ANOVA and R-squared revisited. Multiple regression and r-squared. correlation matrix

28 Pages·2016·1.16 MB·English

by  Jack Davis

Agenda for Week 7, Hour 1
ANOVA and R-squared revisited.
Multiple regression and r-squared.
Week 7, Hour 2
Multiple regression: co-linearity, perturbations,
correlation matrix
Stat 302 Notes. Week 7, Hour 1, Page 1 / 28
Consider this made-up dataset on silicon wafers, wafers.csv.
It’s based on a very common type of quality control analysis
in manufacturing.
A factory manager is interested in reducing the number of
bad wafers the factory produces in a batch.
She sets the factory to make 6 batches of wafers each at 3
levels of cooking temperature and 3 levels of spin speed.
There are 54 batches of wafers in total. The response
variable is number of bad wafers (in a batch of 1000).
Stat 302 Notes. Week 7, Hour 1, Page 2 / 28
Here are select rows from the dataset.
'cooktemp' is the cooking temperature in Celcius
'spinrpm' is the spin rate while cooling, in RPM
'bad' is the number of bad wafers in the batch
Stat 302 Notes. Week 7, Hour 1, Page 3 / 28
Note that even though we can describe temperature
and speed as continuous variables, we are treating
them as categories here.
Essentially we are calling them ‘low’, ‘medium’, and
‘high’ settings.
wafers$spinrpm = as.factor(wafers$spinrpm)
wafers$cooktemp = as.factor(wafers$cooktemp)
Stat 302 Notes. Week 7, Hour 1, Page 4 / 28
Here is the one-way ANOVA of 'bad' using cooking
temperature as an explanatory variable.
mod = lm(bad ~ cooktemp, data=wafers)
anova(mod)
Stat 302 Notes. Week 7, Hour 1, Page 5 / 28
p value is small, so we have strong evidence that cooking
temperature matters.
Without the p-value, we could compare the obtained F to a
critical value for F.
(Recall: F test is one-tailed, we only care about larger variances)
Stat 302 Notes. Week 7, Hour 1, Page 6 / 28
A hypothesis test tells us that there some of variance in bad
wafer count is explained by cooktemp.
It doesn't tell us how much of the variance is explained.
For that we need the Sum of Squares total,
SS + SS = 727 + 2934 = 3661
which is
group resid
Stat 302 Notes. Week 7, Hour 1, Page 7 / 28
Proportion of variance explained, or R-squared
= SS / SS
group total
= 727 / 3661
= 0.1986, or 19.86% of variation explained.
Stat 302 Notes. Week 7, Hour 1, Page 8 / 28
We can also get this information from the summary of the
lm() object that we used to get the ANOVA in the first place.
There's no such thing as a correlation in an ANOVA, but the
sometimes the ANOVA is referred to as having an R-squared
because of this variance explained connection.
Stat 302 Notes. Week 7, Hour 1, Page 9 / 28
A two-armed bird needs a two-way ANOVA
Stat 302 Notes. Week 7, Hour 1, Page 10 / 28