decide between polynomial vs ordered factor model (lme)

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decide between polynomial vs ordered factor model (lme)

Leo Gürtler
Dear alltogether,

two lme's, the data are available at:

http://www.anicca-vijja.de/lg/hlm3_nachw.Rdata

explanations of the data:

nachw = post hox knowledge tests over 6 measure time points (= equally
spaced)
zeitn = time points (n = 6)
subgr = small learning groups (n = 28)
gru = 4 different groups = treatment factor

levels: time (=zeitn) (n=6) within subject (n=4) within smallgroups
(=gru) (n = 28), i.e. n = 4 * 28 = 112 persons and 112 * 6 = 672 data points

library(nlme)
fitlme7 <- lme(nachw ~ I(zeitn-3.5) + I((zeitn-3.5)^2) +
I((zeitn-3.5)^3) + I((zeitn-3.5)^4)*gru, random = list(subgr = ~ 1,
subject = ~ zeitn), data = hlm3)

fit5 <- lme(nachw ~ ordered(I(zeitn-3.5))*gru, random = list(subgr =
~ 1, subject = ~ zeitn), data = hlm3)

anova( update(fit5, method="ML"), update(fitlme7, method="ML") )

 > anova( update(fit5, method="ML"), update(fitlme7, method="ML") )
                                Model df      AIC      BIC    logLik   Test
update(fit5, method = "ML")        1 29 2535.821 2666.619 -1238.911
update(fitlme7, method = "ML")     2 16 2529.719 2601.883 -1248.860 1 vs 2
                                 L.Ratio p-value
update(fit5, method = "ML")
update(fitlme7, method = "ML") 19.89766  0.0978
 >

shows that both are ~ equal, although I know about the uncertainty of ML
tests with lme(). Both models show that the ^2 and the ^4 terms are
important parts of the model.

My question is:

- Is it legitim to choose a model based on these outputs according to
theoretical considerations instead of statistical tests that not really
show a superiority of one model over the other one?

- Is there another criterium I've overseen to decide which model can be
clearly prefered?

- The idea behind that is that in the one model (fit5) the second
contrast of the factor (gru) is statistically significant, although not
the whole factor in the anova output.
In the other model, this is not the case.
Theoretically interesting is of course the significance of the second
contrast of gru, as it shows a tendency of one treatment being slightly
superior. I want to choose this model but I am not sure whether this is
proper action. Both models shows this trend, but only one model clearly
indicates that this trend bears some empirical meaning.

Thanks for any suggestions,

leo


here are the outputs for each model:

> fitlme7 <- lme(nachw ~ I(zeitn-3.5) + I((zeitn-3.5)^2) +
I((zeitn-3.5)^3) + I((zeitn-3.5)^4)*gru, random = list(subgr = ~ 1,
subject = ~ zeitn), data = hlm3)
> plot(augPred(fitlme7), layout=c(14,8))
> summary(fitlme7); anova(fitlme7); intervals(fitlme7)
Linear mixed-effects model fit by REML
Data: hlm3
       AIC      BIC    logLik
  2582.934 2654.834 -1275.467

Random effects:
Formula: ~1 | subgr
        (Intercept)
StdDev:   0.5833797

Formula: ~zeitn | subject %in% subgr
Structure: General positive-definite, Log-Cholesky parametrization
            StdDev    Corr
(Intercept) 0.6881908 (Intr)
zeitn       0.1936087 -0.055
Residual    1.3495785

Fixed effects: nachw ~ I(zeitn - 3.5) + I((zeitn - 3.5)^2) + I((zeitn -
3.5)^3) +      I((zeitn - 3.5)^4) * gru
                            Value  Std.Error  DF   t-value p-value
(Intercept)              4.528757 0.17749012 553 25.515542  0.0000
I(zeitn - 3.5)           0.010602 0.08754449 553  0.121100  0.9037
I((zeitn - 3.5)^2)       0.815693 0.09765075 553  8.353171  0.0000
I((zeitn - 3.5)^3)       0.001336 0.01584169 553  0.084329  0.9328
I((zeitn - 3.5)^4)      -0.089655 0.01405811 553 -6.377486  0.0000
gru1                     0.187181 0.30805090  24  0.607630  0.5491
gru2                     0.532665 0.30805090  24  1.729147  0.0966
gru3                    -0.046305 0.30805090  24 -0.150317  0.8818
I((zeitn - 3.5)^4):gru1 -0.007860 0.00600928 553 -1.307993  0.1914
I((zeitn - 3.5)^4):gru2 -0.001259 0.00600928 553 -0.209516  0.8341
I((zeitn - 3.5)^4):gru3 -0.000224 0.00600928 553 -0.037225  0.9703
Correlation:
                        (Intr) I(-3.5 I((-3.5)^2 I((-3.5)^3 I((z-3.5)^4)
I(zeitn - 3.5)           0.071
I((zeitn - 3.5)^2)      -0.465  0.000
I((zeitn - 3.5)^3)       0.000 -0.914  0.000
I((zeitn - 3.5)^4)       0.401  0.000 -0.977      0.000
gru1                     0.000  0.000  0.000      0.000      0.000
gru2                     0.000  0.000  0.000      0.000      0.000
gru3                     0.000  0.000  0.000      0.000      0.000
I((zeitn - 3.5)^4):gru1  0.000  0.000  0.000      0.000      0.000
I((zeitn - 3.5)^4):gru2  0.000  0.000  0.000      0.000      0.000
I((zeitn - 3.5)^4):gru3  0.000  0.000  0.000      0.000      0.000
                        gru1   gru2   gru3   I((-3.5)^4):1 I((-3.5)^4):2
I(zeitn - 3.5)
I((zeitn - 3.5)^2)
I((zeitn - 3.5)^3)
I((zeitn - 3.5)^4)
gru1
gru2                     0.000
gru3                     0.000  0.000
I((zeitn - 3.5)^4):gru1 -0.287  0.000  0.000
I((zeitn - 3.5)^4):gru2  0.000 -0.287  0.000  0.000
I((zeitn - 3.5)^4):gru3  0.000  0.000 -0.287  0.000         0.000

Standardized Within-Group Residuals:
       Min         Q1        Med         Q3        Max
-3.1326192 -0.5888543  0.0239228  0.6519002  2.1238820

Number of Observations: 672
Number of Groups:
             subgr subject %in% subgr
                28                112
                       numDF denDF   F-value p-value
(Intercept)                1   553 1426.5275  <.0001
I(zeitn - 3.5)             1   553    0.2381  0.6258
I((zeitn - 3.5)^2)         1   553   98.6712  <.0001
I((zeitn - 3.5)^3)         1   553    0.0071  0.9328
I((zeitn - 3.5)^4)         1   553   40.6723  <.0001
gru                        3    24    1.0410  0.3924
I((zeitn - 3.5)^4):gru     3   553    0.5854  0.6248
Approximate 95% confidence intervals

Fixed effects:
                              lower          est.        upper
(Intercept)              4.18011938  4.5287566579  4.877393940
I(zeitn - 3.5)          -0.16135875  0.0106016498  0.182562052
I((zeitn - 3.5)^2)       0.62388162  0.8156933820  1.007505144
I((zeitn - 3.5)^3)      -0.02978133  0.0013359218  0.032453178
I((zeitn - 3.5)^4)      -0.11726922 -0.0896553959 -0.062041570
gru1                    -0.44860499  0.1871808283  0.822966643
gru2                    -0.10312045  0.5326653686  1.168451183
gru3                    -0.68209096 -0.0463051419  0.589480673
I((zeitn - 3.5)^4):gru1 -0.01966389 -0.0078600880  0.003943709
I((zeitn - 3.5)^4):gru2 -0.01306284 -0.0012590380  0.010544759
I((zeitn - 3.5)^4):gru3 -0.01202749 -0.0002236923  0.011580105
attr(,"label")
[1] "Fixed effects:"

Random Effects:
  Level: subgr
                    lower      est.     upper
sd((Intercept)) 0.3459779 0.5833797 0.9836812
  Level: subject
                            lower        est.     upper
sd((Intercept))         0.4388885  0.68819079 1.0791046
sd(zeitn)               0.1320591  0.19360866 0.2838449
cor((Intercept),zeitn) -0.4835884 -0.05541043 0.3941661

Within-group standard error:
   lower     est.    upper
1.267548 1.349579 1.436918

#########################################################
an the other model:

> summary(fit5); anova(fit5); intervals(fit5)
Linear mixed-effects model fit by REML
Data: hlm3
       AIC      BIC    logLik
  2564.135 2693.878 -1253.067

Random effects:
Formula: ~1 | subgr
        (Intercept)
StdDev:   0.5833753

Formula: ~zeitn | subject %in% subgr
Structure: General positive-definite, Log-Cholesky parametrization
            StdDev    Corr
(Intercept) 0.6453960 (Intr)
zeitn       0.1709843 0.13
Residual    1.3497627

Fixed effects: nachw ~ ordered(I(zeitn - 3.5)) + gru + ordered(I(zeitn -
3.5)):gru
                                   Value Std.Error  DF  t-value p-value
(Intercept)                     5.587313 0.1505852 540 37.10400  0.0000
ordered(I(zeitn - 3.5)).L       0.072572 0.1443422 540  0.50278  0.6153
ordered(I(zeitn - 3.5)).Q       1.266731 0.1275406 540  9.93198  0.0000
ordered(I(zeitn - 3.5)).C       0.010754 0.1275406 540  0.08432  0.9328
ordered(I(zeitn - 3.5))^4      -0.813277 0.1275406 540 -6.37662  0.0000
ordered(I(zeitn - 3.5))^5       0.070373 0.1275406 540  0.55177  0.5813
gru1                            0.056700 0.3011704  24  0.18826  0.8523
gru2                            0.679057 0.3011704  24  2.25473  0.0335
gru3                           -0.141425 0.3011704  24 -0.46958  0.6429
ordered(I(zeitn - 3.5)).L:gru1 -0.070352 0.2886844 540 -0.24370  0.8076
ordered(I(zeitn - 3.5)).Q:gru1 -0.360380 0.2550812 540 -1.41281  0.1583
ordered(I(zeitn - 3.5)).C:gru1 -0.162411 0.2550812 540 -0.63670  0.5246
ordered(I(zeitn - 3.5))^4:gru1  0.086343 0.2550812 540  0.33849  0.7351
ordered(I(zeitn - 3.5))^5:gru1 -0.017207 0.2550812 540 -0.06746  0.9462
ordered(I(zeitn - 3.5)).L:gru2  0.788896 0.2886844 540  2.73273  0.0065
ordered(I(zeitn - 3.5)).Q:gru2  0.033386 0.2550812 540  0.13089  0.8959
ordered(I(zeitn - 3.5)).C:gru2  0.089757 0.2550812 540  0.35188  0.7251
ordered(I(zeitn - 3.5))^4:gru2 -0.402616 0.2550812 540 -1.57839  0.1151
ordered(I(zeitn - 3.5))^5:gru2 -0.507855 0.2550812 540 -1.99095  0.0470
ordered(I(zeitn - 3.5)).L:gru3 -0.439200 0.2886844 540 -1.52138  0.1287
ordered(I(zeitn - 3.5)).Q:gru3  0.026105 0.2550812 540  0.10234  0.9185
ordered(I(zeitn - 3.5)).C:gru3 -0.273643 0.2550812 540 -1.07277  0.2839
ordered(I(zeitn - 3.5))^4:gru3 -0.163738 0.2550812 540 -0.64191  0.5212
ordered(I(zeitn - 3.5))^5:gru3  0.204174 0.2550812 540  0.80043  0.4238
Correlation:
                               (Intr) or(I(-3.5)).L or(I(-3.5)).Q
or(I(-3.5)).C or(I(-3.5))^4 or(I(-3.5))^5 gru1 gru2 gru3 o(I(-3.5)).L:1
o(I(-3.5)).Q:1 o(I(-3.5)).C:1
ordered(I(zeitn - 3.5)).L
0.2


ordered(I(zeitn - 3.5)).Q      0.0
0.0


ordered(I(zeitn - 3.5)).C      0.0    0.0
0.0


ordered(I(zeitn - 3.5))^4      0.0    0.0           0.0
0.0


ordered(I(zeitn - 3.5))^5      0.0    0.0           0.0
0.0
0.0


gru1                           0.0    0.0           0.0
0.0           0.0
0.0
gru2                           0.0    0.0           0.0
0.0           0.0           0.0
0.0
gru3                           0.0    0.0           0.0
0.0           0.0           0.0           0.0
0.0
ordered(I(zeitn - 3.5)).L:gru1 0.0    0.0           0.0
0.0           0.0           0.0           0.2  0.0
0.0
ordered(I(zeitn - 3.5)).Q:gru1 0.0    0.0           0.0
0.0           0.0           0.0           0.0  0.0  0.0
0.0
ordered(I(zeitn - 3.5)).C:gru1 0.0    0.0           0.0
0.0           0.0           0.0           0.0  0.0  0.0  0.0
0.0
ordered(I(zeitn - 3.5))^4:gru1 0.0    0.0           0.0
0.0           0.0           0.0           0.0  0.0  0.0  0.0
0.0            0.0
ordered(I(zeitn - 3.5))^5:gru1 0.0    0.0           0.0
0.0           0.0           0.0           0.0  0.0  0.0  0.0
0.0            0.0
ordered(I(zeitn - 3.5)).L:gru2 0.0    0.0           0.0
0.0           0.0           0.0           0.0  0.2  0.0  0.0
0.0            0.0
ordered(I(zeitn - 3.5)).Q:gru2 0.0    0.0           0.0
0.0           0.0           0.0           0.0  0.0  0.0  0.0
0.0            0.0
ordered(I(zeitn - 3.5)).C:gru2 0.0    0.0           0.0
0.0           0.0           0.0           0.0  0.0  0.0  0.0
0.0            0.0
ordered(I(zeitn - 3.5))^4:gru2 0.0    0.0           0.0
0.0           0.0           0.0           0.0  0.0  0.0  0.0
0.0            0.0
ordered(I(zeitn - 3.5))^5:gru2 0.0    0.0           0.0
0.0           0.0           0.0           0.0  0.0  0.0  0.0
0.0            0.0
ordered(I(zeitn - 3.5)).L:gru3 0.0    0.0           0.0
0.0           0.0           0.0           0.0  0.0  0.2  0.0
0.0            0.0
ordered(I(zeitn - 3.5)).Q:gru3 0.0    0.0           0.0
0.0           0.0           0.0           0.0  0.0  0.0  0.0
0.0            0.0
ordered(I(zeitn - 3.5)).C:gru3 0.0    0.0           0.0
0.0           0.0           0.0           0.0  0.0  0.0  0.0
0.0            0.0
ordered(I(zeitn - 3.5))^4:gru3 0.0    0.0           0.0
0.0           0.0           0.0           0.0  0.0  0.0  0.0
0.0            0.0
ordered(I(zeitn - 3.5))^5:gru3 0.0    0.0           0.0
0.0           0.0           0.0           0.0  0.0  0.0  0.0
0.0            0.0
                               o(I(-3.5))^4:1 o(I(-3.5))^5:1
o(I(-3.5)).L:2 o(I(-3.5)).Q:2 o(I(-3.5)).C:2 o(I(-3.5))^4:2
o(I(-3.5))^5:2 o(I(-3.5)).L:3 o(I(-3.5)).Q:3
ordered(I(zeitn -
3.5)).L


ordered(I(zeitn -
3.5)).Q


ordered(I(zeitn -
3.5)).C


ordered(I(zeitn -
3.5))^4


ordered(I(zeitn -
3.5))^5


gru1
 


gru2
 


gru3
 


ordered(I(zeitn -
3.5)).L:gru1


ordered(I(zeitn -
3.5)).Q:gru1


ordered(I(zeitn -
3.5)).C:gru1


ordered(I(zeitn -
3.5))^4:gru1


ordered(I(zeitn - 3.5))^5:gru1
0.0


ordered(I(zeitn - 3.5)).L:gru2 0.0
0.0


ordered(I(zeitn - 3.5)).Q:gru2 0.0            0.0
0.0


ordered(I(zeitn - 3.5)).C:gru2 0.0            0.0
0.0
0.0


ordered(I(zeitn - 3.5))^4:gru2 0.0            0.0
0.0            0.0
0.0
ordered(I(zeitn - 3.5))^5:gru2 0.0            0.0
0.0            0.0            0.0
0.0
ordered(I(zeitn - 3.5)).L:gru3 0.0            0.0
0.0            0.0            0.0            0.0
0.0
ordered(I(zeitn - 3.5)).Q:gru3 0.0            0.0
0.0            0.0            0.0            0.0
0.0            0.0
ordered(I(zeitn - 3.5)).C:gru3 0.0            0.0
0.0            0.0            0.0            0.0
0.0            0.0            0.0
ordered(I(zeitn - 3.5))^4:gru3 0.0            0.0
0.0            0.0            0.0            0.0
0.0            0.0            0.0
ordered(I(zeitn - 3.5))^5:gru3 0.0            0.0
0.0            0.0            0.0            0.0
0.0            0.0            0.0
                               o(I(-3.5)).C:3 o(I(-3.5))^4:3
ordered(I(zeitn - 3.5)).L
ordered(I(zeitn - 3.5)).Q
ordered(I(zeitn - 3.5)).C
ordered(I(zeitn - 3.5))^4
ordered(I(zeitn - 3.5))^5
gru1
gru2
gru3
ordered(I(zeitn - 3.5)).L:gru1
ordered(I(zeitn - 3.5)).Q:gru1
ordered(I(zeitn - 3.5)).C:gru1
ordered(I(zeitn - 3.5))^4:gru1
ordered(I(zeitn - 3.5))^5:gru1
ordered(I(zeitn - 3.5)).L:gru2
ordered(I(zeitn - 3.5)).Q:gru2
ordered(I(zeitn - 3.5)).C:gru2
ordered(I(zeitn - 3.5))^4:gru2
ordered(I(zeitn - 3.5))^5:gru2
ordered(I(zeitn - 3.5)).L:gru3
ordered(I(zeitn - 3.5)).Q:gru3
ordered(I(zeitn - 3.5)).C:gru3
ordered(I(zeitn - 3.5))^4:gru3 0.0
ordered(I(zeitn - 3.5))^5:gru3 0.0            0.0

Standardized Within-Group Residuals:
        Min          Q1         Med          Q3         Max
-3.10206117 -0.62626454  0.02807962  0.64554138  2.13155536

Number of Observations: 672
Number of Groups:
             subgr subject %in% subgr
                28                112
                            numDF denDF   F-value p-value
(Intercept)                     1   540 1426.5315  <.0001
ordered(I(zeitn - 3.5))         5   540   27.9740  <.0001
gru                             3    24    1.0410  0.3924
ordered(I(zeitn - 3.5)):gru    15   540    1.4115  0.1363
Approximate 95% confidence intervals

Fixed effects:
                                    lower        est.        upper
(Intercept)                     5.2915086  5.58731309  5.883117621
ordered(I(zeitn - 3.5)).L      -0.2109689  0.07257212  0.356113124
ordered(I(zeitn - 3.5)).Q       1.0161942  1.26673073  1.517267227
ordered(I(zeitn - 3.5)).C      -0.2397825  0.01075396  0.261290456
ordered(I(zeitn - 3.5))^4      -1.0638138 -0.81327731 -0.562740815
ordered(I(zeitn - 3.5))^5      -0.1801634  0.07037312  0.320909612
gru1                           -0.5648856  0.05669953  0.678284624
gru2                            0.0574723  0.67905739  1.300642487
gru3                           -0.7630097 -0.14142458  0.480160517
ordered(I(zeitn - 3.5)).L:gru1 -0.6374343 -0.07035232  0.496729683
ordered(I(zeitn - 3.5)).Q:gru1 -0.8614532 -0.36038020  0.140692783
ordered(I(zeitn - 3.5)).C:gru1 -0.6634839 -0.16241093  0.338662057
ordered(I(zeitn - 3.5))^4:gru1 -0.4147301  0.08634286  0.587415843
ordered(I(zeitn - 3.5))^5:gru1 -0.5182803 -0.01720729  0.483865692
ordered(I(zeitn - 3.5)).L:gru2  0.2218139  0.78889594  1.355977946
ordered(I(zeitn - 3.5)).Q:gru2 -0.4676866  0.03338637  0.534459352
ordered(I(zeitn - 3.5)).C:gru2 -0.4113159  0.08975711  0.590830099
ordered(I(zeitn - 3.5))^4:gru2 -0.9036894 -0.40261640  0.098456584
ordered(I(zeitn - 3.5))^5:gru2 -1.0089275 -0.50785453 -0.006781542
ordered(I(zeitn - 3.5)).L:gru3 -1.0062815 -0.43919953  0.127882479
ordered(I(zeitn - 3.5)).Q:gru3 -0.4749680  0.02610502  0.527178001
ordered(I(zeitn - 3.5)).C:gru3 -0.7747163 -0.27364336  0.227429629
ordered(I(zeitn - 3.5))^4:gru3 -0.6648114 -0.16373838  0.337334604
ordered(I(zeitn - 3.5))^5:gru3 -0.2968991  0.20417390  0.705246883
attr(,"label")
[1] "Fixed effects:"

Random Effects:
  Level: subgr
                    lower      est.     upper
sd((Intercept)) 0.3464888 0.5833753 0.9822158
  Level: subject
                            lower      est.     upper
sd((Intercept))         0.3640439 0.6453960 1.1441916
sd(zeitn)               0.1000264 0.1709843 0.2922790
cor((Intercept),zeitn) -0.6712236 0.1295558 0.7907922

Within-group standard error:
   lower     est.    upper
1.265702 1.349763 1.439406

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Re: decide between polynomial vs ordered factor model (lme)

Dimitris Rizopoulos
I think that these models are not nested and thus the LRT produced by
anova.lme() will not be valid; AIC and BIC could be more relevant. In
terms of interpretability, I'd say that a model treating 'zeitn' as a
factor is much easier to explain than a model with 4th order
polynomial.

I hope it helps.

Best,
Dimitris

----
Dimitris Rizopoulos
Ph.D. Student
Biostatistical Centre
School of Public Health
Catholic University of Leuven

Address: Kapucijnenvoer 35, Leuven, Belgium
Tel: +32/(0)16/336899
Fax: +32/(0)16/337015
Web: http://www.med.kuleuven.be/biostat/
     http://www.student.kuleuven.be/~m0390867/dimitris.htm



----- Original Message -----
From: "Leo Gürtler" <[hidden email]>
To: <[hidden email]>
Sent: Monday, January 09, 2006 2:59 PM
Subject: [R] decide between polynomial vs ordered factor model (lme)


> Dear alltogether,
>
> two lme's, the data are available at:
>
> http://www.anicca-vijja.de/lg/hlm3_nachw.Rdata
>
> explanations of the data:
>
> nachw = post hox knowledge tests over 6 measure time points (=
> equally
> spaced)
> zeitn = time points (n = 6)
> subgr = small learning groups (n = 28)
> gru = 4 different groups = treatment factor
>
> levels: time (=zeitn) (n=6) within subject (n=4) within smallgroups
> (=gru) (n = 28), i.e. n = 4 * 28 = 112 persons and 112 * 6 = 672
> data points
>
> library(nlme)
> fitlme7 <- lme(nachw ~ I(zeitn-3.5) + I((zeitn-3.5)^2) +
> I((zeitn-3.5)^3) + I((zeitn-3.5)^4)*gru, random = list(subgr = ~ 1,
> subject = ~ zeitn), data = hlm3)
>
> fit5 <- lme(nachw ~ ordered(I(zeitn-3.5))*gru, random = list(subgr =
> ~ 1, subject = ~ zeitn), data = hlm3)
>
> anova( update(fit5, method="ML"), update(fitlme7, method="ML") )
>
> > anova( update(fit5, method="ML"), update(fitlme7, method="ML") )
>                                Model df      AIC      BIC    logLik
> Test
> update(fit5, method = "ML")        1 29 2535.821 2666.619 -1238.911
> update(fitlme7, method = "ML")     2 16 2529.719 2601.883 -1248.860
> 1 vs 2
>                                 L.Ratio p-value
> update(fit5, method = "ML")
> update(fitlme7, method = "ML") 19.89766  0.0978
> >
>
> shows that both are ~ equal, although I know about the uncertainty
> of ML
> tests with lme(). Both models show that the ^2 and the ^4 terms are
> important parts of the model.
>
> My question is:
>
> - Is it legitim to choose a model based on these outputs according
> to
> theoretical considerations instead of statistical tests that not
> really
> show a superiority of one model over the other one?
>
> - Is there another criterium I've overseen to decide which model can
> be
> clearly prefered?
>
> - The idea behind that is that in the one model (fit5) the second
> contrast of the factor (gru) is statistically significant, although
> not
> the whole factor in the anova output.
> In the other model, this is not the case.
> Theoretically interesting is of course the significance of the
> second
> contrast of gru, as it shows a tendency of one treatment being
> slightly
> superior. I want to choose this model but I am not sure whether this
> is
> proper action. Both models shows this trend, but only one model
> clearly
> indicates that this trend bears some empirical meaning.
>
> Thanks for any suggestions,
>
> leo
>
>
> here are the outputs for each model:
>
>> fitlme7 <- lme(nachw ~ I(zeitn-3.5) + I((zeitn-3.5)^2) +
> I((zeitn-3.5)^3) + I((zeitn-3.5)^4)*gru, random = list(subgr = ~ 1,
> subject = ~ zeitn), data = hlm3)
>> plot(augPred(fitlme7), layout=c(14,8))
>> summary(fitlme7); anova(fitlme7); intervals(fitlme7)
> Linear mixed-effects model fit by REML
> Data: hlm3
>       AIC      BIC    logLik
>  2582.934 2654.834 -1275.467
>
> Random effects:
> Formula: ~1 | subgr
>        (Intercept)
> StdDev:   0.5833797
>
> Formula: ~zeitn | subject %in% subgr
> Structure: General positive-definite, Log-Cholesky parametrization
>            StdDev    Corr
> (Intercept) 0.6881908 (Intr)
> zeitn       0.1936087 -0.055
> Residual    1.3495785
>
> Fixed effects: nachw ~ I(zeitn - 3.5) + I((zeitn - 3.5)^2) +
> I((zeitn -
> 3.5)^3) +      I((zeitn - 3.5)^4) * gru
>                            Value  Std.Error  DF   t-value p-value
> (Intercept)              4.528757 0.17749012 553 25.515542  0.0000
> I(zeitn - 3.5)           0.010602 0.08754449 553  0.121100  0.9037
> I((zeitn - 3.5)^2)       0.815693 0.09765075 553  8.353171  0.0000
> I((zeitn - 3.5)^3)       0.001336 0.01584169 553  0.084329  0.9328
> I((zeitn - 3.5)^4)      -0.089655 0.01405811 553 -6.377486  0.0000
> gru1                     0.187181 0.30805090  24  0.607630  0.5491
> gru2                     0.532665 0.30805090  24  1.729147  0.0966
> gru3                    -0.046305 0.30805090  24 -0.150317  0.8818
> I((zeitn - 3.5)^4):gru1 -0.007860 0.00600928 553 -1.307993  0.1914
> I((zeitn - 3.5)^4):gru2 -0.001259 0.00600928 553 -0.209516  0.8341
> I((zeitn - 3.5)^4):gru3 -0.000224 0.00600928 553 -0.037225  0.9703
> Correlation:
>                        (Intr) I(-3.5 I((-3.5)^2 I((-3.5)^3
> I((z-3.5)^4)
> I(zeitn - 3.5)           0.071
> I((zeitn - 3.5)^2)      -0.465  0.000
> I((zeitn - 3.5)^3)       0.000 -0.914  0.000
> I((zeitn - 3.5)^4)       0.401  0.000 -0.977      0.000
> gru1                     0.000  0.000  0.000      0.000      0.000
> gru2                     0.000  0.000  0.000      0.000      0.000
> gru3                     0.000  0.000  0.000      0.000      0.000
> I((zeitn - 3.5)^4):gru1  0.000  0.000  0.000      0.000      0.000
> I((zeitn - 3.5)^4):gru2  0.000  0.000  0.000      0.000      0.000
> I((zeitn - 3.5)^4):gru3  0.000  0.000  0.000      0.000      0.000
>                        gru1   gru2   gru3   I((-3.5)^4):1
> I((-3.5)^4):2
> I(zeitn - 3.5)
> I((zeitn - 3.5)^2)
> I((zeitn - 3.5)^3)
> I((zeitn - 3.5)^4)
> gru1
> gru2                     0.000
> gru3                     0.000  0.000
> I((zeitn - 3.5)^4):gru1 -0.287  0.000  0.000
> I((zeitn - 3.5)^4):gru2  0.000 -0.287  0.000  0.000
> I((zeitn - 3.5)^4):gru3  0.000  0.000 -0.287  0.000         0.000
>
> Standardized Within-Group Residuals:
>       Min         Q1        Med         Q3        Max
> -3.1326192 -0.5888543  0.0239228  0.6519002  2.1238820
>
> Number of Observations: 672
> Number of Groups:
>             subgr subject %in% subgr
>                28                112
>                       numDF denDF   F-value p-value
> (Intercept)                1   553 1426.5275  <.0001
> I(zeitn - 3.5)             1   553    0.2381  0.6258
> I((zeitn - 3.5)^2)         1   553   98.6712  <.0001
> I((zeitn - 3.5)^3)         1   553    0.0071  0.9328
> I((zeitn - 3.5)^4)         1   553   40.6723  <.0001
> gru                        3    24    1.0410  0.3924
> I((zeitn - 3.5)^4):gru     3   553    0.5854  0.6248
> Approximate 95% confidence intervals
>
> Fixed effects:
>                              lower          est.        upper
> (Intercept)              4.18011938  4.5287566579  4.877393940
> I(zeitn - 3.5)          -0.16135875  0.0106016498  0.182562052
> I((zeitn - 3.5)^2)       0.62388162  0.8156933820  1.007505144
> I((zeitn - 3.5)^3)      -0.02978133  0.0013359218  0.032453178
> I((zeitn - 3.5)^4)      -0.11726922 -0.0896553959 -0.062041570
> gru1                    -0.44860499  0.1871808283  0.822966643
> gru2                    -0.10312045  0.5326653686  1.168451183
> gru3                    -0.68209096 -0.0463051419  0.589480673
> I((zeitn - 3.5)^4):gru1 -0.01966389 -0.0078600880  0.003943709
> I((zeitn - 3.5)^4):gru2 -0.01306284 -0.0012590380  0.010544759
> I((zeitn - 3.5)^4):gru3 -0.01202749 -0.0002236923  0.011580105
> attr(,"label")
> [1] "Fixed effects:"
>
> Random Effects:
>  Level: subgr
>                    lower      est.     upper
> sd((Intercept)) 0.3459779 0.5833797 0.9836812
>  Level: subject
>                            lower        est.     upper
> sd((Intercept))         0.4388885  0.68819079 1.0791046
> sd(zeitn)               0.1320591  0.19360866 0.2838449
> cor((Intercept),zeitn) -0.4835884 -0.05541043 0.3941661
>
> Within-group standard error:
>   lower     est.    upper
> 1.267548 1.349579 1.436918
>
> #########################################################
> an the other model:
>
>> summary(fit5); anova(fit5); intervals(fit5)
> Linear mixed-effects model fit by REML
> Data: hlm3
>       AIC      BIC    logLik
>  2564.135 2693.878 -1253.067
>
> Random effects:
> Formula: ~1 | subgr
>        (Intercept)
> StdDev:   0.5833753
>
> Formula: ~zeitn | subject %in% subgr
> Structure: General positive-definite, Log-Cholesky parametrization
>            StdDev    Corr
> (Intercept) 0.6453960 (Intr)
> zeitn       0.1709843 0.13
> Residual    1.3497627
>
> Fixed effects: nachw ~ ordered(I(zeitn - 3.5)) + gru +
> ordered(I(zeitn -
> 3.5)):gru
>                                   Value Std.Error  DF  t-value
> p-value
> (Intercept)                     5.587313 0.1505852 540 37.10400
> 0.0000
> ordered(I(zeitn - 3.5)).L       0.072572 0.1443422 540  0.50278
> 0.6153
> ordered(I(zeitn - 3.5)).Q       1.266731 0.1275406 540  9.93198
> 0.0000
> ordered(I(zeitn - 3.5)).C       0.010754 0.1275406 540  0.08432
> 0.9328
> ordered(I(zeitn - 3.5))^4      -0.813277 0.1275406 540 -6.37662
> 0.0000
> ordered(I(zeitn - 3.5))^5       0.070373 0.1275406 540  0.55177
> 0.5813
> gru1                            0.056700 0.3011704  24  0.18826
> 0.8523
> gru2                            0.679057 0.3011704  24  2.25473
> 0.0335
> gru3                           -0.141425 0.3011704  24 -0.46958
> 0.6429
> ordered(I(zeitn - 3.5)).L:gru1 -0.070352 0.2886844 540 -0.24370
> 0.8076
> ordered(I(zeitn - 3.5)).Q:gru1 -0.360380 0.2550812 540 -1.41281
> 0.1583
> ordered(I(zeitn - 3.5)).C:gru1 -0.162411 0.2550812 540 -0.63670
> 0.5246
> ordered(I(zeitn - 3.5))^4:gru1  0.086343 0.2550812 540  0.33849
> 0.7351
> ordered(I(zeitn - 3.5))^5:gru1 -0.017207 0.2550812 540 -0.06746
> 0.9462
> ordered(I(zeitn - 3.5)).L:gru2  0.788896 0.2886844 540  2.73273
> 0.0065
> ordered(I(zeitn - 3.5)).Q:gru2  0.033386 0.2550812 540  0.13089
> 0.8959
> ordered(I(zeitn - 3.5)).C:gru2  0.089757 0.2550812 540  0.35188
> 0.7251
> ordered(I(zeitn - 3.5))^4:gru2 -0.402616 0.2550812 540 -1.57839
> 0.1151
> ordered(I(zeitn - 3.5))^5:gru2 -0.507855 0.2550812 540 -1.99095
> 0.0470
> ordered(I(zeitn - 3.5)).L:gru3 -0.439200 0.2886844 540 -1.52138
> 0.1287
> ordered(I(zeitn - 3.5)).Q:gru3  0.026105 0.2550812 540  0.10234
> 0.9185
> ordered(I(zeitn - 3.5)).C:gru3 -0.273643 0.2550812 540 -1.07277
> 0.2839
> ordered(I(zeitn - 3.5))^4:gru3 -0.163738 0.2550812 540 -0.64191
> 0.5212
> ordered(I(zeitn - 3.5))^5:gru3  0.204174 0.2550812 540  0.80043
> 0.4238
> Correlation:
>                               (Intr) or(I(-3.5)).L or(I(-3.5)).Q
> or(I(-3.5)).C or(I(-3.5))^4 or(I(-3.5))^5 gru1 gru2 gru3
> o(I(-3.5)).L:1
> o(I(-3.5)).Q:1 o(I(-3.5)).C:1
> ordered(I(zeitn - 3.5)).L
> 0.2
>
>
> ordered(I(zeitn - 3.5)).Q      0.0
> 0.0
>
>
> ordered(I(zeitn - 3.5)).C      0.0    0.0
> 0.0
>
>
> ordered(I(zeitn - 3.5))^4      0.0    0.0           0.0
> 0.0
>
>
> ordered(I(zeitn - 3.5))^5      0.0    0.0           0.0
> 0.0
> 0.0
>
>
> gru1                           0.0    0.0           0.0
> 0.0           0.0
> 0.0
> gru2                           0.0    0.0           0.0
> 0.0           0.0           0.0
> 0.0
> gru3                           0.0    0.0           0.0
> 0.0           0.0           0.0           0.0
> 0.0
> ordered(I(zeitn - 3.5)).L:gru1 0.0    0.0           0.0
> 0.0           0.0           0.0           0.2  0.0
> 0.0
> ordered(I(zeitn - 3.5)).Q:gru1 0.0    0.0           0.0
> 0.0           0.0           0.0           0.0  0.0  0.0
> 0.0
> ordered(I(zeitn - 3.5)).C:gru1 0.0    0.0           0.0
> 0.0           0.0           0.0           0.0  0.0  0.0  0.0
> 0.0
> ordered(I(zeitn - 3.5))^4:gru1 0.0    0.0           0.0
> 0.0           0.0           0.0           0.0  0.0  0.0  0.0
> 0.0            0.0
> ordered(I(zeitn - 3.5))^5:gru1 0.0    0.0           0.0
> 0.0           0.0           0.0           0.0  0.0  0.0  0.0
> 0.0            0.0
> ordered(I(zeitn - 3.5)).L:gru2 0.0    0.0           0.0
> 0.0           0.0           0.0           0.0  0.2  0.0  0.0
> 0.0            0.0
> ordered(I(zeitn - 3.5)).Q:gru2 0.0    0.0           0.0
> 0.0           0.0           0.0           0.0  0.0  0.0  0.0
> 0.0            0.0
> ordered(I(zeitn - 3.5)).C:gru2 0.0    0.0           0.0
> 0.0           0.0           0.0           0.0  0.0  0.0  0.0
> 0.0            0.0
> ordered(I(zeitn - 3.5))^4:gru2 0.0    0.0           0.0
> 0.0           0.0           0.0           0.0  0.0  0.0  0.0
> 0.0            0.0
> ordered(I(zeitn - 3.5))^5:gru2 0.0    0.0           0.0
> 0.0           0.0           0.0           0.0  0.0  0.0  0.0
> 0.0            0.0
> ordered(I(zeitn - 3.5)).L:gru3 0.0    0.0           0.0
> 0.0           0.0           0.0           0.0  0.0  0.2  0.0
> 0.0            0.0
> ordered(I(zeitn - 3.5)).Q:gru3 0.0    0.0           0.0
> 0.0           0.0           0.0           0.0  0.0  0.0  0.0
> 0.0            0.0
> ordered(I(zeitn - 3.5)).C:gru3 0.0    0.0           0.0
> 0.0           0.0           0.0           0.0  0.0  0.0  0.0
> 0.0            0.0
> ordered(I(zeitn - 3.5))^4:gru3 0.0    0.0           0.0
> 0.0           0.0           0.0           0.0  0.0  0.0  0.0
> 0.0            0.0
> ordered(I(zeitn - 3.5))^5:gru3 0.0    0.0           0.0
> 0.0           0.0           0.0           0.0  0.0  0.0  0.0
> 0.0            0.0
>                               o(I(-3.5))^4:1 o(I(-3.5))^5:1
> o(I(-3.5)).L:2 o(I(-3.5)).Q:2 o(I(-3.5)).C:2 o(I(-3.5))^4:2
> o(I(-3.5))^5:2 o(I(-3.5)).L:3 o(I(-3.5)).Q:3
> ordered(I(zeitn -
> 3.5)).L
>
>
> ordered(I(zeitn -
> 3.5)).Q
>
>
> ordered(I(zeitn -
> 3.5)).C
>
>
> ordered(I(zeitn -
> 3.5))^4
>
>
> ordered(I(zeitn -
> 3.5))^5
>
>
> gru1
>
>
>
> gru2
>
>
>
> gru3
>
>
>
> ordered(I(zeitn -
> 3.5)).L:gru1
>
>
> ordered(I(zeitn -
> 3.5)).Q:gru1
>
>
> ordered(I(zeitn -
> 3.5)).C:gru1
>
>
> ordered(I(zeitn -
> 3.5))^4:gru1
>
>
> ordered(I(zeitn - 3.5))^5:gru1
> 0.0
>
>
> ordered(I(zeitn - 3.5)).L:gru2 0.0
> 0.0
>
>
> ordered(I(zeitn - 3.5)).Q:gru2 0.0            0.0
> 0.0
>
>
> ordered(I(zeitn - 3.5)).C:gru2 0.0            0.0
> 0.0
> 0.0
>
>
> ordered(I(zeitn - 3.5))^4:gru2 0.0            0.0
> 0.0            0.0
> 0.0
> ordered(I(zeitn - 3.5))^5:gru2 0.0            0.0
> 0.0            0.0            0.0
> 0.0
> ordered(I(zeitn - 3.5)).L:gru3 0.0            0.0
> 0.0            0.0            0.0            0.0
> 0.0
> ordered(I(zeitn - 3.5)).Q:gru3 0.0            0.0
> 0.0            0.0            0.0            0.0
> 0.0            0.0
> ordered(I(zeitn - 3.5)).C:gru3 0.0            0.0
> 0.0            0.0            0.0            0.0
> 0.0            0.0            0.0
> ordered(I(zeitn - 3.5))^4:gru3 0.0            0.0
> 0.0            0.0            0.0            0.0
> 0.0            0.0            0.0
> ordered(I(zeitn - 3.5))^5:gru3 0.0            0.0
> 0.0            0.0            0.0            0.0
> 0.0            0.0            0.0
>                               o(I(-3.5)).C:3 o(I(-3.5))^4:3
> ordered(I(zeitn - 3.5)).L
> ordered(I(zeitn - 3.5)).Q
> ordered(I(zeitn - 3.5)).C
> ordered(I(zeitn - 3.5))^4
> ordered(I(zeitn - 3.5))^5
> gru1
> gru2
> gru3
> ordered(I(zeitn - 3.5)).L:gru1
> ordered(I(zeitn - 3.5)).Q:gru1
> ordered(I(zeitn - 3.5)).C:gru1
> ordered(I(zeitn - 3.5))^4:gru1
> ordered(I(zeitn - 3.5))^5:gru1
> ordered(I(zeitn - 3.5)).L:gru2
> ordered(I(zeitn - 3.5)).Q:gru2
> ordered(I(zeitn - 3.5)).C:gru2
> ordered(I(zeitn - 3.5))^4:gru2
> ordered(I(zeitn - 3.5))^5:gru2
> ordered(I(zeitn - 3.5)).L:gru3
> ordered(I(zeitn - 3.5)).Q:gru3
> ordered(I(zeitn - 3.5)).C:gru3
> ordered(I(zeitn - 3.5))^4:gru3 0.0
> ordered(I(zeitn - 3.5))^5:gru3 0.0            0.0
>
> Standardized Within-Group Residuals:
>        Min          Q1         Med          Q3         Max
> -3.10206117 -0.62626454  0.02807962  0.64554138  2.13155536
>
> Number of Observations: 672
> Number of Groups:
>             subgr subject %in% subgr
>                28                112
>                            numDF denDF   F-value p-value
> (Intercept)                     1   540 1426.5315  <.0001
> ordered(I(zeitn - 3.5))         5   540   27.9740  <.0001
> gru                             3    24    1.0410  0.3924
> ordered(I(zeitn - 3.5)):gru    15   540    1.4115  0.1363
> Approximate 95% confidence intervals
>
> Fixed effects:
>                                    lower        est.        upper
> (Intercept)                     5.2915086  5.58731309  5.883117621
> ordered(I(zeitn - 3.5)).L      -0.2109689  0.07257212  0.356113124
> ordered(I(zeitn - 3.5)).Q       1.0161942  1.26673073  1.517267227
> ordered(I(zeitn - 3.5)).C      -0.2397825  0.01075396  0.261290456
> ordered(I(zeitn - 3.5))^4      -1.0638138 -0.81327731 -0.562740815
> ordered(I(zeitn - 3.5))^5      -0.1801634  0.07037312  0.320909612
> gru1                           -0.5648856  0.05669953  0.678284624
> gru2                            0.0574723  0.67905739  1.300642487
> gru3                           -0.7630097 -0.14142458  0.480160517
> ordered(I(zeitn - 3.5)).L:gru1 -0.6374343 -0.07035232  0.496729683
> ordered(I(zeitn - 3.5)).Q:gru1 -0.8614532 -0.36038020  0.140692783
> ordered(I(zeitn - 3.5)).C:gru1 -0.6634839 -0.16241093  0.338662057
> ordered(I(zeitn - 3.5))^4:gru1 -0.4147301  0.08634286  0.587415843
> ordered(I(zeitn - 3.5))^5:gru1 -0.5182803 -0.01720729  0.483865692
> ordered(I(zeitn - 3.5)).L:gru2  0.2218139  0.78889594  1.355977946
> ordered(I(zeitn - 3.5)).Q:gru2 -0.4676866  0.03338637  0.534459352
> ordered(I(zeitn - 3.5)).C:gru2 -0.4113159  0.08975711  0.590830099
> ordered(I(zeitn - 3.5))^4:gru2 -0.9036894 -0.40261640  0.098456584
> ordered(I(zeitn - 3.5))^5:gru2 -1.0089275 -0.50785453 -0.006781542
> ordered(I(zeitn - 3.5)).L:gru3 -1.0062815 -0.43919953  0.127882479
> ordered(I(zeitn - 3.5)).Q:gru3 -0.4749680  0.02610502  0.527178001
> ordered(I(zeitn - 3.5)).C:gru3 -0.7747163 -0.27364336  0.227429629
> ordered(I(zeitn - 3.5))^4:gru3 -0.6648114 -0.16373838  0.337334604
> ordered(I(zeitn - 3.5))^5:gru3 -0.2968991  0.20417390  0.705246883
> attr(,"label")
> [1] "Fixed effects:"
>
> Random Effects:
>  Level: subgr
>                    lower      est.     upper
> sd((Intercept)) 0.3464888 0.5833753 0.9822158
>  Level: subject
>                            lower      est.     upper
> sd((Intercept))         0.3640439 0.6453960 1.1441916
> sd(zeitn)               0.1000264 0.1709843 0.2922790
> cor((Intercept),zeitn) -0.6712236 0.1295558 0.7907922
>
> Within-group standard error:
>   lower     est.    upper
> 1.265702 1.349763 1.439406
>
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Re: decide between polynomial vs ordered factor model (lme)

Douglas Bates
In reply to this post by Leo Gürtler
On 1/9/06, Leo Gürtler <[hidden email]> wrote:

> Dear alltogether,
>
> two lme's, the data are available at:
>
> http://www.anicca-vijja.de/lg/hlm3_nachw.Rdata
>
> explanations of the data:
>
> nachw = post hox knowledge tests over 6 measure time points (= equally
> spaced)
> zeitn = time points (n = 6)
> subgr = small learning groups (n = 28)
> gru = 4 different groups = treatment factor
>
> levels: time (=zeitn) (n=6) within subject (n=4) within smallgroups
> (=gru) (n = 28), i.e. n = 4 * 28 = 112 persons and 112 * 6 = 672 data points
>
> library(nlme)
> fitlme7 <- lme(nachw ~ I(zeitn-3.5) + I((zeitn-3.5)^2) +
> I((zeitn-3.5)^3) + I((zeitn-3.5)^4)*gru, random = list(subgr = ~ 1,
> subject = ~ zeitn), data = hlm3)
>
> fit5 <- lme(nachw ~ ordered(I(zeitn-3.5))*gru, random = list(subgr =
> ~ 1, subject = ~ zeitn), data = hlm3)
>
> anova( update(fit5, method="ML"), update(fitlme7, method="ML") )
>
>  > anova( update(fit5, method="ML"), update(fitlme7, method="ML") )
>                                 Model df      AIC      BIC    logLik   Test
> update(fit5, method = "ML")        1 29 2535.821 2666.619 -1238.911
> update(fitlme7, method = "ML")     2 16 2529.719 2601.883 -1248.860 1 vs 2
>                                  L.Ratio p-value
> update(fit5, method = "ML")
> update(fitlme7, method = "ML") 19.89766  0.0978
>  >
>
> shows that both are ~ equal, although I know about the uncertainty of ML
> tests with lme(). Both models show that the ^2 and the ^4 terms are
> important parts of the model.
>
> My question is:
>
> - Is it legitimate to choose a model based on these outputs according to
> theoretical considerations instead of statistical tests that not really
> show a superiority of one model over the other one?
>
> - Is there another criterium I've overlooked to decide which model can be
> clearly preferred?
>
> - The idea behind that is that in the one model (fit5) the second
> contrast of the factor (gru) is statistically significant, although not
> the whole factor in the anova output.
> In the other model, this is not the case.
> Theoretically interesting is of course the significance of the second
> contrast of gru, as it shows a tendency of one treatment being slightly
> superior. I want to choose this model but I am not sure whether this is
> proper action. Both models shows this trend, but only one model clearly
> indicates that this trend bears some empirical meaning.
>
> Thanks for any suggestions,

The comparisons may be more clearly shown if you create the ordered
factor and a second version of the ordered factor what has the
contrasts set so it produces a 4th order polynomial.  That is, set

hlm3$ozeit <- ordered(hlm3$zeitn)
hlm3$ozeit4 <- C(hlm3$ozeit, contr.poly, 4)

then define one model in terms of ozeit and a second model in terms of ozeit4.

I would go further and create a new binary factor from gru that
contrasted level 2 against the other three levels and use that instead
of gru.

For a model fit by lme I would use the one-argument form of anova to
assess the significance of terms in the fixed effects.  (That advice
doesn't hold for models fit by lmer - at least at present.)

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