Course Introduction and Review

Outline

  • What is a regression model?

  • Descriptive statistics -- numerical

  • Descriptive statistics -- graphical

  • Inference about a population mean

  • Difference between two population means

What is course about?

  • It is a course on applied statistics.

  • Hands-on: we use R, an open-source statistics software environment.

  • Course notes will be jupyter notebooks.

  • We will start out with a review of introductory statistics to see R in action.

  • Main topic is (linear) regression models: these are the bread and butter of applied statistics.

What is a regression model?

A regression model is a model of the relationships between some covariates (predictors) and an outcome.

Specifically, regression is a model of the average outcome given or having fixed the covariates.

Heights of mothers and daughters

  • We will consider the heights of mothers and daughters collected by Karl Pearson in the late 19th century.

  • One of our goals is to understand height of the daughter, D, knowing the height of the mother, M.

  • A mathematical model might look like $$ D = f(M) + \varepsilon$$ where $f$ gives the average height of the daughter of a mother of height M and $\varepsilon$ is error: not every daughter has the same height.

  • A statistical question: is there any relationship between covariates and outcomes -- is $f$ just a constant?

Let's create a plot of the heights of the mother/daughter pairs. The data is in an R package that can be downloaded from CRAN with the command:

install.packages("alr3")

If the package is not installed, then you will get an error message when calling library(alr3).

In [1]:
library(alr3)
data(heights)
M = heights$Mheight
D = heights$Dheight
plot(M, D, pch = 23, bg = "red", cex = 2)
Loading required package: car
Warning message:
“package ‘car’ was built under R version 3.3.2”

In the first part of this course we'll talk about fitting a line to this data. Let's do that and remake the plot, including this "best fitting line".

In [2]:
plot(M, D, pch = 23, bg = "red", cex = 2)
height.lm = lm(D ~ M)
abline(height.lm, lwd = 3, col = "yellow")

Linear regression model

  • How do we find this line? With a model.

  • We might model the data as $$ D = \beta_0+ \beta_1 M + \varepsilon. $$

  • This model is linear in $\beta_1$, the coefficient of M (the mother's height), it is a simple linear regression model.

  • Another model: $$ D = \beta_0 + \beta_1 M + \beta_2 M^2 + \beta_3 F + \varepsilon $$ where $F$ is the height of the daughter's father.

  • Also linear (in the coefficients of $M,M^2,F$).

  • Which model is better? We will need a tool to compare models... more to come later.

A more complex model

  • Our example here was rather simple: we only had one independent variable.

  • Independent variables are sometimes called features or covariates.

  • In practice, we often have many more than one independent variable.

Right-to-work

This example considers the effect of right-to-work legislation (which varies by state) on various factors. A description of the data can be found here.

The variables are:

  • Income: income for a four-person family

  • COL: cost of living for a four-person family

  • PD: Population density

  • URate: rate of unionization in 1978

  • Pop: Population

  • Taxes: Property taxes in 1972

  • RTWL: right-to-work indicator

In a study like this, there are many possible questions of interest. Our focus will be on the relationship between RTWL and Income. However, we should recognize that other variables have an effect on Income. Let's look at some of these relationships.

In [3]:
url = "http://www1.aucegypt.edu/faculty/hadi/RABE4/Data4/P005.txt"
rtw.table <- read.table(url, header=TRUE, sep='\t')
print(head(rtw.table))
         City COL   PD URate     Pop Taxes Income RTWL
1     Atlanta 169  414  13.6 1790128  5128   2961    1
2      Austin 143  239  11.0  396891  4303   1711    1
3 Bakersfield 339   43  23.7  349874  4166   2122    0
4   Baltimore 173  951  21.0 2147850  5001   4654    0
5 Baton Rouge  99  255  16.0  411725  3965   1620    1
6      Boston 363 1257  24.4 3914071  4928   5634    0

A graphical way to visualize the relationship between Income and RTWL is the boxplot.

In [4]:
attach(rtw.table) # makes variables accessible in top namespace
boxplot(Income ~ RTWL, col='orange', pch=23, bg='red')

One variable that may have an important effect on the relationship between is the cost of living COL. It also varies between right-to-work states.

In [5]:
boxplot(COL ~ RTWL, col='orange', pch=23, bg='red')

We may want to include more than one plot in a given display. The first line of the code below achieves this.

In [6]:
par(mfrow=c(2,2))
plot(URate, COL, pch=23, bg='red')
plot(URate, Income, pch=23, bg='red')
plot(URate, Pop, pch=23, bg='red')
plot(COL, Income, pch=23, bg='red')

R has a builtin function that will try to display all pairwise relationships in a given dataset, the function pairs.

In [7]:
pairs(rtw.table, pch=23, bg='red')