Discussion 3
1/20/2014
Outline
• How to fill out the table in the appendix in
HW3
• What does the Model statement do in SAS
Proc GLM (please download lab 2 for
reference)
– What is a statistical model in layman’s terms
– What are the residuals and predicted values
• How to calculate Variance Components
• Power questions
Questions?
How to fill out the table in the
The treatments are
appendix in HW3
the herbicides
used and there are
6 different
herbicides
(treatments)
The pots are the
objects that were
applied the
treatment
therefore they are
your experimental
unit
Each pot
contained 4 plants
therefore 4
measurements
were made to
each pot (subsamples)
How to fill out the table in the
appendix in HW3
Specify what
design was
used
Specify what
was measured
Specify your what
was the treatment
applied to
Specify what were
your treatments and
how many levels (i.e.
different treatments)
Specify if sub-samples were
measured and what was measure
as a sub-sample
What does the Model statement do in
SAS Proc GLM
• What is a statistical model in layman’s terms
– A mathematical equation constructed to describe
the makeup of an observation or a group of
observations
– Example: One-way ANOVA model
The One-way ANOVA model describes an observation (Y) as a deviation
from an overall mean (µ) of a group of observations due to a treatment
effect (τ) and the addition of random error (ε).
What does the Model statement do in
SAS Proc GLM
• The overall mean (Ῡ..) is the mean of all the
observations Yij
– Yij read as observation from the ith treatment and
the jth replication.
• The treatment effect is the deviation from the
overall mean Ῡ.. to the treatment mean (Ῡi.)
• The random error is the deviation from the
treatment mean (Ῡi.) to a given replication of
that treatment (Yij )
What does the Model statement do in
SAS Proc GLM
• What are the predicted values?
– The theoretical values obtained based on the
statistical model (the error is excluded from the
model):
Predicted Yij
• What are the residuals?
– The deviation from the expected values to the
observed values:
—
(
)
What does the Model statement do in
SAS Proc GLM
• To do ANOVA in SAS we use Proc GLM and
specify our model:
– Example 3.2 in lab 2:
Proc GLM;
Class Culture;
Model Nlevel = Culture;
Means Culture;
Output Out = Residual R = Res1 P = Pred1;
The Class statement tells SAS that our data is grouped by the variable
Culture in the model statement we tell SAS that we want to explain
Nlevel by the variable Culture; therefore:
SAS calculates the overall mean and the residuals but
what Proc GLM is also calculating is the sums of
squares, mean squares, F – values, and p –values for
each variable we specify in the model see how in the
following slide.
What does the Model statement do in
SAS Proc GLM
• How does SAS calculate the sums of square
(SS)?
Which is equivalent to:
Where: r = number of replications
When you divide the SS by their respective degrees of freedom the
mean squares are obtained (Equivalent to the variances, s2)
What does the Model statement do in
SAS Proc GLM
• In Nested Designs:
– Example 3.4 in lab 2:
Proc GLM;
Class Trtmt Pot;
* We want SAS to calculate the variances between pots because
that will be our error for our ANOVA
Model Growth = Trtmt Pot(Trtmt);
*Pot is not a treatment.
Pot is only an ID variable
Random Pot(Trtmt); *must specify pot as random because we are not interested in
detecting differences between pots
Test H = Trtmt E = Pot(Trtmt);
* Here we request a customized F test
Total SS
Treatment SS
Pot SS (e.u.)
Where i is treatment ID, j is replication ID, k is subsample ID
r = number of replications, s = number of sub-samples
sub-sample SS
How to Calculate Variance
Components
• We analyze nested designs to estimate the
variance components which can be used to
estimate optimal sub-sample size (section
3.5.2.3 in lecture reading topic 3)
• The variance components are the estimate of
variance for a particular variable (e.g.
treatment, experimental unit, and subsample)
– The variance components can be calculate using
Proc VarComp in SAS
How to Calculate Variance
Components
• In the lecture topic 3 reading section 3.5.2.2
an experiment is described where mint plants
are exposed to different treatments of
temperature and daylight and stem length was
measured
• There are a total of 6 treatments, 3 pots
(replications) per treatment, and 4 plants
were measured per pot (subsamples)
How to Calculate Variance
Components
• The sums of squares was calculated for
treatment, pots and subsamples
Total SS
Treatment SS
Pot SS (e.u.)
sub-sample SS
• Then the variances (equivalent to means
squares)
MS Treatment
t -1
MS Pot
t (r -1)
MS sub-sample
rt (s - 1)
Where t = number of treatments, r = number of replications, and s = number of sub-samples
How to Calculate Variance
Components
• The variance due to the sub-sample is the
variance due to error:
MS sub-sample = σδ2
• To estimate the variance component of the
subsample we just calculate the MS of subsample
rt (s - 1)
=
0.93
How to Calculate Variance
Components
• The variance of pots contains the variance of the
sub-samples (NOTE: this is not the variance
component of pots):
MS Pots = σδ2 + 4σε2 = 2.15
• To estimate the variance component of pots we have
to solve for σε2. The variance components of pot is
calculated below:
σε2 = (MS Pots - σδ2) / 4
σε2 = (2.15 - 0.93) / 4 = 0.30
How to Calculate Variance
Components
• The variance of treatments includes the
variance of pots and subsamples (NOTE:):
MS Treatments = σδ2 + 4σε2 + 12Στ2/5 = 35.92
• To estimate the variance component of
treatments only we have to solve for Στ2/5
Στ2/5 = (MS Treatments - σδ2 - 4σε2) / 12
Στ2/5 = (35.92 - 0.93 – 4*0.3) / 12 = 2.81
Power
• How to use Power Charts for ANOVA