将 vline 添加到 geom_density 并为平均 R 的置信区间添加阴影
Add vline to geom_density and shade confidence interval of mean R
阅读不同的帖子后，我发现了如何向密度图添加均值 v 线，如图 here 所示。
使用上面提供的数据link:
1) 如何使用 geom_ribbon 在均值周围添加 95% 的置信区间？
CI 可以计算为
#computation of the standard error of the mean
sem<-sd(x)/sqrt(length(x))
#95% confidence intervals of the mean
c(mean(x)-2*sem,mean(x)+2*sem)
2) 如何将vline限制在曲线下的区域？您将在下图中看到 vline 绘制在曲线之外。
可以在 https://www.dropbox.com/s/bvvfdpgekbjyjh0/test.csv?dl=0
找到非常接近我的实际问题的示例数据
更新
使用上面 link 中的真实数据，我尝试使用@beetroot 的回答进行以下操作。
# Find the mean of each group
dat=me
library(dplyr)
library(plyr)
cdat <- ddply(data,.(direction,cond), summarise, rating.mean=mean(rating,na.rm=T))# summarize by season and variable
cdat
#ggplot
p=ggplot(data,aes(x = rating)) +
geom_density(aes(colour = cond),size=1.3,adjust=4)+
facet_grid(.~direction, scales="free")+
xlab(NULL) + ylab("Density")
p=p+coord_cartesian(xlim = c(0, 130))+scale_color_manual(name="",values=c("blue","#00BA38","#F8766D"))+
scale_fill_manual(values=c("blue", "#00BA38", "#F8766D"))+
theme(legend.title = element_text(colour="black", size=15, face="plain"))+
theme(legend.text = element_text(colour="black", size = 15, face = "plain"))+
theme(title = red.bold.italic.text, axis.title = red.bold.italic.text)+
theme(strip.text.x = element_text(size=20, color="black",face="plain"))+ # facet labels
ggtitle("SAMPLE A") +theme(plot.title = element_text(size = 20, face = "bold"))+
theme(axis.text = blue.bold.italic.16.text)+ theme(legend.position = "none")+
geom_vline(data=cdat, aes(xintercept=rating.mean, color=cond),linetype="dotted",size=1)
p
## implementing @beetroot's code to restrict lines under the curve and shade CIs around the mean
# I will use ddply for mean and CIs
cdat <- ddply(data,.(direction,cond), summarise, rating.mean=mean(rating,na.rm=T),
sem = sd(rating,na.rm=T)/sqrt(length(rating)),
ci.low = mean(rating,na.rm=T) - 2*sem,
ci.upp = mean(rating,na.rm=T) + 2*sem)# summarize by direction and variable
#In order to limit the lines to the outline of the curves you first need to find out which y values
#of the curves correspond to the means, e.g. by accessing the density values with ggplot_build and
#using approx:
cdat.dens <- ggplot_build(ggplot(data, aes(x=rating, colour=cond)) +
facet_grid(.~direction, scales="free")+
geom_density(aes(colour = cond),size=1.3,adjust=4))$data[[1]] %>%
mutate(cond = ifelse(group==1, "A",
ifelse(group==2, "B","C"))) %>%
left_join(cdat) %>%
select(y, x, cond, rating.mean, sem, ci.low, ci.upp) %>%
group_by(cond) %>%
mutate(dens.mean = approx(x, y, xout = rating.mean)[[2]],
dens.cilow = approx(x, y, xout = ci.low)[[2]],
dens.ciupp = approx(x, y, xout = ci.upp)[[2]]) %>%
select(-y, -x) %>%
slice(1)
cdat.dens
#---
#You can then combine everything with various geom_segments:
ggplot(data, aes(x=rating, colour=cond)) +
geom_density(data = data, aes(x = rating, colour = cond),size=1.3,adjust=4) +facet_grid(.~direction, scales="free")+
geom_segment(data = cdat.dens, aes(x = rating.mean, xend = rating.mean, y = 0, yend = dens.mean, colour = cond),
linetype = "dashed", size = 1) +
geom_segment(data = cdat.dens, aes(x = ci.low, xend = ci.low, y = 0, yend = dens.cilow, colour = cond),
linetype = "dotted", size = 1) +
geom_segment(data = cdat.dens, aes(x = ci.upp, xend = ci.upp, y = 0, yend = dens.ciupp, colour = cond),
linetype = "dotted", size = 1)
给出这个：
您会注意到均值和 CI 没有像原始图中那样对齐。 @beetroot 我做错了什么？
使用 link 中的数据，您可以像这样计算平均值、se 和 ci（我建议使用 dplyr，plyr 的继承者） :
set.seed(1234)
dat <- data.frame(cond = factor(rep(c("A","B"), each=200)),
rating = c(rnorm(200),rnorm(200, mean=.8)))
library(ggplot2)
library(dplyr)
cdat <- dat %>%
group_by(cond) %>%
summarise(rating.mean = mean(rating),
sem = sd(rating)/sqrt(length(rating)),
ci.low = mean(rating) - 2*sem,
ci.upp = mean(rating) + 2*sem)
为了将线条限制在曲线的轮廓内，您首先需要找出曲线的哪些 y 值对应于均值，例如通过使用 ggplot_build 访问密度值并使用 approx:
cdat.dens <- ggplot_build(ggplot(dat, aes(x=rating, colour=cond)) + geom_density())$data[[1]] %>%
mutate(cond = ifelse(group == 1, "A", "B")) %>%
left_join(cdat) %>%
select(y, x, cond, rating.mean, sem, ci.low, ci.upp) %>%
group_by(cond) %>%
mutate(dens.mean = approx(x, y, xout = rating.mean)[[2]],
dens.cilow = approx(x, y, xout = ci.low)[[2]],
dens.ciupp = approx(x, y, xout = ci.upp)[[2]]) %>%
select(-y, -x) %>%
slice(1)
> cdat.dens
Source: local data frame [2 x 8]
Groups: cond [2]
cond rating.mean sem ci.low ci.upp dens.mean dens.cilow dens.ciupp
<chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
1 A -0.05775928 0.07217200 -0.2021033 0.08658471 0.3865929 0.403623 0.3643583
2 B 0.87324927 0.07120697 0.7308353 1.01566320 0.3979347 0.381683 0.4096153
然后您可以将所有内容与各种 geom_segment 组合起来：
ggplot() +
geom_density(data = dat, aes(x = rating, colour = cond)) +
geom_segment(data = cdat.dens, aes(x = rating.mean, xend = rating.mean, y = 0, yend = dens.mean, colour = cond),
linetype = "dashed", size = 1) +
geom_segment(data = cdat.dens, aes(x = ci.low, xend = ci.low, y = 0, yend = dens.cilow, colour = cond),
linetype = "dotted", size = 1) +
geom_segment(data = cdat.dens, aes(x = ci.upp, xend = ci.upp, y = 0, yend = dens.ciupp, colour = cond),
linetype = "dotted", size = 1)
正如 Axeman 指出的那样，您可以根据 this answer 中所述的功能区区域创建多边形。
因此，对于您的数据，您可以像这样子集化并添加额外的行：
ribbon <- ggplot_build(ggplot(dat, aes(x=rating, colour=cond)) + geom_density())$data[[1]] %>%
mutate(cond = ifelse(group == 1, "A", "B")) %>%
left_join(cdat.dens) %>%
group_by(cond) %>%
filter(x >= ci.low & x <= ci.upp) %>%
select(cond, x, y)
ribbon <- rbind(data.frame(cond = c("A", "B"), x = c(-0.2021033, 0.7308353), y = c(0, 0)),
as.data.frame(ribbon),
data.frame(cond = c("A", "B"), x = c(0.08658471, 1.01566320), y = c(0, 0)))
并在图中添加 geom_polygon：
ggplot() +
geom_polygon(data = ribbon, aes(x = x, y = y, fill = cond), alpha = .5) +
geom_density(data = dat, aes(x = rating, colour = cond)) +
geom_segment(data = cdat.dens, aes(x = rating.mean, xend = rating.mean, y = 0, yend = dens.mean, colour = cond),
linetype = "dashed", size = 1) +
geom_segment(data = cdat.dens, aes(x = ci.low, xend = ci.low, y = 0, yend = dens.cilow, colour = cond),
linetype = "dotted", size = 1) +
geom_segment(data = cdat.dens, aes(x = ci.upp, xend = ci.upp, y = 0, yend = dens.ciupp, colour = cond),
linetype = "dotted", size = 1)
这是为您的真实数据改编的代码。合并两个组而不是一个组有点棘手：
cdat <- dat %>%
group_by(direction, cond) %>%
summarise(rating.mean = mean(rating, na.rm = TRUE),
sem = sd(rating, na.rm = TRUE)/sqrt(length(rating)),
ci.low = mean(rating, na.rm = TRUE) - 2*sem,
ci.upp = mean(rating, na.rm = TRUE) + 2*sem)
cdat.dens <- ggplot_build(ggplot(dat, aes(x=rating, colour=interaction(direction, cond))) + geom_density())$data[[1]] %>%
mutate(cond = ifelse((group == 1 | group == 2 | group == 3 | group == 4), "A",
ifelse((group == 5 | group == 6 | group == 7 | group == 8), "B", "C")),
direction = ifelse((group == 1 | group == 5 | group == 9), "EAST",
ifelse((group == 2 | group == 6 | group == 10), "NORTH",
ifelse((group == 3 | group == 7 | group == 11), "SOUTH", "WEST")))) %>%
left_join(cdat) %>%
select(y, x, cond, direction, rating.mean, sem, ci.low, ci.upp) %>%
group_by(cond, direction) %>%
mutate(dens.mean = approx(x, y, xout = rating.mean)[[2]],
dens.cilow = approx(x, y, xout = ci.low)[[2]],
dens.ciupp = approx(x, y, xout = ci.upp)[[2]]) %>%
select(-y, -x) %>%
slice(1)
ggplot() +
geom_density(data = dat, aes(x = rating, colour = cond)) +
geom_segment(data = cdat.dens, aes(x = rating.mean, xend = rating.mean, y = 0, yend = dens.mean, colour = cond),
linetype = "dashed", size = 1) +
geom_segment(data = cdat.dens, aes(x = ci.low, xend = ci.low, y = 0, yend = dens.cilow, colour = cond),
linetype = "dotted", size = 1) +
geom_segment(data = cdat.dens, aes(x = ci.upp, xend = ci.upp, y = 0, yend = dens.ciupp, colour = cond),
linetype = "dotted", size = 1) +
facet_wrap(~direction)
如果您想在不构建绘图对象且不在绘图之前处理数据的情况下绘制平均线，您可以使用 stat_summary():
(
ggplot(data = dat, aes(x = rating, colour = cond))
+ geom_density()
+ stat_summary(
aes(y = rating, x = 0),
geom = 'rect',
fun.data = density_mean_line(dat$rating),
key_glyph = "vline",
size = 1
)
)
给予：
其中：
density_mean_line = function(values) {
values_range = range(values, na.rm=TRUE)
function(x) {
density_data = StatDensity$compute_group(
data.frame(x=x),
scales=list(
x=scale_x_continuous(limits = values_range)
)
)
mean_x = mean(x)
data.frame(
xmin=mean_x,
xmax=mean_x,
ymin=0,
ymax=approx(density_data$x, density_data$density, xout=mean_x)$y
)
}
}
并且 dat 定义为 erc 的回答：
set.seed(1234)
dat <- data.frame(cond = factor(rep(c("A","B"), each=200)),
rating = c(rnorm(200),rnorm(200, mean=.8)))
此技术也可用于生成实心区域（与密度轮廓颜色相同）：
(
ggplot(data = dat, aes(x = rating, colour = cond, group = cond))
+ stat_summary(
aes(y = rating, x = 0, fill = cond),
geom = 'rect',
fun.data = density_ci(dat$rating),
size=1
)
+ stat_summary(
aes(y = rating, x = 0),
geom = 'rect',
fun.data = density_mean_line(dat$rating),
key_glyph = "vline",
size = 0.5,
color='grey20'
)
+ geom_density()
)
其中：
density_ci = function(values, resolution=100) {
values_range = range(values, na.rm=TRUE)
function(x) {
density_data = StatDensity$compute_group(
data.frame(x=x),
scales=list(
x=scale_x_continuous(limits = values_range)
)
)
mean_x = mean(x)
sem = sd(x) / sqrt(length(x))
ci_lower = mean_x - 1.96 * sem
ci_upper = mean_x + 1.96 * sem
x_values = seq(ci_lower, ci_upper, length.out=resolution)
data.frame(
xmin=x_values,
xmax=x_values,
ymin=rep(0, resolution),
ymax=approx(density_data$x, density_data$density, xout=x_values)$y
)
}
}
阅读不同的帖子后，我发现了如何向密度图添加均值 v 线，如图 here 所示。 使用上面提供的数据link:
1) 如何使用 geom_ribbon 在均值周围添加 95% 的置信区间？ CI 可以计算为
#computation of the standard error of the mean
sem<-sd(x)/sqrt(length(x))
#95% confidence intervals of the mean
c(mean(x)-2*sem,mean(x)+2*sem)
2) 如何将vline限制在曲线下的区域？您将在下图中看到 vline 绘制在曲线之外。
可以在 https://www.dropbox.com/s/bvvfdpgekbjyjh0/test.csv?dl=0
找到非常接近我的实际问题的示例数据更新
使用上面 link 中的真实数据，我尝试使用@beetroot 的回答进行以下操作。
# Find the mean of each group
dat=me
library(dplyr)
library(plyr)
cdat <- ddply(data,.(direction,cond), summarise, rating.mean=mean(rating,na.rm=T))# summarize by season and variable
cdat
#ggplot
p=ggplot(data,aes(x = rating)) +
geom_density(aes(colour = cond),size=1.3,adjust=4)+
facet_grid(.~direction, scales="free")+
xlab(NULL) + ylab("Density")
p=p+coord_cartesian(xlim = c(0, 130))+scale_color_manual(name="",values=c("blue","#00BA38","#F8766D"))+
scale_fill_manual(values=c("blue", "#00BA38", "#F8766D"))+
theme(legend.title = element_text(colour="black", size=15, face="plain"))+
theme(legend.text = element_text(colour="black", size = 15, face = "plain"))+
theme(title = red.bold.italic.text, axis.title = red.bold.italic.text)+
theme(strip.text.x = element_text(size=20, color="black",face="plain"))+ # facet labels
ggtitle("SAMPLE A") +theme(plot.title = element_text(size = 20, face = "bold"))+
theme(axis.text = blue.bold.italic.16.text)+ theme(legend.position = "none")+
geom_vline(data=cdat, aes(xintercept=rating.mean, color=cond),linetype="dotted",size=1)
p
## implementing @beetroot's code to restrict lines under the curve and shade CIs around the mean
# I will use ddply for mean and CIs
cdat <- ddply(data,.(direction,cond), summarise, rating.mean=mean(rating,na.rm=T),
sem = sd(rating,na.rm=T)/sqrt(length(rating)),
ci.low = mean(rating,na.rm=T) - 2*sem,
ci.upp = mean(rating,na.rm=T) + 2*sem)# summarize by direction and variable
#In order to limit the lines to the outline of the curves you first need to find out which y values
#of the curves correspond to the means, e.g. by accessing the density values with ggplot_build and
#using approx:
cdat.dens <- ggplot_build(ggplot(data, aes(x=rating, colour=cond)) +
facet_grid(.~direction, scales="free")+
geom_density(aes(colour = cond),size=1.3,adjust=4))$data[[1]] %>%
mutate(cond = ifelse(group==1, "A",
ifelse(group==2, "B","C"))) %>%
left_join(cdat) %>%
select(y, x, cond, rating.mean, sem, ci.low, ci.upp) %>%
group_by(cond) %>%
mutate(dens.mean = approx(x, y, xout = rating.mean)[[2]],
dens.cilow = approx(x, y, xout = ci.low)[[2]],
dens.ciupp = approx(x, y, xout = ci.upp)[[2]]) %>%
select(-y, -x) %>%
slice(1)
cdat.dens
#---
#You can then combine everything with various geom_segments:
ggplot(data, aes(x=rating, colour=cond)) +
geom_density(data = data, aes(x = rating, colour = cond),size=1.3,adjust=4) +facet_grid(.~direction, scales="free")+
geom_segment(data = cdat.dens, aes(x = rating.mean, xend = rating.mean, y = 0, yend = dens.mean, colour = cond),
linetype = "dashed", size = 1) +
geom_segment(data = cdat.dens, aes(x = ci.low, xend = ci.low, y = 0, yend = dens.cilow, colour = cond),
linetype = "dotted", size = 1) +
geom_segment(data = cdat.dens, aes(x = ci.upp, xend = ci.upp, y = 0, yend = dens.ciupp, colour = cond),
linetype = "dotted", size = 1)
给出这个：
您会注意到均值和 CI 没有像原始图中那样对齐。 @beetroot 我做错了什么？
使用 link 中的数据，您可以像这样计算平均值、se 和 ci（我建议使用 dplyr，plyr 的继承者） :
set.seed(1234)
dat <- data.frame(cond = factor(rep(c("A","B"), each=200)),
rating = c(rnorm(200),rnorm(200, mean=.8)))
library(ggplot2)
library(dplyr)
cdat <- dat %>%
group_by(cond) %>%
summarise(rating.mean = mean(rating),
sem = sd(rating)/sqrt(length(rating)),
ci.low = mean(rating) - 2*sem,
ci.upp = mean(rating) + 2*sem)
为了将线条限制在曲线的轮廓内，您首先需要找出曲线的哪些 y 值对应于均值，例如通过使用 ggplot_build 访问密度值并使用 approx:
cdat.dens <- ggplot_build(ggplot(dat, aes(x=rating, colour=cond)) + geom_density())$data[[1]] %>%
mutate(cond = ifelse(group == 1, "A", "B")) %>%
left_join(cdat) %>%
select(y, x, cond, rating.mean, sem, ci.low, ci.upp) %>%
group_by(cond) %>%
mutate(dens.mean = approx(x, y, xout = rating.mean)[[2]],
dens.cilow = approx(x, y, xout = ci.low)[[2]],
dens.ciupp = approx(x, y, xout = ci.upp)[[2]]) %>%
select(-y, -x) %>%
slice(1)
> cdat.dens
Source: local data frame [2 x 8]
Groups: cond [2]
cond rating.mean sem ci.low ci.upp dens.mean dens.cilow dens.ciupp
<chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
1 A -0.05775928 0.07217200 -0.2021033 0.08658471 0.3865929 0.403623 0.3643583
2 B 0.87324927 0.07120697 0.7308353 1.01566320 0.3979347 0.381683 0.4096153
然后您可以将所有内容与各种 geom_segment 组合起来：
ggplot() +
geom_density(data = dat, aes(x = rating, colour = cond)) +
geom_segment(data = cdat.dens, aes(x = rating.mean, xend = rating.mean, y = 0, yend = dens.mean, colour = cond),
linetype = "dashed", size = 1) +
geom_segment(data = cdat.dens, aes(x = ci.low, xend = ci.low, y = 0, yend = dens.cilow, colour = cond),
linetype = "dotted", size = 1) +
geom_segment(data = cdat.dens, aes(x = ci.upp, xend = ci.upp, y = 0, yend = dens.ciupp, colour = cond),
linetype = "dotted", size = 1)
正如 Axeman 指出的那样，您可以根据 this answer 中所述的功能区区域创建多边形。
因此，对于您的数据，您可以像这样子集化并添加额外的行：
ribbon <- ggplot_build(ggplot(dat, aes(x=rating, colour=cond)) + geom_density())$data[[1]] %>%
mutate(cond = ifelse(group == 1, "A", "B")) %>%
left_join(cdat.dens) %>%
group_by(cond) %>%
filter(x >= ci.low & x <= ci.upp) %>%
select(cond, x, y)
ribbon <- rbind(data.frame(cond = c("A", "B"), x = c(-0.2021033, 0.7308353), y = c(0, 0)),
as.data.frame(ribbon),
data.frame(cond = c("A", "B"), x = c(0.08658471, 1.01566320), y = c(0, 0)))
并在图中添加 geom_polygon：
ggplot() +
geom_polygon(data = ribbon, aes(x = x, y = y, fill = cond), alpha = .5) +
geom_density(data = dat, aes(x = rating, colour = cond)) +
geom_segment(data = cdat.dens, aes(x = rating.mean, xend = rating.mean, y = 0, yend = dens.mean, colour = cond),
linetype = "dashed", size = 1) +
geom_segment(data = cdat.dens, aes(x = ci.low, xend = ci.low, y = 0, yend = dens.cilow, colour = cond),
linetype = "dotted", size = 1) +
geom_segment(data = cdat.dens, aes(x = ci.upp, xend = ci.upp, y = 0, yend = dens.ciupp, colour = cond),
linetype = "dotted", size = 1)
这是为您的真实数据改编的代码。合并两个组而不是一个组有点棘手：
cdat <- dat %>%
group_by(direction, cond) %>%
summarise(rating.mean = mean(rating, na.rm = TRUE),
sem = sd(rating, na.rm = TRUE)/sqrt(length(rating)),
ci.low = mean(rating, na.rm = TRUE) - 2*sem,
ci.upp = mean(rating, na.rm = TRUE) + 2*sem)
cdat.dens <- ggplot_build(ggplot(dat, aes(x=rating, colour=interaction(direction, cond))) + geom_density())$data[[1]] %>%
mutate(cond = ifelse((group == 1 | group == 2 | group == 3 | group == 4), "A",
ifelse((group == 5 | group == 6 | group == 7 | group == 8), "B", "C")),
direction = ifelse((group == 1 | group == 5 | group == 9), "EAST",
ifelse((group == 2 | group == 6 | group == 10), "NORTH",
ifelse((group == 3 | group == 7 | group == 11), "SOUTH", "WEST")))) %>%
left_join(cdat) %>%
select(y, x, cond, direction, rating.mean, sem, ci.low, ci.upp) %>%
group_by(cond, direction) %>%
mutate(dens.mean = approx(x, y, xout = rating.mean)[[2]],
dens.cilow = approx(x, y, xout = ci.low)[[2]],
dens.ciupp = approx(x, y, xout = ci.upp)[[2]]) %>%
select(-y, -x) %>%
slice(1)
ggplot() +
geom_density(data = dat, aes(x = rating, colour = cond)) +
geom_segment(data = cdat.dens, aes(x = rating.mean, xend = rating.mean, y = 0, yend = dens.mean, colour = cond),
linetype = "dashed", size = 1) +
geom_segment(data = cdat.dens, aes(x = ci.low, xend = ci.low, y = 0, yend = dens.cilow, colour = cond),
linetype = "dotted", size = 1) +
geom_segment(data = cdat.dens, aes(x = ci.upp, xend = ci.upp, y = 0, yend = dens.ciupp, colour = cond),
linetype = "dotted", size = 1) +
facet_wrap(~direction)
如果您想在不构建绘图对象且不在绘图之前处理数据的情况下绘制平均线，您可以使用 stat_summary():
(
ggplot(data = dat, aes(x = rating, colour = cond))
+ geom_density()
+ stat_summary(
aes(y = rating, x = 0),
geom = 'rect',
fun.data = density_mean_line(dat$rating),
key_glyph = "vline",
size = 1
)
)
给予：
其中：
density_mean_line = function(values) {
values_range = range(values, na.rm=TRUE)
function(x) {
density_data = StatDensity$compute_group(
data.frame(x=x),
scales=list(
x=scale_x_continuous(limits = values_range)
)
)
mean_x = mean(x)
data.frame(
xmin=mean_x,
xmax=mean_x,
ymin=0,
ymax=approx(density_data$x, density_data$density, xout=mean_x)$y
)
}
}
并且 dat 定义为 erc 的回答：
set.seed(1234)
dat <- data.frame(cond = factor(rep(c("A","B"), each=200)),
rating = c(rnorm(200),rnorm(200, mean=.8)))
此技术也可用于生成实心区域（与密度轮廓颜色相同）：
(
ggplot(data = dat, aes(x = rating, colour = cond, group = cond))
+ stat_summary(
aes(y = rating, x = 0, fill = cond),
geom = 'rect',
fun.data = density_ci(dat$rating),
size=1
)
+ stat_summary(
aes(y = rating, x = 0),
geom = 'rect',
fun.data = density_mean_line(dat$rating),
key_glyph = "vline",
size = 0.5,
color='grey20'
)
+ geom_density()
)
其中：
density_ci = function(values, resolution=100) {
values_range = range(values, na.rm=TRUE)
function(x) {
density_data = StatDensity$compute_group(
data.frame(x=x),
scales=list(
x=scale_x_continuous(limits = values_range)
)
)
mean_x = mean(x)
sem = sd(x) / sqrt(length(x))
ci_lower = mean_x - 1.96 * sem
ci_upper = mean_x + 1.96 * sem
x_values = seq(ci_lower, ci_upper, length.out=resolution)
data.frame(
xmin=x_values,
xmax=x_values,
ymin=rep(0, resolution),
ymax=approx(density_data$x, density_data$density, xout=x_values)$y
)
}
}