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(This is a question about the case study on page 636 of the book Applied Longitudinal Analysis by Fitzmaurice, Laird and Ware)

The data consists of 1028 weights of mice fetuses, they are clustered in 94 mother mice. The mother mice were assigned to a dose of poison.

I am puzzled as I don't see the advantage of clustering the data. Can you help me comparing the following two approaches:

  • Standard linear regression model of the weight against the treatment. (blue dashed line on the graph).
  • Linear mixed model with a random intercept for mother mice. (red line on the graph, with the random intercept for one mother mice displayed next to the weights of the fetuses of that mother mice).

I thought the advantage would be that the standard errors of treatment effect of the clustered model would be a lot smaller than the one of a linear model, but the opposite is true (the standard errors are 10 times as large as the ones of the linear model)! So what is the true advantage of clustering the data and adding a random intercept?

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You can basically choose between small but wrong standard errors (due to the correlatedness of the error terms) or larger but correct ones. I prefer the honest option ;). –  Michael Mayer May 7 at 11:54
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Expanding on @Michael Mayer's comment:

The advantage isn't in clustering; the data are already clustered - mice are clustered by mother.

The assumptions of the standard linear regression model are violated when the data are clustered. In particular, the assumption that the errors are independent is violated. Mice from one mother are going to be more similar to each other than mice from different mothers.

The linear mixed model (aka multi-level model and various other terms) is a way of dealing with this violation. There are various options. The random intercept model allows each family of mice to have its own starting point; you can also have models with random slopes as well.

Why are the SE's larger for the correct model? First, this isn't always the case. Second, it's because some of the variation in whatever it is that was measured was due to the family that the mouse was in, rather than to being in the treatment or control group.

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Thank you for your answer. The next example in the book answers also the question, comparing a model where the clustering is ignored with a correct model and states: "the model based standard errors (assuming no clustering) are misleadingly small for the randomised intervention effects and lead to substantively different conclusions about the effect." –  Kasper May 7 at 12:23
    
So I lost indeed the assumption of independence out of sight, which lead to wrong inference. Thanks again! –  Kasper May 7 at 12:25
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Thanks@Peter. In Econometry, the "Moulton factor" is sometimes used to quantify how wrong the standard errors are under clustering. –  Michael Mayer May 7 at 12:33
    
I never heard of that @MichaelMayer. But then, I've done almost no econometrics. –  Peter Flom May 7 at 12:35
    
@Peter Flom does the same apply when you don't add a random slope if you have longitudinal data? (Being that inference is not correct as you ignore that particular patients grow for example faster than others?) –  Kasper May 7 at 14:35
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