In deze documenten worden lectures 1 tot en met les 11 van Advanced statistics samengevat. Deze lectures zijn gegeven in de eerste periode van schooljaar 2021/2022. Lecture 6 en 7 zijn samengevoegd tot één document omdat de stof naadloos op elkaar aansluit.
Interest in treatment effects (voorbeeld 1)
quantitative factor = covariance
ANCOVA:
-het gebruiken van extra informatie van een quantitative variabele x
(getallen), dat niet gebruikt wordt in het design.
- We voegen β1x toe aan het ANOVA model om te corrigeren voor verschillen
tussen x
model: yij = β0 + τ1 + β1xij + εij met εij ~ N(0, σ) independent, met τref = 0 (zelf
kiezen welke)
- Assumptions:
linear relationship between response y en covariate x
slope (β1) is the same for all treatments
testen door een full vs reduced model F test
covariate x does not depend on the treatments
β0 = mean yield voor τref = C at x = 0
β0 + τ2 = mean yield for F at x = 0
β0 + τ3 = mean yield for S at x = 0
Adjusted treatment means
y i, adj = y i - ^β 1( x i. - x ..) = ^β 0 + τ^ i + ^β 1 x .. voor elke i = 1,…, t
y 1, adj - y 2, adj = y 1 - y 2 - ^β 1( x 1. - x 2) = τ^ 1 - τ^ 2
F-test for treatment effects
H0 = geen treatment effects τi = 0
Full model vs reduced model
Full model:
- Intercept β0
- Treatment effects τi
- Coefficient β1 van x (3 lijnen -> zie voorbeeld)
Reduced model:
- Intercept β0
- Coefficient β1 van x (1 lijn)
Interest in x-effect(s) (voorbeeld 2)
- Linear model met quantitative en qualitative explanatory variables
interest in the relationship between variable y (response) and x
(explanatory of regressor)
x is nu ook van belang, niet alleen meer om de precisie te verhogen
Assume:
- relatie tussen y en covariabele x is linear
, Lecture 11 – Analysis of covariance
- slopes van covariabele mogen verschillen tussen treatments
interaction term tussen treatment en covariate
Parallel-lines model
model: yij = β0 + τ1 + β1xij + εij
Voordrug A: μy = β0 + β1x
- Intercept = β0
- Slope = β1
Voordrug B: μy = β0 + τ2 + β1x
- Intercept = β0 + τ2 -> τ2 is het verschil tussen de bovenste en onderste
lijn in de grafiek
- Slope = β1
Interaction: non parallel lines
model: yij = β0 + τ1 + β1xij + λixij + εij met τ1 = λ1 = 0, εij ~ N(0, σ) indep
Slopes/hellingen vergelijken met t-test
H0: λ2 = 0 vs Ha: λ2 ≠ 0
TS: t = ^λ 2/se( ^λ 2)
under H0 t~t(dfE)
under Ha t tends to larger of smaller values -> two tailed
uitkomst t =
p = … dus:
Slopes/hellingen vergelijken met F-test
H0: λ2 = 0 vs Ha: λ2 ≠ 0
TS: F = MSDrug*dose/MSE (zie voorbeeld)
under H0 F ~ F(df 1 = df interactie, df 2 = dfE)
Under Ha F tends to lager values (altijd)
Rechter P waarde
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller jitskevanbrink. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $5.88. You're not tied to anything after your purchase.
