Lecture 9: Relationships Between Variables, Part 1

Nick Huntington-Klein

February 5, 2019

Recap

  • Summary statistics are ways of describing the distribution of a variable
  • We can also just look at the variable directly
  • Understanding a variable’s distribution is important if we want to use it

This week

  • We aren’t just interested in looking at variables by themselves!
  • We want to know how variables can be related to each other
  • When X is high, would we expect Y to also be high, or be low?
  • How are variables correlated?
  • How does one variable explain another?
  • How does one variable cause another? (later!)

An Example: Dependence

table(wage1$numdep,wage1$smsa,dnn=c('Num. Dependents','Lives in Metropolitan Area'))
##                Lives in Metropolitan Area
## Num. Dependents   0   1
##               0  60 192
##               1  27  78
##               2  38  61
##               3  13  32
##               4   3  13
##               5   3   4
##               6   2   0

An Example: Dependence

  • What are we looking for here?
  • For dependence, simply see if the distribution of one variable changes for the different values of the other.
  • Does the distribution of Number of Dependents differ based on your SMSA status?
prop.table(table(wage1$numdep,wage1$smsa,dnn=c('Num. Dependents','Lives in Metropolitan Area')),margin=2)
##                Lives in Metropolitan Area
## Num. Dependents          0          1
##               0 0.41095890 0.50526316
##               1 0.18493151 0.20526316
##               2 0.26027397 0.16052632
##               3 0.08904110 0.08421053
##               4 0.02054795 0.03421053
##               5 0.02054795 0.01052632
##               6 0.01369863 0.00000000

An Example: Dependence

  • Does the distribution of SMSA differ based on your Number of Dependents Status?
prop.table(table(wage1$numdep,wage1$smsa,dnn=c('Number of Dependents','Lives in Metropolitan Area')),margin=1)
##                     Lives in Metropolitan Area
## Number of Dependents         0         1
##                    0 0.2380952 0.7619048
##                    1 0.2571429 0.7428571
##                    2 0.3838384 0.6161616
##                    3 0.2888889 0.7111111
##                    4 0.1875000 0.8125000
##                    5 0.4285714 0.5714286
##                    6 1.0000000 0.0000000
  • Looks like it!
  • What do these two results mean?

An Example: Correlation

  • We are interested in whether two variables tend to move together (positive correlation) or move apart (negative correlation)
  • One basic way to do this is to see whether values tend to be high together
  • One way to check in dplyr is to use group_by() to organize the data into groups
  • Then summarize() the data within those groups
wage1 %>% 
  group_by(smsa) %>%
  summarize(numdep=mean(numdep))
## # A tibble: 2 x 2
##    smsa numdep
##   <dbl>  <dbl>
## 1     0  1.24 
## 2     1  0.968
  • When smsa is high, numdep tends to be low - negative correlation!

An Example: Correlation

  • There’s also a summary statistic we can calculate called correlation, this is typically what we mean by “correlation”
  • Ranges from -1 (perfect negative correlation) to 1 (perfect positive correlation)
  • Basically “a one-standard deviation increase in X is associated with a correlation-standard-deviation increase in Y
cor(wage1$numdep,wage1$smsa)
## [1] -0.09636769
cor(wage1$smsa,wage1$numdep)
## [1] -0.09636769

An Example: Explanation

Let’s go back to those different means:

## # A tibble: 2 x 2
##    smsa numdep
##   <dbl>  <dbl>
## 1     0  1.24 
## 2     1  0.968
  • Explanation would be saying that, based on this, if you’re in an SMSA, I predict that you have 0.9684211 dependents, and if you’re not, you have 1.239726 dependents
  • If you are in an SMSA and have 2 dependents, then 0.9684211 of those dependents are explained by SMSA and 2 - 0.9684211 = 1.0315789 of them are unexplained by SMSA
  • We’ll talk a lot more about this later

Coding Recap

  • table(df$var1,df$var2) to look at two variables together
  • prop.table(table(df$var1,df$var2)) for the proportion in each cell
  • prop.table(table(df$var1,df$var2),margin=2) to get proportions within each column
  • prop.table(table(df$var1,df$var2),margin=1) to get proportions within each row
  • df %>% group_by(var1) %>% summarize(mean(var2)) to get mean of var2 for each value of var1
  • cor(df$var1,df$var2) to calculate correlation

Graphing Relationships

  • Relationships between variables can be easier to see graphically
  • And graphs are extremely important to understanding relationships and the “shape” of those relationships

Wage and Education

  • Let’s use plot(xvar,yvar) with two variables
plot(wage1$educ,wage1$wage,xlab="Years of Education",ylab="Wage")

  • As we look at different values of educ, what changes about the values of wage we see?

Graphing Relationships

  • Try to picture the shape of the data
  • Should this be a straight line? A curved line? Positively sloped? Negatively?
plot(wage1$educ,wage1$wage,xlab="Years of Education",ylab="Wage")
abline(-.9,.5,col='red')
plot(function(x) 5.4-.6*x+.05*(x^2),0,18,add=TRUE,col='blue')

Graphing Relationships

  • plot(xvar,yvar) is extremely powerful, and will show you relationships at a glance
  • The previous graph showed a clear positive relationship, and indeed cor(wage1$wage,wage1$educ) = 0.4059033
  • Further, we don’t only see a positive relationship, but we have some sense of how positive it is, what it looks like roughly
  • Let’s look at some more

Graphing Relationships

  • Let’s compare clothing sales volume vs. profit margin for men’s clothing firms
library(Ecdat)
data(Clothing)
plot(Clothing$sales,Clothing$margin,xlab="Gross Sales",ylab="Margin")

  • No clear up-or-down relationship (although the correlation is 0.1373499!) but clearly the variance is higher for low sales

Graphing Relationships

  • Comparing Singapore diamond prices vs. carats
library(Ecdat)
data(Diamond)
plot(Diamond$carat,Diamond$price,xlab="Number of Carats",ylab="Price")