Author: Scott Atchison

The U.S. News and World Report has collected, compiled, and published a list of the top colleges and universities around the country. This report is based on annual surveys sent to each school as well as general opinion surveys of university faculties and administrators who do not belong to the schools on the list. These rankings are among the most widely quoted of their kind in the United States and have played an important role among students making their college decisions. However, other factors may prove to be meaningful when making these decisions.  The data may indicate an interaction between some of the explanatory variables, such as tuition, cost of living, enrollment, and rank, warranting further investigation:

Data Description:

The data consists of 222 observations, with 8 variables that describe the 2017 edition of the US News Universities Rankings as well as the cost of living population by state based on the US Census Bureau predictions for 2017. The databases for this analysis are available on data.world and the US Census Bureau website.

Building the Model:

To determine the model, both stepwise and best subsets were used to determine best fit. Before stepwise regression, the full model was evaluated:

According to the summary of the full model, the adjusted R-squared is 0.6588, indicating that the full model is explaining 65.8% of the variance of the response variable. Since a p-value below .001 (2.2e-16), this association does not appear to have occurred by chance. Based on the results of the ANOVA of the full model, we can predict that there are several explanatory variables, including tuition and enrollment that could possibly be significant predictors for determining the best model fit.

Regression Assumptions of the Final Model:

The next step is to evaluate the regression assumption. The assumptions are listed below:

To analyze linearity and equal variance, residual vs. fitted value plot is used. To evaluate normality, a normal Q-Q plot is generated:

According to the residual vs. fitted value plot, we can see no pattern in the data and conclude that the equal variance assumptions has been met. According to the Q-Q plot, we can see some deviation at the tails of the distrubition, but it appears that the normality assumption has been met.

The Cook’s distance plot indicates three potential outliers influencing the line of best fit. Surprisedly, BYU was not one of the outliers exerting he most leverge on the model. Instead, those were:

Based on these finding, the full model should suffice for concluding that there is a meaningful relationship between tuition, enrollment, and university ranking. However, further exploration is needed to determing whether or not this is the best model to explain the potential relationship between explanatory variables and the response variable.

Model Development

Following both the stepwise and best subsets regression, we see that tuition, enrollment, and population are recommended as predictors in the regression model:

When comparing the reduced model to the full model through an F-test, we see that there is not a significant difference (p-value: .77) between the two models:

The stepwise regression indictes that the model with tuition, enrollment, and population has an AIC of 1628.18, while another model that includes cost of living has an AIC of 1630. A model that accounts for the interaction between population and cost of living is worth exploring. However, after tests for multicollinearity using Variance Inflation Factors we see significant evidence of multicollinearity between population & cost of living:

Finally, a Variance Inflation Factor (VIF) test was conducted on the reduced model, which found no evidence of multicollinearity, suggesting that the reduced model is a better fit. Once the VIF test was conducted, assumptions were checked again finding similar results as the full model, with all conditions being met. The model was then cross validated using k-fold.

Summary & Conclusions:

After the initial exploratory data analysis (EDA) found here , a number of patterns emerged. Clearly, it appears that cost of tuition is strongly associated with ranking in the US News & World Report. What was not clear is the effects of the other variables (enrollment, region, & cost of living) on the response variable. While enrollment is weakly associated with ranking, it is moderately associated with tuition and cost of living. After initially analyzing the full model, we find that there is statistical evidence of a relationship between tuition and enrollment on university ranking. After examining best subsets and stepwise regression, it is suggested that we use a model which included tuition, enrollment, and population as the predictor variables. Comparing this model against the full model did not yield a significant difference between the two and suggested that the smaller model that only examined tuition and enrollment would yield the best results. An additional model (model 3) was investigated to include the interaction of cost of living and population, but found significant evidence of multicollinearity between those two variables. Multicollinearity was examined in the reduced model (model 2) using a VIF test, finding no evidence of multicollinearity within that model. Assumptions of linearity, normality, and equal variance were satisfied after examining a plot of residuals vs. fitted values as well as QQ plot of residuals. With a sample of 221 observations and a p-value of less than .001, we have statistical evidence to suggest that tuition, enrollment, and population are significant predictors of performance in the US News and World Report’s Best College Ranking. The final model is summarized below:

Limitations

As previously stated, cost of living data was available by state instead of by city or country where the university was listed. So, a university that is located in a community with a high cost of living may be in a state with an overall low COL index score, and vice versa. This eliminates some precision in our predictions. In addition, this list consists of the 231 schools which opten to participate in the US News & World Report Ranking Index. According to the US News, there are over 4,000 college and universities in the United States. This raises the concern of non-response bias and limits the generalizability beyond the scope of participating institutions in the US News Rankings.

One example of this is the University of Minnesota, which chose not to participate in the US News Best College Rankings. Minnesota’s in-state undergraduate tuition and fees are $14,142. The enrollment is 19,819, and the state population is 5,568,155 (in 2017):

This results in a 95% confidence interval of 111 to 266 for the University of Minnesota’s US News Best College Ranking. However, when we compare their US News 2019 ranking, we see that UM is ranked #76 (tied with Virginia Tech). This suggests that this model is not an accurate predictor of school ranking, but rather serves as an illustration of overall national trends between tuition, enrollment, and population with regard to university ranking in this report.

R Code Chunks:

# Heat Map Correlation Plot:
heat <- cor(rankingsreduced)
corrplot(heat, type = "upper", order = "hclust", 
         tl.col = "black", tl.srt = 45)

# Full Model with all variables:
fullmodel <- lm(Rank ~ Tuition + Enrollment + Region + CostOfLiving + Population, data = rankings ) 
 
# Model Summary (Full Model):
summary(fullmodel)

# ANOVA Table (Full Model:
anova(fullmodel)

# Residuals vs. Fitted:
mplot(fullmodel, which =1)

#QQ PLot:
mplot(fullmodel, which =2)

# Cook's Distance:
mplot(fullmodel, which =4)

#Stepwise Regression:
step(fullmodel, direction="both")

# Best subsets:
BestSubsets <- regsubsets(Rank ~ Tuition + Enrollment + Region + CostOfLiving + Population, data = rankings, method = "exhaustive", nbest = 2)  
Result <- summary(BestSubsets)

# Append fit statistics to include R^2, adj R^2, Mallows' Cp, BIC:
data.frame(Result$outmat, Result$rsq, Result$adjr2, Result$cp, Result$bic) 

# Model #2 (Based on Stepwise & Best Fits):
mod2 <- lm(Rank ~ Tuition + Enrollment + Population, data = rankings)
  
# Model Summary (Reduced Model):
msummary(mod2)

# Model Comparrison between Full & Reduced Model:
anova(mod2, fullmodel)

# Model with Interaction of Population & Cost of Living:
mod3 <- lm(Rank ~ Tuition + Enrollment + Population + CostOfLiving + Population:CostOfLiving, data = rankings)  

# Model Summary (Model 3):
msummary(mod3)

# Variance inflation factor (Model 3):
VIFtest1 <- lm(formula = Rank ~ Tuition + Enrollment + Population + CostOfLiving + Population:CostOfLiving, data = rankings)
vif(VIFtest1)

VIFtest2 <- lm(formula = Rank ~ Tuition + Enrollment + Population, data = rankings)
#Variance inflation factor (small model)
vif(VIFtest2)

# Checking model accuracy against "real world" data:
minnesota <- data.frame(Tuition = 14142, Enrollment = 29819, Population = 5568155)

#Confidence Interval:
predict(mod2, minnesota, interval="prediction") 

Since 1985, the U.S. News and World Report has collected, compiled, and published a list of the top colleges and universities around the country. This report is based on annual surveys sent to each school as well as general opinion surveys of university faculties and administrators who do not belong to the schools on the list. These rankings are among the most widely quoted of their kind in the United States and have played an important role among students making their college decisions. However, other factors may prove to be meaningful when making these decisions. For example, is cost of tuition associated with the ranking of a university? Said another way, do “better” schools cost more money to attend? Are other factors, such as enrollment, state population, cost of living, and region associated with these rankings? For potential students looking to get ahead in a global economy, these may be important considerations, especially for those who come from lower socioeconomic backgrounds.

Objectives & Variables of Interest

The purpose of this study is to investigate the associations between tuition, enrollment, cost of living, population, and region of the country on the 2017 US News & World Report’s Best College Rankings. The variables of interest are university ranking, undergraduate tuition, undergraduate enrollment, cost of living index by state, state population according to 2017 Census data, and region of the country (Northeast, Midwest, South, & West). The response variable is ranking, and the potential explanatory variables are undergraduate tuition, undergraduate enrollment, cost of living index, state population, and region of the country.

Concerns

Cost of living (COL) data was available by state instead of by community. A university that is located in a community with a high cost of living may be in a state with an overall low COL index score (and vice versa), which eliminates some precision in our prediction. In addition, many schools chose not to participate in this ranking report, which introduces non-response bias into the design.

Exploratory Data Analysis

The first step in exploratory data analysis was to look at the shape of each of the variables from a univariate standpoint (not pictured in this analysis). From there, I explored the associations between continuous variables, represented in the correlation matrix and the correlation plots below:

Looking at the data, we see some moderate to strong associations between rank, tuition, enrollment that warrant further investigation. Before building models, the data was explored by comparing some of these associations by region:

Finally, we can see the strongest association between two variables by visualizing Rank vs Tuition. When coded by Region we see some slight curvature in the data, but a similar negative shape and slope across parts of the United States:

Summary & Initial Observations of EDA:

Within the explanatory variables, we see a strong association between cost of tuition and ranking in the US News and World Report metric. One outlier (Brigham Young University–Provo) reports a relatively high ranking (68) in comparison to its tuition ($5,300). This observation appears to be influencing the line of best fit, lowering the correlation coefficient. Even with that influential data point, we still see a strong negative correlation (-.75) between tuition and rank. Enrollment does not appear to have a significant effect on university ranking, but it does appear to be positively associated with tuition (.37) cost of living (-.19). Cost of Living Index (COL) and state population appear to have a weak, negative assocation with university ranking, and a moderate positive association with the cost of tuition. The data may indicate an interaction between some of the explanatory variables, such as tuition, cost of living, enrollment, and rank, warranting further investigation.

Relationship between variables is summarized in the heat map below:

Through this EDA, we have a better understanding of the shape and relationships among the variables, which should inform model construction and analysis. The second part of this project tackles those objectives, found here. Code for the plots above can be found below:

R Code Chunks

# Correlation Matrix
cor(DATA)

# Plot 1:Corelation Plot
plot(DATA)

# Plot 2: Boxplots of Ranking (by Region):
ggplot(DATA, aes(x=Region, y=Rank, fill=Region)) + 
    geom_boxplot(alpha=0.3) +
    theme(legend.position="none")

# Plot 3:Interactive Plot of Tuition vs Rank (by Region)
ggplot(data = DATA, aes(x = Tuition, y = Rank)) +
  geom_point(aes(text = paste("Enrollment:", Enrollment)), size = .5) +
  geom_smooth(aes(colour = Region, fill = Region))

# Plot 4:Interactive Plot of Enrollment vs Rank (by Region)
p <- ggplot(data = DATA, aes(x = Enrollment, y = Rank)) +
  geom_point(aes(text = paste("Enrollment:", Enrollment)), size = .5) +
  geom_smooth(aes(colour = Region, fill = Region))

# Plot 5: Heat Map Correlation Plot
heat <- cor(DATA)
corrplot(heat, type = "upper", order = "hclust", 
         tl.col = "black", tl.srt = 45)

Who doesn’t want to know what makes a song popular? As a musician, I have spent decades trying to get an answer to that question, with little success. People just seem to like what they like. So, when I enrolled in an applied statistics course that took a deep-dive into regression analysis, I got my chance. We were required to conduct an exploratory data analysis (EDA) on a data set of our choosing, but we had to go out and find it. This led me to Kaggle and Spotify’s million song dataset.

The purpose of this EDA was to investigate what variables may influence song popularity while developing a greater understanding of statistical procedures. More specifically, the following questions were to be addressed:

1. What variables are associated with popularity of song choice by Spotify users? 
2. Is one variable associated with popularity above others? 
3. If there is an association, is it linear? 

My intuition before conducting analysis was that danceability, energy, and valence will be the most highly associated with song popularity, but not necessarily in a linear manner (or in that order).

Description of Data

The original dataset included 228,159 observations and 17 variables that describe the Spotify Tracks Database created by Tim Igolo and posted on Kaggle.com. Data was harvested through Spotify for Developers in April of of 2019. Unfortunately, the data included music from soundtracks in addition to “movie music” in addition to Opera (but not classical) and a number of other musical styles that could make any kind of regression analysis difficult. So, I filtered the data to include only popular music types, resulting in 130,663 observations (i.e. songs), with 11 variables of interest.  Those variables of interest are popularity, acousticness, danceability, duration (in milliseconds), energy, liveness, loudness, mode (major or minor), speechiness, tempo, & valence. The response variable is popularity and the potential explanatory variables are acoustics, danceability, duration, energy, liveness, loudness, mode, speechiness, tempo, & valence. 

Exploratory Data Analysis (EDA)

Originally proposed by John Tukey, the inventory of the Tukey Test,  EDA’s are typically used to summarize summarize the data’s main characteristics. This can be through simple summary statistics (measures of central tendency, five number summary, etc), but often includes data visualization as well. Simply put, we use EDA’s to look for any patterns or problems in the data and right away I found one:

This variable indicates the modality (major or minor) of a track, the type of scale from which its melodic content is derived. Major is represented by 1 and minor is. As you can see, there are more songs in minor (79409), than in major (51254), and the mode does not appear to have an appreciable effect on the popularity of a song:

While these two visualizations seem pretty clear, there is one glaring issue: Most songs in popular music are neither major, nor minor. Instead, they are modal and they sometimes shift tonalities throughout, so categorizing all of these songs dichotomously is a problem. Upon further investigation, you see some songs listed multiple times and in multiple categories. This is almost certainly due to how Spotify classifies their songs, which is fine, but poses problems when trying to investigate the questions stated above. 

Adjusting Plots for Better Visualization:

I was very new to using R at this point and didn’t have the time to sort through these issues, so I ended up choosing another dataset to analyze, but I figured while I had this data I would go ahead and figure out some ways to visualize it. Fortunately, this dataset did provide some interesting obstacles to overcome with data visualization. 

The first problem was the sheer number of observations.  I used the mplot function within the Mosaic package for many of these initial plots, which is super convenient for beginners in R.  However, since the dataset was so large, many of the plot were not helpful, like these two below:

These first two plot show the relationships between continuous variables, but are not the most helpful. We can get a sense on some linear relationships in the data, but it’s pretty tough to really see what is going on simply due to the number of data points. So, a solution was to use a heat plot instead to show correlations across continuous variables instead:

Similarly, the plot below of these two variables is much more clear as to what patterns are in the data simply by changing the size and transparency of the data points themselves:

While this was an early foray for me into R and not a dataset I wanted to investigate further to make inferences from, it did provide some interesting obstacles to overcome on how to use the software to visualize the data. Ultimately, my interests navigated to a different question: Investigating Factors for School Ranking in the US News & World Report, which can be found here and the code I used to create the plots above here:

R Code Chunks

# Plot 1: Correlation plot of Popularity vs Explanatory Variables
library(corrplot)
plot(DATA)

# Plot 2: Scatter Plot of Popularity vs Speechiness
library(mosaic)
library(ggplot2)
gf_point(Popularity ~ Speechiness, data = DATA) %>% 
gf_labs(title = "Popularity vs Speechiness", caption = "Spotify Dataset")

# Plot 3: Heat plot of Popularity vs Explanatory Variables
heat <- cor(DATA)
corrplot(heat, type = "upper", order = "hclust", 
         tl.col = "black", tl.srt = 45)

# Plot 4: Scatterplot of Speechiness vs. Popularity
DATA %>%
  ggplot( aes(x=Speechiness, y=Popularity)) +
    geom_point(color="darkblue", size=1.75, alpha=0.01) +
    theme(legend.position="none")

In 2008, I was hired to teach on the music faculty at Texas A & M – Commerce. Great job. Great people there. On my contract, the final sentence said “other duties as assigned,” and one of those dates ended up being a music technology course. I had no experience teaching a class like that and no formal training in a class like that, but that didn’t change the fact that I was going to teach it. So, I got started on figuring out how to do that and not feel like an idiot in the process.

The first step was to find an existing syllabus, of which there was none to be found. The next step was to ask the faculty what had been covered in the class in the past, which nobody really knew (this is not uncommon). There was a textbook that had been used, but it was not geared towards the population of students I was going to be teaching (according to my boss….and he wasn’t wrong). So, the next logical step was to get a sense from the faculty (and my bosses) what they would like me to cover in the course.

What I got was a laundry list of softwares and technologies students should be able to know, many of which were incredibly outdated or not relevant to the entire class. For example, being able to use a drill writing software like Pyware or EnVision is not going to be a meaningful exercise for people who intent to teach choir, orchestra, elementary, or middle school. It is also a really difficult to learn, because it requires knowledge in other areas besides just how to operate the software. This left me with these questions to resolve:

1. What softwares and skills are transferable across all students in the class?
2. What can we reasonably expect students to be able to know and be able to do within the confines of the semester and technological resources available.
3. What kinds of activities and assessments can we create to achieve these goals?

The result was a series of projects that centered on 3 main areas of being a music teacher and how technology could be used to serve them. Those areas were teaching, creativity, and administration. Projects were designed to scaffold learners in a way that not only helped them gain knowledge and understanding of software, but how to leverage those skills against their own interests to create new knowledge and digital fluency. Some projects were the same across degree tracks (choral, instrumental, general music, etc), while others were specific to each tracks. Examples of projects that remained consistent across tracks included the missing part assignment and the digital audio assignments, while some of the projects that bifurcated were the orchestration projects and the final projects.

The result was something unexpected. Students got really into the projects and we all ended up helping each other learn together. Eventually I became the most knowledge person in the room with respect to the various technologies, but we never stopped learning together. With each passing year, I found that the more interesting and authentic assignments were and the more they allowed for students’ interests and creativity to come out, the more wonderful the projects became.

Years later, I came to understand that this approach was known as Problem Based Learning (PBL) and Authentic Context Learning (ACL). These approaches have become huge interests of mine across all domains of learning and have had some wonderful experiences in classes that use them in areas like the Applied Statistics program at Penn State. Now, as I finish my PhD in the time of Covid-19, I see the importance of creating meaningful projects that students and teachers can both learn from to increase engagement and understanding. Below are some examples of projects I have developed in the past for any of those that are interested:

Music Notation Projects:

1. Recreate a Score
2. Tranform a Score
3. Creating & Composing with Music Notation Software

Digital Audio Projects:

1. Recording & Editing Project
2. Digital Audio Transformation Project
3. Composing & Creating with Digital Audio Workstations

Music Administration:

1. Data Cleaning Project
2. Tracking Expenses
3. Create a Mail Merge