# Can I statistically describe a single case/outlier vs. a distribution?

I have a dataset consisting of body weight and corresponding age for a bunch of healthy subjects (grey triangles below). I fit a nonlinear function to this data and graphed a 95% prediction interval. I then graphed the body weights and ages of some experimental subjects (colored circles).

One of the experimental subjects, the light blue circle, falls outside the 95% prediction interval from the control data. Is there any way to describe this one relationship statistically? E.g. significance testing? I am not interested in comparing the group means of the control vs. experimental subjects; I am only interested in the light blue subject.

Alternatively, I have 13 healthy controls that were the same age as the light blue subject. If I convert their body weights to Z-scores, the Z-score of the light blue subject is ~3.5. Can I just convert it to a p-score (~0.001), or would I be making an invalid statistical assumption?

I really appreciate any help - it's been years since I had even basic statistics.

-Edit-

To clarify, the grey triangles on this graph are healthy controls. The nonlinear regression and 95% prediction interval were created only from this control data.

The colored circles are experimental subjects. The light blue circle corresponds to a subject with numerous health problems; I'm wondering if there's a way to statistically describe this subject's relationship to the control data, or if I am limited to qualitative comparisons.

• The answer depends on how the blue subject was chosen. After all, you appear to have about 45 subjects of whom three lie outside the 95% prediction interval (and not too far outside, at that). This is close to the number you expect to lie outside that interval, equal to $(100 - 95)/100 \times 45 = 2.25.$ Thus, if this subject is merely the most extreme of the three outliers, one valid description is "there are no data that depart unusually from the fit." But if this subject had independently been identified as of interest, the interpretation would be entirely different. – whuber Dec 19 '18 at 22:22
• The 95% prediction interval was constructed from data from the grey triangles only (healthy controls). The light blue subject is independently of interest (many significant health issues). I'm just wondering if I can statistically describe its deviance from the control data, or if I'm restricted to more qualitative comparisons. – Rose Dec 19 '18 at 22:28
• Thank you (+1). It would help future readers if you would edit your post to explain what the symbol shapes and colors mean and to include that clarifying comment. Welcome to our site! – whuber Dec 19 '18 at 22:30
• Why are you calling it a prediction interval? Is this not a confidence interval? You need to give us more information about how you conducted this analysis. Did you build a model on the healthy individuals first (because you knew they were healthy) and then you tested the "colored" individuals? Or did you already know which are which are decided based on your prior knowledge? – user2974951 Dec 20 '18 at 12:04
• I graphed the healthy subjects in Prism, did a nonlinear regression (logistic growth curve) and told Prism to plot a 95% prediction interval. I'm interested in the spread of the control data more than the mean - that's why it's a 95% prediction interval rather than a 95% confidence interval. Then I overlaid the body weights from the experimental subjects (colored circles) to see if any of them were notably heavier or lighter than the controls. The experimental subjects were all infected with a virus, but they weren't all "unhealthy." – Rose Dec 20 '18 at 17:00