Hello Lars John If I were you, I would use both models to see which is better.

Moreover, why do you want to use a linear regression model? A curve estimation gives you a better answer.

Dear Lars John,

I think it is better to use aggregated values; please consider that the amount of samples is more, these results will be more accurate.

Kind regards.

Shakeri.

Hello Lars John. You have multilevel data with individual pedestrians (level 1) clustered within cities (level 2). If you have a sufficient number of cities, and if they are considered a random sample from some larger population of cities, then a random intercept model (with city as the level 2 grouping variable) would be sensible. And city size (population) would be a level 2 explanatory variable. You may also wish to include level 1 explanatory variables (e.g., age & sex of individual pedestrians). If you say what software you use, readers may be able to point to helpful resources specific to that software. Meanwhile, here are some other resources you can look at:

• http://www.bristol.ac.uk/cmm/
• https://www.pilesofvariance.com/

HTH.

PS- How many cities do you have?

For the information of other readers, Lars told me the following in an email:

I have collected samples in 11 Swiss cities. 200 samples per city. The cities were randomly selected.

I'm not sure what current guidelines suggest about the number of clusters that are needed to support estimation of random intercept model. My suspicion is that 11 clusters is a bit on the low side. (Perhaps Kelvyn Jones can weigh in on that.) But given that population size (a level 2 variable) is your primary explanatory variables, treating City as a fixed categorical explanatory variables is not a viable option.

The problem is that you only have a sample of 11 to evaluate between city variation which is quite a small sample . See https://research-information.bris.ac.uk/en/publications/analysing-and-designing-extensive-studies-in-which-place-matters-

Article ANALYZING AND DESIGNING EXTENSIVE STUDIES IN WHICH PLACE MAT...

You could estimate a multilevel model as Bruce Weaver suggested and much will depend on how substantial the variation is (that is how far the between variance is away from zero). I suggest that you need MCMC estimation as the credible intervals are likely to be asymmetric as the variance cannot go below zero; see Bayesian Chapter in this Book Developing multilevel models for analysing contextuality, he...

Hello Lars John, an additional consideration for fitting the multilevel model suggested by Bruce Weaver , which seems to be the best solution:

Because there is a single population size for each city, the fixed effect "population size" and the random effect "variance among cities" will have redundant information. Because random effects are penalized, the model should attribute effects to population size, unless the shape of the relationship is not properly modeled. For example, fitting a linear effect of population size to a nonlinear relationship will result in the random effect explaining more variance (perhaps to the point that population size does not seem to have any effect).

See the excellent argument and demonstration by Thierry Onkelinx here:

* R Mixed Effect mailing list: https://stat.ethz.ch/pipermail/r-sig-mixed-models/2015q3/023927.html

* Demo with R code: https://rpubs.com/INBOstats/both_fixed_random

It's another argument for considering Abolfazl Ghoodjani 's suggestion to fit a non linear model. Of course it might be challenging to find the correct shape with 11 cities. How does the plot look like?

Hope that helps

Hello Gi-Mick Wu. Yes, the sample size is 200 for each of the 11 cities, but the cities do not all have the same size population. E.g., taking info from the site linked below, Lars John might have data like this, with n being the common sample size:

City, n, Population

Zurich, 200, 341730

Geneva, 200, 183981

Basel, 200, 164488

etc.

https://www.geonames.org/CH/largest-cities-in-switzerland.html

That's how I understood it.

There might be a simple way to look at this if walking speed and population size are proportional. This might then be a good application for ratio estimation within each cluster. Each city would be a cluster. For each city, if the 200 cases were drawn at random, you might look at "3.4.1 Considering probability of selection based sampling" under "3.4 ‘Optimal’ sampling" in

https://www.researchgate.net/publication/319434665_Comparison_of_Model-Based_to_Design-Based_Ratio_Estimators

Often an intercept term is not a good idea. However, here I can imagine it would be needed, and you'd go to design-based regression estimation. I generally consider model-based approaches, but I think the n=200 cases in clusters with various N may make a design-based approach advantageous.

For a book on model-assisted design-based techniques, see Särndal, C.-E., Swensson, B., and Wretman, J.(1992), Model Assisted Survey Sampling, Springer-Verlang.

Whatever possible approaches you try, comparison of results and testing is encouraged.

Note: Lars, you said "Other studies work with the average walking speed per city. My guess is because they have an unequal sample size." But please note that equal sample sizes, n, are not helpful when N is different for each city, and within cluster variances likely differ, so you cannot really equate one city's sample of 200 to another's for 'representing' each city. - Others above may have touched on that in a different way that I missed. (In my ratio paper and in the book reference I gave, pi is used to consider sampling weights.) -- Yes, I see Bruce noted this, maybe others.

However, the n=200 cases for each city might be considered needed for the purpose of handling measurement error, and you will then have only one ('average') value to use for each city. If measurement error is a problem, this might not be a bad idea. You do not want to throw away information, on the one hand, but you don't want to make something out of nothing on the other hand. (This reminds me of errors-in-variables regression. But that would generally only be for the predictors, I think.) Still, it's hard to believe that measurement error is enough of a problem to need 200 tries at it in each case.

Like others, the first choice is likely multilevel models, though the number of clusters is small. If you were choosing one or the other, the choice depends a bit on research questions if you just wanted at analyze individual or the group. Google ecological and atomistic fallacies.

You should definitely use weighted regression. The higher the standard deviation in a city, the lower the weight. This is done automatically in this software:

www.lerenisplezant.be/fitting.htm