---
title: "Exposure-Response Model Assessing the Impact of Covariates on Probabilities of Clinical Outcome"
output: 
  rmarkdown::html_vignette:
    toc: true
    df_print: kable
    
vignette: >
  %\VignetteIndexEntry{Exposure_Response_Example}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---
Here we illustrate the approach using a Binary response linked to exposure (AUC) via a saturating EMAX function. Weight is a covariate on Clearance. We also have a disease severity categorical covariate on EMAX where patient with severe disease have a lower EMAX.

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  message =FALSE,
  warning =FALSE,
  fig.width = 7,
  comment = "#>"
)
if (capabilities(("cairo"))) {
  knitr::opts_chunk$set(dev.args = list(png = list(type = "cairo")))
}

library(coveffectsplot)
library(ggplot2)
library(dplyr)
library(tidyr)
library(mrgsolve)
library(ggridges)
library(ggstance)
library(Rcpp)
theme_set(theme_bw())
nsim <- 100 # for vignette to make it run faster otherwise increase to 1000
#utility function to simulate varying one covariate at a time keeping the rest at the reference
expand.modelframe <- function(..., rv, covcol="covname") {
  args <- list(...)
  df <- lapply(args, function(x) x[[1]])
  df[names(rv)] <- rv
  res <- lapply(seq_along(rv), function(i) {
    df[[covcol]] <- names(rv)[i]
    df[[names(rv)[i]]] <- args[[names(rv)[i]]]
    as.data.frame(df)
  })
  do.call(rbind, res)
}

expit <- function(x) exp(x)/ (1 + exp(x) )
 
```

## Specifying an Exposure Response Model using `mrgsolve`
```{r exprespmodel, collapse=TRUE }
# the typical probability from the model parameters will be :
TypicalProb<- 1/(1+exp(-(log(0.1/(1-0.1)) + (5*75/10/(7.5+75/10)))))
MaxProb<- 1/(1+exp(-(log(0.1/(1-0.1)) + (5*750/10/(7.5+750/10)))))
MinProb<- 1/(1+exp(-(log(0.1/(1-0.1)) + (5*0/10/(7.5+0/10)))))

exprespmodel <- '
$PLUGIN Rcpp
$PARAM @annotated
TVCL    : 10  : Clearance CL (L/h)
WTCL    : 0.75: Weight on CL (ref. 70 kg)
TVEMAX  : 5   : Maximum Drug Effect
SEVEMAX : 3   : Severity Reduction of Drug Effect
AUC50   : 7.5 : Area Under the Curve providing half maximal response
BASEP   : 0.1 : Baseline Probability of Response 

$PARAM @annotated // reference values for covariate
WT      : 70  : Weight (kg)
SEV     : 0   : Sex (0=Female, 1=Male)
DOSE    : 75  : Dose (mg)
$OMEGA @annotated @block
nCL     :0.09 : ETA on CL


$PRED
double CL = TVCL *
    pow((WT/70.0), WTCL)*exp(ETA(1)); 
double EMAX = TVEMAX - SEVEMAX*(SEV == 1) ; 
double Intercept = log(BASEP/(1-BASEP));
capture CLi = CL;
capture AUC = DOSE/CL;
capture LGST = Intercept + (EMAX*AUC/(AUC50+AUC));
capture P1 = 1/(1+exp(-LGST));
capture DV = R::runif(0,1)< P1 ? 1 : 0;
'

modexprespsim <- mcode("exprespmodel", exprespmodel)
simdata <-  expand.idata(SEV=c(0),
               DOSE = c(0,75),
               ID = 1:1000) %>% 
  dplyr::mutate(WT = 70) #exp(rnorm(n(),log(70),0.3)
set.seed(466548)
simout <- modexprespsim %>%
  data_set(simdata) %>%
  carry.out(WT, DOSE, SEV) %>%
  mrgsim()%>%
  as.data.frame
```
## Probability of Cure 
This is a plot of the disease being cured versus PK exposure by disease severity and by Weight intervals.

```{r exprespmodeplotl, collapse=TRUE }
WT_names <- c(
  '70'="Weight: 70 kg"
)
SEV_names <- c(
  '0'="Severity: 0 (Not Severe)"
)
probplot<- ggplot(simout, aes(AUC,DV,linetype=factor(SEV))) +
  facet_grid( WT~SEV,labeller=labeller(WT=WT_names,SEV=SEV_names))+
  geom_point(position=position_jitter(height=0.02,width=0.1),
             aes(color=factor(DOSE)),size=1,alpha=0.5)+
  geom_line(aes(y=P1),color="black",size=1.1)+
  geom_label(data=data.frame(
    x=9,y=TypicalProb,label=paste(round(100*TypicalProb,1),"%"),SEV=0),
    aes(x=x,y=y,label=label),fill="transparent")+
  geom_label(data=data.frame(
    x=0.37,y=0.1,label=paste(round(100*0.1,1),"%"),SEV=0),
    aes(x=x,y=y,label=label),fill="transparent")+
  labs(color="Dose (mg)",y="Probability of Response",
       linetype="Severity")+
  theme_bw() + 
  theme(legend.position = "top")
probplot
```

```{r bsvrangeplot, collapse=TRUE }
simoutbsvplacebo <- simout %>% 
  filter(DOSE==0)%>% 
  mutate(LGST =LGST)%>%
  gather(paramname, paramvalue,LGST,P1)%>%
  group_by(paramname)%>%
  dplyr::summarize(P50 = quantile(paramvalue, 0.5)
)
simoutbsv <- simout %>% 
  mutate(logodds =LGST)%>% 
  filter(DOSE==75)
# the probability of response at the typical AUC
simoutbsvlong  <- simoutbsv  %>%
  mutate(P1std=P1/TypicalProb) %>% 
  gather(paramname, paramvalue,P1std,P1)


yvar_names <- c(
  'P1std'="Standardized Probability",
  'P1'="Probability"
)

pbsvranges<-  ggplot(simoutbsvlong, aes(
        x      = paramvalue,
        y      = paramname,
        fill   = factor(..quantile..),
        height = ..ndensity..)) +
  facet_wrap(paramname~. , scales="free", ncol=1,
             labeller=labeller(paramname=yvar_names) ) +
  stat_density_ridges(
    geom="density_ridges_gradient", calc_ecdf=TRUE,
    quantile_lines=TRUE, rel_min_height=0.001, scale=0.9,
    quantiles=c(0.05, 0.25, 0.5, 0.75, 0.95)) +
  scale_fill_manual(
    name="Probability",
    values=c("white", "#FF000050", "#FF0000A0", "#FF0000A0", "#FF000050", "white"),
    labels = c("(0, 0.05]", "(0.05, 0.25]",
             "(0.25, 0.5]", "(0.5, 0.75]",
             "(0.75, 0.95]", "(0.95, 1]")) +
  theme_bw() +
  theme(
    legend.position = "none",
    axis.text.y     = element_blank(),
    axis.ticks.y    = element_blank(),
    axis.title.y    = element_blank()) +
  labs(x="Parameters", y="") +
  scale_x_log10() +
  coord_cartesian(expand=FALSE)

pbsvranges


simoutbsvranges <- simoutbsvlong %>%
  group_by(paramname)%>%
  dplyr::summarize(
   P05 = quantile(paramvalue, 0.05),
    P25 = quantile(paramvalue, 0.25),
    P50 = quantile(paramvalue, 0.5),
    P75 = quantile(paramvalue, 0.75),
    P95 = quantile(paramvalue, 0.95))
simoutbsvranges
```

```{r, fig.width=7 ,message=FALSE, include=FALSE}
# probplot <-  probplot +theme_bw(base_size=18)+
#    theme(legend.position = "top")
# pbsvranges <- pbsvranges+theme_bw(base_size=22)+
#   theme(
#     legend.position = "none",
#     axis.text.y     = element_blank(),
#     axis.ticks.y    = element_blank(),
#     axis.title.y    = element_blank())
#  ggsave(plot = egg::ggarrange(probplot , pbsvranges,ncol=2),
#         "Figure_8_1.png", device="png",type="cairo-png",width = 10, height = 5)
```

## Computing the Odds and Probabilities
Here we show how the odds and probabilities can be computed. We already know that the distribution of AUC depends on the Dose and on the clearance distributions. The model had five parameters shown in <span style="color: red;">red</span>, the dose, disease severity and weight were covariates and are shown in <span style="color: green;">green</span>. A Change in body weight will trigger a change in Clearance which in turn will control the AUC. To define an odds ratio we need to define a reference odds with reference covariate values Severity = 0 and changes in covariate values for example Severity = 1 (everything else being equal). For nonlinear relationships, in addition to the covariate unit change e.g. 25 mg change of dose it is important to define what reference value we are using e.g. A change from Placebo = 0 mg to 25 mg is not the same as a change from the typical dose of 75 mg increasing it to 100 mg.

where:
$$AUC = \left(\frac { \color{green}{Dose}} {\color{red}{CL} \times \left( \frac { \color{green}{Weight}} {70}\right)^{WTCL} \times exp(\eta{CL}) }\right)$$
$$E_{max}= \color{red}{E_{max} \left(intercept \right)} + \color{red}{SevE_{max}}\times\left(\color{green}{Severity} = 1\right) $$
$$log(odds) = \color{red}{intercept} + \left( \frac {E_{max} \times \color{blue}{AUC}} {\color{red}{AUC_{50}} +\color{blue}{AUC} }\right)$$

```{r, collapse=TRUE }
set.seed(678549)
thmeans <- c(10,0.75, #TVCL WTCL
             5,3, # TVEMAX  SEVEMAX
             7.5, # AUC50
              0.1) #BASEP
thvariances<- (thmeans*0.15)^2
thecorrelations <- matrix(ncol=length(thmeans),nrow=length(thmeans))
diag(thecorrelations)<- 1
thecorrelations[lower.tri(thecorrelations, diag = FALSE)]<- 0.2
thecorrelations[upper.tri(thecorrelations, diag = FALSE)]<- 0.2
thevarcovmatrix<- diag(sqrt(thvariances))%*%thecorrelations%*%diag(sqrt(thvariances))
sim_parameters <- MASS::mvrnorm(n = nsim, mu=as.numeric(thmeans),
                                Sigma=thevarcovmatrix, empirical = TRUE)
colnames(sim_parameters) <- colnames(thevarcovmatrix) <- c("TVCL","WTCL",
                                                           "TVEMAX","SEVEMAX","AUC50",
                                                           "BASEP")
sim_parameters<- as.data.frame(sim_parameters)

reference.values <- data.frame(WT = 70, DOSE = 75, SEV = 0 )   
covcomb <- expand.modelframe(
  WT  = c(50,60,70,80,90),
  DOSE = c(0,25,50,75,100,125,150),
  SEV = c(0,1),
  rv = reference.values)
covcomb <- covcomb[!duplicated(
  paste(covcomb$WT,covcomb$WT,covcomb$DOSE,covcomb$SEV)),]
covcomb$ID <- 1:nrow(covcomb)

iter_sims <- NULL
for(i in 1:nsim) {
  idata <- as.data.frame(covcomb)
  idata$covname<- NULL
  data.all <- idata
  data.all$TVCL <- as.numeric(sim_parameters[i,1])
  data.all$WTCL <- as.numeric(sim_parameters[i,2])
  data.all$TVEMAX   <- as.numeric(sim_parameters[i,3])
  data.all$SEVEMAX      <- as.numeric(sim_parameters[i,4])
  data.all$AUC50 <- as.numeric(sim_parameters[i,5])
  data.all$BASEP <- as.numeric(sim_parameters[i,6])
  out <- modexprespsim %>%
    data_set(data.all) %>%
    carry.out(CL,WT, DOSE, SEV, AUC) %>%
    zero_re() %>% 
    mrgsim()
  dfsimunc <- as.data.frame(out%>% mutate(rep = i) )
  iter_sims <- rbind(iter_sims,dfsimunc)
}
```


```{r, collapse=TRUE }
stdprobplot<- ggplot(iter_sims, aes(DOSE,P1,col=factor(SEV) ) )+
  geom_point(aes(group=interaction(ID,rep)),alpha=0.5,size=3)+
  geom_hline(yintercept=TypicalProb)+
  facet_grid(SEV~ WT,labeller = label_both)+
  labs(y="Probability of Response", colour="Severity")
stdprobplot
```

```{r, fig.width=7 ,message=FALSE, include=FALSE}
#  stdprobplot<-  stdprobplot+
#   theme(axis.title.y = element_text(size=15),
#         legend.position = c(0.8,0.2))+
#   scale_y_continuous( sec.axis = sec_axis(~ . /TypicalProb,
#                                           name = "Standardized Probability"))
# stdprobplot
#   ggsave("Figure_8_2.png", device="png",type="cairo-png",
#          width = 7, height = 4)
```


```{r, fig.height= 7, collapse=TRUE }
iter_sims <- iter_sims %>%
  mutate(P1std=P1/TypicalProb)%>%
  gather(paramname,paramvalue,P1std)%>% 
  ungroup() %>% 
  dplyr::mutate( covname = case_when(
    ID== 1 ~ "Weight",
    ID== 2 ~ "Weight",
    ID== 3 ~ "REF",
    ID== 4 ~ "Weight",
    ID== 5 ~ "Weight",
    ID== 6 ~ "DOSE",
    ID== 7 ~ "DOSE",
    ID== 8 ~ "DOSE",
    ID== 9 ~ "DOSE",
    ID== 10 ~ "DOSE",
    ID== 11 ~ "DOSE",
    ID== 12 ~ "SEV"
  ),
  covvalue =case_when(
    ID== 1 ~ paste(WT,"kg"), 
    ID== 2 ~ paste(WT,"kg"),
    ID== 3 ~ "70 kg\nNot Severe\n75 mg",
    ID== 4 ~ paste(WT,"kg"),
    ID== 5 ~ paste(WT,"kg"),
    ID== 6 ~ paste(DOSE,"mg"),
    ID== 7 ~ paste(DOSE,"mg"),
    ID== 8 ~ paste(DOSE,"mg"),
    ID== 9 ~ paste(DOSE,"mg"),
    ID== 10 ~ paste(DOSE,"mg"),
    ID== 11 ~ paste(DOSE,"mg"),
    ID== 12 ~ "Severe"
  ) )
iter_sims$covname <-factor(as.factor(iter_sims$covname ),
                          levels =  c("Weight","DOSE","SEV","REF"))
iter_sims$covvalue <- factor(as.factor(iter_sims$covvalue),
                          levels =  c("0 mg","25 mg","50 mg",
                                      "100 mg","125 mg","150 mg",
                          "50 kg","60 kg","80 kg", "90 kg",
                          "70 kg\nNot Severe\n75 mg",  "Severe"))


ggplot(iter_sims,aes(x=paramvalue,y=covvalue))+
  stat_density_ridges(aes(fill=factor(..quantile..),height=..ndensity..),
    geom = "density_ridges_gradient", calc_ecdf = TRUE,
    quantile_lines = TRUE, rel_min_height = 0.001,scale=0.9,
    quantiles = c(0.025,0.5, 0.975))+
    facet_grid(covname~paramname,scales="free",switch="both",
             labeller = labeller(paramname=yvar_names))+ 
  scale_fill_manual(
    name = "Probability", values = c("white","#0000FFA0", "#0000FFA0","white"),
    labels = c("(0, 0.025]","(0.025, 0.5]","(0.5, 0.975]","(0.975, 1]")
  )+
  theme_bw()+
  theme(axis.title = element_blank(),strip.placement = "outside")
```
```{r, fig.width=7 ,message=FALSE, include=FALSE}
  #   last_plot()+theme_bw(base_size = 16)+
  # theme(legend.position = "none",
  #       axis.title = element_blank(),strip.placement = "outside")
  #  ggsave("Figure_8_3.png", device="png",type="cairo-png",
  #         width= 7, height = 6,dpi=72)
```

```{r plot3, collapse=TRUE }
coveffectsdatacovrep <- iter_sims %>%
  dplyr::group_by(paramname,ID,WT,DOSE,SEV,covname,covvalue) %>% 
  dplyr::summarize(
    mid= median(paramvalue),
    lower= quantile(paramvalue,0.025),
    upper = quantile(paramvalue,0.975))%>% 
  dplyr::filter(!is.na(mid))
simoutbsvranges<-simoutbsvranges[simoutbsvranges$paramname=="P1std",]

coveffectsdatacovrepbsv <- coveffectsdatacovrep[coveffectsdatacovrep$covname=="REF",]
coveffectsdatacovrepbsv$covname <- "BSV"
coveffectsdatacovrepbsv$covvalue <- "90% of patients"
coveffectsdatacovrepbsv$label <-    "90% of patients"
coveffectsdatacovrepbsv$lower <- simoutbsvranges$P05
coveffectsdatacovrepbsv$upper <- simoutbsvranges$P95
coveffectsdatacovrepbsv2 <- coveffectsdatacovrep[coveffectsdatacovrep$covname=="REF",]
coveffectsdatacovrepbsv2$covname <- "BSV"
coveffectsdatacovrepbsv2$covvalue <- "50% of patients"
coveffectsdatacovrepbsv2$label <-    "50% of patients"
coveffectsdatacovrepbsv2$lower <- simoutbsvranges$P25
coveffectsdatacovrepbsv2$upper <- simoutbsvranges$P75

coveffectsdatacovrepbsv<- rbind(coveffectsdatacovrep,coveffectsdatacovrepbsv2,
                                coveffectsdatacovrepbsv)

coveffectsdatacovrepbsv <- coveffectsdatacovrepbsv %>% 
  mutate(
    label= covvalue,
    LABEL = paste0(format(round(mid,2), nsmall = 2),
                   " [", format(round(lower,2), nsmall = 2), "-",
                   format(round(upper,2), nsmall = 2), "]"))
coveffectsdatacovrepbsv<- as.data.frame(coveffectsdatacovrepbsv)

coveffectsdatacovrepbsv$label <-factor(as.factor(coveffectsdatacovrepbsv$label ),
            levels = c("All Subjects","90% of patients","50% of patients",
              "50 kg","60 kg","80 kg","90 kg",
              "0 mg","25 mg","50 mg","100 mg","125 mg","150 mg",
              "Severe","70 kg\nNot Severe\n75 mg"
              ))
    

coveffectsdatacovrepbsv$covname <-factor(as.factor(coveffectsdatacovrepbsv$covname ),
                    levels = c("Weight","DOSE","SEV","REF","BSV"))

ref_legend_text      <- "Reference (vertical line)"
png("./Figure_8_4.png",width =9 ,height = 7,units = "in",res=72)
forest_plot(coveffectsdatacovrepbsv,
                            strip_placement = "outside",
                            show_ref_area = FALSE,
                            show_ref_value=TRUE,
                            ref_legend_text = ref_legend_text,
                            plot_table_ratio = 2,
                            base_size = 12,
                            table_text_size = 4,
                            y_label_text_size      = 12,
                            xlabel= " ",
                            facet_formula = "covname~paramname",
                            facet_labeller      = labeller(paramname=yvar_names),
                            facet_scales           = "free",
                            facet_space ="free",
                            logxscale              = TRUE,
                            major_x_ticks          = c(0.1,0.25, 0.5,1,1.5),
                            x_range                = c(0.1, 1.5))

dev.off()

```
![Covariate Effects Plot.](./Figure_8_4.png)


