r - 为混合模型选择哪个公式
问题描述
编辑：我已经在Cross Validated上发布了一个更详细的问题，因为我被告知以下内容更适合在那里发布。
我正在尝试在重复数据上编写混合模型，但我很难编写公式。
我的数据库由学校不同考试的结果组成。每行包含一个从 0 到 100 的结果。对于每个结果，我都知道学校、就读年份和考试科目。我将学校分为两种（A 和 B）；我的目标是衡量一所学校的类型（A 或 B）是否对其结果有影响。但是，正如我所说，我对每所学校都重复了措施，因为我有 20 个年级，每个年级（2014 年至 2019 年）一个年级，四个科目各一个。这是我的数据示例：
Result School_ID Year Subject School_type
1 19 1 2015 math A
2 35 1 2015 english A
3 4 1 2015 history A
4 16 1 2015 philosophy A
5 55 1 2016 math A
6 62 1 2016 english A
7 74 1 2016 history A
8 66 1 2016 philosophy A
9 32 1 2017 math A
10 16 1 2017 english A
11 42 1 2017 history A
12 52 1 2017 philosophy A
13 95 1 2018 math A
14 8 1 2018 english A
15 35 1 2018 history A
16 41 1 2018 philosophy A
17 12 1 2019 math A
18 40 1 2019 english A
19 56 1 2019 history A
20 65 2 2019 philosophy B
21 12 2 2015 math B
22 23 2 2015 english B
23 45 2 2015 history B
24 90 2 2015 philosophy B
25 3 2 2016 math B
26 66 2 2016 english B
27 51 2 2016 history B
28 26 2 2016 philosophy B
29 4 2 2017 math B
因为我有重复的数据，我知道我必须使用混合模型lmer（例如）。但是，我正在努力寻找一个好的公式。
首先我尝试了这个：
Result ~ School_type * Subject + (1|Year)
但我不确定这是一个好的公式。我也试过这个
Result ~ School_type * Subject + (Year|School_ID)
但它没有给出任何显着的结果。起初我认为我的变量School_type对结果没有显着影响。这真是出乎意料。然后我想，通过将学校视为一个随机变量，这个公式是在抑制学校对我的成绩的影响，这显然会降低 School_type 的影响，因为它是附属于学校的。
换句话说，我想了解学校特征对其结果的影响，但如果我“压制”个人的影响，这里的学校，在随机效应中，我显然不会得到任何结果。
但是，我感觉我的第一个公式：
Result ~ School_type * Subject + (1|Year)
这给了我预期的结果，没有考虑到学校重复数据的事实。
根据要求，这里是 dput(my_data_frame)。
structure(list(result = c(69.9, 72.4, 67.1, 84.4, 84.9, 68, 78.1,
65.1, 69.9, 77.5, 80.2, 84.7, 89.3, 82.6, 63.8, 40.8, 72.2, 71.4,
77.2, 79.9, 93.5, 65, 67.7, 91.8, 79.6, 73.4, 80.9, 85.7, 91.8,
67.1, 66.5, 84.6, 80.8, 87.3, 94, 87.8, 86.2, 36.6, 37.6, 18,
30, 34.8, 32.5, 21.9, 29, 22.7, 47.3, 70, 84.6, 60.8, 42.1, 18.6,
49.2, 33.9, 34.9, 47.1, 29.2, 34.5, 70.3, 56, 67.8, 60.9, 50.3,
40.4, 20.8, 45.4, 57.7, 65, 26.5, 40.1, 36.6, 49.1, 52.4, 22.8,
46.5, 42.4, 54.1, 51.3, 27.2, 42.9, 47.1, 61.6, 45.1, 86.5, 69.2,
58.4, 58.4, 46.8, 77, 46.9, 73.1, 50.1, 61.4, 49.2, 75, 75.4,
53.2, 71.9, 49.5, 27.4, 48.3, 51.9, 68.8, 69, 44.6, 39, 48.3,
77.5, 59.3, 70.9, 80.1, 73.5, 77.9, 57.3, 76.8, 67, 63.2, 89.8,
79.5, 70.8, 78.5, 79.4, 79.4, 80.5, 72, 68.6, 91.7, 75.6, 77.2,
77.8, 73, 85.3, 68.8, 64.6, 88.2, 76.9, 84.7, 88.5, 76.6, 81.8,
26.6, 30.9, 27.9, 33.5, 27.8, 8.7, 27.8, 31.8, 68, 23.5, 80.2,
54.8, 42.7, 46.1, 49.6, 27, 35.2, 30, 32, 62.9, 56.6, 70.2, 44.3,
39.3, 37, 24, 52.2, 44.2, 59.3, 30.4, 33.9, 33.8, 65.7, 42.8,
36.5, 43.1, 49.7, 34.3, 10.4, 47.4, 43.4, 60.8, 58.5, 93.2, 81.7,
79.4, 76.8, 66.3, 76.2, 68.2, 45, 46.7, 57.4, 89.6, 81.6, 73.8,
61.1, 43.2, 41.9, 74.8, 56.9, 71.5, 30.9, 59.6, 39.8, 76.3, 67.3,
54.5, 78, 92.6, 80.9, 81.7, 64, 75.2, 62.3, 75.9, 87.1, 73.1,
64.8, 79.3, 85.4, 81.3, 74.4, 90.6, 68.7, 71.3, 92.4, 79.5, 72.1,
86, 85.7, 90.8, 66.6, 62.8, 84.4, 77.9, 85.6, 86, 80.6, 49.4,
42.1, 28, 35.2, 32.5, 45.6, 35.6, 38.1, 38.8, 48.3, 32.2, 84.6,
61.4, 46.7, 68.7, 65, 26.7, 49.2, 55.2, 26.8, 21.8, 55.4, 43.4,
56, 44, 65.2, 42.4, 41.9, 27.6, 44, 51.1, 63, 32.9, 52.3, 44.8,
58.1, 44, 15.2, 44.2, 68.8, 53.3, 25, 57.8, 51.7, 60.8, 59.4,
88.2, 83.6, 74.2, 81.7, 64.8, 63.4, 61.1, 46.8, 47.2, 69.4, 76.3,
87.9, 78.7, 70.7, 60.8, 34.6, 42.3, 54.7, 62.1, 33.6, 51, 31.1,
72.5, 66.9, 69.4, 79.5, 80.6, 80.8, 70.5, 72.5, 63.6, 96.6, 78.4,
90.8, 75.9, 57.1, 76.2, 74.8, 59.9, 61.7, 82.6, 65.9, 73.8, 90.1,
83.2, 78, 83.5, 83.8, 87, 67.2, 60.7, 79.9, 76.1, 82, 82.2, 84.6,
48.1, 33, 13.5, 35.7, 42.6, 23.6, 35.4, 40.4, 41.6, 46.5, 24.6,
78.8, 73.6, 41.1, 68.5, 45.1, 38.6, 45, 78.4, 23.9, 35.6, 55.6,
53.3, 57.1, 63.2, 67.9, 55.2, 23.5, 41.3, 56, 50.6, 50.6, 44.2,
44.9, 46.4, 54.6, 41.5, 30, 53.6, 81, 37.2, 48, 56.8, 77.4, 59.2,
91.8, 77.3, 78.1, 80.8, 63.4, 54.6, 51.3, 63.5, 53.2, 76.3, 72.7,
79.6, 57.8, 40.2, 66, 58.3, 80.4, 72.8, 53.6, 54.2, 54, 65.3,
68.4, 79, 75.6, 81.8, 54.4, 68.5, 68.5, 77.9, 84.8, 87.1, 74.9,
64.9, 74.1, 63.8, 76, 86, 67.9, 81.8, 91.2, 81.2, 72.2, 74.9,
85.4, 90.8, 71, 64.5, 84.3, 83.3, 84.3, 85.1, 38.5, 46, 44.1,
49.3, 37.9, 26.9, 36.9, 32.3, 45.2, 51.9, 84.8, 65.9, 53.3, 43.3,
42.1, 71.4, 42.6, 29.4, 38.6, 49.5, 40.5, 63.8, 46.3, 28.7, 31.7,
57.1, 66.1, 21.8, 37.6, 49.5, 49.9, 55.7, 53.2, 53.2, 53.2, 40.2,
57.3, 64.4, 88.3, 74.2, 68.6, 53.4, 70.5, 52.4, 60.6, 49.6, 57.2,
62.6, 65.5, 66.7, 52.7, 46.7, 51.1, 53.4, 56.3, 73.7, 34.7, 65.1,
50, 54.8, 48.2, 59.8, 79, 50.1, 53, 54.8, 53.7, 52.8, 72.8, 55.4,
90, 64.1, 50.3, 2.5, 49.3, 46, 57.9, 42.2, 93.2, 53.1, 51.5,
80.7, 49.7, 70.9, 63.6, 63.2, 61.8, 59.3, 47.4, 54.9, 77.4, 69.7,
66.1, 48.8, 72.1, 59.5, 58.7, 54.3, 64.4, 83.9, 52.1, 53.1, 55.1,
63.3, 61.4, 79.5, 57.5, 68.3, 47.7, 70.4, 61, 53, 55.1, 89.2,
60.3, 50.2, 83.1, 76.9, 69.7, 64.7, 58.8, 62, 48.5, 46, 66.1,
74.1, 56, 69, 64.3, 54, 33.9, 58, 78.9, 53.4, 59.7, 59.1, 63.4,
52.2, 68.5, 56, 96.5, 53.3, 47.9, 60, 66.9, 56.6, 39.7, 93.5,
47.4, 40.1, 78.1, 63.1, 62.6, 64.3, 57.5, 56.8, 51.9, 48.8, 44.7,
56.4, 65.2, 70, 56.2, 63.2, 53.4, 44.6, 42, 56.2, 50.3, 50.2,
53.3, 47.5, 46.7, 64.1, 49.5, 71.9, 57.8, 37.2, 47.9, 46.9, 42.3,
38.4, 42.9, 46.4, 73.6, 60.6, 53.2, 61, 55.4, 41.7, 47.9, 49.6,
58.5, 63, 42.3, 65.7, 48.4, 53.5, 53.2, 51.5, 53.9, 47.8, 52.3,
58.5, 59.9, 59.9, 74, 57.8, 44.1, 61.5, 47, 59.4, 48.9, 40.5,
74.6, 56.6, 52.9, 72.2, 59.8, 50.1, 46.5, 68.7, 64.1, 64.6, 53.1,
64.8), school_ID = structure(c(1L, 2L, 3L, 4L, 5L, 6L, 7L, 8L,
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35L, 36L, 37L, 1L, 2L, 3L, 4L, 5L, 6L, 7L, 8L, 9L, 10L, 11L,
13L, 14L, 15L, 16L, 17L, 18L, 19L, 20L, 22L, 23L, 24L, 25L, 26L,
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19L, 20L, 21L, 22L, 23L, 24L, 25L, 26L, 27L, 28L, 29L, 30L, 31L,
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10L, 12L, 13L, 14L, 15L, 17L, 18L, 19L, 20L, 22L, 23L, 24L, 25L,
26L, 27L, 28L, 29L, 30L, 31L, 32L, 33L, 34L, 35L, 36L, 37L, 1L,
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34L, 35L, 36L, 37L, 1L, 2L, 3L, 4L, 6L, 7L, 8L, 9L, 10L, 11L,
12L, 13L, 14L, 15L, 17L, 18L, 19L, 20L, 22L, 23L, 24L, 26L, 27L,
28L, 29L, 30L, 31L, 32L, 33L, 34L, 35L, 36L, 37L, 1L, 2L, 3L,
4L, 5L, 6L, 7L, 8L, 9L, 10L, 12L, 13L, 14L, 15L, 17L, 18L, 19L,
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26L, 27L, 28L, 29L, 30L, 31L, 32L, 33L, 34L, 35L, 36L, 37L, 1L,
2L, 3L, 5L, 6L, 7L, 8L, 9L, 10L, 11L, 12L, 13L, 14L, 15L, 17L,
18L, 19L, 21L, 22L, 23L, 24L, 25L, 26L, 27L, 28L, 29L, 30L, 32L,
33L, 34L, 35L, 36L, 37L, 1L, 2L, 3L, 4L, 6L, 7L, 8L, 9L, 10L,
11L, 12L, 13L, 14L, 15L, 17L, 18L, 19L, 20L, 21L, 22L, 23L, 24L,
25L, 26L, 27L, 28L, 29L, 30L, 31L, 32L, 33L, 34L, 36L, 37L, 1L,
2L, 3L, 4L, 6L, 7L, 8L, 9L, 10L, 11L, 12L, 13L, 14L, 15L, 17L,
18L, 19L, 20L, 21L, 22L, 23L, 24L, 25L, 26L, 27L, 28L, 29L, 30L,
31L, 32L, 33L, 34L, 35L, 36L, 37L, 1L, 2L, 3L, 4L, 6L, 7L, 9L,
10L, 11L, 12L, 13L, 14L, 15L, 17L, 18L, 19L, 22L, 23L, 24L, 26L,
27L, 28L, 29L, 30L, 31L, 32L, 33L, 34L, 35L, 36L, 37L, 1L, 2L,
3L, 6L, 7L, 8L, 9L, 10L, 11L, 12L, 13L, 14L, 15L, 17L, 18L, 19L,
21L, 22L, 23L, 24L, 25L, 26L, 27L, 28L, 29L, 30L, 31L, 32L, 33L,
34L, 36L, 1L, 2L, 3L, 6L, 7L, 8L, 9L, 10L, 11L, 12L, 13L, 14L,
15L, 17L, 18L, 19L, 22L, 23L, 25L, 26L, 27L, 28L, 29L, 30L, 31L,
32L, 34L, 35L, 36L, 1L, 2L, 3L, 6L, 7L, 9L, 10L, 11L, 12L, 13L,
14L, 17L, 18L, 21L, 22L, 23L, 24L, 25L, 26L, 27L, 28L, 29L, 30L,
31L, 32L, 33L, 34L, 35L, 36L, 1L, 2L, 3L, 4L, 5L, 6L, 7L, 8L,
9L, 10L, 11L, 12L, 13L, 14L, 15L, 16L, 17L, 18L, 19L, 20L, 21L,
22L, 23L, 24L, 25L, 26L, 27L, 28L, 29L, 30L, 31L, 32L, 33L, 34L,
35L, 36L, 37L, 1L, 2L, 3L, 4L, 5L, 6L, 7L, 8L, 9L, 10L, 11L,
12L, 14L, 15L, 17L, 18L, 19L, 20L, 21L, 22L, 23L, 24L, 26L, 27L,
28L, 29L, 30L, 31L, 33L, 34L, 35L, 36L, 37L, 1L, 2L, 3L, 4L,
5L, 6L, 7L, 8L, 9L, 10L, 11L, 12L, 13L, 14L, 15L, 17L, 18L, 19L,
20L, 21L, 22L, 23L, 24L, 25L, 26L, 27L, 28L, 29L, 30L, 31L, 32L,
33L, 34L, 35L, 36L, 37L, 1L, 2L, 3L, 4L, 6L, 7L, 8L, 9L, 10L,
11L, 12L, 13L, 14L, 15L, 17L, 18L, 19L, 20L, 22L, 23L, 24L, 26L,
27L, 28L, 29L, 30L, 31L, 33L, 34L, 35L, 36L, 37L, 1L, 2L, 3L,
6L, 7L, 8L, 9L, 10L, 11L, 12L, 13L, 14L, 15L, 17L, 18L, 19L,
22L, 23L, 24L, 26L, 27L, 28L, 29L, 30L, 31L, 33L, 34L, 35L, 36L,
37L), .Label = c("1", "2", "3", "4", "5", "6", "7", "8", "9",
"10", "11", "12", "13", "14", "15", "16", "17", "18", "19", "20",
"21", "22", "23", "24", "25", "26", "27", "28", "29", "30", "31",
"32", "33", "34", "35", "36", "37"), class = "factor"), year = structure(c(5L,
5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L,
5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L,
5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L,
5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L,
5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L,
5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L,
5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 4L, 4L, 4L, 4L, 4L, 4L,
4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L,
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4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L,
4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L,
4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L,
4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 3L, 3L, 3L, 3L,
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L,
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L,
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3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L,
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2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L,
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L,
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L,
2L, 2L, 2L, 2L, 2L, 2L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
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3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L,
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 2L, 2L, 2L, 2L, 2L, 2L, 2L,
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L,
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L), .Label = c("2015", "2016", "2017",
"2018", "2019"), class = "factor"), subject = structure(c(1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L,
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L,
2L, 2L, 2L, 2L, 2L, 2L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L,
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L,
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 2L, 2L, 2L, 2L, 2L,
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L,
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 3L, 3L, 3L, 3L, 3L,
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L,
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 2L, 2L,
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L,
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L,
2L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L,
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L,
3L, 3L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L,
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L,
2L, 2L, 2L, 2L, 2L, 2L, 2L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L,
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L,
3L, 3L, 3L, 3L, 3L, 3L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L,
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L,
2L, 2L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L,
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 4L,
4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L,
4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L,
4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L,
4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L,
4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L,
4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L,
4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L,
4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L,
4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L,
4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L,
4L, 4L, 4L, 4L, 4L, 4L, 4L), .Label = c("Math", "History", "Philosiphy",
"English"), class = "factor"), school_type = structure(c(1L,
1L, 1L, 3L, 3L, 2L, 2L, 1L, 2L, 2L, 3L, 2L, 3L, 3L, 1L, 1L, 2L,
2L, 3L, 2L, 1L, 1L, 2L, 3L, 1L, 1L, 2L, 1L, 3L, 1L, 1L, 2L, 3L,
1L, 3L, 3L, 3L, 1L, 1L, 1L, 3L, 3L, 2L, 2L, 1L, 2L, 2L, 3L, 3L,
3L, 1L, 1L, 2L, 2L, 3L, 2L, 1L, 2L, 3L, 1L, 1L, 2L, 1L, 3L, 1L,
2L, 3L, 1L, 3L, 3L, 3L, 1L, 1L, 1L, 3L, 3L, 2L, 2L, 1L, 2L, 2L,
3L, 2L, 3L, 3L, 1L, 2L, 2L, 3L, 2L, 1L, 1L, 2L, 3L, 1L, 1L, 2L,
1L, 3L, 1L, 1L, 2L, 3L, 1L, 3L, 3L, 3L, 1L, 1L, 1L, 3L, 2L, 2L,
1L, 2L, 2L, 2L, 3L, 3L, 1L, 2L, 2L, 3L, 2L, 1L, 2L, 3L, 1L, 1L,
2L, 1L, 3L, 1L, 1L, 2L, 3L, 1L, 3L, 3L, 3L, 1L, 1L, 1L, 3L, 2L,
1L, 2L, 2L, 3L, 2L, 3L, 3L, 1L, 2L, 2L, 3L, 2L, 1L, 2L, 3L, 1L,
1L, 2L, 1L, 3L, 1L, 2L, 3L, 1L, 3L, 3L, 3L, 1L, 1L, 1L, 3L, 2L,
2L, 1L, 2L, 2L, 3L, 2L, 3L, 3L, 1L, 2L, 2L, 3L, 2L, 1L, 2L, 3L,
1L, 2L, 1L, 3L, 1L, 1L, 2L, 3L, 1L, 3L, 3L, 3L, 1L, 1L, 1L, 3L,
3L, 2L, 2L, 1L, 2L, 2L, 2L, 3L, 3L, 1L, 2L, 2L, 3L, 2L, 1L, 1L,
2L, 3L, 1L, 1L, 2L, 1L, 3L, 1L, 1L, 2L, 3L, 1L, 3L, 3L, 1L, 1L,
1L, 3L, 3L, 2L, 2L, 2L, 2L, 3L, 2L, 3L, 3L, 1L, 2L, 2L, 3L, 2L,
1L, 1L, 2L, 3L, 1L, 1L, 2L, 1L, 3L, 1L, 1L, 2L, 3L, 1L, 3L, 3L,
3L, 1L, 1L, 1L, 3L, 2L, 2L, 1L, 2L, 2L, 3L, 2L, 3L, 3L, 1L, 2L,
2L, 3L, 1L, 1L, 2L, 3L, 1L, 1L, 2L, 1L, 3L, 1L, 2L, 3L, 1L, 3L,
3L, 3L, 1L, 1L, 1L, 3L, 2L, 2L, 1L, 2L, 2L, 3L, 2L, 3L, 3L, 1L,
2L, 2L, 3L, 2L, 1L, 1L, 2L, 3L, 1L, 1L, 2L, 1L, 3L, 1L, 1L, 2L,
3L, 1L, 3L, 3L, 1L, 1L, 1L, 3L, 2L, 2L, 1L, 2L, 2L, 3L, 2L, 3L,
3L, 1L, 2L, 2L, 3L, 2L, 1L, 1L, 2L, 3L, 1L, 1L, 2L, 1L, 3L, 1L,
1L, 2L, 3L, 1L, 3L, 3L, 3L, 1L, 1L, 1L, 3L, 2L, 2L, 2L, 2L, 3L,
2L, 3L, 3L, 1L, 2L, 2L, 3L, 1L, 2L, 3L, 1L, 2L, 1L, 3L, 1L, 1L,
2L, 3L, 1L, 3L, 3L, 3L, 1L, 1L, 1L, 2L, 2L, 1L, 2L, 2L, 3L, 2L,
3L, 3L, 1L, 2L, 2L, 3L, 1L, 1L, 2L, 3L, 1L, 1L, 2L, 1L, 3L, 1L,
1L, 2L, 3L, 1L, 3L, 1L, 1L, 1L, 2L, 2L, 1L, 2L, 2L, 3L, 2L, 3L,
3L, 1L, 2L, 2L, 3L, 1L, 2L, 1L, 1L, 2L, 1L, 3L, 1L, 1L, 2L, 1L,
3L, 3L, 1L, 1L, 1L, 2L, 2L, 2L, 2L, 3L, 2L, 3L, 3L, 2L, 2L, 1L,
1L, 2L, 3L, 1L, 1L, 2L, 1L, 3L, 1L, 1L, 2L, 3L, 1L, 3L, 3L, 1L,
1L, 1L, 3L, 3L, 2L, 2L, 1L, 2L, 2L, 3L, 2L, 3L, 3L, 1L, 1L, 2L,
2L, 3L, 2L, 1L, 1L, 2L, 3L, 1L, 1L, 2L, 1L, 3L, 1L, 1L, 2L, 3L,
1L, 3L, 3L, 3L, 1L, 1L, 1L, 3L, 3L, 2L, 2L, 1L, 2L, 2L, 3L, 2L,
3L, 1L, 2L, 2L, 3L, 2L, 1L, 1L, 2L, 3L, 1L, 2L, 1L, 3L, 1L, 1L,
3L, 1L, 3L, 3L, 3L, 1L, 1L, 1L, 3L, 3L, 2L, 2L, 1L, 2L, 2L, 3L,
2L, 3L, 3L, 1L, 2L, 2L, 3L, 2L, 1L, 1L, 2L, 3L, 1L, 1L, 2L, 1L,
3L, 1L, 1L, 2L, 3L, 1L, 3L, 3L, 3L, 1L, 1L, 1L, 3L, 2L, 2L, 1L,
2L, 2L, 3L, 2L, 3L, 3L, 1L, 2L, 2L, 3L, 2L, 1L, 2L, 3L, 1L, 2L,
1L, 3L, 1L, 1L, 3L, 1L, 3L, 3L, 3L, 1L, 1L, 1L, 2L, 2L, 1L, 2L,
2L, 3L, 2L, 3L, 3L, 1L, 2L, 2L, 3L, 1L, 2L, 3L, 1L, 2L, 1L, 3L,
1L, 1L, 3L, 1L, 3L, 3L, 3L), .Label = c("A", "B", "C"), class = "factor")), row.names = c(NA,
-664L), class = c("tbl_df", "tbl", "data.frame"))
解决方案
这可能更适合CrossValidated，因为它更多地与您应该建立什么样的模型有关，而不是如何去做，但这里是这样。
让我们来看看你的模型。
result ~ school_type * subject + (1|year)
这表示结果取决于学校类型、学科及其相互作用（即学科之间的差异因学校类型而异），并且结果作为跨年份的随机效应而变化（即，您只对年份之间的可变性感兴趣，而不是在年份之间进行统计比较）。
问题：
- 这警告你你有一个“奇异拟合”——年际变化估计为零。这可能是因为您的样本中没有很多年，并且不一定是致命的，但确实表明该模型可能不是最佳的（参见例如此处和此处）
- 正如您暗示/建议的那样，此模型没有考虑到您每所学校有多个测量值的事实。它假设学校之间没有差异/学校内部没有相关性，这几乎肯定是一个问题。
result ~ school_type * subject + (year|school_ID)
除了上面列出的学校类型/学科相互作用之外，该模型还假设学校之间存在差异和年份之间的差异，并且学校之间的年份效应差异可能是相关的（例如，效应高于平均水平的学校）第 1 年的平均影响可能在第 2 年也高于平均水平）。
- 这个模型也是奇异的
- 它（几乎）是这种观察设计的最大模型；理论上，您还可以添加
(1|year)以允许跨学校保持一致的逐年变化（当前模型仅允许跨学校的逐年变化） - 你说“没有什么是重要的”。这不是我所看到的（而且这不是确定您是否拥有正确模型的好方法！令人失望但完全有可能拥有正确的模型并且没有什么重要的）。你的意思是学校类型的影响不显着（在传统的p<=0.05）水平？
例如，以下是car::Anova()应用于此拟合的结果：
Analysis of Deviance Table (Type II Wald chisquare tests)
Response: result
Chisq Df Pr(>Chisq)
school_type 5.3273 2 0.069692 .
subject 1030.9423 3 < 2.2e-16 ***
school_type:subject 22.1616 6 0.001132 **
请注意，您在解释主效应测试时应该非常小心?car::Anova：阅读“详细信息”部分（如果您要进行类型 3 测试，请考虑设置总和为零的对比）。
一般来说，我还建议您在确定您对模型的结构感到满意之前不要查看显着性检验。
我将推荐另外两个模型：
result ~ school_type * subject + (1|year) + (1|school_ID) + (1|year:school_ID)
这允许不同年份、不同学校之间的差异以及不同年份之间的独立差异。这是明智的（并且比上面的最大模型更简洁；它忽略了学校之间逐年变化的可能相关性），但它仍然给出了一个奇异的模型。
result ~ school_type * subject + year + (1|school_ID/year)
这会将年份效应从随机更改为固定，当您没有很多级别来估计方差时，这是一个合理的策略。（该(1|school_ID/year)术语的意思是“校内变异和校内嵌套的年间变异”，与之前模型中其他两个随机效应项的组合相同）
这个模型不是单一的。
除了第一个模型（忽略学校之间的变化）之外，所有模型都给出了模型部分几乎相同的结果school_type*subject，这并不奇怪，因为我们基本上只是将（学校×年份）的变化分割成稍微不同的部分方法。
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