STAT 371 F18 Lec 5 (20-09-2018) SLR and ANOVA examples.pdf...

Lecture 5
20
th
Sept 2018

ANOVA
Source of
Variability
Degrees of
Freedom (df)
Sum of Squares Mean Square F-Ratio P-value
Regression
(Model)
(n-1)-(n-2)=1
??? = ෍
?=1
𝑛
(ො 𝜇
?
− ത?)
2
𝑀?? =
???
1
? =
𝑀??
𝑀??
Residual
(Error)
n-2
??? = ෍
?=1
𝑛
𝑒
?
2
= ෍
?=1
𝑛
(?
?
− ො 𝜇
?
)
2
𝑀?? =
???
𝑛 − 2
Total
(corrected)
n-1
??? = ෍
?=1
𝑛
(?
?
− ത?)
2

• Why is this F ratio informative?
? =
൘
???
𝜎 2
1
൘
???
𝜎 2
(𝑛 − 2)
=
𝑴??
𝑴??
=
(explained variation)
(unexplained variation)

• Coefficient of determination: ?
?
• Consider again
SST = SSR + SSE
• The ratio
?
?
=
???
???
=
(explained variation)
(Total variation)
= ? −
???
???
• This is also used to assess the “fit” of the regression model. In
particular it gives us the proportion of the total variation in the
response that is explained by the regression model .
• 0 ≤ ?
2
≤ 1, ?
2
= 0 indicates that none of the variability in ? is
explained by the linear regression model. Hence the higher ?
2
the
better.

• Adjusted ?
?
:
• Adding a new explanatory variable to the model will always keep the
?
2
value the same or increase it. But just because the value has
increased, this does not mean that the addition of the variable is
resulting in a better model (interpret with caution).
• Adjusted ?
?
imposes a “penalty” for each new variable that is added
to the model in an effort to make models of different sizes
comparable. The adjusted ?
2
can decrease when a new variable is
added to the model, or increase when a variable is removed from the
model.
?
𝒂𝒅𝒋
?
= ?
2
−
𝑝
𝑛 − 1
𝑛 − 1
𝑛 − 𝑝 − 1
= ? −
???/(𝒏 − 𝒑 − ?)
???/(𝒏 − ?)

Example
• A car dealer is interested in modeling the relationship between the
weekly number of cars sold and the daily average number of
salespeople who work on the showroom floor during that week.