---
title: "Continuous data MCP-Mod"
output: rmarkdown::html_vignette
bibliography: refs.bib
csl: american-statistical-association.csl
link-citations: yes
vignette: >
  %\VignetteIndexEntry{Analysis template MCP-Mod for continuous data}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, child="children/settings.txt"}
```

## 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](sample_size.html).

We will use data from @verkindre2010,
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](https://clinicaltrials.gov/ct2/show/NCT00501852)
and on the R help page `?glycobrom`.

The main purpose @verkindre2010 was to provide safety and efficacy data on Glycopyrronium Bromide (NVA237)
in patients with stable Chronic Obstructive Pulmonary Disease ([COPD](https://en.wikipedia.org/wiki/Chronic_obstructive_pulmonary_disease)).
The primary endpoint in this study was the mean of two measurements of forced expiratory volume in 1 second
([FEV1](https://en.wikipedia.org/wiki/FEV1#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.

```{r, load_data}
library(DoseFinding)
data(glycobrom)
print(glycobrom)
```

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.

```{r, simulate_dataset}
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 $\delta = \beta_2/\lvert\beta_1\rvert$ in the quadratic model $f(d, \theta) = E_0 + \beta_1 d + \beta_2 d^2$

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.

```{r, models}
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.

```{r, plot_models, fig.asp = 0.42}
plot(mods, ylab = "FEV1", layout = c(4, 1))
```

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.

```{r, contrasts}
optC <- optContr(mods, w=1)
print(optC)
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 $\mu^0_m$ 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 @pinheiro2014 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.

### Multiple comparisons
We run the multiple contrast test with the pre-specified models.
Note that the `type` parameter defaults to `type="normal"`,
which means that we assume a homoscedastic ANOVA model for `FEV1`,
i.e. critical values are taken from a multivariate t distribution.
Further note that when `data` is supplied,
the first two arguments `dose` and `FEV1` are _not evaluated_,
but symbolically refer to the columns in `data=NVA`.
```{r, mctest_normal}
test_normal <- MCTtest(dose = dose, resp = FEV1, models = mods, data = NVA)
print(test_normal)
```

The test results suggest a clear dose-response trend.

Alternatively we can use generalized MCP-Mod
(see the FAQ for the [difference](faq.html)).
We use R's builtin `lm()` function to manually fit the ANOVA model
and extract estimates for the model coefficients and their covariance matrix.
We also need the model degrees of freedom.

```{r, fit_lm_1}
fitlm <- lm(FEV1 ~ factor(dose) - 1, data = NVA)
mu_hat <- coef(fitlm)
S_hat <- vcov(fitlm)
anova_df <- fitlm$df.residual
```

Next we supply them to the `MCTtest` function together with `type="general"`.
Note that in contrast to the invocation above we here supply the `doses` and the estimates `mu_hat` and `S_hat` directly and not within a `data.frame`.

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

For the simple ANOVA case at hand the results of the original and the generalized MCP-Mod approaches actually coincide.
The p-values differ due to the numerical methods used for obtaining them.

```{r, compare_normal_generalized}
cbind(normal = test_normal$tStat, generalized = test_general$tStat)
cbind(normal = attr(test_normal$tStat, "pVal"), generalized = attr(test_general$tStat, "pVal"))
```

## 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):
```{r, fit_single}
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 @schorning2016 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.

```{r, fit_lm_2}
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.

```{r, bootstrap_draw}
one_bootstrap_prediction <- function(mu_hat, S_hat, doses, bounds, dose_seq) {
  sim <- drop(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.

```{r, bootstrap_summarize}
# 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.

```{r, bootstrap_plot}
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](binary_data.html), 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.

## References
