Lecture 7 (Week 4, Monday)

Review: Empty Model

empty model gives us a hint to how much variation we need to explain
every data point can be partitioned into model + error
error doesn't really exist unless you have a model
last week:
- lm(), predict(), resid()
- using the mean as the model
notation: what GLM notation would you use to represent the numbers inside the rectangle? Y_i

aggregating error to assess model fit
- once you have the data, asking the question: how well does the model fit the data?
- quantifying how much error is around the model
- how much error is reduced when using a more complicated model
modeling the distribution of errors to make prediction

is the study a correlational design or experimental design?
- correlational because we're not manipulating any variables
- data that naturally arose in a season of basketball
what are the rows in the data frame?
- games
let's use Points as the outcome variable
- where's the variation in points? (horizontal)
- where's the variation in games? (vertical)
- word equation: Points = FG + other stuff
- explore the relationship graphically: does your explanatory variable appear to explain variation in points?
  - scatterplot, faceted histogram, boxplot
- what would the empty/null model be that we would compare to our complex model?
  - write as a word equation: Points = Mean + other stuff
  - write in GLM notation: Y_i = b₀ + e_i
- fit the empty model (save the empty model in an object called Empty.model) lm()

Empty.model <- lm(Points ~ NULL, data = MiamiHeat)

example:
assists model: points = assists + error
empty/null model: points = mean + error
what is the mean here the mean of? the mean is the mean of points (mean of outcome variable)
which model would have the most error? the empty model (that's all the error in the outcome variable)
model + error = 100% of variation in outcome
- assists model explains ?% of error
- empty model explains 0% of error
use data to create model of DGP

histogram: where is the model? where is the error?
where would you draw the empty model on the scatterplot?
- horizontal line at the mean of Points

empty model is a single number, but not just any number - a number that will give us the lowest amount of error possible

mean as an estimator is: unbiased, efficient, consistent
and: lots of techniques have been worked out using the mean as an estimator (i.e. lm())
but:
- assumes that errors are distributed normally
- and we have new techniques for computational modeling that make it easier to use other estimators (e.g. the median)
what indicator of total is minimized at the mean?

fit the empty model to Points and save in Empty.model
save the residuals from the empty model into a new variable: MiamiHeat$Resid.points

MiamiHeat$Points.resid <- resid(Empty.model)
head(select(MiamiHeat, Points, Points.resid))

sum(MiamiHeat$Points.resid)

we can't use this as an indicator of an aggregation of error because it is too close to zero
sum of squares
how many methods can you find to use R to calculate the sum of squared errors?
- sum(MiamiHeat$Points.resid^2)
- sum((resid(Empty.model)^2))
- anova(Empty.model)
confirm: is SSE really the smallest at the mean?
- SSE at mean was 10,485
- calculate SSE for other possible models
- start with 102.06
- sum((MiamiHeat$Points-102.06)^2) gives the SSE 102.06
- sum((MiamiHeat$Points-103)^2) gives the SSE 10557
- sum((MiamiHeat$Points-104)^2) gives the SSE 10793

##$ Sum of Absolute Errors

sum(abs(MiamiHeat$Points-mean(MiamiHeat$Points)))
sum(abs(MiamiHeat$Points-median(MiamiHeat$Points)))

SSE is minimized at the mean
no other number will produce a lower Sum of Squares than the mean of the distribution
SAE is minimized at the median