A second issue that arises specifically in the case of using prior
information is that the more relative weight is put onto the prior
information, the greater the bias caused if the prior information does
not completely align with the ‘true’ distribution for the trial
requiring event prediction. Worse, as more weight is put on it, so the
confidence in prediction increases, so the coverage of the prediction
intervals becomes increasingly poor. It is important therefore to not
put excessive weight on prior information, and when setting weightings,
to account for not only the size of the previous trial(s), but also the
degree of their similarity to the current trial. This needs to account
not just for similarity of patient population, but also differences in
standard of care that can arise over even relatively short periods of
time in certain clinical conditions.