# cleaning
library(tidyverse)
# visualization
theme_set(theme_classic() +
theme(panel.grid = element_blank(),
axis.text = element_text(size = 18),
axis.title = element_text(size = 18),
text = element_text(family = "Lato")))
library(patchwork)
# cohen's d
library(effectsize)
# power
library(pwr)\[ Cohen's \; d = \frac{\bar{y_A} - \bar{y_B}}{\sqrt{(s^2_A + s^2_B)/2}} \]
\[ Cohen's \; d = \frac{\bar{y_A} - \bar{y_B}}{\sqrt{\frac{(n_A - 1)\times s^2_A + (n_B - 1)\times s^2_B}{n_A + n_B - 2}}} \]
\[ CI = (\bar{y_A} - \bar{y_B}) \pm t \times \sqrt{\frac{(n_A - 1)s_A^2 + (n_B - 1)s_B^2}{n_A+n_B-2}} \times \sqrt{\frac{1}{n_A}+\frac{1}{n_B}} \]
\[ t_s = \frac{\bar{y}_d - \mu_0}{s_d - \sqrt{n}} \]
If variances are not equal:
\[ SE_{\bar{y_A} - \bar{y_B}} = \sqrt{\frac{s_A^2}{n_A}+\frac{s_B^2}{n_B}} \]
You have two raised beds in which you’re growing tomatoes. One bed is in the sun, but the other is in shade. You want to know if the weight of the tomatoes is different between beds. You measure 33 tomatoes from each bed.
tomatoes <- cbind(sunny = rnorm(n = 33, mean = 150, sd = 20),
shaded = rnorm(n = 33, mean = 130, sd = 10)) %>%
as_tibble() %>%
pivot_longer(cols = 1:2, names_to = "sun_level", values_to = "weight_g")
ggplot(data = tomatoes,
aes(x = sun_level,
y = weight_g)) +
geom_jitter(width = 0.1)
var.test(weight_g ~ sun_level,
data = tomatoes)
F test to compare two variances
data: weight_g by sun_level
F = 0.21215, num df = 32, denom df = 32, p-value = 3.181e-05
alternative hypothesis: true ratio of variances is not equal to 1
95 percent confidence interval:
0.1047793 0.4295538
sample estimates:
ratio of variances
0.2121517 t.test(weight_g ~ sun_level,
data = tomatoes,
var.equal = FALSE)
Welch Two Sample t-test
data: weight_g by sun_level
t = -4.8692, df = 44.993, p-value = 1.42e-05
alternative hypothesis: true difference in means between group shaded and group sunny is not equal to 0
95 percent confidence interval:
-27.55166 -11.42803
sample estimates:
mean in group shaded mean in group sunny
129.5924 149.0822 # higher power
pwr.t.test(n = NULL, d = 0.7, sig.level = 0.05, power = 0.95)
Two-sample t test power calculation
n = 54.01938
d = 0.7
sig.level = 0.05
power = 0.95
alternative = two.sided
NOTE: n is number in *each* group# lower power
pwr.t.test(n = NULL, d = 0.7, sig.level = 0.05, power = 0.80)
Two-sample t test power calculation
n = 33.02457
d = 0.7
sig.level = 0.05
power = 0.8
alternative = two.sided
NOTE: n is number in *each* groupYou have two worm compost bins: one in which you throw citrus peels, and the other in which you don’t. You’re curious to see if the citrus worms are bigger than the non-citrus worms. You measure 34 worms from each bin and find this result:
worms <- cbind(citrus = rnorm(n = 34, mean = 140, sd = 20),
non_citrus = rnorm(n = 34, mean = 160, sd = 15)) %>%
as_tibble() %>%
pivot_longer(cols = 1:2, names_to = "compost_bin", values_to = "weight_g")
ggplot(data = worms,
aes(x = compost_bin,
y = weight_g)) +
geom_boxplot()
var.test(weight_g ~ compost_bin,
data = worms)
F test to compare two variances
data: weight_g by compost_bin
F = 2.2021, num df = 33, denom df = 33, p-value = 0.02627
alternative hypothesis: true ratio of variances is not equal to 1
95 percent confidence interval:
1.099766 4.409198
sample estimates:
ratio of variances
2.202064 t.test(weight_g ~ compost_bin,
data = worms,
var.equal = FALSE)
Welch Two Sample t-test
data: weight_g by compost_bin
t = -4.8755, df = 57.848, p-value = 8.84e-06
alternative hypothesis: true difference in means between group citrus and group non_citrus is not equal to 0
95 percent confidence interval:
-27.86552 -11.64356
sample estimates:
mean in group citrus mean in group non_citrus
136.0371 155.7917 \[ \bar{y_a} - \bar{y_b} \]
set.seed(1)
data <- cbind(a = rnorm(n = 100, mean = 10, sd = 2),
b = rnorm(n = 100, mean = 11, sd = 2)) %>%
as_tibble() %>%
pivot_longer(cols = 1:2, names_to = "group", values_to = "value") %>%
group_by(group)
set.seed(1)
large <- data %>%
slice_sample(n = 40)
ggplot(data = large,
aes(x = group,
y = value)) +
stat_summary(geom = "pointrange",
fun.data = mean_se) +
scale_y_continuous(limits = c(5, 16)) 
t.test(value ~ group,
data = large,
var.equal = TRUE)
Two Sample t-test
data: value by group
t = -2.9666, df = 78, p-value = 0.003996
alternative hypothesis: true difference in means between group a and group b is not equal to 0
95 percent confidence interval:
-2.1892413 -0.4308888
sample estimates:
mean in group a mean in group b
9.819199 11.129264 set.seed(1)
small <- data %>%
slice_sample(n = 20)
ggplot(data = small,
aes(x = group,
y = value)) +
stat_summary(geom = "pointrange",
fun.data = mean_se) +
scale_y_continuous(limits = c(5, 16)) 
t.test(value ~ group,
data = small,
var.equal = TRUE)
Two Sample t-test
data: value by group
t = -1.5737, df = 38, p-value = 0.1238
alternative hypothesis: true difference in means between group a and group b is not equal to 0
95 percent confidence interval:
-1.9408171 0.2431029
sample estimates:
mean in group a mean in group b
10.23925 11.08811 set.seed(1)
small <- cbind(a = MASS::mvrnorm(n = 10, mu = 16.5, Sigma = 2, empirical = TRUE),
b = MASS::mvrnorm(n = 10, mu = 17, Sigma = 2, empirical = TRUE)) %>%
as.data.frame() %>%
rename("a" = "V1", "b" = "V2") %>%
pivot_longer(cols = 1:2, names_to = "group", values_to = "value")
# small <- cbind(a = rnorm(n = 10, mean = 14, sd = 3),
# b = rnorm(n = 10, mean = 17, sd = 3)) %>%
# as_tibble() %>%
# pivot_longer(cols = 1:2, names_to = "group", values_to = "value")
ggplot(data = small,
aes(x = group,
y = value)) +
geom_point(size = 3,
shape = 21,
aes(color = group),
position = position_jitter(
width = 0.15,
height = 0,
seed = 10
)) +
stat_summary(geom = "point",
fun = mean,
color = "red",
size = 4) +
# scale_y_continuous(limits = c(5, 16),
# breaks = c(6, 9, 12, 15)) +
scale_color_manual(values = c("darkblue", "darkgreen")) +
theme(legend.position = "none")
ggplot(data = small,
aes(x = value)) +
geom_density(aes(fill = group),
alpha = 0.4)
t.test(value ~ group,
data = small,
var.equal = TRUE)
Two Sample t-test
data: value by group
t = -0.79057, df = 18, p-value = 0.4395
alternative hypothesis: true difference in means between group a and group b is not equal to 0
95 percent confidence interval:
-1.8287398 0.8287398
sample estimates:
mean in group a mean in group b
16.5 17.0 cohens_d(value ~ group,
data = small,
pooled_sd = FALSE)Cohen's d | 95% CI
-------------------------
-0.35 | [-1.23, 0.54]
- Estimated using un-pooled SD.set.seed(1)
large <- cbind(a = MASS::mvrnorm(n = 100, mu = 16.5, Sigma = 2, empirical = TRUE),
b = MASS::mvrnorm(n = 100, mu = 17, Sigma = 2, empirical = TRUE)) %>%
as.data.frame() %>%
rename("a" = "V1", "b" = "V2") %>%
pivot_longer(cols = 1:2, names_to = "group", values_to = "value")
ggplot(data = large,
aes(x = group,
y = value)) +
geom_point(size = 3,
shape = 21,
aes(color = group),
position = position_jitter(
width = 0.15,
height = 0,
seed = 10
)) +
stat_summary(geom = "point",
fun = mean,
color = "red",
size = 4) +
# scale_y_continuous(limits = c(5, 16),
# breaks = c(6, 9, 12, 15)) +
scale_color_manual(values = c("darkblue", "darkgreen")) +
theme(legend.position = "none")
ggplot(data = large,
aes(x = value)) +
geom_histogram(aes(fill = group),
alpha = 0.4)
t.test(value ~ group,
data = large,
var.equal = TRUE)
Two Sample t-test
data: value by group
t = -2.5, df = 198, p-value = 0.01323
alternative hypothesis: true difference in means between group a and group b is not equal to 0
95 percent confidence interval:
-0.8944035 -0.1055965
sample estimates:
mean in group a mean in group b
16.5 17.0 cohens_d(value ~ group,
data = large,
pooled_sd = FALSE)Cohen's d | 95% CI
--------------------------
-0.35 | [-0.63, -0.07]
- Estimated using un-pooled SD.set.seed(1)
needlegrass <- cbind(ungrazed = rnorm(n = 35, mean = 82, sd = 6),
grazed = rnorm(n = 35, mean = 74, sd = 5)) %>%
as_tibble() %>%
pivot_longer(cols = 1:2, names_to = "plot_type", values_to = "height_cm")
# boxplot
ggplot(data = needlegrass,
aes(x = plot_type,
y = height_cm,
color = plot_type)) +
geom_boxplot(outliers = FALSE,
linewidth = 1) +
geom_point(position = position_jitter(
width = 0.1,
height = 0.1,
seed = 10
),
alpha = 0.4,
size = 1) +
scale_color_manual(values = c("darkgreen", "cornflowerblue")) 
# plot without all the adjustments
ggplot(data = needlegrass,
aes(x = plot_type,
y = height_cm,
color = plot_type)) +
geom_point(position = position_jitter(width = 0.1, height = 0, seed = 10),
alpha = 0.2) +
stat_summary(geom = "pointrange",
fun.data = mean_cl_normal) 
# "finalized" plot
ggplot(data = needlegrass,
aes(x = plot_type,
y = height_cm,
color = plot_type)) +
geom_point(position = position_jitter(width = 0.1, height = 0, seed = 10),
alpha = 0.2) +
scale_color_manual(values = c("darkgreen", "cornflowerblue")) +
stat_summary(geom = "pointrange",
fun.data = mean_cl_normal,
size = 1,
linewidth = 1) +
labs(x = "Plot type",
y = "Height (cm)") +
theme(legend.position = "none")
Doing a t-test:
var.test(
height_cm ~ plot_type,
data = needlegrass
)
F test to compare two variances
data: height_cm by plot_type
F = 0.72641, num df = 34, denom df = 34, p-value = 0.3559
alternative hypothesis: true ratio of variances is not equal to 1
95 percent confidence interval:
0.366669 1.439115
sample estimates:
ratio of variances
0.7264149 t.test(
height_cm ~ plot_type,
data = needlegrass,
var.equal = TRUE
)
Two Sample t-test
data: height_cm by plot_type
t = -5.9471, df = 68, p-value = 1.052e-07
alternative hypothesis: true difference in means between group grazed and group ungrazed is not equal to 0
95 percent confidence interval:
-9.719086 -4.835512
sample estimates:
mean in group grazed mean in group ungrazed
75.18323 82.46053 vect <- dt(x = seq(from = -10, to = 10, by = 0.5), df = 66.334) %>%
enframe()
ggplot(data = vect,
aes(x = name,
y = value)) +
geom_line() +
geom_vline(xintercept = -5.9471)
Cohen’s d:
# pooled SD
cohens_d(
height_cm ~ plot_type,
data = needlegrass,
pooled_sd = TRUE
)Cohen's d | 95% CI
--------------------------
-1.42 | [-1.94, -0.89]
- Estimated using pooled SD.# unpooled SD
cohens_d(
height_cm ~ plot_type,
data = needlegrass,
pooled_sd = FALSE
)Cohen's d | 95% CI
--------------------------
-1.42 | [-1.94, -0.89]
- Estimated using un-pooled SD.# by hand
needlegrass_sum <- needlegrass %>%
group_by(plot_type) %>%
reframe(
mean = mean(height_cm),
var = var(height_cm),
n = length(height_cm)
)
ya <- pluck(needlegrass_sum, 2, 1)
yb <- pluck(needlegrass_sum, 2, 2)
vara <- pluck(needlegrass_sum, 3, 1)
varb <- pluck(needlegrass_sum, 3, 2)
na <- pluck(needlegrass_sum, 4, 1)
nb <- pluck(needlegrass_sum, 4, 2)
# by hand
(ya - yb)/sqrt((vara + varb)/2)[1] -1.421636(ya - yb)/sqrt(
((na - 1)*vara + (nb - 1)*varb)/(na + nb - 2)
)[1] -1.421636managed <- rnorm(n = 33, mean = 5, sd = 1) %>%
enframe() %>%
mutate(treatment = "managed")
nonintervention <- rnorm(n = 30, mean = 7, sd = 1) %>%
enframe() %>%
mutate(treatment = "non-intervention")
pools <- rbind(managed, nonintervention) %>%
select(treatment, value) %>%
rename(temp = value)
var.test(temp ~ treatment,
data = pools)
F test to compare two variances
data: temp by treatment
F = 1.287, num df = 32, denom df = 29, p-value = 0.4953
alternative hypothesis: true ratio of variances is not equal to 1
95 percent confidence interval:
0.6197933 2.6377877
sample estimates:
ratio of variances
1.286962 t.test(temp ~ treatment,
data = pools)
Welch Two Sample t-test
data: temp by treatment
t = -11.206, df = 60.948, p-value < 2.2e-16
alternative hypothesis: true difference in means between group managed and group non-intervention is not equal to 0
95 percent confidence interval:
-2.558536 -1.783706
sample estimates:
mean in group managed mean in group non-intervention
4.952439 7.123560 cohens_d(temp ~ treatment,
data = pools)Cohen's d | 95% CI
--------------------------
-2.81 | [-3.51, -2.10]
- Estimated using pooled SD.Statement: Our data suggest a difference in water temperature between managed (n = 33) and non-intervention (i.e. control, n = 30) vernal pools, with a strong (Cohen’s d = 2.19) effect of management.
Temperatures in managed pools were different from those in non-intervention pools (two-tailed two-sample t-test, t(60.9) = -8.7, p < 0.001, ⍺ = 0.05); on average, managed pools were 5.3 °C, while control pools were 7.1 °C.
set.seed(666)
data1 <- rnorm(n = 30, mean = 15, sd = 2) |>
enframe()
ggplot(data = data1,
aes(x = value)) +
geom_histogram(bins = 6,
fill = "cornflowerblue",
color = "black") +
scale_y_continuous(expand = c(0, 0))
ggplot(data = data1,
aes(sample = value)) +
geom_qq_line(color = "black") +
geom_qq(color = "cornflowerblue",
size = 4)
set.seed(666)
data2 <- rgamma(n = 30, shape = 0.5) |>
enframe()
ggplot(data = data2,
aes(x = value)) +
geom_histogram(bins = 6,
fill = "orange",
color = "black") +
scale_y_continuous(expand = c(0, 0))
ggplot(data = data2,
aes(sample = value)) +
geom_qq_line(color = "black") +
geom_qq(color = "orange",
size = 4)
@online{bui2026,
author = {Bui, An},
title = {Week 4 Figures - {Lectures} 7 and 8},
date = {2026-01-26},
url = {https://winter-2026.envs-193ds.com/lecture/lecture_week-04.html},
langid = {en}
}