# ICC Value of 0. Boundary singular fit? Interpretation?

In this case I have 7 samples in which the concentration of a particular analyte was measured and the measurement was repeated 3 times for each sample. Here's the data

1   2.17    3.36    3.54
2   2.97    2.16    2.85
3   2.75    2.99    2.07
4   2.46    1.92    2.83
5   2.63    2.56    2.48
6   2.44    2.47    2.49
7   2.53    3.81    2.32


I want to calculate the test-retest reliability using ICC but the results I get are this

> ICC(df,missing=FALSE)
boundary (singular) fit: see ?isSingular
Call: ICC(x = df, missing = FALSE)

Intraclass correlation coefficients
type ICC F df1 df2    p lower bound upper bound
Single_raters_absolute   ICC1   0 1   6  14 0.46       -0.31        0.59
Single_random_raters     ICC2   0 1   6  12 0.47       -0.32        0.59
Single_fixed_raters      ICC3   0 1   6  12 0.47       -0.32        0.59
Average_raters_absolute ICC1k   0 1   6  14 0.46       -2.50        0.81
Average_random_raters   ICC2k   0 1   6  12 0.47       -2.73        0.81
Average_fixed_raters    ICC3k   0 1   6  12 0.47       -2.73        0.81

Number of subjects = 7     Number of Judges =  3


Why would the ICC be 0 and what does the warning boundary (singular) fit mean in layman terms?

• I can reproduce the output exactly--but the "singular fit" error does not appear.
– whuber
Dec 5 '19 at 18:44
• The ICC function internally uses lmer to fit a random effects model. Specifically, it fits the model lmer(y~1 + (1|samples) + (1|repeats)). The random intercepts of samples and repetitions are estimated to be nearly $0$, hence the error message. To see this, save the results of ICC in an object, e.g. res and inspect res\$summary. Dec 5 '19 at 18:56
• There's some information on this 0 ICC phenomenon in my answer here Dec 5 '19 at 19:01