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I am trying to estimate a linear relation between body temperature and body mass, and I have a sample of measurements from subjects, with most subjects having one measurement, but several subjects having two measurements in different times.

In order to estimating the slope 95% confidence interval, I could have used the usual formula, but since my data is partially correlated (due to having two measurements from the same subjects, for some of the subjects), I understand this will not be correct. So I was told it is better to use bootstraping technique.

My question is: How do I use bootstrapping while accounting for the fact that some of the subjects were sampled twice? Should I just take those subjects with 1/2 the probability of other subjects? Is that a common practice, and if so - how is it called? (possibly "Inverse Probability Bootstrap Sampling"?)

(In fact, there is an additional twist here, since I am using Passing-Bablok linear model, due to the nature of data; but answers related to ordinary linear regression, or even bootstrapping of 1 variable alone, are welcome as well).

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  • $\begingroup$ In another comment you write that the dataset is "not large." This ought to lead you away from bootstrapping, whose theoretical legitimacy holds only for large datasets (that is, asymptotically). Instead, use an appropriate model. A simple and likely effective one is to suppose independent, homoscedastic measurement errors, permitting direct application of standard (ordinary least squares) modeling. $\endgroup$
    – whuber ♦
    Commented Aug 3, 2021 at 13:35

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We use the bootstrap when we want to know how some function will behave when applied to different datasets drawn from a particular population. Obtaining new datasets is expensive, so instead the bootstrap pretends that the dataset you have is the population, and simulates new sampled datasets by resampling from the dataset you have.

Therefore, you want your resampling procedure to mimic how you obtained your original dataset in the first place. In your case, your original dataset was obtained by sampling a person, and then measuring them some number of times. Your resampling procedure should mimic this by resampling people, and including all of a person's measurements each time they are resampled.

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If the dataset is large, it shouldn't be a problem. In fact, if your dataset is large, repeated observations may even help you to get a better idea of the underlying true distributions.

It is worth noting that even with the regular bootstrap, sooner or later there are bound to be repeated obsevations in the bootstraped dataset anyway.

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  • $\begingroup$ 1. The dataset is not large. 2. Only part of the measurements are repeated. 3. Obviously this helps narrowing the CI, the question is - by how much? 4. In regular bootstrap, you have euqal probability for every point to be sampled repeatedly, while in my data only some of the points are. I guess I can make bootstrap with lower probability for repeated subjects (so that all subjects have equal probability). But is this a common practice? $\endgroup$
    – Dani
    Commented Jun 11, 2019 at 8:37

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