Analyzing results through time
using a multivariate approach
Heidi A. Lewis and Christopher C. Kohler
Fisheries and Illinois Aquaculture Center
Southern Illinois University
Presentation outline
• MANOVA vs ANOVA
• How MANOVA works
• Profile analysis
• Parallelism
• Levels
• Flatness
• Example with SAS code
MANOVA vs ANOVA
ANOVA—Analysis of variance
• Two or more means are compared to find
reliable differences among them
MANOVA—Multivariate analysis of variance
• Two or more centroids are compared to find
reliable differences among them
• Centroid—average of the combined set of
dependent variables
Assumptions to be met…
• Normality of each variable distribution
• Independence
• Data represent random samples from the
population
• Observations on each subject are independent
• Equal covariance
• Variances must be equal across all groups
• Correlations between any two variables must
be the same across all groups
When to chose MANOVA
• Several correlated dependent variables and
ONE statistical test is desirable
• Influence of independent variables on the
pattern of response observed within
dependent variables
• MANOVA may reveal differences not shown
by separate ANOVAs
How MANOVA works
• MANOVA creates a new DV
• Maximizes group differences
• Linear combination of all measured DV
• New variable for each main effect and
interaction—Factorial MANOVA
How MANOVA works
• MANOVA creates a new DV
• Maximizes group differences
• Linear combination of all measured DV
• New variable for each main effect and
interaction—Factorial MANOVA
• ANOVA is performed on the new DV
• Hypotheses about means are tested by
comparing variances
Power
Large sample size
More variables
Differences between means are large
Test Statistics
Pillai’s Trace—best for general use
• Uneven sample sizes—OK
• Data is correlated—OK
• Reject H0: large values
Test Statistics
Pillai’s Trace—best for general use
• Uneven sample sizes—OK
• Data is correlated—OK
• Reject H0: large values
Wilk’s Lambda
• Measures amount of variability not explained by
levels of the IV
• Reject H0: small values
Test Statistics
Pillai’s Trace—best for general use
• Uneven sample sizes—OK
• Data is correlated—OK
• Reject H0: large values
Wilk’s Lambda
• Measures amount of variability not explained by
levels of the IV
• Reject H0: small values
Roy’s Largest/Greatest Root
• Best when variables are not correlated
Test Statistics
Pillai’s Trace—best for general use
• Uneven sample sizes—OK
• Data is correlated—OK
• Reject H0: large values
Wilk’s Lambda
• Measures amount of variability not explained by
levels of the IV
• Reject H0: small values
Roy’s Largest/Greatest Root
• Best when variables are not correlated
Hotelling-Lawley’s Trace
Test Statistics
•
•
•
•
Pillai’s Trace
Wilk’s Lambda
Roy’s Largest/Greatest Root
Hotelling-Lawley’s Trace
All translated into F statistics to test the null
hypothesis
Profile Analysis
• All DV are measured on the same scale
• Determine if groups differ on the scale
• Three types
1. Types of IV differ for a particular group of DV
Protein retention
Lipid retention
Energy retention
Profile Analysis
• Three types
2. A particular group of DV differ over repeated
measurements or time
Protein retention
Profile Analysis
• Three types
3. Types of IV differ for a particular group of DV
over repeated measurements or time
Protein retention
Profile Analysis
2. A particular group of DV differ over
repeated measurements or time
Protein retention
Each successive measurement taken over
time is converted into a new DV
Larval sunshine bass C18 PUFA
100%
Flax
66%
Flax
33%
Flax
100%
Fish
3 DPH
5 DPH
Profile Analysis
• Parallelism of profiles
• Levelness among groups
• Flatness of profiles
C18 PUFA
120
90
100% Fish
mg FAME/
g dry
60
matter
33% Flax
30
100% Flax
66% Flax
0
Egg
3 DPH
5 DPH
Profile Analysis
• Parallelism of profiles
• Do the different groups have parallel profiles?
• Test of interaction among groups
• Primary research question
C18 PUFA
120
90
100% Fish
mg FAME/
g dry
60
matter
33% Flax
30
100% Flax
66% Flax
0
Egg
3 DPH
5 DPH
Profile Analysis
• Flatness of profiles
• Do all DV elicit the same average response?
• Equivalent to the within-subjects main effect
C18 PUFA
120
90
100% Fish
mg FAME/
g dry
60
matter
33% Flax
30
100% Flax
66% Flax
0
Egg
3 DPH
5 DPH
Profile Analysis
• Levelness among groups
• Do the groups produce similar results?
• Equivalent to between-subjects main effect
C18 PUFA
120
90
100% Fish
mg FAME/
g dry
60
matter
33% Flax
30
100% Flax
66% Flax
0
Egg
3 DPH
5 DPH
Profile Analysis—SAS Syntax
DATA WHB_FLAX;
input diet $ female $ time1 time2 time3;
datalines;
66_Flax
1
89.30
40.32
51.38
66_Flax
2
83.96
53.90
28.76
.
.
Each measurement period is considered
.
a correlated dependent variable
;
Profile Analysis—SAS Syntax
PROC GLM data=<data set>;
class <independent classification variables>;
model <dependent variables>=<independent
variable>/ nouni;
repeated <factor name> profile;
run;
Profile Analysis—SAS Syntax
PROC GLM data=<data set>;
class <independent classification variables>;
model <dependent variables>=<independent
variable>/ nouni;
repeated <factor name> profile;
run;
No separate
Identifies which factor
univariate tests for
is repeated within the
each DV
profile analysis
Profile Analysis—SAS Results
The GLM Procedure
Class Level Information
Class
Levels Values
diet
4 0_Flax 33_Flax 66_Flax
100_Flax
Number of Observations Read
23
Number of Observations Used
17
Profile Analysis—SAS Results
MANOVA Test Criteria and Exact F Statistics for the
Hypothesis of no time effect—(FLATNESS)
H = Type III SSCP Matrix for time
E = Error SSCP Matrix
S=1 M=0 N=5
Statistic
Wilks' Lambda
Pillai's Trace
H-L Trace
Roy's Greatest Root
Value F Value Num DF Den DF
0.12 44.03
2
12
0.88 44.03
2
12
7.34 44.03
2
12
7.34 44.03
2
12
Some variation through time
Pr > F
<.0001
<.0001
<.0001
<.0001
Profile Analysis—SAS Results
MANOVA Test Criteria and F Approximations for the
Hypothesis of no time*trt effect—(PARALLELISM)
H = Type III SSCP Matrix for time*trt
E = Error SSCP Matrix for time*trt
S=2 M=0 N=5
Statistic
Wilks' Lambda
Pillai's Trace
H-L Trace
Roy's Greatest Root
Value
0.71
0.30
0.39
0.34
F Value Num DF Den DF
0.75
6
24
0.78
6
26
0.76
6
14.4
1.49
3
13
Parallel response
Pr > F
<.6136
<.5956
<.6101
<.2637
Profile Analysis—SAS Results
Repeated measures egg- 5 DPH larvae
The GLM Procedure
Repeated Measures Analysis of Variance
Tests of Hypotheses for Between Subjects Effects
Source DF Type III SS Mean Square F Value Pr > F
diet
3
10389
3463
15.33 0.0001
Error 13
2937
226
Levels differ—treatment
differences
Larval sunshine bass C18 PUFA
120
*
*
90
mg FAME/
g dry
matter
100% Fish
33% Flax
66% Flax
100% Flax
*
60
30
0
Egg
3 DPH
5 DPH
Diet
p<0.0001
Time
p<0.0001
DxT
p=0.5956
When to chose profile analysis for repeated
measures analysis…
• Many replicates are available—more reps
than DVs
• Often “clean” data sets with balanced designs
fit univariate designs better—test for
homogeneity of covariance first
• If assumption of homogeneity of covariance is
violated—use multivariate approach to
repeated measures
Multivariate References
• Tabachnich, BG and LS Fidell. 1989. Using
multivariate statistics.
HarperCollinsPublishers, Inc. New York, NY.
• Zar, JH. 1999. Biostatistical analysis.
Prentice Hall, Inc. Upper Saddle River, NJ.