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<P DIR=LTR><SPAN LANG="en-ca"><FONT FACE="Calibri">I want to compare tweaks to stock assessment models using AIC</FONT></SPAN><SPAN LANG="en-ca"><FONT FACE="Calibri">, but am challenged by deviation parameters.</FONT></SPAN><SPAN LANG="en-ca"> <FONT FACE="Calibri">I found a (draft?) PhD thesis by Ian Stewart (reviewers indicated as</FONT></SPAN><SPAN LANG="en-ca"> <FONT FACE="Calibri">Ray</FONT></SPAN><SPAN LANG="en-ca"> <FONT FACE="Calibri">Hilborn</FONT></SPAN><SPAN LANG="en-ca"><FONT FACE="Calibri">, Andr</FONT></SPAN><SPAN LANG="en-ca"><FONT FACE="Calibri">e</FONT></SPAN><SPAN LANG="en-ca"><FONT FACE="Calibri">$B!-(J</FONT></SPAN><SPAN LANG="(Jen-ca(J"><FONT FACE="Calibri"> Punt and Richard Methot</FONT></SPAN><SPAN LANG="(Jen-ca(J"><FONT FACE="Calibri">) that tackles the issue on pages 208-210, excerpted below.</FONT></SPAN><SPAN LANG="(Jen-ca(J"><FONT FACE="Calibri"> But I$B!G(Jm not clear on</FONT></SPAN><SPAN LANG="(Jen-ca(J"> <FONT FACE="Calibri">how</FONT></SPAN><SPAN LANG="(Jen-ca(J"><FONT FACE="Calibri"> the data</FONT></SPAN><SPAN LANG="(Jen-ca(J"> <FONT FACE="Calibri">(</FONT></SPAN><SPAN LANG="(Jen-ca(J"></SPAN><SPAN LANG="(Jen-ca(J"></SPAN><SPAN LANG="(Jen-ca(J"><FONT SIZE=5 FACE="Symbol">f</FONT></SPAN><SPAN LANG="(Jen-ca(J"><I></I></SPAN><SPAN LANG="(Jen-ca(J"><I></I></SPAN><SPAN LANG="(Jen-ca(J"><I><FONT FACE="Times New Roman">data</FONT></I></SPAN><SPAN LANG="(Jen-ca(J"><FONT FACE="Calibri">)</FONT></SPAN><SPAN LANG="(Jen-ca(J"><FONT FACE="Calibri"></FONT></SPAN><SPAN LANG="(Jen-ca(J"> <FONT FACE="Calibri">and parameter</FONT></SPAN><SPAN LANG="(Jen-ca(J"><FONT FACE="Calibri"></FONT></SPAN><SPAN LANG="(Jen-ca(J"> <FONT FACE="Calibri">(</FONT></SPAN><SPAN LANG="(Jen-ca(J"></SPAN><SPAN LANG="(Jen-ca(J"></SPAN><SPAN LANG="(Jen-ca(J"><FONT SIZE=5 FACE="Symbol">f</FONT></SPAN><SPAN LANG="(Jen-ca(J"><I></I></SPAN><SPAN LANG="(Jen-ca(J"><I></I></SPAN><SPAN LANG="(Jen-ca(J"><I><FONT FACE="Times New Roman">parameter</FONT></I></SPAN><SPAN LANG="(Jen-ca(J"><FONT FACE="Calibri">)</FONT></SPAN><SPAN LANG="(Jen-ca(J"> <FONT FACE="Calibri"> scalars are</FONT></SPAN><SPAN LANG="(Jen-ca(J"><FONT FACE="Calibri"> derived</FONT></SPAN><SPAN LANG="(Jen-ca(J"><FONT FACE="Calibri">.</FONT></SPAN><SPAN LANG="(Jen-ca(J"> <FONT FACE="Calibri">T</FONT></SPAN><SPAN LANG="(Jen-ca(J"><FONT FACE="Calibri">he</FONT></SPAN><SPAN LANG="(Jen-ca(J"><FONT FACE="Calibri"></FONT></SPAN><SPAN LANG="(Jen-ca(J"> <FONT FACE="Calibri">data</FONT></SPAN><SPAN LANG="(Jen-ca(J"><FONT FACE="Calibri"> scalar</FONT></SPAN><SPAN LANG="(Jen-ca(J"><FONT FACE="Calibri">, and I expect the parameter scalar,</FONT></SPAN><SPAN LANG="(Jen-ca(J"><FONT FACE="Calibri"> should</FONT></SPAN><SPAN LANG="(Jen-ca(J"><FONT FACE="Calibri"> range between 0 and 1</FONT></SPAN><SPAN LANG="(Jen-ca(J"><FONT FACE="Calibri">. Anybody</FONT></SPAN><SPAN LANG="(Jen-ca(J"> <FONT FACE="Calibri">know</FONT></SPAN><SPAN LANG="(Jen-ca(J"> <FONT FACE="Calibri">more about</FONT></SPAN><SPAN LANG="(Jen-ca(J"><FONT FACE="Calibri"> these?</FONT></SPAN><SPAN LANG="(Jen-ca(J"></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"><FONT FACE="Times New Roman">Both AIC</FONT></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"><FONT SIZE=1 FACE="Times New Roman">c</FONT></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"> <FONT FACE="Times New Roman">and BIC require specification of the number of data points, an area</FONT></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"><FONT FACE="Times New Roman">of some disagreement when diverse data sets are included with multiple error</FONT></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"><FONT FACE="Times New Roman">assumptions in the same model. Specifically, the sample sizes of multinomial likelihoods</FONT></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"><FONT FACE="Times New Roman">applied to compositional data are often tuned during model fitting to be more consistent</FONT></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"><FONT FACE="Times New Roman">with observed residual error (although no tuning of multinomial sample sizes was</FONT></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"><FONT FACE="Times New Roman">employed in the 2005 English sole assessment). Therefore, the number of categories has</FONT></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"><FONT FACE="Times New Roman">been used as a proxy for the dimension of these data in some studies (Helu et al., 2000).</FONT></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"><FONT FACE="Times New Roman">Assuming only that the likelihoods and error structures are correctly specified, the</FONT></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"><FONT FACE="Times New Roman">effective dimension of these data should lie somewhere between the number of</FONT></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"><FONT FACE="Times New Roman">multinomially distributed observations and the sum of the input sample sizes for all such</FONT></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"><FONT FACE="Times New Roman">distributions used in the model. Therefore, for the purposes of calculating measures of</FONT></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"><FONT FACE="Times New Roman">model fit, a range in the dimension of the data is explored via a multiplicative scalar</FONT></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"><FONT FACE="Times New Roman">(bounded by zero and one) describing the additional dimension added through the</FONT></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"><FONT FACE="Times New Roman">number of observations greater than one per multinomial component. All other data</FONT></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"><FONT FACE="Times New Roman">sources, including survey indices, discard and mean weight observations are enumerated</FONT></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"><FONT FACE="Times New Roman">directly for a total count of data points equal to:</FONT></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I><FONT SIZE=4 FACE="Times New Roman">D</FONT></I></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"> <FONT SIZE=4 FACE="Symbol">=</FONT></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"><FONT SIZE=5 FACE="TTE10CDB68t00">$B&2(J</FONT></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"><FONT SIZE=4 FACE="Times New Roman">(</FONT></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I><FONT SIZE=4 FACE="Times New Roman">N</FONT></I></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I><FONT FACE="Times New Roman">index obs</FONT></I></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"><FONT FACE="Times New Roman">,</FONT></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I><FONT FACE="Times New Roman">discard obs</FONT></I></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"><FONT FACE="Times New Roman">,...</FONT></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"><FONT SIZE=4 FACE="Times New Roman">)</FONT></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"> <FONT SIZE=4 FACE="Symbol">+ </FONT></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I><FONT SIZE=4 FACE="Times New Roman">M</FONT></I></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"> <FONT SIZE=4 FACE="Symbol">+</FONT></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"><FONT SIZE=5 FACE="Symbol">f</FONT></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I><FONT FACE="Times New Roman">data</FONT></I></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"> <FONT SIZE=4 FACE="TTE2E3B688t00">i</FONT></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"><FONT SIZE=4 FACE="Times New Roman">(</FONT></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"><FONT SIZE=5 FACE="TTE10CDB68t00">$B&2(J</FONT></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I><FONT SIZE=4 FACE="Times New Roman">N</FONT></I></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I><FONT FACE="Times New Roman">mult</FONT></I></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"> <FONT SIZE=4 FACE="Symbol">-</FONT></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I><FONT SIZE=4 FACE="Times New Roman">M</FONT></I></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"><FONT SIZE=4 FACE="Times New Roman">)</FONT></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"> <FONT FACE="Times New Roman">,</FONT></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"><FONT FACE="Times New Roman">where</FONT></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I> <FONT FACE="Times New Roman">D</FONT></I></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"> <FONT FACE="Times New Roman">is the dimension of the data,</FONT></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I> <FONT FACE="Times New Roman">N</FONT></I></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"><FONT SIZE=1 FACE="Times New Roman">$B!D(J</FONT></SPAN><SPAN LANG="(Jen-ca(J"></SPAN><SPAN LANG="(Jen-ca(J"></SPAN><SPAN LANG="(Jen-ca(J"> <FONT FACE="Times New Roman">is the number of individual data points, or input</FONT></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"><FONT FACE="Times New Roman">sample size,</FONT></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I> <FONT FACE="Times New Roman">M</FONT></I></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"> <FONT FACE="Times New Roman">is the number of multinomially distributed length- or age-frequency</FONT></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"><FONT FACE="Times New Roman">distributions, and</FONT></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"> <FONT SIZE=5 FACE="Symbol">f</FONT></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I><FONT FACE="Times New Roman">data</FONT></I></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"> <FONT FACE="Times New Roman">is the multiplicative data scalar. This generalized approach</FONT></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"><FONT FACE="Times New Roman">allows full exploration of the role of sample size in model comparison.</FONT></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"><FONT FACE="Times New Roman">209</FONT></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"><FONT FACE="Times New Roman">Both metrics also require specifying the number of parameters estimated in</FONT></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"><FONT FACE="Times New Roman">the model, however, the dimension of model parameters differs from many statistical</FONT></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"><FONT FACE="Times New Roman">applications because many parameters (e.g., recruitment deviations) are explicitly</FONT></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"><FONT FACE="Times New Roman">constrained. This means that the effective number of parameters is a function of the</FONT></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"><FONT FACE="Times New Roman">relative constraint. Although Bayesian approaches for calculating the effective number of</FONT></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"><FONT FACE="Times New Roman">parameters have been developed (e.g., Spiegelhalter et al., 2002), there are currently no</FONT></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"><FONT FACE="Times New Roman">maximum likelihood based tools available. However, the effective number of parameters</FONT></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"><FONT FACE="Times New Roman">must lie within a definable range, so an approach similar the data scalar is employed here.</FONT></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"><FONT FACE="Times New Roman">As the limit as the variance (</FONT></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"><FONT FACE="TTE10C6448t00">$B&R(J</FONT></SPAN><SPAN LANG="(Jen-ca(J"></SPAN><SPAN LANG="(Jen-ca(J"></SPAN><SPAN LANG="(Jen-ca(J"><FONT SIZE=1 FACE="Times New Roman">2</FONT></SPAN><SPAN LANG="(Jen-ca(J"></SPAN><SPAN LANG="(Jen-ca(J"></SPAN><SPAN LANG="(Jen-ca(J"><FONT FACE="Times New Roman">) of the distribution constraining deviation parameters</FONT></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"><FONT FACE="Times New Roman">goes to infinity, the number of constrained parameters is equal to the number of</FONT></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"><FONT FACE="Times New Roman">deviations estimated, while as</FONT></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"> <FONT FACE="TTE10C6448t00">$B&R(J</FONT></SPAN><SPAN LANG="(Jen-ca(J"></SPAN><SPAN LANG="(Jen-ca(J"></SPAN><SPAN LANG="(Jen-ca(J"> <FONT SIZE=1 FACE="Times New Roman">2</FONT></SPAN><SPAN LANG="(Jen-ca(J"></SPAN><SPAN LANG="(Jen-ca(J"></SPAN><SPAN LANG="(Jen-ca(J"> <FONT FACE="Times New Roman">goes to 0, there are effectively zero estimated</FONT></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"><FONT FACE="Times New Roman">parameters. Another multiplicative scalar is used to describe this range:</FONT></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I><FONT SIZE=4 FACE="Times New Roman">P</FONT></I></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I><FONT FACE="Times New Roman">eff</FONT></I></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"> <FONT SIZE=4 FACE="Symbol">=</FONT></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"><FONT SIZE=5 FACE="TTE10CDB68t00">$B&2(J</FONT></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"><FONT SIZE=4 FACE="Times New Roman">(</FONT></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I><FONT SIZE=4 FACE="Times New Roman">N</FONT></I></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I><FONT FACE="Times New Roman">non</FONT></I></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"><FONT FACE="Symbol">-</FONT></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I><FONT FACE="Times New Roman">dev</FONT></I></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"> <FONT SIZE=4 FACE="Times New Roman">)</FONT></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"> <FONT SIZE=4 FACE="Symbol">+</FONT></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"><FONT SIZE=5 FACE="Symbol">f </FONT></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I><FONT FACE="Times New Roman">parameter</FONT></I></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"> <FONT SIZE=4 FACE="TTE2E3B688t00">i</FONT></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"><FONT SIZE=4 FACE="Times New Roman">(</FONT></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"><FONT SIZE=5 FACE="TTE10CDB68t00">$B&2(J</FONT></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I><FONT SIZE=4 FACE="Times New Roman">N</FONT></I></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I><FONT FACE="Times New Roman">dev</FONT></I></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"> <FONT SIZE=4 FACE="Times New Roman">)</FONT></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"> <FONT FACE="Times New Roman">,</FONT></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"><FONT FACE="Times New Roman">where</FONT></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I> <FONT FACE="Times New Roman">P</FONT></I></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I><FONT SIZE=1 FACE="Times New Roman">eff</FONT></I></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"> <FONT FACE="Times New Roman">is the effective number of parameters, N is the raw number of parameters,</FONT></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"><FONT FACE="Times New Roman">either deviations or non-deviation parameters, and</FONT></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"> <FONT SIZE=5 FACE="Symbol">f </FONT></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I><FONT FACE="Times New Roman">parameter</FONT></I></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"> <FONT FACE="Times New Roman">is the multiplicative</FONT></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"><FONT FACE="Times New Roman">parameter scalar. This approach allows direct exploration of the relative model weights</FONT></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"><FONT FACE="Times New Roman">conditioned on both the data and parameter dimensions (via the scalars); allowing a</FONT></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"><FONT FACE="Times New Roman">determination of the conditions under which alternate model weighting might occur.</FONT></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"><FONT FACE="Times New Roman">To draw inference from more than one model, it is necessary to obtain the relative</FONT></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"><FONT FACE="Times New Roman">model weights. First, the metrics (AIC</FONT></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"><FONT SIZE=1 FACE="Times New Roman">c</FONT></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"> <FONT FACE="Times New Roman">or BIC) for all candidate models are standardized</FONT></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"><FONT FACE="Times New Roman">to the relative difference between the value for each model and the minimum value for</FONT></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"><FONT FACE="Times New Roman">any model considered (Burnham and Anderson, 2002):</FONT></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"><FONT FACE="Times New Roman">210</FONT></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"><FONT SIZE=4 FACE="TTE2E3B688t00">$B"$(J</FONT></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I><FONT FACE="Times New Roman">i</FONT></I></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"><FONT SIZE=4 FACE="Symbol">= </FONT></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I><FONT SIZE=4 FACE="Times New Roman">AIC</FONT></I></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I><FONT FACE="Times New Roman">ci</FONT></I></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"> <FONT SIZE=4 FACE="Symbol">- </FONT></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I><FONT SIZE=4 FACE="Times New Roman">AIC</FONT></I></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I><FONT FACE="Times New Roman">c</FONT></I></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"><FONT FACE="Times New Roman">min</FONT></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"><FONT FACE="Times New Roman">The likelihood or probability of the data (</FONT></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I><FONT FACE="Times New Roman">D</FONT></I></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"><FONT FACE="Times New Roman">) given the model (</FONT></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I><FONT FACE="Times New Roman">M</FONT></I></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I><FONT SIZE=1 FACE="Times New Roman">i</FONT></I></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"><FONT FACE="Times New Roman">), or relative strength of</FONT></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"><FONT FACE="Times New Roman">evidence for each model, is then proportional (after renormalizing) to</FONT></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I> <FONT FACE="Times New Roman">w</FONT></I></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I><FONT SIZE=1 FACE="Times New Roman">i</FONT></I></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"><FONT FACE="Times New Roman">:</FONT></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I><FONT SIZE=4 FACE="Times New Roman">p</FONT></I></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"><FONT SIZE=6 FACE="Symbol">(</FONT></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I><FONT SIZE=4 FACE="Times New Roman">D M</FONT></I></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I><FONT FACE="Times New Roman">i</FONT></I></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"> <FONT SIZE=6 FACE="Symbol">)</FONT></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"><FONT SIZE=4 FACE="Symbol">µ </FONT></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I><FONT SIZE=4 FACE="Times New Roman">w</FONT></I></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I><FONT FACE="Times New Roman">i</FONT></I></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"> <FONT SIZE=4 FACE="Symbol">= </FONT></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"><FONT SIZE=4 FACE="Times New Roman">exp(</FONT></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"><FONT SIZE=4 FACE="Symbol">-</FONT></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"><FONT SIZE=4 FACE="Times New Roman">0.5</FONT></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"><FONT SIZE=4 FACE="TTE2E3B688t00">$B"$(J</FONT></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I></I></SPAN><SPAN LANG="en-ca"><I><FONT FACE="Times New Roman">i</FONT></I></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"> <FONT SIZE=4 FACE="Times New Roman">)</FONT></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"><FONT FACE="Times New Roman">Model weights were calculated for each model under values of the data and parameter</FONT></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"><FONT FACE="Times New Roman">scalars ranging from 0 to 1.0. Resulting sensitivity to the assumed dimension of the data</FONT></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"><FONT FACE="Times New Roman">and parameters are then jointly explored.</FONT></SPAN><SPAN LANG="en-ca"></SPAN></P>
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<P DIR=LTR><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"><FONT SIZE=2 FACE="Arial">Mark Fowler</FONT></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"><FONT SIZE=2 FACE="Arial">Population Ecology Division</FONT></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"><FONT SIZE=2 FACE="Arial">Bedford Inst of Oceanography</FONT></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"><FONT SIZE=2 FACE="Arial">Dept Fisheries & Oceans</FONT></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"><FONT SIZE=2 FACE="Arial">Dartmouth NS Canada</FONT></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"><FONT SIZE=2 FACE="Arial">B2Y 4A2</FONT></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"><FONT SIZE=2 FACE="Arial">Tel. (902) 426-3529</FONT></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"><FONT SIZE=2 FACE="Arial">Fax (902) 426-9710</FONT></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"><FONT SIZE=2 FACE="Arial">Email Mark.Fowler@dfo-mpo.gc.ca</FONT></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"></SPAN><SPAN LANG="en-ca"><FONT SIZE=2 FACE="Arial">Home Tel. (902) 461-0708</FONT></SPAN></P>
<P DIR=LTR><SPAN LANG="en-ca"><FONT SIZE=2 FACE="Arial">Home Email mark.fowler@ns.sympatico.ca</FONT></SPAN></P>
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