# Fixed Effects Regression Models Allison Pdf 27

Results of latent growth curve regression with individual-level fixed effects, controlling for all time-invariant characteristics of the patient. Confidence intervals are in parentheses. The absence of association would be indicated by an odds ratio of 1.

## fixed effects regression models allison pdf 27

Results of latent growth curve regression with individual-level fixed effects, controlling for all time-invariant characteristics of the patient. The within-patient R2 were 0.25 for the no limitation group, 0.37 for those with mild/moderate baseline limitations, and 0.45 for those with severe baseline limitations. Confidence intervals are in parentheses. The absence of association would be indicated by the acquisition of 0 new functional limitations.

In addition to this synthesis of the inter-disciplinary methodological literature on FE and RE models (information that, whilst often misunderstood, is not new), we present an original simulation study showing how various forms of these models respond in the presence of some plausible model mis-specifications. The simulations show that estimated standard errors are anti-conservative when random-slope variation exists but a model does not allow for it. They also show the robustness of estimation results to mis-specification of random effects as Normally distributed, when they are not; substantial biases are confined to variance and random effect estimates in models with a non-continuous response variable.

Both the Mundlak model and the within-between random effects (REWB) models (Eqs. 2 and 3 respectively) are easy to fit in all major software packages (e.g. R, Stata, SAS, as well as more specialist software like HLM and MLwiN). They are simply random effects models with the mean of \(x_it\) included as an additional explanatory variable (Howard 2015).

We hope the discussion above has convinced readers of the superiority of the REWB model, except perhaps when the within and between effects are approximately equal, in which case the standard RE model (without separated within and between effects) might be preferable for reasons of efficiency.Footnote 8 Even then, the REWB model should be considered first, or as an alternative, since the equality of the within and between coefficients should not be assumed. As for FE, except for simplicity there is nothing that such models offer that a REWB model does not.

The only difference between RE and FE lies in the assumption they make about the relationship between υ [the unobserved time-constant fixed/random effects] and the observed predictors: RE models assume that the observed predictors in the model are not correlated with υ while FE models allow them to be correlated.

Further, using the REWB model as if it were a FE model leads researchers to use it without taking full advantage of the benefits that RE models can offer. The RE framework allows a wider range of research questions to be investigated: involving time-invariant variables, shrunken random effects, additional hierarchical (e.g. geographical) levels and, as we discuss in the next section, random slopes estimates that allow relationships to vary across individuals, or allow variances at any level to vary with variables. As well as yielding new, substantively interesting results, such actions can alter the average associations found. Describing the REWB, or Hybrid, model as falling under a FE framework therefore undersells and misrepresents its value and capabilities.

These results support the strong critique by Barr et al. (2013) that not to include random slopes is anticonservative. On the other hand, Matuschek et al. (2017) counter that analytical models should also be parsimonious, and fitting models with many random effects quickly multiplies the number of parameters to be estimated, particularly since random slopes are generally given covariances as well as variances. Sometimes the data available will not be sufficient to estimate such a model. Still, it will make sense in much applied work to test whether a statistically significant coefficient remains so when allowed to vary randomly. We discuss this further in the conclusions.

Datasets often have structures that span more than two levels. A further advantage of the multilevel/random effects framework over fixed effects is its allowing for complex data structures of this kind. Fixed effects models are not problematic when additional higher levels exist (insofar as they can still estimate a within effect), but they are unable to include a third level (if the levels are hierarchically structured), because the dummy variables at the second level will automatically use up all degrees of freedom for any levels further up the hierarchy. Multilevel models allow competing explanations to be considered, specifically at which level in a hierarchy matters the most, with a highly parsimonious specification (estimating a variance parameter at each level).Footnote 12

In sum, even substantial violations of the Normality assumption of the higher-level random effects do not have much impact on estimates in the fixed part of the model, nor the standard errors. Such violations can however affect the random effects estimates, particularly in models with a non-continuous response.

Third, and in contrast to much of the applied literature, we argue that researchers should not use a Hausman test to decide between fixed and random effects models. Rather, they can use this test, or models equivalent to it, to verify the equivalence of the within and between relationships. A lack of equality should be in itself of interest and worthy of further investigation through the REWB model.

Note though that, in the longitudinal setting, between effects will only be fully controlled if those effects do not change over time (this is the case with the REWB/Mundlak models as well, unless such heterogeneity is explicitly modelled).

That is, the random effects were in all cases uncorrelated. We also generated binary data based on similar models (both random intercept-only and random intercept, random slope models), using a logit link. In all cases, \(\sigma_\upsilon 0^2\) and \(\sigma_\upsilon 1^2\) were set to 4, and (for the Normally distributed data) the variance of \(\epsilon_it\) to 1. The overall intercept \(\beta_0\) and the overall slope \(\beta_1\) were also set to 1. The \(x_it\) data were drawn from a Normal distribution with a mean of 0 and a variance of 0.25^2.

We then fitted three different models to each simulated dataset: a fixed effects model (with naïve and clustered standard errors), a random intercepts-only model, and a random intercepts-random slopes model.

We conducted the simulations in R. For fitting multilevel models we used the package lme4 (Bates et al. 2015). For deriving clustered standard errors from the fixed effects models, we used the plm package (Croissant and Millo 2008). We caught false or questionable convergences and simply removed them, simulating a new dataset instead (this should not bias the results, although it should be noted as an advantage of FE is that it is unlikely to show convergence problems due to being estimated by OLS). We tried multiple runs of simulations, and found stable results beyond about 200 simulations per DGP.

Abstract:Objectives: To quantify the odds of fatal injuries associated with drivers involved in single-vehicle, run-off-road (SVROR), injury crashes. Methods: An in-service safety evaluation was carried out using multivariate logistic regression models. Results: The odds of motorist death was lower for w-beam guardrail crashes as compared to tree, pole, and concrete barrier crashes. On the other hand, there was no statistically significant difference between the odds of motorist death in concrete barrier crashes as compared to tree or pole crashes. The odds of motorist death were lower for curbs and collision-free crashes as compared to tree, pole, and barrier crashes. Thus, obstacles should be removed whenever possible and barriers installed only whenever absolutely necessary. The lack of vehicle containment (in barrier crashes) was found: (i) to tend to occur on higher-posted-speed-limit roads and result in a higher percentage of fatal crashes, (ii) to be more prevalent with the less rigid barrier type, and (iii) to result in a consistently higher percentage of fatal crashes under the concrete barrier category. Conclusions: Findings not only support state-of-the-art roadside design guidelines and crash-testing criteria, but they may also be useful in evaluating proposed roadside safety improvements.Keywords: run-off-road crashes; fatal injuries; logistic regression

In sensitivity analysis, treating cantons and institutions as fixed effects alluded to a diminished association of the protective effect of FFP2 use (but only for the outcome of self-reported SARS-CoV-2 swab and not for seroconversion). This could be explained by differences in testing of HCW between institutions. However, the occurrence of regional differences and institutional factors contributing to the observed effect cannot be excluded. Similarly, excluding participants with positive household contact resulted in a non-significant association for the outcome of self-reported positive swabs. Yet again, the consistent result for seroconversion (which is a more objective outcome than self-reported swabs) strengthens the validity of our data.