written by Junvie Pailden
# install the necessary package if it doesn't exist
if (!require(mosaic)) install.packages(`mosaic`)
# load the packages
library(mosaic)Consider the data below on six-year graduation rate (%), student-related expenditure per full-time student, and median SAT score for the 38 primarily undergraduate public universities and colleges in the United States with enrollments between 10,000 and 20,000.
gradrates <- read.csv("https://www.siue.edu/~jpailde/gradrates.csv")
gradrates
# Graduation.Rate Expenditure Median.SAT
# 1 81.2 7462 1160
# 2 66.8 7310 1115
# 3 66.4 6959 1070
# 4 66.1 8810 1205
# 5 64.9 7657 1135
# 6 63.7 8063 1060
# 7 62.6 8352 1130
# 8 62.5 7789 1200
# 9 61.2 8106 1015
# 10 59.8 7776 1100
# 11 56.6 8515 990
# 12 54.8 7037 1085
# 13 52.7 8715 1040
# 14 52.4 7780 1040
# 15 52.4 7198 1105
# 16 50.5 7429 975
# 17 49.9 7551 1030
# 18 48.9 8112 1030
# 19 48.1 8149 950
# 20 46.5 6744 1010
# 21 45.3 8842 1223
# 22 45.2 7743 990
# 23 43.7 5587 1010
# 24 43.5 7166 1010
# 25 42.9 5749 950
# 26 42.1 6268 955
# 27 42.0 8477 985
# 28 38.9 7076 990
# 29 38.8 8153 990
# 30 38.3 7342 910
# 31 35.9 8444 1075
# 32 32.8 7245 885
# 33 32.6 6408 1060
# 34 32.3 4981 990
# 35 31.8 7333 970
# 36 31.3 7984 905
# 37 31.0 5811 1010
# 38 26.0 7410 1005Let's start by summarizing the data by computing all relevant one-sample statistics for each variable.
favstats(~ Graduation.Rate, data = gradrates)
# min Q1 median Q3 max mean sd n missing
# 26 38.8 47.3 59 81.2 48.5 12.9 38 0
favstats(~ Expenditure, data = gradrates)
# min Q1 median Q3 max mean sd n missing
# 4981 7098 7506 8110 8842 7461 919 38 0
favstats(~ Median.SAT, data = gradrates)
# min Q1 median Q3 max mean sd n missing
# 885 990 1012 1082 1223 1036 81.9 38 0We can use the xyplot function under the mosaic to create scatterplots between quantitative variables. Adding an extra argument type = c("p", "r") superimposes a regression line on top of the points in the plot.
xyplot( dependent.var ~ independent.var, # assign variables
data = DATA, # data set
type = c("p", "r"), # add regression line
)The graph below displays scatterplots of graduation rate versus student-related expenditure and graduation rate versus median SAT score. It seems that there is a linear relationship between the variables so its wise to fit a regression line to summarize these relationships.
xyplot(Graduation.Rate ~ Expenditure,
data = gradrates)
# add a regression line
xyplot(Graduation.Rate ~ Expenditure,
data = gradrates,
type = c("p", "r"))
xyplot(Graduation.Rate ~ Median.SAT,
data = gradrates)
# add a regression line
xyplot(Graduation.Rate ~ Median.SAT,
data = gradrates,
type = c("p", "r"))
Notice that the points on the scatterplot of graduation rate versus median SAT score seems to hug the regression line better than the points on the scatterplot of graduation rate versus student-related expenditure. This visual feature (how well the line fits the data) summarizes how two variables are strongly related with each other. We will describe a measure that quantifies the strength of this relationship.
We can compute the equation of the regression line using the command lm.
model1 <- lm(Graduation.Rate ~ Expenditure,
data = gradrates)
model1
#
# Call:
# lm(formula = Graduation.Rate ~ Expenditure, data = gradrates)
#
# Coefficients:
# (Intercept) Expenditure
# 12.41357 0.00483
model2 <- lm(Graduation.Rate ~ Median.SAT,
data = gradrates)
model2
#
# Call:
# lm(formula = Graduation.Rate ~ Median.SAT, data = gradrates)
#
# Coefficients:
# (Intercept) Median.SAT
# -57.433 0.102The equation of the estimated regression line that describes the relationship of graduation rate versus student-related expenditure is y = 12.414 + 0.005x. From this equation, we can say that graduation rate (on average) increases by .005 percent for every dollar increase in student expenditure. Equivalently, we say that graduation rate (on average) increases by 5 percent (.005 x 1000) for every 1000 dollars increase in student expenditure.
On the other hand, the equation of the estimated regression line that describes the relationship of graduation rate versus median SAT score is y = -57.433 + 0.102x. We can say that graduation rate (on average) increases by .102 percent for every point increase in median SAT score. Equivalently, we say that graduation rate (on average) increases by 5.1 percent (.102 x 50) for every 50 point increase in median SAT score.
In other words, simple regression analysis on graduation rates of public universities and colleges in this data increases (on average) by 5% when either student-related expenditure increases by 1000 dollars or when median SAT score increases by 50 points.
We can compute linear correlations between pairs of quantitative variables, or for a matrix of quantitative variables using the function cor.
# `with` command another way to select variables, for use in computations, within a data set
with(gradrates, cor(Graduation.Rate, Expenditure))
# [1] 0.344
# or, using the `$` way of calling a variable in a data
cor(gradrates$Graduation.Rate, gradrates$Expenditure)
# [1] 0.344
# correlation for a matrix of variables
cor(gradrates)
# Graduation.Rate Expenditure Median.SAT
# Graduation.Rate 1.000 0.344 0.649
# Expenditure 0.344 1.000 0.320
# Median.SAT 0.649 0.320 1.000The estimated linear correlation between graduation rate versus expenditure is 0.344 while the estimated linear correlation between graduation rate versus median SAT score is 0.649.
There is positive (direct or increasing) linear relationship between graduation rates versus expenditure and median SAT score. Also, it seems that median SAT score is more strongly related to graduation rates than student expenditure.
The coefficient of determination R2 is a natural extension to correlation coefficients as measures of quantitative relationships. R2 is computed by squaring the estimated correlation values. R2 also plays an important role in regression analysis since it describes the proportion of variation in the dependent variable that the regression model was able to capture.
We can compute the coefficient of determination R2 in multiple ways in R as shown below.
# coefficient of determination
cor(gradrates)^2
# Graduation.Rate Expenditure Median.SAT
# Graduation.Rate 1.000 0.119 0.421
# Expenditure 0.119 1.000 0.102
# Median.SAT 0.421 0.102 1.000
# using lm output model1 for gradrates vs expenditure
summary(model1)$r.squared
# [1] 0.119
# using lm output model2 for gradrates vs SAT score
summary(model2)$r.squared
# [1] 0.421The estimated coefficient of determination R2 for graduation rate versus median SAT score is 0.421 while that for graduation rate versus expenditure is 0.119.
Notice that that the regression model when using median SAT score as independent variable was able to capture more variation in graduation rates than when using student expenditure as independent variable. This observation also supports our initial observation earlier when looking at the fitted regression line in the scatterplots.