Processing math: 47%

R Package `CoRpower`:
Power Calculations for
Correlates of Risk

Based on Gilbert, Janes, Huang (2016, Stat Med)

Michal Juraska

September 24???26, 2018

Outline

Correlates of risk power calculations

Without replacement sampling for retrospective case-control or 2-phase designs
- Trichotomous biomarker
  - Dichotomous biomarker
- Continuous biomarker
Bernoulli sampling for prospective case-cohort designs

Set-up and notation

Without replacement sampling

Assume a randomized vaccine vs. placebo/control vaccine efficacy trial
$N=$ number of vaccine recipients observed to be at risk at $\tau$
ncases (ncontrols) = number of observed cases (controls) between τ and τmax, where cases have ΔY=1 and controls have Δ(1−Y)=1
- $n_{cases} + n_{controls} \leq N$
$n^S_{cases}$ ( $n^S_{controls}$ ) $=$ number of observed cases (controls) between $\tau$ and $\tau_{\mathrm{max}}$ with measured $S$ (or $S^{\ast}$ )

- If calculations done at design stage, then $N$ , $n_{cases}$ , $n_{controls}$ , $n^S_{cases}$ , and $n^S_{controls}$ projected numbers

Algorithm for trichotomous biomarker $\, S$

Without replacement sampling

Specify overall VE between τ and τmax
- Protocol-specified design alternative or $\widehat{VE}$
Specify risk0
- Protocol-specified placebo-group endpoint rate or $\widehat{risk}_0$
Select a grid of VElat0 values
- E.g., ranging from $VE$ ( $H_0$ ) to 0 (maximal $H_1$ not allowing harm by vaccination)
Select a grid of VElat1≥VElat0 values
- E.g., $VE^{lat}_1 = VE$

Algorithm for trichotomous biomarker $\, S$

Without replacement sampling

Specify Plat0 and Plat2, then Plat1=1−Plat0−Plat2
- Assuming $risk^{lat}_0(x) = risk_0$ , $VE = VE^{lat}_0 P^{lat}_0 + VE^{lat}_1 P^{lat}_1 + VE^{lat}_2 P^{lat}_2$ yields $VE^{lat}_2$
- If $VE^{lat}_0$ varies from $VE$ to 0, then $VE^{lat}_2$ varies from VE to $VE\, (P^{lat}_0 + P^{lat}_2)/P^{lat}_2$
Specify P0 and P2, then P1=1−P0−P2
- E.g., $P_0 = P^{lat}_0$ and $P_2 = P^{lat}_2$

Algorithm for trichotomous biomarker $\, S$

Without replacement sampling

Approach 1: Specify two of the three (Sens,FP0,FP1) and two of the three (Spec,FN2,FN1)
- E.g., specify $Sens$ and $Spec$ and $FP^0 = FN^2 = 0$
Approach 2: Specify σ2obs and ρ=σ2tr/σ2obs and determine (Sens,Spec,FP0,FP1,FN1,FN2) (subsequent slides)
- Typically, $\sigma^2_{obs} = 1$

Algorithm for trichotomous biomarker $\, S$

Without replacement sampling

Calculation of $(Sens, Spec, FP^0, FP^1, FN^1, FN^2)$ in Approach 2

Assuming the classical measurement error model, where $X^{\ast} \sim \mathsf{N}(0,\rho\sigma^2_{obs})$ , solve $P^{lat}_0 = P(X^{\ast} \leq \theta_0) \quad \textrm{and} \quad P^{lat}_2 = P(X^{\ast} > \theta_2)$ for $\theta_0$ and $\theta_2$

Algorithm for trichotomous biomarker $\, S$

Without replacement sampling

Calculation of $(Sens, Spec, FP^0, FP^1, FN^1, FN^2)$ in Approach 2

Generate B realizations of X∗ and S∗=X∗+e, where e∼N(0,(1−ρ)σ2obs), and X∗ independent of e
- $B = 20,000$ by default

Algorithm for trichotomous biomarker $\, S$

Without replacement sampling

Calculation of $(Sens, Spec, FP^0, FP^1, FN^1, FN^2)$ in Approach 2

Using $\theta_0$ and $\theta_2$ from Step i., define $\begin{align} Spec(\phi_0) &= P(S^{\ast} \leq \phi_0 \mid X^{\ast} \leq \theta_0)\\ FN^1(\phi_0) &= P(S^{\ast} \leq \phi_0 \mid X^{\ast} \in (\theta_0,\theta_2])\\ FN^2(\phi_0) &= P(S^{\ast} \leq \phi_0 \mid X^{\ast} > \theta_2)\\ Sens(\phi_2) &= P(S^{\ast} > \phi_2 \mid X^{\ast} > \theta_2)\\ FP^1(\phi_2) &= P(S^{\ast} > \phi_2 \mid X^{\ast} \in (\theta_0,\theta_2])\\ FP^0(\phi_2) &= P(S^{\ast} > \phi_2 \mid X^{\ast} \leq \theta_0) \end{align}$

Algorithm for trichotomous biomarker $\, S$

Without replacement sampling

Calculation of $(Sens, Spec, FP^0, FP^1, FN^1, FN^2)$ in Approach 2

Estimate $Spec(\phi_0)$ by $\widehat{Spec}(\phi_0) = \frac{\#\{S^{\ast}_b \leq \phi_0, X^{\ast}_b \leq \theta_0\}}{\#\{X^{\ast}_b \leq \theta_0\}}\,,$ etc.

Algorithm for trichotomous biomarker $\, S$

Without replacement sampling

Calculation of $(Sens, Spec, FP^0, FP^1, FN^1, FN^2)$ in Approach 2

Find $\phi_0 = \phi^{\ast}_0$ and $\phi_2 = \phi^{\ast}_2$ that numerically solve $\begin{align} P_0 &= \widehat{Spec}(\phi_0)P^{lat}_0 + \widehat{FN}^1(\phi_0)P^{lat}_1 + \widehat{FN}^2(\phi_0)P^{lat}_2\\ P_2 &= \widehat{Sens}(\phi_2)P^{lat}_2 + \widehat{FP}^1(\phi_2)P^{lat}_1 + \widehat{FP}^0(\phi_2)P^{lat}_0 \end{align}$ and compute $Spec = \widehat{Spec}(\phi^{\ast}_0),\; Sens = \widehat{Sens}(\phi^{\ast}_2),\; \textrm{etc.}$

Algorithm for trichotomous biomarker $\, S$

Without replacement sampling

In Approach 2, plot $RR_t \quad \textrm{vs.} \quad RR^{lat}_2\big/ RR^{lat}_0,$ where $RR_t := \frac{risk_1(2)}{risk_1(0)} = \frac{\sum_{x=0}^2 RR^{lat}_x P(X=x|S=2)}{\sum_{x=0}^2 RR^{lat}_x P(X=x|S=0)}$
For $\rho < 1$ , $RR_t$ closer to 1 than $RR^{lat}_2\big/ RR^{lat}_0$

Algorithm for trichotomous biomarker $\, S$

Without replacement sampling

Simulate M data sets under the true parameter values:
1. $N_x = (n_{controls} + n_{cases}) P^{lat}_x$
2. $\left(n_{cases}(0),n_{cases}(1),n_{cases}(2)\right) \sim \mathsf{Mult}(n_{cases},(p_0,p_1,p_2))$ , where $p_k=P(X=k|Y=1,Y^{\tau}=0,Z=1)$
3. For each of the $N_x$ subjects, generate $S_i$ by a draw from $\mathsf{Mult}(1,(p_0,p_1,p_2))$ , where $p_k=P(S=k|X=x)$
4. Sample without replacement $n^S_{cases}$ and $n^S_{controls} = K n^S_{cases}$ controls with measured $S$ $(R=1)$ , i.e., the control:case ratio is not fixed within subgroup $X=x$

Algorithm for trichotomous biomarker $\, S$

Without replacement sampling

For each observed data set, compute the 1-sided one-degree-of-freedom Wald test statistic for H0⇔{˜H0:βS≥0} from IPW logistic regression model that accounts for biomarker sampling design (function tps in R package osDesign)
- Alternatively, use the generalized two-degree-of-freedom Wald test
Compute power as proportion of data sets with 1-sided Wald test $p \leq \alpha/2$ for specified $\alpha$
Repeat power calculation varying control:case ratio, $n_{cases}$ , $n^S_{cases}$ , $(P^{lat}_0, P^{lat}_2, P_0, P_2)$ , $(Sens, Spec)$ , $\rho$

Illustration: hypothetical randomized placebo-controlled VE trial

Trial design

$N_{rand} = 4,100$ participants randomized to each of the vaccine and placebo group and followed for $\tau_{\mathrm{max}}=$ 24 months
Samples for measurement of $S$ at $\tau=$ 3.5 months stored in all vaccine recipients
Cumulative endpoint rate between $\tau$ and $\tau_{\mathrm{max}}$ in placebo group $=3.4\%$ $(=risk_0)$
$VE_{\tau-\tau_{\mathrm{max}}}=75\%$ , $\quad VE_{0-\tau}=VE_{\tau-\tau_{\mathrm{max}}}/2$
Cumulative dropout rate between 0 and $\tau_{\mathrm{max}}$ in both groups $=10\%$

Illustration: calculation of input parameters

Assumptions

Failure time $T$ and censoring time $C$ are independent
$T\mid Z=0 \sim \mathsf{Exp}(\lambda_T)$ and $C\mid Z=0 \sim \mathsf{Exp}(\lambda_C)$
$RR_{\tau-\tau_{\mathrm{max}}} := \frac{P(T \leq \tau_{\mathrm{max}} \mid T>\tau, Z=1)}{P(T \leq \tau_{\mathrm{max}} \mid T>\tau, Z=0)} = \frac{P(T \leq t\mid T>\tau, Z=1)}{P(T \leq t\mid T>\tau, Z=0)}$ for all $t \in (\tau,\tau_{\mathrm{max}}]$

Illustration: calculation of input parameters

Number of vaccine recipients observed to be at risk at $\tau$ $\begin{align} N &= N_{rand}\, P(T>\tau, C>\tau \mid Z=1)\\ &= N_{rand}\, P(T>\tau \mid Z=1)\, P(C>\tau \mid Z=1)\\ &= N_{rand}\, \{1 - RR_{0-\tau}\, P(T \leq \tau \mid Z=0)\}\, P(C>\tau \mid Z=1)\\ &\approx 4,023 \end{align}$ where $RR_{0-\tau} := \frac{P(T \leq \tau \mid Z=1)}{P(T \leq \tau \mid Z=0)}$

Illustration: calculation of input parameters

Number of observed cases between $\tau$ and $\tau_{\mathrm{max}}$ $\begin{align} n_{cases} &= N\, P(T\leq \tau_{\mathrm{max}}, T\leq C \mid T>\tau, C>\tau, Z=1)\\ &= N\, P(T\leq \min(C,\tau_{\mathrm{max}}) \mid T>\tau, C>\tau, Z=1)\\ &= N\, \frac{\int_{\tau}^{\infty}P(\tau < T \leq \min(c,\tau_{\mathrm{max}})\mid Z=1)f_C(c)\mathop{\mathrm{d}}\!c}{P(T>\tau, C>\tau \mid Z=1)}\\ &= N\, \bigg\{\int_{\tau}^{\tau_{\mathrm{max}}}P(\tau < T \leq c\mid Z=1)f_C(c)\mathop{\mathrm{d}}\!c +\\ &\quad+ P(\tau < T \leq \tau_{\mathrm{max}}\mid Z=1)\, P(C>\tau_{\mathrm{max}})\bigg\}\Big/ \\ &\quad\Big/ P(T>\tau, C>\tau \mid Z=1)\\ &\approx 32 \end{align}$

Illustration: calculation of input parameters

Number of observed controls between $\tau$ and $\tau_{\mathrm{max}}$ $\begin{align} n_{controls} &= N \, P(T > \tau_{\mathrm{max}}, C > \tau_{\mathrm{max}} \mid T>\tau, C>\tau, Z=1)\\ &\approx 3,654 \end{align}$ Number of observed cases (controls) between $\tau$ and $\tau_{\mathrm{max}}$ with measured $S$ $\begin{align} n^S_{cases} &= n_{cases}\\ n^S_{controls} &= K n^S_{cases} \end{align}$

Illustration: calculation of input parameters

library(CoRpower)

computeN(Nrand = 4100,          # participants randomized to vaccine arm
         tau = 3.5,             # biomarker sampling timepoint
         taumax = 24,           # end of follow-up
         VEtauToTaumax = 0.75,  # VE between 'tau' and 'taumax'
         VE0toTau = 0.75/2,     # VE between 0 and 'tau'
         risk0 = 0.034,         # placebo-group endpoint risk between 'tau' and 'taumax'
         dropoutRisk = 0.1,     # dropout risk between 0 and 'taumax'
         propCasesWithS = 1)    # proportion of observed cases with measured S

## $N
## [1] 4023
## 
## $nCases
## [1] 32
## 
## $nControls
## [1] 3654
## 
## $nCasesWithS
## [1] 32

Illustration: `CoRpower` for trichotomous $\, S$

Without replacement sampling

Approach 1 $(Sens, Spec, FP^0, FN^2$ specified $)$

Scenario 1: vary control:case ratio
Scenario 2: vary $Sens$ , $Spec$
Scenario 3: vary $P_0^{lat}$ , $P_2^{lat}$ , $P_0$ , $P_2$

Approach 2 $(\sigma^2_{obs}$ and $\rho$ specified $)$

Scenario 4: vary $\rho$
Scenario 5: vary $P_0^{lat}$ , $P_2^{lat}$ , $P_0$ , $P_2$
Scenario 6: vary $n_{cases}$

Scenario 1: vary control:case ratio (Approach 1)

Trichotomous $\, S$ , without replacement sampling

pwr <- computePower(nCases = 32,                 
                    nControls = 3654,            
                    nCasesWithS = 32,           
                    controlCaseRatio = 5,                # scalar, i.e., separate calls for 5, 3, 1
                    VEoverall = 0.75,                    # overall VE
                    risk0 = 0.034,                       # placebo-group endpoint risk between tau-tau_max
                    VElat0 = seq(0, VEoverall, len=100), # grid of VE (V/P) among lower protected
                    VElat1 = rep(VEoverall, 100),        # grid of VE (V/P) among medium protected
                    Plat0 = 0.2,                         # prevalence of lower protected
                    Plat2 = 0.6,                         # prevalence of higher protected
                    P0 = 0.2,                            # probability of low biomarker response
                    P2 = 0.6,                            # probability of high biomarker response
                    sens = 0.8, spec = 0.8, FP0 = 0, FN2 = 0,
                    M = 1000,                            # number of simulated clinical trials
                    alpha = 0.05,                        # two-sided Wald test Type 1 error rate
                    biomType = "trichotomous")           # "continuous" by default

Scenario 1: vary control:case ratio (Approach 1)

Trichotomous $\, S$ , without replacement sampling

Output list for controlCaseRatio = 5:

pwr$power[1, c(1, 25, 50, 75, 100)]

## [1] 0.995 0.933 0.597 0.197 0.022

pwr$RRt[1, c(1, 25, 50, 75, 100)]

## [1] 0.05882353 0.13978495 0.26940639 0.49685535 1.00000000

pwr$VElat2[c(1, 25, 50, 75, 100)]

## [1] 1.0000000 0.9393939 0.8762626 0.8131313 0.7500000

Other components: risk1_2, risk1_0, VElat0, Plat2, Plat0, P2, P0, alphaLat, betaLat, sens, spec, FP0, FN2, rho, N, nCases, nCasesPhase2, controlCaseRatio, VEoverall, risk0, alpha,

Scenario 1: vary control:case ratio (Approach 1)

Trichotomous $\, S$ , without replacement sampling

Plotting of power curves

plotPowerTri(outComputePower = list(pwr5to1, pwr3to1, pwr1to1),    # output lists from 'computePower'
             legendText = paste0("Control:Case = ", c("5:1", "3:1", "1:1")))

Alternatively,

computePower(..., saveDir = "myDir", saveFile = "myFile.RData")
plotPowerTri(outComputePower = c("myFile1", "myFile2", "myFile3"), # files with 'computePower' output
             outDir = rep("~/myDir", 3),                           # path to each myFilex.RData
             legendText = paste0("Control:Case = ", c("5:1", "3:1", "1:1")))

Scenario 2: vary $\, Sens$ and $\, Spec$ (Approach 1)

Trichotomous $\, S$ , without replacement sampling

pwr <- computePower(nCases = 32,                 
                    nControls = 3654,            
                    nCasesWithS = 32,           
                    controlCaseRatio = 5,                # n^S_controls : n^S_cases ratio
                    VEoverall = 0.75,                    # overall VE
                    risk0 = 0.034,                       # placebo-group endpoint risk between tau-tau_max
                    VElat0 = seq(0, VEoverall, len=100), # grid of VE (V/P) among lower protected
                    VElat1 = rep(VEoverall, 100),        # grid of VE (V/P) among medium protected
                    Plat0 = 0.2,                         # prevalence of lower protected
                    Plat2 = 0.6,                         # prevalence of higher protected
                    P0 = 0.2,                            # probability of low biomarker response
                    P2 = 0.6,                            # probability of high biomarker response
                    sens = c(1, 0.9, 0.8, 0.7), spec = c(1, 0.9, 0.8, 0.7), 
                    FP0 = 0, FN2 = 0,
                    M = 1000,                            # number of simulated clinical trials
                    alpha = 0.05,                        # two-sided Wald test Type 1 error rate
                    biomType = "trichotomous")           # "continuous" by default

Scenario 2: vary $\, Sens$ and $\, Spec$ (Approach 1)

Scenario 3: vary $\, P^{lat}_0, P^{lat}_2, P_0, P_2$ (Approach 1)

Trichotomous $\, S$ , without replacement sampling

pwr <- computePower(nCases = 32,                 
                    nControls = 3654,            
                    nCasesWithS = 32,           
                    controlCaseRatio = 5,                # n^S_controls : n^S_cases ratio
                    VEoverall = 0.75,                    # overall VE
                    risk0 = 0.034,                       # placebo-group endpoint risk between tau-tau_max
                    VElat0 = seq(0, VEoverall, len=100), # grid of VE (V/P) among lower protected
                    VElat1 = rep(VEoverall, 100),        # grid of VE (V/P) among medium protected
                    Plat0 = 0.05,                        # scalar, ie separate calls for 0.05, 0.1, 0.15, 0.2
                    Plat2 = 0.15,                        # scalar, ie separate calls for 0.15, 0.3, 0.45, 0.6
                    P0 = 0.05,                           # scalar, ie separate calls for 0.05, 0.1, 0.15, 0.2
                    P2 = 0.15,                           # scalar, ie separate calls for 0.15, 0.3, 0.45, 0.6
                    sens = 0.8, spec = 0.8, FP0 = 0, FN2 = 0,
                    M = 1000,                            # number of simulated clinical trials
                    alpha = 0.05,                        # two-sided Wald test Type 1 error rate
                    biomType = "trichotomous")           # "continuous" by default

Scenario 3: vary $\, P^{lat}_0, P^{lat}_2, P_0, P_2$ (Approach 1)

Scenario 4: vary $\, \rho$ (Approach 2)

Trichotomous $\, S$ , without replacement sampling

pwr <- computePower(nCases = 32,                 
                    nControls = 3654,            
                    nCasesWithS = 32,           
                    controlCaseRatio = 5,                # n^S_controls : n^S_cases ratio
                    VEoverall = 0.75,                    # overall VE
                    risk0 = 0.034,                       # placebo-group endpoint risk between tau-tau_max 
                    VElat0 = seq(0, VEoverall, len=100), # grid of VE (V/P) among lower protected
                    VElat1 = rep(VEoverall, 100),        # grid of VE (V/P) among medium protected
                    Plat0 = 0.2,                         # prevalence of lower protected
                    Plat2 = 0.6,                         # prevalence of higher protected
                    P0 = 0.2,                            # probability of low biomarker response
                    P2 = 0.6,                            # probability of high biomarker response
                    sigma2obs = 1,                       # variance of observed biomarker S
                    rho = c(1, 0.9, 0.7, 0.5),           # protection-relevant fraction of the variance of S
                    M = 1000,                            # number of simulated clinical trials
                    alpha = 0.05,                        # two-sided Wald test Type 1 error rate
                    biomType = "trichotomous")           # "continuous" by default

Scenario 4: vary $\, \rho$ (Approach 2)

Trichotomous $\, S$ , without replacement sampling

Scenario 4: vary $\, \rho$ (Approach 2)

Plotting of $RR_t$ vs. $RR_2^{lat}/RR_0^{lat}$

plotRRgradVE(outComputePower = pwr,         # output list from 'computePower'
             legendText = paste0("rho = ", c(1, 0.9, 0.7, 0.5)))

Alternatively,

computePower(..., saveDir = "myDir", saveFile = "myFile.RData")
plotRRgradVE(outComputePower = "myFile",    # file with 'computePower' output
             outDir = "~/myDir",            # path to myFile.RData
             legendText = paste0("rho = ", c(1, 0.9, 0.7, 0.5)))

Scenario 5: vary $\, P^{lat}_0, P_0, P^{lat}_2, P_2$ (Approach 2)

Trichotomous $\, S$ , without replacement sampling

pwr <- computePower(nCases = 32,                 
                    nControls = 3654,            
                    nCasesWithS = 32,           
                    controlCaseRatio = 5,                # n^S_controls : n^S_cases ratio
                    VEoverall = 0.75,                    # overall VE
                    risk0 = 0.034,                       # placebo-group endpoint risk between tau-tau_max 
                    VElat0 = seq(0, VEoverall, len=100), # grid of VE (V/P) among lower protected
                    VElat1 = rep(VEoverall, 100),        # grid of VE (V/P) among medium protected
                    Plat0 = 0.05,                        # scalar, ie separate calls for 0.05, 0.1, 0.15, 0.2
                    Plat2 = 0.15,                        # scalar, ie separate calls for 0.15, 0.3, 0.45, 0.6
                    P0 = 0.05,                           # scalar, ie separate calls for 0.05, 0.1, 0.15, 0.2
                    P2 = 0.15,                           # scalar, ie separate calls for 0.15, 0.3, 0.45, 0.6
                    sigma2obs = 1,                       # variance of observed biomarker S
                    rho = 0.9,                           # protection-relevant fraction of the variance of S
                    M = 1000,                            # number of simulated clinical trials
                    alpha = 0.05,                        # two-sided Wald test Type 1 error rate
                    biomType = "trichotomous")           # "continuous" by default

Scenario 5: vary $\, P^{lat}_0, P_0, P^{lat}_2, P_2$ (Approach 2)

Trichotomous $\, S$ , without replacement sampling

Scenario 5: vary $\, P^{lat}_0, P_0, P^{lat}_2, P_2$ (Approach 2)

Scenario 6: vary $\, n_{cases}$ (Approach 2)

Trichotomous $\, S$ , without replacement sampling

pwr <- computePower(nCases = c(25, 32, 35, 40),             
                    nControls = c(3661, 3654, 3651, 3646),  
                    nCasesWithS = c(25, 32, 35, 40),       
                    controlCaseRatio = 5,                # n^S_controls : n^S_cases ratio
                    VEoverall = 0.75,                    # overall VE
                    risk0 = 0.034,                       # placebo-group endpoint risk between tau-tau_max
                    VElat0 = seq(0, VEoverall, len=100), # grid of VE (V/P) among lower protected
                    VElat1 = rep(VEoverall, 100),        # grid of VE (V/P) among medium protected
                    Plat0 = 0.2,                         # prevalence of lower protected
                    Plat2 = 0.6,                         # prevalence of higher protected
                    P0 = 0.2,                            # probability of low biomarker response
                    P2 = 0.6,                            # probability of high biomarker response
                    sigma2obs = 1,                       # variance of observed biomarker S
                    rho = 0.9,                           # protection-relevant fraction of the variance of S
                    M = 1000,                            # number of simulated clinical trials
                    alpha = 0.05,                        # two-sided Wald test Type 1 error rate
                    biomType = "trichotomous")           # "continuous" by default

Scenario 6: vary $\, n_{cases}$ (Approach 2)

Trichotomous $\, S$ , without replacement sampling

Scenario 6: vary $\, n_{cases}$ (Approach 2)

Illustration: `CoRpower` for binary $\, S$

Without replacement sampling

Achieved by selecting $P_0^{lat}$ , $P_2^{lat}$ , $P_0$ , $P_2$ such that $\begin{align} P_0^{lat} + P_2^{lat} &= 1\\ P_0 + P_2 &= 1 \end{align}$

Approach 2 $(\sigma^2_{obs}$ and $\rho$ specified $)$

Scenario 7: vary $n_{cases}$

Scenario 7: vary $\, n_{cases}$ (Approach 2)

Binary $\, S$ , without replacement sampling

pwr <- computePower(nCases = c(25, 32, 35, 40),             
                    nControls = c(3661, 3654, 3651, 3646),  
                    nCasesWithS = c(25, 32, 35, 40),       
                    controlCaseRatio = 5,                # n^S_controls : n^S_cases ratio
                    VEoverall = 0.75,                    # overall VE
                    risk0 = 0.034,                       # placebo-group endpoint risk between tau-tau_max
                    VElat0 = seq(0, VEoverall, len=100), # grid of VE (V/P) among lower protected
                    VElat1 = rep(VEoverall, 100),        # grid of VE (V/P) among medium protected
                    Plat0 = 0.2,                         # prevalence of lower protected
                    Plat2 = 0.8,                         # prevalence of higher protected
                    P0 = 0.2,                            # probability of low biomarker response
                    P2 = 0.8,                            # probability of high biomarker response
                    sigma2obs = 1,                       # variance of observed biomarker S
                    rho = 0.9,                           # protection-relevant fraction of the variance of S
                    M = 1000,                            # number of simulated clinical trials
                    alpha = 0.05,                        # two-sided Wald test Type 1 error rate
                    biomType = "binary")                 # "continuous" by default

Scenario 7: vary $\, n_{cases}$ (Approach 2)

Binary $\, S$ , without replacement sampling

Scenario 7: vary $\, n_{cases}$ (Approach 2)

Algorithm for continuous biomarker $\, S^{\ast}$

Without replacement sampling

Specify overall VE between \tau and \tau_{\mathrm{max}}
- Protocol-specified design alternative or $\widehat{VE}$
Specify risk_0
- Protocol-specified placebo-group endpoint rate or $\widehat{risk}_0$

Algorithm for continuous biomarker $\, S^{\ast}$

Without replacement sampling

Specify P^{lat}_{lowestVE}, \rho, and a grid of VE_{lowest} values (e.g., ranging from VE to 0)
- Fixed $(VE, risk_0, P^{lat}_{lowestVE}, VE_{lowest}, \rho)$ and $\begin{align} risk^{lat}_1(x^{\ast}) &= (1 - VE_{lowest}) risk_0,\quad x^{\ast} \leq \nu\\ \mathop{\mathrm{logit}}\{risk^{lat}_1(x^{\ast})\} &= \alpha^{lat}+\beta^{lat}x^{\ast},\quad x^{\ast} \geq \nu\\ VE &= 1 - \frac{\int risk^{lat}_1(x^{\ast})\phi(x^{\ast}/\sqrt{\rho} \sigma_{obs})\mathop{\mathrm{d}}\!x^{\ast}}{risk_0} \end{align}$ yield $\alpha^{lat}$ and $\beta^{lat}$
- Plot $VE^{lat}_{x^{\ast}}$ vs. $x^{\ast}$ and calculate the pertaining CoR effect size $\exp(\beta^{lat})$ for each level of $VE_{lowest}$

Algorithm for continuous biomarker $\, S^{\ast}$

Without replacement sampling

Simulate M data sets under the true parameter values:
1. Sample $X^{\ast}$ for $n_{cases}$ cases from $f_{X^{\ast}}(x^{\ast}|Y=1,Y^{\tau}=0,Z=1)$ using Bayes rule
2. Sample $X^{\ast}$ for $n_{controls}$ controls from $f_{X^{\ast}}(x^{\ast}|Y=0,Y^{\tau}=0,Z=1)$ using Bayes rule
  - How? Use a fine grid of $\widetilde{x}^{\ast}$ values and then
  $\;\;$ sample( $\widetilde{x}^{\ast}$ , prob= $f_{X^{\ast}}(\widetilde{x}^{\ast}|Y=\cdot,Y^{\tau}=0,Z=1)$ , replace=TRUE)
3. Sample $S^{\ast}$ following $S^{\ast} = X^{\ast} + e$
4. Sample without replacement $n^S_{cases}$ and $n^S_{controls} = K n^S_{cases}$ controls with measured $S^{\ast}$ $(R=1)$

Algorithm for continuous biomarker $\, S^{\ast}$

Without replacement sampling

For each observed data set, compute the 1-sided one-degree-of-freedom Wald test statistic for $H_0 \Leftrightarrow \{\widetilde{H}_0: \beta_{S^{\ast}} \geq 0\}$ from IPW logistic regression model that accounts for biomarker sampling design (function tps in R package osDesign)
Compute power as proportion of data sets with 1-sided Wald test $p \leq \alpha/2$ for specified $\alpha$
Repeat power calculation varying control:case ratio, $n_{cases}$ , $n^S_{cases}$ , $P^{lat}_{lowestVE}$ , $\rho$

Illustration: `CoRpower` for continuous $\, S^{\ast}$

Without replacement sampling

Scenario 8: vary $\rho$
Scenario 9: vary $P_{lowestVE}^{lat}$

Scenario 8: vary $\, \rho$

Continuous $\, S^{\ast}$ , without replacement sampling

pwr <- computePower(nCases = 32,             
                    nControls = 3654,         
                    nCasesWithS = 32,         
                    controlCaseRatio = 5,               # n^S_controls : n^S_cases ratio
                    VEoverall = 0.75,                   # overall VE
                    risk0 = 0.034,                      # placebo-group endpoint risk between tau-tau_max
                    PlatVElowest = 0.2,                 # prevalence of VE_lowest
                    VElowest = seq(0, VEoverall, len=100), # lowest VE level for true biomarker X* <= nu
                    sigma2obs = 1,                      # variance of observed biomarker S
                    rho = c(1, 0.9, 0.7, 0.5)           # protection-relevant fraction of the variance of S
                    M = 1000,                           # number of simulated clinical trials
                    alpha = 0.05,                       # two-sided Wald test Type 1 error rate
                    biomType = "continuous")            # "continuous" by default

Scenario 8: vary $\, \rho$

Continuous $\, S^{\ast}$ , without replacement sampling

pwr$power[1, c(1, 25, 50, 75, 100)]

## [1] 1.000 1.000 0.983 0.620 0.023

pwr$RRc[c(1, 25, 50, 75, 100)]

## [1] 0.001707853 0.040854962 0.178181573 0.455111981 0.999997688

pwr$VElowest[c(1, 25, 50, 75, 100)]

## [1] 0.0000000 0.1818182 0.3712121 0.5606061 0.7500000

Other components: betaLat, alphaLat, PlatVElowest, N, nCases, nCasesPhase2, controlCaseRatio, VEoverall, risk0, alpha, rho,

Scenario 8: vary $\, \rho$

Continuous $\, S^{\ast}$ , without replacement sampling

Scenario 8: vary $\, \rho$

Continuous $\, S^{\ast}$ , without replacement sampling

Plotting of $VE^{lat}_{x^{\ast}}$ curves

plotVElatCont(outComputePower = pwr)  # output list from 'computePower'

Alternatively,

computePower(..., saveDir = "myDir", saveFile = "myFile.RData")
plotVElatCont(outComputePower = "myFile",   # file with 'computePower' output
              outDir = "~/myDir")           # path to myFile.RData

Scenario 8: vary $\, \rho$

Plotting of power curves

plotPowerCont(outComputePower = pwr,          # output list from 'computePower'
              legendText = paste0("rho = ", c(1, 0.9, 0.7, 0.5)))

Alternatively,

computePower(..., saveDir = "myDir", saveFile = "myFile.RData")
plotPowerCont(outComputePower = "myFile",     # file with 'computePower' output
              outDir = "~/myDir",             # path to myFile.RData
              legendText = paste0("rho = ", c(1, 0.9, 0.7, 0.5)))

Scenario 9: vary $\, P_{lowestVE}^{lat}$

Continuous $\, S^{\ast}$ , without replacement sampling

pwr <- computePower(nCases = 32,               
                    nControls = 3654,         
                    nCasesWithS = 32,         
                    controlCaseRatio = 5,               # n^S_controls : n^S_cases ratio
                    VEoverall = 0.75,                   # overall VE
                    risk0 = 0.034,                      # placebo-group endpoint risk between tau-tau_max
                    PlatVElowest = 0.05,                # scalar, ie separate calls for 0.05, 0.1, 0.15, 0.2
                    VElowest = seq(0, VEoverall, len=100), # lowest VE level for true biomarker X* <= nu
                    sigma2obs = 1,                      # variance of observed biomarker S
                    rho = 0.9                           # protection-relevant fraction of the variance of S
                    M = 1000,                           # number of simulated clinical trials
                    alpha = 0.05,                       # two-sided Wald test Type 1 error rate
                    biomType = "continuous")            # "continuous" by default

Scenario 9: vary $\, P_{lowestVE}^{lat}$

Bernoulli / case-cohort sampling of $\, S$ (or $\, S^{\ast}$ )

Bernoulli sample at baseline with sampling probability $p$
S (or S^{\ast}) measured at \tau in
- a subset of the sample with $Y^{\tau}=0$ , and
- all cases with $Y^{\tau}=0$
Implications:
- $n_{cases} = n^S_{cases}$
- design parameter $n^S_{controls}$ replaced by probability $p$ because $n^S_{controls}$ is a random variable in case-cohort designs

Illustration: `CoRpower` for trichotomous $\, S$ and continuous $\, S^{\ast}$

Bernoulli sampling

Trichotomous $S$ (Approach 1)

Scenario 10: vary $p$

Continuous $S^{\ast}$

Scenario 11: vary $p$

Scenario 10: vary $\, p$ (Approach 1)

Trichotomous $\, S$ , Bernoulli sampling

pwr <- computePower(nCases = 32,             
                    nControls = 3654,       
                    nCasesWithS = 32,       
                    cohort = TRUE,                    # FALSE by default
                    p = 0.01,                         # scalar, ie separate calls for 0.01, 0.02, 0.03, 0.05
                    VEoverall = 0.75,                 # overall VE
                    risk0 = 0.034,                    # placebo-group endpoint risk between tau-tau_max
                    VElat0 = seq(0, VEoverall, len=100), # grid of VE (V/P) among lower protected
                    VElat1 = rep(VEoverall, 100),     # grid of VE (V/P) among medium protected
                    Plat0 = 0.2,                      # prevalence of lower protected
                    Plat2 = 0.6,                      # prevalence of higher protected
                    P0 = 0.2,                         # probability of low biomarker response
                    P2 = 0.6,                         # probability of high biomarker response
                    sens = 0.8, spec = 0.8, FP0 = 0, FN2 = 0,
                    M = 1000,                         # number of simulated clinical trials
                    alpha = 0.05,                     # two-sided Wald test Type 1 error rate
                    biomType = "trichotomous")        # "continuous" by default

Scenario 10: vary $\, p$ (Approach 1)

Scenario 11: vary $\, p$

Continuous $\, S^{\ast}$ , Bernoulli sampling

pwr <- computePower(nCases = 32,             
                    nControls = 3654,        
                    nCasesWithS = 32,       
                    cohort = TRUE,                    # FALSE by default
                    p = 0.01,                         # scalar, ie separate calls for 0.01, 0.02, 0.03, 0.05
                    VEoverall = 0.75,                 # overall VE
                    risk0 = 0.034,                    # placebo-group endpoint risk between tau and tau_max
                    PlatVElowest = 0.2,               # prevalence of VE_lowest
                    VElowest = seq(0, VEoverall, len=100), # lowest VE level for true biomarker X* <= nu
                    sigma2obs = 1,                    # variance of observed biomarker S
                    rho = 0.9                         # protection-relevant fraction of the variance of S
                    M = 1000,                         # number of simulated clinical trials
                    alpha = 0.05,                     # two-sided Wald test Type 1 error rate
                    biomType = "continuous")          # "continuous" by default

Scenario 11: vary $\, p$

Key R functions to design and conduct a CoR power analysis, and display analysis results

computeN
computePower
plotPowerTri
plotRRgradVE
plotPowerCont
plotVElatCont

R package CoRpower to be submitted to CRAN this week

Currently downloadable at:

https://github.com/smwu/CoRpower

Implementation of Gilbert, Janes, Huang (2016, Stat Med)

Code and slide contributions:

Stephanie Wu

R Package CoRpower: Power Calculations for Correlates of Risk

Based on Gilbert, Janes, Huang (2016, Stat Med)

Outline

Correlates of risk power calculations

Set-up and notation

Without replacement sampling

Algorithm for trichotomous biomarker S\, S

Without replacement sampling

Algorithm for trichotomous biomarker S\, S

Without replacement sampling

Algorithm for trichotomous biomarker S\, S

Without replacement sampling

Algorithm for trichotomous biomarker S\, S

Without replacement sampling

Calculation of (Sens,Spec,FP0,FP1,FN1,FN2)(Sens, Spec, FP^0, FP^1, FN^1, FN^2) in Approach 2

Algorithm for trichotomous biomarker S\, S

Without replacement sampling

Calculation of (Sens,Spec,FP0,FP1,FN1,FN2)(Sens, Spec, FP^0, FP^1, FN^1, FN^2) in Approach 2

Algorithm for trichotomous biomarker S\, S

Without replacement sampling

Calculation of (Sens,Spec,FP0,FP1,FN1,FN2)(Sens, Spec, FP^0, FP^1, FN^1, FN^2) in Approach 2

Algorithm for trichotomous biomarker S\, S

Without replacement sampling

Calculation of (Sens,Spec,FP0,FP1,FN1,FN2)(Sens, Spec, FP^0, FP^1, FN^1, FN^2) in Approach 2

Algorithm for trichotomous biomarker S\, S

Without replacement sampling

Calculation of (Sens,Spec,FP0,FP1,FN1,FN2)(Sens, Spec, FP^0, FP^1, FN^1, FN^2) in Approach 2

Algorithm for trichotomous biomarker S\, S

Without replacement sampling

Algorithm for trichotomous biomarker S\, S

Without replacement sampling

Algorithm for trichotomous biomarker S\, S

Without replacement sampling

Illustration: hypothetical randomized placebo-controlled VE trial

Trial design

Illustration: calculation of input parameters

Assumptions

Illustration: calculation of input parameters

Illustration: calculation of input parameters

Illustration: calculation of input parameters

Illustration: calculation of input parameters

Illustration: CoRpower for trichotomous \, S\, S

Without replacement sampling

Scenario 1: vary control:case ratio (Approach 1)

Trichotomous \, S\, S, without replacement sampling

Scenario 1: vary control:case ratio (Approach 1)

Trichotomous \, S\, S, without replacement sampling

Scenario 1: vary control:case ratio (Approach 1)

Scenario 1: vary control:case ratio (Approach 1)

Trichotomous \, S\, S, without replacement sampling

Plotting of power curves

Scenario 2: vary \, Sens\, Sens and \, Spec\, Spec (Approach 1)

Trichotomous \, S\, S, without replacement sampling

Scenario 2: vary \, Sens\, Sens and \, Spec\, Spec (Approach 1)

Scenario 3: vary \, P^{lat}_0, P^{lat}_2, P_0, P_2\, P^{lat}_0, P^{lat}_2, P_0, P_2 (Approach 1)

Trichotomous \, S\, S, without replacement sampling

Scenario 3: vary \, P^{lat}_0, P^{lat}_2, P_0, P_2\, P^{lat}_0, P^{lat}_2, P_0, P_2 (Approach 1)

Scenario 4: vary \, \rho \, \rho (Approach 2)

Trichotomous \, S\, S, without replacement sampling

Scenario 4: vary \, \rho \, \rho (Approach 2)

Trichotomous \, S\, S, without replacement sampling

Scenario 4: vary \, \rho \, \rho (Approach 2)

Scenario 4: vary \, \rho \, \rho (Approach 2)

Scenario 4: vary \, \rho \, \rho (Approach 2)

Plotting of RR_tRR_t vs. RR_2^{lat}/RR_0^{lat}RR_2^{lat}/RR_0^{lat}

Scenario 5: vary \, P^{lat}_0, P_0, P^{lat}_2, P_2 \, P^{lat}_0, P_0, P^{lat}_2, P_2 (Approach 2)

Trichotomous \, S\, S, without replacement sampling

Scenario 5: vary \, P^{lat}_0, P_0, P^{lat}_2, P_2 \, P^{lat}_0, P_0, P^{lat}_2, P_2 (Approach 2)

Trichotomous \, S\, S, without replacement sampling

Scenario 5: vary \, P^{lat}_0, P_0, P^{lat}_2, P_2 \, P^{lat}_0, P_0, P^{lat}_2, P_2 (Approach 2)

Scenario 6: vary \, n_{cases}\, n_{cases} (Approach 2)

Trichotomous \, S\, S, without replacement sampling

Scenario 6: vary \, n_{cases}\, n_{cases} (Approach 2)

Trichotomous \, S\, S, without replacement sampling

Scenario 6: vary \, n_{cases}\, n_{cases} (Approach 2)

Illustration: CoRpower for binary \, S\, S

Without replacement sampling

Scenario 7: vary \, n_{cases}\, n_{cases} (Approach 2)

Binary \, S\, S, without replacement sampling

Scenario 7: vary \, n_{cases}\, n_{cases} (Approach 2)

R Package `CoRpower`:
Power Calculations for
Correlates of Risk

Algorithm for trichotomous biomarker $\, S$

Algorithm for trichotomous biomarker $\, S$

Algorithm for trichotomous biomarker $\, S$

Algorithm for trichotomous biomarker $\, S$

Calculation of $(Sens, Spec, FP^0, FP^1, FN^1, FN^2)$ in Approach 2

Algorithm for trichotomous biomarker $\, S$

Calculation of $(Sens, Spec, FP^0, FP^1, FN^1, FN^2)$ in Approach 2

Algorithm for trichotomous biomarker $\, S$

Calculation of $(Sens, Spec, FP^0, FP^1, FN^1, FN^2)$ in Approach 2

Algorithm for trichotomous biomarker $\, S$

Calculation of $(Sens, Spec, FP^0, FP^1, FN^1, FN^2)$ in Approach 2

Algorithm for trichotomous biomarker $\, S$

Calculation of $(Sens, Spec, FP^0, FP^1, FN^1, FN^2)$ in Approach 2

Algorithm for trichotomous biomarker $\, S$

Algorithm for trichotomous biomarker $\, S$

Algorithm for trichotomous biomarker $\, S$

Illustration: `CoRpower` for trichotomous $\, S$

Trichotomous $\, S$ , without replacement sampling

Trichotomous $\, S$ , without replacement sampling

Trichotomous $\, S$ , without replacement sampling

Scenario 2: vary $\, Sens$ and $\, Spec$ (Approach 1)

Trichotomous $\, S$ , without replacement sampling

Scenario 2: vary $\, Sens$ and $\, Spec$ (Approach 1)

Scenario 3: vary $\, P^{lat}_0, P^{lat}_2, P_0, P_2$ (Approach 1)

Trichotomous $\, S$ , without replacement sampling

Scenario 3: vary $\, P^{lat}_0, P^{lat}_2, P_0, P_2$ (Approach 1)

Scenario 4: vary $\, \rho$ (Approach 2)

Trichotomous $\, S$ , without replacement sampling

Scenario 4: vary $\, \rho$ (Approach 2)

Trichotomous $\, S$ , without replacement sampling

Scenario 4: vary $\, \rho$ (Approach 2)

Scenario 4: vary $\, \rho$ (Approach 2)

Scenario 4: vary $\, \rho$ (Approach 2)

Plotting of $RR_t$ vs. $RR_2^{lat}/RR_0^{lat}$

Scenario 5: vary $\, P^{lat}_0, P_0, P^{lat}_2, P_2$ (Approach 2)

Trichotomous $\, S$ , without replacement sampling

Scenario 5: vary $\, P^{lat}_0, P_0, P^{lat}_2, P_2$ (Approach 2)

Trichotomous $\, S$ , without replacement sampling

Scenario 5: vary $\, P^{lat}_0, P_0, P^{lat}_2, P_2$ (Approach 2)

Scenario 6: vary $\, n_{cases}$ (Approach 2)

Trichotomous $\, S$ , without replacement sampling

Scenario 6: vary $\, n_{cases}$ (Approach 2)

Trichotomous $\, S$ , without replacement sampling

Scenario 6: vary $\, n_{cases}$ (Approach 2)

Illustration: `CoRpower` for binary $\, S$

Scenario 7: vary $\, n_{cases}$ (Approach 2)

Binary $\, S$ , without replacement sampling

Scenario 7: vary $\, n_{cases}$ (Approach 2)

Binary $\, S$ , without replacement sampling

Scenario 7: vary $\, n_{cases}$ (Approach 2)

Algorithm for continuous biomarker $\, S^{\ast}$

Algorithm for continuous biomarker $\, S^{\ast}$

Algorithm for continuous biomarker $\, S^{\ast}$

Algorithm for continuous biomarker $\, S^{\ast}$

Illustration: `CoRpower` for continuous $\, S^{\ast}$

Scenario 8: vary $\, \rho$

Continuous $\, S^{\ast}$ , without replacement sampling

Scenario 8: vary $\, \rho$

Continuous $\, S^{\ast}$ , without replacement sampling

Scenario 8: vary $\, \rho$

Continuous $\, S^{\ast}$ , without replacement sampling

Scenario 8: vary $\, \rho$

Continuous $\, S^{\ast}$ , without replacement sampling

Plotting of $VE^{lat}_{x^{\ast}}$ curves

Scenario 8: vary $\, \rho$

Scenario 8: vary $\, \rho$

Scenario 9: vary $\, P_{lowestVE}^{lat}$

Continuous $\, S^{\ast}$ , without replacement sampling

Scenario 9: vary $\, P_{lowestVE}^{lat}$

Scenario 9: vary $\, P_{lowestVE}^{lat}$

Bernoulli / case-cohort sampling of $\, S$ (or $\, S^{\ast}$ )

Illustration: `CoRpower` for trichotomous $\, S$ and continuous $\, S^{\ast}$

Scenario 10: vary $\, p$ (Approach 1)

Trichotomous $\, S$ , Bernoulli sampling

Scenario 10: vary $\, p$ (Approach 1)

Scenario 11: vary $\, p$

Continuous $\, S^{\ast}$ , Bernoulli sampling

Scenario 11: vary $\, p$