Application of repeated measurement ANOVA models using …

Transcript Application of repeated measurement ANOVA models using …

Application of repeated measurement ANOVA models using SAS and SPSS: examination of the effect of intravenous lactate infusion in Alzheimer's disease

Krisztina Boda 1 , János Kálmán 2 , Zoltán Janka 2 Department of Medical Informatics 1 , Department of Psychiatry 2 University of Szeged, Hungary

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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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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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 MIE '2002 4

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) 0’ 10’ 20’ Na-lactate (day 2) 0’ 10’ 20’ MIE '2002 5

The data „multivariate” or „wide” form

Proband

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

9.00

10.00

11.00

12.00

13.00

14.00

15.00

16.00

17.00

18.00

19.00

20.00

PH1_0

7.43

7.39

7.37

7.43

7.39

7.36

7.38

7.39

7.34

7.32

7.40

7.32

7.42

7.37

7.39

7.43

PH1_10

7.42

7.39

7.38

7.42

7.39

7.40

7.39

7.34

7.38

7.35

7.41

7.36

7.39

7.38

7.41

PH1_20

7.43

7.39

7.38

7.42

7.39

7.41

7.38

7.39

7.41

7.35

7.39

7.33

7.39

7.40

7.36

7.39

7.37

7.48

PH2_0

7.42

7.36

7.40

7.43

7.38

7.32

7.37

7.36

7.34

7.31

7.34

7.37

7.42

7.46

7.37

7.45

7.42

7.41

20 .

PH2_10

7.36

7.45

7.40

7.39

7.41

7.44

7.41

7.32

7.40

7.43

7.42

7.47

7.36

7.40

7.39

7.46

PH2_20

7.46

7.43

7.46

7.48

7.42

7.45

7.46

7.48

7.45

7.37

7.47

7.43

7.48

7.43

7.51

7.41

7.48

7.44

7.37

7.45

PCO1_0

34.60

50.40

45.60

48.20

44.50

47.20

48.10

44.40

50.10

57.20

42.70

51.40

43.20

45.80

53.60

45.80

42.50

41.60

18 .

18 MIE '2002

PCO1_10

34.50

48.70

46.90

47.10

44.60

48.00

49.50

46.60

49.80

58.10

45.00

54.90

49.20

45.50

55.10

48.60

43.10

44.30

The data „univariate” or „long” form

Proband

1.00 1.00 1.00 1.00 1.00 1.00 2.00 2.00 2.00 2.00 2.00 2.00

Treatment Time

Saline .00 Saline Saline Lactate Lactate 10.00 20.00 .00 10.00 Lactate Saline Saline Saline 20.00 .00 10.00 20.00 Lactate Lactate Lactate .00 10.00 20.00

7.43 7.42 7.43 7.42 . 7.46 7.39 7.39 7.39 7.36 7.36 7.43

pCO2

34.60 34.50 34.80 41.10 . 33.60 50.40 48.70 49.70 55.90 58.70 49.60

pO2

25.80 26.50 26.50 20.20 . 27.60 43.90 42.00 38.10 28.40 21.80 29.90

Lactate

2.10 1.80 1.70 2.40 . 7.00 1.60 1.40 1.30 1.20 3.40 4.00

Proband

1.00

PH1_0

7.43

PH1_10

7.42

PH1_20

7.43

PH2_0

7.42

PH2_10

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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 MIE '2002 8

7.3

7.4

7.5

7.6

Venous blood PH levels

•sample size •interaction 111 7.2

N = 18 18 Saline 18 69 59 20 19 Lactate 20 TIme .00

10.00

20.00

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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 MIE '2002 10

Statistical models and programs



-tests  the repeated use of the

-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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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 MIE '2002 12

Statistical models

 

GLM

(General Linear Model)

y= X



   

MIXED



: a vector of observed data 

an unknown vector of fixed-effects parameters with known design matrix

 : an unknown random error vector – assumed to be independently and identically distributed

(0,  2 ) Model

y= X



+ Z



  : an unknown vector of random-effects parameters with known design matrix

   : an unknown random error vector – whose elements are no longer required to be independent and homogenous. Assume that  expectations

and  are Gaussian random variables and have and variances

and

, respectively.

 The variance of

ZGZ

’ +

 For

and

some covariance structure must be selected MIE '2002 13

The within-subjects covariance matrix covariance patterns for 3 time periods

 UN-Unstructured     1 2 12  13  12  2 2  23   13 23    3 2  VC-Variance Components    0 1 2  2 2 0 0 0  3 2 0 0    CS-Compound Symmetry  AR(1) - First-Order Autoregressive     1 2 2  1 2   1 2  2  1 2   1 2  1 2  1 2  2  1 2   1 2    1      2  1   2  1    MIE '2002 14

GLM MIXED

  Requires balanced data; subjects with missing observations are deleted Assumes special form of the within-subject covariance matrix:  Type H (Sphericity) approach – univariate  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)-1 … th order autoregressive Estimates covariance parameters using restricted maximum likelihood,… ….

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Topics of Discussion

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 MIE '2002 17

Paired t-test

(only for demonstration –not recommended)

Comparison 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-Day2 10’, Day1-Day2 20’, Day1-Day2 Sig. (2-tailed) 0.140

0.164

0.607

0.009

0.000

0.788

0.018

0.000

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Correlation of PH measurements

PH1_0 PH1_10 PH1_20 PH2_0 PH2_10 PH2_20 PH1_0 PH1_10 PH1_20 PH2_0 PH2_10 1 .874

.874

1 .691

.820

.658

.600

.512

.677

.691

.658

.512

.243

.820

.600

.677

.407

1 .381

.296.

.006

.381

1 635 .399

.296

.635

.720

D1 T0 D1 T10 D1 T20 D2 T0 D2 T10 D2 T20

PH2_20 .243

.407

.006

.399

720 1

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Repeated measures ANOVA

Effects:  between-subjects effects -none  within-subjects effects • • • Treatment (Saline - Lactate) Time (0’-10’-10’) Patient -

random fixed fixed

 Interactions  Treatment*time interactions will be examined MIE '2002 20

GLM Univariate commands (data must be in „wide” form)



SPSS 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.



SAS PROC GLM

; model ph1_0 ph1_10 ph1_20 ph2_0 ph2_10 ph2_20=; repeated treat

, time

polynomial / summary ;

Run

; MIE '2002 21

Tests of Within-Subjects Effects

Measure: MEASURE_1 Source TREAT Sphericity As sumed Greenhouse-Geiss er Type III Sum of Squares 1.569E-02 1.569E-02 df 1 1.000

Mean Square 1.569E-02 1.569E-02 F 11.277

11.277

Sig.

.004

GLM univariate assumptions and results (SPSS)

Lower-bound 1.569E-02 1.000

1.569E-02 11.277

.004

Error(TREAT) Sphericity As sumed 2.226E-02 16 1.391E-03    Greenhouse-Geiss er Huynh-Feldt 2.226E-02 2.226E-02 16.000

16.000

1.391E-03 1.391E-03

3 subjects are deleted because of missing value

TIME Measure: MEASURE_1 Sphericity As sumed 2.109E-02 2 1.054E-02 20.718

TREATMENT*TIME interaction is significant

1 Mean Square Error(TIME) TREAT * TIME Error(TREAT*TIME) Huynh-Feldt Greenhouse-Geisser 2.109E-02 1.430

1.569E-02 1.569E-02 1.000

1.000

Lower-bound Error(TREAT) Lower-bound 2.109E-02 1.000

1.569E-02 2.226E-02 1.000

16 Sphericity As sumed Greenhouse-Geisser 1.629E-02 Huynh-Feldt 32 Greenhouse-Geiss er TIME Huynh-Feldt 21.596

Sphericity Assumed Huynh-Feldt Lower-bound 16.000

Error(TIME) Sphericity As sumed Sphericity Assumed 1.227E-02 2 Greenhouse-Geiss er Huynh-Feldt 1.227E-02 1.501

2.226E-02 2.109E-02 2.109E-02 16.000

16.000

2 1.350

1.430

1.000

1.629E-02 1.629E-02 32 21.596

1.629E-02 1.629E-02 22.875

16.000

Huynh-Feldt Lower-bound Greenhouse-Geiss er Huynh-Feldt TREAT * TIME Sphericity Assumed 1.227E-02 Greenhouse-Geisser Lower-bound 1.622

1.000

Greenhouse-Geisser 32 24.012

Lower-bound 1.385E-02 25.947

1.227E-02 7.564E-03 1.227E-02 2 1.501

1.227E-02 1.385E-02 1.385E-02 1.622

1.000

32 24.012

25.947

16.000

5.338E-04 1.569E-02 20.718

1.569E-02 20.718

1.391E-03 1.391E-03 1.391E-03 1.054E-02 1.562E-02 1.475E-02 2.109E-02 5.089E-04 14.171

7.119E-04 14.171

6.133E-03 14.171

8.174E-03 1.227E-02 4.328E-04 5.768E-04 5.338E-04 Lower-bound .000

22 1.385E-02 16.000

8.656E-04 F 11.277

11.277

.000

Sig.

.000

.004

.000

.004

.000

.004

20.718

.000

14.171

.000

.002

.000

.002

GLM multivariate results (SPSS)

Multivariate Tests b

Effect TREAT TIME TREAT * TIME Pillai's Trace Wilks' Lambda Hotelling's Trace Roy's Larges t Root Pillai's Trace Wilks' Lambda Hotelling's Trace Roy's Larges t Root Pillai's Trace Wilks' Lambda Hotelling's Trace Roy's Larges t Root Value .413

.587

.705

.724

.276

2.620

.537

.463

1.160

F 11.277

a 11.277

a 19.651

a 8.702

a a. Exact statistic b. Des ign: Intercept Within Subjects Design: TREAT+TIME+TREAT*TIME Hypothesis df 1.000

1.000

2.000

Error df 16.000

16.000

15.000

MIE '2002 Sig.

.004

.000

.003

Plot in SPSS

7.45

Estimated Marginal Means of MEASURE_1 7.44

7.43

7.42

7.41

7.40

7.39

7.38

7.37

1 TIME 2 3 TREAT 1 2 MIE '2002 24

Mixed models commands (Data must be in „long” form)

 SAS 8.02

proc mixed covtest

; class name treat time; model ph = treat time treat*time; repeated /type=un sub=name r rcorr; lsmeans treat*time / pdiff ;  SPSS 11.0

run

; MIXED ph BY treat time /CRITERIA = CIN(95) MXITER(100) MXSTEP(10) SCORING(1) SINGULAR(0.000000000001) HCONVERGE(0, ABSOLUTE) LCONVERGE(0, ABSOLUTE) PCONVERGE(0.000001, 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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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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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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Selecting the covariance structure

Correlation matrix for a block using UN covariance structure Row COL1 COL2 COL3 COL4 COL5 COL6 1 1.00000000 0.87572288 0.69284518 0.64717994 0.54989785 0.24762745

2 0.87572288 1.00000000 0.82563330 0.59373944 0.69760124 0.39606241

3 0.69284518 0.82563330 1.00000000 0.37105322 0.36385899 0.00696442

4 0.64717994 0.59373944 0.37105322 1.00000000 0.64283355 0.39879596

5 0.54989785 0.69760124 0.36385899 0.64283355 1.00000000 0.71658187

6 0.24762745 0.39606241 0.00696442 0.39879596 0.71658187 1.00000000

Correlation matrix for a block using AR(1) covariance structure Row Col1 Col2 Col3 Col4 Col5 Col6 1 1.0000 0.6169 0.3805 0.2348 0.1448 0.08933

2 0.6169 1.0000 0.6169 0.3805 0.2348 0.1448

3 0.3805 0.6169 1.0000 0.6169 0.3805 0.2348

4 0.2348 0.3805 0.6169 1.0000 0.6169 0.3805

5 0.1448 0.2348 0.3805 0.6169 1.0000 0.6169

6 0.08933 0.1448 0.2348 0.3805 0.6169 1.0000

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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 UN@AR(1), we assume equal correlation between treatments (UN) and AR(1) covariance structure between the three time  points.

UN@AR(1): we assume the UN covariance matrix for the treatments and the AR(1) covariance matrix for the time effects MIE '2002 29

The UN@AR(1) composite covariance model in SAS

 For each subject, we have the following covariance matrix:        1 2  12     1      1   2 2  1   1   2  1     2  1    1  12      2 1  2 2      2  1 2  12   1  1   12  2 2  2  1  2  1              1   @      2 1   1  1 2  1 2           1 1 2 2 12  12    2    12   1 2 1 2  12  12  2  12  2      1 2  1 2    12   12 2  12  12   1 2  12  2  12  2  2 2   2 2  2 2  2   12   2 2    12  12 2 2 2  2    12   12  2 12  2 2  2 2  2 2  2        MIE '2002 30

Selecting the covariance structure

Correlation matrix for a block using UN@AR(1) covariance structure Row COL1 COL2 COL3 COL4 COL5 COL6 1 1.00000000 0.73001496 0.53292185 0.22698641 0.16570348 0.12096602

2 0.73001496 1.00000000 0.73001496 0.16570348 0.22698641 0.16570348

3 0.53292185 0.73001496 1.00000000 0.12096602 0.16570348 0.22698641

4 0.22698641 0.16570348 0.12096602 1.00000000 0.73001496 0.53292185

5 0.16570348 0.22698641 0.16570348 0.73001496 1.00000000 0.73001496

6 0.12096602 0.16570348 0.22698641 0.53292185 0.73001496 1.00000000

Correlation between time Time 0 Time 10 Time 20 Time 0 1.00000000 0.73001496 0.53292185 Time 10 0.73001496 1.00000000 0.73001496 Time 20 0.53292185 0.73001496 1.00000000 R=0.227 (correlation between treatments)

MIE '2002 31

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 MIE '2002 32

Comparison of covariance structures for PH data

Information criteria (smaller-is-better forms). Covariance structure Number of parameters -2 Restricted Log Likelihood UN 21 VC 6 CS 2 AR(1) 2 UN@AR(1) 4 -500.416 -405.118 -433.927 -429.208 -454.181 Akaike's Information Criterion (AIC) Likelihood ratio test (comparison to UN) df diff -458.416 -393.118 -429.927 -425.208 -446.2 15 95.294 19 66.496 Models VC,CS are significantly different (worse) from model with UN covariace structure. However, UN@AR(1) model will be used, -because this is a doubly repeated model, -the covariance structure is simpler MIE '2002 33

Results using mixed model (SAS)

Tests of Fixed Effects (Type=UN@AR) Source NDF DDF Type III F TREAT 1 88 8.77 TIME 2 88 15.86 TREAT*TIME 2 88 14.22 Pr > F 0.0039

0.0001

MIE '2002 34

Differences of Least Squares Means

Differences of Least Squares Means Effect TREAT TIME _TREAT _TIME Difference Std Error DF t Pr > |t| TREAT*TIME 1.00 0.00 1.00 10.00 -0.00668 0.004978 33 -1.34 0.1886

TREAT*TIME 1.00 0.00 1.00 20.00 -0.00888 0.006548 33 -1.36 0.1844

TREAT*TIME 1.00 10.00 1.00 20.00 -0.00219 0.004978 33 -0.44 0.6624

TREAT*TIME 2.00 0.00 2.00 10.00 -0.02260 0.006852 33 -3.30 0.0023

TREAT*TIME 2.00 0.00 2.00 20.00 -0.05895 0.008888 33 -6.63 <.0001

TREAT*TIME 2.00 10.00 2.00 20.00 -0.03635 0.006852 33 -5.30 <.0001 TREAT*TIME 1.00 0.00 2.00 0.00 -0.00459 0.01018 33 -0.45 0.6551

TREAT*TIME 1.00 10.00 2.00 10.00 -0.02051 0.01024 33 -2.00 0.0536

TREAT*TIME 1.00 20.00 2.00 20.00 -0.05466 0.01018 33 -5.37 <.0001

7.6

7.5

7.4

7.3

7.2

N = 18 18 Saline 18 111 69 59 20 19 Lactate 20 TIme .00

10.00

20.00

Paired t-test: Comparison 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-Day2 10’, Day1-Day2 20’, Day1-Day2 Sig. (2-tailed) MIE '2002 0.140

0.164

0.607

0.009

0.000

0.788

0.018

0.000

Distribution of residuals using UN@AR(1) covariance structure

MIE '2002 36

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.

MIE '2002 37

Examination of another parameter: systolic blood pressure (RRS)

200 180 160 140 120 100 80 N = 19 19 Saline 19 19 18 Lactate 19 Time 120 0 100 10 80 200 180 160 140 RRS 1-10 RRS 1-20 RRS 2-0 RRS 2-10 RRS 2-20 MIE '2002 38

Mean and SD of systolic blood pressure 180.00

160.00

140.00

120.00

100.00

80.00

60.00

40.00

20.00

0.00

N 19 19 19 19 0 10 20 Time (min) Saline Lactate

MIE '2002 39

The same figure with different scaling

Mean of systolic blood pressure 150.00

148.00

146.00

144.00

142.00

146.89

142.05

140.00

138.00

136.00

138.89

139.74

140.26

140.61

134.00

132.00

130.00

N 19 19 19 19 18 19 0 10 20 Saline Lactate Time (min) Mean of systolic blood pressure 150.00

148.00

146.00

144.00

142.00

140.00

138.00

136.00

134.00

132.00

130.00

138.17

138.50

138.67

140.61

144.83

141.11

N 18 18 18 18 18 18 0 10 20 Saline Lactate Time (min) Different sample size Equal sample size

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GLM results (2 cases are deleted)

 GLM Multivariate (Wilks’ Lambda Sig):  TREAT 0.868

  TIME TREAT*TIME 0.270

 GLM Univariate (Spericity assumptions met)  TREAT 0.868

  TIME TREAT*TIME 0.253

 Is there a significant time effect?

MIE '2002 41

Plot in SPSS GLM

148 Estimated Marginal Means 146 144 142 140 138 0 10 TIME 20 TREATMENT Saline Lactate MIE '2002 42

Correlation matrix of systolic blood pressures

BP1_0 BP1_10 BP1_20 BP2_0 BP2_10 BP2_20 BP1_0 BP1_10 BP1_20 BP2_0 BP2_10 BP2_20 1 .954

.893

.884

.619

.790

.954

.893

1 .908

.908

1 .842

.825

.569

.566

.776

.778

.884

.619

.790

.842

.569

.776

.825

.566

.778

1 .755

.791

.755

1 .825

.791

.825

Paired t-tests RRS 1-0 - RRS 1-10 RRS 1-0 - RRS 1-20 RRS 1-10 - RRS 1-20 RRS 2-0 - RRS 2-10 RRS 2-0 - RRS 2-20 RRS 2-10 - RRS 2-20 RRS 1-0 - RRS 2-0 RRS 1-10 - RRS 2-10 RRS 1-20 - RRS 2-20 Sig. (2-tailed) .409

.003

.009

.515

.439

.845

.715

.672

.155

MIE '2002 43

MIXED: Comparison of covariance structures for BP data

Information criteria (smaller-is-better forms).

UN VC CS HF Covariance structure Number of parameters -2 Restricted Log Likelihood Akaike's Information Criterion (AIC) 21 815.637

857.637

6 968.25

978.1

2 858.587

862.587

7 853.88

868.337

Likelihood ratio test (comparison to UN) df diff p 15 152.613

<0.0001

19 42.95

.001317

14 38.243

.000477

AR(1) UN@AR(1) 2 4 848.541 860.93

852.546 868.9

19 17 32.9

46.29

.024686 .000156

UN covariance structure is significantly better than the other models examined

MIE '2002 44

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

-values are close. There is a significant increase in time for BP data.

MIE '2002 45

 Using mixed models, an increasing time effect could be shown.

MIE '2002 46

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.

MIE '2002 47

Individual regression lines

200 TREAT: 1.00 Saline 180 160 140 120 100 -10 Time 0 10 20 30 200 TREAT: 2.00 Lactate 180 160 140 120 100 80 -10 Time 0 10 20 30 MIE '2002 48

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) proc mixed

; class name treat; model rrs= time /s; random int time /sub=name type=un solution;

run

;

Mixed models with two additional effects (with random coefficients for patients and patients*time) proc mixed

; class name treat; model random

run

; rrs=treat time treat*time/s; int time /sub=name type =un solution ; MIE '2002 49

Regression lines by averaged by treatments

200 180 160 140 120 100 -10 Time 0 10 20 30 Treatment Lactate Saline MIE '2002 50

Results I: fixed effects (linear regression)

Covariance Parameter Estimates: Residual 410.02

Residual variance: 410.02

Fit Statistics -2 Res Log Likelihood 996.5

Solution for Fixed Effects Effect Estimate Standard Error DF t Value Pr > |t| Intercept 138.84 3.0039 111 46.22 <.0001

TIME 0.2579 0.2323 111 1.11 0.2693

RRS=0.2579*time + 138.84

Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F TIME 1 111 1.23 0.2693

The time-effect is not significant

MIE '2002 51

Results I: fixed effects (linear regression)

200 180 160 140 120 100 80 -10 Time 0 10

RRS=0.2579*time + 138.84

20 30 Treatment Lactate Saline Total Population

The time-effect is not significant

MIE '2002 52

Results II: mixed model: fixed and random effects (linear regression)

Covariance Parameter Estimates UN(1,1) NAME 346.73

UN(2,1) NAME -0.5609

UN(2,2) NAME 0 Residual 88.3869

Fit Statistics: -2 Res Log Likelihood 882.9

Solution for Fixed Effects

Residual variance: 88.38

Effect Estimate Standard Error DF t Value Pr > |t| Intercept 139.03 4.4939 18 30.94 <.0001

TIME 0.2579 0.1078 18 2.39 0.0279

Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F TIME 1 18 5.72 0.0279

RRS=0.2579*time + 139.03

The time-effect is significant

MIE '2002 53

Results III: mixed model: two fixed effects and random effects

Covariance Parameter Estimates UN(1,1) NAME 346.73

UN(2,1) NAME -0.5609

UN(2,2) NAME 0 Residual 88.3869 Fit Statistics -2 Res Log Likelihood 879.2

Type 3 Tests of Fixed Effects Effect Num Den DF DF F Value Pr > F TREAT TIME 1 73 0.53 0.4703

1 18 5.71 0.0280

TIME*TREAT 1 73 1.74 0.1919

Residual variance: 88.38

The time-effect is significant The other two effects are not significant We decide to use MODEL II

MIE '2002 54

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.

MIE '2002 55

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. MIE '2002 56

References

H. Brown and R. Prescott, Applied Mixed Models in Medicine. Wiley, 2001.

T. Park, and Y.J. Lee,: Covariance models for nested repeated measures data: analysis of ovarian steroid secretion data.

Statistics in

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) 442-449.

R. Wolfinger and M. Chang, Comparing the SAS Procedures for Repeated Measures, SAS Institute Inc., Cary, NC. http://www.ats.ucla.edu/stat/sas/library/  GLM and MIXED MIE '2002 57

Application of repeated measurement ANOVA models using …

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