<h2>File List</h2><blockquote>
<p><b>Supplement 1, original</b> </p>
<p><i><b>All files at once</b></i></p>
<blockquote>
<p><a href="Spencer_code.tar">Spencer_code.tar</a></p>
</blockquote>
<p><i><b>Matlab code</b></i></p>
<blockquote>
<p><a href="get_Q_like.m">get_Q_like.m</a></p>
<p><a href="ML_Q_est5.m">ML_Q_est5.m</a></p>
<p><a href="mlq_ufun3.m">mlq_ufun3.m</a></p>
<p><a href="multinomial.m">multinomial.m</a></p>
<p><a href="plike.m">plike.m</a></p>
<p><a href="Qfromest.m">Qfromest.m</a></p>
<p><a href="Qmatrix_bootstrap.m">Qmatrix_bootstrap.m</a></p>
<p><a href="stationary_probt.m">stationary_probt.m</a></p>
<p><a href="sorted_eigs.m">sorted_eigs.m</a></p>
<p> </p>
</blockquote>
<p><i><b>Example data</b></i></p>
<blockquote>
<p><a href="example_data.mat">example_data.mat</a></p>
<p> </p>
</blockquote>
<p><b>Supplement 1, Revision 1</b></p>
<blockquote>
<p><a href="suppl-1R1.htm">Download files</a> </p>
</blockquote>
</blockquote><h2>Description</h2><blockquote><p> </p>
Matlab code for the bootstrap analysis in Spencer and Susko, Continuous-time Markov models for species interactions. Tested under Matlab release 14. Requires the optimization toolbox. <br>
The main function, ML_Q_est5.m, tests the hypothesis that the data can be explained by a homogeneous continuous-time Markov model against the more general alternative that rates are not homogeneous. It first estimates parameters for a homogeneous continuous-time Markov model, then calculates the likelihood ratio between this model and the maximum likelihood discrete-time model. It then generates parametric bootstrap samples from the estimated homogeneous continuous-time model to generate a distribution of the likelihood ratio under the hypothesis of time homogeneity. For detailed instructions on usage, type help ML_Q_est5 in Matlab. Typical usage:
<p> [Q,negl,nobs,neglobs,neglz,twodelta,p,timetaken]=ML_Q_est5(P,obs_csum,reps,nsites,ntimes);</p>
<p> The inputs are:</p>
<p>P is the observed transition matrix for time step of 1 unit.</p>
<p>obs_csum is the number of observed transitions from each state.</p>
<p>reps is number of parametric bootstrap reps to run.</p>
<p>ntimes is number of time periods sampled in original data.</p>
<p>nsites is number of sites sampled in original data.</p>
<p> The outputs are:</p>
<p>Q is the estimated homogeneous instantaneous rate matrix, with all off-diagonal elements constrained to be positive.</p>
<p>negl is partial negative log likelihood for this Q matrix.</p>
<p>nobs is an array of number of observations of each transition (with some rounding if the P matrix wasn't reported exactly).</p>
<p>neglobs is partial negative log likelihood if we use observed transition frequencies (which are the ML estimates in a discrete-time model, not requiring homogeneity in continuous time).</p>
<p>neglz is partial negative log likelihood if we set any tiny elements of Q (<2*eps, where eps is machine precision) to zero. This is usually almost identical to negl.</p>
<p>twodelta is twice difference in partial log likelihoods between the Q matrix and the empirical P matrix: first row is observed, remaining reps rows are from parametric bootstrap. First col is for the Q matrix as estimated, second is with tiny elements of Q set to zero.</p>
<p>p is proportion of reps with twodelta >= observed (first col Q as estimated, second col with tiny elements of Q set to zero).</p>
<p>timetaken is clock time used.</p>
<p>The other functions (mlq_ufun3.m, get_Q_like.m, Qmatrix_bootstrap.m, stationary_probt.m, sorted_eigs.m, Qfromest.m, plike.m, multinomial.m) are called by ML_Q_est5.m.</p>
<p>example_data.mat contains the data described in the paper (Matlab format, compatible with version 6 and later). The original source for these data is Tanner, J., T. Hughes, and J. Connell. 1994. Species coexistence, keystone species, and succession: a sensitivity analysis. Ecology <b>75</b>:2204–2219 (Exposed Crest site, their Table 2, with additional information from J. Tanner, <i>personal communication</i>)</p>
<p></p>
</blockquote
