ElMaestro ★★★ Denmark, 20210804 10:18 (1190 d 16:35 ago) Posting: # 22502 Views: 4,009 

Hi all, here's a paper that discusses the funky issue with fixed effects versus random effects. Enjoy. In particular, who can come up with a quantitative relevant measure of the difference any estimator makes if you have two model alternatives? One thing is of course to judge if estimate A is closer to the true value that estimator B, or if its variance is smaller, but another is to judge practical relevance. I was told that it is the latter that is of interest to the author. — Pass or fail! ElMaestro 
Helmut ★★★ Vienna, Austria, 20210804 12:16 (1190 d 14:37 ago) @ ElMaestro Posting: # 22503 Views: 3,347 

Hi ElMaestro, ❝ here's a paper that discusses the funky issue with fixed effects versus random effects. ❝ Enjoy. THX, I did. ❝ In particular, who can come up with a quantitative relevant measure of the difference any estimator makes if you have two model alternatives? Define relevant. ❝ One thing is of course to judge if estimate A is closer to the true value that estimator B, or if its variance is smaller, … \(\small{\delta_{\,estimatetrue}\small}\) or – likely better – \(\small{\,\textrm{RE}\,(\%)\,=\,100(estimatetrue)/true\,}\) is commonly used. Example for \(\small{\sigma_\textrm{wR}^2=0.2025}\) of the paper’s Table 2: $$\textbf{Table I:}\;\textrm{Comparison of models' estimates}$$$$\small{\begin{array}{cccccccl} \hline \text{Scenario} & \hat\sigma_\textrm{wR,REML}^2 & \delta & \,RE\,(\%)\, & \hat\sigma_\textrm{wR,Lin.model}^2 & \delta & \,RE\,(\%)\, & \textrm{Comparison of }\,RE\,(\%)\,\\ \hline 1 & 0.2023 & 0.0002 & 0.0988 & 0.2024 & 0.0001 & 0.0494 & \text{Linear model "better"} \\ 2 & 0.2036 & +0.0011 & 0.5432 & 0.2036 & +0.0011 & 0.5432 & \text{Equal} \\ 3 & 0.2014 & 0.0011 & 0.5432 & 0.2013 & 0.0012 & 0.5926 & \text{REML "better"} \\ 4 & 0.2022 & 0.0003 & 0.1481 & 0.2023 & 0.0002 & 0.0988 & \text{Linear model "better"} \\ 5 & 0.2025 & \pm 0.0000 & 0.0000 & 0.2024 & 0.0001 & 0.0494 & \text{REML "better"} \\ 6 & 0.2027 & +0.0002 & 0.0988 & 0.2027 & +0.0002 & 0.0988 & \text{Equal} \\ \hline \end{array}}$$There is no clear winner. IMHO, it boils down to the question which of the scenarios is most likely occurring in practice. No idea. We can look at the expanded limits \(\small{\left\{L\,,U\right\}}=100\exp(\mp0.76\,\hat{\sigma}_\textrm{wR})\) and the backcalculated ‘clinically not relevant difference’ \(\small{\Delta^{\star}=100L}\) as well: $$\textbf{Table II:}\;\textrm{Comparison of ABEL}$$$$\small{\begin{array}{ccccccc} \hline \text{Scenario} & \left\{\textit{L},\,\textit{U}\right\}_\textrm{REML} & \Delta^{\star} & \left\{\textit{L},\,\textit{U}\right\}_\textrm{LM} & \Delta^{\star} & \left\{\textit{L}\textit{U}\right\} & \Delta^{\star} \\ \hline 1 & 71.05,\,140.75 & 28.95\% & 71.04,\,140.76 & 28.96\% & \textrm{REML}>\textrm{LM} & \textrm{REML}<\textrm{LM}\\ 2 & 70.97,\,140.91 & 29.03\% & 70.97,\,140.91 & 29.03\% & \textrm{REML}=\textrm{LM} & \textrm{REML}=\textrm{LM}\\ 3 & 70.10,\,140.65 & 28.90\% & 71.11,\,140.63 & 28.89\% & \textrm{REML}>\textrm{LM} & \textrm{REML}>\textrm{LM}\\ 4 & 71.05,\,140.74 & 28.95\% & 71.05,\,140.75 & 28.95\% & \textrm{REML}=\textrm{LM} & \textrm{REML}=\textrm{LM}\\ 5 & 71.03,\,140.78 & 28.97\% & 71.04,\,140.76 & 28.96\% & \textrm{REML}>\textrm{LM} & \textrm{REML}>\textrm{LM}\\ 6 & 71.02,\,140.80 & 28.98\% & 71.02,\,140.80 & 28.98\% & \textrm{REML}=\textrm{LM} & \textrm{REML}=\textrm{LM}\\ \hline \end{array}}$$A sponsor would prefer \(\small{\left\{L\,,U\right\}}\) as wide as possible. Hence, \(\small{\textrm{REML}}\) is the way to go. From the patient’s – and therefore, regulatory? – perspective it is obviously the other way ’round \(\small{(\Delta^{\star}}\) as small as possible). Bonus question: What about the Type I Error? Of course, discussed in the paper… ❝ … but another is to judge practical relevance. I was told that it is the latter that is of interest to the author. Understandable. However, even if not practically relevant I prefer the “better” one. The paper’s Table 4 is interesting. — Diftor heh smusma 🖖🏼 Довге життя Україна! _{} Helmut Schütz The quality of responses received is directly proportional to the quality of the question asked. 🚮 Science Quotes 
ElMaestro ★★★ Denmark, 20210804 14:51 (1190 d 12:02 ago) @ Helmut Posting: # 22504 Views: 3,222 

Hi Hötzi, ❝ Define relevant. Please go right ahead help me define it. Relevance is not part of the inventory in my department. One thing that is not explicit in the paper but which is really funny is that:
I'd like to study the phenomenon a bit more with e.g. datasets having missing values. But on my old toaster these things run slow. Bonus: Someone tell me how to control bobyqa convergence in a paradigm that fits into things like tolerance or relative tolerance. I don't quite get the convergence principles for that optimzer. It would help speed up experiments. — Pass or fail! ElMaestro 
Helmut ★★★ Vienna, Austria, 20210804 17:22 (1190 d 09:31 ago) @ ElMaestro Posting: # 22505 Views: 3,253 

Hi ElMaestro, ❝ ❝ Define relevant. ❝ ❝ Please go right ahead help me define it. ❝ Relevance is not part of the inventory in my department. In mine not either… ❝ One thing that is not explicit in the paper but which is really funny is that: ❝ 1. The individual treatment estimates are completely, utterly, terribly off with the lm. ❝ 2. The individual treatment estimates are very close to true with the mm. ❝ 3. Both give the same treatment differences as you can see in the paper. #1 & #2 hints towards the mixed model. #3 is interesting – meaning out? BTW, from the Introduction: EU regulators did not present arguments for their proposal, and at the time of introduction they expressed that their approach is straightforward to calculate. Exactly. [I still think that the “all effects fixed” approach is questionable, at least. It assumes homoscedasticity of withinsubject variances \(\small{(\sigma_\textrm{wT}^2\equiv\sigma_\textrm{wR}^2})\), a condition, which is – more often than not – outright false. I have seen many full replicate studies, where \(\small{s_\textrm{wT}^2<s_\textrm{wR}^2}\). Rarely it was the other way ’round. Biopharmaceutical technology improves. Of note, a mixedmodel is mandatory for the FDA^{1,2} and Health Canada^{3} writes: By definition the cross‐over design is a mixed effects model with fixed and random effects. (my emphasis)The confidence interval of the mixedeffects model can be wider than the one of the Linear Model (LM). It depends on how much information can be recovered and moved to the additional terms (in the LM all remains in the residual error). If little can be recovered, the CI of the mixedeffects model will be wider due to less degrees of freedom. Even observed in the omniscient oracle’s fabricated data sets.^{4} $$\small{\begin{array}{ccrcccc} \hline \text{Data set} & \text{Model} & df & t_\textrm{crit} & \text{PE} & \text{90% CI} & \textrm{Width} \\ \hline \text{I } & \text{LM} & 217.0 & 1.6519 & 115.65{\color{red}9} & \text{107.106, 124.895} & {\color{Red}{17.789\%}} \\ \text{I } & \text{REML} & 207.7 & 1.6522 & 115.65{\color{red}8} & \text{107.104, 124.894} & {\color{DarkGreen}{17.790\%}} \\ \hline \text{II} & \text{LM} & 45.00 & 1.6794 & 102.264 & \text{ 97.316, 107.465} & {\color{Red}{10.149\%}} \\ \text{II} & \text{REML} & 19.89 & 1.7252 & 102.264 & \text{ 97.053, 107.755} & {\color{DarkGreen}{10.702\%}} \\\hline \end{array}}$$ ❝ I'd like to study the phenomenon a bit more with e.g. datasets having missing values. But on my old toaster these things run slow. Can imagine. Even the point estimates might be different, which is not so obvious in ‘data set I’ with missings above. In one of my studies (partial replicate, unbalanced: \(\small{n_1=n_2=21}\), \(\small{n_3=20}\) and incomplete: one missing in the 2^{nd} period and seven in the 3^{rd}) I got: $$\small{ \begin{array}{cccccc} \hline \text{Model} & df & t_\textrm{crit} &\text{PE} & \text{90% CI} & \textrm{Width} \\ \hline \text{LM} & 113.00 & 1.6585 & 107.194 & \text{98.307, 116.884} & 18.577 \\ \text{REML} & \text{ 57.81} & 1.6716 & 106.933 & \text{98.616, 115.951} & 17.335 \\\hline \end{array}}$$In this case information could be recovered and hence, the confidence interval of REML was narrower than the LM’s, despite its fewer degrees of freedom. Endrényi and Tóthfalusi noted^{5} that this lack of harmonization of statistical approaches might lead – in the hypothetical situation of submitting the same study to different agencies – to acceptance in one jurisdiction and rejection in another. See also a desperate attempt from the real world. ❝ Bonus: Someone tell me how to control bobyqa convergence … Sorry, never met uncle Bob in person.
— Diftor heh smusma 🖖🏼 Довге життя Україна! _{} Helmut Schütz The quality of responses received is directly proportional to the quality of the question asked. 🚮 Science Quotes 