pCR Meta Analysis
Michael Sachs
2015-10-28
Figure 1
library(pcrmeta)
library(gridExtra)
library(ggplot2)
set.seed(420)
a <- plot_surrogates(TRUE, TRUE) + ggtitle("A")
b <- plot_surrogates(TRUE, FALSE) + ggtitle("B")
c <- plot_surrogates(FALSE, TRUE) + ggtitle("C")
pdf("~/Desktop/Figure1.pdf", width = 8, height = 3.5)
grid.arrange(a, b, c, nrow = 1)
dev.off()## quartz_off_screen
## 2Data
This is an attempt to reanalze the data from the meta analyses described in Cortazar et al. (2014). Cortazar et al. (2014) includes the results from 10 trials in their trial-level analysis. Some of the comparisons are not included in the trial-level analysis because they were randomized and some were single-arm trials.
| LancetID | Patients | OR.PCR | HR.DFS | HR.OS | PCR.SE | DFS.SE | OS.SE | Trial | |
|---|---|---|---|---|---|---|---|---|---|
| 14 | A | 950 | 1.0316 | 1.0472 | 0.8105 | 0.1559 | 0.1322 | 0.1715 | GeparQuatro1 |
| 15 | B | 913 | 2.1002 | 1.1153 | 1.0186 | 0.2306 | 0.1180 | 0.1618 | GeparDuo |
| 16 | C | 942 | 0.9608 | 0.8795 | 0.8396 | 0.1605 | 0.1357 | 0.1771 | GeparQuatro2 |
| 17 | D | 1856 | 1.1110 | 0.8396 | 0.8900 | 0.1704 | 0.0772 | 0.1069 | EORTC |
| 18 | E | 733 | 1.4997 | 0.8688 | 0.7994 | 0.2040 | 0.1431 | 0.1948 | PREPARE |
| 19 | F | 2411 | 2.2416 | 0.8877 | 0.9192 | 0.1364 | 0.0819 | 0.1035 | NSABP |
| 20 | G | 1390 | 1.2510 | 0.7494 | 0.7609 | 0.1292 | 0.0987 | 0.1451 | GeparTrio1 |
| 21 | H | 622 | 0.8806 | 0.6906 | 0.8203 | 0.3481 | 0.1287 | 0.2049 | GeparTrio2 |
| 22 | I | 671 | 2.1719 | 0.7092 | 0.6904 | 0.2323 | 0.1378 | 0.1825 | AGO |
| 23 | J | 235 | 3.0383 | 0.6689 | 0.5999 | 0.3002 | 0.2177 | 0.3066 | NOAH |
We have included an addtional 13 trials that were analyzed by Berruti et al. (2014). That paper included 29 trials total, but we excluded the small ones (less than 20 events), and ones that did not have sufficient information for the analysis.
| LancetID | Patients | OR.PCR | HR.DFS | HR.OS | PCR.SE | DFS.SE | OS.SE | Trial |
|---|---|---|---|---|---|---|---|---|
| NA | 199 | 1.528 | 0.713 | 0.892 | 0.459 | 0.218 | 0.217 | Arun |
| NA | 150 | 1.585 | 0.773 | 0.871 | 0.928 | 0.248 | 0.295 | Bald |
| NA | 96 | 3.000 | 0.413 | 0.397 | 0.701 | 0.399 | 0.925 | Chen |
| NA | 451 | 1.003 | 1.180 | 1.410 | 0.310 | 0.195 | 0.252 | Chua |
| NA | 211 | 3.165 | 0.730 | 0.770 | 0.830 | 0.233 | 0.273 | Cocc |
| NA | 372 | 1.232 | 1.030 | 1.190 | 0.255 | 0.158 | 0.196 | Ellis |
| NA | 200 | 2.940 | 0.723 | 0.644 | 0.504 | 0.203 | 0.243 | Frasci |
| NA | 209 | 2.472 | 0.967 | 0.180 | 0.411 | 0.392 | 0.791 | Lee |
| NA | 363 | 0.716 | 0.818 | 0.818 | 0.317 | 0.157 | 0.171 | Mansi |
| NA | 426 | 1.021 | 1.050 | 0.760 | 0.368 | 0.161 | 0.203 | Smith |
| NA | 448 | 0.677 | 1.051 | 1.010 | 0.368 | 0.120 | 0.136 | Theras |
| NA | 477 | 0.928 | 0.907 | 0.671 | 0.215 | 0.256 | 0.408 | Toi |
| NA | 89 | 0.661 | 0.900 | 1.260 | 0.529 | 0.437 | 0.540 | Walker |
Methods
Here we are going to fit model (7) from Korn, Albert, and McShane (2005). This is using the reported standard errors on the log OR and log HR, instead of the variances of the arm-specific summaries. The model is
\[ \log(OR_i) = \mu + m_i + \epsilon_i \]
\[ \log(HR_i) = \alpha + \beta * (\mu + m_i) + g_i + \delta_i \]
where for trial \(i\) , \(OR_i\) is the observed odds ratio comparing pCR for the two treatment arms, \(HR_i\) is the observed hazard ratio comparing the treatment arms, \(\mu + m_i\) is the true log odds ratio for pCR, \(\mu\) is a fixed effect representing the average log odds ratio across trials, and \(m_i\) is a random effect with mean 0 and variance \(\sigma^2_m\), \(\epsilon_i\) is a random error with standard deviation equal to standard error of the estimate of the log odds ratio. In the second equation \(HR_i\) is the observed hazard ratio comparing two treatments for trial \(i\). This equation specifies a linear relationship, with intercept \(\alpha\) and slope \(\beta\) between the true log hazard ratio and the true log odds ratio for pCR. Here \(g_i\) is a random effect with mean 0 and variance \(\sigma_g^2\) and \(\delta_i\) is a random error with mean 0 and standard deviation equal to the standard error of the estimate of the log hazard ratio.
This model implies a bivariate normal likelihood for \(OR_i\) and \(HR_i\). The likelihood of this model is coded as follows:
like.hood.model4## function (Yeff, Xeff, Yse, Xse, param)
## {
## stopifnot(length(param) == 5)
## stopifnot(length(Xeff) == length(Yeff))
## stopifnot(length(Yse) == length(Yeff))
## stopifnot(length(Xse) == length(Xeff))
## vmg <- param[5]
## beta <- param[3]
## muy <- param[2]
## vmx <- param[4]
## mux <- param[1]
## n <- length(Yeff)
## c11 <- vmx + Xse^2
## c22 <- beta^2 * vmx + vmg + Yse^2
## c12 <- beta * vmx
## sigma <- rbind(cbind(diag(c11), diag(c12, nrow = n)), cbind(diag(c12,
## nrow = n), diag(c22)))
## XX <- c(Xeff, Yeff)
## mu = c(rep(mux, n), rep(muy + beta * mux, n))
## hood <- mvtnorm::dmvnorm(XX, mean = mu, sigma = sigma, log = TRUE)
## -hood
## }
## <environment: namespace:pcrmeta>The estimates are obtained by maximum likelihood, with model based standard errors. Due to the small sample size, the maximum likelihood estimates of the variance components will be on average too small. General procedures for handling this problem (e.g., REML), are not readily applicable in the measurement error model being considered here. Instead, we have used an ad hoc adjustment of multiplying \(\hat{\sigma}^2_m\) by \(n/(n-1)\) and \(\hat{\sigma}^2_g\) by \(n/(n-3)\). The adjusted MLEs are denoted \(\tilde{\sigma^2}_g\) and \(\tilde{\sigma^2}_m\).
For a new trial with \(i = 0\) in which pCR is assessed but not the definitive outcome, it is of interest to predict the true hazard ratio for a new trial based on its observed pCR results and sample size. Let \(\log(OR_0) = v\) be the observed log odds ratio and \(\sigma^2_0\) be the variance of \(v\). The quantity of interest is
\[ \Delta = E[\alpha + \beta(\mu + m_0) + g_0 | \mu + m_0 + \epsilon_0 = v] = \]
\[ \alpha + \beta \mu + E[m_0 | \epsilon_0 = v - \mu] = \]
\[ \alpha + \beta v \left(\frac{\sigma^2_m}{\sigma^2_m + \sigma_0^2}\right) + \beta \mu \left(1 - \frac{\sigma^2_m}{\sigma^2_m + \sigma_0^2}\right) \]
The estimator of \(\Delta\), \(\tilde{\Delta}\) is obtained by plugging in the adjusted maximum likelihood estimators. The variance of \(\tilde{\Delta}\) is obtained using the delta method.
To obtain a prediction interval that contains the true log hazard ratio for a new trial, that is, that contains \(\alpha + \beta(\mu + m_0) + g_0\), let
\[ V = Var(\tilde{\Delta}) + Var(g_0) + \beta^2\sigma^2_m \left(1 - \frac{\sigma_m}{\sqrt{\sigma^2_m + \sigma^2_0}}\right), \]
and let \(\tilde{V}\) denote its plug-in estimate. Then an approximate 95% prediction interval for the true log hazard ratio is
\[ \tilde{\Delta} \pm t_{n - 3} \sqrt{\tilde{V}}. \]
The code to estimate these predictions and standard errors (using the delta method) is as follows:
predict_klmfit## function (fit4, v, sig0)
## {
## delta <- fit4$par["muy"] + fit4$par["beta"] * v * (fit4$par["vmx"]/(fit4$par["vmx"] +
## sig0)) + fit4$par["beta"] * fit4$par["mux"] * (1 - fit4$par["vmx"]/(fit4$par["vmx"] +
## sig0))
## mu.deriv <- deriv(expression(muy + beta * v * (vmx/(vmx +
## sig0)) + beta * mux * (1 - vmx/(vmx + sig0))), namevec = c("mux",
## "muy", "beta", "vmx", "vmg"), function.arg = function(mux,
## muy, beta, vmx, vmg, v, sig0) {
## })
## delta <- do.call(mu.deriv, as.list(c(fit4$par, v, sig0)))
## mu.grad <- attr(delta, "gradient")
## var.delta <- c(mu.grad %*% solve(fit4$hessian) %*% t(mu.grad))
## a3 <- fit4$par["beta"]^2 * fit4$par["vmx"] * (1 - sqrt(fit4$par["vmx"])/sqrt(fit4$par["vmx"] +
## sig0))
## tilde.v <- var.delta + fit4$par["vmg"] + a3
## list(delta = delta, se.delta = sqrt(var.delta), se.pred = sqrt(tilde.v),
## a3 = a3)
## }
## <environment: namespace:pcrmeta>The performance of this method, including prediction interval coverage, is assessed in the simulation study vignette.
Results
Event Free Survival
Ten trials only
ten.dfs <- with(trials10, fit_model4(log(HR.DFS), log(OR.PCR), DFS.SE, PCR.SE))
kable(ten.dfs$summary)| Est | SE | |
|---|---|---|
| mux | 0.3873 | 0.1198 |
| muy | -0.1828 | 0.0805 |
| beta | 0.0452 | 0.1836 |
| vmx | 0.1119 | 0.0730 |
| vmg | 0.0075 | 0.0152 |
Prediction table
library(magrittr)
library(tidyr)##
## Attaching package: 'tidyr'
##
## The following object is masked from 'package:magrittr':
##
## extractparams <- data.frame(p1 = rep(1:4/10, 3),
p0 = rep(c(.1, .1, .1, .2), 3),
n = sort(rep(c(100, 300, 1000), 4)))
params$CI <- sapply(1:nrow(params), function(i){
with(params[i, ], fill_table(p1, p0, n, ten.dfs)$a1)
})
params %>%
spread(n, CI) %>% kable| p1 | p0 | 100 | 300 | 1000 |
|---|---|---|---|---|
| 0.1 | 0.1 | 0.84 (0.67, 1.07) | 0.84 (0.66, 1.07) | 0.84 (0.65, 1.08) |
| 0.2 | 0.1 | 0.85 (0.67, 1.08) | 0.86 (0.67, 1.10) | 0.86 (0.65, 1.13) |
| 0.3 | 0.1 | 0.86 (0.66, 1.11) | 0.87 (0.63, 1.19) | 0.88 (0.58, 1.32) |
| 0.4 | 0.2 | 0.86 (0.67, 1.10) | 0.86 (0.65, 1.15) | 0.87 (0.63, 1.20) |
Prediction table for a new observation
params$CI <- sapply(1:nrow(params), function(i){
with(params[i, ], fill_table(p1, p0, n, ten.dfs)$a3)
})
params %>%
spread(n, CI) %>% kable| p1 | p0 | 100 | 300 | 1000 |
|---|---|---|---|---|
| 0.1 | 0.1 | 0.84 (0.35, 2.07) | 0.84 (0.48, 1.46) | 0.84 (0.58, 1.22) |
| 0.2 | 0.1 | 0.85 (0.35, 2.08) | 0.86 (0.49, 1.49) | 0.86 (0.58, 1.27) |
| 0.3 | 0.1 | 0.86 (0.35, 2.11) | 0.87 (0.48, 1.57) | 0.88 (0.54, 1.43) |
| 0.4 | 0.2 | 0.86 (0.35, 2.10) | 0.86 (0.49, 1.53) | 0.87 (0.57, 1.32) |
All 23 trials
all.dfs <- with(rbind(trials10, trials13),
fit_model4(log(HR.DFS), log(OR.PCR), DFS.SE, PCR.SE))
kable(all.dfs$summary)| Est | SE | |
|---|---|---|
| mux | 0.3057 | 0.0942 |
| muy | -0.1072 | 0.0582 |
| beta | -0.1230 | 0.1506 |
| vmx | 0.1120 | 0.0610 |
| vmg | 0.0076 | 0.0093 |
Prediction table
params$CI <- sapply(1:nrow(params), function(i){
with(params[i, ], fill_table(p1, p0, n, all.dfs)$a1)
})
params %>%
spread(n, CI) %>% kable| p1 | p0 | 100 | 300 | 1000 |
|---|---|---|---|---|
| 0.1 | 0.1 | 0.87 (0.71, 1.07) | 0.88 (0.71, 1.08) | 0.89 (0.72, 1.10) |
| 0.2 | 0.1 | 0.85 (0.69, 1.05) | 0.84 (0.67, 1.05) | 0.83 (0.65, 1.04) |
| 0.3 | 0.1 | 0.84 (0.67, 1.05) | 0.81 (0.62, 1.06) | 0.78 (0.56, 1.09) |
| 0.4 | 0.2 | 0.84 (0.67, 1.05) | 0.82 (0.64, 1.05) | 0.81 (0.62, 1.06) |
Prediction table for a new trial
params$CI <- sapply(1:nrow(params), function(i){
with(params[i, ], fill_table(p1, p0, n, all.dfs)$a3)
})
params %>%
spread(n, CI) %>% kable| p1 | p0 | 100 | 300 | 1000 |
|---|---|---|---|---|
| 0.1 | 0.1 | 0.87 (0.40, 1.92) | 0.88 (0.54, 1.43) | 0.89 (0.65, 1.22) |
| 0.2 | 0.1 | 0.85 (0.39, 1.88) | 0.84 (0.51, 1.37) | 0.83 (0.59, 1.16) |
| 0.3 | 0.1 | 0.84 (0.38, 1.85) | 0.81 (0.48, 1.36) | 0.78 (0.52, 1.18) |
| 0.4 | 0.2 | 0.84 (0.38, 1.86) | 0.82 (0.50, 1.36) | 0.81 (0.56, 1.16) |
Overall Survival
Ten trials only
ten.os <- with(trials10, fit_model4(log(HR.OS), log(OR.PCR), OS.SE, PCR.SE))## Warning in sqrt(diag(solve(fit4$hessian))): NaNs producedkable(ten.os$summary)| Est | SE | |
|---|---|---|
| mux | 0.3882 | 0.1188 |
| muy | -0.1866 | 0.0827 |
| beta | 0.0525 | 0.1607 |
| vmx | 0.1100 | 0.0718 |
| vmg | 0.0000 | NaN |
Prediction table
params$CI <- sapply(1:nrow(params), function(i){
with(params[i, ], fill_table(p1, p0, n, ten.os)$a1)
})
params %>%
spread(n, CI) %>% kable| p1 | p0 | 100 | 300 | 1000 |
|---|---|---|---|---|
| 0.1 | 0.1 | 0.84 (0.75, 0.94) | 0.84 (0.74, 0.96) | 0.83 (0.71, 0.98) |
| 0.2 | 0.1 | 0.85 (0.77, 0.94) | 0.86 (0.77, 0.95) | 0.86 (0.75, 0.98) |
| 0.3 | 0.1 | 0.86 (0.76, 0.97) | 0.87 (0.72, 1.05) | 0.88 (0.67, 1.16) |
| 0.4 | 0.2 | 0.86 (0.77, 0.95) | 0.86 (0.75, 1.00) | 0.87 (0.72, 1.05) |
Prediction table for a new trial
params$CI <- sapply(1:nrow(params), function(i){
with(params[i, ], fill_table(p1, p0, n, ten.os)$a3)
})
params %>%
spread(n, CI) %>% kable| p1 | p0 | 100 | 300 | 1000 |
|---|---|---|---|---|
| 0.1 | 0.1 | 0.84 (0.35, 2.01) | 0.84 (0.50, 1.41) | 0.83 (0.61, 1.15) |
| 0.2 | 0.1 | 0.85 (0.36, 2.03) | 0.86 (0.51, 1.43) | 0.86 (0.64, 1.17) |
| 0.3 | 0.1 | 0.86 (0.36, 2.05) | 0.87 (0.51, 1.48) | 0.88 (0.60, 1.30) |
| 0.4 | 0.2 | 0.86 (0.36, 2.04) | 0.86 (0.51, 1.45) | 0.87 (0.62, 1.21) |
All 23 trials
all.os <- with(rbind(trials10, trials13),
fit_model4(log(HR.OS), log(OR.PCR), OS.SE, PCR.SE))
kable(all.os$summary)| Est | SE | |
|---|---|---|
| mux | 0.3011 | 0.0937 |
| muy | -0.1214 | 0.0575 |
| beta | -0.0740 | 0.2146 |
| vmx | 0.1110 | 0.0611 |
| vmg | 0.0000 | 0.0381 |
Prediction table
params$CI <- sapply(1:nrow(params), function(i){
with(params[i, ], fill_table(p1, p0, n, all.os)$a1)
})
params %>%
spread(n, CI) %>% kable| p1 | p0 | 100 | 300 | 1000 |
|---|---|---|---|---|
| 0.1 | 0.1 | 0.87 (0.77, 0.98) | 0.87 (0.79, 0.97) | 0.88 (0.79, 0.98) |
| 0.2 | 0.1 | 0.86 (0.73, 1.01) | 0.85 (0.69, 1.05) | 0.84 (0.64, 1.10) |
| 0.3 | 0.1 | 0.85 (0.68, 1.06) | 0.83 (0.60, 1.16) | 0.82 (0.52, 1.28) |
| 0.4 | 0.2 | 0.85 (0.69, 1.05) | 0.84 (0.63, 1.11) | 0.83 (0.59, 1.17) |
Prediction table for a new trial
params$CI <- sapply(1:nrow(params), function(i){
with(params[i, ], fill_table(p1, p0, n, all.os)$a3)
})
params %>%
spread(n, CI) %>% kable| p1 | p0 | 100 | 300 | 1000 |
|---|---|---|---|---|
| 0.1 | 0.1 | 0.87 (0.40, 1.88) | 0.87 (0.56, 1.37) | 0.88 (0.68, 1.15) |
| 0.2 | 0.1 | 0.86 (0.39, 1.87) | 0.85 (0.52, 1.39) | 0.84 (0.59, 1.20) |
| 0.3 | 0.1 | 0.85 (0.38, 1.88) | 0.83 (0.48, 1.44) | 0.82 (0.49, 1.36) |
| 0.4 | 0.2 | 0.85 (0.39, 1.87) | 0.84 (0.50, 1.41) | 0.83 (0.55, 1.26) |
References
Berruti, Alfredo, Vito Amoroso, Fabio Gallo, Valentina Bertaglia, Edda Simoncini, Rebecca Pedersini, Laura Ferrari, Alberto Bottini, Paolo Bruzzi, and Maria Pia Sormani. 2014. “Pathologic Complete Response as a Potential Surrogate for the Clinical Outcome in Patients with Breast Cancer After Neoadjuvant Therapy: A Meta-Regression of 29 Randomized Prospective Studies.” Journal of Clinical Oncology 32 (34). American Society of Clinical Oncology: 3883–91.
Cortazar, Patricia, Lijun Zhang, Michael Untch, Keyur Mehta, Joseph P Costantino, Norman Wolmark, Hervé Bonnefoi, et al. 2014. “Pathological Complete Response and Long-Term Clinical Benefit in Breast Cancer: The CTNeoBC Pooled Analysis.” The Lancet 384 (9938). Elsevier: 164–72.
Korn, Edward L, Paul S Albert, and Lisa M McShane. 2005. “Assessing Surrogates as Trial Endpoints Using Mixed Models.” Statistics in Medicine 24 (2). Wiley Online Library: 163–82.