# Strange result with GLMM (binomial)

I'm analyzing some data, using GLMM and obtain very strange results. The data is of student passing a test, each group of students belong to a different school. So I analyzed the data using glmer(is_pass ~ (1|school), data, family = 'binomial'). The schools are the random effect.

Now, the proportion of passing is very high. The average across all schools is 0.99. However, the confidence interval obtained from the GLMM is between 0.68 - 1. Furthermore, if I construct the Clopper-Pearson CI for each of the schools individually each CI is actually shorter (with the minimal one, the only school where students failed its 0.75 - 0.99).

  Confidence.Interval Lower.limit Upper.limit alpha
two.sided   0.8765639   1.0000000  0.05
1           two.sided   0.8765639   1.0000000  0.05
2           two.sided   0.8765639   1.0000000  0.05
3           two.sided   0.8842967   1.0000000  0.05
4           two.sided   0.8765639   1.0000000  0.05
5           two.sided   0.8518149   1.0000000  0.05
6           two.sided   0.8628148   1.0000000  0.05
7           two.sided   0.8575264   1.0000000  0.05
8           two.sided   0.7486971   0.9905446  0.05


The glmer function returns no error or warnings. Why does it happen? and how can I circumvent it?

Data attached in dput structure as well as code for analysis.

library(lme4)
library(tidyverse)

data <- structure(list(
student = c("1004", "1007", "1008", "1009", "1011",
"1012", "1014", "1015", "1016", "1017", "1018", "1020", "1021",
"1022", "1023", "1024", "1025", "1026", "1029", "1030", "1031",
"1032", "1033", "1034", "1035", "1036", "1037", "1038", "1039",
"1040", "1041", "1042", "1043", "1044", "1045", "1046", "1047",
"1048", "1049", "1050", "1051", "1052", "1053", "1054", "1055",
"1056", "1057", "1058", "1059", "1060", "1061", "1062", "1063",
"1064", "1065", "1066", "1067", "1068", "1069", "1070", "1071",
"1072", "1073", "1074", "1075", "1076", "1077", "1078", "1080",
"1081", "1082", "1083", "1084", "1085", "1086", "1087", "1088",
"1089", "1090", "1092", "1093", "1094", "1095", "1096", "2003",
"2004", "2006", "2007", "2008", "2009", "2010", "2011", "2012",
"2013", "2014", "2015", "2016", "2017", "2018", "2019", "2020",
"2021", "2022", "2023", "2024", "2025", "2026", "2027", "2028",
"2029", "2030", "2031", "2032", "2033", "2034", "2035", "2036",
"2037", "2038", "2039", "2040", "2041", "2042", "2043", "2044",
"2045", "2046", "2047", "2048", "2049", "2050", "2052", "2053",
"2054", "2055", "2056", "2057", "2058", "2059", "2060", "2061",
"2062", "2063", "2064", "2065", "2066", "2067", "2068", "2069",
"2071", "2072", "2073", "2075", "2076", "2077", "2078", "2079",
"2080", "2081", "2082", "2083", "2084", "2085", "2086", "2087",
"5004", "5008", "5009", "5010", "5011", "5012", "5013", "5014",
"5015", "5016", "5018", "5019", "5020", "5022", "5024", "5025",
"5026", "5027", "5028", "5030", "5031", "5032", "5033", "5034",
"5035", "5036", "5037", "5038", "5039", "5040", "5041", "5042",
"5043", "5044", "5045", "5046", "5047", "5048", "5049", "5050",
"5051", "5052", "5053", "5054", "5055", "5056", "5057", "5058",
"5059", "5060", "5061", "5062", "5063", "5064", "5065", "5066",
"5067", "5068", "5069", "5071", "5072", "5073", "5074", "5075",
"5076", "5077", "5078", "5079", "5080", "5081", "5082", "5083",
"5084", "5085", "5086"),
school = c("153", "152", "153", "154",
"152", "154", "153", "152", "153", "154", "152", "153", "152",
"153", "152", "153", "152", "153", "152", "153", "152", "152",
"154", "154", "154", "154", "152", "152", "153", "152", "153",
"152", "153", "152", "154", "152", "154", "152", "154", "154",
"152", "154", "152", "152", "154", "154", "152", "152", "152",
"153", "152", "152", "153", "153", "153", "153", "153", "154",
"154", "154", "153", "153", "154", "154", "154", "154", "153",
"154", "153", "154", "153", "154", "154", "153", "153", "153",
"153", "153", "152", "152", "152", "154", "154", "154", "252",
"253", "251", "253", "252", "252", "251", "253", "251", "252",
"251", "251", "253", "251", "251", "251", "251", "253", "252",
"252", "252", "251", "251", "253", "253", "252", "251", "252",
"252", "253", "253", "253", "253", "252", "252", "253", "252",
"251", "252", "251", "253", "253", "252", "252", "251", "253",
"251", "251", "251", "252", "252", "252", "252", "251", "252",
"252", "253", "253", "251", "252", "253", "252", "251", "253",
"252", "251", "252", "253", "253", "253", "253", "251", "252",
"252", "251", "251", "251", "251", "251", "251", "251", "553",
"554", "554", "554", "553", "553", "553", "553", "553", "553",
"552", "552", "552", "552", "554", "554", "553", "552", "552",
"554", "553", "553", "553", "554", "552", "552", "552", "552",
"552", "552", "554", "554", "554", "554", "553", "553", "553",
"553", "552", "552", "552", "552", "553", "553", "553", "553",
"553", "552", "552", "553", "553", "553", "553", "552", "552",
"552", "552", "552", "552", "552", "554", "554", "554", "554",
"554", "554", "554", "554", "554", "554", "554", "554", "554",
"554", "554"),
is_pass = c(TRUE, TRUE, TRUE, TRUE, TRUE, TRUE,
TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE,
TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE,
TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE,
TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE,
TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE,
TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE,
TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE,
TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE,
TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE,
TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE,
TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE,
TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE,
TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE,
TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE,
TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, FALSE, TRUE, TRUE, TRUE,
TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE,
TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE,
TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE,
TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE,
TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE,
TRUE, TRUE, TRUE, TRUE, TRUE, FALSE, TRUE, TRUE, TRUE, TRUE,
TRUE, TRUE, TRUE, TRUE, TRUE)),
class = "data.frame",
row.names = c(NA, -240L))

res        <- glmer(is_pass ~ (1|school), data, family = 'binomial')
coef_logis <- coef(summary(res))

est_p      <- boot::inv.logit(coef_logis[1])
lb         <- boot::inv.logit(coef_logis[1] - coef_logis[2] * qnorm(1 - 0.05 / 2))
ub         <- boot::inv.logit(coef_logis[1] + coef_logis[2] * qnorm(1 - 0.05 / 2))