Transcript Mixed Models Workshop

Mixed Models
Example Data Sets
Example
1. Apple Size
Type of Example
SAS
R
data file: Example?.DAT
Example?.SAS
Example?.R
Linear (clustered longitudinal)
Yes
Yes
Yes
Yes
Yes
Yes
(Schabenberger and Pierce 2002, page 464)
2.Orange Trees
Nonlinear (balanced longitudinal)
(Draper and Smith 1981, page 524)
3. HT-DBH
Linear (multi-level)
(based on Robinson and Wykoff 2004)
4. Yield By Age
Nonlinear (unbalanced longitudinal)
Yes
Yes
5. Yield By Age
Nonlinear (clustered unbalanced
longitudinal)
No
Yes
Example 1: Apples
(Schabenberger and Pierce 2002, page 464)
Linear Mixed Model
Clustered Longitudinal Data
•3-12 apples per tree
•10 trees
•6 measurements (weekly)
Yij = (Bo + boi) + (B1 + b1i)Xij + eij
Y = apple diameter (inches)
X = time (week)
Example 2: Orange Trees
(Draper and Smith 1981, page 524)
Nonlinear Mixed Model
Longitudinal Data (balanced)
•5 orange trees
•8 measurements
Yi = (B1+b1i)/(1+exp(-(Xi-B2)/B3))
Y = tree circumference (mm)
X = time (days)
Example 3: HT imputation
(based on Robinson and Wykoff 2004)
Linear Mixed Model
Multi-level Data
•5 stands in compartment (j)
•3-11 plots per stand (i)
•Multiple trees per plot (k)
ln(Ykij-4.5) = (Bo + boi + boj)
+ (B1 + b1i + b1j)/(1+Xkij) + eij
Y = Total Height (feet)
X = DBH (inches)
Example 4: Yield/Age
Nonlinear Mixed Model
Longitudinal Data
(unbalanced)
•9 installations (locations)
•5-10 measurements
Yi = (B1+b1i)/(1+exp(-(Xi-B2)/B3))
Y = Merchantable Cunits (ft3/acre)
X = Total Age (years)
Example 5: Yield/Age
Nonlinear Mixed Model
Clustered Longitudinal Data
(unbalanced)
•9 installations (locations) (j)
•3 plots per installation (i)
•5-10 measurements
Yi = (B1+b1i+b1j)/(1+exp(-(Xi-B2)/B3))
Y = Merchantable Cunits (ft3/acre)
X = Total Age (years)