1

我有以下纵向（3 Timepoints）数据集：

# A tibble: 504 x 6
      ID   Age Age_group Sex    Timepoint  outcome
   <int> <int> <fct>     <chr>  <chr>        <dbl>
 1 33714    15 Young     Male   bl        0.00103 
 2 35377    15 Young     Female bl        0.00106 
 3 38623    45 Older     Female bl        0.00103 
 4 38806    66 Older     Female bl        0.00114 
 5 39593    69 Older     Female bl        0.00113 
 6 39820    60 Older     Female bl        0.00113 
 7 39951    46 Older     Male   bl        0.000986
 8 40286    68 Older     Female bl        0.00107 
 9 40556     9 Young     Male   bl        0.00114 
10 40798    11 Young     Male   bl        0.00111 
# ... with 494 more rows

数据：

39820L, 39951L, 40286L, 40556L, 40798L, 40800L, 40815L, 43762L, 
50848L, 52183L, 52461L, 52577L, 53320L, 53873L, 54153L, 54206L, 
54581L, 55122L, 55267L, 55462L, 55612L, 55920L, 56022L, 56307L, 
56420L, 56679L, 57405L, 57445L, 57480L, 57725L, 57809L, 58004L, 
58215L, 58229L, 58503L, 59326L, 59327L, 59344L, 59361L, 59865L, 
60099L, 60100L, 60280L, 60384L, 60429L, 60493L, 60503L, 60575L, 
60603L, 60664L, 60846L, 61415L, 61656L, 61749L, 61883L, 62081L, 
62210L, 62285L, 62937L, 62983L, 63327L, 63329L, 64081L, 64328L, 
64418L, 64507L, 64596L, 65178L, 65250L, 65302L, 65478L, 65480L, 
65487L, 65572L, 65802L, 65935L, 65974L, 65975L, 65978L, 65991L, 
65995L, 66013L, 66154L, 66237L, 66245L, 66389L, 66396L, 66460L, 
66572L, 66589L, 67174L, 73230L, 73525L, 73539L, 73677L, 73942L, 
73953L, 74034L, 74113L, 74114L, 74427L, 74439L, 74607L, 74641L, 
74657L, 74794L, 74800L, 74836L, 74942L, 74952L, 74962L, 74969L, 
74977L, 74985L, 74989L, 75220L, 75229L, 75407L, 75653L, 75732L, 
75735L, 75757L, 75895L, 75898L, 76381L, 76559L, 76574L, 76594L, 
76595L, 76746L, 76751L, 76755L, 76759L, 76775L, 77088L, 77091L, 
77099L, 77134L, 77188L, 77203L, 77252L, 77304L, 77413L, 77453L, 
77528L, 77556L, 77585L, 77668L, 78262L, 79724L, 79730L, 79850L, 
79977L, 80052L, 80819L, 80901L, 80932L, 81064L, 81065L, 81071L, 
81098L, 81142L, 81175L, 33714L, 35377L, 38623L, 38806L, 39593L, 
39820L, 39951L, 40286L, 40556L, 40798L, 40800L, 40815L, 43762L, 
50848L, 52183L, 52461L, 52577L, 53320L, 53873L, 54153L, 54206L, 
54581L, 55122L, 55267L, 55462L, 55612L, 55920L, 56022L, 56307L, 
56420L, 56679L, 57405L, 57445L, 57480L, 57725L, 57809L, 58004L, 
58215L, 58229L, 58503L, 59326L, 59327L, 59344L, 59361L, 59865L, 
60099L, 60100L, 60280L, 60384L, 60429L, 60493L, 60503L, 60575L, 
60603L, 60664L, 60846L, 61415L, 61656L, 61749L, 61883L, 62081L, 
62210L, 62285L, 62937L, 62983L, 63327L, 63329L, 64081L, 64328L, 
64418L, 64507L, 64596L, 65178L, 65250L, 65302L, 65478L, 65480L, 
65487L, 65572L, 65802L, 65935L, 65974L, 65975L, 65978L, 65991L, 
65995L, 66013L, 66154L, 66237L, 66245L, 66389L, 66396L, 66460L, 
66572L, 66589L, 67174L, 73230L, 73525L, 73539L, 73677L, 73942L, 
73953L, 74034L, 74113L, 74114L, 74427L, 74439L, 74607L, 74641L, 
74657L, 74794L, 74800L, 74836L, 74942L, 74952L, 74962L, 74969L, 
74977L, 74985L, 74989L, 75220L, 75229L, 75407L, 75653L, 75732L, 
75735L, 75757L, 75895L, 75898L, 76381L, 76559L, 76574L, 76594L, 
76595L, 76746L, 76751L, 76755L, 76759L, 76775L, 77088L, 77091L, 
77099L, 77134L, 77188L, 77203L, 77252L, 77304L, 77413L, 77453L, 
77528L, 77556L, 77585L, 77668L, 78262L, 79724L, 79730L, 79850L, 
79977L, 80052L, 80819L, 80901L, 80932L, 81064L, 81065L, 81071L, 
81098L, 81142L, 81175L, 33714L, 35377L, 38623L, 38806L, 39593L, 
39820L, 39951L, 40286L, 40556L, 40798L, 40800L, 40815L, 43762L, 
50848L, 52183L, 52461L, 52577L, 53320L, 53873L, 54153L, 54206L, 
54581L, 55122L, 55267L, 55462L, 55612L, 55920L, 56022L, 56307L, 
56420L, 56679L, 57405L, 57445L, 57480L, 57725L, 57809L, 58004L, 
58215L, 58229L, 58503L, 59326L, 59327L, 59344L, 59361L, 59865L, 
60099L, 60100L, 60280L, 60384L, 60429L, 60493L, 60503L, 60575L, 
60603L, 60664L, 60846L, 61415L, 61656L, 61749L, 61883L, 62081L, 
62210L, 62285L, 62937L, 62983L, 63327L, 63329L, 64081L, 64328L, 
64418L, 64507L, 64596L, 65178L, 65250L, 65302L, 65478L, 65480L, 
65487L, 65572L, 65802L, 65935L, 65974L, 65975L, 65978L, 65991L, 
65995L, 66013L, 66154L, 66237L, 66245L, 66389L, 66396L, 66460L, 
66572L, 66589L, 67174L, 73230L, 73525L, 73539L, 73677L, 73942L, 
73953L, 74034L, 74113L, 74114L, 74427L, 74439L, 74607L, 74641L, 
74657L, 74794L, 74800L, 74836L, 74942L, 74952L, 74962L, 74969L, 
74977L, 74985L, 74989L, 75220L, 75229L, 75407L, 75653L, 75732L, 
75735L, 75757L, 75895L, 75898L, 76381L, 76559L, 76574L, 76594L, 
76595L, 76746L, 76751L, 76755L, 76759L, 76775L, 77088L, 77091L, 
77099L, 77134L, 77188L, 77203L, 77252L, 77304L, 77413L, 77453L, 
77528L, 77556L, 77585L, 77668L, 78262L, 79724L, 79730L, 79850L, 
79977L, 80052L, 80819L, 80901L, 80932L, 81064L, 81065L, 81071L, 
81098L, 81142L, 81175L), Age = c(15L, 15L, 45L, 66L, 69L, 60L, 
46L, 68L, 9L, 11L, 16L, 9L, 56L, 16L, 16L, 14L, 53L, 8L, 6L, 
63L, 14L, 10L, 15L, 13L, 15L, 8L, 9L, 9L, 8L, 9L, 9L, 13L, 58L, 
10L, 7L, 8L, 8L, 6L, 15L, 43L, 8L, 11L, 44L, 70L, 14L, 12L, 10L, 
16L, 12L, 10L, 6L, 13L, 67L, 11L, 12L, 13L, 10L, 66L, 13L, 14L, 
12L, 45L, 52L, 64L, 17L, 9L, 12L, 44L, 69L, 11L, 10L, 12L, 10L, 
10L, 70L, 54L, 45L, 43L, 54L, 14L, 42L, 44L, 16L, 15L, 43L, 45L, 
50L, 53L, 53L, 49L, 69L, 14L, 65L, 14L, 13L, 67L, 59L, 52L, 54L, 
44L, 62L, 69L, 10L, 63L, 57L, 12L, 62L, 9L, 53L, 54L, 66L, 49L, 
63L, 51L, 9L, 45L, 49L, 49L, 61L, 62L, 57L, 67L, 65L, 45L, 16L, 
55L, 64L, 67L, 56L, 52L, 63L, 10L, 62L, 14L, 66L, 68L, 15L, 13L, 
43L, 47L, 55L, 69L, 67L, 52L, 15L, 64L, 55L, 44L, 13L, 48L, 71L, 
64L, 13L, 50L, 61L, 70L, 57L, 51L, 46L, 57L, 69L, 46L, 8L, 11L, 
46L, 71L, 38L, 56L, 16L, 16L, 46L, 67L, 70L, 61L, 47L, 69L, 11L, 
13L, 18L, 10L, 57L, 18L, 18L, 15L, 54L, 10L, 8L, 64L, 15L, 12L, 
16L, 14L, 16L, 9L, 11L, 11L, 10L, 10L, 11L, 14L, 59L, 12L, 8L, 
9L, 9L, 8L, 16L, 44L, 9L, 13L, 45L, 71L, 16L, 13L, 12L, 18L, 
13L, 11L, 8L, 14L, 68L, 12L, 13L, 14L, 11L, 67L, 14L, 15L, 14L, 
46L, 53L, 65L, 18L, 11L, 14L, 46L, 70L, 12L, 12L, 13L, 11L, 11L, 
71L, 55L, 46L, 44L, 55L, 15L, 43L, 45L, 17L, 16L, 44L, 46L, 51L, 
55L, 54L, 50L, 70L, 15L, 66L, 15L, 14L, 68L, 60L, 53L, 55L, 46L, 
63L, 70L, 11L, 64L, 58L, 13L, 63L, 10L, 54L, 55L, 67L, 50L, 64L, 
52L, 11L, 46L, 50L, 50L, 62L, 63L, 58L, 68L, 66L, 46L, 18L, 56L, 
65L, 68L, 57L, 53L, 64L, 11L, 63L, 15L, 67L, 69L, 16L, 14L, 44L, 
48L, 56L, 70L, 68L, 53L, 17L, 65L, 56L, 45L, 14L, 49L, 73L, 65L, 
14L, 50L, 62L, 71L, 58L, 52L, 47L, 58L, 70L, 47L, 9L, 12L, 47L, 
72L, 39L, 57L, 18L, 18L, 47L, 68L, 71L, 62L, 48L, 70L, 12L, 14L, 
19L, 11L, 58L, 19L, 19L, 16L, 55L, 11L, 9L, 65L, 17L, 13L, 18L, 
16L, 18L, 11L, 12L, 12L, 11L, 11L, 12L, 16L, 60L, 13L, 9L, 11L, 
10L, 9L, 17L, 45L, 11L, 14L, 46L, 72L, 17L, 14L, 13L, 19L, 15L, 
12L, 9L, 15L, 69L, 14L, 14L, 15L, 12L, 68L, 16L, 17L, 15L, 47L, 
54L, 66L, 20L, 12L, 15L, 47L, 71L, 13L, 13L, 14L, 12L, 12L, 72L, 
56L, 47L, 45L, 56L, 16L, 44L, 46L, 19L, 18L, 44L, 47L, 52L, 56L, 
55L, 51L, 71L, 16L, 67L, 16L, 15L, 69L, 60L, 54L, 56L, 46L, 64L, 
71L, 12L, 65L, 59L, 14L, 64L, 11L, 55L, 57L, 68L, 51L, 65L, 53L, 
11L, 47L, 51L, 51L, 63L, 64L, 59L, 69L, 67L, 48L, 19L, 57L, 66L, 
69L, 59L, 54L, 65L, 12L, 64L, 16L, 68L, 70L, 17L, 15L, 45L, 48L, 
57L, 71L, 69L, 54L, 18L, 66L, 57L, 50L, 15L, 50L, 74L, 66L, 15L, 
51L, 63L, 72L, 59L, 53L, 48L, 59L, 71L, 48L, 10L, 13L, 48L, 73L, 
40L, 58L), Age_group = structure(c(1L, 1L, 2L, 2L, 2L, 2L, 2L, 
2L, 1L, 1L, 1L, 1L, 2L, 1L, 1L, 1L, 2L, 1L, 1L, 2L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 2L, 1L, 1L, 1L, 1L, 1L, 1L, 
2L, 1L, 1L, 2L, 2L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 2L, 1L, 1L, 
1L, 1L, 2L, 1L, 1L, 1L, 2L, 2L, 2L, 1L, 1L, 1L, 2L, 2L, 1L, 1L, 
1L, 1L, 1L, 2L, 2L, 2L, 2L, 2L, 1L, 2L, 2L, 1L, 1L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 1L, 2L, 1L, 1L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 1L, 
2L, 2L, 1L, 2L, 1L, 2L, 2L, 2L, 2L, 2L, 2L, 1L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 1L, 2L, 2L, 2L, 2L, 2L, 2L, 1L, 2L, 1L, 2L, 
2L, 1L, 1L, 2L, 2L, 2L, 2L, 2L, 2L, 1L, 2L, 2L, 2L, 1L, 2L, 2L, 
2L, 1L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 1L, 1L, 2L, 2L, 2L, 
2L, 1L, 1L, 2L, 2L, 2L, 2L, 2L, 2L, 1L, 1L, 1L, 1L, 2L, 1L, 1L, 
1L, 2L, 1L, 1L, 2L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 2L, 1L, 1L, 1L, 1L, 1L, 1L, 2L, 1L, 1L, 2L, 2L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 2L, 1L, 1L, 1L, 1L, 2L, 1L, 1L, 1L, 2L, 2L, 
2L, 1L, 1L, 1L, 2L, 2L, 1L, 1L, 1L, 1L, 1L, 2L, 2L, 2L, 2L, 2L, 
1L, 2L, 2L, 1L, 1L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 1L, 2L, 1L, 1L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 1L, 2L, 2L, 1L, 2L, 1L, 2L, 2L, 2L, 
2L, 2L, 2L, 1L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 1L, 2L, 2L, 
2L, 2L, 2L, 2L, 1L, 2L, 1L, 2L, 2L, 1L, 1L, 2L, 2L, 2L, 2L, 2L, 
2L, 1L, 2L, 2L, 2L, 1L, 2L, 2L, 2L, 1L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 1L, 1L, 2L, 2L, 2L, 2L, 1L, 1L, 2L, 2L, 2L, 2L, 2L, 
2L, 1L, 1L, 1L, 1L, 2L, 1L, 1L, 1L, 2L, 1L, 1L, 2L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 2L, 1L, 1L, 1L, 1L, 1L, 1L, 
2L, 1L, 1L, 2L, 2L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 2L, 1L, 1L, 
1L, 1L, 2L, 1L, 1L, 1L, 2L, 2L, 2L, 1L, 1L, 1L, 2L, 2L, 1L, 1L, 
1L, 1L, 1L, 2L, 2L, 2L, 2L, 2L, 1L, 2L, 2L, 1L, 1L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 1L, 2L, 1L, 1L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 1L, 
2L, 2L, 1L, 2L, 1L, 2L, 2L, 2L, 2L, 2L, 2L, 1L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 1L, 2L, 2L, 2L, 2L, 2L, 2L, 1L, 2L, 1L, 2L, 
2L, 1L, 1L, 2L, 2L, 2L, 2L, 2L, 2L, 1L, 2L, 2L, 2L, 1L, 2L, 2L, 
2L, 1L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 1L, 1L, 2L, 2L, 2L, 
2L), .Label = c("Young", "Older"), class = "factor"), Sex = c("Male", 
"Female", "Female", "Female", "Female", "Female", "Male", "Female", 
"Male", "Male", "Male", "Male", "Female", "Male", "Male", "Male", 
"Female", "Male", "Male", "Male", "Male", "Male", "Male", "Male", 
"Male", "Female", "Female", "Male", "Female", "Male", "Female", 
"Female", "Female", "Male", "Female", "Female", "Male", "Female", 
"Male", "Female", "Male", "Female", "Male", "Male", "Female", 
"Male", "Male", "Male", "Female", "Female", "Female", "Male", 
"Female", "Male", "Female", "Female", "Male", "Male", "Female", 
"Male", "Male", "Female", "Female", "Male", "Male", "Female", 
"Female", "Female", "Female", "Female", "Male", "Male", "Male", 
"Male", "Female", "Female", "Female", "Female", "Female", "Male", 
"Female", "Female", "Male", "Male", "Female", "Male", "Female", 
"Female", "Female", "Female", "Female", "Male", "Male", "Female", 
"Female", "Male", "Female", "Female", "Female", "Female", "Female", 
"Female", "Female", "Female", "Female", "Female", "Female", "Female", 
"Female", "Female", "Male", "Female", "Male", "Female", "Male", 
"Female", "Female", "Female", "Female", "Female", "Male", "Male", 
"Female", "Female", "Male", "Female", "Male", "Female", "Female", 
"Male", "Female", "Female", "Female", "Male", "Female", "Male", 
"Male", "Male", "Female", "Female", "Male", "Male", "Male", "Female", 
"Female", "Male", "Female", "Female", "Male", "Female", "Female", 
"Male", "Female", "Male", "Male", "Male", "Male", "Female", "Male", 
"Male", "Female", "Male", "Male", "Male", "Male", "Male", "Male", 
"Female", "Male", "Female", "Female", "Female", "Female", "Female", 
"Male", "Female", "Male", "Male", "Male", "Male", "Female", "Male", 
"Male", "Male", "Female", "Male", "Male", "Male", "Male", "Male", 
"Male", "Male", "Male", "Female", "Female", "Male", "Female", 
"Male", "Female", "Female", "Female", "Male", "Female", "Female", 
"Male", "Female", "Male", "Female", "Male", "Female", "Male", 
"Male", "Female", "Male", "Male", "Male", "Female", "Female", 
"Female", "Male", "Female", "Male", "Female", "Female", "Male", 
"Male", "Female", "Male", "Male", "Female", "Female", "Male", 
"Male", "Female", "Female", "Female", "Female", "Female", "Male", 
"Male", "Male", "Male", "Female", "Female", "Female", "Female", 
"Female", "Male", "Female", "Female", "Male", "Male", "Female", 
"Male", "Female", "Female", "Female", "Female", "Female", "Male", 
"Male", "Female", "Female", "Male", "Female", "Female", "Female", 
"Female", "Female", "Female", "Female", "Female", "Female", "Female", 
"Female", "Female", "Female", "Female", "Male", "Female", "Male", 
"Female", "Male", "Female", "Female", "Female", "Female", "Female", 
"Male", "Male", "Female", "Female", "Male", "Female", "Male", 
"Female", "Female", "Male", "Female", "Female", "Female", "Male", 
"Female", "Male", "Male", "Male", "Female", "Female", "Male", 
"Male", "Male", "Female", "Female", "Male", "Female", "Female", 
"Male", "Female", "Female", "Male", "Female", "Male", "Male", 
"Male", "Male", "Female", "Male", "Male", "Female", "Male", "Male", 
"Male", "Male", "Male", "Male", "Female", "Male", "Female", "Female", 
"Female", "Female", "Female", "Male", "Female", "Male", "Male", 
"Male", "Male", "Female", "Male", "Male", "Male", "Female", "Male", 
"Male", "Male", "Male", "Male", "Male", "Male", "Male", "Female", 
"Female", "Male", "Female", "Male", "Female", "Female", "Female", 
"Male", "Female", "Female", "Male", "Female", "Male", "Female", 
"Male", "Female", "Male", "Male", "Female", "Male", "Male", "Male", 
"Female", "Female", "Female", "Male", "Female", "Male", "Female", 
"Female", "Male", "Male", "Female", "Male", "Male", "Female", 
"Female", "Male", "Male", "Female", "Female", "Female", "Female", 
"Female", "Male", "Male", "Male", "Male", "Female", "Female", 
"Female", "Female", "Female", "Male", "Female", "Female", "Male", 
"Male", "Female", "Male", "Female", "Female", "Female", "Female", 
"Female", "Male", "Male", "Female", "Female", "Male", "Female", 
"Female", "Female", "Female", "Female", "Female", "Female", "Female", 
"Female", "Female", "Female", "Female", "Female", "Female", "Male", 
"Female", "Male", "Female", "Male", "Female", "Female", "Female", 
"Female", "Female", "Male", "Male", "Female", "Female", "Male", 
"Female", "Male", "Female", "Female", "Male", "Female", "Female", 
"Female", "Male", "Female", "Male", "Male", "Male", "Female", 
"Female", "Male", "Male", "Male", "Female", "Female", "Male", 
"Female", "Female", "Male", "Female", "Female", "Male", "Female", 
"Male", "Male", "Male", "Male", "Female", "Male", "Male", "Female", 
"Male", "Male", "Male", "Male", "Male", "Male", "Female"), Timepoint = c("bl", 
"bl", "bl", "bl", "bl", "bl", "bl", "bl", "bl", "bl", "bl", "bl", 
"bl", "bl", "bl", "bl", "bl", "bl", "bl", "bl", "bl", "bl", "bl", 
"bl", "bl", "bl", "bl", "bl", "bl", "bl", "bl", "bl", "bl", "bl", 
"bl", "bl", "bl", "bl", "bl", "bl", "bl", "bl", "bl", "bl", "bl", 
"bl", "bl", "bl", "bl", "bl", "bl", "bl", "bl", "bl", "bl", "bl", 
"bl", "bl", "bl", "bl", "bl", "bl", "bl", "bl", "bl", "bl", "bl", 
"bl", "bl", "bl", "bl", "bl", "bl", "bl", "bl", "bl", "bl", "bl", 
"bl", "bl", "bl", "bl", "bl", "bl", "bl", "bl", "bl", "bl", "bl", 
"bl", "bl", "bl", "bl", "bl", "bl", "bl", "bl", "bl", "bl", "bl", 
"bl", "bl", "bl", "bl", "bl", "bl", "bl", "bl", "bl", "bl", "bl", 
"bl", "bl", "bl", "bl", "bl", "bl", "bl", "bl", "bl", "bl", "bl", 
"bl", "bl", "bl", "bl", "bl", "bl", "bl", "bl", "bl", "bl", "bl", 
"bl", "bl", "bl", "bl", "bl", "bl", "bl", "bl", "bl", "bl", "bl", 
"bl", "bl", "bl", "bl", "bl", "bl", "bl", "bl", "bl", "bl", "bl", 
"bl", "bl", "bl", "bl", "bl", "bl", "bl", "bl", "bl", "bl", "bl", 
"bl", "bl", "flu1", "flu1", "flu1", "flu1", "flu1", "flu1", "flu1", 
"flu1", "flu1", "flu1", "flu1", "flu1", "flu1", "flu1", "flu1", 
"flu1", "flu1", "flu1", "flu1", "flu1", "flu1", "flu1", "flu1", 
"flu1", "flu1", "flu1", "flu1", "flu1", "flu1", "flu1", "flu1", 
"flu1", "flu1", "flu1", "flu1", "flu1", "flu1", "flu1", "flu1", 
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), class = c("tbl_df", "tbl", "data.frame"))

outcomeID是在timepoint-bl(1)、timepoint-flu1(2) 和timepoint-flu2(3)对每个受试者 ( ) 进行采样的生理测量。每个主题age，age_group并sex记录为好。Age是一个预测变量，outcome并sex用作交互变量。

目标：

  1. 我试图估计3和成人之间的outcome变量斜率是否存在显着差异。 YoungOlderTimepoints

  2. 我正在尝试估计和之间的个体( ID) 方差。outcometimepoints age_group

这是为我的目标建模的正确方法吗？情节在我看来并不准确。我是线性混合模型的新手，所以任何建议都值得赞赏。

#mixed
DF_mixed <- lmer(outcome ~ Age* Sex + (1 + Age| Age_group:Timepoint) + (1|ID), data = DF) 
# identify age*sex interactions to predict outcome value between timepoints, nested in age_group that have random slope and intercepts

#plot
ggplot(DF, aes(x = Age, y = outcome, color = Timepoint)) +
      geom_point(alpha = 0.7) +
      theme_classic() +
      geom_line(data = cbind(DF, pred = predict(DF_mixed)), aes(y = pred), size = 1)   # adding predicted line from mixed model
4

1 回答 1

2

下面，我将介绍有关 (1) 准备数据、(2) 指定模型和 (3) 检查模型结果的建议。

除了tidyverse包 和lme4，我将使用broom.mixed::glance()和tidy()来检查模型参数，并ggeffects::ggpredict()生成用于绘图的模型预测。

library(tidyverse)
library(lme4)
library(broom.mixed)
library(ggeffects)

数据准备和变量选择

Timepoint是一个字符，将被视为模型中的无序类别，这可能不是您想要的。使用自基线以来经过的实际时间通常比使用通用“时间点”更好，尤其是在时间点之间的距离可变的情况下。Timepoint_yrs对于这个答案，我根据每个受试者相对于基线的年龄变化计算了一个替代值；但如果你有更精确的时间记录，你可能想用它来代替。

此外，outcome由小范围的值组成，大约为 1e-4 - 1e-3；这可能会导致 lmer 行为不端，因此我对其进行了缩放。

DF <- DF %>% 
  group_by(ID) %>% 
  mutate(Timepoint_yrs = Age - min(Age)) %>% 
  ungroup() %>% 
  mutate(outcome_scaled = as.numeric(scale(outcome)))

Age将两者都包含Age_group在模型中会导致多重共线性问题，因为一个是另一个的函数。鉴于 的极端双峰分布Age，我只使用了Age_group. 如上所述，年龄的变化现在由Timepoint_yrs.

型号规格

“我正在尝试估计结果变量的斜率是否存在显着差异Young和Older成人之间的 3 Timepoints” - 即，Age_group通过Timepoint_yrs交互。您还写道“sex用作交互变量”。因此，将这些放在一起，您正在通过交互查看 3 Age_group- way Sex。Timepoint_yrs因为Age_group并且Sex仅在主题之间有所不同，所以我们将在固定效果的级别上指定这一点：

outcome_scaled ~ Age_group * Sex * Timepoint_yrs

因为每个人有多个测量值 ( ID)，我们将包含以下的随机截距ID：

outcome_scaled ~ Age_group * Sex * Timepoint_yrs + (1 | ID)

我们还可以允许 的斜率Timepoint_yrs在个体内变化。从你的帖子中，我看不出是否包含随机斜率的强有力的理论理由——所以我们会同时做这两件事，看看哪种模型最适合。

# model 1 - random intercepts
rnd_intercept <- lmer(outcome_scaled ~ Age_group * Sex * Timepoint_yrs + (1 | ID), data = DF) 

# model 2 - random intercepts and slopes
rnd_int_slope <- lmer(outcome_scaled ~ Age_group * Sex * Timepoint_yrs + (Timepoint_yrs | ID), data = DF)

模型结果

glance(rnd_intercept)
# # A tibble: 1 x 6
# sigma logLik   AIC   BIC REMLcrit df.residual
#   <dbl>  <dbl> <dbl> <dbl>    <dbl>       <int>
# 1 0.510  -575. 1170. 1212.    1150.         494

glance(rnd_int_slopes)
# A tibble: 1 x 6
#   sigma logLik   AIC   BIC REMLcrit df.residual
#   <dbl>  <dbl> <dbl> <dbl>    <dbl>       <int>
# 1 0.487  -575. 1173. 1224.    1149.         492

看看 AIC 和 BIC，随机截距模型似乎更好/更简约；没有任何理论上的理由偏爱随机斜率模型，我们将坚持使用该模型。

tidy(rnd_intercept, conf.int = TRUE)
# # A tibble: 10 x 8
#    effect   group    term               estimate std.error statistic conf.low conf.high
#    <chr>    <chr>    <chr>                 <dbl>     <dbl>     <dbl>    <dbl>     <dbl>
#  1 fixed    NA       (Intercept)         -0.123     0.122    -1.01    -0.361     0.116 
#  2 fixed    NA       Age_groupYoung      -0.0142    0.221    -0.0643  -0.448     0.419 
#  3 fixed    NA       SexMale              0.248     0.214     1.16    -0.171     0.667 
#  4 fixed    NA       Timepoint_yrs        0.131     0.0407    3.21     0.0509    0.210 
#  5 fixed    NA       Age_groupYoung:Se~  -0.0704    0.317    -0.222   -0.692     0.551 
#  6 fixed    NA       Age_groupYoung:Ti~  -0.152     0.0654   -2.32    -0.280    -0.0234
#  7 fixed    NA       SexMale:Timepoint~  -0.0547    0.0771   -0.709   -0.206     0.0964
#  8 fixed    NA       Age_groupYoung:Se~   0.0192    0.101     0.190   -0.178     0.217 
#  9 ran_pars ID       sd__(Intercept)      0.862    NA        NA       NA        NA     
# 10 ran_pars Residual sd__Observation      0.510    NA        NA       NA        NA

最后，我们可以从模型中绘制预测值。您的原始代码为原始数据中的每一行预测了一个数据点，然后基本上在每行之间追踪了一条线，正如您所指出的，这不是很有信息量。相反，我们希望为您的预测变量的每个独特组合仅基于一个预测结果值（+/- 误差）绘制图表。ggeffects::ggpredict()是一个很好的捷径；emmeans::emmeans()将是另一种选择。

plot_data <- ggpredict(rnd_intercept, terms = c("Timepoint_yrs", "Age_group", "Sex"))
# col names: x = Timepoint_yrs, group = Age_group, facet = Sex, predicted = observed_scaled

ggplot(plot_data, aes(x, predicted)) +
  geom_line(aes(color = group)) +
  geom_ribbon(aes(ymin = conf.low, ymax = conf.high, fill = group), alpha = .1) +
  facet_wrap(vars(facet)) +
  labs(x = "Years Since Baseline", y = "Outcome\n(scaled, model-predicted)") +
  theme_classic()

一个多面图，按年龄、性别和时间点显示预测值，置信度为 95%。 预测值通常会随着时间的推移而增加

Age_group根据您的模型系数，您可能还想查看 crossTimepoint_yrs以及Timepoint_yrs折叠其他变量的主要影响。您可以通过将terms参数更改为ggpredict()并相应地修改ggplot规范来做到这一点。

于 2022-02-09T03:33:48.000 回答