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4

Indeed a paired $t$-test is equivalent to a linear mixed model that you formulated as $Y_{ij} = β_0 + β_1t + a_i + ε_{ij}; \\a_i ∼ N(0, σ^2_{subject}), ~ε_{ij} ∼ N(0, σ^2_{res}); \\i=1,2,...,n; j=1,2;$ where $i$ indices the subjects and $j$ codes the two paired conditions. why wouldn't it make sense to include a random slope? The dummy variable $t$ ...


2

First of all, the 'random effects' can be viewed in different ways and the approaches to them and associated definitions may seem conflicting but it is just a different viewpoint. The 'random effect' term in a model can be seen as both a term in the deterministic part of the model as a term in the random part of the model. Basically, in general, the ...


5

Trying to find single "authoritative" definition is always tempting in cases like this, but the variety of different definitions shows that this term simply is not used in consistent manner. Andrew Gelman seems to have reached same conclusions, you can look as his blog posts here and here, or into his handbook Data Analysis Using Regression and Multilevel/...


0

The output looks perfectly fine to me - your estimate divided by its standard error (-1.7223/1.2) is -1.44, which gives you the non-significant result you obtained. The ratio of the estimate of NEURO to the estimate for the intercept has nothing to do with the significance of NEURO. Also, in plots, differences can look important although they are not. If you ...


0

Welcome to the site, Marco. The random slope is necessary for multiple reasons. Among the most important is recent methodological work by Heisig & Schaeffer, which shows that for a level 1 variable involved in a cross-level interaction with a level 2 variable, that interaction is more likely to be significant if the level 1 variable is not specified as ...


2

Yes, formally speaking teacher should a random effect but with only three levels estimation will be extremely problematic (i.e. how much we would trust a standard deviation out of a sample with just 3 items). Yes, it is hypothesis dependent. But based on the initial information, teacher assignment was not explicitly determined. We can model students as ...


0

Just in case it would be helpful, I have tried to illustrate how my data would be nested (in addition to the example in my table above). Here, each timepoint (t1,t2,t3,etc.) gets "observed" by two different methods of calculating Outcome, i.e., Method A and Method B. Each set of Outcome values across time points for each Method are nested within a given ...


3

Indeed, because the model only includes random intercepts terms, the marginal mean of your Poisson outcome will be $$E(Y_{ijk}) = \exp \bigl (\beta_0^* + \beta_1 \texttt{time}_{ijk} + \beta_2 \texttt{x2}_{ijk} + v_i + w_{ij}\bigr ),$$ where $$\beta_0^* = \beta_0 + \frac{\sigma_v^2}{2} + \frac{\sigma_w^2}{2},$$ with $\sigma_v^2$ and $\sigma_w^2$ the ...


5

As you have correctly observed, both in meta-analysis and beyond, a frequentist mixed model does something similar to a Bayesian approach. Namely, it assumes a parameter not to have a fixed value, but rather to have been randomly drawn from some probability distribution (in practice: almost always a normal distribution with mean $0$ and unknown variance). ...


1

Nb; don't use ti() for univariate smooths: it currently works but Simon Wood, maintainer of mgcv has remarked that this may be removed in a future version of the package. I think the main problem is that you have the factor and continuous variable back to front in the fs smooth. time is the continuous covariate so you want a smooth of it for each level of ...


0

You would trust the Fixed Effects because it is consistent under weaker assumptions. This is not about the efficiency so the fact that variance is smaller is secondary. If the estimates are different, chances are strict exogeneity is violated and RE becomes biased, while FE remains consistent even without strict exogeneity.


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