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Statistical Models: Full Course Summary

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Course
X_400418

Institution
Vrije Universiteit Amsterdam (VU)

This is a summary of all material from the course statistical models and contains everything you need to know for the exam. It contains all 4 topics: analysis of variance, generalized linear models, nonlinear regression and time series.

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Uploaded on December 8, 2022
Number of pages 14
Written in 2022/2023
Type Summary

business
analytics
master
statistics
statistical
models
anova
linear regression
business analyics
full course
lecture
time series
generalized linear models
mathematics
summary

Institution
Vrije Universiteit Amsterdam (VU)
Education
Business Analytics
Course
X_400418

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LECTURE NOTES STATISTICAL MODELS

§ I Analysis of Variance

◦ ne -

way anova model :
Y =
Ni t ei
Observation from level
"
*
Y :
jt I

*
Ni = E[
Y ] is the mean of the response variable (level i
)
•
e
:
random error
it ( i, j ) =/ (Kl)

02 (e en)
±0
model assumption :
Eleij] = 0 ,
Varfe ] =
,
COV ,

let Mi =
te + ✗ i 5. t .
Y =

µ + ✗ ite
*
µ : mean across alt levels
* ✗i : Main effect of level I on response variable

balanced design when ni is the same for alt levels i
= ( H ,✗ i , . ..
.
X, )T
←
matrix design : full model R : Y = ✗B te
+ tirst column = '
↑
design matrix (n 11=+1 ) )

LSE :
minimize the error iv.rt .
B
SCB) =

? ? ez =
?? (Y -
µ -
✗i)
≥
= 11 Y -
13112
✗

Â that satisties the normalequation XTXB = XTY St .
Â -
_

(XTXÏXTY

For XTX to be invertible , ✗ must be 1-411 rank

However , this is often not the cases ,
so we add a constraint :

① Standard parametrizationtt-onis.t.li =
Hi
BI ( 0 Â À) Â;
, . . ..
.
and = In
§ Yij =
Yi .

I

② Sum parametrization
£✗ i =
0

Û È
=
?Ë ? Y = Ì . . and & .
= vii. -
Ï . .

③ Treatment parametrization ✗ 0 ,
=
S.t.li =
Hi -
µ,
Û YÌ and Ô Yi -41
=
( default
.
=
. . in R)
ĳĳĳ
ĳ
ĳ
ĳĳ
ĳ ĳ

, Assume we dea.de µ= 0

Within sum of residual sum of squares
groups squares =

≥
= sum of Square errors (SSE) Sr IIY ✗ÂII ( Y YÌ )
= = - =

§ -
.

Sr
02
~ 22 =
In I
sit E[ ¥] =
n I
-
and hence E[ { ] 02 =

rank (x)
.
-
n -

*
Sr
50 we define Ô? n -
I (unbiased)

Consi der Null model WCR that implies that the mean response

is the same for AN factors ,
It .
the factor has no Influence

on the response .
This redeuced model is : w : Y -
. late where

✗, = ✗2 = = ✗I = ✗ -

ÌÎÏY
. . .

1×11 ?
≥
*
Sw = 11 Y -
- F ) . .

i j
In
2
*
Sw
/ ~ -
RANK(x)
=
In -1

• = Sw
N - I

I
≥

Between groups sum of squares Sw sr-
=
§ ,
ni (Ì . .
-
YT )
.

* Sw is independent of Sw Er -

is Sw -
Sr ~
TE -1

02

F- statistici . F = (Sw -
Sr) ( I T) -
~ F-→ in ±
-

Sr / ( n I) -

#

* F is deviation Of w -
fit from 1- fit normalized b
*
If F is
big ,
then W -
fit is bad

Hypothesis test Ho :
✗i =
✗ for alt i (wnolds )
H , :X ; ≠ ×; for some (ij) ( W does not had )

> we
reject Ho it f >
FI Or p value
- < ✗
-
1. n ± ; ,
- _
×

norm al
Model diagnostic : ↓
* Plot Û against ê ( symmetrie + no pattern) + QQ -
plot of ê
* Barlett -
test for constancy of error variante

*
Shapiro -
Wilk test for normality ( Kruska1- Wallis it normalit does not had )
ĳĳ Ĳ
ĳ

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