psc-vignette
Richard Jackson
2025-03-31
Introduction
The psc.R package implements the methods for applying Personalised
Synthetic Controls, which allows for patients receiving some
experimental treatment to be compared against a model which predicts
their reponse to some control. This is a form of causal inference which
differes from other approaches in that
Data are only required on a single treatment - all counterfactual
evidence is supplied by a parametric model
Causal inference, in theory at least, is estimated at a patient level
- as opposed to estimating average effects over a population
The causal estimand obtained is the Average Treatment Effect of the
Treated (ATT) which differs from the Average Treatment Effect (ATE)
obtained in other settings and addresses the question of whether
treatments are effective in the population of patients who are treated.
This estimand then targets efficacy over effectivness.
In its basic form, this method creates a likelihood to compare a
cohort of data to a parametric model. See (X) for disucssion on it’s use
as a causal inference tool. To use this package, two basic peices of
information are required, a dataset and a model against which they can
be compared.
In this vignette, we will detail how the psc.r package is constructed
and give some examples for it’s application in practice.
Methodology
The pscfit function compares a dataset (‘DC’) against a
parametric model. This is done by selecting a likelihood which is
identified by the type of CFM that is supplied. At present, two types of
model are supported, a flexible parmaeteric survival model of type
‘flexsurvreg’ and a geleneralised linear model of type ‘glm’.
Where the CFM is of type ‘flexsurvreg’ the likeihood supplied is of
the form:
\[L(D∣\Lambda,\Gamma_i)=\prod_{i=1}^{n}
f(t_i∣\Lambda,\Gamma_i)^{c_i}
S(t_i∣\Gamma,\Lambda_i)^{(1−c_i)}\]
Where \(\Gamma\) defines the
cumulative baseline hazard function, \(\Lambda\) is the linear predictor and \(t\) and \(c\) are the event time and indicator
variables.
Where the CFM is of the type ‘glm’ the likelihood supplied is of the
form:
\[L(x∣\Gamma_i) = \prod_{i=1}^{n} b (x∣
\Gamma_i )\exp\{\Gamma_i t(x)−
c(\Gamma_i)\}\]
Where \(b(.)\), \(t(.)\) and \(c(.)\) represent the functions of the
exponential family. In both cases, \(\Gamma\) is defiend as:
\[ \Gamma_i = \gamma x_i+\beta
\]
Where \(\gamma\) are the model
coefficients supplied by the CFM and \(\beta\) is the parameter set to measure the
difference between the CFM and the DC.
Estimation is performed using a Bayesian MCMC procedure. Prior
distributions for \(\Gamma\) (&
\(\Lambda\)) are derived directly from
the model coefficients (mean and variance covariance matrix) or the CFM.
A bespoke MCMC routine is performed to estimate \(\beta\). Please see ‘?mcmc’ for more
detials.
For the standard example where the DC contains information from only
a single treatment, trt need not be specified. Where comparisons between
the CFM and multiple treatments are require, a covariate of treamtne
allocations must be specified sperately (using the ‘trt’ option).
Package Structure
The main function for using applying Personal Synthetic Controls is
the pscfit() function which has two inputs, a Counter-Factual Model
(CFM) and a data cohort (DC). Further arguments include
- nsim which sets the number of MCMC iterations (defaults to
5000)
- ‘id’ if the user wishes to restrict estimation to a sub-set (or
individual) within the DC
- ‘trt’ to be used as an initial identifier if mulitple treatment
comparisons are to be made (please see the Mulitple Treatment Comparison
below)
psc object
The output of the “pscfit()” function is an object of class ‘psc’.
This class contains the following attributes
- A definition of the calss of the model supplied
- A ‘cleaned’ dataset including extracted components of the CFM and
the cleaned DC included in the procedure
- An object defingin the class of model (and therefore the procedure
applied - see above)
- A matrix containing the draws of the posterior distributions
Postestimation functions
basic post estimation functions have been developed to work with the
psc object, namely “print()”, “coef()”, “summary()” and “plot()”. For
the first three of these these provided basic summaries of the efficacy
parameter obtained from the posterior distribution.
Motivating Example
The psc.r package includes as example, a dataset which is derived
from patients with advanced Hepatocellular Carcinoma (aHCC) who have all
received some experimental treatment. The dataset is simply named ‘data’
and is loaded into the enviroment using the “data()” function
#install.packages("psc")
library(psc)
Included is a list of prognostic covariates:
- vi - Vascular Invasion
- age60 - Age - centered at 60
- ecog - ECOG performance Status
- logafp - AFP on the natural log scale
- alb - Albumin
- logcreat - Creatinine on the log scale
- logast - AST on the natuarl log scale
- allmets -Presence of Metastesis
- aet - Ateiology; HBV, HCV or Other
Also included are the following structures
- time - survival time
- cen - censoring indictor
- os - time to be used as a continuous outcome
- event - binary event for use as a binary outcome
- count - count data to be used for a count outcome
Lastly the dataset also inlclude a ‘trt’ variable to be used in the
estimation of multiple treatment comparisons.
We give esamples of how the ‘pscfit()’ function can be used to
comapre data against models with survival outcomes (with a ‘flexsurvreg’
model) along with binary, continuous and count outcomes (with a ‘glm’
model).
Survival Example
For an example with a survival outcome a model must be supplied which
is contructed ont he basis of flexible parametric splines. This is
contructed using the “flexsurvreg” function within the “flexsurv”
package. An example is included within the ‘psc.r’ package names
‘surv.mod’ and is loaded using the ’data()” function:
surv.mod <- psc::surv.mod
surv.mod
#> Call:
#> flexsurvspline(formula = Surv(time, cen) ~ vi/age60 + ecog +
#> allmets + logafp + alb + logcreat + logast + aet, data = pros,
#> k = 3)
#>
#> Estimates:
#> data mean est L95% U95% se exp(est)
#> gamma0 NA -7.57573 -9.90310 -5.24836 1.18746 NA
#> gamma1 NA 2.23506 1.37951 3.09060 0.43651 NA
#> gamma2 NA -0.13758 -0.74322 0.46807 0.30901 NA
#> gamma3 NA 0.26981 -0.72101 1.26063 0.50553 NA
#> gamma4 NA -0.03031 -0.60738 0.54676 0.29443 NA
#> viyes 0.28400 0.32856 0.08489 0.57224 0.12433 1.38897
#> ecog1 0.40400 0.45602 0.23429 0.67776 0.11313 1.57779
#> allmetsyes 0.69400 0.29767 0.03666 0.55869 0.13317 1.34672
#> logafp 5.42436 0.08320 0.05038 0.11602 0.01674 1.08676
#> alb 38.55600 -0.05539 -0.07809 -0.03269 0.01158 0.94612
#> logcreat 4.30929 0.71084 0.26457 1.15712 0.22770 2.03571
#> logast 4.08274 0.35000 0.15915 0.54086 0.09738 1.41907
#> aetHCV 0.21200 -0.52754 -0.86299 -0.19210 0.17115 0.59005
#> aetOther 0.37000 -0.01847 -0.26319 0.22624 0.12486 0.98170
#> vino:age60 0.06400 -0.02312 -0.03479 -0.01144 0.00596 0.97715
#> viyes:age60 -0.37000 0.00716 -0.00790 0.02221 0.00768 1.00718
#> L95% U95%
#> gamma0 NA NA
#> gamma1 NA NA
#> gamma2 NA NA
#> gamma3 NA NA
#> gamma4 NA NA
#> viyes 1.08859 1.77223
#> ecog1 1.26401 1.96946
#> allmetsyes 1.03734 1.74838
#> logafp 1.05167 1.12301
#> alb 0.92488 0.96784
#> logcreat 1.30286 3.18076
#> logast 1.17251 1.71748
#> aetHCV 0.42190 0.82523
#> aetOther 0.76860 1.25388
#> vino:age60 0.96581 0.98862
#> viyes:age60 0.99214 1.02246
#>
#> N = 500, Events: 360, Censored: 140
#> Total time at risk: 5357.664
#> Log-likelihood = -1221.857, df = 16
#> AIC = 2475.714
In this example you can see that this is a model constructed with 3
internal knots and hence 5 parameters to describe the baseline
cumulative hazard function. There are also prognostic covariates which
match with the prognostic covaraites in the data cohort.
To begin it is worth looking at the performance of the model and
looking at how the survival of patietns in the data cohort compare
library(survival)
sfit <- survfit(Surv(data$time,data$cen)~1)
plot(sfit)
Comparing the dataset to the model is then performed using
and we can view the attributes of the psc object that is created
attributes(surv.psc)
#> $names
#> [1] "model.type" "DC_clean" "posterior"
#>
#> $class
#> [1] "psc"
For example to view the matrix contianing the draws of the posterior
distribution we use
surv.post <- surv.psc$posterior
head(surv.post)
#> gamma0 gamma1 gamma2 gamma3 gamma4 viyes ecog1
#> 1 -7.575727 2.235057 -0.1375763 0.26981088 -0.03030718 0.3285628 0.4560236
#> 2 -8.783084 1.154787 -0.7917518 1.17484806 -0.39178557 0.1089544 0.2950701
#> 3 -7.377666 1.527198 -0.5578965 0.67967559 -0.01884522 0.1541632 0.4686588
#> 4 -8.423701 2.596120 0.2640484 -0.37565961 0.28780878 0.3311778 0.6712726
#> 5 -7.333020 2.482688 0.1627321 -0.09504227 0.02781444 0.2422736 0.4361815
#> 6 -6.646425 1.167473 -0.6193367 0.89893162 -0.28631740 0.2041348 0.4247801
#> allmetsyes logafp alb logcreat logast aetHCV aetOther
#> 1 0.2976742 0.08319787 -0.05538982 0.7108430 0.3500019 -0.5275438 -0.018472762
#> 2 0.2598015 0.08193497 -0.04748843 1.0134077 0.4230066 -0.2610741 0.132297786
#> 3 0.3527148 0.09998786 -0.04470036 0.6041287 0.3559206 -0.7411918 -0.018722686
#> 4 0.3453000 0.06144966 -0.05094966 0.9471375 0.2623956 -0.4036805 -0.178470218
#> 5 0.3051844 0.11317526 -0.04498086 0.4847662 0.3860891 -0.6284135 0.169416154
#> 6 0.3496431 0.08953432 -0.05072497 0.5772301 0.3629893 -0.4255817 -0.007269523
#> vino:age60 viyes:age60 beta DIC
#> 1 -0.02311823 0.0071560470 0.3784481 NA
#> 2 -0.02358952 -0.0054254177 0.3784481 283.1303
#> 3 -0.02230334 0.0100966750 0.3784481 281.0624
#> 4 -0.01198174 0.0070481039 0.3784481 277.9096
#> 5 -0.02955431 0.0074607266 0.4613659 284.6571
#> 6 -0.01832453 0.0005143405 0.2797051 279.3641
Inspection will show that there is a column for each parameter in the
original model as well as ‘beta’ and ‘DIC’ vcolumns which give teh
posterior estiamtes for \(\beta\) and
the Deviance Informaiton Criterion respectively.
We can inspect the poterior distribution using the autocorrelation
function, trace and stardard summary statistics:
Autocorrelation
Trace
plot(surv.post$beta,typ="s")
Summary
summary(surv.post$beta)
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> 0.0363 0.2998 0.3708 0.3679 0.4339 0.7049
Standard ‘summary()’ function wil summarise the model fit
summary(surv.psc)
#> Summary:
#>
#> 100 observations selected from the data cohort for comparison
#> CFM of type flexsurvreg identified
#> linear predictor succesfully obtained with median:
#> trt: 3.15
#> Average expected response:
#> trt: 9.1
#> Average observed response: 6.366
#>
#> Counterfactual Model (CFM):
#> A model of class 'flexsurvreg'
#> Fit with 3 internal knots
#>
#> Formula:
#> Surv(time, cen) ~ vi/age60 + ecog + allmets + logafp + alb +
#> logcreat + logast + aet
#> <environment: 0x1238cc7f0>
#>
#> Call:
#> CFM model + beta
#>
#> Coefficients:
#> median 2.5% 97.5% Pr(x<0) Pr(x>0)
#> beta 0.3708 0.1702 0.5552 0.0000 1.0000
#> DIC 280.6209 273.5226 291.8364 NA NA
To visualise the original model and the fit of the data, the plot
function has been developed
GLM
The “pscfit()” object uses class of the model supplied to derive the
likelihoiod and estimation procedure that isde required. In this
example, the “enrichwith” package is utilised to extract from the model
the parameters of the exponential family. Important from the attributes
of the GLM are the “family” statements which dictates the form of the
likelihood. For each of the binary, continuous and Count data outcomes
then, the syntax and the DC remains the same and it is the form of the
CFM that dictates the analysis
Binary
Step 1: Load Model
bin.mod <- psc::bin.mod
bin.mod
#>
#> Call: glm(formula = event ~ vi/age60 + ecog + allmets + logafp + alb +
#> logcreat, family = "binomial", data = pros)
#>
#> Coefficients:
#> (Intercept) viyes ecog1 allmetsyes logafp alb
#> -1.65694 0.80007 0.38279 0.40733 0.12941 -0.04993
#> logcreat vino:age60 viyes:age60
#> 0.76346 -0.01901 0.01380
#>
#> Degrees of Freedom: 570 Total (i.e. Null); 562 Residual
#> (17 observations deleted due to missingness)
#> Null Deviance: 683
#> Residual Deviance: 616 AIC: 634
Step 2: Fit psc object
Step 3: Review summary statistics
summary(psc.bin)
#> Summary:
#>
#> 100 observations selected from the data cohort for comparison
#> CFM of type glm lm identified
#> linear predictor succesfully obtained with median:
#> trt: 1.2
#> Average expected response:
#> trt: 0.769
#> Average observed response: 0.48
#>
#> Counterfactual Model (CFM):
#> A model of class 'GLM'
#> Family: binomial
#> Link: logit
#>
#> Formula:
#> event ~ vi/age60 + ecog + allmets + logafp + alb + logcreat
#>
#> Call:
#> CFM model + beta
#>
#> Coefficients:
#> median 2.5% 97.5% Pr(x<0) Pr(x>0)
#> beta -1.2891 -1.7228 -0.8779 1.0000 0.0000
#> DIC 56.6123 52.7736 63.8423 NA NA
Step 4: Plot Output
Count
Step 1: Load Model
count.mod <- psc::count.mod
count.mod
#>
#> Call: glm(formula = count ~ ecog + logafp + alb + logcreat, family = "poisson",
#> data = data)
#>
#> Coefficients:
#> (Intercept) ecog logafp alb logcreat
#> 1.414244 -0.170884 0.017803 -0.007255 -0.056183
#>
#> Degrees of Freedom: 99 Total (i.e. Null); 95 Residual
#> Null Deviance: 127.2
#> Residual Deviance: 125.1 AIC: 382.1
#data("count.mod")
Step 2: Fit psc object
Step 3: Review summary statistics
summary(psc.count)
#> Summary:
#>
#> 100 observations selected from the data cohort for comparison
#> CFM of type glm lm identified
#> linear predictor succesfully obtained with median:
#> trt: 0.919
#> Average expected response:
#> trt: 2.507
#> Average observed response: 2.49
#>
#> Counterfactual Model (CFM):
#> A model of class 'GLM'
#> Family: poisson
#> Link: log
#>
#> Formula:
#> count ~ ecog + logafp + alb + logcreat
#>
#> Call:
#> CFM model + beta
#>
#> Coefficients:
#> median 2.5% 97.5% Pr(x<0) Pr(x>0)
#> beta -0.00495 -0.14972 0.13618 0.52880 0.47120
#> DIC 13.62680 10.12471 22.41417 NA NA
Step 4: Plot Output
Continuous
Step 1: Load Model
cont.mod <- psc::cont.mod
cont.mod
#>
#> Call: glm(formula = os ~ vi/age60 + ecog + allmets + logafp + alb,
#> data = pros)
#>
#> Coefficients:
#> (Intercept) viyes ecog1 allmetsyes logafp alb
#> 2.16581 -2.57037 -2.97107 -2.13909 -0.37175 0.36754
#> vino:age60 viyes:age60
#> 0.09174 0.02106
#>
#> Degrees of Freedom: 570 Total (i.e. Null); 563 Residual
#> (17 observations deleted due to missingness)
#> Null Deviance: 33570
#> Residual Deviance: 25300 AIC: 3803
Step 2: Fit psc object
Step 3: Review summary statistics
summary(psc.con)
#> Summary:
#>
#> 100 observations selected from the data cohort for comparison
#> CFM of type glm lm identified
#> linear predictor succesfully obtained with median:
#> trt: 10.8
#> Average expected response:
#> trt: 10.8
#> Average observed response: 10.743
#>
#> Counterfactual Model (CFM):
#> A model of class 'GLM'
#> Family: gaussian
#> Link: identity
#>
#> Formula:
#> os ~ vi/age60 + ecog + allmets + logafp + alb
#>
#> Call:
#> CFM model + beta
#>
#> Coefficients:
#> median 2.5% 97.5% Pr(x<0) Pr(x>0)
#> beta 0.1201 -0.2154 0.4408 0.2200 0.7800
#> DIC -6800.6159 -6897.0307 -6671.5675 NA NA
Step 4: Plot Output
Sub group Effects
An attractive feature of Personalised Synthetic Controls is its use
in fitting sub-group effects. Whereas other casual inference tools
typically make ssumptions about population levels of balance and then
further assume that this balance holds at sub-group levels, Personalised
Synthetic Controls differ in that they estimate treatment effects at a
patient level and then average across populations. To estimate sub-group
effects then we need only to restrict estimation over some sub-group of
the population. This can be achived by directly slecting the subgroup
you wish to evaluate
Using an example where we wnat to see if the treatment effect is
consistent by patients with ECOG=0 and ECOG =1
Sub group effects by restricting the population
PSC fit for pateitns with ECOG=0
id1 <- which(data$ecog==0)
sub1 <- pscfit(surv.mod,data,id=id1)
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PSC fit for pateitns with ECOG=1
id2 <- which(data$ecog==1)
sub2 <- pscfit(surv.mod,data,id=id2)
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We can then easily compare the model coefficients
summary(sub1)
#> Summary:
#>
#> 53 observations selected from the data cohort for comparison
#> CFM of type flexsurvreg identified
#> linear predictor succesfully obtained with median:
#> trt: 2.84
#> Average expected response:
#> trt: 12.605
#> Average observed response: 8.135
#>
#> Counterfactual Model (CFM):
#> A model of class 'flexsurvreg'
#> Fit with 3 internal knots
#>
#> Formula:
#> Surv(time, cen) ~ vi/age60 + ecog + allmets + logafp + alb +
#> logcreat + logast + aet
#> <environment: 0x1239ffe78>
#>
#> Call:
#> CFM model + beta
#>
#> Coefficients:
#> median 2.5% 97.5% Pr(x<0) Pr(x>0)
#> beta 2.979e-01 1.685e-03 5.732e-01 2.460e-02 9.754e-01
#> DIC 1.503e+02 1.449e+02 1.593e+02 NA NA
cat("\n")
summary(sub2)
#> Summary:
#>
#> 47 observations selected from the data cohort for comparison
#> CFM of type flexsurvreg identified
#> linear predictor succesfully obtained with median:
#> trt: 3.423
#> Average expected response:
#> trt: 8.099
#> Average observed response: 5.856
#>
#> Counterfactual Model (CFM):
#> A model of class 'flexsurvreg'
#> Fit with 3 internal knots
#>
#> Formula:
#> Surv(time, cen) ~ vi/age60 + ecog + allmets + logafp + alb +
#> logcreat + logast + aet
#> <environment: 0x1120bb6a0>
#>
#> Call:
#> CFM model + beta
#>
#> Coefficients:
#> median 2.5% 97.5% Pr(x<0) Pr(x>0)
#> beta 0.4500 0.1365 0.7375 0.0016 0.9984
#> DIC 100.6344 95.1574 109.6594 NA NA
And look at the plots for each outcome
PSC plot for ECOG=0
PSC plot for ECOG=1
