1. Application of repeated measurement ANOVA models using SAS and SPSS: examination of the effect of intravenous lactate infusion in Alzheimer's disease
Krisztina Boda1, János Kálmán2, Zoltán Janka2
Department of Medical Informatics1, Department of Psychiatry2
University of Szeged, Hungary

2. Introduction
Repeated measures analysis of variance (ANOVA) generalizes Student's t-test for paired samples. It is used when an outcome variable of interest is measured repeatedly over time or under different experimental conditions on the same subject.
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3. The purpose of the discussion
to show the application of different statistical models to investigate the effect of intravenous Na-lactate on cerebral blood flow and on venous blood parameters in Alzheimer's dementia (AD) probands using SAS and SPSS programs.
to show the most important properties of these statistical models.
to show that different models on the same data set may give different results.
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4. Topics of Discussion The medical experiment
The data table
Statistical models and programs
Statistical analysis of two parameters (venous blood PH and systoloc blood pressure) using different models and programs
GLM models
Mixed models
Comparison of the results
Summary of the key points
Medical results and discussion
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5. The medical experiment
Patients: 20 patients having moderate-severe dementia syndrome (AD).
Experimental design: self-control study
measurements were performed on the same patient at 0, 10 and 20 minutes after 0.9 % NaCl (Saline) or 0.5 M Na-lactate infusion on two different days
NaCl (Saline) (day 1) Na-lactate (day 2)
0’ 10’ 20’ 0’ 10’ 20’
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6. The data „multivariate” or „wide” form
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7. The data „univariate” or „long” form
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8. Statistical model
The statistical models will be shown using one chosen parameter the venous blood PH.
2 repeated measures factors:
days (treatments) with 2 levels (Saline or Lactate)
time with 3 levels (0, 10 and 20 minutes)
both factors are fixed
values of interest are all represented in the data file
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9. Venous blood PH levels
sample size
interaction
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10. Topics of Discussion The medical experiment
The data table
Statistical models and programs
Statistical analysis of one parameter (venous blood PH) using different models and programs
GLM models
Mixed models
Comparison of the results
Summary of the key points
Medical results and discussion
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11. Statistical models and programs
t-tests
the repeated use of the t-tests may increase the experiment wise probability of Type I error.
ANOVA
GLM
Mixed
Programs used
SAS 6.12, 8.02
SPSS 9.0, 11.0
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12. Repeated measures ANOVA
Observations on the same subject are usually correlated and often exhibit heterogeneous variability
a covariance pattern across time periods can be specified within the residual matrix.
Effects:
between-subjects effects
within-subjects effects
Interactions
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13. Statistical models GLM (General Linear Model) y= X  + 
y: a vector of observed data
: an unknown vector of fixed-effects parameters with known design matrix X
: an unknown random error vector – assumed to be independently and identically distributed N(0,2)
MIXED Model y= X  + Z  + 
: an unknown vector of random-effects parameters with known design matrix Z
: an unknown random error vector – whose elements are no longer required to be independent and homogenous.
Assume that  and  are Gaussian random variables and have expectations 0 and variances G and R, respectively.
The variance of y is V=ZGZ’ + R
For G and R some covariance structure must be selected
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14. The within-subjects covariance matrix - covariance patterns for 3 time periods
UN-Unstructured
CS-Compound Symmetry
VC-Variance Components
AR(1) - First-Order Autoregressive
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15. GLM MIXED
Requires balanced data; subjects with missing observations are deleted
Assumes special form of the within-subject covariance matrix:
Type H (Sphericity) – univariate approach
Unstructured –multivariate approach
Estimates covariance parameters using a method of moments ….
Allows data that are missing at random
Allows a wide variety of within-subject covariance matrix
UN-Unstructured
VC-Variance Components
CS-Compound Symmetry
AR(1)-1th order autoregressive …
Estimates covariance parameters using restricted maximum likelihood,… ….
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16. Topics of Discussion The medical experiment
The data table
Statistical models and programs
Statistical analysis of one parameter (venous blood PH) using different models and programs
GLM models
Mixed models
Comparison of the results
Summary of the key points
Medical results and discussion
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17. Statistical analysis of venous blood PH using different models and programs
Examination of univariate statistics and correlation structure
GLM univariate and multivariate results, verifying assumptions
Mixed models
Create the model
Examine and choose the covariance structure
Compare fixed effects
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18. Paired t-test (only for demonstration –not recommended)
Comparison Sig. (2-tailed)
Day 1, 0’-10’
Day 1, 0’-20’
Day 1, 10’-20’
Day 2, 0’-10’
Day 2, 0’-20’
Day 2, 10’-20’
0’, Day1-Day
10’, Day1-Day
20’, Day1-Day
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19. Correlation of PH measurements
PH1_0 PH1_10 PH1_20 PH2_0 PH2_10 PH2_20
PH1_
PH1_
PH1_
PH2_
PH2_
PH2_
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20. Repeated measures ANOVA
Effects:
between-subjects effects -none
within-subjects effects
Treatment (Saline - Lactate) - fixed
Time (0’-10’-10’) - fixed
Patient -random
Interactions
Treatment*time interactions will be examined
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21. GLM Univariate commands (data must be in „wide” form)
SPSS
SAS
GLM ph1_0 ph1_10 ph1_20 ph2_0 ph2_10 ph2_20
/WSFACTOR = treat 2 Polynomial time 3 Polynomial
/METHOD = SSTYPE(3)
/PLOT = PROFILE( time*treat )
/WSDESIGN = treat time treat*time.
PROC GLM ;
model ph1_0 ph1_10 ph1_20 ph2_0 ph2_10 ph2_20=;
repeated treat 2, time 3 polynomial / summary ;
Run;
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22. GLM univariate assumptions and results (SPSS)
Sphericity test failed, a correction can be applied
3 subjects are deleted because of missing value
TREATMENT*TIME interaction is significant
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23. GLM multivariate results (SPSS)
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24. Plot in SPSS
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25. Mixed models commands (Data must be in „long” form)
SAS 8.02
SPSS 11.0
proc mixed covtest;
class name treat time;
model ph = treat time treat*time;
repeated /type=un sub=name r rcorr;
lsmeans treat*time /pdiff;
run;
MIXED ph BY treat time
/CRITERIA = CIN(95) MXITER(100) MXSTEP(10) SCORING(1)
SINGULAR( ) HCONVERGE(0, ABSOLUTE)
LCONVERGE(0, ABSOLUTE) PCONVERGE( , ABSOLUTE)
/FIXED = treat time time*treat | SSTYPE(3)
/METHOD = REML
/PRINT = G LMATRIX R SOLUTION TESTCOV
/REPEATED = treat time | SUBJECT(name ) COVTYPE(UN)
/SAVE = RESID .
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26. Selecting the covariance structure
Using SAS command, replacing “UN” in type=UN with CS, VC, HF , AR(1) and others defines Unstructured, Variance Components, Huynh-Feldt and First Order Autoregressive, etc… variance-covariance structures of the fixed effects. The default is VC.
Using SPSS command, replacing “UN” in COVTYPE(UN) with ID, CS, VC, HF , AR(1) defines the above covariance structures. No other types are available.
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27. Selecting the covariance structure
The unstructured covariance is overly complex. In our example we have 6 levels for treat*time effects, so the unstructured covariance has 6 variances and 15 covariances (6*5)/2 ), for a total of 21 variances and covariances being estimated. The other structures use less covariance parameter for the repeated effects.
Another problem with CS, HF and AR(1) structures that they do not take into account the double repeated nature of our model.
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28. Selecting the covariance structure
Correlation matrix for a block using UN covariance structure
Row COL COL COL COL COL COL6
Correlation matrix for a block using AR(1) covariance structure
Row Col Col Col Col Col Col6
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29. Selecting the covariance structure: a composite covariance model
Under a composite covariance model separate covariance structures are specified for each of two repeat factors. Using we assume equal correlation between treatments (UN) and AR(1) covariance structure between the three time points.
we assume the UN covariance matrix for the treatments and the AR(1) covariance matrix for the time effects
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30. The UN@AR(1) composite covariance model in SAS
For each subject, we have the following covariance matrix:
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31. Selecting the covariance structure
Correlation matrix for a block using covariance structure
Row COL COL COL COL COL COL6
Correlation between time
Time Time Time 20
Time
Time
Time
R=0.227 (correlation between treatments)
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32. Comparison of mixed models with different covariance structures
Based on information criteria about the model fit
Akaike's Information Criterion (AIC)
-2 Restricted Log Likelihood:
Likelihood ratio test (for nested models)
Smaller values indicate better models
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33. Comparison of covariance structures for PH data
Models VC,CS are significantly different (worse) from model with UN covariace structure.
However, model will be used,
-because this is a doubly repeated model,
-the covariance structure is simpler
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34. Results using mixed model (SAS)
Tests of Fixed Effects
Source NDF DDF Type III F Pr > F
TREAT
TIME
TREAT*TIME
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35. Differences of Least Squares Means
Effect TREAT TIME _TREAT _TIME Difference Std Error DF t Pr > |t|
TREAT*TIME
TREAT*TIME
TREAT*TIME
TREAT*TIME
TREAT*TIME <.0001
TREAT*TIME <.0001
TREAT*TIME
TREAT*TIME
TREAT*TIME <.0001
Paired t-test: Comparison Sig. (2-tailed)
Day 1, 0’-10’
Day 1, 0’-20’
Day 1, 10’-20’
Day 2, 0’-10’
Day 2, 0’-20’
Day 2, 10’-20’
0’, Day1-Day
10’, Day1-Day
20’, Day1-Day
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36. Distribution of residuals using UN@AR(1) covariance structure
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37. Summary of statistical results for venous blood PH
Changing models might give different results.
GLM models are useful in case of balanced data satisfying special assumptions.
Using mixed model, the covariance structure of repeated effects can be taken into account, and cases with missing values are not deleted.
The presence of a treatment*time interaction is obvious by any model.
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38. Examination of another parameter: systolic blood pressure (RRS)
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40. The same figure with different scaling
Different sample size
Equal sample size
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41. GLM results (2 cases are deleted)
GLM Multivariate (Wilks’ Lambda Sig):
TREAT
TIME
TREAT*TIME 0.270
GLM Univariate (Spericity assumptions met)
TIME
TREAT*TIME 0.253
Is there a significant time effect?
0.095
0.042
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42. Plot in SPSS GLM
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43. Correlation matrix of systolic blood pressures
BP1_0 BP1_10 BP1_20 BP2_0 BP2_10 BP2_20
BP1_
BP1_
BP1_
BP2_
BP2_
BP2_
Paired t-tests Sig. (2-tailed)
RRS RRS
RRS RRS
RRS RRS
RRS RRS
RRS RRS
RRS RRS
RRS RRS
RRS RRS
RRS RRS
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44. MIXED: Comparison of covariance structures for BP data
UN covariance structure is significantly better than the other models examined
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45. Results for time-trend using mixed model
GLM: based on data of 18 patients, univariate results seem to be acceptable, showing a significant time-trend. However, assumptions of the multivariate approach are more realistic.
Multivariate (UN): 2, 16, p=0.095
Univariate (CS): 2, 34 p=0.042.
MIXED: based on data of 20 patients, UN covariance structure has to be used.
UN: 2, 18, p=0.045
CS: 2, 89 p=0.0587
The p-values are close. There is a significant increase in time for BP data.
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46. Using mixed models, an increasing time effect could be shown.
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47. Covariance pattern model vs. random coefficients model
When correlation between observations on the same patients is not constant, a covariance pattern model can be used.
When the relationship of the response variable with time is of interest, a random coefficients model is more appropriate. Here, regression curves are fitted for each patient and the regression coefficients are allowed to vary randomly between the patients.
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48. Individual regression lines
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49. SAS commands
Fixed effects approach (linear regression with one independent variable). The effect of patient is ignored – all observations are treated as independent.
proc mixed;
model rrs= time /s;
run;
Mixed models (with random coefficients for patients and patients*time)
class name treat;
model rrs= time /s;
random int time /sub=name type=un solution;
Mixed models with two additional effects (with random coefficients for patients and patients*time)
model rrs=treat time treat*time/s;
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50. Regression lines by averaged by treatments
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51. Results I: fixed effects (linear regression)
Covariance Parameter Estimates: Residual
Fit Statistics
-2 Res Log Likelihood
Solution for Fixed Effects
Effect Estimate Standard Error DF t Value Pr > |t|
Intercept <.0001
TIME
Type 3 Tests of Fixed Effects
Num Den
Effect DF DF F Value Pr > F
TIME
Residual variance:
RRS=0.2579*time
The time-effect is not significant
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52. Results I: fixed effects (linear regression)
RRS=0.2579*time
The time-effect is not significant
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53. Results II: mixed model: fixed and random effects (linear regression)
Covariance Parameter Estimates
UN(1,1) NAME
UN(2,1) NAME
UN(2,2) NAME
Residual
Fit Statistics: Res Log Likelihood
Solution for Fixed Effects
Effect Estimate Standard Error DF t Value Pr > |t|
Intercept <.0001
TIME
Type 3 Tests of Fixed Effects
Num Den
Effect DF DF F Value Pr > F
TIME
Residual variance: 88.38
RRS=0.2579*time
The time-effect is significant
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54. Results III: mixed model: two fixed effects and random effects
Covariance Parameter Estimates
UN(1,1) NAME
UN(2,1) NAME
UN(2,2) NAME
Residual
Fit Statistics
-2 Res Log Likelihood
Type 3 Tests of Fixed Effects
Num Den
Effect DF DF F Value Pr > F
TREAT
TIME
TIME*TREAT
Residual variance: 88.38
The time-effect is significant
The other two effects
are not significant
We decide to use MODEL II
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55. Discussion
Using statistical software without knowing their main properties or using only their default parameters may lead to spurious results.
Using only the default parameters means that simple models are supposed (i.e. VC covariance pattern in mixed procedure).
Medical experiments often result in repeated measures data, nested repeated measures data. The use of carefully chosen statistical model may improve the quality of statistical evaluation of medical data.
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56. Medical consequences
The main results are that the diminished elevation of serum cortisol levels indicates blunted stress response to Na-lactate in AD. The decreased vascular responsiveness of the majority of AD cases reflects impaired vasoreactivity and disturbed vasoregulation. Since the catecholaminerg system and cholinergic mechanisms are also involved in the regulation of reactivity of the brain microvasculature, these alterations might be the consequences of the general cholinergic deficit in AD.
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57. References
H. Brown and R. Prescott, Applied Mixed Models in Medicine. Wiley, 2001.
SAS Institute, Inc: The MIXED procedure in SAS/STAT Software: Changes and Enhancements through Release Copyright © 1996 by SAS Institute Inc., Cary, NC
T. Park, and Y.J. Lee,: Covariance models for nested repeated measures data: analysis of ovarian steroid secretion data. Statistics in Medicine 21 (2002)
SPSS Advanced Models Copyright © 1996 by SPSS Inc P.
R. S. Stewart, M. D. Devous, A. J. Rush, L. Lane, F. J. Bonte, Cerebral blood flow changes during sodium-lactate induced panic attacks. Am. J. Psych., 145 (1988)
R. Wolfinger and M. Chang, Comparing the SAS GLM and MIXED Procedures for Repeated Measures, SAS Institute Inc., Cary, NC.
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