Application of repeated measurement ANOVA models using SAS and SPSS: examination of the effect of intravenous lactate infusion in Alzheimer's disease Krisztina Boda1, Jnos Klmn2, Zoltn Janka2 Department of Medical Informatics1, Department of Psychiatry2 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. MIE '2002 2 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. MIE '2002 3 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 0 10 Na-lactate (day 2) 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 20

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 7.42 7.37 7.37 7.39 7.43 . 18 PH1_10 7.42 7.39 7.38 7.42 7.39 7.39 7.39 7.40 7.39 7.34 7.38 7.35 . 7.41 7.41 7.36 7.39 7.38 7.41 . 18

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 . 18 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 7.42 7.46 7.37 7.45 7.42 7.42 7.41 20 PH2_10

. 7.36 7.45 7.45 7.40 7.39 7.41 7.44 7.41 7.32 7.40 7.40 7.43 7.42 7.47 7.36 7.40 7.40 7.39 7.46 19 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 20 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 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 . 18 MIE '2002 6

The data univariate or long form MIE '2002 7 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 Venous blood PH levels 7.6 sample size 7.5 interaction PH 111 7.4 69 TIme 59 7.3 .00 10.00 7.2 N=

20.00 18 18 Saline 18 20 19 20 Lactate MIE '2002 9 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 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 MIE '2002 11 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 + 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 MIE '2002

13 The within-subjects covariance matrix covariance patterns for 3 time periods UN-Unstructured 12 12 13 2 12 2 23 13 23 32 CS-Compound Symmetry 2 12 12 12 2 2 2 2 1 1 1 2 2 2 2 1 1 1 VC-Variance Components

12 0 0 2 0 2 0 0 0 32 AR(1) - First-Order Autoregressive 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) univariate approach Unstructured multivariate approach

Allows data that are missing at random Estimates covariance parameters using a method of moments . Allows a wide variety of withinsubject covariance matrix UN-Unstructured VC-Variance Components CS-Compound Symmetry AR(1)-1th order autoregressive Estimates covariance parameters using restricted maximum likelihood, . MIE '2002 15 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 16 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 0.000 0.788 0.018 0.000

MIE '2002 18 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 .691 .658 .512 .874 1 .820 .600 .677 .691 .820 1 .381 .296 .658 .600 .381 1 .635 .512 .677 .296. 635 1. .243 .407 .006 .399 .720 PH2_20 .243

.407 .006 .399 720 1 D1 T0 D1 T10 D1 T20 D2 T0 D2 T10 D2 T20 MIE '2002 19 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 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 2, time 3 polynomial / summary ; Run; MIE '2002 21 TREAT Sphericity Assumed 1.569E-02 1 1.569E-02 11.277 Greenhouse-Geisser 1.569E-02 1.000 1.569E-02 11.277 Huynh-Feldt 1.569E-02 1.000 1.569E-02 11.277 Lower-bound 1.569E-02 1.000 1.569E-02 11.277 Error(TREAT) Sphericity Assumed 2.226E-02 16 1.391E-03 Greenhouse-Geisser 2.226E-02 16.000 1.391E-03 SphericityHuynh-Feldt test failed, a 2.226E-02 correction

be applied 16.000can1.391E-03 Tests of Within-Subj ects Effects Lower-bound 2.226E-02 16.000 1.391E-03 3 subjects are deleted because of missing value Measure: MEASURE_1 TIME Sphericity Assumed 2.109E-02 2Type III Sum 1.054E-02 20.718 Source of Squares df Mean Square F TREATMENT*TIME interaction is1.350 significant Greenhouse-Geisser 2.109E-02 1.562E-021 1.569E-02 20.718 11.277 TREAT Sphericity Assumed 1.569E-02 Greenhouse-Geisser 1.569E-02 1.000 1.569E-02 11.277 Huynh-Feldt 2.109E-02 1.430 1.569E-02 1.475E-02 20.718 11.277 Huynh-Feldt 1.000 1.569E-02 Lower-bound

1.000 1.569E-02 Lower-bound 2.109E-02 1.000 1.569E-02 2.109E-02 20.718 11.277 Error(TREAT) Sphericity Assumed 2.226E-02 16 1.391E-03 Greenhouse-Geisser 16.000 1.391E-03 Error(TIME) Sphericity Assumed 1.629E-02 32 2.226E-02 5.089E-04 Huynh-Feldt 2.226E-02 16.000 1.391E-03 Lower-bound 2.226E-02 16.000 Greenhouse-Geisser 1.629E-02 21.596 7.541E-04 1.391E-03 TIME Sphericity Assumed 2.109E-02 2 1.054E-02 20.718 Greenhouse-Geisser 1.350 1.562E-02 20.718 Huynh-Feldt 1.629E-02 22.875 2.109E-02 7.119E-04 Huynh-Feldt 2.109E-02

1.430 1.475E-02 20.718 Lower-bound Lower-bound 16.000 2.109E-02 1.629E-02 1.018E-03 1.000 2.109E-02 20.718 Error(TIME) Sphericity Assumed 1.629E-02 32 5.089E-04 TREAT * TIME Sphericity Assumed 1.227E-02 6.133E-03 14.171 Greenhouse-Geisser 2 1.629E-02 21.596 7.541E-04 Huynh-Feldt 1.629E-02 22.875 7.119E-04 Greenhouse-Geisser 1.227E-02 1.501 1.629E-02 8.174E-03 14.171 Lower-bound 16.000 1.018E-03 TREAT * TIME Sphericity Assumed 2 6.133E-03 Huynh-Feldt 1.227E-02 1.622 1.227E-02 7.564E-03 14.171 14.171 Greenhouse-Geisser 1.227E-02 1.501

8.174E-03 14.171 Huynh-Feldt 1.227E-02 1.622 7.564E-03 Lower-bound 1.227E-02 1.000 1.227E-02 1.227E-02 14.171 14.171 Lower-bound 1.000 1.227E-02 14.171 Error(TREAT*TIME) Sphericity Assumed 1.385E-02 32 4.328E-04 Error(TREAT*TIME) Sphericity Assumed 1.385E-02 32 4.328E-04 Greenhouse-Geisser 1.385E-02 24.012 5.768E-04 Huynh-Feldt Greenhouse-Geisser 1.385E-02 25.947 1.385E-02 24.012 5.768E-04 5.338E-04 Lower-bound 1.385E-02 16.000 8.656E-04 MIE '2002 Huynh-Feldt 1.385E-02 25.947 5.338E-04 .004

.004 .004 .004 GLM univariate assumptions and results (SPSS) .000 Sig. .000.004 .004 .000.004 .000.004 .000 .000 .000 .000 .000 .000 .000.000 .000 .002.000 .002 22 GLM multivariate results (SPSS) Multivariate Testsb Effect TREAT Pillai's Trace Wilks' Lambda Hotelling's Trace Roy's Largest Root TIME Pillai's Trace Wilks' Lambda Hotelling's Trace Roy's Largest Root TREAT * TIME Pillai's Trace Wilks' Lambda

Hotelling's Trace Roy's Largest Root Value .413 .587 .705 .705 .724 .276 2.620 2.620 .537 .463 1.160 1.160 F Hypothesis df a 11.277 1.000 11.277a 1.000 11.277a 1.000 11.277a 1.000 a 19.651 2.000 a 19.651 2.000 19.651a 2.000 19.651a 2.000 8.702a 2.000 8.702a 2.000 a 8.702 2.000 a 8.702

2.000 Error df 16.000 16.000 16.000 16.000 15.000 15.000 15.000 15.000 15.000 15.000 15.000 15.000 Sig. .004 .004 .004 .004 .000 .000 .000 .000 .003 .003 .003 .003 a. Exact statistic b. Design: Intercept Within Subjects Design: TREAT+TIME+TREAT*TIME MIE '2002 23 Plot in SPSS Estimated Marginal Means of MEASURE_1 Estimated Marginal Means of MEASURE_1 7.45 7.45 7.44 7.44

7.43 7.43 Estimated Marginal Means Estimated Marginal Means 7.42 7.42 7.41 7.41 7.40 7.40 TREAT TREAT 7.39 7.39 7.38 7.38 1 7.37 7.37 1 2 1 2 2 3 1 2 3 TIME TIME 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; run; SPSS 11.0 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 . MIE '2002 25 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. MIE '2002 26 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. MIE '2002 27 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 MIE '2002 28 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 [email protected](1), we assume equal correlation between treatments (UN) and AR(1) covariance structure between the three time points. [email protected](1): we assume the UN covariance matrix for the treatments and the AR(1) covariance matrix for the time effects MIE '2002 29

The [email protected](1) composite covariance model in SAS For each subject, we have the following covariance matrix: 12 12 1 12 1 2 1 12 1 2 1 2 12 1 2 1 1 2 2 2 1 2 1 1 2 12 @ 1 22 2 1

2 1 2 1 12 12 12 2 12 12 12 12 12 12 12 2 2 2 2 2 1 1 1 12 12 2 2 2 12 12 2 2 12 12 12 12 22 22 2 22 2 22 12 12 12 MIE '2002 12 2

12 12 22 2 22 2 2 30 Selecting the covariance structure Correlation matrix for a block using [email protected](1) covariance structure Row 1 2 3 4 5 6 COL1 1.00000000 0.73001496 0.53292185 0.22698641 0.16570348 0.12096602 COL2 0.73001496 1.00000000 0.73001496 0.16570348 0.22698641 0.16570348 Correlation between time Time 0 Time 0 1.00000000 Time 10 0.73001496 Time 20 0.53292185 COL3

COL4 0.53292185 0.73001496 1.00000000 0.12096602 0.16570348 0.22698641 Time 10 0.73001496 1.00000000 0.73001496 COL5 0.22698641 0.16570348 0.12096602 1.00000000 0.73001496 0.53292185 COL6 0.16570348 0.22698641 0.16570348 0.73001496 1.00000000 0.73001496 0.12096602 0.16570348 0.22698641 0.53292185 0.73001496 1.00000000 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 Covariance structure Information criteria (smaller-is-better forms). UN VC CS Number of parameters 21 -2 Restricted Log Likelihood -500.416 Akaike's Information Criterion (AIC) -458.416 Likelihood ratio test (comparison to UN) df diff AR(1) [email protected](1) 6 2 2 4 -405.118 -433.927 -429.208 -454.181 -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, [email protected](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 ([email protected]) 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 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 TREAT*TIME TREAT*TIME TREAT*TIME TREAT*TIME TREAT*TIME 1.00 1.00 1.00 2.00 2.00 2.00 0.00 0.00 10.00 0.00 0.00 10.00 1.00 1.00 1.00 2.00 2.00 2.00 10.00 20.00 20.00 10.00 20.00

20.00 -0.00668 -0.00888 -0.00219 -0.02260 -0.05895 -0.03635 0.004978 0.006548 0.004978 0.006852 0.008888 0.006852 33 33 33 33 33 33 -1.34 -1.36 -0.44 -3.30 -6.63 -5.30 0.1886 0.1844 0.6624 0.0023 <.0001 <.0001 TREAT*TIME TREAT*TIME TREAT*TIME 1.00 1.00 1.00 0.00 10.00

20.00 2.00 2.00 2.00 0.00 10.00 20.00 -0.00459 -0.02051 -0.05466 0.01018 0.01024 0.01018 33 33 33 -0.45 -2.00 -5.37 0.6551 0.0536 <.0001 7.6 7.5 PH 111 7.4 69 TIme 59 7.3

.00 10.00 7.2 N= 20.00 18 18 Saline 18 20 19 Lactate 20 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) 0.140 0.164 0.607 0.009 0.000 0.000 0.788 0.018 0.000 MIE '2002 35

Distribution of residuals using [email protected](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) 200 200 180 180 160 160 140 140 120 120 Time 0100

100 RRs 10 80 80 N= 20RRS 1-0 19 19 Saline 19 19 18 RRS 1-10 RRS 1-20 RRS 2-0 RRS 2-10 RRS 2-20 19 Lactate MIE '2002 38 Mean and SD of systolic blood pressure 180.00 160.00 140.00 Hgmm

120.00 100.00 Saline Lactate 80.00 60.00 40.00 20.00 0.00 N 19 0 19 19 10 19 20 Time (min) MIE '2002 39 The same figure with different scaling Mean of systolic blood pressure Mean of systolic blood pressure 150.00 150.00 148.00

146.89 148.00 146.00 146.00 144.00 Hgmm 140.00 138.89 139.74 140.26 140.61 Saline Lactate 138.00 Hgmm 142.05 142.00 142.00 134.00 134.00 132.00 132.00 19 0 19 19

10 19 Different sample size 18 20 130.00 Time (min) 138.17 138.50 141.11 Saline Lactate 138.67 138.00 136.00 N 19 140.61 140.00 136.00 130.00 144.83 144.00 N 18 0 18

18 10 18 18 20 18 Time (min) Equal sample size MIE '2002 40 GLM results (2 cases are deleted) GLM Multivariate (Wilks Lambda Sig): TREAT TIME TREAT*TIME 0.868 0.095 0.270 GLM Univariate (Spericity assumptions met) TREAT TIME TREAT*TIME 0.868 0.042 0.253 Is there a significant time effect? MIE '2002 41 Plot in SPSS GLM Estimated Marginal Means 148

Estimated Marginal Means 146 144 142 TREATMENT 140 Saline Lactate 138 0 10 20 TIME MIE '2002 42 Correlation matrix of systolic blood pressures BP1_0 BP1_10 BP1_20 BP2_0 BP2_10 BP2_20 BP1_0 1 .954 .893 .884 .619 .790 BP1_10 .954 1

.908 .842 .569 .776 BP1_20 BP2_0 .893 .884 .908 .842 1 .825 .825 1 .566 .755 .778 .791 BP2_10 BP2_20 .619 .790 .569 .776 .566 .778 .755 .791 1 .825 .825 1 Paired t-tests Sig. (2-tailed) 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 .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 21 -2 Restricted Log Likelihood 815.637 Akaike's Information Criterion (AIC) 857.637 Likelihood ratio test (comparison to UN) df diff p 6 968.25 978.1 AR(1) [email protected](1) 2 858.587 862.587 7 853.88 868.337 2

4 848.541 860.93 852.546 868.9 15 19 152.613 42.95 <0.0001 .001317 14 38.243 .000477 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 p-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 TREAT: 1.00 Saline TREAT: 200 2.00 Lactate 200 180 180 160 160 140 140 120 100

RRs RRs 120 100 -10 Time 0 10 20 30 80 -10 0 10 20 30 Time MIE '2002 48 SAS commands 1. 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 run; /s;

2. 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; 3. Mixed models with two additional effects (with random coefficients for patients and patients*time) proc mixed; class name treat; model rrs=treat time treat*time/s; random int time /sub=name type=un solution; run; MIE '2002 49 Regression lines by averaged by treatments 200 180 160 140 Treatment 120 RRs Lactate 100 Saline -10 0 10 20

30 Time 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 Intercept TIME Estimate 138.84 0.2579 Standard Error 3.0039 0.2323 DF t Value Pr > |t| 111 111 46.22 1.11

<.0001 0.2693 RRS=0.2579*time + 138.84 Type 3 Tests of Fixed Effects Effect TIME Num Den DF DF F Value 1 111 1.23 Pr > F 0.2693 The time-effect is not significant MIE '2002 51 Results I: fixed effects (linear regression) 200 180 160 RRS=0.2579*time + 138.84 140 120 Treatment Lactate 100 RRs

Saline 80 Total Population -10 Time 0 10 20 30 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: Residual variance: 88.38 -2 Res Log Likelihood 882.9 Solution for Fixed Effects Effect Intercept

TIME Estimate 139.03 0.2579 Standard Error 4.4939 0.1078 Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value TIME 1 18 5.72 DF 18 18 t Value 30.94 2.39 Pr > |t| <.0001 0.0279 RRS=0.2579*time + 139.03 Pr > F 0.0279 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 Num Den Effect DF DF F Value TREAT TIME TIME*TREAT 1 1 1 Residual variance: 88.38 73 18 73 0.53 5.71 1.74 Pr > F The time-effect is significant 0.4703 0.0280 0.1919

We decide to use MODEL II The other two effects are not significant 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 1. H. Brown and R. Prescott, Applied Mixed Models in Medicine. Wiley, 2001.

2. SAS Institute, Inc: The MIXED procedure in SAS/STAT Software: Changes and Enhancements through Release 6.11. Copyright 1996 by SAS Institute Inc., Cary, NC 27513. 3. T. Park, and Y.J. Lee,: Covariance models for nested repeated measures data: analysis of ovarian steroid secretion data. Statistics in Medicine 21 (2002) 134-164 4. SPSS Advanced Models 9.0. Copyright 1996 by SPSS Inc P. 5. 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. 6. R. Wolfinger and M. Chang, Comparing the SAS GLM and MIXED Procedures for Repeated Measures, SAS Institute Inc., Cary, NC. http:// www.ats.ucla.edu/stat/sas/library/ MIE '2002 57