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Oct
14
comment Variance along the regression line
Beautiful charts! :-)
Oct
9
awarded  Benefactor
Oct
9
comment Can correlated random effects “steal” the variability (and the significance) from the regression coefficient?
Dear Lars, sorry for period of silence. I will award the bounty to you, though your answer is not clear to me yet. Could you please rewrite indices so that at least $x_i$ and $\epsilon_i$ doesn't use the same index $i$ when in fact they are indexed differently? I am still not able to get it even after your comment. You can use the same indices as I used in my answer.
Oct
9
comment Can correlated random effects “steal” the variability (and the significance) from the regression coefficient?
"is $\sigma$ significantly larger than zero?" Peter, well, I think $\sigma$ is always significantly larger than zero because it simply can't be negative...
Oct
3
comment Can correlated random effects “steal” the variability (and the significance) from the regression coefficient?
Lars, exactly, $x_{i,j}$ are known environmental conditions, this is the independent predictor variable
Oct
3
comment Can correlated random effects “steal” the variability (and the significance) from the regression coefficient?
I don't understand your model and the notation. Are you sure the indices are ok? If $i$ is the observation, where is the index for the group the random effect applies to? $\epsilon_i$ is the residual then, not the random effect, but what is the $x_i$? Why does it have the same index as $\epsilon_i$? What does the $(y_i)_i$ notation mean? Please make sure the indices are precise because otherwise I have troubles understanding it.
Oct
3
comment Can correlated random effects “steal” the variability (and the significance) from the regression coefficient?
Lars, but the values of $x_{i,j}$ have approx normal distribution with the mean = zero, so what is the "actual value of $\beta_i x_{i,j}$" then? Mean of this term is actually zero.
Oct
3
comment Can correlated random effects “steal” the variability (and the significance) from the regression coefficient?
Peter, 1) I am using random effects because the residuals of the model where correlated within years. This means there is some part of between-year population growth present over all sites which is not explained by $x_{i,j}$. This is not handled by the time series model. You speak of autocorrelation on the population size, while I speak of the autocorrelation of population growth, this is a different thing. 2) Sometimes it can happen that adding random effect will in fact enlarge the confidence intervals.
Oct
2
comment Can correlated random effects “steal” the variability (and the significance) from the regression coefficient?
And you say $V$ is the "overall variance", don't you mean "overall residual variance" instead?
Oct
2
comment Can correlated random effects “steal” the variability (and the significance) from the regression coefficient?
Peter, thank you for the answer! I have to go so I will have to look at your answer in more detail later. Now just a small comment - $\epsilon$ is not the only residual variability. $\epsilon$ is only the overdispersion - main residual variability is present within the $N_{i,j} \sim \mbox{Poiss} (\mu_{i,j})$ term.
Oct
2
revised Can correlated random effects “steal” the variability (and the significance) from the regression coefficient?
Replacing $\gamma_i$ with $\gamma_j$ as it is in my question
Oct
2
suggested suggested edit on Can correlated random effects “steal” the variability (and the significance) from the regression coefficient?
Oct
2
comment Can correlated random effects “steal” the variability (and the significance) from the regression coefficient?
I have replaced $\gamma_i$ with $\gamma_j$ in your answer - I guess this is what you meant.
Oct
2
comment Can correlated random effects “steal” the variability (and the significance) from the regression coefficient?
Note to point 2) above - I am not comparing "variance of $\sigma^2$" as you suggested but instead I am comparing $sd(\beta_ix_{i,j}) = 0.03$ with $\sigma$. I hope this is what you meant?
Oct
2
comment Can correlated random effects “steal” the variability (and the significance) from the regression coefficient?
Lars, thank you for your answer! I have a few questions: 1) why is it "cheaper to put the variability in $\beta$?" 2) Regarding the size of $\sigma^2$ vs $\beta_ix_{i,j}$, how do I compare it properly? Unfortunatelly there is a convergence problem and I have only one model run where estimate of $\sigma$ converged. The estimate is $\sigma = 0.17$, $\beta = -0.47$, $sd(x_{i,j}) = 0.06$ (should I compute sd of $x$? mean is zero), so this gives me $sd(\beta_ix_{i,j}) = 0.03$ which is much less than the $\sigma$! Does it mean that the variability is then consumed by the random effect?
Oct
1
revised Can correlated random effects “steal” the variability (and the significance) from the regression coefficient?
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Oct
1
revised Can correlated random effects “steal” the variability (and the significance) from the regression coefficient?
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Sep
30
awarded  Explainer
Sep
27
awarded  Popular Question
Sep
26
asked Can correlated random effects “steal” the variability (and the significance) from the regression coefficient?