Analysis of Repeated Measures / Longitudinal Data
The term repeated measures refer to multiple responses taken in sequence on the same experimental unit, such as an animal or a field plot. Generally, the responses are taken over time, as in weekly weights to estimate growth curves. However, the repeated measures could also be taken in spatial sequence, such as observations along a transect in a plot. The typical repeated measures experiment consists of experimental units (subjects, animals, trees, plots, etc.) randomly assigned to treatment groups, and with responses measured on each unit over time. This is a factorial experiment with two factors: treatment (“between subjects” factor) and time (“within subjects” factor). These experiments have been used commonly in animal, plant and human research for a long time, but in the last two decades there have been big developments in methodological and computational issues that permit to analyze them effectively and efficiently.
The main objectives of repeated measures analysis are to examine and compare responses over time for different treatments. This could involve comparisons of treatments at specific times, or averaged over time; or could involve comparisons of times in a given treatment. Like in any factorial experiment, these types of research questions can be written as interaction, main effects or simple effects hypotheses.
The main difference with other factorial experiments is the fact that observations taken on the same experimental unit over time are correlated, and thus the assumption of independent observations cannot be used. Therefore, besides the usual hypotheses tested in factorials, we need to pay attention to the correlation pattern among observations from the same unit, and estimate these correlations.
Responses measured on the same unitcould be correlated because they contain a common contribution from the unit. Furthermore, responses from the same unitwhich are closer in time tend be more highly correlated than responses farther apart in time. Another feature that is common in repeated measures of growth, etc. is the fact that variances tend to increase over time. All these features are combined to produce a complicated correlation structure of the repeated measures. This structure needs to be estimated and considered in the tests for the fixed effects.
The simplest model that we can use for repeated measures considers the effects of treatment, time and their interaction (usually fixed), the (random) effect of the experimental unit (typically the animal, tree, plot, etc.) and the (random) experimental error from different observations at the same unit (at different times):
wherei represents treatments, j represents time and k is the replicate. The main difference between this model and the one used in the split-plot design is the fact that the are not independent with constant variance, but the errors from the same individual (same indices i and k) have a covariance matrix that needs to be estimated. This covariance matrix has the following structure:
There are several statistical methods used for analyzing repeated measures data:
- separate analysis at each time point.
- factorial analysis ignoring correlation.
- factorial analysis assuming a split-plot design.
- factorial analysis assuming other correlation structures.
- The separate analysis at each time point does not give any information about the time effects nor the interaction time-by-treatment, and thus is very limited. It is used sometimes to analyze the effect of the treatment at the last time point observed (for example, weight gain after 4 months).
- The factorial analysis ignoring correlation is incorrect, and will generally result in finding incorrectly too many significant differences.
- The factorial analysis assuming a split-plot design was used before the availability of software for modeling the covariance structure. It considers that treatment is the factor applied to “whole plots” (the experimental units) and time is the factor applied to “subplots”. This analysis would be correct if we could randomize time, but since time comes in a sequence, it is generally not correct. The analysis is the same that we would obtain if we assume that the variances are the same and the correlations are all equal. This structure is called “compound symmetry”:
- The factorial analysis assuming correlation structures is the most efficient analysis, and permits the use different covariance patterns. Its main problem is that there are (in general) no exact procedures, and several approximations must be used. The method for estimating the covariance structure is the same used for mixed models: REML. For some of these structures we need to pool both random effects (d and e) into one effect (e*) and specify the covariance structure for this “error”. The reason for this is that there is no way of distinguishing between the effects (this is called an identifiability problem). This is what happens, for example, in the compound symmetry structure or the unstructured covariance: if we use one of these, we do not specify a unit effectd.
The most commonly used covariance structures are:
- Compound symmetry:
- First order autoregressive:
- First order autoregressive with heterogeneous variances:
4. Unstructured:
Since we do not want to use an incorrect structure, and an unnecessarily complex structure will decrease the power of the tests, we need to be sure that we are using a reasonable structure before proceeding with the tests of interest. One simple way of deciding between alternative structures, is the use of penalized likelihood criteria, like AIC (Akaike criterion) and BIC (Schawarz criterion), to select the most appropriate structure. Typically we fit models with different structures and then choose the one having the smallest AIC or BIC. Once we have this model, we make inferences for treatments, times, and the interaction time x treatment.
/* The following data are weights of turkeys (lb) under
4 different diets (control, control + lysine, commercial
formulation A, commercial formulation B), during 8 months */
odsrtf;
data growth;
input rep treat @@;
do time=1to8;
input weight @@;
output;
end;
datalines;
1 1 3.93 7.76 10.81 12.69 12.90 13.35 13.12 18.72
2 1 4.29 8.38 10.54 11.02 11.23 11.62 10.76 18.10
3 1 4.62 8.60 11.09 12.18 13.29 11.84 11.42 17.19
4 1 5.69 10.27 13.88 14.78 13.95 13.99 13.28 18.98
5 1 4.67 7.75 11.15 11.86 11.93 12.80 11.95 19.12
6 1 5.57 9.03 11.16 13.66 14.44 14.08 12.69 17.27
7 1 5.64 7.49 9.50 10.56 11.42 11.57 11.54 17.97
8 1 6.14 9.97 11.93 13.09 13.17 11.49 10.72 15.93
1 2 6.64 9.31 12.20 13.37 13.70 13.87 11.80 19.03
2 2 6.05 10.23 12.83 14.08 13.86 13.14 11.39 16.18
3 2 6.87 10.82 11.62 12.26 13.49 13.31 11.54 17.03
4 2 5.08 7.90 10.73 12.36 13.56 13.20 14.05 21.91
5 2 6.17 9.86 12.01 12.99 13.16 12.80 12.16 17.69
6 2 5.32 8.76 11.97 13.55 14.09 13.12 12.57 18.94
7 2 5.59 9.89 12.75 13.52 15.44 14.26 13.36 19.71
8 2 4.75 8.69 11.47 12.60 13.27 12.07 11.29 18.25
1 3 7.04 13.23 17.56 20.10 21.08 22.32 21.45 33.85
2 3 8.52 14.51 19.22 20.76 22.53 23.33 23.60 40.01
3 3 8.10 13.33 17.15 18.90 19.70 21.07 20.29 31.65
4 3 8.24 13.96 18.41 20.21 22.10 23.13 23.91 37.18
5 3 7.71 13.42 17.35 18.75 20.35 21.03 21.54 34.19
6 3 8.03 14.61 19.38 21.49 22.43 23.42 22.34 36.19
7 3 8.64 13.74 17.73 20.08 21.05 21.57 21.01 34.67
8 3 7.46 13.34 17.72 20.48 21.92 23.24 22.29 35.12
1 4 8.22 14.15 19.35 22.58 24.48 24.96 25.50 42.50
2 4 8.22 13.77 18.92 21.33 22.68 23.22 23.37 38.22
3 4 8.35 14.65 19.32 21.87 23.25 23.46 23.69 37.37
4 4 8.73 14.72 19.58 22.56 24.30 24.87 24.87 40.64
5 4 8.04 14.75 18.68 20.99 22.11 23.25 23.32 38.42
6 4 8.31 14.03 18.77 21.79 23.60 24.15 23.78 39.49
7 4 7.63 13.88 18.56 21.95 22.96 23.44 23.96 37.78
8 4 8.40 14.01 18.01 20.49 22.49 24.09 23.85 39.01
run;
procglimmixdata=growth;
class treat time rep;
model weight = treat time treat*time;
random time / residualsubject=rep(treat) type=un vvcorr;
*OBSERVAR QUE LA VARIABLE time NO ES ALEATORIA, SINO QUE
SE PONE AQUÍ PARA INDICAR EL ORDEN DE LAS OBSERVACIONES;
odsselectFitStatistics v vcorr;
title'Matriz de covarianza sin estructura'; run;
procglimmixdata=growth;
class treat time rep;
model weight = treat time treat*time;
random time / residualsubject=rep(treat) type=csvvcorr;
odsselectFitStatistics v vcorr;
title'Simetríacompuesta'; run;
procglimmixdata=growth;
class treat time rep;
model weight = treat time treat*time;
random time / residualsubject=rep(treat) type=cshvvcorr;
odsselectFitStatistics v vcorr;
title'Simetría compuesta heteroscedástica'; run;
procglimmixdata=growth;
class treat time rep;
model weight = treat time treat*time;
random time / residualsubject=rep(treat) type=ar(1) vvcorr;
odsselectFitStatistics v vcorr;
title'Autorregresivo de orden 1'; run;
procglimmixdata=growth;
class treat time rep;
model weight = treat time treat*time;
random time / residualsubject=rep(treat) type=arh(1) vvcorr;
odsselectFitStatistics v vcorr;
title'Autorregresivo de orden 1 Heteroscedástico'; run;
procglimmixdata=growth plots=studentpanel;
class treat time rep;
model weight = treat time treat*time;
random time / residualsubject=rep(treat) type=arh(1);
slice treat*time / sliceby=time adjust=tukeylines ;
lsmeans treat*time / plot=meanplot(sliceby=treat join) ;
odsrtfexcludediffs;
title'Análisis final Autorregresivo de orden 1 Heteroscedástico '; run;
odsrtfclose;
01:53 Sunday, September 16, 2018 1
Matriz de covarianza sin estructuraThe GLIMMIX Procedure
Fit Statistics
-2 Res Log Likelihood / 451.52
AIC (smaller is better) / 523.52
AICC (smaller is better) / 537.77
BIC (smaller is better) / 576.29
CAIC (smaller is better) / 612.29
HQIC (smaller is better) / 541.01
Generalized Chi-Square / 224.00
Gener. Chi-Square / DF / 1.00
Estimated V Matrix for rep(treat) 1 1
Row / Col1 / Col2 / Col3 / Col4 / Col5 / Col6 / Col7 / Col8
1 / 0.3947 / 0.3216 / 0.1983 / 0.1371 / 0.1256 / 0.08706 / -0.03833 / -0.07821
2 / 0.3216 / 0.6090 / 0.5162 / 0.4220 / 0.3406 / 0.2118 / 0.01196 / -0.1584
3 / 0.1983 / 0.5162 / 0.7615 / 0.7217 / 0.5758 / 0.4432 / 0.3234 / 0.4017
4 / 0.1371 / 0.4220 / 0.7217 / 0.9223 / 0.7710 / 0.6058 / 0.4622 / 0.4383
5 / 0.1256 / 0.3406 / 0.5758 / 0.7710 / 0.9104 / 0.6843 / 0.6551 / 0.8260
6 / 0.08706 / 0.2118 / 0.4432 / 0.6058 / 0.6843 / 0.8040 / 0.7047 / 0.9698
7 / -0.03833 / 0.01196 / 0.3234 / 0.4622 / 0.6551 / 0.7047 / 1.0297 / 1.5234
8 / -0.07821 / -0.1584 / 0.4017 / 0.4383 / 0.8260 / 0.9698 / 1.5234 / 3.3392
Estimated V Correlation Matrix for rep(treat) 1 1
Row / Col1 / Col2 / Col3 / Col4 / Col5 / Col6 / Col7 / Col8
1 / 1.0000 / 0.6559 / 0.3617 / 0.2272 / 0.2096 / 0.1545 / -0.06012 / -0.06813
2 / 0.6559 / 1.0000 / 0.7579 / 0.5632 / 0.4575 / 0.3027 / 0.01510 / -0.1111
3 / 0.3617 / 0.7579 / 1.0000 / 0.8612 / 0.6916 / 0.5664 / 0.3652 / 0.2519
4 / 0.2272 / 0.5632 / 0.8612 / 1.0000 / 0.8414 / 0.7035 / 0.4743 / 0.2498
5 / 0.2096 / 0.4575 / 0.6916 / 0.8414 / 1.0000 / 0.7998 / 0.6766 / 0.4737
6 / 0.1545 / 0.3027 / 0.5664 / 0.7035 / 0.7998 / 1.0000 / 0.7744 / 0.5919
7 / -0.06012 / 0.01510 / 0.3652 / 0.4743 / 0.6766 / 0.7744 / 1.0000 / 0.8216
8 / -0.06813 / -0.1111 / 0.2519 / 0.2498 / 0.4737 / 0.5919 / 0.8216 / 1.0000
01:53 Sunday, September 16, 2018 1
SimetríacompuestaThe GLIMMIX Procedure
Fit Statistics
-2 Res Log Likelihood / 660.76
AIC (smaller is better) / 664.76
AICC (smaller is better) / 664.82
BIC (smaller is better) / 667.70
CAIC (smaller is better) / 669.70
HQIC (smaller is better) / 665.74
Generalized Chi-Square / 147.95
Gener. Chi-Square / DF / 0.66
Estimated V Matrix for rep(treat) 1 1
Row / Col1 / Col2 / Col3 / Col4 / Col5 / Col6 / Col7 / Col8
1 / 1.0964 / 0.4358 / 0.4358 / 0.4358 / 0.4358 / 0.4358 / 0.4358 / 0.4358
2 / 0.4358 / 1.0964 / 0.4358 / 0.4358 / 0.4358 / 0.4358 / 0.4358 / 0.4358
3 / 0.4358 / 0.4358 / 1.0964 / 0.4358 / 0.4358 / 0.4358 / 0.4358 / 0.4358
4 / 0.4358 / 0.4358 / 0.4358 / 1.0964 / 0.4358 / 0.4358 / 0.4358 / 0.4358
5 / 0.4358 / 0.4358 / 0.4358 / 0.4358 / 1.0964 / 0.4358 / 0.4358 / 0.4358
6 / 0.4358 / 0.4358 / 0.4358 / 0.4358 / 0.4358 / 1.0964 / 0.4358 / 0.4358
7 / 0.4358 / 0.4358 / 0.4358 / 0.4358 / 0.4358 / 0.4358 / 1.0964 / 0.4358
8 / 0.4358 / 0.4358 / 0.4358 / 0.4358 / 0.4358 / 0.4358 / 0.4358 / 1.0964
Estimated V Correlation Matrix for rep(treat) 1 1
Row / Col1 / Col2 / Col3 / Col4 / Col5 / Col6 / Col7 / Col8
1 / 1.0000 / 0.3975 / 0.3975 / 0.3975 / 0.3975 / 0.3975 / 0.3975 / 0.3975
2 / 0.3975 / 1.0000 / 0.3975 / 0.3975 / 0.3975 / 0.3975 / 0.3975 / 0.3975
3 / 0.3975 / 0.3975 / 1.0000 / 0.3975 / 0.3975 / 0.3975 / 0.3975 / 0.3975
4 / 0.3975 / 0.3975 / 0.3975 / 1.0000 / 0.3975 / 0.3975 / 0.3975 / 0.3975
5 / 0.3975 / 0.3975 / 0.3975 / 0.3975 / 1.0000 / 0.3975 / 0.3975 / 0.3975
6 / 0.3975 / 0.3975 / 0.3975 / 0.3975 / 0.3975 / 1.0000 / 0.3975 / 0.3975
7 / 0.3975 / 0.3975 / 0.3975 / 0.3975 / 0.3975 / 0.3975 / 1.0000 / 0.3975
8 / 0.3975 / 0.3975 / 0.3975 / 0.3975 / 0.3975 / 0.3975 / 0.3975 / 1.0000
01:53 Sunday, September 16, 2018 1
SimetríacompuestaheteroscedásticaThe GLIMMIX Procedure
Fit Statistics
-2 Res Log Likelihood / 598.06
AIC (smaller is better) / 616.06
AICC (smaller is better) / 616.90
BIC (smaller is better) / 629.26
CAIC (smaller is better) / 638.26
HQIC (smaller is better) / 620.44
Generalized Chi-Square / 224.00
Gener. Chi-Square / DF / 1.00
Estimated V Matrix for rep(treat) 1 1
Row / Col1 / Col2 / Col3 / Col4 / Col5 / Col6 / Col7 / Col8
1 / 0.5182 / 0.2736 / 0.2790 / 0.3047 / 0.2981 / 0.2855 / 0.3431 / 0.6596
2 / 0.2736 / 0.6656 / 0.3162 / 0.3453 / 0.3379 / 0.3236 / 0.3889 / 0.7475
3 / 0.2790 / 0.3162 / 0.6923 / 0.3522 / 0.3446 / 0.3300 / 0.3966 / 0.7623
4 / 0.3047 / 0.3453 / 0.3522 / 0.8258 / 0.3764 / 0.3604 / 0.4331 / 0.8326
5 / 0.2981 / 0.3379 / 0.3446 / 0.3764 / 0.7906 / 0.3527 / 0.4238 / 0.8147
6 / 0.2855 / 0.3236 / 0.3300 / 0.3604 / 0.3527 / 0.7252 / 0.4059 / 0.7802
7 / 0.3431 / 0.3889 / 0.3966 / 0.4331 / 0.4238 / 0.4059 / 1.0472 / 0.9376
8 / 0.6596 / 0.7475 / 0.7623 / 0.8326 / 0.8147 / 0.7802 / 0.9376 / 3.8691
Estimated V Correlation Matrix for rep(treat) 1 1
Row / Col1 / Col2 / Col3 / Col4 / Col5 / Col6 / Col7 / Col8
1 / 1.0000 / 0.4658 / 0.4658 / 0.4658 / 0.4658 / 0.4658 / 0.4658 / 0.4658
2 / 0.4658 / 1.0000 / 0.4658 / 0.4658 / 0.4658 / 0.4658 / 0.4658 / 0.4658
3 / 0.4658 / 0.4658 / 1.0000 / 0.4658 / 0.4658 / 0.4658 / 0.4658 / 0.4658
4 / 0.4658 / 0.4658 / 0.4658 / 1.0000 / 0.4658 / 0.4658 / 0.4658 / 0.4658
5 / 0.4658 / 0.4658 / 0.4658 / 0.4658 / 1.0000 / 0.4658 / 0.4658 / 0.4658
6 / 0.4658 / 0.4658 / 0.4658 / 0.4658 / 0.4658 / 1.0000 / 0.4658 / 0.4658
7 / 0.4658 / 0.4658 / 0.4658 / 0.4658 / 0.4658 / 0.4658 / 1.0000 / 0.4658
8 / 0.4658 / 0.4658 / 0.4658 / 0.4658 / 0.4658 / 0.4658 / 0.4658 / 1.0000
01:53 Sunday, September 16, 2018 1
Autorregresivo de orden 1The GLIMMIX Procedure
Fit Statistics
-2 Res Log Likelihood / 552.07
AIC (smaller is better) / 556.07
AICC (smaller is better) / 556.12
BIC (smaller is better) / 559.00
CAIC (smaller is better) / 561.00
HQIC (smaller is better) / 557.04
Generalized Chi-Square / 304.09
Gener. Chi-Square / DF / 1.36
Estimated V Matrix for rep(treat) 1 1
Row / Col1 / Col2 / Col3 / Col4 / Col5 / Col6 / Col7 / Col8
1 / 1.3576 / 1.1131 / 0.9126 / 0.7482 / 0.6135 / 0.5030 / 0.4124 / 0.3381
2 / 1.1131 / 1.3576 / 1.1131 / 0.9126 / 0.7482 / 0.6135 / 0.5030 / 0.4124
3 / 0.9126 / 1.1131 / 1.3576 / 1.1131 / 0.9126 / 0.7482 / 0.6135 / 0.5030
4 / 0.7482 / 0.9126 / 1.1131 / 1.3576 / 1.1131 / 0.9126 / 0.7482 / 0.6135
5 / 0.6135 / 0.7482 / 0.9126 / 1.1131 / 1.3576 / 1.1131 / 0.9126 / 0.7482
6 / 0.5030 / 0.6135 / 0.7482 / 0.9126 / 1.1131 / 1.3576 / 1.1131 / 0.9126
7 / 0.4124 / 0.5030 / 0.6135 / 0.7482 / 0.9126 / 1.1131 / 1.3576 / 1.1131
8 / 0.3381 / 0.4124 / 0.5030 / 0.6135 / 0.7482 / 0.9126 / 1.1131 / 1.3576
Estimated V Correlation Matrix for rep(treat) 1 1
Row / Col1 / Col2 / Col3 / Col4 / Col5 / Col6 / Col7 / Col8
1 / 1.0000 / 0.8199 / 0.6722 / 0.5512 / 0.4519 / 0.3705 / 0.3038 / 0.2491
2 / 0.8199 / 1.0000 / 0.8199 / 0.6722 / 0.5512 / 0.4519 / 0.3705 / 0.3038
3 / 0.6722 / 0.8199 / 1.0000 / 0.8199 / 0.6722 / 0.5512 / 0.4519 / 0.3705
4 / 0.5512 / 0.6722 / 0.8199 / 1.0000 / 0.8199 / 0.6722 / 0.5512 / 0.4519
5 / 0.4519 / 0.5512 / 0.6722 / 0.8199 / 1.0000 / 0.8199 / 0.6722 / 0.5512
6 / 0.3705 / 0.4519 / 0.5512 / 0.6722 / 0.8199 / 1.0000 / 0.8199 / 0.6722
7 / 0.3038 / 0.3705 / 0.4519 / 0.5512 / 0.6722 / 0.8199 / 1.0000 / 0.8199
8 / 0.2491 / 0.3038 / 0.3705 / 0.4519 / 0.5512 / 0.6722 / 0.8199 / 1.0000
01:53 Sunday, September 16, 2018 1
Autorregresivo de orden 1 HeteroscedásticoThe GLIMMIX Procedure
Fit Statistics
-2 Res Log Likelihood / 486.03
AIC (smaller is better) / 504.03
AICC (smaller is better) / 504.87
BIC (smaller is better) / 517.22
CAIC (smaller is better) / 526.22
HQIC (smaller is better) / 508.40
Generalized Chi-Square / 224.00
Gener. Chi-Square / DF / 1.00
Estimated V Matrix for rep(treat) 1 1
Row / Col1 / Col2 / Col3 / Col4 / Col5 / Col6 / Col7 / Col8
1 / 0.4881 / 0.4659 / 0.3684 / 0.3013 / 0.2390 / 0.1824 / 0.1623 / 0.2287
2 / 0.4659 / 0.7197 / 0.5692 / 0.4654 / 0.3692 / 0.2817 / 0.2507 / 0.3533
3 / 0.3684 / 0.5692 / 0.7287 / 0.5959 / 0.4727 / 0.3607 / 0.3209 / 0.4523
4 / 0.3013 / 0.4654 / 0.5959 / 0.7887 / 0.6257 / 0.4774 / 0.4248 / 0.5987
5 / 0.2390 / 0.3692 / 0.4727 / 0.6257 / 0.8035 / 0.6131 / 0.5455 / 0.7687
6 / 0.1824 / 0.2817 / 0.3607 / 0.4774 / 0.6131 / 0.7572 / 0.6738 / 0.9495
7 / 0.1623 / 0.2507 / 0.3209 / 0.4248 / 0.5455 / 0.6738 / 0.9705 / 1.3676
8 / 0.2287 / 0.3533 / 0.4523 / 0.5987 / 0.7687 / 0.9495 / 1.3676 / 3.1197
Estimated V Correlation Matrix for rep(treat) 1 1
Row / Col1 / Col2 / Col3 / Col4 / Col5 / Col6 / Col7 / Col8
1 / 1.0000 / 0.7860 / 0.6178 / 0.4856 / 0.3816 / 0.3000 / 0.2358 / 0.1853
2 / 0.7860 / 1.0000 / 0.7860 / 0.6178 / 0.4856 / 0.3816 / 0.3000 / 0.2358
3 / 0.6178 / 0.7860 / 1.0000 / 0.7860 / 0.6178 / 0.4856 / 0.3816 / 0.3000
4 / 0.4856 / 0.6178 / 0.7860 / 1.0000 / 0.7860 / 0.6178 / 0.4856 / 0.3816
5 / 0.3816 / 0.4856 / 0.6178 / 0.7860 / 1.0000 / 0.7860 / 0.6178 / 0.4856
6 / 0.3000 / 0.3816 / 0.4856 / 0.6178 / 0.7860 / 1.0000 / 0.7860 / 0.6178
7 / 0.2358 / 0.3000 / 0.3816 / 0.4856 / 0.6178 / 0.7860 / 1.0000 / 0.7860
8 / 0.1853 / 0.2358 / 0.3000 / 0.3816 / 0.4856 / 0.6178 / 0.7860 / 1.0000
Model Information
Data Set / WORK.GROWTH
Response Variable / weight
Response Distribution / Gaussian
Link Function / Identity
Variance Function / Default
Variance Matrix Blocked By / rep(treat)
Estimation Technique / Restricted Maximum Likelihood
Degrees of Freedom Method / Between-Within
Class Level Information
Class / Levels / Values
treat / 4 / 1 2 3 4
time / 8 / 1 2 3 4 5 6 7 8
rep / 8 / 1 2 3 4 5 6 7 8
Number of Observations Read / 256
Number of Observations Used / 256
Dimensions
R-side Cov. Parameters / 9
Columns in X / 45
Columns in Z per Subject / 0
Subjects (Blocks in V) / 32
Max Obs per Subject / 8
Fit Statistics
-2 Res Log Likelihood / 486.03
AIC (smaller is better) / 504.03
AICC (smaller is better) / 504.87
BIC (smaller is better) / 517.22
CAIC (smaller is better) / 526.22
HQIC (smaller is better) / 508.40
Generalized Chi-Square / 224.00
Gener. Chi-Square / DF / 1.00
Covariance Parameter Estimates
CovParm / Subject / Estimate / Standard
Error
Var(1) / rep(treat) / 0.4881 / 0.1242
Var(2) / rep(treat) / 0.7197 / 0.1765
Var(3) / rep(treat) / 0.7287 / 0.1734
Var(4) / rep(treat) / 0.7887 / 0.1839
Var(5) / rep(treat) / 0.8035 / 0.1873
Var(6) / rep(treat) / 0.7572 / 0.1769
Var(7) / rep(treat) / 0.9705 / 0.2229
Var(8) / rep(treat) / 3.1197 / 0.7334
ARH(1) / rep(treat) / 0.7860 / 0.03770
Type III Tests of Fixed Effects
Effect / Num DF / Den DF / F Value / PrF
treat / 3 / 28 / 396.26 / <.0001
time / 7 / 196 / 1150.92 / <.0001
treat*time / 21 / 196 / 41.40 / <.0001
treat*time Least Squares Means
treat / time / Estimate / Standard
Error / DF / tValue / Pr > |t|
1 / 1 / 5.0687 / 0.2470 / 196 / 20.52 / <.0001
1 / 2 / 8.6562 / 0.2999 / 196 / 28.86 / <.0001
1 / 3 / 11.2575 / 0.3018 / 196 / 37.30 / <.0001
1 / 4 / 12.4800 / 0.3140 / 196 / 39.75 / <.0001
1 / 5 / 12.7913 / 0.3169 / 196 / 40.36 / <.0001
1 / 6 / 12.5925 / 0.3076 / 196 / 40.93 / <.0001
1 / 7 / 11.9350 / 0.3483 / 196 / 34.27 / <.0001
1 / 8 / 17.9100 / 0.6245 / 196 / 28.68 / <.0001
2 / 1 / 5.8087 / 0.2470 / 196 / 23.52 / <.0001
2 / 2 / 9.4325 / 0.2999 / 196 / 31.45 / <.0001
2 / 3 / 11.9475 / 0.3018 / 196 / 39.59 / <.0001
2 / 4 / 13.0913 / 0.3140 / 196 / 41.69 / <.0001
2 / 5 / 13.8213 / 0.3169 / 196 / 43.61 / <.0001
2 / 6 / 13.2213 / 0.3076 / 196 / 42.98 / <.0001
2 / 7 / 12.2700 / 0.3483 / 196 / 35.23 / <.0001
2 / 8 / 18.5925 / 0.6245 / 196 / 29.77 / <.0001
3 / 1 / 7.9675 / 0.2470 / 196 / 32.26 / <.0001
3 / 2 / 13.7675 / 0.2999 / 196 / 45.90 / <.0001
3 / 3 / 18.0650 / 0.3018 / 196 / 59.86 / <.0001
3 / 4 / 20.0962 / 0.3140 / 196 / 64.00 / <.0001
3 / 5 / 21.3950 / 0.3169 / 196 / 67.51 / <.0001
3 / 6 / 22.3888 / 0.3076 / 196 / 72.77 / <.0001
3 / 7 / 22.0538 / 0.3483 / 196 / 63.32 / <.0001
3 / 8 / 35.3575 / 0.6245 / 196 / 56.62 / <.0001
4 / 1 / 8.2375 / 0.2470 / 196 / 33.35 / <.0001
4 / 2 / 14.2450 / 0.2999 / 196 / 47.49 / <.0001
4 / 3 / 18.8988 / 0.3018 / 196 / 62.62 / <.0001
4 / 4 / 21.6950 / 0.3140 / 196 / 69.09 / <.0001
4 / 5 / 23.2337 / 0.3169 / 196 / 73.31 / <.0001
4 / 6 / 23.9300 / 0.3076 / 196 / 77.78 / <.0001
4 / 7 / 24.0425 / 0.3483 / 196 / 69.03 / <.0001
4 / 8 / 39.1787 / 0.6245 / 196 / 62.74 / <.0001
F Test for treat*time Least Squares Means Slice
Slice / Num DF / Den DF / F Value / PrF
time 1 / 3 / 196 / 40.46 / <.0001
SimpleDifferencesoftreat*timeLeastSquaresMeans
AdjustmentforMultipleComparisons:Tukey
Slice / treat / _treat / Estimate / Standard Error / DF / tValue / Pr > |t| / Adj P
time 1 / 1 / 2 / -0.7400 / 0.3493 / 196 / -2.12 / 0.0354 / 0.1509
time 1 / 1 / 3 / -2.8988 / 0.3493 / 196 / -8.30 / <.0001 / <.0001
time 1 / 1 / 4 / -3.1688 / 0.3493 / 196 / -9.07 / <.0001 / <.0001
time 1 / 2 / 3 / -2.1588 / 0.3493 / 196 / -6.18 / <.0001 / <.0001
time 1 / 2 / 4 / -2.4288 / 0.3493 / 196 / -6.95 / <.0001 / <.0001
time 1 / 3 / 4 / -0.2700 / 0.3493 / 196 / -0.77 / 0.4405 / 0.8666
TukeyGroupingfortreat*timeLeastSquaresMeansSlice(Alpha=0.05)
LS-means with the same letter are not significantly different.
Slice / treat / Estimate
time 1 / 4 / 8.2375 / A
time 1 / A
time 1 / 3 / 7.9675 / A
time 1
time 1 / 2 / 5.8087 / B
time 1 / B
time 1 / 1 / 5.0687 / B
F Test for treat*time Least Squares Means Slice
Slice / Num DF / Den DF / F Value / PrF
time 2 / 3 / 196 / 92.76 / <.0001
SimpleDifferencesoftreat*timeLeastSquaresMeans
AdjustmentforMultipleComparisons:Tukey
Slice / treat / _treat / Estimate / Standard Error / DF / tValue / Pr > |t| / Adj P
time 2 / 1 / 2 / -0.7763 / 0.4242 / 196 / -1.83 / 0.0688 / 0.2624
time 2 / 1 / 3 / -5.1113 / 0.4242 / 196 / -12.05 / <.0001 / <.0001
time 2 / 1 / 4 / -5.5888 / 0.4242 / 196 / -13.18 / <.0001 / <.0001
time 2 / 2 / 3 / -4.3350 / 0.4242 / 196 / -10.22 / <.0001 / <.0001
time 2 / 2 / 4 / -4.8125 / 0.4242 / 196 / -11.35 / <.0001 / <.0001
time 2 / 3 / 4 / -0.4775 / 0.4242 / 196 / -1.13 / 0.2617 / 0.6741
TukeyGroupingfortreat*timeLeastSquaresMeansSlice(Alpha=0.05)
LS-means with the same letter are not significantly different.
Slice / treat / Estimate
time 2 / 4 / 14.2450 / A
time 2 / A
time 2 / 3 / 13.7675 / A
time 2
time 2 / 2 / 9.4325 / B
time 2 / B
time 2 / 1 / 8.6562 / B
F Test for treat*time Least Squares Means Slice
Slice / Num DF / Den DF / F Value / PrF
time 3 / 3 / 196 / 175.33 / <.0001
SimpleDifferencesoftreat*timeLeastSquaresMeans
AdjustmentforMultipleComparisons:Tukey
Slice / treat / _treat / Estimate / Standard Error / DF / tValue / Pr > |t| / Adj P
time 3 / 1 / 2 / -0.6900 / 0.4268 / 196 / -1.62 / 0.1076 / 0.3717
time 3 / 1 / 3 / -6.8075 / 0.4268 / 196 / -15.95 / <.0001 / <.0001
time 3 / 1 / 4 / -7.6413 / 0.4268 / 196 / -17.90 / <.0001 / <.0001
time 3 / 2 / 3 / -6.1175 / 0.4268 / 196 / -14.33 / <.0001 / <.0001
time 3 / 2 / 4 / -6.9513 / 0.4268 / 196 / -16.29 / <.0001 / <.0001
time 3 / 3 / 4 / -0.8338 / 0.4268 / 196 / -1.95 / 0.0522 / 0.2095
TukeyGroupingfortreat*timeLeastSquaresMeansSlice(Alpha=0.05)
LS-means with the same letter are not significantly different.
Slice / treat / Estimate
time 3 / 4 / 18.8988 / A
time 3 / A
time 3 / 3 / 18.0650 / A
time 3
time 3 / 2 / 11.9475 / B
time 3 / B
time 3 / 1 / 11.2575 / B
F Test for treat*time Least Squares Means Slice
Slice / Num DF / Den DF / F Value / PrF
time 4 / 3 / 196 / 227.33 / <.0001
SimpleDifferencesoftreat*timeLeastSquaresMeans
AdjustmentforMultipleComparisons:Tukey
Slice / treat / _treat / Estimate / Standard Error / DF / tValue / Pr > |t| / Adj P
time 4 / 1 / 2 / -0.6112 / 0.4440 / 196 / -1.38 / 0.1702 / 0.5156
time 4 / 1 / 3 / -7.6162 / 0.4440 / 196 / -17.15 / <.0001 / <.0001
time 4 / 1 / 4 / -9.2150 / 0.4440 / 196 / -20.75 / <.0001 / <.0001
time 4 / 2 / 3 / -7.0050 / 0.4440 / 196 / -15.78 / <.0001 / <.0001
time 4 / 2 / 4 / -8.6038 / 0.4440 / 196 / -19.38 / <.0001 / <.0001
time 4 / 3 / 4 / -1.5988 / 0.4440 / 196 / -3.60 / 0.0004 / 0.0023
TukeyGroupingfortreat*timeLeastSquaresMeansSlice(Alpha=0.05)
LS-means with the same letter are not significantly different.
Slice / treat / Estimate
time 4 / 4 / 21.6950 / A
time 4
time 4 / 3 / 20.0962 / B
time 4
time 4 / 2 / 13.0913 / C
time 4 / C
time 4 / 1 / 12.4800 / C
F Test for treat*time Least Squares Means Slice
Slice / Num DF / Den DF / F Value / PrF
time 5 / 3 / 196 / 276.70 / <.0001
SimpleDifferencesoftreat*timeLeastSquaresMeans
AdjustmentforMultipleComparisons:Tukey
Slice / treat / _treat / Estimate / Standard Error / DF / tValue / Pr > |t| / Adj P
time 5 / 1 / 2 / -1.0300 / 0.4482 / 196 / -2.30 / 0.0226 / 0.1019
time 5 / 1 / 3 / -8.6037 / 0.4482 / 196 / -19.20 / <.0001 / <.0001
time 5 / 1 / 4 / -10.4425 / 0.4482 / 196 / -23.30 / <.0001 / <.0001
time 5 / 2 / 3 / -7.5737 / 0.4482 / 196 / -16.90 / <.0001 / <.0001
time 5 / 2 / 4 / -9.4125 / 0.4482 / 196 / -21.00 / <.0001 / <.0001
time 5 / 3 / 4 / -1.8387 / 0.4482 / 196 / -4.10 / <.0001 / 0.0003
TukeyGroupingfortreat*timeLeastSquaresMeansSlice(Alpha=0.05)
LS-means with the same letter are not significantly different.
Slice / treat / Estimate
time 5 / 4 / 23.2337 / A
time 5
time 5 / 3 / 21.3950 / B
time 5
time 5 / 2 / 13.8213 / C
time 5 / C
time 5 / 1 / 12.7913 / C
F Test for treat*time Least Squares Means Slice
Slice / Num DF / Den DF / F Value / PrF
time 6 / 3 / 196 / 375.07 / <.0001
SimpleDifferencesoftreat*timeLeastSquaresMeans
AdjustmentforMultipleComparisons:Tukey
Slice / treat / _treat / Estimate / Standard Error / DF / tValue / Pr > |t| / Adj P
time 6 / 1 / 2 / -0.6287 / 0.4351 / 196 / -1.45 / 0.1500 / 0.4728
time 6 / 1 / 3 / -9.7962 / 0.4351 / 196 / -22.52 / <.0001 / <.0001
time 6 / 1 / 4 / -11.3375 / 0.4351 / 196 / -26.06 / <.0001 / <.0001
time 6 / 2 / 3 / -9.1675 / 0.4351 / 196 / -21.07 / <.0001 / <.0001
time 6 / 2 / 4 / -10.7087 / 0.4351 / 196 / -24.61 / <.0001 / <.0001
time 6 / 3 / 4 / -1.5412 / 0.4351 / 196 / -3.54 / 0.0005 / 0.0028
TukeyGroupingfortreat*timeLeastSquaresMeansSlice(Alpha=0.05)
LS-means with the same letter are not significantly different.
Slice / treat / Estimate
time 6 / 4 / 23.9300 / A
time 6
time 6 / 3 / 22.3888 / B
time 6
time 6 / 2 / 13.2213 / C
time 6 / C
time 6 / 1 / 12.5925 / C
F Test for treat*time Least Squares Means Slice
Slice / Num DF / Den DF / F Value / PrF
time 7 / 3 / 196 / 334.78 / <.0001
SimpleDifferencesoftreat*timeLeastSquaresMeans
AdjustmentforMultipleComparisons:Tukey
Slice / treat / _treat / Estimate / Standard Error / DF / tValue / Pr > |t| / Adj P
time 7 / 1 / 2 / -0.3350 / 0.4926 / 196 / -0.68 / 0.4972 / 0.9046
time 7 / 1 / 3 / -10.1187 / 0.4926 / 196 / -20.54 / <.0001 / <.0001
time 7 / 1 / 4 / -12.1075 / 0.4926 / 196 / -24.58 / <.0001 / <.0001
time 7 / 2 / 3 / -9.7837 / 0.4926 / 196 / -19.86 / <.0001 / <.0001
time 7 / 2 / 4 / -11.7725 / 0.4926 / 196 / -23.90 / <.0001 / <.0001
time 7 / 3 / 4 / -1.9887 / 0.4926 / 196 / -4.04 / <.0001 / 0.0004
TukeyGroupingfortreat*timeLeastSquaresMeansSlice(Alpha=0.05)
LS-means with the same letter are not significantly different.
Slice / treat / Estimate
time 7 / 4 / 24.0425 / A
time 7
time 7 / 3 / 22.0538 / B
time 7
time 7 / 2 / 12.2700 / C
time 7 / C
time 7 / 1 / 11.9350 / C
F Test for treat*time Least Squares Means Slice
Slice / Num DF / Den DF / F Value / PrF
time 8 / 3 / 196 / 315.57 / <.0001
SimpleDifferencesoftreat*timeLeastSquaresMeans
AdjustmentforMultipleComparisons:Tukey
Slice / treat / _treat / Estimate / Standard Error / DF / tValue / Pr > |t| / Adj P
time 8 / 1 / 2 / -0.6825 / 0.8831 / 196 / -0.77 / 0.4406 / 0.8667
time 8 / 1 / 3 / -17.4475 / 0.8831 / 196 / -19.76 / <.0001 / <.0001
time 8 / 1 / 4 / -21.2687 / 0.8831 / 196 / -24.08 / <.0001 / <.0001
time 8 / 2 / 3 / -16.7650 / 0.8831 / 196 / -18.98 / <.0001 / <.0001
time 8 / 2 / 4 / -20.5862 / 0.8831 / 196 / -23.31 / <.0001 / <.0001
time 8 / 3 / 4 / -3.8212 / 0.8831 / 196 / -4.33 / <.0001 / 0.0001
TukeyGroupingfortreat*timeLeastSquaresMeansSlice(Alpha=0.05)
LS-means with the same letter are not significantly different.
Slice / treat / Estimate
time 8 / 4 / 39.1787 / A
time 8
time 8 / 3 / 35.3575 / B
time 8
time 8 / 2 / 18.5925 / C
time 8 / C
time 8 / 1 / 17.9100 / C