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<h1 class="title toc-ignore">PK/PD, Exposure-Response - Binary</h1>
<h4 class="author">Alison Margolskee, Andy Stein</h4>
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<div id="overview" class="section level2">
<h2>Overview</h2>
<!--START_EXPLANATION-->
<p>This document contains exploratory plots for binary response data as
well as the R code that generates these graphs.The plots presented here
are based on simulated data (<a href="PKPD_Datasets.html">see: PKPD
Datasets</a>). Data specifications can be accessed on <a
href="Datasets.html">Datasets</a> and Rmarkdown template to generate
this page can be found on <a
href="Rmarkdown/Multiple_Ascending_Dose_PKPD_binary.Rmd">Rmarkdown-Template</a>.
You may also download the Multiple Ascending Dose PK/PD dataset for your
reference (<a href="Data/Multiple_Ascending_Dose_Dataset2.csv">download
dataset</a>). <!--END_EXPLANATION--></p>
</div>
<div id="setup" class="section level2">
<h2>Setup</h2>
<pre class="r"><code>library(ggplot2)
library(dplyr)
library(xgxr)
library(tidyr)
#set seed for random number generator (for plots with jitter)
set.seed(12345)
#flag for labeling figures as draft
status = "DRAFT"
# ggplot settings
xgx_theme_set()
#directories for saving individual graphs
dirs = list(
parent_dir = "Parent_Directory",
rscript_dir = "./",
rscript_name = "Example.R",
results_dir = "./",
filename_prefix = "",
filename = "Example.png")</code></pre>
</div>
<div id="load-dataset" class="section level2">
<h2>Load Dataset</h2>
<!--START_EXPLANATION-->
<p>The plots presented here are based on simulated data (<a
href="PKPD_Datasets.html">see: PKPD Datasets</a>). You may also download
the Multiple Ascending Dose PK/PD dataset for your reference (<a
href="Data/Multiple_Ascending_Dose_Dataset2.csv">download dataset</a>).
<!--END_EXPLANATION--></p>
<pre class="r"><code>#load dataset
pkpd_data <- read.csv("../Data/Multiple_Ascending_Dose_Dataset2.csv")
DOSE_CMT = 1
PK_CMT = 2
PD_CMT = 6
SS_PROFDAY = 6 # steady state prof day
PD_PROFDAYS <- c(0, 2, 4, 6)
TAU = 24 # time between doses, units should match units of TIME, e.g. 24 for QD, 12 for BID, 7*24 for Q1W (when units of TIME are h)
#ensure dataset has all the necessary columns
pkpd_data = pkpd_data %>%
mutate(ID = ID, #ID column
TIME = TIME, #TIME column name
NOMTIME = NOMTIME,#NOMINAL TIME column name
PROFDAY = case_when(
NOMTIME < (SS_PROFDAY - 1)*24 ~ 1 + floor(NOMTIME / 24),
NOMTIME >= (SS_PROFDAY - 1)*24 ~ SS_PROFDAY
), #PROFILE DAY day associated with profile, e.g. day of dose administration
LIDV = LIDV, #DEPENDENT VARIABLE column name
CENS = CENS, #CENSORING column name
CMT = CMT, #COMPARTMENT column
DOSE = DOSE, #DOSE column here (numeric value)
TRTACT = TRTACT, #DOSE REGIMEN column here (character, with units),
LIDV_NORM = LIDV/DOSE,
LIDV_UNIT = EVENTU,
DAY_label = ifelse(PROFDAY > 0, paste("Day", PROFDAY), "Baseline"),
BINARY_LEVELS = factor(case_when(
CMT != PD_CMT ~ as.character(NA),
LIDV == 0 ~ "Nonresponder",
LIDV == 1 ~ "Responder"
), levels = c("Nonresponder", "Responder"))
)
#create a factor for the treatment variable for plotting
pkpd_data = pkpd_data %>%
arrange(DOSE) %>%
mutate(TRTACT_low2high = factor(TRTACT, levels = unique(TRTACT)),
TRTACT_high2low = factor(TRTACT, levels = rev(unique(TRTACT)))) %>%
select(-TRTACT)
#create pk dataset
pk_data <- pkpd_data %>%
filter(CMT==PK_CMT)
#create pd binary dataset
pd_data <- pkpd_data %>%
filter(CMT==PD_CMT) %>%
mutate(LIDV_jitter = jitter(LIDV, amount = 0.1))
#create wide pkpd dataset for plotting PK vs PD
pkpd_data_wide = pd_data %>%
select(ID, NOMTIME, PD = LIDV, BINARY_LEVELS) %>%
right_join(pk_data %>% select(-BINARY_LEVELS), by = c("ID", "NOMTIME")) %>%
rename(CONC = LIDV) %>%
filter(!is.na(PD))
#perform NCA, for additional plots
NCA = pk_data %>%
group_by(ID, DOSE) %>%
filter(!is.na(LIDV)) %>%
summarize(AUC_0 = ifelse(length(LIDV[NOMTIME > 0 & NOMTIME <= TAU]) > 1,
caTools::trapz(TIME[NOMTIME > 0 & NOMTIME <= TAU],
LIDV[NOMTIME > 0 & NOMTIME <= TAU]),
NA),
Cmax_0 = ifelse(length(LIDV[NOMTIME > 0 & NOMTIME <= TAU]) > 1,
max(LIDV[NOMTIME > 0 & NOMTIME <= TAU]),
NA),
AUC_tau = ifelse(length(LIDV[NOMTIME > (SS_PROFDAY-1)*24 &
NOMTIME <= ((SS_PROFDAY-1)*24 + TAU)]) > 1,
caTools::trapz(TIME[NOMTIME > (SS_PROFDAY-1)*24 &
NOMTIME <= ((SS_PROFDAY-1)*24 + TAU)],
LIDV[NOMTIME > (SS_PROFDAY-1)*24 &
NOMTIME <= ((SS_PROFDAY-1)*24 + TAU)]),
NA),
Cmax_tau = ifelse(length(LIDV[NOMTIME > (SS_PROFDAY-1)*24 &
NOMTIME <= ((SS_PROFDAY-1)*24 + TAU)]) > 1,
max(LIDV[NOMTIME > (SS_PROFDAY-1)*24 &
NOMTIME <= ((SS_PROFDAY-1)*24 + TAU)]),
NA),
SEX = SEX[1], #this part just keeps the SEX and WEIGHTB covariates
WEIGHTB = WEIGHTB[1]) %>%
gather(PARAM, VALUE,-c(ID, DOSE, SEX, WEIGHTB)) %>%
ungroup() %>%
mutate(VALUE_NORM = VALUE/DOSE,
PROFDAY = ifelse(PARAM %in% c("AUC_0", "Cmax_0"), 1, SS_PROFDAY))
#add response data at day 1 and at steady state to NCA for additional plots
NCA <- pd_data %>% subset(PROFDAY %in% c(1, SS_PROFDAY),) %>%
select(ID, PROFDAY, DAY_label, PD = LIDV, TRTACT_low2high, TRTACT_high2low) %>%
merge(NCA, by = c("ID", "PROFDAY"))
#units and labels
time_units_dataset = "hours"
time_units_plot = "days"
trtact_label = "Dose"
dose_units = unique((pkpd_data %>% filter(CMT == DOSE_CMT))$LIDV_UNIT) %>% as.character()
dose_label = paste0("Dose (", dose_units, ")")
conc_units = unique(pk_data$LIDV_UNIT) %>% as.character()
conc_label = paste0("Concentration (", conc_units, ")")
AUC_units = paste0("h.", conc_units)
concnorm_label = paste0("Normalized Concentration (", conc_units, ")/", dose_units)
pd_binary_label = "Response"
pd_response_label = "Responder Rate (%)"</code></pre>
<!--START_EXPLANATION-->
<p>Binary data is data that can take on one of two values. This often
happens when there is a characteristic/event/response, etc. of interest
that subjects either have/achieve or they don’t. Binary response data
can also come out of dichotomizing continuous data. For example in the
psoriasis indication the binary response variable PASI90 (yes/no) is
defined as subjects achieving a PASI score of at least 90.</p>
<p>There are two broad categories of PK/PD exploratory plots covered on
this page</p>
<ol style="list-style-type: decimal">
<li><em>Exposure and Response vs Time</em>, stratified by dose. You may
also have heard these referred to as longitudinal (meaning over
time).</li>
<li><em>Response vs Exposure</em> at a particular time. For binomial
response vs exposure plots, fitting a logistic regression is often
helpful, as you will see below.</li>
</ol>
<p>These plots are displayed below. <!--END_EXPLANATION--></p>
</div>
<div id="provide-an-overview-of-the-data" class="section level2">
<h2>Provide an overview of the data</h2>
<!--START_EXPLANATION-->
<p>We start with <em>Expsoure and Response vs Time</em>, or longitduinal
plots</p>
<p>Summarize data with Mean +/- 95% confidence intervals for the percent
responders over time. Confidence intervals should be claculated with
<code>binom::binom.exact()</code>. You should either color or facet by
dose group.<br />
<!--END_EXPLANATION--></p>
<div
id="pk-and-pd-marker-over-time-colored-by-dose-mean---95-ci-by-nominal-time"
class="section level3">
<h3>PK and PD marker over time, colored by Dose, mean +/- 95% CI by
nominal time</h3>
<!--START_EXPLANATION-->
<p>Questions to ask:</p>
<ul>
<li>How quickly does effect occur?<br />
</li>
<li>Do the PK and PD profiles have the same time scale, or does the PD
seem delayed?<br />
</li>
<li>Is there clear separation between the profiles for different
doses?</li>
<li>Does the effect appear to increase (decrease) with increasing
dose?</li>
<li>Do you detect a saturation of the effect?</li>
</ul>
<!--END_EXPLANATION-->
<pre class="r"><code>#PK data
gg <- ggplot(data = pk_data, aes(x = NOMTIME, y = LIDV, color = TRTACT_high2low))
gg <- gg + xgx_stat_ci()
gg <- gg + xgx_scale_y_log10()
gg <- gg + xgx_scale_x_time_units(units_dataset = time_units_dataset,
units_plot = time_units_plot)
gg <- gg + labs(y=conc_label,color = trtact_label)
gg <- gg + xgx_annotate_status(status)
gg <- gg + xgx_annotate_filenames(dirs)
#if saving copy of figure, replace xgx_annotate lines with xgx_save() shown below:
#xgx_save(width,height,dirs,"filename_main",status)
print(gg)</code></pre>
<p><img src="Multiple_Ascending_Dose_PKPD_binary_files/figure-html/unnamed-chunk-3-1.png" width="960" /></p>
<pre class="r"><code>#PD data
gg <- ggplot(data = pd_data, aes(x = NOMTIME, y = LIDV, color = TRTACT_high2low))
gg <- gg + xgx_stat_ci(distribution = "binomial", position = position_dodge(width = 12), alpha = 0.5)
gg <- gg + xgx_scale_x_time_units(units_dataset = time_units_dataset,
units_plot = time_units_plot)
gg <- gg + scale_y_continuous(labels=scales::percent)
gg <- gg + labs(y = pd_response_label, color = trtact_label)
gg <- gg + xgx_annotate_status(status)
gg <- gg + xgx_annotate_filenames(dirs)
#if saving copy of figure, replace xgx_annotate lines with xgx_save() shown below:
#xgx_save(width,height,dirs,"filename_main",status)
print(gg)</code></pre>
<p><img src="Multiple_Ascending_Dose_PKPD_binary_files/figure-html/unnamed-chunk-3-2.png" width="960" /></p>
</div>
<div
id="pk-and-pd-marker-over-time-faceted-by-dose-mean---95-ci-by-nominal-time"
class="section level3">
<h3>PK and PD marker over time, faceted by Dose, mean +/- 95% CI by
nominal time</h3>
<!--START_EXPLANATION-->
<p>If coloring by dose makes a messy plot, you can try faceting by dose
instead. Here we show the same data, but faceted by dose.
<!--END_EXPLANATION--></p>
<pre class="r"><code>#PK data
gg <- ggplot(data = pk_data, aes(x = NOMTIME, y = LIDV))
gg <- gg + xgx_stat_ci()
gg <- gg + xgx_scale_y_log10()
gg <- gg + xgx_scale_x_time_units(units_dataset = time_units_dataset,
units_plot = time_units_plot)
gg <- gg + labs(y=conc_label,color = trtact_label)
gg <- gg + facet_grid(~TRTACT_low2high)
gg <- gg + xgx_annotate_status(status)
gg <- gg + xgx_annotate_filenames(dirs)
#if saving copy of figure, replace xgx_annotate lines with xgx_save() shown below:
#xgx_save(width,height,dirs,"filename_main",status)
print(gg)</code></pre>
<p><img src="Multiple_Ascending_Dose_PKPD_binary_files/figure-html/unnamed-chunk-4-1.png" width="960" /></p>
<pre class="r"><code>#PD data
gg <- ggplot(data = pd_data, aes(x = NOMTIME, y = LIDV))
gg <- gg + xgx_stat_ci(distribution = "binomial")
gg <- gg + xgx_scale_x_time_units(units_dataset = time_units_dataset,
units_plot = time_units_plot)
gg <- gg + scale_y_continuous(labels=scales::percent)
gg <- gg + labs(y = pd_response_label, color = trtact_label)
gg <- gg + facet_grid(~TRTACT_low2high)
gg <- gg + xgx_annotate_status(status)
gg <- gg + xgx_annotate_filenames(dirs)
#if saving copy of figure, replace xgx_annotate lines with xgx_save() shown below:
#xgx_save(width,height,dirs,"filename_main",status)
print(gg)</code></pre>
<p><img src="Multiple_Ascending_Dose_PKPD_binary_files/figure-html/unnamed-chunk-4-2.png" width="960" /></p>
</div>
<div id="pk-over-time-dots-colored-by-response-type"
class="section level3">
<h3>PK over time, dots colored by response type</h3>
<!--START_EXPLANATION-->
<p>Another longitudinal plot that can be useful for binary data is
plotting the PK over time for all individuals, and coloring by response
type. This plot is often used for adverse events plotting. For studies
with few PK observations, a PK model may be needed in order to produce
concentrations at each time point where response is measured.
<!--END_EXPLANATION--></p>
<pre class="r"><code>gg <- ggplot(data = pkpd_data_wide, aes(x = NOMTIME, y = CONC))
gg <- gg + geom_boxplot(aes(group = factor(NOMTIME)), width = 0.5*24)
gg <- gg + geom_jitter(aes( color = factor(BINARY_LEVELS), alpha = factor(BINARY_LEVELS)), width = 0.1*24, height = 0)
gg <- gg + scale_color_manual(values = c("black","blue")) + scale_alpha_manual(values = c(0.2,1))
gg <- gg + labs(y = conc_label, color = "Response", alpha = "Response")
gg <- gg + xgx_scale_y_log10()
gg <- gg + xgx_scale_x_time_units(units_dataset = time_units_dataset,
units_plot = time_units_plot)
print(gg)</code></pre>
<p><img src="Multiple_Ascending_Dose_PKPD_binary_files/figure-html/unnamed-chunk-5-1.png" width="768" /></p>
</div>
</div>
<div id="explore-variability" class="section level2">
<h2>Explore Variability</h2>
<div
id="pk-and-pd-marker-over-time-colored-by-dose-dots-lines-grouped-by-individuals"
class="section level3">
<h3>PK and PD marker over time, colored by Dose, dots & lines
grouped by individuals</h3>
<!--START_EXPLANATION-->
<p>Use spaghetti plots to visualize the extent of variability between
individuals. The wider the spread of the profiles, the higher the
between subject variability. Distinguish different doses by color, or
separate into different panels.</p>
<p>When plotting individual binomial data, it is often helpful to
stagger the dots and use transparency, so that it is easier to detect
individual dots. <!--END_EXPLANATION--></p>
<pre class="r"><code>gg <- ggplot(data = pk_data, aes(x = TIME, y = LIDV))
gg <- gg + geom_line(aes(group = ID, color = factor(TRTACT_high2low)), size = 1, alpha = 0.5)
gg <- gg + geom_point(data = pk_data %>% filter(CENS==0), aes(color = TRTACT_high2low), size = 2, alpha = 0.5)
gg <- gg + geom_point(data = pk_data %>% filter(CENS==1), color="red", shape=8, size = 2, alpha = 0.5)
gg <- gg + xgx_scale_y_log10()
gg <- gg + xgx_scale_x_time_units(units_dataset = time_units_dataset,
units_plot = time_units_plot)
gg <- gg + labs(y = conc_label, color = trtact_label)
gg <- gg + xgx_annotate_status(status)
gg <- gg + xgx_annotate_filenames(dirs)
#if saving copy of figure, replace xgx_annotate lines with xgx_save() shown below:
#xgx_save(width,height,dirs,"filename_main",status)
print(gg)</code></pre>
<p><img src="Multiple_Ascending_Dose_PKPD_binary_files/figure-html/unnamed-chunk-6-1.png" width="960" /></p>
<pre class="r"><code>gg <- ggplot(data = pd_data, aes(x = TIME, y = LIDV_jitter))
gg <- gg + geom_line(aes(group = ID, color = factor(TRTACT_high2low)), size = 1, alpha = 0.5)
gg <- gg + xgx_scale_x_time_units(units_dataset = time_units_dataset,
units_plot = time_units_plot)
gg <- gg + scale_y_continuous(labels=scales::percent)
gg <- gg + labs(y = conc_label, color = pd_response_label)
gg <- gg + xgx_annotate_status(status)
gg <- gg + xgx_annotate_filenames(dirs)
#if saving copy of figure, replace xgx_annotate lines with xgx_save() shown below:
#xgx_save(width,height,dirs,"filename_main",status)
print(gg)</code></pre>
<p><img src="Multiple_Ascending_Dose_PKPD_binary_files/figure-html/unnamed-chunk-6-2.png" width="960" /></p>
</div>
<div
id="pk-and-pd-marker-over-time-faceted-by-dose-dots-lines-grouped-by-individuals"
class="section level3">
<h3>PK and PD marker over time, faceted by Dose, dots & lines
grouped by individuals</h3>
<pre class="r"><code>gg <- ggplot(data = pk_data, aes(x = TIME, y = LIDV))
gg <- gg + geom_line(aes(group = ID), size = 1, alpha = 0.2)
gg <- gg + geom_point(aes(color = factor(CENS), shape = factor(CENS), alpha = 0.3), size = 2, alpha = 0.2)
gg <- gg + scale_shape_manual(values=c(1,8))
gg <- gg + scale_color_manual(values=c("grey50","red"))
gg <- gg + xgx_scale_y_log10()
gg <- gg + xgx_scale_x_time_units(units_dataset = time_units_dataset,
units_plot = time_units_plot)
gg <- gg + labs(y = conc_label, shape = "BLQ", color = "BLQ")
gg <- gg + facet_grid(.~TRTACT_low2high)
gg <- gg + xgx_annotate_status(status)
gg <- gg + xgx_annotate_filenames(dirs)
#if saving copy of figure, replace xgx_annotate lines with xgx_save() shown below:
#xgx_save(width,height,dirs,"filename_main",status)
print(gg)</code></pre>
<p><img src="Multiple_Ascending_Dose_PKPD_binary_files/figure-html/unnamed-chunk-7-1.png" width="960" /></p>
<pre class="r"><code>gg <- ggplot(data = pd_data, aes(x = TIME, y = LIDV_jitter))
gg <- gg + geom_line(aes(group = ID), alpha = 0.2)
gg <- gg + geom_point(size = 2, alpha = 0.2)
gg <- gg + xgx_scale_x_time_units(units_dataset = time_units_dataset,
units_plot = time_units_plot)
gg <- gg + labs(y = pd_binary_label, shape = "BLQ", color = "BLQ")
gg <- gg + facet_grid(.~TRTACT_low2high)
gg <- gg + xgx_annotate_status(status)
gg <- gg + xgx_annotate_filenames(dirs)
#if saving copy of figure, replace xgx_annotate lines with xgx_save() shown below:
#xgx_save(width,height,dirs,"filename_main",status)
print(gg)</code></pre>
<p><img src="Multiple_Ascending_Dose_PKPD_binary_files/figure-html/unnamed-chunk-7-2.png" width="960" /></p>
</div>
</div>
<div id="explore-exposure-response-relationship" class="section level2">
<h2>Explore Exposure-Response Relationship</h2>
<pre class="r"><code>gg <- ggplot(data = pkpd_data_wide, aes(y = CONC, x = PD))
gg <- gg + geom_boxplot(aes(group = PD), width = 0.5, outlier.shape=NA)
gg <- gg + geom_jitter(data = pkpd_data_wide %>% filter(CENS == 0, PROFDAY == 5),
aes(color = TRTACT_high2low), shape=19, width = 0.1, height = 0.0, alpha = 0.5)
gg <- gg + geom_jitter(data = pkpd_data_wide %>% filter(CENS == 1, PROFDAY == 5),
color = "red", shape=8, width = 0.1, height = 0.0, alpha = 0.5)
gg <- gg + xgx_scale_y_log10()
gg <- gg + labs(x = pd_binary_label, y = conc_label, color = trtact_label)
gg <- gg + coord_flip()
print(gg)</code></pre>
<p><img src="Multiple_Ascending_Dose_PKPD_binary_files/figure-html/unnamed-chunk-8-1.png" width="768" /></p>
<!--START_EXPLANATION-->
<p>Plot response against exposure. Include a logistic regression for
binary data to help determine the shape of the exposure-respone
relationship. Summary information such as mean and 95% confidence
intervals by quartiles of exposure can also be plotted. The exposure
metric that you use in these plots could be either raw concentrations,
or NCA or model-derived exposure metrics (e.g. Cmin, Cmax, AUC), and may
depend on the level of data that you have available.
<!--END_EXPLANATION--></p>
<pre class="r"><code>pkpd_data_wide_plot = pkpd_data_wide %>%
filter(PROFDAY %in% c(1,3,5))
gg <- ggplot(data = pkpd_data_wide_plot, aes(x = CONC, y = PD))
gg <- gg + geom_jitter(aes(color = TRTACT_high2low), width = 0, height = 0.05, alpha = 0.5)
gg <- gg + geom_smooth(method = "glm", method.args = list(family=binomial(link = logit)), color = "black")
gg <- gg + xgx_stat_ci(bins = 4, distribution = "binomial", geom = "errorbar", size = 0.5)
gg <- gg + xgx_stat_ci(bins = 4, distribution = "binomial", geom = "point", shape = 0, size = 4)
gg <- gg + scale_y_continuous(labels=scales::percent)
gg <- gg + labs(x = conc_label, y = pd_response_label, color = trtact_label)
gg <- gg + xgx_scale_x_log10()
gg <- gg + facet_grid(~DAY_label)
print(gg)</code></pre>
<p><img src="Multiple_Ascending_Dose_PKPD_binary_files/figure-html/unnamed-chunk-9-1.png" width="960" /></p>
<!--START_EXPLANATION-->
<p>Plotting AUC vs response instead of concentration vs response may
make more sense in some situations. For example, when there is a large
delay between PK and PD it would be diffcult to relate the time-varying
concentration with the response. If rich sampling is only done at a
particular point in the study, e.g. at steady state, then the AUC
calculated on the rich profile could be used as the exposure variable
for a number of PD visits. If PK samples are scarce, average Cmin could
also be used as the exposure metric. <!--END_EXPLANATION--></p>
<pre class="r"><code>gg <- ggplot(NCA, aes(x = VALUE, y = PD))
gg <- gg + geom_jitter(aes( color = TRTACT_high2low), width = 0, height = 0.05, alpha = 0.5)
gg <- gg + geom_smooth(method = "glm", method.args = list(family=binomial(link = logit)), color = "black")
gg <- gg + xgx_stat_ci(bins = 4, conf_level = 0.95, distribution = "binomial", geom = c("point"), shape = 0, size = 4)
gg <- gg + xgx_stat_ci(bins = 4, conf_level = 0.95, distribution = "binomial", geom = c("errorbar"), size = 0.5)
gg <- gg + facet_wrap(~DAY_label + PARAM, scales = "free_x")
gg <- gg + labs(color = trtact_label, x = "NCA parameter", y = pd_response_label)
gg <- gg + xgx_scale_x_log10()
gg <- gg + scale_y_continuous(breaks=c(0,.5,1), labels = scales::percent)
print(gg)</code></pre>
<p><img src="Multiple_Ascending_Dose_PKPD_binary_files/figure-html/unnamed-chunk-10-1.png" width="768" /></p>
<div id="explore-covariate-effects-on-exposure-response-relationship"
class="section level3">
<h3>Explore covariate effects on Exposure-Response Relationship</h3>
<!--START_EXPLANATION-->
<p>Stratify exposure-response plots by covariates of interest to explore
whether any key covariates impact response independent of exposure. For
examples of plots and code startifying by covariate, see <a
href="Multiple_Ascending_Dose_PKPD_continuous.html#explore_covariate_effects_on_exposure-response_relationship">Continuous
PKPD Covariate Section</a> <!--END_EXPLANATION--></p>
</div>
</div>
<div id="r-session-info" class="section level2">
<h2>R Session Info</h2>
<pre class="r"><code>sessionInfo()</code></pre>
<pre><code>## R version 4.1.0 (2021-05-18)
## Platform: x86_64-pc-linux-gnu (64-bit)
## Running under: Red Hat Enterprise Linux Server 7.9 (Maipo)
##
## Matrix products: default
## BLAS/LAPACK: /CHBS/apps/EB/software/imkl/2019.1.144-gompi-2019a/compilers_and_libraries_2019.1.144/linux/mkl/lib/intel64_lin/libmkl_gf_lp64.so
##
## locale:
## [1] LC_CTYPE=en_US.UTF-8 LC_NUMERIC=C LC_TIME=en_US.UTF-8 LC_COLLATE=en_US.UTF-8 LC_MONETARY=en_US.UTF-8
## [6] LC_MESSAGES=en_US.UTF-8 LC_PAPER=en_US.UTF-8 LC_NAME=C LC_ADDRESS=C LC_TELEPHONE=C
## [11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C
##
## attached base packages:
## [1] stats graphics grDevices utils datasets methods base
##
## other attached packages:
## [1] survminer_0.4.9 ggpubr_0.4.0 survival_3.4-0 knitr_1.40 broom_1.0.1 caTools_1.18.2 DT_0.26 forcats_0.5.2 stringr_1.4.1
## [10] purrr_0.3.5 readr_2.1.3 tibble_3.1.8 tidyverse_1.3.2 xgxr_1.1.1 zoo_1.8-11 gridExtra_2.3 tidyr_1.2.1 dplyr_1.0.10