# [R] model specification using lme

li li hannah.hlx at gmail.com
Sun May 29 21:23:33 CEST 2016

```Hi all,
For the following data, I consider the following random intercept and
random slope model. Denote as y_ijk the response value from *j*th
individual within *i*th method at time point *k*. Assume the following
model for y_ijk:

y_ijk= (alpha_0+ tau_i +a_j(i))+(beta_i+b_j(i)) T_k + e_ijk

Here alpha_0 is the grand mean;
tau_i is the fixed effect for ith method;
a_j(i) is random intercept corresponding to the *j*th individual
within *i*th method, assumed to be common for all three methods;
beta_i is the fixed slope corresponding to the ith method;
b_j(i) is the random slope corresponding to jth individual for
the ith method, assumed to be different for different methods;
T_k is the time corresponding to y_ijk;
e_ijk is the residual.

For this model, I consider the three specification using  the lme function
as follows:

mod1 <- lme(fixed= reponse ~ method*time, random=~ 1 +time | individual,
data=one, weights= varIdent(form=~1|method),
control = lmeControl(opt = "optim"))

mod2 <- lme(fixed= reponse ~ method*time, random=~ 0 +time | individual,
data=one, weights= varIdent(form=~1|method),
control = lmeControl(opt = "optim"))

mod3 <- lme(fixed= reponse ~ method*time, random=~ method +time |
individual, data=one, weights= varIdent(form=~1|method),
control = lmeControl(opt = "optim"))

I think mod1 is the correct one. However, I am kind of confused with the
right usage of lme function. Can someone familiar with this give some help
here?

Another question is regarding the fixed effect   tau_1, tau_2 and tau_3
(corresponding to the three methods). One main question I am interested in
is whether each of them are statistically different from zero. In the
summary results below (shaded part), it looks that the result for method 2
and 3 are given with reference to method 1). Is there a way to obtain
specific result separately for alpha_0 (the overall mean) and also tau_1,
tau_2 and tau3?

Thanks very much for the help!
Hanna

> summary(mod1)Linear mixed-effects model fit by REML
Data: one
AIC      BIC    logLik
304.4703 330.1879 -140.2352

Random effects:
Formula: ~1 + time | individual
Structure: General positive-definite, Log-Cholesky parametrization
StdDev       Corr
(Intercept) 0.2487869075 (Intr)
time        0.0001841179 -0.056
Residual    0.3718305953

Variance function:
Structure: Different standard deviations per stratum
Formula: ~1 | method
Parameter estimates:
3        1        2
1.00000 26.59750 24.74476
Fixed effects: reponse ~ method * time
Value Std.Error DF   t-value p-value(Intercept)
96.65395  3.528586 57 27.391694  0.0000
method2       1.17851  4.856026 57  0.242689  0.8091
method3       5.87505  3.528617 57  1.664973  0.1014time
0.07010  0.250983 57  0.279301  0.7810
method2:time -0.12616  0.360585 57 -0.349877  0.7277
method3:time -0.08010  0.251105 57 -0.318999  0.7509
Correlation:
(Intr) methd2 methd3 time   mthd2:
method2      -0.726
method3      -0.999  0.726
time         -0.779  0.566  0.779
method2:time  0.542 -0.712 -0.542 -0.696
method3:time  0.778 -0.566 -0.779 -0.999  0.696

Standardized Within-Group Residuals:
Min          Q1         Med          Q3         Max
-2.67575293 -0.51633192  0.06742723  0.59706762  2.81061874

Number of Observations: 69
Number of Groups: 7 >

> one   response individual time method
1    102.9          3    0      3
2    103.0          3    3      3
3    103.0          3    6      3
4    102.8          3    9      3
5    102.2          3   12      3
6    102.5          3   15      3
7    103.0          3   18      3
8    102.0          3   24      3
9    102.8          1    0      3
10   102.7          1    3      3
11   103.0          1    6      3
12   102.2          1    9      3
13   103.0          1   12      3
14   102.8          1   15      3
15   102.8          1   18      3
16   102.9          1   24      3
17   102.2          2    0      3
18   102.6          2    3      3
19   103.4          2    6      3
20   102.3          2    9      3
21   101.3          2   12      3
22   102.1          2   15      3
23   102.1          2   18      3
24   102.2          2   24      3
25   102.7          4    0      3
26   102.3          4    3      3
27   102.6          4    6      3
28   102.7          4    9      3
29   102.8          4   12      3
30   102.5          5    0      3
31   102.4          5    3      3
32   102.1          5    6      3
33   102.3          6    0      3
34   102.3          6    3      3
35   101.9          7    0      3
36   102.0          7    3      3
37   107.4          3    0      1
38   101.3          3   12      1
39    92.8          3   15      1
40    73.7          3   18      1
41   104.7          3   24      1
42    92.6          1    0      1
43   101.9          1   12      1
44   106.3          1   15      1
45   104.1          1   18      1
46    95.6          1   24      1
47    79.8          2    0      1
48    89.7          2   12      1
49    97.0          2   15      1
50   108.4          2   18      1
51   103.5          2   24      1
52    96.4          4    0      1
53    89.3          4   12      1
54   112.6          5    0      1
55    93.3          6    0      1
56    99.6          7    0      1
57   109.5          3    0      2
58    98.5          3   12      2
59   103.5          3   24      2
60   113.5          1    0      2
61    94.5          1   12      2
62    88.5          1   24      2
63    99.5          2    0      2
64    97.5          2   12      2
65    98.5          2   24      2
66   103.5          4    0      2
67    89.5          5    0      2
68    87.5          6    0      2
69    82.5          7    0      2

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