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205
206<ul class="summary">
207<li><a href='/'>Statistics for Ecologists</a></li>
208
209<li class="divider"></li>
210<li><a href='/#section' id='toc-section'></a></li>
211<li class="chapter" data-level="" data-path="about-the-author.html"><a href='/about-the-author'><i class="fa fa-check"></i>About the Author</a></li>
212<li class="chapter" data-level="" data-path="preface.html"><a href='/preface'><i class="fa fa-check"></i>Preface</a>
213<ul>
214<li class="chapter" data-level="" data-path="preface.html"><a href='/preface#formatting-and-conventions'><i class="fa fa-check"></i>Formatting and conventions</a></li>
215<li class="chapter" data-level="" data-path="preface.html"><a href='/preface#pre-requisites'><i class="fa fa-check"></i>Pre-requisites</a></li>
216<li class="chapter" data-level="" data-path="preface.html"><a href='/preface#learning-objectives'><i class="fa fa-check"></i>Learning objectives</a></li>
217<li class="chapter" data-level="" data-path="preface.html"><a href='/preface#skills-objectives'><i class="fa fa-check"></i>Skills objectives</a></li>
218<li class="chapter" data-level="" data-path="preface.html"><a href='/preface#real-versus-simulated-data'><i class="fa fa-check"></i>Real versus simulated data</a></li>
219<li class="chapter" data-level="" data-path="preface.html"><a href='/preface#standing-on-the-shoulders-of'><i class="fa fa-check"></i>Standing on the shoulders of…</a></li>
220<li class="chapter" data-level="" data-path="preface.html"><a href='/preface#how-to-use-this-book-and-associated-resources-for-teaching'><i class="fa fa-check"></i>How to use this book and associated resources for teaching</a></li>
221<li class="chapter" data-level="" data-path="preface.html"><a href='/preface#feedback'><i class="fa fa-check"></i>Feedback</a></li>
222<li class="chapter" data-level="" data-path="preface.html"><a href='/preface#acknowledgements'><i class="fa fa-check"></i>Acknowledgements</a></li>
223</ul></li>
224<li class="part"><span><b>I Models for Normally Distributed Responses</b></span></li>
225<li class="chapter" data-level="1" data-path="linreg.html"><a href='/linreg'><i class="fa fa-check"></i><b>1</b> Linear regression review</a>
226<ul>
227<li class="chapter" data-level="1.1" data-path="linreg.html"><a href='/linreg#r-packages'><i class="fa fa-check"></i><b>1.1</b> R Packages</a></li>
228<li class="chapter" data-level="1.2" data-path="linreg.html">
228<a href='/linreg#data-example-sustainable-trophy-hunting-of-male-african-lions'><i class="fa fa-check"></i><b>1.2</b> Data example: Sustainable trophy hunting of male African lions</a></li>
229<li class="chapter" data-level="1.3" data-path="linreg.html"><a href='/linreg#intslopeinterp'><i class="fa fa-check"></i><b>1.3</b> Interpreting slope and intercept</a></li>
230<li class="chapter" data-level="1.4" data-path="linreg.html"><a href='/linreg#lmassumptions'><i class="fa fa-check"></i><b>1.4</b> Linear regression assumptions</a></li>
231<li class="chapter" data-level="1.5" data-path="linreg.html"><a href='/linreg#evaluating-assumptions'><i class="fa fa-check"></i><b>1.5</b> Evaluating assumptions</a>
232<ul>
233<li class="chapter" data-level="1.5.1" data-path="linreg.html"><a href='/linreg#exploring-assumptions-using-r'><i class="fa fa-check"></i><b>1.5.1</b> Exploring assumptions using R</a></li>
234</ul></li>
235<li class="chapter" data-level="1.6" data-path="linreg.html"><a href='/linreg#sampdist'><i class="fa fa-check"></i><b>1.6</b> Statistical inference: Sampling distributions</a>
236<ul>
237<li class="chapter" data-level="1.6.1" data-path="linreg.html"><a href='/linreg#exploring-sampling-distributions-through-simulation'><i class="fa fa-check"></i><b>1.6.1</b> Exploring sampling distributions through simulation</a></li>
238<li class="chapter" data-level="1.6.2" data-path="linreg.html"><a href='/linreg#writing-a-for-loop'><i class="fa fa-check"></i><b>1.6.2</b> Writing a for loop</a></li>
239<li class="chapter" data-level="1.6.3" data-path="linreg.html"><a href='/linreg#exercise1'><i class="fa fa-check"></i><b>1.6.3</b> Exercise: Sampling distribution</a></li>
240</ul></li>
241<li class="chapter" data-level="1.7" data-path="linreg.html"><a href='/linreg#tdist'><i class="fa fa-check"></i><b>1.7</b> Sampling distribution of the t-statistic</a>
242<ul>
243<li class="chapter" data-level="1.7.1" data-path="linreg.html"><a href='/linreg#class-exercise-sampling-distribution-of-the-t-statistic'><i class="fa fa-check"></i><b>1.7.1</b> Class exercise: sampling distribution of the t-statistic</a></li>
244</ul></li>
245<li class="chapter" data-level="1.8" data-path="linreg.html"><a href='/linreg#confidence-intervals'><i class="fa fa-check"></i><b>1.8</b> Confidence intervals</a>
246<ul>
247<li class="chapter" data-level="1.8.1" data-path="linreg.html"><a href='/linreg#CI'><i class="fa fa-check"></i><b>1.8.1</b> Exercise: explore confidence intervals through simulation</a></li>
248</ul></li>
249<li class="chapter" data-level="1.9" data-path="linreg.html"><a href='/linreg#confpred'><i class="fa fa-check"></i><b>1.9</b> Confidence intervals versus prediction intervals</a></li>
250<li class="chapter" data-level="1.10" data-path="linreg.html"><a href='/linreg#hypothesis-tests-and-p-values'><i class="fa fa-check"></i><b>1.10</b> Hypothesis tests and p-values</a>
251<ul>
252<li class="chapter" data-level="1.10.1" data-path="linreg.html"><a href='/linreg#formal-decision-process'><i class="fa fa-check"></i><b>1.10.1</b> Formal decision process</a></li>
253<li class="chapter" data-level="1.10.2" data-path="linreg.html"><a href='/linreg#issues-with-null-hypothesis-testing'><i class="fa fa-check"></i><b>1.10.2</b> Issues with null hypothesis testing</a></li>
254<li class="chapter" data-level="1.10.3" data-path="linreg.html"><a href='/linreg#exemplification-testing-whether-the-slope-is-different-from-0'><i class="fa fa-check"></i><b>1.10.3</b> Exemplification: Testing whether the slope is different from 0</a></li>
255</ul></li>
256<li class="chapter" data-level="1.11" data-path="linreg.html"><a href='/linreg#r2-amount-of-variance-explained'><i class="fa fa-check"></i><b>1.11</b> <span class="math inline">\(R^2\)</span>: Amount of variance explained</a></li>
257<li class="chapter" data-level="1.12" data-path="linreg.html"><a href='/linreg#coded-answers-to-exercises'><i class="fa fa-check"></i><b>1.12</b> Coded answers to exercises</a></li>
258</ul></li>
259<li class="chapter" data-level="2" data-path="boot.html"><a href='/boot'><i class="fa fa-check"></i><b>2</b> Bootstrapping</a>
260<ul>
261<li class="chapter" data-level="2.1" data-path="boot.html"><a href='/boot#r-packages-1'><i class="fa fa-check"></i><b>2.1</b> R Packages</a></li>
262<li class="chapter" data-level="2.2" data-path="boot.html"><a href='/boot#motivating-data-example-rikz-dutch-governmental-institute-data'><i class="fa fa-check"></i><b>2.2</b> Motivating data example: RIKZ [Dutch governmental institute] data</a></li>
263<li class="chapter" data-level="2.3" data-path="boot.html"><a href='/boot#consequences-of-assumption-violations'><i class="fa fa-check"></i><b>2.3</b> Consequences of assumption violations</a></li>
264<li class="chapter" data-level="2.4" data-path="boot.html"><a href='/boot#introduction-to-the-bootstrap'><i class="fa fa-check"></i><b>2.4</b> Introduction to the Bootstrap</a>
265<ul>
266<li class="chapter" data-level="2.4.1" data-path="boot.html"><a href='/boot#bootstrap-procedure'><i class="fa fa-check"></i><b>2.4.1</b> Bootstrap procedure</a></li>
267<li class="chapter" data-level="2.4.2" data-path="boot.html"><a href='/boot#bootstrap-example-simple-linear-regression'><i class="fa fa-check"></i><b>2.4.2</b> Bootstrap example: Simple linear regression</a></li>
268</ul></li>
269<li class="chapter" data-level="2.5" data-path="boot.html"><a href='/boot#replicating-the-sampling-design'><i class="fa fa-check"></i><b>2.5</b> Replicating the sampling design</a></li>
270<li class="chapter" data-level="2.6" data-path="boot.html"><a href='/boot#utility-of-the-bootstrap'><i class="fa fa-check"></i><b>2.6</b> Utility of the bootstrap</a></li>
271</ul></li>
272<li class="chapter" data-level="3" data-path="MatrixReg.html"><a href='/matrixreg'><i class="fa fa-check"></i><b>3</b> Multiple regression</a>
273<ul>
274<li class="chapter" data-level="3.1" data-path="MatrixReg.html"><a href='/matrixreg#r-packages-2'><i class="fa fa-check"></i><b>3.1</b> R Packages</a></li>
275<li class="chapter" data-level="3.2" data-path="MatrixReg.html"><a href='/matrixreg#introduction-to-multiple-regression'><i class="fa fa-check"></i><b>3.2</b> Introduction to multiple regression</a></li>
276<li class="chapter" data-level="3.3" data-path="MatrixReg.html"><a href='/matrixreg#matrix-notation-for-regression'><i class="fa fa-check"></i><b>3.3</b>
276 Matrix notation for regression</a></li>
277<li class="chapter" data-level="3.4" data-path="MatrixReg.html"><a href='/matrixreg#r2'><i class="fa fa-check"></i><b>3.4</b> Parameter estimation, sums-of-squares, and R<span class="math inline">\(^2\)</span></a></li>
278<li class="chapter" data-level="3.5" data-path="MatrixReg.html"><a href='/matrixreg#parameter-interpretation-multiple-regression-with-rikz-data'><i class="fa fa-check"></i><b>3.5</b> Parameter interpretation: Multiple regression with RIKZ data</a></li>
279<li class="chapter" data-level="3.6" data-path="MatrixReg.html"><a href='/matrixreg#categorical-predictors'><i class="fa fa-check"></i><b>3.6</b> Categorical predictors</a>
280<ul>
281<li class="chapter" data-level="3.6.1" data-path="MatrixReg.html"><a href='/matrixreg#ttest'><i class="fa fa-check"></i><b>3.6.1</b> T-test as a regression</a></li>
282<li class="chapter" data-level="3.6.2" data-path="MatrixReg.html"><a href='/matrixreg#dummy-variables-reference-or-effects-coding'><i class="fa fa-check"></i><b>3.6.2</b> Dummy variables: Reference (or effects) coding</a></li>
283<li class="chapter" data-level="3.6.3" data-path="MatrixReg.html"><a href='/matrixreg#dummy-variables-cell-means-coding'><i class="fa fa-check"></i><b>3.6.3</b> Dummy variables: Cell means coding</a></li>
284<li class="chapter" data-level="3.6.4" data-path="MatrixReg.html"><a href='/matrixreg#comparing-assumptions-linear-model-and-t-test'><i class="fa fa-check"></i><b>3.6.4</b> Comparing assumptions: Linear model and t-test</a></li>
285</ul></li>
286<li class="chapter" data-level="3.7" data-path="MatrixReg.html"><a href='/matrixreg#ancova'><i class="fa fa-check"></i><b>3.7</b> Categorical variables with <span class="math inline">\(&gt; 2\)</span> levels or categories</a>
287<ul>
288<li class="chapter" data-level="3.7.1" data-path="MatrixReg.html"><a href='/matrixreg#effects-coding'><i class="fa fa-check"></i><b>3.7.1</b> Effects coding</a></li>
289<li class="chapter" data-level="3.7.2" data-path="MatrixReg.html"><a href='/matrixreg#means-coding'><i class="fa fa-check"></i><b>3.7.2</b> Means coding</a></li>
290</ul></li>
291<li class="chapter" data-level="3.8" data-path="MatrixReg.html"><a href='/matrixreg#models-with-interactions'><i class="fa fa-check"></i><b>3.8</b> Models with interactions</a>
292<ul>
293<li class="chapter" data-level="3.8.1" data-path="MatrixReg.html"><a href='/matrixreg#intereffects'><i class="fa fa-check"></i><b>3.8.1</b> Effects coding</a></li>
294<li class="chapter" data-level="3.8.2" data-path="MatrixReg.html"><a href='/matrixreg#means-model'><i class="fa fa-check"></i><b>3.8.2</b> Means model</a></li>
295<li class="chapter" data-level="3.8.3" data-path="MatrixReg.html"><a href='/matrixreg#custom'><i class="fa fa-check"></i><b>3.8.3</b> Creating flexible models with dummy variables</a></li>
296<li class="chapter" data-level="3.8.4" data-path="MatrixReg.html"><a href='/matrixreg#improving-parameter-interpretation-through-centering'><i class="fa fa-check"></i><b>3.8.4</b> Improving parameter interpretation through centering</a></li>
297</ul></li>
298<li class="chapter" data-level="3.9" data-path="MatrixReg.html"><a href='/matrixreg#pairwise-comparisons'><i class="fa fa-check"></i><b>3.9</b> Pairwise comparisons</a></li>
299<li class="chapter" data-level="3.10" data-path="MatrixReg.html"><a href='/matrixreg#multipledf'><i class="fa fa-check"></i><b>3.10</b> Multiple degree-of-freedom hypothesis tests</a></li>
300<li class="chapter" data-level="3.11" data-path="MatrixReg.html"><a href='/matrixreg#regression-f-statistic'><i class="fa fa-check"></i><b>3.11</b> Regression F-statistic</a></li>
301<li class="chapter" data-level="3.12" data-path="MatrixReg.html"><a href='/matrixreg#contrasts'><i class="fa fa-check"></i><b>3.12</b> Contrasts: Estimation of linear combinations of parameters</a></li>
302<li class="chapter" data-level="3.13" data-path="MatrixReg.html"><a href='/matrixreg#aside-revisiting-f-tests-and-comparing-them-to-wald-chi2-tests'><i class="fa fa-check"></i><b>3.13</b> Aside: Revisiting F-tests and comparing them to Wald <span class="math inline">\(\chi^2\)</span> tests</a></li>
303<li class="chapter" data-level="3.14" data-path="MatrixReg.html"><a href='/matrixreg#partialsoft'><i class="fa fa-check"></i><b>3.14</b> Visualizing multiple regression models</a>
304<ul>
305<li class="chapter" data-level="3.14.1" data-path="MatrixReg.html"><a href='/matrixreg#addedv'><i class="fa fa-check"></i><b>3.14.1</b> Added variable plots</a></li>
306<li class="chapter" data-level="3.14.2" data-path="MatrixReg.html"><a href='/matrixreg#partialresid'><i class="fa fa-check"></i><b>3.14.2</b> Component + residual plots or partial residual plots</a></li>
307<li class="chapter" data-level="3.14.3" data-path="MatrixReg.html"><a href='/matrixreg#Othereffects'><i class="fa fa-check"></i><b>3.14.3</b> Effect plots</a></li>
308</ul></li>
309</ul></li>
310<li class="chapter" data-level="4" data-path="nonlin.html"><a href='/nonlin'><i class="fa fa-check"></i><b>4</b> Modeling Non-linear relationships</a>
311<ul>
312<li class="chapter" data-level="4.1" data-path="nonlin.html"><a href='/nonlin#r-packages-3'><i class="fa fa-check"></i><b>4.1</b> R Packages</a></li>
313<li class="chapter" data-level="4.2" data-path="nonlin.html"><a href='/nonlin#modeling-non-linear-relationships'><i class="fa fa-check"></i><b>4.2</b> Modeling non-linear relationships</a></li>
314<li class="chapter" data-level="4.3" data-path="nonlin.html"><a href='/nonlin#polys'><i class="fa fa-check"></i><b>4.3</b> Polynomials</a></li>
315<li class="chapter" data-level="4.4" data-path="nonlin.html"><a href='/nonlin#basis-functionsvectors'><i class="fa fa-check"></i><b>4.4</b> Basis functions/vectors</a></li>
316<li class="chapter" data-level="4.5" data-path="nonlin.html"><a href='/nonlin#splines'><i class="fa fa-check"></i><b>4.5</b> Splines</a>
317<ul>
318<li class="chapter" data-level="4.5.1" data-path="nonlin.html"><a href='/nonlin#linear-splines'><i class="fa fa-check"></i><b>4.5.1</b> Linear splines</a></li>
319<li class="chapter" data-level="4.5.2" data-path="nonlin.html"><a href='/nonlin#cubic-regression-splines'><i class="fa fa-check"></i><b>4.5.2</b> Cubic regression splines</a></li>
320<li class="chapter" data-level="4.5.3" data-path="nonlin.html"><a href='/nonlin#natural-cubic-regression-splines'><i class="fa fa-check"></i><b>4.5.3</b> Natural cubic regression splines</a></li>
321<li class="chapter" data-level="4.5.4" data-path="nonlin.html"><a href='/nonlin#knotloc'><i class="fa fa-check"></i><b>4.5.4</b> Number of knots and their locations</a></li>
322</ul></li>
323<li class="chapter" data-level="4.6" data-path="nonlin.html"><a href='/nonlin#splines-versus-polynomials'><i class="fa fa-check"></i><b>4.6</b> Splines versus polynomials</a></li>
324<li class="chapter" data-level="4.7" data-path="nonlin.html"><a href='/nonlin#visualizing-spline-models'><i class="fa fa-check"></i><b>4.7</b> Visualizing non-linear relationships</a></li>
325<li class="chapter" data-level="4.8" data-path="nonlin.html"><a href='/nonlin#generalized-additive-models-gams'><i class="fa fa-check"></i><b>4.8</b> Generalized additive models (GAMS)</a></li>
326<li class="chapter" data-level="4.9" data-path="nonlin.html"><a href='/nonlin#generalizations-to-multiple-non-linear-relationships'><i class="fa fa-check"></i><b>4.9</b> Generalizations to multiple non-linear relationships</a></li>
327<li class="chapter" data-level="4.10" data-path="nonlin.html"><a href='/nonlin#non-linear-models-with-a-mechanistic-basis'><i class="fa fa-check"></i><b>4.10</b> Non-Linear models with a mechanistic basis</a></li>
328<li class="chapter" data-level="4.11" data-path="nonlin.html"><a href='/nonlin#ortho'><i class="fa fa-check"></i><b>4.11</b> Aside: Orthogonal polynomials</a></li>
329<li class="chapter" data-level="4.12" data-path="nonlin.html"><a href='/nonlin#aside-segmented-and-piecewise-regression'><i class="fa fa-check"></i><b>4.12</b> Aside: Segmented and piecewise regression</a></li>
330</ul></li>
331<li class="chapter" data-level="5" data-path="gls.html"><a href='/gls'><i class="fa fa-check"></i><b>5</b> Generalized least squares (GLS)</a>
332<ul>
333<li class="chapter" data-level="5.1" data-path="gls.html"><a href='/gls#r-packages-4'><i class="fa fa-check"></i><b>5.1</b> R Packages</a></li>
334<li class="chapter" data-level="5.2" data-path="gls.html"><a href='/gls#why-cover-models-fit-using-generalized-least-squares'><i class="fa fa-check"></i><b>5.2</b> Why cover models fit using generalized least squares?</a></li>
335<li class="chapter" data-level="5.3" data-path="gls.html"><a href='/gls#generalized-least-squares-gls-non-constant-variance'><i class="fa fa-check"></i><b>5.3</b> Generalized least squares (GLS): Non-constant variance</a></li>
336<li class="chapter" data-level="5.4" data-path="gls.html"><a href='/gls#variance-heterogeneity-among-groups-t-test-with-unequal-variances'><i class="fa fa-check"></i><b>5.4</b> Variance heterogeneity among groups: T-test with unequal variances</a>
337<ul>
338<li class="chapter" data-level="5.4.1" data-path="gls.html"><a href='/gls#comparing-the-gls-model-to-a-linear-model'><i class="fa fa-check"></i><b>5.4.1</b> Comparing the GLS model to a linear model</a></li>
339<li class="chapter" data-level="5.4.2" data-path="gls.html"><a href='/gls#aside-parameterization-of-the-model'><i class="fa fa-check"></i><b>5.4.2</b> Aside: Parameterization of the model</a></li>
340</ul></li>
341<li class="chapter" data-level="5.5" data-path="gls.html"><a href='/gls#contgls'><i class="fa fa-check"></i><b>5.5</b> Variance increasing with <span class="math inline">\(X_i\)</span> or <span class="math inline">\(\mu_i\)</span></a>
342<ul>
343<li class="chapter" data-level="5.5.1" data-path="gls.html"><a href='/gls#variance-increasing-with-x_i'><i class="fa fa-check"></i><b>5.5.1</b> Variance increasing with <span class="math inline">\(X_i\)</span></a></li>
344<li class="chapter" data-level="5.5.2" data-path="gls.html"><a href='/gls#extension-to-multiple-groups'><i class="fa fa-check"></i><b>5.5.2</b> Extension to multiple groups</a></li>
345<li class="chapter" data-level="5.5.3" data-path="gls.html"><a href='/gls#varpowmu'><i class="fa fa-check"></i><b>5.5.3</b> Variance depending on <span class="math inline">\(\mu_i\)</span></a></li>
346</ul></li>
347<li class="chapter" data-level="5.6" data-path="gls.html"><a href='/gls#Sockeye'><i class="fa fa-check"></i><b>5.6</b> Approximate confidence and prediction intervals</a></li>
348<li class="chapter" data-level="5.7" data-path="gls.html"><a href='/gls#glsselect'><i class="fa fa-check"></i><b>5.7</b> Modeling strategy</a></li>
349</ul></li>
350<li class="part"><span><b>II What Variables to Include in a Model?</b></span></li>
351<li class="chapter" data-level="6" data-path="multicol.html"><a href='/multicol'><i class="fa fa-check"></i><b>6</b> Multicollinearity</a>
352<ul>
353<li class="chapter" data-level="6.1" data-path="multicol.html"><a href='/multicol#r-packages-5'><i class="fa fa-check"></i><b>6.1</b> R Packages</a></li>
354<li class="chapter" data-level="6.2" data-path="multicol.html"><a href='/multicol#multicollinearity'><i class="fa fa-check"></i><b>6.2</b> Multicollinearity</a></li>
355<li class="chapter" data-level="6.3" data-path="multicol.html"><a href='/multicol#motivating-example-what-factors-predict-how-long-mammals-sleep'><i class="fa fa-check"></i><b>6.3</b> Motivating example: what factors predict how long mammals sleep?</a></li>
356<li class="chapter" data-level="6.4" data-path="multicol.html"><a href='/multicol#variance-inflation-factors-vif'><i class="fa fa-check"></i><b>6.4</b> Variance inflation factors (VIF)</a></li>
357<li class="chapter" data-level="6.5" data-path="multicol.html"><a href='/multicol#multicollsim'><i class="fa fa-check"></i><b>6.5</b>
357 Understanding confounding using DAGs: A simulation example</a></li>
358<li class="chapter" data-level="6.6" data-path="multicol.html"><a href='/multicol#strategies-for-addressing-multicollinearity'><i class="fa fa-check"></i><b>6.6</b> Strategies for addressing multicollinearity</a></li>
359<li class="chapter" data-level="6.7" data-path="multicol.html"><a href='/multicol#applied-example-modeling-the-effect-of-correlated-environmental-factors-on-the-distribution-of-subtidal-kelp'><i class="fa fa-check"></i><b>6.7</b> Applied example: Modeling the effect of correlated environmental factors on the distribution of subtidal kelp</a></li>
360<li class="chapter" data-level="6.8" data-path="multicol.html"><a href='/multicol#seqreg'><i class="fa fa-check"></i><b>6.8</b> Residual and sequential regression</a></li>
361<li class="chapter" data-level="6.9" data-path="multicol.html"><a href='/multicol#pca'><i class="fa fa-check"></i><b>6.9</b> Principal components regression</a></li>
362<li class="chapter" data-level="6.10" data-path="multicol.html"><a href='/multicol#otherindex'><i class="fa fa-check"></i><b>6.10</b> Other methods</a></li>
363</ul></li>
364<li class="chapter" data-level="7" data-path="causal.html"><a href='/causal'><i class="fa fa-check"></i><b>7</b> Causal Inference</a>
365<ul>
366<li class="chapter" data-level="7.1" data-path="causal.html"><a href='/causal#r-packages-6'><i class="fa fa-check"></i><b>7.1</b> R Packages</a></li>
367<li class="chapter" data-level="7.2" data-path="causal.html"><a href='/causal#introduction-to-causal-inference'><i class="fa fa-check"></i><b>7.2</b> Introduction to causal inference</a></li>
368<li class="chapter" data-level="7.3" data-path="causal.html"><a href='/causal#directed-acyclical-graphs-and-conditional-independencies'><i class="fa fa-check"></i><b>7.3</b> Directed acyclical graphs and conditional independencies</a>
369<ul>
370<li class="chapter" data-level="7.3.1" data-path="causal.html"><a href='/causal#collider-bias'><i class="fa fa-check"></i><b>7.3.1</b> Collider bias</a></li>
371</ul></li>
372<li class="chapter" data-level="7.4" data-path="causal.html"><a href='/causal#d-separation'><i class="fa fa-check"></i><b>7.4</b> <em>d</em>-separation</a></li>
373<li class="chapter" data-level="7.5" data-path="causal.html"><a href='/causal#estimating-causal-effects-direct-indirect-and-total-effects'><i class="fa fa-check"></i><b>7.5</b> Estimating causal effects (direct, indirect, and total effects)</a></li>
374<li class="chapter" data-level="7.6" data-path="causal.html"><a href='/causal#some-summary-comments'><i class="fa fa-check"></i><b>7.6</b> Some (summary) comments</a></li>
375</ul></li>
376<li class="chapter" data-level="8" data-path="mmi.html"><a href='/mmi'><i class="fa fa-check"></i><b>8</b> Modeling Strategies</a>
377<ul>
378<li class="chapter" data-level="8.1" data-path="mmi.html"><a href='/mmi#r-packages-7'><i class="fa fa-check"></i><b>8.1</b> R Packages</a></li>
379<li class="chapter" data-level="8.2" data-path="mmi.html"><a href='/mmi#goals-of-multivariable-regression-modeling'><i class="fa fa-check"></i><b>8.2</b> Goals of multivariable regression modeling</a></li>
380<li class="chapter" data-level="8.3" data-path="mmi.html"><a href='/mmi#myexp'><i class="fa fa-check"></i><b>8.3</b> My experience</a></li>
381<li class="chapter" data-level="8.4" data-path="mmi.html"><a href='/mmi#stepwise-algorithms'><i class="fa fa-check"></i><b>8.4</b> Stepwise selection algorithms</a>
382<ul>
383<li class="chapter" data-level="8.4.1" data-path="mmi.html"><a href='/mmi#steps'><i class="fa fa-check"></i><b>8.4.1</b> Backwards elimination and forward-stepwise selection</a></li>
384<li class="chapter" data-level="8.4.2" data-path="mmi.html"><a href='/mmi#augmented-backward-elimination'><i class="fa fa-check"></i><b>8.4.2</b> Augmented backward elimination</a></li>
385</ul></li>
386<li class="chapter" data-level="8.5" data-path="mmi.html"><a href='/mmi#fullm'><i class="fa fa-check"></i><b>8.5</b> Degrees of freedom (df) spending: One model to rule them all</a></li>
387<li class="chapter" data-level="8.6" data-path="mmi.html"><a href='/mmi#modav'><i class="fa fa-check"></i><b>8.6</b> AIC and model-averaging</a></li>
388<li class="chapter" data-level="8.7" data-path="mmi.html"><a href='/mmi#lasso'><i class="fa fa-check"></i><b>8.7</b> Regularization using penalization</a></li>
389<li class="chapter" data-level="8.8" data-path="mmi.html"><a href='/mmi#evaluating-model-performance'><i class="fa fa-check"></i><b>8.8</b> Evaluating model performance</a>
390<ul>
391<li class="chapter" data-level="8.8.1" data-path="mmi.html"><a href='/mmi#crossval'><i class="fa fa-check"></i><b>8.8.1</b> Cross-validation</a></li>
392<li class="chapter" data-level="8.8.2" data-path="mmi.html"><a href='/mmi#stability'><i class="fa fa-check"></i><b>8.8.2</b> Boostrapping to evaluate model stability</a></li>
393</ul></li>
394<li class="chapter" data-level="8.9" data-path="mmi.html"><a href='/mmi#summary'><i class="fa fa-check"></i><b>8.9</b> Summary</a></li>
395</ul></li>
396<li class="part"><span><b>III Frequentist and Bayesian Inferential Frameworks</b></span></li>
397<li class="chapter" data-level="9" data-path="probability.html"><a href='/probability'><i class="fa fa-check"></i><b>9</b> Introduction to probability distributions</a>
398<ul>
399<li class="chapter" data-level="9.1" data-path="probability.html"><a href='/probability#statistical-distributions-and-regression'><i class="fa fa-check"></i><b>9.1</b> Statistical distributions and regression</a></li>
400<li class="chapter" data-level="9.2" data-path="probability.html"><a href='/probability#probrules'><i class="fa fa-check"></i><b>9.2</b> Probability rules</a>
401<ul>
402<li class="chapter" data-level="9.2.1" data-path="probability.html"><a href='/probability#application-of-bayes-theorem-to-diagnostic-screening-tests'><i class="fa fa-check"></i><b>9.2.1</b> Application of Bayes theorem to diagnostic screening tests</a></li>
403<li class="chapter" data-level="9.2.2" data-path="probability.html"><a href='/probability#bayes-rule-and-problems-with-multiple-testing'><i class="fa fa-check"></i><b>9.2.2</b> Bayes rule and problems with multiple testing</a></li>
404</ul></li>
405<li class="chapter" data-level="9.3" data-path="probability.html"><a href='/probability#probability-trees'><i class="fa fa-check"></i><b>
4059.3</b> Probability Trees</a>
406<ul>
407<li class="chapter" data-level="9.3.1" data-path="probability.html"><a href='/probability#just-for-fun-monte-hall-problem'><i class="fa fa-check"></i><b>9.3.1</b> Just for fun: Monte Hall problem</a></li>
408</ul></li>
409<li class="chapter" data-level="9.4" data-path="probability.html"><a href='/probability#sample-space-random-variables-probability-distributions'><i class="fa fa-check"></i><b>9.4</b> Sample space, random variables, probability distributions</a>
410<ul>
411<li class="chapter" data-level="9.4.1" data-path="probability.html"><a href='/probability#discrete-random-variables'><i class="fa fa-check"></i><b>9.4.1</b> Discrete random variables</a></li>
412<li class="chapter" data-level="9.4.2" data-path="probability.html"><a href='/probability#continuous-random-variables'><i class="fa fa-check"></i><b>9.4.2</b> Continuous random variables</a></li>
413</ul></li>
414<li class="chapter" data-level="9.5" data-path="probability.html"><a href='/probability#cumulative-distribution-functions'><i class="fa fa-check"></i><b>9.5</b> Cumulative distribution functions</a></li>
415<li class="chapter" data-level="9.6" data-path="probability.html"><a href='/probability#expected-value-and-variance-of-a-random-variable'><i class="fa fa-check"></i><b>9.6</b> Expected value and variance of a random variable</a></li>
416<li class="chapter" data-level="9.7" data-path="probability.html"><a href='/probability#variancesums'><i class="fa fa-check"></i><b>9.7</b> Expected value and variance of sums and products</a></li>
417<li class="chapter" data-level="9.8" data-path="probability.html"><a href='/probability#jointmargcond'><i class="fa fa-check"></i><b>9.8</b> Joint, marginal, and conditional distributions</a></li>
418<li class="chapter" data-level="9.9" data-path="probability.html"><a href='/probability#probability-distributions-in-r'><i class="fa fa-check"></i><b>9.9</b> Probability distributions in R</a></li>
419<li class="chapter" data-level="9.10" data-path="probability.html"><a href='/probability#a-sampling-of-discrete-random-variables'><i class="fa fa-check"></i><b>9.10</b> A sampling of discrete random variables</a>
420<ul>
421<li class="chapter" data-level="9.10.1" data-path="probability.html"><a href='/probability#bernoulli-distribution-y-sim-bernoullip'><i class="fa fa-check"></i><b>9.10.1</b> Bernoulli distribution: <span class="math inline">\(Y \sim\)</span> Bernoulli(<span class="math inline">\(p\)</span>)</a></li>
422<li class="chapter" data-level="9.10.2" data-path="probability.html"><a href='/probability#binom'><i class="fa fa-check"></i><b>9.10.2</b> Binomial distribution <span class="math inline">\(Y \sim\)</span> Binomial<span class="math inline">\((n,p)\)</span></a></li>
423<li class="chapter" data-level="9.10.3" data-path="probability.html"><a href='/probability#geometric-distribution'><i class="fa fa-check"></i><b>9.10.3</b> Geometric distribution</a></li>
424<li class="chapter" data-level="9.10.4" data-path="probability.html"><a href='/probability#multinomial-distribution-y-sim-multinomialn-p_1-p_2-ldots-p_k'><i class="fa fa-check"></i><b>9.10.4</b> Multinomial distribution <span class="math inline">\(Y \sim\)</span> Multinomial<span class="math inline">\((n, p_1, p_2, \ldots, p_k)\)</span></a></li>
425<li class="chapter" data-level="9.10.5" data-path="probability.html"><a href='/probability#poisson-distribution-y-sim-poissonlambda'><i class="fa fa-check"></i><b>9.10.5</b> Poisson distribution: <span class="math inline">\(Y \sim Poisson(\lambda)\)</span></a></li>
426<li class="chapter" data-level="9.10.6" data-path="probability.html"><a href='/probability#negative-binomial'><i class="fa fa-check"></i><b>9.10.6</b> Negative Binomial</a></li>
427</ul></li>
428<li class="chapter" data-level="9.11" data-path="probability.html"><a href='/probability#a-sampling-of-continuous-probability-distributions'><i class="fa fa-check"></i><b>9.11</b> A sampling of continuous probability distributions</a>
429<ul>
430<li class="chapter" data-level="9.11.1" data-path="probability.html"><a href='/probability#normaldist'><i class="fa fa-check"></i><b>9.11.1</b> Normal Distribution <span class="math inline">\(Y \sim N(\mu, \sigma^2)\)</span></a></li>
431<li class="chapter" data-level="9.11.2" data-path="probability.html"><a href='/probability#lognorm'><i class="fa fa-check"></i><b>9.11.2</b> log-normal Distribution: <span class="math inline">\(Y \sim\)</span> Lognormal<span class="math inline">\((\mu,\sigma)\)</span></a></li>
432<li class="chapter" data-level="9.11.3" data-path="probability.html"><a href='/probability#continuous-uniform-distribution-y-sim-uab'><i class="fa fa-check"></i><b>9.11.3</b> Continuous Uniform distribution: <span class="math inline">\(Y \sim U(a,b)\)</span></a></li>
433<li class="chapter" data-level="9.11.4" data-path="probability.html"><a href='/probability#beta-distribution-y-sim-betaalpha-beta'><i class="fa fa-check"></i><b>9.11.4</b> Beta Distribution: <span class="math inline">\(Y \sim\)</span> Beta(<span class="math inline">\(\alpha, \beta\)</span>)</a></li>
434<li class="chapter" data-level="9.11.5" data-path="probability.html"><a href='/probability#exponential-y-sim-explambda'><i class="fa fa-check"></i><b>9.11.5</b> Exponential: <span class="math inline">\(Y \sim\)</span> Exp(<span class="math inline">\(\lambda\)</span>)</a></li>
435<li class="chapter" data-level="9.11.6" data-path="probability.html"><a href='/probability#gamma-distribution-y-sim-gammaalpha-beta'><i class="fa fa-check"></i><b>9.11.6</b> Gamma Distribution: <span class="math inline">\(Y \sim\)</span> Gamma(<span class="math inline">\(\alpha, \beta\)</span>)</a></li>
436</ul></li>
437<li class="chapter" data-level="9.12" data-path="probability.html"><a href='/probability#some-probability-distributions-used-for-hypotheses-testing'><i class="fa fa-check"></i><b>9.12</b> Some probability distributions used for hypotheses testing</a>
438<ul>
439<li class="chapter" data-level="9.12.1" data-path="probability.html"><a href='/probability#chi2-distribution-y-sim-chi2_dfk'><i class="fa fa-check"></i><b>9.12.1</b> <span class="math inline">\(\chi^2\)</span> distribution: <span class="math inline">\(Y \sim \chi^2_{df=k}\)</span></a></li>
440<li class="chapter" data-level="9.12.2" data-path="probability.html"><a href='/probability#students-t-distribution-y-sim-t_dfk'><i class="fa fa-check"></i><b>9.12.2</b> Student’s t distribution <span class="math inline">\(Y \sim t_{df=k}\)</span></a></li>
441<li class="chapter" data-level="9.12.3" data-path="probability.html"><a href='/probability#f-distribution-y-sim-f_df_1k_1-df_2k_2'><i class="fa fa-check"></i><b>9.12.3</b> F distribution, <span class="math inline">\(Y \sim F_{df_1=k_1, df_2=k_2}\)</span></a></li>
442</ul></li>
443<li class="chapter" data-level="9.13" data-path="probability.html">
443<a href='/probability#choosing-an-appropriate-distribution'><i class="fa fa-check"></i><b>9.13</b> Choosing an appropriate distribution</a></li>
444<li class="chapter" data-level="9.14" data-path="probability.html"><a href='/probability#summary-of-statistical-distributions'><i class="fa fa-check"></i><b>9.14</b> Summary of statistical distributions</a></li>
445</ul></li>
446<li class="chapter" data-level="10" data-path="mle.html"><a href='/mle'><i class="fa fa-check"></i><b>10</b> Maximum likelihood</a>
447<ul>
448<li class="chapter" data-level="10.1" data-path="mle.html"><a href='/mle#r-packages-8'><i class="fa fa-check"></i><b>10.1</b> R packages</a></li>
449<li class="chapter" data-level="10.2" data-path="mle.html"><a href='/mle#parameter-estimation'><i class="fa fa-check"></i><b>10.2</b> Parameter estimation</a></li>
450<li class="chapter" data-level="10.3" data-path="mle.html"><a href='/mle#intromle'><i class="fa fa-check"></i><b>10.3</b> Introductory example: Estimating slug densities</a></li>
451<li class="chapter" data-level="10.4" data-path="mle.html"><a href='/mle#probability-to-the-likelihood'><i class="fa fa-check"></i><b>10.4</b> Probability to the likelihood</a></li>
452<li class="chapter" data-level="10.5" data-path="mle.html"><a href='/mle#maximizing-the-likelihood'><i class="fa fa-check"></i><b>10.5</b> Maximizing the likelihood</a>
453<ul>
454<li class="chapter" data-level="10.5.1" data-path="mle.html"><a href='/mle#calculus'><i class="fa fa-check"></i><b>10.5.1</b> Calculus</a></li>
455<li class="chapter" data-level="10.5.2" data-path="mle.html"><a href='/mle#gridsearch'><i class="fa fa-check"></i><b>10.5.2</b> Grid search</a></li>
456<li class="chapter" data-level="10.5.3" data-path="mle.html"><a href='/mle#using-numerical-optimizers'><i class="fa fa-check"></i><b>10.5.3</b> Using numerical optimizers</a></li>
457<li class="chapter" data-level="10.5.4" data-path="mle.html"><a href='/mle#fitting-the-alternative-data-generating-model-site-specific-poisson-distributions'><i class="fa fa-check"></i><b>10.5.4</b> Fitting the alternative data-generating model (site-specific Poisson distributions)</a></li>
458</ul></li>
459<li class="chapter" data-level="10.6" data-path="mle.html"><a href='/mle#mleproperties'><i class="fa fa-check"></i><b>10.6</b> Properties of maximum likelihood estimators</a></li>
460<li class="chapter" data-level="10.7" data-path="mle.html"><a href='/mle#weight-at-age-model'><i class="fa fa-check"></i><b>10.7</b> A more complicated example: Fitting a weight-at-age model</a></li>
461<li class="chapter" data-level="10.8" data-path="mle.html"><a href='/mle#confidence-intervals-for-functions-of-parameters'><i class="fa fa-check"></i><b>10.8</b> Confidence intervals for functions of parameters</a>
462<ul>
463<li class="chapter" data-level="10.8.1" data-path="mle.html"><a href='/mle#delta-method'><i class="fa fa-check"></i><b>10.8.1</b> Delta Method</a></li>
464</ul></li>
465<li class="chapter" data-level="10.9" data-path="mle.html"><a href='/mle#likelihood-ratio-test'><i class="fa fa-check"></i><b>10.9</b> Likelihood ratio test</a></li>
466<li class="chapter" data-level="10.10" data-path="mle.html"><a href='/mle#profile'><i class="fa fa-check"></i><b>10.10</b> Profile likelihood confidence intervals</a></li>
467<li class="chapter" data-level="10.11" data-path="mle.html"><a href='/mle#leastsqnorm'><i class="fa fa-check"></i><b>10.11</b> Aside: Least squares and maximum likelihood</a></li>
468</ul></li>
469<li class="chapter" data-level="11" data-path="Bayes.html">
469<a href='/bayes'><i class="fa fa-check"></i><b>11</b> Introduction to Bayesian statistics</a>
470<ul>
471<li class="chapter" data-level="11.1" data-path="Bayes.html"><a href='/bayes#r-packages-9'><i class="fa fa-check"></i><b>11.1</b> R packages</a></li>
472<li class="chapter" data-level="11.2" data-path="Bayes.html"><a href='/bayes#review-of-frequentist-statistics'><i class="fa fa-check"></i><b>11.2</b> Review of frequentist statistics</a></li>
473<li class="chapter" data-level="11.3" data-path="Bayes.html"><a href='/bayes#bayesian-statistics'><i class="fa fa-check"></i><b>11.3</b> Bayesian statistics</a></li>
474<li class="chapter" data-level="11.4" data-path="Bayes.html"><a href='/bayes#comparing-frequentist-and-bayesian-inference-for-a-simple-model'><i class="fa fa-check"></i><b>11.4</b> Comparing frequentist and Bayesian inference for a simple model</a>
475<ul>
476<li class="chapter" data-level="11.4.1" data-path="Bayes.html"><a href='/bayes#maximum-likelihood'><i class="fa fa-check"></i><b>11.4.1</b> Maximum likelihood</a></li>
477<li class="chapter" data-level="11.4.2" data-path="Bayes.html">
477<a href='/bayes#coverage'><i class="fa fa-check"></i><b>11.4.2</b> Aside: how well do these frequentist intervals work</a></li>
478<li class="chapter" data-level="11.4.3" data-path="Bayes.html"><a href='/bayes#bayesmoose'><i class="fa fa-check"></i><b>11.4.3</b> Bayesian inference for <span class="math inline">\(p\)</span></a></li>
479<li class="chapter" data-level="11.4.4" data-path="Bayes.html"><a href='/bayes#comparing-interpretations-frequentist-vs.-bayesian'><i class="fa fa-check"></i><b>11.4.4</b> Comparing Interpretations: frequentist vs. Bayesian</a></li>
480</ul></li>
481</ul></li>
482<li class="chapter" data-level="12" data-path="BayesMCMC.html"><a href='/bayesmcmc'><i class="fa fa-check"></i><b>12</b> A Brief introduction to MCMC sampling and JAGS</a>
483<ul>
484<li class="chapter" data-level="12.1" data-path="BayesMCMC.html"><a href='/bayesmcmc#r-packages-10'><i class="fa fa-check"></i><b>12.1</b> R packages</a></li>
485<li class="chapter" data-level="12.2" data-path="BayesMCMC.html"><a href='/bayesmcmc#introduction-to-mcmc-sampling'><i class="fa fa-check"></i><b>12.2</b> Introduction to MCMC sampling</a></li>
486<li class="chapter" data-level="12.3" data-path="BayesMCMC.html">
486<a href='/bayesmcmc#metropolis-algorithm'><i class="fa fa-check"></i><b>12.3</b> Metropolis algorithm</a></li>
487<li class="chapter" data-level="12.4" data-path="BayesMCMC.html"><a href='/bayesmcmc#aside-sampler-performance'><i class="fa fa-check"></i><b>12.4</b> Aside: Sampler performance</a></li>
488<li class="chapter" data-level="12.5" data-path="BayesMCMC.html"><a href='/bayesmcmc#bayesjackals'><i class="fa fa-check"></i><b>12.5</b> Specifying a model in JAGS</a></li>
489<li class="chapter" data-level="12.6" data-path="BayesMCMC.html"><a href='/bayesmcmc#fitting-a-model-using-jags'><i class="fa fa-check"></i><b>12.6</b> Fitting a model using JAGS</a></li>
490<li class="chapter" data-level="12.7" data-path="BayesMCMC.html"><a href='/bayesmcmc#diagnostic-plots'><i class="fa fa-check"></i><b>12.7</b> Density and traceplots for assessing convergence</a></li>
491<li class="chapter" data-level="12.8" data-path="BayesMCMC.html"><a href='/bayesmcmc#tips-on-running-models-in-jags'><i class="fa fa-check"></i><b>12.8</b> Tips on running models in JAGS</a></li>
492</ul></li>
493<li class="chapter" data-level="13" data-path="bayesian-linear-regression.html">
493<a href='/bayesian-linear-regression'><i class="fa fa-check"></i><b>13</b> Bayesian linear regression</a>
494<ul>
495<li class="chapter" data-level="13.1" data-path="bayesian-linear-regression.html"><a href='/bayesian-linear-regression#r-packages-11'><i class="fa fa-check"></i><b>13.1</b> R packages</a></li>
496<li class="chapter" data-level="13.2" data-path="bayesian-linear-regression.html"><a href='/bayesian-linear-regression#bayesian-linear-regression-1'><i class="fa fa-check"></i><b>13.2</b> Bayesian linear regression</a></li>
497<li class="chapter" data-level="13.3" data-path="bayesian-linear-regression.html"><a href='/bayesian-linear-regression#testing-assumptions-of-the-model'><i class="fa fa-check"></i><b>13.3</b> Testing assumptions of the model</a></li>
498<li class="chapter" data-level="13.4" data-path="bayesian-linear-regression.html"><a href='/bayesian-linear-regression#gofbayes'><i class="fa fa-check"></i><b>13.4</b> Goodness-of-fit testing: Bayesian p-values</a></li>
499<li class="chapter" data-level="13.5" data-path="bayesian-linear-regression.html">
499<a href='/bayesian-linear-regression#credible-intervals'><i class="fa fa-check"></i><b>13.5</b> Credible intervals</a></li>
500<li class="chapter" data-level="13.6" data-path="bayesian-linear-regression.html"><a href='/bayesian-linear-regression#prediction-intervals'><i class="fa fa-check"></i><b>13.6</b> Prediction intervals</a></li>
501<li class="chapter" data-level="13.7" data-path="bayesian-linear-regression.html"><a href='/bayesian-linear-regression#credible-and-prediction-intervals-the-easy-way'><i class="fa fa-check"></i><b>13.7</b> Credible and prediction intervals the easy way</a></li>
502</ul></li>
503<li class="part"><span><b>IV Models for Non-Normal Data</b></span></li>
504<li class="chapter" data-level="14" data-path="glm.html"><a href='/glm'><i class="fa fa-check"></i><b>14</b> Introduction to generalized linear models (GLMs)</a>
505<ul>
506<li class="chapter" data-level="14.1" data-path="glm.html"><a href='/glm#r-packages-12'><i class="fa fa-check"></i><b>14.1</b> R packages</a></li>
507<li class="chapter" data-level="14.2" data-path="glm.html"><a href='/glm#the-normal-distribution-as-a-data-generating-model'><i class="fa fa-check"></i><b>14.2</b> The Normal distribution as a data-generating model</a></li>
508<li class="chapter" data-level="14.3" data-path="glm.html"><a href='/glm#generalized-linear-models'><i class="fa fa-check"></i><b>14.3</b> Generalized linear models</a>
509<ul>
510<li class="chapter" data-level="14.3.1" data-path="glm.html"><a href='/glm#nomenclature'><i class="fa fa-check"></i><b>14.3.1</b> Nomenclature</a></li>
511</ul></li>
512<li class="chapter" data-level="14.4" data-path="glm.html"><a href='/glm#describing-the-data-generating-mechanism'><i class="fa fa-check"></i><b>14.4</b> Describing the data-generating mechanism</a>
513<ul>
514<li class="chapter" data-level="14.4.1" data-path="glm.html"><a href='/glm#link-functions-g'><i class="fa fa-check"></i><b>14.4.1</b> Link functions, <span class="math inline">\(g()\)</span></a></li>
515</ul></li>
516<li class="chapter" data-level="14.5" data-path="glm.html"><a href='/glm#example-poisson-regression-for-pheasant-counts'><i class="fa fa-check"></i><b>14.5</b> Example: Poisson regression for pheasant counts</a>
517<ul>
518<li class="chapter" data-level="14.5.1" data-path="glm.html"><a href='/glm#poisson-regression-assumptions'><i class="fa fa-check"></i><b>14.5.1</b> Poisson regression assumptions</a></li>
519</ul></li>
520</ul></li>
521<li class="chapter" data-level="15" data-path="PoissonR.html"><a href='/poissonr'><i class="fa fa-check"></i><b>15</b> Regression models for count data</a>
522<ul>
523<li class="chapter" data-level="15.1" data-path="PoissonR.html"><a href='/poissonr#r-packages-13'><i class="fa fa-check"></i><b>15.1</b> R packages</a></li>
524<li class="chapter" data-level="15.2" data-path="PoissonR.html"><a href='/poissonr#introductory-examples-slugs-and-fish'><i class="fa fa-check"></i><b>15.2</b> Introductory examples: Slugs and Fish</a>
525<ul>
526<li class="chapter" data-level="15.2.1" data-path="PoissonR.html"><a href='/poissonr#maximum-likelihood-revisiting-the-slug-data'><i class="fa fa-check"></i><b>15.2.1</b> Maximum likelihood: revisiting the slug data</a></li>
527<li class="chapter" data-level="15.2.2" data-path="PoissonR.html"><a href='/poissonr#modeling-the-distribution-of-longnose-dace'><i class="fa fa-check"></i><b>15.2.2</b> Modeling the distribution of longnose dace</a></li>
528</ul></li>
529<li class="chapter" data-level="15.3" data-path="PoissonR.html"><a href='/poissonr#parameter-interpretation'><i class="fa fa-check"></i><b>15.3</b> Parameter interpretation</a></li>
530<li class="chapter" data-level="15.4" data-path="PoissonR.html"><a href='/poissonr#evaluating-assumptions-and-fit'><i class="fa fa-check"></i><b>15.4</b> Evaluating assumptions and fit</a>
531<ul>
532<li class="chapter" data-level="15.4.1" data-path="PoissonR.html"><a href='/poissonr#glmresid'><i class="fa fa-check"></i><b>15.4.1</b> Residuals</a></li>
533<li class="chapter" data-level="15.4.2" data-path="PoissonR.html"><a href='/poissonr#Dev'><i class="fa fa-check"></i><b>15.4.2</b> Aside: Deviance and pseudo-R<span class="math inline">\(^2\)</span></a></li>
534<li class="chapter" data-level="15.4.3" data-path="PoissonR.html"><a href='/poissonr#overdispersion-what-is-it'><i class="fa fa-check"></i><b>15.4.3</b> Overdispersion: What is it?</a></li>
535<li class="chapter" data-level="15.4.4" data-path="PoissonR.html"><a href='/poissonr#simple-tests-for-overdispersion'><i class="fa fa-check"></i><b>15.4.4</b> Simple tests for overdispersion</a></li>
536<li class="chapter" data-level="15.4.5" data-path="PoissonR.html"><a href='/poissonr#GOF'><i class="fa fa-check"></i><b>15.4.5</b> Goodness-of-fit test using predictive simulation</a></li>
537</ul></li>
538<li class="chapter" data-level="15.5" data-path="PoissonR.html"><a href='/poissonr#quasi-poisson-model-for-overdispersed-data'><i class="fa fa-check"></i><b>15.5</b> Quasi-Poisson model for overdispersed data</a></li>
539<li class="chapter" data-level="15.6" data-path="PoissonR.html"><a href='/poissonr#NBR'><i class="fa fa-check"></i><b>15.6</b> Negative Binomial regression</a></li>
540<li class="chapter" data-level="15.7" data-path="PoissonR.html"><a href='/poissonr#model-comparisons'><i class="fa fa-check"></i><b>15.7</b> Model comparisons</a></li>
541<li class="chapter" data-level="15.8" data-path="PoissonR.html"><a href='/poissonr#bayesian-implementation-of-count-based-regression-models'><i class="fa fa-check"></i><b>15.8</b> Bayesian implementation of count-based regression models</a>
542<ul>
543<li class="chapter" data-level="15.8.1" data-path="PoissonR.html"><a href='/poissonr#poisson-regression'><i class="fa fa-check"></i><b>15.8.1</b> Poisson regression</a></li>
544<li class="chapter" data-level="15.8.2" data-path="PoissonR.html"><a href='/poissonr#negative-binomial-regression'><i class="fa fa-check"></i><b>15.8.2</b> Negative Binomial regression</a></li>
545<li class="chapter" data-level="15.8.3" data-path="PoissonR.html"><a href='/poissonr#poisson-normal-model'><i class="fa fa-check"></i><b>15.8.3</b> Poisson-Normal model</a></li>
546<li class="chapter" data-level="15.8.4" data-path="PoissonR.html"><a href='/poissonr#aside-bayesian-model-selection'><i class="fa fa-check"></i><b>15.8.4</b> Aside: Bayesian model selection</a></li>
547<li class="chapter" data-level="15.8.5" data-path="PoissonR.html"><a href='/poissonr#cautionary-statements-regarding-dic'><i class="fa fa-check"></i><b>15.8.5</b> Cautionary statements regarding DIC</a></li>
548</ul></li>
549<li class="chapter" data-level="15.9" data-path="PoissonR.html"><a href='/poissonr#modeling-rates-and-densities-using-an-offset'><i class="fa fa-check"></i><b>15.9</b> Modeling rates and densities using an offset</a></li>
550</ul></li>
551<li class="chapter" data-level="16" data-path="Logistic.html"><a href='/logistic'><i class="fa fa-check"></i><b>16</b> Logistic regression</a>
552<ul>
553<li class="chapter" data-level="16.1" data-path="Logistic.html"><a href='/logistic#r-packages-14'><i class="fa fa-check"></i><b>16.1</b> R packages</a></li>
554<li class="chapter" data-level="16.2" data-path="Logistic.html"><a href='/logistic#introduction-to-logistic-regression'><i class="fa fa-check"></i><b>16.2</b> Introduction to logistic regression</a></li>
555<li class="chapter" data-level="16.3" data-path="Logistic.html"><a href='/logistic#LRmod1'><i class="fa fa-check"></i><b>16.3</b> Parameter Interpretation: Application to moose detection data</a>
556<ul>
557<li class="chapter" data-level="16.3.1" data-path="Logistic.html"><a href='/logistic#model-with-continuous-and-categorical-variables'><i class="fa fa-check"></i><b>16.3.1</b> Model with continuous and categorical variables</a></li>
558<li class="chapter" data-level="16.3.2" data-path="Logistic.html"><a href='/logistic#interaction-model'><i class="fa fa-check"></i><b>16.3.2</b> Interaction model</a></li>
559</ul></li>
560<li class="chapter" data-level="16.4" data-path="Logistic.html"><a href='/logistic#evaluating-assumptions-and-fit-1'><i class="fa fa-check"></i><b>16.4</b> Evaluating assumptions and fit</a>
561<ul>
562<li class="chapter" data-level="16.4.1" data-path="Logistic.html"><a href='/logistic#residual-plots'><i class="fa fa-check"></i><b>16.4.1</b>
562 Residual plots</a></li>
563<li class="chapter" data-level="16.4.2" data-path="Logistic.html"><a href='/logistic#goodness-of-fit-test'><i class="fa fa-check"></i><b>16.4.2</b> Goodness-of-fit test</a></li>
564<li class="chapter" data-level="16.4.3" data-path="Logistic.html"><a href='/logistic#aside-alternative-methods-for-goodness-of-fit-testing'><i class="fa fa-check"></i><b>16.4.3</b> Aside: alternative methods for goodness-of-fit testing</a></li>
565</ul></li>
566<li class="chapter" data-level="16.5" data-path="Logistic.html"><a href='/logistic#model-comparisons-1'><i class="fa fa-check"></i><b>16.5</b> Model comparisons</a></li>
567<li class="chapter" data-level="16.6" data-path="Logistic.html"><a href='/logistic#effectsglm'><i class="fa fa-check"></i><b>16.6</b> Effect plots: Visualizing generalized linear models</a>
568<ul>
569<li class="chapter" data-level="16.6.1" data-path="Logistic.html"><a href='/logistic#predictions-and-confidence-intervals'><i class="fa fa-check"></i><b>16.6.1</b> Predictions and confidence intervals</a></li>
570<li class="chapter" data-level="16.6.2" data-path="Logistic.html"><a href='/logistic#aside-revisiting-predictions-using-matrix-algebra'><i class="fa fa-check"></i><b>16.6.2</b> Aside: Revisiting predictions using matrix algebra</a></li>
571<li class="chapter" data-level="16.6.3" data-path="Logistic.html"><a href='/logistic#additive-effects-on-logit-scale-ne-additive-effects-on-probability-scale'><i class="fa fa-check"></i><b>16.6.3</b> Additive effects on logit scale <span class="math inline">\(\ne\)</span> additive effects on probability scale</a></li>
572<li class="chapter" data-level="16.6.4" data-path="Logistic.html"><a href='/logistic#effectsp'><i class="fa fa-check"></i><b>16.6.4</b> Effect plots for complex models: Using the effects and ggpredict packages</a></li>
573<li class="chapter" data-level="16.6.5" data-path="Logistic.html"><a href='/logistic#effectplots1'><i class="fa fa-check"></i><b>16.6.5</b> Understanding effect estimates from the effects and ggeffects packages</a></li>
574<li class="chapter" data-level="16.6.6" data-path="Logistic.html"><a href='/logistic#predictions-using-the-ggpredict-function'><i class="fa fa-check"></i><b>16.6.6</b> Predictions using the <code>ggpredict</code> function</a></li>
575</ul></li>
576<li class="chapter" data-level="16.7" data-path="Logistic.html"><a href='/logistic#logistic-regression-bayesian-implementations'><i class="fa fa-check"></i><b>16.7</b> Logistic regression: Bayesian implementations</a>
577<ul>
578<li class="chapter" data-level="16.7.1" data-path="Logistic.html"><a href='/logistic#fitting-the-bayesian-logistic-regression-model-to-moose-sightability-data'><i class="fa fa-check"></i><b>16.7.1</b> Fitting the Bayesian logistic regression model to moose sightability data</a></li>
579</ul></li>
580<li class="chapter" data-level="16.8" data-path="Logistic.html"><a href='/logistic#binomlogit'><i class="fa fa-check"></i><b>16.8</b> Aside: Logistic regression with multiple trials</a></li>
581<li class="chapter" data-level="16.9" data-path="Logistic.html"><a href='/logistic#aside-complete-separation'><i class="fa fa-check"></i><b>16.9</b> Aside: Complete separation</a></li>
582</ul></li>
583<li class="chapter" data-level="17" data-path="zimodels.html"><a href='/zimodels'><i class="fa fa-check"></i><b>17</b> Models for zero-inflated data</a>
584<ul>
585<li class="chapter" data-level="17.1" data-path="zimodels.html"><a href='/zimodels#r-packages-15'><i class="fa fa-check"></i><b>17.1</b> R packages</a></li>
586<li class="chapter" data-level="17.2" data-path="zimodels.html"><a href='/zimodels#zero-inflation'><i class="fa fa-check"></i><b>17.2</b> Zero-inflation</a></li>
587<li class="chapter" data-level="17.3" data-path="zimodels.html"><a href='/zimodels#GOFzero'><i class="fa fa-check"></i><b>17.3</b> Testing for excess zeros</a></li>
588<li class="chapter" data-level="17.4" data-path="zimodels.html"><a href='/zimodels#zeros-and-the-negative-binomial-model'><i class="fa fa-check"></i><b>17.4</b> Zeros and the Negative Binomial model</a></li>
589<li class="chapter" data-level="17.5" data-path="zimodels.html"><a href='/zimodels#hurdle-models'><i class="fa fa-check"></i><b>17.5</b> Hurdle models</a>
590<ul>
591<li class="chapter" data-level="17.5.1" data-path="zimodels.html"><a href='/zimodels#trunc'><i class="fa fa-check"></i><b>17.5.1</b> Truncated distributions</a></li>
592<li class="chapter" data-level="17.5.2" data-path="zimodels.html"><a href='/zimodels#hurdle-models-for-continuous-response-data'><i class="fa fa-check"></i><b>17.5.2</b> Hurdle models for continuous response data</a></li>
593</ul></li>
594<li class="chapter" data-level="17.6" data-path="zimodels.html">
594<a href='/zimodels#zero-inflated-mixture-models'><i class="fa fa-check"></i><b>17.6</b> Zero-inflated mixture models</a>
595<ul>
596<li class="chapter" data-level="17.6.1" data-path="zimodels.html"><a href='/zimodels#marginal-distribution'><i class="fa fa-check"></i><b>17.6.1</b> Marginal distribution</a></li>
597<li class="chapter" data-level="17.6.2" data-path="zimodels.html"><a href='/zimodels#zero-inflated-mixture-models-defined-using-a-latent-variable'><i class="fa fa-check"></i><b>17.6.2</b> Zero-inflated mixture models defined using a latent variable</a></li>
598</ul></li>
599<li class="chapter" data-level="17.7" data-path="zimodels.html"><a href='/zimodels#example-fishing-success-in-state-parks'><i class="fa fa-check"></i><b>17.7</b> Example: Fishing success in state parks</a></li>
600<li class="chapter" data-level="17.8" data-path="zimodels.html"><a href='/zimodels#goodness-of-fit'><i class="fa fa-check"></i><b>17.8</b> Goodness-of-fit</a></li>
601<li class="chapter" data-level="17.9" data-path="zimodels.html"><a href='/zimodels#zipjags'><i class="fa fa-check"></i><b>17.9</b> Bayesian implementation</a></li>
602<li class="chapter" data-level="17.10" data-path="zimodels.html"><a href='/zimodels#implementation-using-glmmtmb'><i class="fa fa-check"></i><b>17.10</b> Implementation using glmmTMB</a></li>
603</ul></li>
604<li class="part"><span><b>V Models for Correlated Data</b></span></li>
605<li class="chapter" data-level="18" data-path="mixed.html"><a href='/mixed'><i class="fa fa-check"></i><b>18</b> Linear Mixed Effects Models</a>
606<ul>
607<li class="chapter" data-level="18.1" data-path="mixed.html"><a href='/mixed#r-packages-16'><i class="fa fa-check"></i><b>18.1</b> R packages</a></li>
608<li class="chapter" data-level="18.2" data-path="mixed.html"><a href='/mixed#revisiting-the-independence-assumption'><i class="fa fa-check"></i><b>18.2</b> Revisiting the independence assumption</a>
609<ul>
610<li class="chapter" data-level="18.2.1" data-path="mixed.html"><a href='/mixed#what-if-sample-sizes-are-random'><i class="fa fa-check"></i><b>18.2.1</b> What if sample sizes are random?</a></li>
611<li class="chapter" data-level="18.2.2" data-path="mixed.html"><a href='/mixed#what-if-sample-sizes-are-not-completely-random'><i class="fa fa-check"></i><b>18.2.2</b> What if sample sizes are not completely random?</a></li>
612</ul></li>
613<li class="chapter" data-level="18.3" data-path="mixed.html"><a href='/mixed#alt'><i class="fa fa-check"></i><b>18.3</b> Optional: Mixed-effect models versus simple alternatives</a></li>
614<li class="chapter" data-level="18.4" data-path="mixed.html"><a href='/mixed#what-are-random-effects-mixed-effect-models-and-when-should-they-be-considered'><i class="fa fa-check"></i><b>18.4</b> What are random effects, mixed-effect models, and when should they be considered?</a></li>
615<li class="chapter" data-level="18.5" data-path="mixed.html"><a href='/mixed#two-step-approach-to-building-a-mixed-effects-model'><i class="fa fa-check"></i><b>18.5</b> Two-step approach to building a mixed-effects model</a>
616<ul>
617<li class="chapter" data-level="18.5.1" data-path="mixed.html"><a href='/mixed#stage-1-level-1-model'><i class="fa fa-check"></i><b>18.5.1</b> Stage 1 (level-1 model)</a></li>
618<li class="chapter" data-level="18.5.2" data-path="mixed.html"><a href='/mixed#stage-2-level-2-model'><i class="fa fa-check"></i><b>18.5.2</b> Stage 2 (level-2 model)</a></li>
619<li class="chapter" data-level="18.5.3" data-path="mixed.html"><a href='/mixed#putting-things-together-composite-equation'><i class="fa fa-check"></i><b>18.5.3</b> Putting things together: Composite equation</a></li>
620</ul></li>
621<li class="chapter" data-level="18.6" data-path="mixed.html"><a href='/mixed#random-intercept-versus-random-intercept-and-slope-model'><i class="fa fa-check"></i><b>18.6</b> Random-intercept versus random intercept and slope model</a></li>
622<li class="chapter" data-level="18.7" data-path="mixed.html"><a href='/mixed#fittinglmer'><i class="fa fa-check"></i><b>18.7</b> Fitting mixed-effects models in R</a>
623<ul>
624<li class="chapter" data-level="18.7.1" data-path="mixed.html"><a href='/mixed#random-intercept-model'><i class="fa fa-check"></i><b>18.7.1</b> Random intercept model</a></li>
625<li class="chapter" data-level="18.7.2" data-path="mixed.html"><a href='/mixed#rc'><i class="fa fa-check"></i><b>18.7.2</b> Random intercepts and slopes model</a></li>
626</ul></li>
627<li class="chapter" data-level="18.8" data-path="mixed.html"><a href='/mixed#blups'><i class="fa fa-check"></i><b>18.8</b> Site-specific parameters (BLUPs)</a>
628<ul>
629<li class="chapter" data-level="18.8.1" data-path="mixed.html">
629<a href='/mixed#digging-deeper-how-are-the-blups-calculated'><i class="fa fa-check"></i><b>18.8.1</b> Digging deeper, how are the BLUPS calculated?</a></li>
630</ul></li>
631<li class="chapter" data-level="18.9" data-path="mixed.html"><a href='/mixed#fixed-effects-versus-random-effects-and-shrinkage'><i class="fa fa-check"></i><b>18.9</b> Fixed effects versus random effects and shrinkage</a></li>
632<li class="chapter" data-level="18.10" data-path="mixed.html"><a href='/mixed#predmixed'><i class="fa fa-check"></i><b>18.10</b> Fitted/predicted values from mixed-effects models</a></li>
633<li class="chapter" data-level="18.11" data-path="mixed.html"><a href='/mixed#mixeddiag'><i class="fa fa-check"></i><b>18.11</b> Model assumptions</a>
634<ul>
635<li class="chapter" data-level="18.11.1" data-path="mixed.html"><a href='/mixed#evaluating-assumptions-1'><i class="fa fa-check"></i><b>18.11.1</b> Evaluating assumptions</a></li>
636</ul></li>
637<li class="chapter" data-level="18.12" data-path="mixed.html"><a href='/mixed#lmecompare'><i class="fa fa-check"></i><b>18.12</b> Model comparisons, hypothesis tests, and confidence intervals for fixed-effects and variance parameters</a>
638<ul>
639<li class="chapter" data-level="18.12.1" data-path="mixed.html"><a href='/mixed#estimation-using-restricted-maximum-likelihood-reml-or-maximum-likelihood-ml'><i class="fa fa-check"></i><b>18.12.1</b> Estimation using restricted maximum likelihood (REML) or maximum likelihood (ML)</a></li>
640<li class="chapter" data-level="18.12.2" data-path="mixed.html"><a href='/mixed#hypothesis-tests-for-fixed-effects-parameters-and-degrees-of-freedom'><i class="fa fa-check"></i><b>18.12.2</b> Hypothesis tests for fixed-effects parameters and degrees of freedom</a></li>
641<li class="chapter" data-level="18.12.3" data-path="mixed.html"><a href='/mixed#alternative-methods-for-testing-hypotheses-about-fixed-effects'><i class="fa fa-check"></i><b>18.12.3</b> Alternative methods for testing hypotheses about fixed effects</a></li>
642<li class="chapter" data-level="18.12.4" data-path="mixed.html"><a href='/mixed#confidence-intervals-for-fixed-effects-and-variance-parameters'><i class="fa fa-check"></i><b>18.12.4</b> Confidence intervals for fixed effects and variance parameters</a></li>
643<li class="chapter" data-level="18.12.5" data-path="mixed.html"><a href='/mixed#simtesting'><i class="fa fa-check"></i><b>18.12.5</b> Simulation-based testing: Comparing models with different random-effect structures</a></li>
644<li class="chapter" data-level="18.12.6" data-path="mixed.html"><a href='/mixed#AICmixed'><i class="fa fa-check"></i><b>18.12.6</b> Comparisons using AIC</a></li>
645</ul></li>
646<li class="chapter" data-level="18.13" data-path="mixed.html"><a href='/mixed#lmestrat'><i class="fa fa-check"></i><b>18.13</b> Modeling strategies revisited</a>
647<ul>
648<li class="chapter" data-level="18.13.1" data-path="mixed.html"><a href='/mixed#fixed-versus-random-effects'><i class="fa fa-check"></i><b>18.13.1</b> Fixed versus random effects</a></li>
649<li class="chapter" data-level="18.13.2" data-path="mixed.html"><a href='/mixed#convergeissues'><i class="fa fa-check"></i><b>18.13.2</b> Maximal model</a></li>
650<li class="chapter" data-level="18.13.3" data-path="mixed.html"><a href='/mixed#top-down-model-selection'><i class="fa fa-check"></i><b>18.13.3</b> Top-down model-selection</a></li>
651<li class="chapter" data-level="18.13.4" data-path="mixed.html"><a href='/mixed#a-multi-level-bottom-up-approach-to-model-selection'><i class="fa fa-check"></i><b>18.13.4</b> A multi-level bottom-up approach to model selection</a></li>
652</ul></li>
653<li class="chapter" data-level="18.14" data-path="mixed.html"><a href='/mixed#marginalgls'><i class="fa fa-check"></i><b>18.14</b> Induced correlation, marginal model, generalized least squares</a></li>
654<li class="chapter" data-level="18.15" data-path="mixed.html"><a href='/mixed#multiplemixed'><i class="fa fa-check"></i><b>18.15</b> Random effects specified using multiple grouping variables</a>
655<ul>
656<li class="chapter" data-level="18.15.1" data-path="mixed.html"><a href='/mixed#Dickiemods'><i class="fa fa-check"></i><b>18.15.1</b> Aside: Digging deeper into models fit by <span class="citation">Dickie et al. (2022)</span></a></li>
657</ul></li>
658<li class="chapter" data-level="18.16" data-path="mixed.html"><a href='/mixed#lmeJAGs'><i class="fa fa-check"></i><b>18.16</b> Implementing mixed-effects models in JAGS</a>
659<ul>
660<li class="chapter" data-level="18.16.1" data-path="mixed.html"><a href='/mixed#random-intercept-model-1'><i class="fa fa-check"></i><b>18.16.1</b> Random intercept model</a></li>
661<li class="chapter" data-level="18.16.2" data-path="mixed.html"><a href='/mixed#random-intercept-and-slope-model'><i class="fa fa-check"></i><b>18.16.2</b> Random intercept and slope model</a></li>
662</ul></li>
663</ul></li>
664<li class="chapter" data-level="19" data-path="glmm.html"><a href='/glmm'><i class="fa fa-check"></i><b>19</b> Generalized linear mixed effects models (GLMMs)</a>
665<ul>
666<li class="chapter" data-level="19.1" data-path="glmm.html"><a href='/glmm#r-packages-17'><i class="fa fa-check"></i><b>19.1</b> R packages</a></li>
667<li class="chapter" data-level="19.2" data-path="glmm.html"><a href='/glmm#case-study-a-comparison-of-single-and-double-cylinder-nest-structures'><i class="fa fa-check"></i><b>19.2</b> Case study: A comparison of single and double-cylinder nest structures</a></li>
668<li class="chapter" data-level="19.3" data-path="glmm.html"><a href='/glmm#review-of-approaches-for-modeling-correlated-data-for-normally-distributed-response-variables.'><i class="fa fa-check"></i><b>19.3</b> Review of approaches for modeling correlated data for Normally distributed response variables.</a>
669<ul>
670<li class="chapter" data-level="19.3.1" data-path="glmm.html"><a href='/glmm#pilme'><i class="fa fa-check"></i><b>19.3.1</b> Parameter interpretation</a></li>
671<li class="chapter" data-level="19.3.2" data-path="glmm.html"><a href='/glmm#margrecap'><i class="fa fa-check"></i><b>19.3.2</b> Marginal models</a></li>
672</ul></li>
673<li class="chapter" data-level="19.4" data-path="glmm.html"><a href='/glmm#nomarg'><i class="fa fa-check"></i><b>19.4</b> Extentions to Count and Binary Data</a></li>
674<li class="chapter" data-level="19.5" data-path="glmm.html"><a href='/glmm#glmer'><i class="fa fa-check"></i><b>19.5</b> Generalized linear mixed effects models</a></li>
675<li class="chapter" data-level="19.6" data-path="glmm.html"><a href='/glmm#modeling-nest-occupancy-using-generalized-linear-mixed-effects-models'><i class="fa fa-check"></i><b>19.6</b> Modeling nest occupancy using generalized linear mixed effects models</a></li>
676<li class="chapter" data-level="19.7" data-path="glmm.html"><a href='/glmm#glmmest'><i class="fa fa-check"></i><b>19.7</b> Parameter estimation</a></li>
677<li class="chapter" data-level="19.8" data-path="glmm.html"><a href='/glmm#glmminterp'><i class="fa fa-check"></i><b>19.8</b> Parameter interpretation</a>
678<ul>
679<li class="chapter" data-level="19.8.1" data-path="glmm.html"><a href='/glmm#condmarg'><i class="fa fa-check"></i><b>19.8.1</b> Approximating marginal (population-average) means from fitted GLMMs</a></li>
680<li class="chapter" data-level="19.8.2" data-path="glmm.html"><a href='/glmm#specialcases'><i class="fa fa-check"></i><b>19.8.2</b> Special cases</a></li>
681<li class="chapter" data-level="19.8.3" data-path="glmm.html"><a href='/glmm#glmmnumint'><i class="fa fa-check"></i><b>19.8.3</b> Numerical integration</a></li>
682<li class="chapter" data-level="19.8.4" data-path="glmm.html"><a href='/glmm#Glmmsim'><i class="fa fa-check"></i><b>19.8.4</b> Simulation</a></li>
683<li class="chapter" data-level="19.8.5" data-path="glmm.html"><a href='/glmm#approximating-marginal-means-using-glmmadpative'><i class="fa fa-check"></i><b>19.8.5</b> Approximating marginal means using GLMMadpative</a></li>
684</ul></li>
685<li class="chapter" data-level="19.9" data-path="glmm.html"><a href='/glmm#hypothesis-testing-and-modeling-strategies'><i class="fa fa-check"></i><b>19.9</b> Hypothesis testing and modeling strategies</a></li>
686</ul></li>
687<li class="chapter" data-level="20" data-path="gee.html"><a href='/gee'><i class="fa fa-check"></i><b>20</b> Generalized Estimating Equations (GEE)</a>
688<ul>
689<li class="chapter" data-level="20.1" data-path="gee.html"><a href='/gee#r-packages-18'><i class="fa fa-check"></i><b>20.1</b> R packages</a></li>
690<li class="chapter" data-level="20.2" data-path="gee.html"><a href='/gee#introduction-to-generalized-estimating-equations'><i class="fa fa-check"></i><b>20.2</b> Introduction to Generalized Estimating Equations</a></li>
691<li class="chapter" data-level="20.3" data-path="gee.html"><a href='/gee#motivating-gees'><i class="fa fa-check"></i><b>20.3</b> Motivating GEEs</a></li>
692<li class="chapter" data-level="20.4" data-path="gee.html"><a href='/gee#gee-applied-to-our-previous-simulation-examples'><i class="fa fa-check"></i><b>20.4</b>
692 GEE applied to our previous simulation examples</a></li>
693<li class="chapter" data-level="20.5" data-path="gee.html"><a href='/gee#gees-versus-glmms'><i class="fa fa-check"></i><b>20.5</b> GEEs versus GLMMs</a></li>
694</ul></li>
695<li class="appendix"><span><b>Appendix</b></span></li>
696<li class="chapter" data-level="A" data-path="projects-and-reproducible-reports-in-r.html"><a href='/projects-and-reproducible-reports-in-r'><i class="fa fa-check"></i><b>A</b> Projects and Reproducible Reports in R</a>
697<ul>
698<li class="chapter" data-level="A.1" data-path="projects-and-reproducible-reports-in-r.html"><a href='/projects-and-reproducible-reports-in-r#introduction-to-r-and-rstudio'><i class="fa fa-check"></i><b>A.1</b> Introduction to R and RStudio</a>
699<ul>
700<li class="chapter" data-level="A.1.1" data-path="projects-and-reproducible-reports-in-r.html"><a href='/projects-and-reproducible-reports-in-r#installing-packages-and-accessing-data'><i class="fa fa-check"></i><b>A.1.1</b> Installing packages and accessing data</a></li>
701</ul></li>
702<li class="chapter" data-level="A.2" data-path="projects-and-reproducible-reports-in-r.html"><a href='/projects-and-reproducible-reports-in-r#reproducible-projects-with-rstudio-and-r-markdown'><i class="fa fa-check"></i><b>A.2</b> Reproducible projects with RStudio and R markdown</a></li>
703<li class="chapter" data-level="A.3" data-path="projects-and-reproducible-reports-in-r.html"><a href='/projects-and-reproducible-reports-in-r#rstudio-projects'><i class="fa fa-check"></i><b>A.3</b> RStudio projects</a></li>
704<li class="chapter" data-level="A.4" data-path="projects-and-reproducible-reports-in-r.html"><a href='/projects-and-reproducible-reports-in-r#r-markdown'><i class="fa fa-check"></i><b>A.4</b> R markdown</a>
705<ul>
706<li class="chapter" data-level="A.4.1" data-path="projects-and-reproducible-reports-in-r.html"><a href='/projects-and-reproducible-reports-in-r#the-header'><i class="fa fa-check"></i><b>A.4.1</b> The header</a></li>
707<li class="chapter" data-level="A.4.2" data-path="projects-and-reproducible-reports-in-r.html"><a href='/projects-and-reproducible-reports-in-r#r-code-chunks'><i class="fa fa-check"></i><b>A.4.2</b> R code chunks</a></li>
708<li class="chapter" data-level="A.4.3" data-path="projects-and-reproducible-reports-in-r.html"><a href='/projects-and-reproducible-reports-in-r#global-options'><i class="fa fa-check"></i><b>A.4.3</b> Global options</a></li>
709<li class="chapter" data-level="A.4.4" data-path="projects-and-reproducible-reports-in-r.html"><a href='/projects-and-reproducible-reports-in-r#knitr'><i class="fa fa-check"></i><b>A.4.4</b> knitR</a></li>
710</ul></li>
711<li class="chapter" data-level="A.5" data-path="projects-and-reproducible-reports-in-r.html"><a href='/projects-and-reproducible-reports-in-r#markdownresources'><i class="fa fa-check"></i><b>A.5</b> More on R markdown</a></li>
712</ul></li>
713<li class="chapter" data-level="" data-path="references.html"><a href='/references'><i class="fa fa-check"></i>References</a></li>
714<li class="divider"></li>
715<li><a href="https://github.com/rstudio/bookdown" target="blank">Published with bookdown</a></li>
716collapse: subsection
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727            <i class="fa fa-circle-o-notch fa-spin"></i><a href="./">Statistics for Ecologists</a>
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733
734            <section class="normal" id="section-">
735<div class = rmdreview>
736  This book is in <b><a href="open.html#open">Open Review</a></b>. I want your feedback to make the book better for you and other readers. To add your annotation, <span style="background-color: #3297FD; color: white">select some text</span> and then click the <i class="h-icon-annotate"></i> on the pop-up menu. To see the annotations of others, click the <i class="h-icon-chevron-left"></i> in the upper right hand corner of the page <i class="fa fa-arrow-circle-right  fa-rotate-315" aria-hidden="true"></i>
737    </div>
738<div id="header">
739<h1 class="title">Statistics for Ecologists</h1>
740<h1 class="title">A newer version of the book can be found <a href="https://statistics4ecologists-v3.netlify.app/"> here</a></h1>
741<p class="author"><em>John Fieberg</em></p>
742<p class="date"><em>Last compiled on 31 January, 2024</em></p>
743</div>
744<div id="section" class="section level1 unnumbered">
745<h1 class="unnumbered"></h1>
746<p><img src="cover2.jpg" width="50%" style="display: block; margin: auto;" /></p>
747<br>
748<b>Cover photograph</b>: 
749<br>
750Caroline Fieberg at Arches National Park, summer 2021. Photo taken by John Fieberg. 
751<br>
752<br>
753<b>Suggested Citation:</b>
754<br>
755Fieberg, John. (2024). Statistics for Ecologists: A Frequentist and Bayesian Treatment of Modern Regression Models. University of Minnesota Libraries Publishing. Retrieved from the University of Minnesota Digital Conservancy. https://doi.org/10.24926/9781959870029 <br>
756<br>
757<b>License:</b>
758<br>
759This work is licensed under a <a rel="license" href="http://creativecommons.org/licenses/by/4.0/">
759Creative Commons Attribution 4.0 International License</a>.
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