text: # Lecture 9: Relationships Between Variables, Part 1
  
  ## 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
  
  ## 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')
  
  ## 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")``````