13 Topic Random Effects
Background Reading:
Parts of Chapters 17 and 19
1
Reading Summary
Section 17.1 (particularly page 422)
Section 17.6 (pages 438-440)
Section 19.7 (pages 538-541)
2
Random Effects
So far we have really only dealt with fixed
effects we set the levels (or they are at
least predetermined) and are only interested
in those particular levels.
Often we have factors with lots of levels,
and to get observations at all of them would
be inconvenient if we could do it at all. So
we take a random sample of levels hence
random effects.
3
Fixed vs. Random
Recall:
A factor is fixed if the levels under consideration
are the only ones of interest. Selected by a nonrandom process. (e.g. gender, hair color)
A factor is random if the levels under
consideration may be regarded as a sample
from a larger population. Want to draw
inference on this larger population of levels.
(e.g. subject / person)
4
Why care about fixed/random?
Affects the expected mean squares on
which F-tests are based.
No major differences for 1-way ANOVA.
For ANOVA with multiple factors and interactions,
we will see that different ratios of mean squares
will be needed for the significance tests. The
denominator will no longer always be the MSE.
Affects interpretations as well (we wouldnt do
multiple comparisons for a random factor).
5
CLG Discussion
For each of three examples:
1.
Identify the response variable and
factors involved in each study.
2.
Identify a research question of
interest.
3.
Determine the number of levels for
each factor, and decide whether each
factor is fixed or random.
6
Example 1
Auto manufacturer wants to study the
effects of differences between drivers (A)
and differences between cars (B) on
gasoline consumption. Four drivers were
selected at random, and additionally five
cars of the same model with manual
transmissions were randomly taken from the
assembly line. Each driver drove each car
twice over a 40-mile test course and the
MPG were recorded.
7
Example 1
Response:
Research Question:
Factor
Levels
Fixed/Random
8
Example 2
A researcher studied the sodium content of
six brands of U.S. beers sold in a
metropolitan area. For each beer, both the
regular and light versions were examined.
9
Example 2
Response:
Research Question:
Factor
Levels
Fixed/Random
10
Example 3
Twelve job applicants were rated by each of
the three personnel officers for a company.
Each applicant was rated by each officer.
We want to explore whether there are
differences among the personnel officers.
11
Example 3
Response:
Research Question:
Factor
Levels
Fixed/Random
12
Random Effects - Variances
When we have random effects, we have
multiple variances:
Variance associated with the effect itself (e.g.
subject there is a variance associated to
the population of subjects from which we
sample a few)
Variance associated with taking an
observation (otherwise known as error).
13
Random Effects - Variances
In two factor model, we have 3 possibilities:
Fixed (both factors fixed)
Random (both factors random)
Mixed (one fixed and one random)
If we have three or more factors, the number
of possibilities grows even more.
Well examine the two-factor model in depth,
but also learn what to use SAS for three or
more factors.
14
One-factor Random Model
Model (use capital English letters to denote a
random component in a model):
Yij = + Ai + ij
Independent
{
i = 1, 2, ..., k
j = 1, 2, ..., n
= grand mean
2 DIFFERENCE!
Ai ~ N 0, A
IID
2
~ N 0,
ijk
(
)
()
15
One-factor Random Model
Expected Mean Squares (balanced design)
E ( MSA ) = E
(
)
2
n ( Yi2 ) NY2 = n A + 2
E ( MSE ) = 2
We want to test whether there is a treatment
effect. There is one if and only if we have
variability among the treatments. The null:
2
H0 : A = 0
16
One-factor Random Model (2)
Variances are never negative, so we test
against the one-sided alternative:
2
Ha : A > 0
Under the null hypothesis, the of MSA to
MSE is 1. Such a ratio provides the F-test,
and we note that it is the exact same F-test
that we would have if the treatment was
fixed.
17
One-factor Random Model (3)
So there is no difference in the actual test
when we have only one variable. But there
is a difference in the interpretation:
FIXED: There is a significant effect of Factor A,
and we may go on to further analyze this effect
using multiple comparisons.
RANDOM: There is variability in the response
that may be attributed to differences in Factor A.
No multiple comparisons here since not all
levels of interest are involved in the study,
multiple comparisons wont be relevant.
18
Two-factor Random Model
Model:
Yijk = + Ai + B j + ( AB ) ij + ijk
= grand mean
2
Ai ~ N 0, A
i = 1, 2, ..., a
j = 1, 2, ..., b
k = 1, 2, ..., n
()
B ~ N ( 0, )
( AB ) ~ N ( 0, )
2
DIFFERENCES!
j
B
Independent
2
ij
AB
~ N 0, 2 , independent
ijk
(
)
19
Two-factor Random Model
The first thing we must do is to consider
Expected Mean Squares. For a balanced
design, they are:
2
2
E ( MSA) = bn A + n AB + 2
E ( MSB ) = an + n
2
B
2
AB
+
2
2
E ( MSAB ) = n AB + 2
E ( MSE ) = 2
20
Testing (Two Random Factors)
EMS determine what MS to use in the
denominator of our test. Suppose we want
to test:
2
H0 : A = 0
We consider the expectation under the null:
0
2
2
E ( MSA ) = bn A + n AB + 2
Z
= n
2
AB
+ = E ( MSAB )
2
21
Testing (Two Random Factors)
So to test the effect of Factor A, we must
use the ratio: F = MSA / MSAB.
The DF for this test come from the ANOVA
table as well:
Numerator: Associated to MSA
Denominator: Associated to MSAB
22
Two-factor Random Model
In general, EMS determine what MS to use
in the denominator of our test. For the twofactor random model, the complete set of
tests is:
2
H 0 : A = 0 F = MS A / MS AB
H 0 : = 0 F = MS B / MS AB
2
B
H0 :
2
AB
= 0 F = MS AB / MS ERR
23
Two-factor Mixed Model
Model (A fixed, B random):
Yijk = + i + B j + ( B ) ij + ijk
= grand mean
i =0
(
i = 1, 2, ..., a
j = 1, 2, ..., b
k = 1, 2, ..., n
SAME!
)
2
B j ~ N 0, B
DIFFERENCES!
( AB ) ~ N 0, 2
Independent
ij
AB
~ N 0, 2 , independent
ijk
(
(
)
)
24
Another Wrinkle
When one factor is fixed and the other
random, it is not clear how to treat the
interaction. Two possibilities:
Unrestricted Model Interaction is simply
taken to be random (SAS employs this model)
Restricted Model For the fixed component
of the interaction, the following restriction is still
made: a
( B)
i =1
ij
= 0 for any fixed j
25
Mixed Model(s)
Depending on which model is chosen, the
EMS will be slightly different.
There are numerous high-level statistically
based arguments as to why one is better than
the other.
We will tend toward the unrestricted model
simply it because is the one utilized in SAS.
This stuff gets even more complicated when
a third variable is added!
26
Two-factor Mixed Model
Unrestricted Model (PROC GLM)
Expected Mean Squares
E ( MSA ) = Q ( A ) + n
2
AB
+
2
2
2
E ( MSB ) = an B + n AB + 2
2
E ( MSAB ) = n AB + 2
E ( MSE ) =
2
Tests similar to random effects model. Note: Q(A)
just denotes the fixed effects for A (zero if no effect).
27
Two-factor Mixed Model
Restricted Model Assume i ( AB ) ij = 0 for all j.
Expected Mean Squares
E ( MSA ) = Q ( A ) + n
2
AB
2
E ( MSB ) = an B + 2
E ( MSAB ) = n
2
AB
+
+
2
DIFFERENCE!
2
E ( MSE ) = 2
Factor B for this model will be tested over MSE
the rest of the tests are the same.
28
Generalizing...
With three or more factors, tests must be
based on EMS.
You look for a pair of EMS for which the ratio will
be 1 under your null hypothesis.
Fortunately, SAS will print a table of EMS (for
the unrestricted model) when random effects are
indicated.
29
Example (A Random, BC Fixed)
Source
A
B
A*B
C
A*C
B*C
A*B*C
Type III Expected Mean Square
Var(Error) + 4 Var(A*B*C) + 8 Var(A*C) + 12 Var(A*B) + 24 Var(A)
Var(Error) + 4 Var(A*B*C) + 12 Var(A*B) + Q(B,B*C)
Var(Error) + 4 Var(A*B*C) + 12 Var(A*B)
Var(Error) + 4 Var(A*B*C) + 8 Var(A*C) + Q(C,B*C)
Var(Error) + 4 Var(A*B*C) + 8 Var(A*C)
Var(Error) + 4 Var(A*B*C) + Q(B*C)
Var(Error) + 4 Var(A*B*C)
30
Mixed Model Multiple Comparisons
Remember that for random factors, it would
not be useful to do multiple comparisons
because
Only some of the possible levels for the random
factor are involved in the study.
We want the results to extend to all possible
levels of the random factor.
We may, however, want to consider multiple
comparisons for the fixed factor.
31
Mixed Model Multiple Comparisons (2)
The variance associated to the fixed effect
2
will generally not be any more. So MSE
is the wrong estimate.
E ( MSA ) = Q ( A ) + n
2
AB
+
2
2
E ( MSAB ) = n AB + 2
The correct estimate for the above will be
MSAB (except in cases where MSE is larger
than MSAB then those two effects should
be combined in an additive model)
32
Mixed Model Multiple Comparisons (3)
The GLM procedure always uses MSE to
compare LSMeans, unless you tell it to do
different.
If MSAB > MSE, then in the LSMeans statement
you should specify E=A*B as an option.
If MSE > MSAB, simply remove the interaction
term from the model.
The MIXED procedure (new) does make this
change automatically.
33
Example (Gas Mileage)
See CLG Handout for SAS coding and full
output.
Also SAS file with the data, etc. online.
Goal of the example is to illustrate random
effects using SAS GLM and MIXED
procedures
34
Example (Gas Mileage)
Response is Gas Consuption
Two Random Factors
Four Drivers
Five Cars
Two observations per Driver*Car combination.
35
SAS Code
proc glm;
class driver car;
model gas=driver|car;
random driver car driver*car /test;
run;
Note: You must include ALL random effects
INCLUDING INTERACTIONS in the random
statement for GLM. Failing to include the
interaction term will mean that it will improperly
be treated as fixed.
36
Output
ANOVA output is similar but not all of the
Type I/III F-tests will be valid. These tests
assume all factors are fixed.
After ANOVA table comes Expected Mean
Squares. You should be able to use
these to figure out the appropriate tests.
After the EMS come the results of /test.
These are the correct tests for each effect.
37
Output
Source
driver
car
driver*car
Type III Expected Mean Square
Var(Error) + 2 Var(driver*car) + 10 Var(driver)
Var(Error) + 2 Var(driver*car) + 8 Var(car)
Var(Error) + 2 Var(driver*car)
Tests of Hypotheses for Random Model Analysis of Variance
Source
DF
Type III SS
driver
3
280.284750
car
4
94.713500
Error
12
2.446500
Error: MS(driver*car)
Source
DF
driver*car
12
Error: MS(Error) 20
Type III SS
2.446500
3.515000
Mean Squar e
93.428250
23.678375
0.203875
Mean Square
0.203875
0.175750
F Value
458.26
116.14
F Value
1.16
Pr > F
<.0001
<.0001
Pr > F
0.3715
38
Using the EMS
Suppose you want to test the effect of driver.
No effect if Var(driver) = 0
If no effect, then EMS(driver) = EMS(driver*car)
Test statistic should be MS(driver) divided by
MS(driver*car).
Note that you could do this by hand using
Type I SS; the results also come from the
/test option.
39
Studying Main Effects
Interaction was insignificant, so we want to
study main effects.
Note: MS(car*driver) > MSE
Used as error term for main effects, so we should
leave it in the model (though taking it out would
simplify things a bit and in this case wouldnt
change much).
The factors are random, so instead of doing
pairwise comparisons, we instead estimate
and compare variance components.
40
Variance Components
The EMS can be used to get estimates for
2
2
A and B .
both
2
2
E ( MSdriver ) = s 2 + 2sdr *car + 10sdr
2
E ( MSdr * car ) = s + 2s
2
dr *car
2
dr
10s = E ( MSdriver ) - E ( MSdr * car )
41
Estimated Driver Variance
The actual MS are estimates for the EMS.
So we may estimate the variance for the
driver:
MSdriver - MSdr * car
s=
10
93.4 - 0.2
=
= 9.32
10
2
dr
42
Estimated Car Variance
We may also estimate the variance for car:
s
2
car
MScar - MSdr * car
=
8
23.7 - 0.2
=
= 2.93
8
43
Conclusions
There are both car and driver effects, but no
significant interaction.
Based on the estimated variances, we see
clearly that the driver has a greater effect on
gas mileage than the car.
So how do we save gas?
Train drivers to drive economically.
There is likely little we could do about the car
effect, so thankfully it was small.
44
The MIXED procedure
Very similar to PROC GLM in coding
Quite different in output
45
PROC MIXED
proc mixed;
class driver car;
model gas=;
random driver car driver*car;
run;
Note: Only fixed effects appear in the model
statement; random effects appear ONLY in
the random statement.
46
Advantages to MIXED
Will provide correct LSMeans output for a
model that contains random effects.
Not relevant to this example.
Will provide estimates for the variance
components.
Theyll be exactly what we just calculated.
47
Variance Components
Covariance Parameter
Estimates
Cov Parm
driver
car
driver*car
Residual
Estimate
9.3224
2.9343
0.01406
0.1757
48
Variance Components (2)
It is sometimes possible that the estimates
for the variance components may be
negative.
Since negative variances are not possible,
what can we do?
It makes sense to estimate the variance by zero.
We can effectively do this by removing the term
from the model which is reasonable unless
higher order terms are significant.
49
CLG Activities
We now have three more examples.
Please do them in small groups;
then well discuss them as a class.
50