Overview
Area-under-the-curve summaries are useful when the goal is to
condense an entire disease progress curve into a single number that
represents epidemic intensity over time.
epifitter currently provides three related
functions:
AUDPC() for the area under the disease progress
curve.AUDPC_2_points() for estimating AUDPC from just two
observations under a logistic assumption.AUDPS() for the area under the disease progress stairs,
which gives more balanced weight to the first and last
observations.
For AUDPC() and AUDPS(), repeated
observations at the same time point are now handled explicitly through
the aggregate argument. By default, replicated observations
are averaged per time point before the area is calculated.
Build an example curve
We start with a simulated epidemic and use the disease intensity
values across time.
set.seed(1)
epi <- sim_logistic(
N = 40,
y0 = 0.01,
dt = 5,
r = 0.25,
alpha = 0.2,
n = 1
)
knitr::kable(epi, digits = 4)
| 1 |
0 |
0.0100 |
0.0100 |
| 1 |
5 |
0.0341 |
0.0353 |
| 1 |
10 |
0.1096 |
0.0933 |
| 1 |
15 |
0.3005 |
0.3675 |
| 1 |
20 |
0.5999 |
0.6157 |
| 1 |
25 |
0.8396 |
0.8175 |
| 1 |
30 |
0.9481 |
0.9529 |
| 1 |
35 |
0.9846 |
0.9868 |
| 1 |
40 |
0.9955 |
0.9960 |
ggplot(epi, aes(time, y)) +
geom_point(size = 2, color = "#15616d") +
geom_line(linewidth = 0.9, color = "#15616d") +
labs(
title = "Example disease progress curve",
x = "Time",
y = "Disease intensity"
)
Compute AUDPC
AUDPC() uses the trapezoidal method to summarize the
curve over the observed period.
audpc_abs <- AUDPC(
time = epi$time,
y = epi$y,
y_proportion = TRUE,
type = "absolute"
)
audpc_abs
## [1] 21.62241
When the interest is in a scale-free measure, use
type = "relative".
audpc_rel <- AUDPC(
time = epi$time,
y = epi$y,
y_proportion = TRUE,
type = "relative"
)
audpc_rel
## [1] 0.5405602
The relative version is useful for comparing epidemics observed over
the same time span but with different scales.
Multiple observations at the same time
Many experiments include replicated plots assessed on the same dates.
In that setting, the area summary should be computed from one value per
time point, not from the raw row order.
By default, AUDPC() and AUDPS() aggregate
replicated observations using the mean at each time.
time_rep <- c(0, 0, 5, 5, 10, 10)
y_rep <- c(0.10, 0.30, 0.40, 0.60, 0.70, 0.90)
AUDPC(time = time_rep, y = y_rep)
## [1] 5.75
AUDPS(time = time_rep, y = y_rep)
## [1] 8.25
This is equivalent to calculating the area from the per-time
means.
time_mean <- c(0, 5, 10)
y_mean <- c(mean(c(0.10, 0.30)), mean(c(0.40, 0.60)), mean(c(0.70, 0.90)))
AUDPC(time = time_mean, y = y_mean, aggregate = "none")
## [1] 5.75
AUDPS(time = time_mean, y = y_mean, aggregate = "none")
## [1] 8.25
If a more robust summary is preferred, use the median instead of the
mean.
AUDPC(time = time_rep, y = y_rep, aggregate = "median")
## [1] 5.75
AUDPS(time = time_rep, y = y_rep, aggregate = "median")
## [1] 8.25
If you want to require unique time values and catch duplicated
assessments as an error, set aggregate = "none".
AUDPC(time = time_rep, y = y_rep, aggregate = "none")
AUDPS(time = time_rep, y = y_rep, aggregate = "none")
One summary curve versus one value per replicate
This distinction is important in experimental data.
If you pass all replicated observations together to
AUDPC() or AUDPS(), the functions now assume
you want a single summary curve, and they aggregate the
repeated observations at each time point before computing the area.
If instead you want one AUDPC or AUDPS value for each
experimental replicate, compute the area separately within each
replicate.
epi_rep <- sim_logistic(
N = 30,
y0 = 0.01,
dt = 5,
r = 0.3,
alpha = 0.2,
n = 4
)
knitr::kable(head(epi_rep), digits = 4)
| 1 |
0 |
0.0100 |
0.0100 |
| 1 |
5 |
0.0433 |
0.0558 |
| 1 |
10 |
0.1687 |
0.1796 |
| 1 |
15 |
0.4763 |
0.4453 |
| 1 |
20 |
0.8030 |
0.7329 |
| 1 |
25 |
0.9481 |
0.9592 |
A single treatment-level summary can be obtained by pooling all rows
and using the default aggregation by time:
AUDPC(time = epi_rep$time, y = epi_rep$random_y)
## [1] 14.70536
AUDPS(time = epi_rep$time, y = epi_rep$random_y)
## [1] 17.20195
To obtain one value per replicate, group the data and compute the
areas within each experimental unit. In this case, use
aggregate = "none" because each replicate should contribute
only one observation per time point.
epi_rep %>%
group_by(replicates) %>%
summarise(
audpc = AUDPC(time = time, y = random_y, aggregate = "none"),
audps = AUDPS(time = time, y = random_y, aggregate = "none"),
.groups = "drop"
) %>%
knitr::kable(digits = 4)
| 1 |
14.4302 |
16.9247 |
| 2 |
15.2590 |
17.7543 |
| 3 |
14.5306 |
17.0279 |
| 4 |
14.6017 |
17.1009 |
In practice:
- use the default aggregation when you want one area summary for the
mean epidemic across replicates;
- group by replicate when you want one area summary per experimental
unit.
Compute AUDPS
AUDPS() is closely related to AUDPC(), but
uses a staircase correction that often improves discrimination among
curves in comparative studies.
audps_abs <- AUDPS(
time = epi$time,
y = epi$y,
y_proportion = TRUE,
type = "absolute"
)
audps_abs
## [1] 24.13622
audps_rel <- AUDPS(
time = epi$time,
y = epi$y,
y_proportion = TRUE,
type = "relative"
)
audps_rel
## [1] 0.5363605
Estimate AUDPC from two observations
AUDPC_2_points() is helpful when only the initial and
final disease intensities are available and a logistic epidemic shape is
assumed.
audpc_two_points <- AUDPC_2_points(
time = epi$time[7],
y0 = epi$y[1],
yT = epi$y[7]
)
audpc_two_points
## [1] 11.79275
For simulated logistic data, this estimate should be close to the
full-curve AUDPC computed from all intermediate observations.
full_curve_audpc <- AUDPC(
time = epi$time,
y = epi$y,
y_proportion = TRUE
)
c(
AUDPC_full_curve = full_curve_audpc,
AUDPC_two_points = audpc_two_points
)
## AUDPC_full_curve AUDPC_two_points
## 21.62241 11.79275
Absolute versus relative measures
Use absolute values when duration and measurement scale are already
part of the interpretation.
Use relative values when you want a normalized measure between
epidemics observed on the same conceptual scale.