Continuous data MCP-Mod

Background and Data

In this vignette we will illustrate the usage of the DoseFinding package for analyzing continuously distributed data. There is a separate vignette with details on sample size and power calculation.

We will use data from Verkindre et al. (2010), who actually use a cross-over design and utilize MCP-Mod in a supportive analysis. More information can be found at the corresponding clinicaltrials.gov page and on the R help page ?glycobrom.

The main purpose Verkindre et al. (2010) was to provide safety and efficacy data on Glycopyrronium Bromide (NVA237) in patients with stable Chronic Obstructive Pulmonary Disease (COPD). The primary endpoint in this study was the mean of two measurements of forced expiratory volume in 1 second (FEV1) at 23h 15min and 23h 45min post dosing, following 7 days of treatment.

In order to keep this exposition simple, we will ignore the active control and focus on the placebo group and the four dose groups (12.5, 25, 50, and 100μg).

For the purpose here, we recreate a dataset that mimicks a parallel group design, based on the published summary statistics. These can be found in the glycobrom dataset coming with the DoseFinding package. Here fev1 and sdev contain the mean and standard deviation of the mean (standard error) of the primary endpoint for each group, while n denotes the number of participants.

library(DoseFinding)
data(glycobrom)
print(glycobrom)

   dose  fev1   sdev  n
1   0.0 1.243 0.0156 49
2  12.5 1.317 0.0145 55
3  25.0 1.333 0.0151 51
4  50.0 1.374 0.0148 53
5 100.0 1.385 0.0148 53

We want to create a dataset with 60 participants in each of the five groups. Noticing that the standard errors are essentially equal across all groups, we draw five vectors of measurement errors centered at 0 with identical variances 60 * 0.015^2 which we add to the observed means.

Note that here we use MASS::mvrnorm instead of rnorm because it lets us generate random numbers with the specified sample mean and sd.

set.seed(1, kind = "Mersenne-Twister", sample.kind = "Rejection", normal.kind = "Inversion")
rand <- rep(MASS::mvrnorm(60, 0, 60 * 0.015^2, empirical = TRUE), 5)
NVA <- data.frame(dose = rep(glycobrom$dose, each = 60),
                  FEV1 = rep(glycobrom$fev1, each = 60) + rand)
ggplot(NVA) + geom_jitter(aes(dose, FEV1), height = 0, width = 4) +
  labs(title = "Simulated FEV1 by dose (jittered horizontally)") +
  xlab("dose [μg]") + ylab("FEV1 [l]")

Design stage

Now let’s forget we already saw the data and imagine we had to design this trial with MCP-Mod.

First we decide that we want to include two Emax models, one sigmoid Emax model and one quadratic model in the analysis (see ?drmodels for other choices). While the (sigmoid) Emax type covers monotonic dose-response-relationships, the quadratic model is there to accommodate a potentially decreasing effect at high doses.

Next we have to supply guesstimates for the nonlinear parameters:

• ED50 for an Emax model
• ED50 and the Hill parameter h for a sigmoid emax model
• coefficient ratio δ = β₂/|β₁| in the quadratic model f(d, θ) = E₀ + β₁d + β₂d²

The following choices cover a range of plausible relationships:

• ED50 = 2.6 and ED25 = 12.5 for the Emax models (all doses have substantive effects)
• ED50 = 30.5 and h = 3.5 for the sigEmax model (first dose has a negligible effect)
• delta = -0.00776 for the quadratic model (downturn for the fourth dose)

We also fix the effect of placebo at an FEV1 of 1.25 liters and the maximum effect at 0.15 liters above placebo. This implicitly sets the common linear parameters of all the models.

Note the syntax of the arguments to the Mods function: emax = c(2.6, 12.5) specifies two Emax models, but sigEmax = c(30.5, 3.5) only specifies one Sigmoid Emax model.

doses <- c(0, 12.5, 25, 50, 100)
mods <- Mods(emax = c(2.6, 12.5), sigEmax = c(30.5, 3.5), quadratic = -0.00776,
             placEff = 1.25, maxEff = 0.15, doses = doses)

It’s always a good idea to perform a visual sanity check of the functional relationships implied by the guesstimates.

plotMods(mods, ylab = "FEV1")

This concludes the design phase.

We can also take a look at the calculated optimal contrasts. Each contrast has maximum power to detect a non-flat effect profile in the hypothetical world where the particular guesstimate is actually the true value.

optC <- optContr(mods, w=1)
print(optC)

Optimal contrasts
      emax1  emax2 sigEmax quadratic
0    -0.886 -0.813  -0.486    -0.723
12.5  0.116 -0.101  -0.439    -0.240
25    0.211  0.136  -0.120     0.140
50    0.265  0.326   0.448     0.587
100   0.294  0.452   0.597     0.236

plot(optC)

It can be seen that in the balanced sample size case and equal variance assumed for each dose group, the optimal contrasts reflect the underlying assumed mean dose-response shape. This is no surprise, given that the optimal contrasts are given by $$ c^{\textrm{opt}} \propto S^{-1}\biggl(\mu^0_m - \frac{(\mu^0_m)^T S^{-1}1_K}{1_K^T S^{-1} 1_K}\biggr) $$ where μ_m⁰ is the standardized mean response, K is the number doses, and 1_K is an all-ones vector of length K and S is the covariance matrix of the estimates at the doses (see Pinheiro et al. 2014 for a detailed account). As we have equal variance in all dose groups in our case and no correlation, the optimal contrasts are all proportional to the shapes of the candidate model mean vectors. As the standardized model is used in the formula, the values of the linear parameters of the models do not impact the optimal contrasts.

Analysis stage

Now fast-forward to the time when we have collected the data.

test_normal <- MCTtest(dose = dose, resp = FEV1, models = mods, data = NVA)
print(test_normal)

Multiple Contrast Test

Contrasts:
      emax1  emax2 sigEmax quadratic
0    -0.886 -0.813  -0.486    -0.723
12.5  0.116 -0.101  -0.439    -0.240
25    0.211  0.136  -0.120     0.140
50    0.265  0.326   0.448     0.587
100   0.294  0.452   0.597     0.236

Contrast Correlation:
          emax1 emax2 sigEmax quadratic
emax1     1.000 0.957   0.648     0.867
emax2     0.957 1.000   0.839     0.929
sigEmax   0.648 0.839   1.000     0.844
quadratic 0.867 0.929   0.844     1.000

Multiple Contrast Test:
          t-Stat  adj-p
emax2      7.443 <0.001
quadratic  7.016 <0.001
emax1      6.937 <0.001
sigEmax    6.676 <0.001

fitlm <- lm(FEV1 ~ factor(dose) - 1, data = NVA)
mu_hat <- coef(fitlm)
S_hat <- vcov(fitlm)
anova_df <- fitlm$df.residual

test_general <- MCTtest(dose = doses, resp = mu_hat, S = S_hat, df = anova_df,
                        models = mods, type = "general")
print(test_general)

Multiple Contrast Test

Contrasts:
      emax1  emax2 sigEmax quadratic
0    -0.886 -0.813  -0.486    -0.723
12.5  0.116 -0.101  -0.439    -0.240
25    0.211  0.136  -0.120     0.140
50    0.265  0.326   0.448     0.587
100   0.294  0.452   0.597     0.236

Contrast Correlation:
          emax1 emax2 sigEmax quadratic
emax1     1.000 0.957   0.648     0.867
emax2     0.957 1.000   0.839     0.929
sigEmax   0.648 0.839   1.000     0.844
quadratic 0.867 0.929   0.844     1.000

Multiple Contrast Test:
          t-Stat  adj-p
emax2      7.443 <0.001
quadratic  7.016 <0.001
emax1      6.937 <0.001
sigEmax    6.676 <0.001

cbind(normal = test_normal$tStat, generalized = test_general$tStat)

            normal generalized
emax1     6.937000    6.937000
emax2     7.442849    7.442849
sigEmax   6.675739    6.675739
quadratic 7.016303    7.016303

cbind(normal = attr(test_normal$tStat, "pVal"), generalized = attr(test_general$tStat, "pVal"))

           normal  generalized
[1,] 1.221279e-11 1.224099e-11
[2,] 5.375700e-13 6.030731e-13
[3,] 5.924716e-11 6.039769e-11
[4,] 8.166245e-12 7.672307e-12

Dose-response estimation

In the simplest case we would now proceed to fit only a single model type, for example the one with the largest t-statistic (or alternatively smallest AIC or BIC):

fit_single <- fitMod(dose, FEV1, NVA, model = "emax")
plot(fit_single)

But actually we want to use a more robust approach that combines bootstrapping with model averaging in the generalized MCP-Mod framework.

First we draw bootstrap samples from the multivariate normal distribution of the estimates originating from the first-stage model. Next, for each bootstrapped data set we fit our candidate models, select the one with lowest AIC and save the corresponding estimated quantities of interest. This selection step implies that the bootstrap samples potentially come from different models. Finally we use these bootstrapped estimates for inference. For example, we can estimate a dose-response curve by using the median over the bootstrapped means at each dose. Similarly we can derive confidence intervals based on bootstrap quantiles. Inference for other quantities of interest can be performed in an analogous way.

As different models contribute to the bootstrap resamples, the approach can be considered more robust than simple model selection (see also Schorning et al. 2016 for simulations on this topic).

Now let’s apply this general idea to the case at hand. Our first-stage model is an ANOVA, and we’re interested in an estimate of the dose-response curve plus confidence intervals. Our set of candidate model types consists of Emax, sigEmax and quadratic.

We us R’s builtin lm() function to fit an ANOVA model without intercept and extract estimates for the model coefficients and their covariance matrix.

fitlm <- lm(FEV1 ~ factor(dose) - 1, data = NVA)
mu_hat <- coef(fitlm)
S_hat <- vcov(fitlm)

In the following function we simulate a vector of mean FEV1 values, fit our set of candidate models (generalized MCP-Mod is indicated by supplying type = "general") and select the one with lowest AIC. From the selected model we predict the mean response at the doses supplied in the dose_seq argument.

Note that for technical reasons we have to supply boundaries to the fitting algorithm via the bnds argument to fitMod (see ?fitMod and ?defBnds for details). We also don’t need to supply the degrees of freedom here, as they are used neither for fitting nor prediction.

one_bootstrap_prediction <- function(mu_hat, S_hat, doses, bounds, dose_seq) {
  sim <- drop(mvtnorm::rmvnorm(1, mu_hat, S_hat))
  fit <- lapply(c("emax", "sigEmax", "quadratic"), function(mod)
    fitMod(doses, sim, model = mod, S = S_hat, type = "general", bnds = bounds[[mod]]))
  index <- which.min(sapply(fit, gAIC))
  pred <- predict(fit[[index]], doseSeq = dose_seq, predType = "ls-means")
  return(pred)
}

Now we need a function to calculate medians and other quantiles on a bootstrap sample. In principle we could also look a the mean instead of the median.

# bs_predictions is a doses x replications matrix,
# probs is a 4-element vector of increasing probabilities for the quantiles
#   that will be used in the plotting code for outer and inner confidence intervals
summarize_predictions <- function(bs_predictions, probs) {
  stopifnot(length(probs) == 4)
  med <- apply(bs_predictions, 1, median)
  quants <- apply(bs_predictions, 1, quantile, probs = probs)
  bs_df <- as.data.frame(cbind(med, t(quants)))
  names(bs_df) <- c("median", "outer_low", "inner_low", "inner_high", "outer_high")
  return(bs_df)
}

Finally we plot the bootstrap quantiles together with point estimates and confidence intervals from the first-stage ANOVA fit.

dose_seq <- 0:100
bs_rep <- replicate(1000, one_bootstrap_prediction(mu_hat, S_hat, doses, defBnds(max(doses)), dose_seq))
bs_summary <- summarize_predictions(bs_rep, probs = c(0.025, 0.25, 0.75, 0.975))
ci_half_width <- qt(0.975, fitlm$df.residual) * sqrt(diag(S_hat))
lm_summary <- data.frame(dose = doses, mu_hat = mu_hat,
                         low = mu_hat - ci_half_width, high = mu_hat + ci_half_width)

ggplot(cbind(bs_summary, dose_seq = dose_seq)) + geom_line(aes(dose_seq, median)) +
  geom_ribbon(aes(x = dose_seq, ymin = inner_low, ymax = inner_high), alpha = 0.2) +
  geom_ribbon(aes(x = dose_seq, ymin = outer_low, ymax = outer_high), alpha = 0.2) +
  geom_point(aes(dose, mu_hat), lm_summary) +
  geom_errorbar(aes(dose, ymin = low, ymax = high), lm_summary, width = 0, alpha = 0.5) +
  scale_y_continuous(breaks = seq(1.2,1.45,by=0.02)) +
  xlab("Dose") + ylab("FEV1") +
  labs(title = "ANOVA and bootstrap estimates for FEV1 population average",
       subtitle = "confidence levels 50% and 95%")

How to adjust for covariates?

In all practical situations covariates will be used to adjust for in the analysis. The MCP step can then be performed for example by including the covariates in the addCovars argument. Another approach to perform the MCP step is based on the differences to placebo: In a first stage lm(.) is fit with an intercept included. Then the treatment differences and corresponding covariance matrix would be extracted. This could then be fed into the MCTtest function, with the placAdj = TRUE argument, see ?MCTtest for an example. Both approaches will give the same result.

A third alternative is to calculate the adjusted means (and corresponding covariance matrix) and then perform generalized MCP-Mod based on these estimates (following the same steps as in the unadjusted analysis above, but adding the type = "general" argument as well as the estimated covariance matrix via S). The procedure is very similar to the situation explained in detail in the vignette for the analysis of binary data, so not repeated here.

For the case of normally distributed data adjusted means are calculated by predicting the outcome (using the covariate adjusted model) of each patient in the study under every dose, and then averaging over all patients per dose.