Introduction

In this document, I describe the process of matching the MSDS students to capstones. First I clean the raw data from the Qualtrics survey. Then I report on the distribution of rankings for each capstone. Finally I assign students to capstones, and report on the rankings that each student gave to their assigned capstone.

In this report, we require the following packages:

library(knitr)
library(summarytools)
library(lpSolve)
library(lubridate)
library(tidyverse)
library(DT)

The Qualtrics Survey

I wrote a survey for students to report their rankings of 17 capstones, and disseminated it using this link: https://virginia.az1.qualtrics.com/jfe/form/SV_88EAacOODg7OccZ. The survey consists of three questions. First, students provide their UVA computing ID and full name. Then they rank all 17 capstones from the one they most want to work on (1) to the one they least want to work on (17). The survey looks like this:

The desktop and mobile versions of the survey allow a student to drag the different capstones to positions higher and lower on this list. When they do so, numbers appear next to each capstone, with the capstone on top labeled 1. The capstones on top represent the students’ top preferences.

Cleaning the Raw Qualtrics Data

I downloaded the raw data from the Qualtrics website in CSV format and loaded it into R:

data <- read_csv("Residential+MSDS+Capstones+2020-2021_August+28,+2020_11.01.csv")
## Parsed with column specification:
## cols(
##   .default = col_character()
## )
## See spec(...) for full column specifications.

To deal with missing values in the student rankings, I replace the missing values with the least preferred rank:

data[is.na(data)] <- 17

Because the raw data uses numeric codes for capstones, we also input the names of the capstones in the order they are recognized by the Qualtrics survey:

capnames <- c("LMI", "Clarabridge", "UVA Pediatrics", "InnovateEDU", 
              "dbS Productions", "North American ALMA Science Center at NRAO", 
              "UVA Psychology", "UVA Physics", "UVA Department of Anesthesiology", 
              "UVA Biocomplexity Institute & Initiative", "Department of Defense ", 
              "UVA Astronomy", 
              "UVA Oncology: Integrating clinical and genomic data in cancer patients", 
              "UVA Oncology: Mutational processes in cancer genomes", 
              "UVA School of Education - Special Education Research Accelerator (SERA)", 
              "UVA School of Education", "UVA Environmental Science")

We need to remove the first two rows of metadata, and isolate the columns that refer to the students’ rankings, which all begin with the letter “Q”. We also save the time and date each set of responses was submitted to address the students who submitted more than one set of rankings.

data <- data[-c(1,2),] %>%
  dplyr::select(RecordedDate, starts_with("Q")) %>%
  filter(Q1 != "ajg3eh") # one student has a special circumstance

It is possible that some students submitted more than one set of rankings. For these students, we keep only the most recent rankings:

data <- data %>%
  mutate(RecordedDate = ymd_hms(RecordedDate),
         Q1 = str_to_lower(Q1)) %>%
  group_by(Q1) %>%
  slice(which.max(RecordedDate)) %>%
  ungroup() %>%
  select(-RecordedDate)

The data at this point are coded as character. We convert every column to numeric class:

data <- data %>%
  mutate(Q1 = as.factor(Q1),
         Q3 = as.factor(Q3)) %>%
  mutate_if(is.character,as.numeric) 
colnames(data) <- c("student", "fullname", capnames)

We save these data as a CSV:

write_csv(data, path="student_rankings2021.csv")

Understanding the Rankings

To better understand the distribution of students’ rankings for each capstone we create a data frame that places the capstones in the columns and orders these columns from the lowest to the highest average rank:

capstone.ranks <- data[,-c(1,2)]
capstone.ranks <- capstone.ranks[,order(colMeans(capstone.ranks))]

The following table lists the capstones from most popular, at the top, to least popular, on the bottom. For each capstone, the table lists the mean and standard deviation of the students’ rankings, as well as minimum, median, maximum, and interquartile range. The bar graph on the right is a histogram of the rankings: high bars to the left indicate a lot of high rankings (1, 2, etc.) and high bars to the right indicate a lot of low rankings (20, 21, etc.)

dfSummary(capstone.ranks, plain.ascii = FALSE, style = "grid", 
          graph.magnif = 0.75, valid.col = FALSE, 
          tmp.img.dir = "/tmp", headings = FALSE)
No Variable Stats / Values Freqs (% of Valid) Graph Missing
1 UVA Psychology
[numeric]		 Mean (sd) : 6.4 (4.5)
min < med < max:
1 < 6 < 17
IQR (CV) : 7.2 (0.7)		 15 distinct values 0
(0%)
2 UVA Pediatrics
[numeric]		 Mean (sd) : 7.3 (4.7)
min < med < max:
1 < 6 < 17
IQR (CV) : 7 (0.6)		 15 distinct values 0
(0%)
3 dbS Productions
[numeric]		 Mean (sd) : 7.5 (4.5)
min < med < max:
1 < 7 < 17
IQR (CV) : 9 (0.6)		 16 distinct values 0
(0%)
4 UVA Biocomplexity Institute & Initiative
[numeric]		 Mean (sd) : 7.6 (4.8)
min < med < max:
1 < 8 < 17
IQR (CV) : 8.2 (0.6)		 16 distinct values 0
(0%)
5 UVA Oncology: Integrating clinical and genomic data in cancer patients
[numeric]		 Mean (sd) : 8.3 (4.8)
min < med < max:
1 < 8 < 16
IQR (CV) : 9 (0.6)		 15 distinct values 0
(0%)
6 Clarabridge
[numeric]		 Mean (sd) : 8.4 (4.9)
min < med < max:
1 < 9 < 17
IQR (CV) : 9 (0.6)		 16 distinct values 0
(0%)
7 UVA Department of Anesthesiology
[numeric]		 Mean (sd) : 8.4 (4.8)
min < med < max:
1 < 8 < 17
IQR (CV) : 7 (0.6)		 16 distinct values 0
(0%)
8 UVA Oncology: Mutational processes in cancer genomes
[numeric]		 Mean (sd) : 8.5 (5.1)
min < med < max:
1 < 8 < 17
IQR (CV) : 10 (0.6)		 16 distinct values 0
(0%)
9 InnovateEDU
[numeric]		 Mean (sd) : 8.6 (4.4)
min < med < max:
1 < 8.5 < 17
IQR (CV) : 7 (0.5)		 16 distinct values 0
(0%)
10 LMI
[numeric]		 Mean (sd) : 9.3 (4.3)
min < med < max:
1 < 9.5 < 17
IQR (CV) : 5.2 (0.5)		 17 distinct values 0
(0%)
11 North American ALMA Science Center at NRAO
[numeric]		 Mean (sd) : 9.3 (3.9)
min < med < max:
2 < 9 < 17
IQR (CV) : 5.2 (0.4)		 15 distinct values 0
(0%)
12 Department of Defense
[numeric]		 Mean (sd) : 9.8 (5.1)
min < med < max:
2 < 10 < 17
IQR (CV) : 10 (0.5)		 16 distinct values 0
(0%)
13 UVA School of Education
[numeric]		 Mean (sd) : 10.1 (5.2)
min < med < max:
1 < 11 < 17
IQR (CV) : 10 (0.5)		 16 distinct values 0
(0%)
14 UVA Environmental Science
[numeric]		 Mean (sd) : 10.3 (5.4)
min < med < max:
1 < 10 < 17
IQR (CV) : 10.5 (0.5)		 16 distinct values 0
(0%)
15 UVA Astronomy
[numeric]		 Mean (sd) : 10.5 (4.1)
min < med < max:
1 < 11 < 17
IQR (CV) : 5 (0.4)		 17 distinct values 0
(0%)
16 UVA School of Education - Special Education Research Accelerator (SERA)
[numeric]		 Mean (sd) : 11.1 (5.3)
min < med < max:
1 < 14 < 17
IQR (CV) : 8.2 (0.5)		 15 distinct values 0
(0%)
17 UVA Physics
[numeric]		 Mean (sd) : 11.5 (4.3)
min < med < max:
2 < 12 < 17
IQR (CV) : 7 (0.4)		 16 distinct values 0
(0%)

Next we count, for every capstone, the number of students who ranked the capstone as their first, second, third, through seventh choice:

capstone.ranks2 <- capstone.ranks %>%
  gather(colnames(capstone.ranks), key="capstone", value="rank") %>%
  group_by(capstone) %>%
  dplyr::summarize(`Ranked 1st` = sum(rank==1),
            `Ranked 2nd` = sum(rank==2),
            `Ranked 3rd` = sum(rank==3),
            `Ranked 4th` = sum(rank==4),
            `Ranked 5th` = sum(rank==5),
            `Ranked 6th` = sum(rank==6),
            `Ranked 7th` = sum(rank==7)) %>%
  arrange(desc(`Ranked 1st`))
datatable(capstone.ranks2)

We can also use this data to get a sense of the correlations between capstones and whether there exists clusters of capstones which get interest from the same students. We build a Euclidean distance matrix between the capstones, and pass this distance matrix to a multidimensional scaling algorithm with two dimensions:

d <- dist(t(capstone.ranks)) 
fit <- cmdscale(d,eig=TRUE, k=2)

Next we plot the capstones in two-dimensional space.

x <- fit$points[,1]
y <- fit$points[,2]
plot(x, y, 
     xlab="Dimension 1", 
     ylab="Dimension 2",
     main="A Map of Our Capstones", 
     xlim = c(-50, 50))
text(x, y, labels = row.names(t(capstone.ranks)), cex=.7, pos=3)

The Method for Matching Students to Capstones

I define an \((N \times C)\) matrix \(R\), where \(N\) is the number of students, \(C\) is the number of capstones, and each element \(r_{nc}\) is the rank that student \(n\) has given to capstone \(c\). We define variables \(X_{nc}\), \(\forall n \in \{1,2,. . . ,N\}\) and \(\forall c \in \{1,2,. . . ,C\}\) that are equal to 1 if student \(n\) is assigned to capstone \(c\), and 0 otherwise.

We define an objective function \[ F = \sum_{n=1}^N \sum_{c=1}^C r_{nc}X_{nc}, \] that we minimize with respect to the variables \(X_{nc}\).

To state the problem less formally: we are trying to assign students to capstones in a way that minimizes the sum total of the ranks the students have given to the capstones they’ve been assigned to. If we are able to assign all \(N\) students to their most preferred capstone, then all of the students’ rankings are 1, and \(F = N\). If any students are assigned to a capstone other than their most preferred capstone, then \(F > N\). We are trying to choose the assignments \(X_{nc}\) such that \(F\) is as close as possible to \(N\) as it can be given the constraints we deal with, which are that

  1. (\(L_s\)) Every student must be assigned to one, and only one, capstone, and

  2. (\(L_c\)) Every capstone must have either zero, three, or four students.

The student-constraint \(L_s\) can be expressed with this equation: \[ L_s: \sum_{c=1}^C X_{nc} = 1. \] In other words, the sum of all assignments across capstones for a student must equal 1. The capstone-constraint \(L_c\) can be expressed as \[ L_c: \sum_{n=1}^N X_{nc} \in \{0,3,4\}, \] which means the sum of all assignments across students for a capstone must be either 0, 3, or 4.

sortinghat()

I wrote a function as a wrapper for the lp () function from the lpSolve package to perform this optimization. It takes as input data in which the rows represent students, the columns represent capstones, and the cells contain rankings. The data cannot include a column for student IDs.

sortinghat <- function(X){
  
  require(tidyverse)
  require(lpSolveAPI)
  
  N <- nrow(X)
  C <- ncol(X)
  
  # Build constraint matrix
  data <- expand_grid(student = 1:N, capstone = 1:C)
  for(n in 1:N){
    data <- mutate(data, x = (student == n))
    colnames(data)[ncol(data)] <- paste(c("student",n), collapse ="")
  }
  for(i in 1:C){
    data <- mutate(data, x = (capstone == i))
    colnames(data)[ncol(data)] <- paste(c("capstone",i), collapse ="")
  }
  data <- select(data, -student, -capstone)
  data <- t(data)
  sumcap <- matrix(0, N, C)
  sumcap <- rbind(sumcap, -1 * diag(C))
  data <- cbind(data, sumcap)
  
  # Make an LP solve model
  lpmodel <- make.lp(nrow(data), ncol(data))
  for(i in 1:ncol(data)){
    set.column(lpmodel, i, data[,i])
  }
  
  # Build objective function 
  set.objfn(lpmodel, obj = c(c(t(X)), rep(0, C)))
  
  # Set constraints right-hand side
  set.rhs(lpmodel, b = c(rep(1, N), rep(0, C)))
  
  # Set constraint types
  set.constr.type(lpmodel, types = rep("=", N+C))
  
  # Set the sum variables as semi-continuous, bounded
  set.semicont(lpmodel, columns = c((N*C + 1):(N*C + C)))
  set.bounds(lpmodel, 
             lower = c(rep(0, N*C), rep(3,C)), 
             upper = c(rep(1, N*C), 4,4,4,4,4,4,7,4,4,4,4,4,4,4,4,4,4))
  
  # Solve the LP model
  lp.control(lpmodel, sense = "min") 
  solve(lpmodel)
  results <- matrix(get.variables(lpmodel)[1:(N*C)], N, C, byrow=TRUE)
  return(results)
}

The data frame needs to place students in the rows and capstones in the columns, which is how we cleaned the data. But we need to remove the student name variable, which we save as a separate object, and we need to coerce the data to matrix class. We pass the data to sortinghat():

students <- data$fullname
matches <-sortinghat(as.matrix(data[,-c(1,2)]))
## Loading required package: lpSolveAPI

The matches are expressed in binary format. To make these results easier to use, we include the student names and collapse the data to one column for the matches.

results <- data.frame(fullname = students, 
                      capstone = colnames(data[,-c(1,2)])[apply(matches, 1, which.max)],
                      stringsAsFactors = FALSE)

How Happy are the Students with These Assignments?

We merge these matches with the rankings so that we can see how highly each student ranked the capstone to which they’ve been assigned:

final.assign.df <- data %>%
  gather(-student, -fullname, key = "capstone", value = "rank") %>%
  right_join(results, by = c("capstone", "fullname")) %>%
  select(fullname, capstone, rank)

The final data is as follows:

datatable(arrange(final.assign.df, capstone))

In general, students are very happy with their matches, as the average ranking across students for the capstones to which they’ve been assigned is 1.33. The worst ranking is 3. The overall distribution of the rankings is illustrated below:

g <- ggplot(final.assign.df, aes(x=rank)) +
        geom_histogram(binwidth=1, col="red", fill="blue", alpha=.2) +
        xlab("Students' ranking of their assigned capstone") +
        ylab("Number of students") +
        theme(legend.position = "none") +
        scale_x_continuous(breaks=1:max(final.assign.df$rank)) +
        geom_text(stat='count', aes(label=..count..), vjust=-.5)
g

table(final.assign.df$capstone)
## 
##                                                             Clarabridge 
##                                                                       4 
##                                                         dbS Productions 
##                                                                       4 
##                                                  Department of Defense  
##                                                                       3 
##                                                             InnovateEDU 
##                                                                       4 
##                                                                     LMI 
##                                                                       4 
##                              North American ALMA Science Center at NRAO 
##                                                                       3 
##                                                           UVA Astronomy 
##                                                                       3 
##                                UVA Biocomplexity Institute & Initiative 
##                                                                       4 
##                                        UVA Department of Anesthesiology 
##                                                                       4 
##                                               UVA Environmental Science 
##                                                                       4 
##  UVA Oncology: Integrating clinical and genomic data in cancer patients 
##                                                                       4 
##                    UVA Oncology: Mutational processes in cancer genomes 
##                                                                       4 
##                                                          UVA Pediatrics 
##                                                                       4 
##                                                          UVA Psychology 
##                                                                       7 
##                                                 UVA School of Education 
##                                                                       4 
## UVA School of Education - Special Education Research Accelerator (SERA) 
##                                                                       4

The following capstones were dropped:

capnames[!is.element(capnames, unique(final.assign.df$capstone))]
## [1] "UVA Physics"

Finally, we save these matches in a separate CSV file.

write_csv(final.assign.df, path="capstone_assignments.csv")