Hands-on: we use R, an open-source statistics software environment.
Course notes will be available as 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 fathers and sons
We will consider the of fathers and sons collected by Karl Pearson in the late 19th century. Perhaps the first regression model!
One of our goals is to understand height of the son, S, knowing the height of the father, F.
A mathematical model might look like
\[
S = g(F) + \varepsilon\]
Above \(g\) gives the average height of the son of a father of height F and \(\varepsilon\) is error: not every son whose fathers have the same height themselves have the same height.
A statistical question: is there any relationship between covariates and outcomes – is \(g\) just a constant?
Line of best fit
Linear regression model
How do we find this line? With a model.
We might model the data as
\[
S = \beta_0+ \beta_1 \cdot F + \varepsilon.
\]
This model is linear in \((\beta_0, \beta_1)\), the intercept and the coefficient of F (the father’s height), it is a simple linear regression model.
Another model
\[
S = \beta_0 + \beta_1 F + \beta_2 F^2 + \varepsilon
\]
Also linear (in \((\beta_0, \beta_1, \beta_2)\), the coefficients of \(1, F, F^2\)).
Which model is better? We will need a tool to compare models… more to come later.
Our example here was rather simple: we only had one independent variable: F.
Multiple linear regression: brain size in mammals
Features and response
Response in brains
Brain: average brain weight (in grams)
Features in brains
Body: average body weight (in kilograms)
Gestation: gestation period (in days)
Litter: average litter size
Graphical exploration
Building a model
Some of the main goals of this course:
Build a statistical model describing the effect of Gestation on Brain.
This model should recognize that other variables also affect Brain.
What sort of statistical conclusions can we make based on our model?
Is the model we choose adequate describe this dataset?
Are there other (simpler, more complicated) better models?
