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Introduction: methodology and implementation

2026-06-26

Source: vignettes/Intro.qmd

The CompCausal package provides tools for estimating causal effects in comprehensive cohort studies in the presence of unmeasured confounding and missing outcomes. The primary target of inference is the comprehensive cohort causal effect (CCCE), defined as the average causal effect in the entire study cohort.

This vignette is organized into two sections:

  1. Methodology: an overview of the comprehensive cohort study design, causal estimands, identification assumptions, and estimation procedures.

  2. Implementation: practical guidance on using the package functions to estimate the CCCE under user-specified sensitivity parameters.

Methodology

Comprehensive Cohort Study design

The Comprehensive Cohort Study (CCS) design is a nested study design in which eligible individuals may choose to participate either in a randomized controlled trial (RCT) or in a parallel observational study (OBS). Participants in the RCT are randomly assigned to treatment, whereas participants in the OBS select their preferred treatment from the same treatment options available in the RCT. All participants are followed over time, and baseline covariates, treatment assignments, and outcomes are collected for the entire cohort.

Causal estimands

Let XX be baseline covariates, RR randomization consent indicator (11 for RCT, 00 for OBS), TT binary treatment indicator, YY as outcome and MM the missing outcome indicator (M=1M=1 if YY is observed, M=0M=0 if YY is missing). Let O=(X,R,T,Y)O = (X, R, T, Y) be the full data without outcome missingness, and Õ=(X,R,T,M,Y:M=1)\widetilde{O} = (X,R,T,M,Y:M=1) the observed data. Let PP and P̃\widetilde{P} be the distribution of OO and Õ\widetilde{O}.

We further introduce the following notations:

  • F̃t,r(y|x)=P(Y≤y∣T=t,R=r,X=x,M=1)\widetilde{F}_{t, r}(y|x)=P(Y\leq y\mid T=t, R=r, X=x, M=1)
  • μ̃t,r(h(Y);x)=E[h(Y)∣T=t,R=r,X=x,M=1]\widetilde{\mu}_{t,r}(h(Y); x)=E[h(Y) \mid T=t, R=r, X=x, M=1]
  • πt,r(x)=P(T=t∣R=r,X=x)\pi_{t,r}(x)=P(T = t \mid R=r, X=x)
  • gr(x)=P(R=r∣X=x)g_r(x)=P(R = r \mid X=x)
  • ηm(x,r,t)=P(M=m∣X=x,R=r,T=t)\eta_m(x, r, t)=P(M =m \mid X=x, R=r, T=t)

Our causal estimand of interest is the comprehensive cohort causal effect (CCCE), E[Y(1)]−E[Y(0)]E[Y(1)]-E[Y(0)], which represents the average causal effect in the entire study population. Estimation of the CCCE is based on inference for the marginal potential outcome mean E[Y(t)]E[Y(t)], for t∈{0,1}t\in\{0, 1\}.

Assumptions

  • Consistency: Y(t)=YY(t) = Y if T=tT=t, for t=0,1t=0,1.
  • Positivity: 0<P(T=1∣R=1)<10 < P(T=1 \mid R=1) < 1 and 0<P(T=1∣R=0,X=x)<10 < P(T=1 \mid R=0, X=x) < 1 for all xx in the support of XX.
  • Randomization in RCT: T⟂{Y(t),X}∣R=1T \perp \{ Y(t), X \} \mid R=1, for t=0,1t = 0, 1.
  • Sensitivity Model for Unmeasured Confounding in the OBS: dF(y(t)∣T=1−t,R=0,X)=dF(y(t)∣T=t,R=0,X)×exp⁡{γtst(y(t))}μt,0(exp⁡{γtst(Y(t))};X)dF(y(t) \mid T=1-t,R=0,X) = dF(y(t) \mid T=t,R=0,X) \times \frac{ \exp\{ \gamma_t s_t(y(t)) \} }{ \mu_{t, 0} (\exp\{ \gamma_t s_t(Y(t)) \}; X)}
  • Missing-At-Random Assumption for Outcome: M⟂Y∣T,X,RM \perp Y \mid T, X, R.

Under the preceding assumptions, the marginal potential outcome mean E[Y(t)]E[Y(t)] is identified by

ψt(P̃;γt)=E[μ̃t,1(Y;X)|R=1]⏟ψt,1(P̃)P(R=1)+E[μ̃t,0(Y;X)πt,0(X)+μ̃t,0(Yexp⁡{γtst(Y)};X)μ̃t,0(exp⁡{γtst(Y)};X)π1−t,0(X)|R=0]⏟ψt,0(P̃;γt)P(R=0)\begin{aligned} \psi_t(\widetilde{P}; \gamma_t) =& \underbrace{E[ \widetilde{\mu}_{t,1}(Y;X) \big| R=1]}_{\psi_{t,1}(\widetilde{P})} P(R=1)\\ &+ \underbrace{E \bigg[ \widetilde{\mu}_{t,0}(Y;X) \pi_{t,0}(X) +\frac{ \widetilde{\mu}_{t,0}(Y \! \exp\{ \gamma_t s_t(Y)\} ;X) }{\widetilde{\mu}_{t,0}( \exp\{ \gamma_t s_t(Y)\} ;X) } \pi_{1-t,0}(X) \bigg| R=0\bigg]}_{\psi_{t,0}(\widetilde{P}; \gamma_t)} P(R=0) \end{aligned}

ψt,1(P̃)\psi_{t, 1}(\widetilde{P}) and ψt,0(P̃;γt)\psi_{t, 0}(\widetilde{P}; \gamma_t) are the identifications of E[Y(t)|R=1]E[Y(t)|R=1] and E[Y(t)|R=0]E[Y(t)|R=0]. Details of estimations for E[Y(t)|R=1]E[Y(t)|R=1] and E[Y(t)|R=0]E[Y(t)|R=0] can be found in the paper. We define the randomized trial causal effect (RTCE) as RTCE=ψ1,1(P̃)−ψ0,1(P̃)\mbox{RTCE}=\psi_{1, 1}(\widetilde{P})-\psi_{0, 1}(\widetilde{P}); and the patient preference causal effect (PPCE) as PPCE(γ1,γ0)=ψ1,0(P̃;γ1)−ψ0,0(P̃;γ0)\mbox{PPCE}(\gamma_1,\gamma_0) = \psi_{1, 0}(\widetilde{P}; \gamma_1)-\psi_{0, 0}(\widetilde{P}; \gamma_0).

Methods

Estimation of ψt(P̃;γt)\psi_{t}(\widetilde{P}; \gamma_t) proceeds in four steps:

  1. Derive the efficient influence function.
  2. Estimate the nuisance functions that characterize P̃\widetilde{P}.
  3. Construct a one-step estimator using sample splitting and truncation.
  4. Estimate the asymptotic variance and construct 95%95\% confidence intervals.

(1). Derive the efficient influence function for ψt(P̃;γt)\psi_t(\widetilde{P}; \gamma_t), denoted ϕt(P̃;γt)(Õ)\phi_t(\widetilde{P}; \gamma_t)(\widetilde{O})

ϕt(P̃;γt)(Õ)=υt,1(P̃)(Õ)×P(R=1)+υt,0(P̃;γt)(Õ)×P(R=0)−ψt(P̃;γt),\phi_t(\widetilde{P}; \gamma_t)(\widetilde{O}) = \upsilon_{t, 1}(\widetilde{P})(\widetilde{O})\times P(R=1)+\upsilon_{t, 0}(\widetilde{P}; \gamma_t)(\widetilde{O})\times P(R=0) - \psi_t(\widetilde{P}; \gamma_t) \ ,

where the exact form of υt,1(P̃)(Õ)\upsilon_{t, 1}(\widetilde{P})(\widetilde{O}) and υt,0(P̃;γt)(Õ)\upsilon_{t, 0}(\widetilde{P}; \gamma_t)(\widetilde{O}) can be found in the paper.

(2). Estimate P̃\widetilde{P}

To construct a estimator for E[Y(t)]E[Y(t)], we need to estimate πt,r(X)\pi_{t, r}(X), η1(X,r,t)\eta_1(X, r, t) and F̃t,r(y|X)\widetilde{F}_{t, r}(y|X) for r=0,1r=0, 1. The nuisance functions are estimated using methods that achieve sufficiently fast convergence rates to ensure asymptotic normality of the resulting estimator at the n\sqrt{n} rates. Specifically, we apply the generalized additive model (GAM) with a logistic link function (Hastie et al. 1990) for πt,r(X)\pi_{t, r}(X) and η1(X,r,t)\eta_1(X, r, t) for r=0,1r=0, 1. We apply the single index model (Chiang and Huang 2012) for F̃t,r(y|X)\widetilde{F}_{t, r}(y|X) for r=0,1r=0, 1. The conditional expectations μ̃t,0(Yexp⁡(γtst(Y));X)\widetilde{\mu}_{t, 0}\big(Y\exp(\gamma_t s_t( Y)); X\big), μ̃t,0(exp⁡(γtst(Y));X)\widetilde{\mu}_{t, 0}\big(\exp(\gamma_t s_t(Y)); X\big) and μ̃t,r(Y;X)\widetilde{\mu}_{t, r}\big(Y; X\big) are subsequently computed via numerical integration.

(3). Compute the one-step, truncated, split sample estimator

We employ K-fold sample splitting together with tuning-free Huberization procedure (Wang et al. 2021) to truncate the estimated influence-function components: υt,1(P̃̂(−k))(Õi)\upsilon_{t, 1} \left(\widehat{\widetilde{P}}^{(-k)}\right) (\widetilde{O}_i) and υt,0(P̃̂(−k);γt)(Õi)\upsilon_{t, 0}\left(\widehat{\widetilde{P}}^{(-k)} ; \gamma_t\right) (\widetilde{O}_i).

ψ̂t†(γt)=1K∑k=1K{1nk∑i:Si=k[υt,1†(P̃̂(−k))(Õi)×Pn/nk(−k)(R=1)+υt,0†(P̃̂(−k);γt)(Õi)×Pn/nk(−k)(R=0)⏟υt†(P̃̂(−k);γt)(Õi)]}⏟ψ̂t(k)†(γt)\begin{equation*} \widehat{\psi}^\dagger_t(\gamma_t) = \frac{1}{K} \sum_{k=1}^K \underbrace{\left\{\frac{1}{n_k} \sum_{i: S_i=k}\Bigg[ \underbrace{\upsilon_{t, 1}^\dagger \left(\widehat{\widetilde{P}}^{(-k)}\right) (\widetilde{O}_i)\times P_{n/n_k}^{(-k)}(R=1) +\upsilon_{t, 0}^\dagger \left(\widehat{\widetilde{P}}^{(-k)} ; \gamma_t\right) (\widetilde{O}_i)\times P_{n/n_k}^{(-k)}(R=0)}_{\upsilon_t^\dagger\left(\widehat{\widetilde{P}}^{(-k)} ; \gamma_t\right)(\widetilde{O}_i)}\Bigg]\right\}}_{\widehat{\psi}_t^{(k)\dagger}(\gamma_t)} \end{equation*} where SiS_i is the split membership of the iith observation (i.e, Si∈{1,…,K}S_i \in \{1, \ldots, K\}).

(4). Compute variance and 95%95\% confidence interval

The asymptotic variance of ψ̂t†(γt)\widehat{\psi}_t^\dagger(\gamma_t) is estimated by σt†2=1nK∑k=1K{1nk−1∑i:Si=k{υt†(P̃̂(−k);γt)(Õi)−ψ̂t(k)†(γt)}2}\sigma_t^{\dagger 2}=\frac{1}{nK}\sum_{k=1}^K\left\{\frac{1}{n_k-1}\sum_{i: S_i=k}\left\{\upsilon_t^\dagger\left(\widehat{\widetilde{P}}^{(-k)} ; \gamma_t\right)(\widetilde{O}_i)-\widehat{\psi}_t^{(k)\dagger}(\gamma_t)\right\}^2\right\} A two-sided 95%95\% Wald confidence interval for ψt(P̃;γt)\psi_t(\widetilde{P}; \gamma_t) is given by ψ̂t†(γt)±z0.975σt†n\widehat{\psi}^\dagger_t(\gamma_t)\pm z_{0.975}\frac{\sigma_t^\dagger}{\sqrt{n}}.

Implementation

We now illustrate the use of CompCausal for estimating the CCCE, RTCE, and PPCE using a simulated dataset included with the package.

Data

The package includes a simulated dataset based on the TOIB study (Underwood et al. 2008). The TOIB study is a comprehensive cohort study on the effect of topical versus oral non-steroidal anti-inflammatory drugs (NSAIDs) in managing knee pain among older adults with chronic knee pain. In the analysis, the outcome of interest is Western Ontario and McMaster Universities Osteoarthritis Index (WOMAC) pain score (0−1000-100) at 12 months. Baseline covariates include age, baseline WOMAC pain score, expected pain one year later, and chronic pain grade.

library(CompCausal)
## load data
data(ccohort)
## data structure
str(ccohort)
'data.frame':   563 obs. of  8 variables:
 $ Y           : num  26.7 72.1 55.3 NA 46 ...
 $ M           : num  1 1 1 0 1 1 1 0 1 1 ...
 $ R           : int  1 0 1 1 0 1 1 1 1 1 ...
 $ t           : num  1 1 1 0 0 0 0 1 0 0 ...
 $ age         : num  54 63 70 74 55 78 76 61 58 52 ...
 $ womac_bq    : num  22.2 100 25.8 30.4 27.4 31.6 45.2 73 39.4 39.4 ...
 $ expectationb: Factor w/ 3 levels "Much/A little worse",..: 3 2 1 2 3 1 1 3 1 1 ...
 $ ChronicPainb: Factor w/ 2 levels "1-2","3-4": 1 2 1 2 1 1 1 1 2 1 ...

The dataset has 563 observations and 8 variables, which include:

  • Y: WOMAC pain score at 12 months, continuous variable ranging from 0 to 100. NA indicate missing outcome.
  • M: outcome missingness indicator, binary variable. 11 if YY is observed, 00 if YY is missing.
  • R: randomization consent indicator, binary variable. 11 for RCT, 00 for OBS.
  • t: treatment indicator, binary variable. 11 for topical NSAIDs treatment group, 00 for oral NSAIDs treatment group.
  • age: age, continuous variable.
  • womac_bq: WOMAC pain score at baseline, continuous variable ranging from 0 to 100. No missing value.
  • expectationb: expected pain a year from now, categorical variable. Three levels: much/a little worse, about the same, a little/much better/free of pain.
  • ChronicPainb: chronic pain grade, categorical variable. Two levels: 1-2, 3-4.

Application

The primary function is est_psi(), which returns point estimates, estimated variance and 95%95\% Wald confidence interval for ψt(P̃;γt)\psi_t(\widetilde{P}; \gamma_t), ψt,0(P̃;γt)\psi_{t, 0}(\widetilde{P}; \gamma_t) and ψt,1(P̃)\psi_{t, 1}(\widetilde{P}) under one or more pre-specified γt\gamma_t.

Users need to input data and specify several parameters:

  • Data: Y, M, R, t, X (baseline covariates as a data frame)
  • Estimand of interest: trt=1 if estimating E[Y(1)]E[Y(1)], E[Y(1)|R=0]E[Y(1)|R=0] and E[Y(1)|R=1]E[Y(1)|R=1]; trt=0 if estimating E[Y(0)]E[Y(0)], E[Y(0)|R=0]E[Y(0)|R=0] and E[Y(0)|R=1]E[Y(0)|R=1]
  • Sensitivity parameters: gamma, a vector of γt\gamma_t value.
  • Single index model settings (Redd et al. 2025): kernel, kernel smoothing, choices of Gaussian and Epanechnikov kernel; single_index_method, types of constraints in estimation, choices of setting first coefficient to 1, norm of coefficients to 1, and bandwidth to 1; method, optimization methods, default to optim; use_mave, whether to apply MAVE (Xia et al. 2002; Wang and Xia 2008) or cumulative sliced inverse regression method (Zhu et al. 2010) to estimate initial value of the coefficients, default to TRUE.
  • Truncation methods: simple_trunc=TRUE to apply quantile truncation of 1πt,r(X)\frac{1}{\pi_{t, r}(X)} and 1ηm(X,r,t)\frac{1}{\eta_m(X, r, t)}, simple_trunc=FALSE to apply tuning-free Huberization procedure (Wang et al. 2021) to influence functions. If simple_trunc=TRUE, specify quantile truncation by setting quant from 0 to 1.
  • K-fold sample splitting: fold=K, an integer. seed, set.seed(seed).

Here is an example of inferences for ψ1(P̃;γ1)\psi_1(\widetilde{P}; \gamma_1), ψ1,0(P̃;γ1)\psi_{1, 0}(\widetilde{P}; \gamma_1) and ψ1,1(P̃)\psi_{1, 1}(\widetilde{P}) under γ1=0,0.5\gamma_1=0, 0.5, using 5-fold sample splitting, influence function truncation procedure and specific single index model settings.

## data adaptive truncation
out <- with(ccohort, {
  est_psi(Y, M, R, X = data.frame(age, womac_bq, expectationb, ChronicPainb), 
          t, trt = 1, gamma = c(0, 0.5), fold = 5, seed = 1, IF_output = FALSE,
          simple_trunc = FALSE, quant = NULL, kernel="dnorm", 
          single_index_method="norm1coef", method="optim")
})

Results are organized into a table:

## organize result into data frame
res_out <- data.frame(est=c(out$est_trunc, out$est_trunc_R1[1], out$est_trunc_R0), 
                      var=c(out$var_trunc, out$var_trunc_R1[1], out$var_trunc_R0), 
                      lowerCI=c(out$lowerCI_trunc, out$lowerCI_trunc_R1[1], out$lowerCI_trunc_R0), 
                      upperCI=c(out$upperCI_trunc, out$upperCI_trunc_R1[1], out$upperCI_trunc_R0), 
                      gamma=c(0, 0.5, NA, 0, 0.5), type=c("CCCE", "CCCE", "RTCE", "PPCE", "PPCE"))
res_out2 <- as.matrix(round(res_out[, 1:4], 2))
rownames(res_out2) <- c("$\\psi_1(\\widetilde{P}; 0)$", "$\\psi_1(\\widetilde{P}; 0.5)$", "$\\psi_{1,1}(\\widetilde{P})$",
                        "$\\psi_{1,0}(\\widetilde{P}; 0)$", "$\\psi_{1,0}(\\widetilde{P}; 0.5)$")
## present result
knitr::kable(res_out2, col.names=c("Estimates", "Var", "Lower CI", "Upper CI"))
Estimates Var Lower CI Upper CI
ψ1(P̃;0)\psi_1(\widetilde{P}; 0) 40.12 2.16 37.24 43.00
ψ1(P̃;0.5)\psi_1(\widetilde{P}; 0.5) 40.44 2.19 37.54 43.34
ψ1,1(P̃)\psi_{1,1}(\widetilde{P}) 38.50 4.72 34.24 42.76
ψ1,0(P̃;0)\psi_{1,0}(\widetilde{P}; 0) 41.70 3.95 37.80 45.60
ψ1,0(P̃;0.5)\psi_{1,0}(\widetilde{P}; 0.5) 42.33 4.04 38.39 46.26

To estimate treatment effects such as ψ1(P̃;γ1)−ψ0(P̃;γ0)\psi_1(\widetilde{P}; \gamma_1)-\psi_0(\widetilde{P}; \gamma_0), the estimated influence functions must be retained so that the covariance between the two estimators can be properly accounted for when computing standard errors. Note that to make sure the influence functions align for each individuals under t=1t=1 and t=0t=0, it is advised that seed is set to the same value under t=1t=1 and t=0t=0.

## t=1
out_t1 <- with(ccohort, {
  est_psi(Y, M, R, X = data.frame(age, womac_bq, expectationb, ChronicPainb), 
          t, trt = 1, gamma = c(0, 0.5), fold = 5, seed = 1, IF_output = TRUE,
          simple_trunc = FALSE, quant = NULL, kernel="dnorm", 
          single_index_method="norm1coef", method="optim")
})
## t=0
out_t0 <- with(ccohort, {
  est_psi(Y, M, R, X = data.frame(age, womac_bq, expectationb, ChronicPainb), 
          t, trt = 0, gamma = c(0, 0.5), fold = 5, seed = 1, IF_output = TRUE,
          simple_trunc = FALSE, quant = NULL, kernel="dnorm", 
          single_index_method="norm1coef", method="optim")
})

Results are organized into a table:

## data frame for results
res_out_diff <- expand.grid(gamma1=c(0, 0.5), gamma0=c(0, 0.5), type=c("CCCE", "PPCE"))
res_out_diff <- rbind(res_out_diff, data.frame(gamma1=NA, gamma0=NA, type="RTCE"))
res_out_diff$est <- 0
res_out_diff$var <- 0

## compute variance
scale_n <- 1 / dim(ccohort)[1]

fold_idx <- split(seq_along(out_t1$fold_index_l), out_t1$fold_index_l)

t1_IF_mat <- t(do.call(rbind, out_t1$IF_trunc))
t1_IF_R0_mat <- t(do.call(rbind, out_t1$IF_trunc_R0))
t1_IF_R1_mat <- t(do.call(rbind, out_t1$IF_trunc_R1))

res_out_diff$est[which(res_out_diff$type=="RTCE")] <- out_t1$est_trunc_R1[1]-out_t0$est_trunc_R1[1]

IF_R1_diff <- t1_IF_R1_mat - out_t0$IF_trunc_R1[[1]]
var_R1_temp <- vapply(fold_idx, function(id) colVars(IF_R1_diff[id, ]), numeric(2))
res_out_diff$var[which(res_out_diff$type=="RTCE")] <- rowMeans(var_R1_temp)[1]*scale_n

for (g_0 in seq_along(c(0, 0.5))) {
  
  IF_diff    <- t1_IF_mat - out_t0$IF_trunc[[g_0]]
  IF_R0_diff <- t1_IF_R0_mat - out_t0$IF_trunc_R0[[g_0]]

  var_temp    <- vapply(fold_idx, function(id) colVars(IF_diff[id, ]),    numeric(2))
  var_R0_temp <- vapply(fold_idx, function(id) colVars(IF_R0_diff[id, ]), numeric(2))

  indx_CCCE <- which(res_out_diff$gamma0==c(0, 0.5)[g_0]&res_out_diff$type=="CCCE")
  res_out_diff$est[indx_CCCE] <- out_t1$est_trunc-out_t0$est_trunc
  res_out_diff$var[indx_CCCE] <- rowMeans(var_temp)*scale_n
  
  indx_PPCE <- which(res_out_diff$gamma0==c(0, 0.5)[g_0]&res_out_diff$type=="PPCE")
  res_out_diff$est[indx_PPCE] <- out_t1$est_trunc_R0-out_t0$est_trunc_R0
  res_out_diff$var[indx_PPCE] <- rowMeans(var_R0_temp)*scale_n

}
## 95% CI
res_out_diff$lowerCI <- res_out_diff$est-qnorm(0.975)*sqrt(res_out_diff$var)
res_out_diff$upperCI <- res_out_diff$est+qnorm(0.975)*sqrt(res_out_diff$var)

## present result
res_out_diff2 <- as.matrix(round(res_out_diff[, 4:7], 2))
rownames(res_out_diff2) <- c("$\\psi_1(\\widetilde{P}; 0)-\\psi_0(\\widetilde{P}; 0)$", "$\\psi_1(\\widetilde{P}; 0.5)-\\psi_0(\\widetilde{P}; 0)$", 
                             "$\\psi_1(\\widetilde{P}; 0)-\\psi_0(\\widetilde{P}; 0.5)$", "$\\psi_1(\\widetilde{P}; 0.5)-\\psi_0(\\widetilde{P}; 0.5)$",
                             "$\\psi_{1,0}(\\widetilde{P}; 0)-\\psi_{0,0}(\\widetilde{P}; 0)$", "$\\psi_{1,0}(\\widetilde{P}; 0.5)-\\psi_{0,0}(\\widetilde{P}; 0)$",
                             "$\\psi_{1,0}(\\widetilde{P}; 0)-\\psi_{0,0}(\\widetilde{P}; 0.5)$", "$\\psi_{1,0}(\\widetilde{P}; 0.5)-\\psi_{0,0}(\\widetilde{P}; 0.5)$", 
                             "$\\psi_{1,1}(\\widetilde{P})-\\psi_{0,1}(\\widetilde{P})$")
knitr::kable(res_out_diff2, col.names=c("Estimates", "Var", "Lower CI", "Upper CI"))
Estimates Var Lower CI Upper CI
ψ1(P̃;0)−ψ0(P̃;0)\psi_1(\widetilde{P}; 0)-\psi_0(\widetilde{P}; 0) -0.69 4.14 -4.68 3.30
ψ1(P̃;0.5)−ψ0(P̃;0)\psi_1(\widetilde{P}; 0.5)-\psi_0(\widetilde{P}; 0) -0.88 4.16 -4.88 3.12
ψ1(P̃;0)−ψ0(P̃;0.5)\psi_1(\widetilde{P}; 0)-\psi_0(\widetilde{P}; 0.5) -0.69 4.15 -4.68 3.30
ψ1(P̃;0.5)−ψ0(P̃;0.5)\psi_1(\widetilde{P}; 0.5)-\psi_0(\widetilde{P}; 0.5) -0.88 4.17 -4.88 3.12
ψ1,0(P̃;0)−ψ0,0(P̃;0)\psi_{1,0}(\widetilde{P}; 0)-\psi_{0,0}(\widetilde{P}; 0) -1.45 8.40 -7.13 4.23
ψ1,0(P̃;0.5)−ψ0,0(P̃;0)\psi_{1,0}(\widetilde{P}; 0.5)-\psi_{0,0}(\widetilde{P}; 0) -1.82 8.45 -7.52 3.88
ψ1,0(P̃;0)−ψ0,0(P̃;0.5)\psi_{1,0}(\widetilde{P}; 0)-\psi_{0,0}(\widetilde{P}; 0.5) -1.45 8.43 -7.14 4.24
ψ1,0(P̃;0.5)−ψ0,0(P̃;0.5)\psi_{1,0}(\widetilde{P}; 0.5)-\psi_{0,0}(\widetilde{P}; 0.5) -1.82 8.49 -7.53 3.89
ψ1,1(P̃)−ψ0,1(P̃)\psi_{1,1}(\widetilde{P})-\psi_{0,1}(\widetilde{P}) 0.22 8.13 -5.37 5.81

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