[R] decide between polynomial vs ordered factor model (lme)

From: Leo Grtler <leog_at_anicca-vijja.de>
Date: Tue 10 Jan 2006 - 00:59:06 EST


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:

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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