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This might be a basic question, but I struggle to find a clear answer to this online.

Let's assume I am measuring heart rate under different activities such as running, walking, and resting. I have made these measurements on 100 different people, and for each individual, I have 20 measurements per activity. That means I have 100 x 20 (measurements) x 3 (activities) observations. I am of course interested in the effect of activity on heart rate.

My usual inclination would be to fit a general linear model like this:

HR ~ Act + Person

I would include the Person variable as a dummy to account for differences between individuals that affect heart rates, such as fitness and age. However, I could also fit a mixed effects model like this:

HR ~ Act + (1|Person)

In the second case, I am fitting one fixed intercept plus 100 random intercepts, one for each individual. In the first case, I am fitting one intercept for individual 1 plus 99 dummies for all the other individuals. It would expect the second approach to be superior, but I can't quite put into words what the difference would be. I am not very familiar with mixed models, so if I am making a mistake in the specification of the model, please let me know.

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For the specific situation you've described, with 100 people, the linear model will, if using default contrast coding, produce 101 estimates: 1 for Act, 99 for person and 1 for the intercept (which will include the reference level of person). On the other hand, the mixed model will produce 1 estmate for Act, one for the intercept and a variance of the random intercept for person.

For these reason alone I would be inclined towards a mixed model.

However, on a more general level this is really just a question about when to treat a factor as fixed and when to treat it as random, and this has been asked and answered a number of times:

What is the difference between fixed effect, random effect and mixed effect models?

Random and fixed factors

Are anxiety measure fixed or random factors in this scenario?

Should I consider time as a fixed or random effect in GLMM?

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