Latent Variables
Isaiah C. Mireles
2026-03-12
Background :
This data is of the midterm performance of students spring 2024 on the Stats 10 Midterm. There are a total of 34 questions, Sample size (n = 155).
Research Question :
- Is there evidence to suggest 3 latent variables (Regression, Design & Probability) structure to the performance of students
Read in Data :
rm(list=ls())
df <- read.csv("/Users/isaiahmireles/Desktop/Misconceptions/24S-STATS-10-LEC-4_Midterm/S24_Midterm_student_responses copy.csv")
library(tidyverse)Anonymization :
df$Student.ID <- paste0("Student", seq_len(nrow(df)))
df <- df |> select(-c(First.Name, Last.Name, Email))Data Basics :
Population : Spring 2024, Stats 10 MCQ Midterm
df$Total.Score <- as.numeric(df$Total.Score)
s2 <- sd(df$Total.Score, na.rm = T)
ybar <- mean(df$Total.Score, na.rm=T)
grd_hist_plt <-
df |>
ggplot(aes(x = Total.Score)) +
geom_histogram(
aes(y = after_stat(density)),
bins = 12,
fill = "lightblue",
color = "black"
) +
stat_function(
fun = dnorm,
args = list(mean = ybar, sd = s2),
color = "blue",
linewidth = 1.2
) +
labs(
title = "Hist of Exam Performance",
x = "Total Score",
y = "Density"
)
box_plt <-
df |>
ggplot(aes(x = Total.Score)) +
geom_boxplot(fill = "lightblue", color = "black") +
labs(
title = "Boxplot of Exam Performance",
x = "Total Score"
)
library(patchwork)
grd_hist_plt/box_plt
Histogram Displays heavy right skew
Normal dist displays a moderate fit
Pct Performance :
df$Max.Points <- as.numeric(df$Max.Points)## Warning: NAs introduced by coercionperformance_df <-
df |>
mutate(pct = round( Total.Score/Max.Points, 3)) |>
select(Total.Score, pct) |>
arrange(desc(pct)) |>
mutate(passed = if_else(pct >= .7, 1, 0))
passed_failed_bar_plt <-
performance_df |>
filter(!is.na(passed)) |>
mutate(
passed_label = factor(
if_else(passed == 1, "Passed", "Failed"),
levels = c("Failed", "Passed")
)
) |>
count(passed_label) |>
mutate(
percent = n / sum(n)
) |>
ggplot(aes(x = passed_label, y = percent, fill = passed_label)) +
geom_col() +
scale_fill_manual(
values = c("Failed" = "red", "Passed" = "green")
) +
scale_y_continuous(labels = scales::percent) +
labs(
title = "Percent Passed vs Failed",
x = "",
y = "Percent"
)
passed_failed_bar_plt
summary(performance_df$passed) # 0.7248 pct passed ## Min. 1st Qu. Median Mean 3rd Qu. Max. NA's
## 0.0000 0.0000 1.0000 0.7248 1.0000 1.0000 6summary(performance_df$Total.Score) # 26.33## Min. 1st Qu. Median Mean 3rd Qu. Max. NA's
## 13.00 23.00 27.00 26.33 30.00 34.00 6Correlations
Notice that our data is not continuous so we cannot use our typical correlation (pearsons’ correlation); but rather the binary version : Tetrachoric Correlation.
binary <-
df |>
select(ends_with("Score")) |>
select(-Total.Score)
library(psych)
tetrachoric(binary)## Warning in cor.smooth(mat): Matrix was not positive definite, smoothing was
## done## Call: tetrachoric(x = binary)
## tetrachoric correlation
## Q.1.S Q.2.S Q.3.S Q.4.S Q.5.S Q.6.S Q.7.S Q.8.S Q.9.S Q.10.
## Question.1.Score 1.00
## Question.2.Score 0.50 1.00
## Question.3.Score 0.20 0.52 1.00
## Question.4.Score 0.15 -0.05 0.02 1.00
## Question.5.Score 0.03 0.27 0.16 -0.10 1.00
## Question.6.Score -0.08 0.11 0.39 -0.12 0.31 1.00
## Question.7.Score 0.52 0.31 0.24 0.00 0.19 0.36 1.00
## Question.8.Score 0.21 0.23 0.28 0.26 0.20 0.03 0.23 1.00
## Question.9.Score 0.18 0.42 0.30 0.16 0.18 0.15 0.36 0.28 1.00
## Question.10.Score 0.35 0.50 0.52 0.36 0.25 0.12 0.11 0.40 0.48 1.00
## Question.11.Score 0.12 0.19 0.14 0.03 0.30 0.09 0.36 0.43 0.49 0.27
## Question.12.Score 0.19 0.30 0.16 0.22 0.17 0.23 0.15 0.34 0.39 0.42
## Question.13.Score -0.04 -0.01 0.33 0.47 0.07 0.39 0.13 0.26 0.09 0.19
## Question.14.Score 0.01 0.25 0.27 0.02 0.25 0.18 0.12 0.23 0.35 0.05
## Question.15.Score 0.02 -0.09 0.05 0.05 0.02 0.24 0.33 -0.05 -0.01 -0.15
## Question.16.Score 0.20 0.15 -0.09 0.15 0.05 0.06 0.18 0.20 0.53 0.24
## Question.17.Score 0.24 0.30 0.09 0.40 -0.12 -0.03 0.17 0.31 0.14 0.29
## Question.18.Score -0.04 -0.03 0.37 0.35 0.26 0.22 0.03 0.37 0.27 0.36
## Question.19.Score 0.19 -0.08 0.22 0.31 0.30 0.23 0.13 0.29 0.18 0.25
## Question.20.Score 0.14 0.14 0.29 0.27 0.43 0.23 0.31 0.53 0.49 0.49
## Question.21.Score -0.12 0.13 0.21 0.29 0.48 0.12 -0.07 0.11 0.06 0.25
## Question.22.Score 0.05 0.32 0.02 0.01 0.09 -0.13 0.11 0.08 0.07 0.00
## Question.23.Score 0.13 0.07 0.34 0.26 0.30 0.42 -0.09 0.19 0.00 0.34
## Question.24.Score 0.32 0.17 0.21 -0.28 0.15 0.02 0.09 -0.12 0.10 -0.01
## Question.25.Score 0.04 -0.06 0.27 0.10 0.22 0.41 0.36 0.32 0.22 0.22
## Question.26.Score -0.12 0.15 0.15 -0.14 0.46 0.33 0.14 0.16 0.23 0.17
## Question.27.Score -0.04 0.24 0.44 0.15 0.44 0.63 0.31 0.31 0.53 0.50
## Question.28.Score 0.30 0.23 0.32 0.29 0.18 0.02 0.01 0.06 0.16 0.45
## Question.29.Score -0.07 0.16 0.25 0.18 0.06 0.46 0.13 0.05 0.36 0.32
## Question.30.Score 0.32 0.52 0.36 0.22 0.08 0.12 0.16 0.31 0.27 0.47
## Question.31.Score 0.30 0.31 0.32 0.07 0.38 0.31 0.23 0.54 0.23 0.39
## Question.32.Score 0.31 0.22 -0.12 0.05 0.09 0.20 0.18 0.04 -0.04 0.25
## Question.33.Score 0.18 0.29 0.07 0.17 0.35 0.35 0.05 0.03 0.18 0.34
## Question.34.Score 0.05 0.28 0.20 0.23 0.23 0.21 -0.01 0.22 0.01 0.37
## Q.11.
## Question.1.Score
## Question.2.Score
## Question.3.Score
## Question.4.Score
## Question.5.Score
## Question.6.Score
## Question.7.Score
## Question.8.Score
## Question.9.Score
## Question.10.Score
## Question.11.Score 1.00
## Question.12.Score 0.40
## Question.13.Score 0.23
## Question.14.Score 0.33
## Question.15.Score -0.06
## Question.16.Score 0.47
## Question.17.Score 0.09
## Question.18.Score 0.26
## Question.19.Score 0.19
## Question.20.Score 0.48
## Question.21.Score 0.24
## Question.22.Score -0.14
## Question.23.Score -0.02
## Question.24.Score 0.14
## Question.25.Score 0.18
## Question.26.Score 0.42
## Question.27.Score 0.15
## Question.28.Score -0.09
## Question.29.Score 0.14
## Question.30.Score 0.03
## Question.31.Score 0.07
## Question.32.Score -0.06
## Question.33.Score -0.16
## Question.34.Score 0.20
## Q.12. Q.13. Q.14. Q.15. Q.16. Q.17. Q.18. Q.19. Q.20. Q.21.
## Question.12.Score 1.00
## Question.13.Score 0.45 1.00
## Question.14.Score 0.16 0.13 1.00
## Question.15.Score 0.13 0.25 0.33 1.00
## Question.16.Score 0.47 0.21 0.08 0.05 1.00
## Question.17.Score 0.41 -0.07 0.12 -0.03 -0.08 1.00
## Question.18.Score 0.28 0.38 -0.05 0.07 0.36 0.04 1.00
## Question.19.Score 0.36 0.43 -0.01 -0.08 0.41 0.13 0.66 1.00
## Question.20.Score 0.51 0.49 0.06 -0.07 0.48 0.10 0.65 0.73 1.00
## Question.21.Score 0.34 0.34 0.22 0.04 0.27 0.01 0.39 0.27 0.21 1.00
## Question.22.Score 0.35 -0.02 0.20 0.08 0.20 0.26 -0.19 0.09 0.11 0.15
## Question.23.Score 0.31 0.43 0.34 0.13 0.14 0.03 0.34 0.36 0.32 0.43
## Question.24.Score -0.06 -0.22 0.27 0.05 0.22 -0.21 0.02 0.04 0.01 0.12
## Question.25.Score 0.30 0.38 0.13 0.13 0.11 0.04 0.38 0.48 0.56 0.09
## Question.26.Score 0.01 0.12 -0.05 -0.18 0.31 -0.17 0.22 0.26 0.40 0.04
## Question.27.Score 0.23 0.31 0.29 0.14 0.13 0.08 0.45 0.33 0.46 0.20
## Question.28.Score -0.07 -0.02 0.08 0.07 0.24 -0.06 0.42 0.32 0.08 0.23
## Question.29.Score 0.44 0.16 0.23 0.13 0.27 0.22 0.06 0.06 0.06 0.27
## Question.30.Score 0.16 -0.19 0.21 -0.25 -0.03 0.63 0.20 0.17 0.02 0.17
## Question.31.Score 0.32 0.17 0.09 0.07 0.30 -0.02 0.37 0.25 0.38 0.33
## Question.32.Score 0.24 0.10 0.03 0.22 0.27 0.20 0.07 0.22 0.12 -0.02
## Question.33.Score 0.35 0.25 0.05 0.33 0.27 0.21 0.31 0.40 0.30 0.20
## Question.34.Score 0.44 0.20 0.06 0.07 0.05 0.39 0.34 0.16 0.28 0.27
## Q.22.
## Question.12.Score
## Question.13.Score
## Question.14.Score
## Question.15.Score
## Question.16.Score
## Question.17.Score
## Question.18.Score
## Question.19.Score
## Question.20.Score
## Question.21.Score
## Question.22.Score 1.00
## Question.23.Score 0.01
## Question.24.Score 0.02
## Question.25.Score 0.01
## Question.26.Score -0.04
## Question.27.Score -0.11
## Question.28.Score 0.11
## Question.29.Score 0.17
## Question.30.Score 0.14
## Question.31.Score 0.02
## Question.32.Score 0.08
## Question.33.Score 0.14
## Question.34.Score -0.09
## Q.23. Q.24. Q.25. Q.26. Q.27. Q.28. Q.29. Q.30. Q.31. Q.32.
## Question.23.Score 1.00
## Question.24.Score 0.03 1.00
## Question.25.Score 0.24 -0.15 1.00
## Question.26.Score 0.04 -0.10 0.19 1.00
## Question.27.Score 0.34 -0.20 0.38 0.33 1.00
## Question.28.Score 0.19 0.18 0.01 0.13 0.33 1.00
## Question.29.Score 0.29 -0.18 0.46 0.26 0.36 0.15 1.00
## Question.30.Score 0.14 -0.12 0.05 -0.08 0.45 0.44 0.27 1.00
## Question.31.Score 0.42 0.04 0.30 0.25 0.54 0.26 0.31 0.30 1.00
## Question.32.Score 0.30 -0.28 0.23 0.11 0.33 0.24 0.21 0.32 0.30 1.00
## Question.33.Score 0.38 -0.09 0.37 0.18 0.36 0.30 0.30 0.16 0.24 0.58
## Question.34.Score 0.11 0.00 0.30 0.19 0.27 0.05 0.32 0.25 0.41 0.30
## Q.33.
## Question.23.Score
## Question.24.Score
## Question.25.Score
## Question.26.Score
## Question.27.Score
## Question.28.Score
## Question.29.Score
## Question.30.Score
## Question.31.Score
## Question.32.Score
## Question.33.Score 1.00
## Question.34.Score 0.32
## [1] 1.00
##
## with tau of
## Question.1.Score Question.2.Score Question.3.Score Question.4.Score
## -1.675 -1.748 -1.498 0.093
## Question.5.Score Question.6.Score Question.7.Score Question.8.Score
## 0.212 -0.299 -2.214 -0.597
## Question.9.Score Question.10.Score Question.11.Score Question.12.Score
## -1.929 -0.246 -1.929 -1.107
## Question.13.Score Question.14.Score Question.15.Score Question.16.Score
## -1.358 -1.047 -1.107 -1.401
## Question.17.Score Question.18.Score Question.19.Score Question.20.Score
## -1.551 -0.861 -1.107 -1.831
## Question.21.Score Question.22.Score Question.23.Score Question.24.Score
## -0.723 0.519 -0.558 -0.281
## Question.25.Score Question.26.Score Question.27.Score Question.28.Score
## -1.171 -1.241 -0.937 -0.937
## Question.29.Score Question.30.Score Question.31.Score Question.32.Score
## -0.352 -1.317 -0.659 -0.519
## Question.33.Score Question.34.Score
## 0.042 0.110