-
Notifications
You must be signed in to change notification settings - Fork 0
/
linear_model_4_exercise_solutions.html
1223 lines (1134 loc) · 82.6 KB
/
linear_model_4_exercise_solutions.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
<!DOCTYPE html>
<html>
<head>
<meta charset="utf-8" />
<meta name="generator" content="pandoc" />
<meta http-equiv="X-UA-Compatible" content="IE=EDGE" />
<title>Exercises</title>
<script src="site_libs/header-attrs-2.26/header-attrs.js"></script>
<script src="site_libs/jquery-3.6.0/jquery-3.6.0.min.js"></script>
<meta name="viewport" content="width=device-width, initial-scale=1" />
<link href="site_libs/bootstrap-3.3.5/css/flatly.min.css" rel="stylesheet" />
<script src="site_libs/bootstrap-3.3.5/js/bootstrap.min.js"></script>
<script src="site_libs/bootstrap-3.3.5/shim/html5shiv.min.js"></script>
<script src="site_libs/bootstrap-3.3.5/shim/respond.min.js"></script>
<style>h1 {font-size: 34px;}
h1.title {font-size: 38px;}
h2 {font-size: 30px;}
h3 {font-size: 24px;}
h4 {font-size: 18px;}
h5 {font-size: 16px;}
h6 {font-size: 12px;}
code {color: inherit; background-color: rgba(0, 0, 0, 0.04);}
pre:not([class]) { background-color: white }</style>
<script src="site_libs/navigation-1.1/tabsets.js"></script>
<script src="site_libs/navigation-1.1/codefolding.js"></script>
<link href="site_libs/font-awesome-6.4.2/css/all.min.css" rel="stylesheet" />
<link href="site_libs/font-awesome-6.4.2/css/v4-shims.min.css" rel="stylesheet" />
<style type="text/css">
code{white-space: pre-wrap;}
span.smallcaps{font-variant: small-caps;}
span.underline{text-decoration: underline;}
div.column{display: inline-block; vertical-align: top; width: 50%;}
div.hanging-indent{margin-left: 1.5em; text-indent: -1.5em;}
ul.task-list{list-style: none;}
</style>
<style type="text/css">
code {
white-space: pre;
}
.sourceCode {
overflow: visible;
}
</style>
<style type="text/css" data-origin="pandoc">
pre > code.sourceCode { white-space: pre; position: relative; }
pre > code.sourceCode > span { display: inline-block; line-height: 1.25; }
pre > code.sourceCode > span:empty { height: 1.2em; }
.sourceCode { overflow: visible; }
code.sourceCode > span { color: inherit; text-decoration: inherit; }
div.sourceCode { margin: 1em 0; }
pre.sourceCode { margin: 0; }
@media screen {
div.sourceCode { overflow: auto; }
}
@media print {
pre > code.sourceCode { white-space: pre-wrap; }
pre > code.sourceCode > span { text-indent: -5em; padding-left: 5em; }
}
pre.numberSource code
{ counter-reset: source-line 0; }
pre.numberSource code > span
{ position: relative; left: -4em; counter-increment: source-line; }
pre.numberSource code > span > a:first-child::before
{ content: counter(source-line);
position: relative; left: -1em; text-align: right; vertical-align: baseline;
border: none; display: inline-block;
-webkit-touch-callout: none; -webkit-user-select: none;
-khtml-user-select: none; -moz-user-select: none;
-ms-user-select: none; user-select: none;
padding: 0 4px; width: 4em;
color: #aaaaaa;
}
pre.numberSource { margin-left: 3em; border-left: 1px solid #aaaaaa; padding-left: 4px; }
div.sourceCode
{ background-color: #f8f8f8; }
@media screen {
pre > code.sourceCode > span > a:first-child::before { text-decoration: underline; }
}
code span.al { color: #ef2929; } /* Alert */
code span.an { color: #8f5902; font-weight: bold; font-style: italic; } /* Annotation */
code span.at { color: #204a87; } /* Attribute */
code span.bn { color: #0000cf; } /* BaseN */
code span.cf { color: #204a87; font-weight: bold; } /* ControlFlow */
code span.ch { color: #4e9a06; } /* Char */
code span.cn { color: #8f5902; } /* Constant */
code span.co { color: #8f5902; font-style: italic; } /* Comment */
code span.cv { color: #8f5902; font-weight: bold; font-style: italic; } /* CommentVar */
code span.do { color: #8f5902; font-weight: bold; font-style: italic; } /* Documentation */
code span.dt { color: #204a87; } /* DataType */
code span.dv { color: #0000cf; } /* DecVal */
code span.er { color: #a40000; font-weight: bold; } /* Error */
code span.ex { } /* Extension */
code span.fl { color: #0000cf; } /* Float */
code span.fu { color: #204a87; font-weight: bold; } /* Function */
code span.im { } /* Import */
code span.in { color: #8f5902; font-weight: bold; font-style: italic; } /* Information */
code span.kw { color: #204a87; font-weight: bold; } /* Keyword */
code span.op { color: #ce5c00; font-weight: bold; } /* Operator */
code span.ot { color: #8f5902; } /* Other */
code span.pp { color: #8f5902; font-style: italic; } /* Preprocessor */
code span.sc { color: #ce5c00; font-weight: bold; } /* SpecialChar */
code span.ss { color: #4e9a06; } /* SpecialString */
code span.st { color: #4e9a06; } /* String */
code span.va { color: #000000; } /* Variable */
code span.vs { color: #4e9a06; } /* VerbatimString */
code span.wa { color: #8f5902; font-weight: bold; font-style: italic; } /* Warning */
.sourceCode .row {
width: 100%;
}
.sourceCode {
overflow-x: auto;
}
.code-folding-btn {
margin-right: -30px;
}
</style>
<script>
// apply pandoc div.sourceCode style to pre.sourceCode instead
(function() {
var sheets = document.styleSheets;
for (var i = 0; i < sheets.length; i++) {
if (sheets[i].ownerNode.dataset["origin"] !== "pandoc") continue;
try { var rules = sheets[i].cssRules; } catch (e) { continue; }
var j = 0;
while (j < rules.length) {
var rule = rules[j];
// check if there is a div.sourceCode rule
if (rule.type !== rule.STYLE_RULE || rule.selectorText !== "div.sourceCode") {
j++;
continue;
}
var style = rule.style.cssText;
// check if color or background-color is set
if (rule.style.color === '' && rule.style.backgroundColor === '') {
j++;
continue;
}
// replace div.sourceCode by a pre.sourceCode rule
sheets[i].deleteRule(j);
sheets[i].insertRule('pre.sourceCode{' + style + '}', j);
}
}
})();
</script>
<style type = "text/css">
.main-container {
max-width: 940px;
margin-left: auto;
margin-right: auto;
}
img {
max-width:100%;
}
.tabbed-pane {
padding-top: 12px;
}
.html-widget {
margin-bottom: 20px;
}
button.code-folding-btn:focus {
outline: none;
}
summary {
display: list-item;
}
details > summary > p:only-child {
display: inline;
}
pre code {
padding: 0;
}
</style>
<style type="text/css">
.dropdown-submenu {
position: relative;
}
.dropdown-submenu>.dropdown-menu {
top: 0;
left: 100%;
margin-top: -6px;
margin-left: -1px;
border-radius: 0 6px 6px 6px;
}
.dropdown-submenu:hover>.dropdown-menu {
display: block;
}
.dropdown-submenu>a:after {
display: block;
content: " ";
float: right;
width: 0;
height: 0;
border-color: transparent;
border-style: solid;
border-width: 5px 0 5px 5px;
border-left-color: #cccccc;
margin-top: 5px;
margin-right: -10px;
}
.dropdown-submenu:hover>a:after {
border-left-color: #adb5bd;
}
.dropdown-submenu.pull-left {
float: none;
}
.dropdown-submenu.pull-left>.dropdown-menu {
left: -100%;
margin-left: 10px;
border-radius: 6px 0 6px 6px;
}
</style>
<script type="text/javascript">
// manage active state of menu based on current page
$(document).ready(function () {
// active menu anchor
href = window.location.pathname
href = href.substr(href.lastIndexOf('/') + 1)
if (href === "")
href = "index.html";
var menuAnchor = $('a[href="' + href + '"]');
// mark the anchor link active (and if it's in a dropdown, also mark that active)
var dropdown = menuAnchor.closest('li.dropdown');
if (window.bootstrap) { // Bootstrap 4+
menuAnchor.addClass('active');
dropdown.find('> .dropdown-toggle').addClass('active');
} else { // Bootstrap 3
menuAnchor.parent().addClass('active');
dropdown.addClass('active');
}
// Navbar adjustments
var navHeight = $(".navbar").first().height() + 15;
var style = document.createElement('style');
var pt = "padding-top: " + navHeight + "px; ";
var mt = "margin-top: -" + navHeight + "px; ";
var css = "";
// offset scroll position for anchor links (for fixed navbar)
for (var i = 1; i <= 6; i++) {
css += ".section h" + i + "{ " + pt + mt + "}\n";
}
style.innerHTML = "body {" + pt + "padding-bottom: 40px; }\n" + css;
document.head.appendChild(style);
});
</script>
<!-- tabsets -->
<style type="text/css">
.tabset-dropdown > .nav-tabs {
display: inline-table;
max-height: 500px;
min-height: 44px;
overflow-y: auto;
border: 1px solid #ddd;
border-radius: 4px;
}
.tabset-dropdown > .nav-tabs > li.active:before, .tabset-dropdown > .nav-tabs.nav-tabs-open:before {
content: "\e259";
font-family: 'Glyphicons Halflings';
display: inline-block;
padding: 10px;
border-right: 1px solid #ddd;
}
.tabset-dropdown > .nav-tabs.nav-tabs-open > li.active:before {
content: "\e258";
font-family: 'Glyphicons Halflings';
border: none;
}
.tabset-dropdown > .nav-tabs > li.active {
display: block;
}
.tabset-dropdown > .nav-tabs > li > a,
.tabset-dropdown > .nav-tabs > li > a:focus,
.tabset-dropdown > .nav-tabs > li > a:hover {
border: none;
display: inline-block;
border-radius: 4px;
background-color: transparent;
}
.tabset-dropdown > .nav-tabs.nav-tabs-open > li {
display: block;
float: none;
}
.tabset-dropdown > .nav-tabs > li {
display: none;
}
</style>
<!-- code folding -->
<style type="text/css">
.code-folding-btn { margin-bottom: 4px; }
</style>
</head>
<body>
<div class="container-fluid main-container">
<div class="navbar navbar-default navbar-fixed-top" role="navigation">
<div class="container">
<div class="navbar-header">
<button type="button" class="navbar-toggle collapsed" data-toggle="collapse" data-bs-toggle="collapse" data-target="#navbar" data-bs-target="#navbar">
<span class="icon-bar"></span>
<span class="icon-bar"></span>
<span class="icon-bar"></span>
</button>
<a class="navbar-brand" href="index.html">PGR-LM</a>
</div>
<div id="navbar" class="navbar-collapse collapse">
<ul class="nav navbar-nav">
<li>
<a href="index.html">
<span class="fa fa-home"></span>
Home
</a>
</li>
<li>
<a href="setup.html">
<span class="fa fa-cog"></span>
Setup
</a>
</li>
<li class="dropdown">
<a href="#" class="dropdown-toggle" data-toggle="dropdown" role="button" data-bs-toggle="dropdown" aria-expanded="false">
<span class="fa fa-book"></span>
R Book
<span class="caret"></span>
</a>
<ul class="dropdown-menu" role="menu">
<li>
<a href="https://intro2r.com">
<span class="fa fa-firefox"></span>
Web book
</a>
</li>
<li class="divider"></li>
<li>
<a href="https://github.com/alexd106/Rbook/raw/master/docs/Rbook.pdf">
<span class="fa fa-file-pdf"></span>
PDF book
</a>
</li>
</ul>
</li>
<li class="dropdown">
<a href="#" class="dropdown-toggle" data-toggle="dropdown" role="button" data-bs-toggle="dropdown" aria-expanded="false">
<span class="fa fa-book"></span>
Exercises
<span class="caret"></span>
</a>
<ul class="dropdown-menu" role="menu">
<li>
<a href="exercises.html">
<span class="fa fa-book"></span>
Exercises
</a>
</li>
<li class="divider"></li>
<li>
<a href="exercise_solutions.html">
<span class="fa fa-book"></span>
Exercise Solutions
</a>
</li>
</ul>
</li>
<li>
<a href="data.html">
<span class="fa fa-download"></span>
Data
</a>
</li>
<li class="dropdown">
<a href="#" class="dropdown-toggle" data-toggle="dropdown" role="button" data-bs-toggle="dropdown" aria-expanded="false">
<span class="fa fa-question-circle"></span>
Info
<span class="caret"></span>
</a>
<ul class="dropdown-menu" role="menu">
<li>
<a href="syllabus.html">
<span class="fa fa-graduation-cap"></span>
Syllabus
</a>
</li>
<li>
<a href="People.html">
<span class="fa fa-user-friends"></span>
People
</a>
</li>
<li class="divider"></li>
<li>
<a href="resources.html">
<span class="fa fa-book"></span>
Resources
</a>
</li>
<li>
<a href="https://forms.gle/8xYAqv19x8SSAdfUA">
<span class="fa fa-commenting"></span>
Feedback
</a>
</li>
<li>
<a href="People.html">
<span class="fa fa-envelope fa-lg"></span>
Contact
</a>
</li>
<li class="divider"></li>
<li>
<a href="https://github.com/alexd106/PGR-LM">
<span class="fa fa-github fa-lg"></span>
Source code
</a>
</li>
<li>
<a href="https://twitter.com/Scedacity">
<span class="fa fa-twitter fa-lg"></span>
Twitter
</a>
</li>
</ul>
</li>
</ul>
<ul class="nav navbar-nav navbar-right">
</ul>
</div><!--/.nav-collapse -->
</div><!--/.container -->
</div><!--/.navbar -->
<div id="header">
<div class="btn-group pull-right float-right">
<button type="button" class="btn btn-default btn-xs btn-secondary btn-sm dropdown-toggle" data-toggle="dropdown" data-bs-toggle="dropdown" aria-haspopup="true" aria-expanded="false"><span>Code</span> <span class="caret"></span></button>
<ul class="dropdown-menu dropdown-menu-right" style="min-width: 50px;">
<li><a id="rmd-show-all-code" href="#">Show All Code</a></li>
<li><a id="rmd-hide-all-code" href="#">Hide All Code</a></li>
</ul>
</div>
<h1 class="title toc-ignore">Exercises</h1>
</div>
<p> </p>
<div id="model-selection-with-the-loyn-data" class="section level2">
<h2>Model selection with the Loyn data</h2>
<p> </p>
<p>In the previous exercise you fitted a pre-conceived model which
included the main effects of the area of the forest patch
(<code>LOGAREA</code>), the grazing intensity (<code>FGRAZE</code>) and
the interaction between these two explanatory variables
(<code>FGRAZE:LOGAREA</code>). This was useful as a training exercise,
and might be a viable approach when analysing these data if an
experiment had been designed to test these effects only. However, if
other potentially important variables are not included in the model this
may lead to biased inferences (interpretation). Additionally, if the
goal of the analysis is to explore what models explain the data in a
parsimonious way (as opposed to formally testing hypotheses), we would
also want to include relevant additional explanatory variables.</p>
<p> </p>
<p>Here we revisit the previous loyn data analysis, and ask if a
‘better’ model for these data could be achieved by including additional
explanatory variables and by performing model selection. Because we
would like to test the significance of the interaction between
<code>LOGAREA</code>, and <code>FGRAZE</code>, whilst accounting for the
potential effects of other explanatory variables, we will also include
<code>LOGAREA</code>, <code>FGRAZE</code> and their interaction
(<code>FGRAZE:LOGAREA</code>) in the model as before. Including other
interaction terms between other variables may be reasonable, but we will
focus only on the <code>FGRAZE:LOGAREA</code> interaction as we have
relatively little information in this data set (67 observations). This
will hopefully avoid fitting an overly complex model which will estimate
many parameters for which we have very little data. This is a balance
you will all have to maintain with your own data and analyses (or better
still, perform a power analysis before you even collect your data). No
4-way interaction terms in your models please!</p>
<p> </p>
<p>It’s also important to note that we will assume that all the
explanatory variables were collected by the researchers because <em>they
believed</em> them to be biologically relevant for explaining bird
abundance (i.e. data were collected for a reason). Of course, this is
probably not your area of expertise but it is nevertheless a good idea
to pause and think what might be relevant or not-so relevant and why.
This highlights the importance of knowing your study organism / study
area and discussing research designs with colleagues and other experts
in the field before you collect your data. What you should try to avoid
is collecting heaps of data across many variables (just because you can)
and then expecting your statistical models to make sense of it for you.
As mentioned in the lecture, model selection is a relatively
controversial topic and should not be treated as a purely mechanical
process (chuck everything in and see what comes out). Your expertise
needs to be woven into this process otherwise you may end up with a
model that is implausible or not very useful (and all models need to be
useful!).</p>
<p> </p>
<p>1. Import the ‘loyn.txt’ data file into RStudio and assign it to a
variable called <code>loyn</code>. Here we will be using all the
explanatory variables to explain the variation in bird density. If
needed, remind yourself of your data exploration you conducted
previously. Do any of the remaining variables need transforming
(i.e. <code>AREA</code>, <code>DIST</code>, <code>LDIST</code>) or
converting to a factor type variable (i.e. <code>GRAZE</code>)? Add the
transformed variables to the <code>loyn</code> dataframe.</p>
<div class="sourceCode" id="cb1"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1-1"><a href="#cb1-1" tabindex="-1"></a>loyn <span class="ot"><-</span> <span class="fu">read.table</span>(<span class="st">"data/loyn.txt"</span>, <span class="at">header =</span> <span class="cn">TRUE</span>)</span>
<span id="cb1-2"><a href="#cb1-2" tabindex="-1"></a><span class="fu">str</span>(loyn)</span>
<span id="cb1-3"><a href="#cb1-3" tabindex="-1"></a><span class="do">## 'data.frame': 67 obs. of 8 variables:</span></span>
<span id="cb1-4"><a href="#cb1-4" tabindex="-1"></a><span class="do">## $ SITE : int 1 60 2 3 61 4 5 6 7 8 ...</span></span>
<span id="cb1-5"><a href="#cb1-5" tabindex="-1"></a><span class="do">## $ ABUND : num 5.3 10 2 1.5 13 17.1 13.8 14.1 3.8 2.2 ...</span></span>
<span id="cb1-6"><a href="#cb1-6" tabindex="-1"></a><span class="do">## $ AREA : num 0.1 0.2 0.5 0.5 0.6 1 1 1 1 1 ...</span></span>
<span id="cb1-7"><a href="#cb1-7" tabindex="-1"></a><span class="do">## $ DIST : int 39 142 234 104 191 66 246 234 467 284 ...</span></span>
<span id="cb1-8"><a href="#cb1-8" tabindex="-1"></a><span class="do">## $ LDIST : int 39 142 234 311 357 66 246 285 467 1829 ...</span></span>
<span id="cb1-9"><a href="#cb1-9" tabindex="-1"></a><span class="do">## $ YR.ISOL: int 1968 1961 1920 1900 1957 1966 1918 1965 1955 1920 ...</span></span>
<span id="cb1-10"><a href="#cb1-10" tabindex="-1"></a><span class="do">## $ GRAZE : int 2 2 5 5 2 3 5 3 5 5 ...</span></span>
<span id="cb1-11"><a href="#cb1-11" tabindex="-1"></a><span class="do">## $ ALT : int 160 180 60 140 185 160 140 130 90 60 ...</span></span>
<span id="cb1-12"><a href="#cb1-12" tabindex="-1"></a></span>
<span id="cb1-13"><a href="#cb1-13" tabindex="-1"></a>loyn<span class="sc">$</span>LOGAREA <span class="ot"><-</span> <span class="fu">log10</span>(loyn<span class="sc">$</span>AREA)</span>
<span id="cb1-14"><a href="#cb1-14" tabindex="-1"></a>loyn<span class="sc">$</span>LOGDIST <span class="ot"><-</span> <span class="fu">log10</span>(loyn<span class="sc">$</span>DIST)</span>
<span id="cb1-15"><a href="#cb1-15" tabindex="-1"></a>loyn<span class="sc">$</span>LOGLDIST <span class="ot"><-</span> <span class="fu">log10</span>(loyn<span class="sc">$</span>LDIST)</span>
<span id="cb1-16"><a href="#cb1-16" tabindex="-1"></a></span>
<span id="cb1-17"><a href="#cb1-17" tabindex="-1"></a><span class="co"># create factor GRAZE as it was originally coded as an integer</span></span>
<span id="cb1-18"><a href="#cb1-18" tabindex="-1"></a>loyn<span class="sc">$</span>FGRAZE <span class="ot"><-</span> <span class="fu">factor</span>(loyn<span class="sc">$</span>GRAZE)</span></code></pre></div>
<p> </p>
<p>2. Let’s start with a very quick graphical exploration of any
potential relationships between each explanatory variable (collinearity)
and also between our response and explanatory variables (what we’re
interested in). Create a pairs plot using the function
<code>pairs()</code>of your variables of interest. Hint: restrict the
plot to the variables you actually need. An effective way of doing this
is to store the names of the variables of interest in a vector
<code>VOI <- c("Var1", "Var2", ...)</code> and then use the naming
method for subsetting the data set <code>Mydata[, VOI]</code>. If you
feel like it, you can also add the correlations to the lower triangle of
the plot as you did previously (don’t forget to define the function
first).</p>
<div class="sourceCode" id="cb2"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb2-1"><a href="#cb2-1" tabindex="-1"></a><span class="co"># define the panel.cor function from ?pairs</span></span>
<span id="cb2-2"><a href="#cb2-2" tabindex="-1"></a>panel.cor <span class="ot"><-</span> <span class="cf">function</span>(x, y, <span class="at">digits =</span> <span class="dv">2</span>, <span class="at">prefix =</span> <span class="st">""</span>, cex.cor, ...)</span>
<span id="cb2-3"><a href="#cb2-3" tabindex="-1"></a>{</span>
<span id="cb2-4"><a href="#cb2-4" tabindex="-1"></a> usr <span class="ot"><-</span> <span class="fu">par</span>(<span class="st">"usr"</span>)</span>
<span id="cb2-5"><a href="#cb2-5" tabindex="-1"></a> <span class="fu">par</span>(<span class="at">usr =</span> <span class="fu">c</span>(<span class="dv">0</span>, <span class="dv">1</span>, <span class="dv">0</span>, <span class="dv">1</span>))</span>
<span id="cb2-6"><a href="#cb2-6" tabindex="-1"></a> r <span class="ot"><-</span> <span class="fu">abs</span>(<span class="fu">cor</span>(x, y))</span>
<span id="cb2-7"><a href="#cb2-7" tabindex="-1"></a> txt <span class="ot"><-</span> <span class="fu">format</span>(<span class="fu">c</span>(r, <span class="fl">0.123456789</span>), <span class="at">digits =</span> digits)[<span class="dv">1</span>]</span>
<span id="cb2-8"><a href="#cb2-8" tabindex="-1"></a> txt <span class="ot"><-</span> <span class="fu">paste0</span>(prefix, txt)</span>
<span id="cb2-9"><a href="#cb2-9" tabindex="-1"></a> <span class="cf">if</span>(<span class="fu">missing</span>(cex.cor)) cex.cor <span class="ot"><-</span> <span class="fl">0.8</span><span class="sc">/</span><span class="fu">strwidth</span>(txt)</span>
<span id="cb2-10"><a href="#cb2-10" tabindex="-1"></a> <span class="fu">text</span>(<span class="fl">0.5</span>, <span class="fl">0.5</span>, txt, <span class="at">cex =</span> cex.cor <span class="sc">*</span> r)</span>
<span id="cb2-11"><a href="#cb2-11" tabindex="-1"></a>}</span>
<span id="cb2-12"><a href="#cb2-12" tabindex="-1"></a></span>
<span id="cb2-13"><a href="#cb2-13" tabindex="-1"></a><span class="co"># subset the variables of interest</span></span>
<span id="cb2-14"><a href="#cb2-14" tabindex="-1"></a>VOI<span class="ot"><-</span> <span class="fu">c</span>(<span class="st">"ABUND"</span>, <span class="st">"LOGAREA"</span>, <span class="st">"LOGDIST"</span>, <span class="st">"LOGLDIST"</span>, <span class="st">"YR.ISOL"</span>, <span class="st">"ALT"</span>, <span class="st">"FGRAZE"</span>)</span>
<span id="cb2-15"><a href="#cb2-15" tabindex="-1"></a><span class="fu">pairs</span>(loyn[, VOI], <span class="at">lower.panel =</span> panel.cor)</span></code></pre></div>
<p><img src="linear_model_4_exercise_solutions_files/figure-html/Q2-1.png" width="672" /></p>
<div class="sourceCode" id="cb3"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb3-1"><a href="#cb3-1" tabindex="-1"></a></span>
<span id="cb3-2"><a href="#cb3-2" tabindex="-1"></a><span class="co"># There are varying degrees of correlation between explanatory variables which</span></span>
<span id="cb3-3"><a href="#cb3-3" tabindex="-1"></a><span class="co"># might indicate some collinearity, i.e. LOGAREA and FGRAZE (0.48), LOGDIST and </span></span>
<span id="cb3-4"><a href="#cb3-4" tabindex="-1"></a><span class="co"># LOGLDIST (0.59) and YR.ISOL and FGRAZE (0.56). However, the relationships</span></span>
<span id="cb3-5"><a href="#cb3-5" tabindex="-1"></a><span class="co"># between these explanatory variables are quite weak so we can probably </span></span>
<span id="cb3-6"><a href="#cb3-6" tabindex="-1"></a><span class="co"># include these variables in the same model (but keep an eye on things). </span></span>
<span id="cb3-7"><a href="#cb3-7" tabindex="-1"></a><span class="co"># There also seems to be a reasonable spread of observations across these </span></span>
<span id="cb3-8"><a href="#cb3-8" tabindex="-1"></a><span class="co"># pairs of explanatory variables which is a good thing.</span></span>
<span id="cb3-9"><a href="#cb3-9" tabindex="-1"></a></span>
<span id="cb3-10"><a href="#cb3-10" tabindex="-1"></a><span class="co"># The relationship between the response variable ABUND and all the explanatory</span></span>
<span id="cb3-11"><a href="#cb3-11" tabindex="-1"></a><span class="co"># variables is visible in the top row:</span></span>
<span id="cb3-12"><a href="#cb3-12" tabindex="-1"></a><span class="co"># Some potential relationships present like with LOGAREA (positive), </span></span>
<span id="cb3-13"><a href="#cb3-13" tabindex="-1"></a><span class="co"># maybe ALT (positive) and FGRAZE (negative).</span></span></code></pre></div>
<p> </p>
<p>3. Now, let’s fit our maximal model. Start with a model of
<code>ABUND</code> and include all explanatory variables as main
effects. Also include the interaction <code>LOGAREA:FGRAZE</code> but no
other interaction terms as justified in the preamble above. Don’t forget
to include the transformed versions of the variables where appropriate
(but not the untransformed variables as well otherwise you will have
very strong collinearity between these variables!). Perhaps, call this
model <code>M1</code>.</p>
<div class="sourceCode" id="cb4"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb4-1"><a href="#cb4-1" tabindex="-1"></a>M1 <span class="ot"><-</span> <span class="fu">lm</span>(ABUND <span class="sc">~</span> LOGDIST <span class="sc">+</span> LOGLDIST <span class="sc">+</span> YR.ISOL <span class="sc">+</span> ALT <span class="sc">+</span> LOGAREA <span class="sc">+</span> FGRAZE <span class="sc">+</span> </span>
<span id="cb4-2"><a href="#cb4-2" tabindex="-1"></a> FGRAZE<span class="sc">:</span>LOGAREA, <span class="at">data =</span> loyn)</span></code></pre></div>
<p> </p>
<p>4. Have a look at the summary table of the model using the
<code>summary()</code> function. You’ll probably find this summary is
quite complicated with lots of parameter estimates (14) and P values
testing lots of hypotheses. Are all the P values less than our cut-off
of 0.05? If not, then this suggests that some form of model selection is
warranted to simplify our model.</p>
<div class="sourceCode" id="cb5"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb5-1"><a href="#cb5-1" tabindex="-1"></a><span class="fu">summary</span>(M1)</span>
<span id="cb5-2"><a href="#cb5-2" tabindex="-1"></a><span class="do">## </span></span>
<span id="cb5-3"><a href="#cb5-3" tabindex="-1"></a><span class="do">## Call:</span></span>
<span id="cb5-4"><a href="#cb5-4" tabindex="-1"></a><span class="do">## lm(formula = ABUND ~ LOGDIST + LOGLDIST + YR.ISOL + ALT + LOGAREA + </span></span>
<span id="cb5-5"><a href="#cb5-5" tabindex="-1"></a><span class="do">## FGRAZE + FGRAZE:LOGAREA, data = loyn)</span></span>
<span id="cb5-6"><a href="#cb5-6" tabindex="-1"></a><span class="do">## </span></span>
<span id="cb5-7"><a href="#cb5-7" tabindex="-1"></a><span class="do">## Residuals:</span></span>
<span id="cb5-8"><a href="#cb5-8" tabindex="-1"></a><span class="do">## Min 1Q Median 3Q Max </span></span>
<span id="cb5-9"><a href="#cb5-9" tabindex="-1"></a><span class="do">## -14.976 -1.972 0.142 2.388 10.631 </span></span>
<span id="cb5-10"><a href="#cb5-10" tabindex="-1"></a><span class="do">## </span></span>
<span id="cb5-11"><a href="#cb5-11" tabindex="-1"></a><span class="do">## Coefficients:</span></span>
<span id="cb5-12"><a href="#cb5-12" tabindex="-1"></a><span class="do">## Estimate Std. Error t value Pr(>|t|) </span></span>
<span id="cb5-13"><a href="#cb5-13" tabindex="-1"></a><span class="do">## (Intercept) 33.963217 93.824477 0.362 0.71880 </span></span>
<span id="cb5-14"><a href="#cb5-14" tabindex="-1"></a><span class="do">## LOGDIST 0.882214 2.204341 0.400 0.69061 </span></span>
<span id="cb5-15"><a href="#cb5-15" tabindex="-1"></a><span class="do">## LOGLDIST -0.253543 1.709295 -0.148 0.88264 </span></span>
<span id="cb5-16"><a href="#cb5-16" tabindex="-1"></a><span class="do">## YR.ISOL -0.008283 0.047374 -0.175 0.86186 </span></span>
<span id="cb5-17"><a href="#cb5-17" tabindex="-1"></a><span class="do">## ALT 0.016979 0.018630 0.911 0.36622 </span></span>
<span id="cb5-18"><a href="#cb5-18" tabindex="-1"></a><span class="do">## LOGAREA 3.733668 1.914379 1.950 0.05643 . </span></span>
<span id="cb5-19"><a href="#cb5-19" tabindex="-1"></a><span class="do">## FGRAZE2 -6.757424 4.132084 -1.635 0.10790 </span></span>
<span id="cb5-20"><a href="#cb5-20" tabindex="-1"></a><span class="do">## FGRAZE3 -12.488020 4.542801 -2.749 0.00816 **</span></span>
<span id="cb5-21"><a href="#cb5-21" tabindex="-1"></a><span class="do">## FGRAZE4 -16.133695 4.838990 -3.334 0.00157 **</span></span>
<span id="cb5-22"><a href="#cb5-22" tabindex="-1"></a><span class="do">## FGRAZE5 -17.191221 4.991340 -3.444 0.00113 **</span></span>
<span id="cb5-23"><a href="#cb5-23" tabindex="-1"></a><span class="do">## LOGAREA:FGRAZE2 4.877440 2.565757 1.901 0.06275 . </span></span>
<span id="cb5-24"><a href="#cb5-24" tabindex="-1"></a><span class="do">## LOGAREA:FGRAZE3 9.410212 3.223378 2.919 0.00514 **</span></span>
<span id="cb5-25"><a href="#cb5-25" tabindex="-1"></a><span class="do">## LOGAREA:FGRAZE4 14.166912 4.304081 3.292 0.00178 **</span></span>
<span id="cb5-26"><a href="#cb5-26" tabindex="-1"></a><span class="do">## LOGAREA:FGRAZE5 2.617845 3.347001 0.782 0.43761 </span></span>
<span id="cb5-27"><a href="#cb5-27" tabindex="-1"></a><span class="do">## ---</span></span>
<span id="cb5-28"><a href="#cb5-28" tabindex="-1"></a><span class="do">## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1</span></span>
<span id="cb5-29"><a href="#cb5-29" tabindex="-1"></a><span class="do">## </span></span>
<span id="cb5-30"><a href="#cb5-30" tabindex="-1"></a><span class="do">## Residual standard error: 5.009 on 53 degrees of freedom</span></span>
<span id="cb5-31"><a href="#cb5-31" tabindex="-1"></a><span class="do">## Multiple R-squared: 0.8034, Adjusted R-squared: 0.7551 </span></span>
<span id="cb5-32"><a href="#cb5-32" tabindex="-1"></a><span class="do">## F-statistic: 16.66 on 13 and 53 DF, p-value: 2.644e-14</span></span></code></pre></div>
<p> </p>
<p>5. Let’s perform a first step in model selection using the
<code>drop1()</code> function and use an <em>F</em> test based model
selection approach. This will allow us to decide which explanatory
variables may be suitable for removal from the model. Remember to use
the <code>test = "F"</code> argument to perform <em>F</em> tests when
using <code>drop1()</code>. Which explanatory variable is the best
candidate for removal and why?</p>
<p>What hypothesis is being tested when we do this model selection
step?</p>
<div class="sourceCode" id="cb6"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb6-1"><a href="#cb6-1" tabindex="-1"></a><span class="co"># Wait: why can't we use information from the 'summary(M1)' or 'anova(M1)' functions</span></span>
<span id="cb6-2"><a href="#cb6-2" tabindex="-1"></a><span class="co"># to do this?</span></span>
<span id="cb6-3"><a href="#cb6-3" tabindex="-1"></a></span>
<span id="cb6-4"><a href="#cb6-4" tabindex="-1"></a><span class="co"># the 'summary' table tests if the coefficient for each explanatory variable </span></span>
<span id="cb6-5"><a href="#cb6-5" tabindex="-1"></a><span class="co"># is significantly different from zero.</span></span>
<span id="cb6-6"><a href="#cb6-6" tabindex="-1"></a></span>
<span id="cb6-7"><a href="#cb6-7" tabindex="-1"></a><span class="co"># the 'anova' tests for the significance of the proportion of variation explained</span></span>
<span id="cb6-8"><a href="#cb6-8" tabindex="-1"></a><span class="co"># by a particular term in the model. </span></span>
<span id="cb6-9"><a href="#cb6-9" tabindex="-1"></a></span>
<span id="cb6-10"><a href="#cb6-10" tabindex="-1"></a><span class="co"># The ANOVA table also allows testing the overall significance of a categorical explanatory</span></span>
<span id="cb6-11"><a href="#cb6-11" tabindex="-1"></a><span class="co"># variable (like FGRAZE) which involves several parameters together (one for each level), </span></span>
<span id="cb6-12"><a href="#cb6-12" tabindex="-1"></a><span class="co"># which is quite is handy. But the results of this ANOVA are based on sequential </span></span>
<span id="cb6-13"><a href="#cb6-13" tabindex="-1"></a><span class="co"># sums of squares and therefore the order of the variables in the model</span></span>
<span id="cb6-14"><a href="#cb6-14" tabindex="-1"></a><span class="co"># (which is arbitrary here) matters.</span></span>
<span id="cb6-15"><a href="#cb6-15" tabindex="-1"></a></span>
<span id="cb6-16"><a href="#cb6-16" tabindex="-1"></a><span class="co"># We could change the order but there are too many possible permutations.</span></span>
<span id="cb6-17"><a href="#cb6-17" tabindex="-1"></a><span class="co"># Summary P values don't suffer from this problem but tests different hypotheses.</span></span>
<span id="cb6-18"><a href="#cb6-18" tabindex="-1"></a><span class="co"># It would be useful to use an ANOVA that doesn't depend on the order</span></span>
<span id="cb6-19"><a href="#cb6-19" tabindex="-1"></a><span class="co"># of inclusion of the variables, this is effectively what 'drop1' does.</span></span>
<span id="cb6-20"><a href="#cb6-20" tabindex="-1"></a></span>
<span id="cb6-21"><a href="#cb6-21" tabindex="-1"></a><span class="fu">drop1</span>(M1, <span class="at">test =</span> <span class="st">"F"</span>)</span>
<span id="cb6-22"><a href="#cb6-22" tabindex="-1"></a><span class="do">## Single term deletions</span></span>
<span id="cb6-23"><a href="#cb6-23" tabindex="-1"></a><span class="do">## </span></span>
<span id="cb6-24"><a href="#cb6-24" tabindex="-1"></a><span class="do">## Model:</span></span>
<span id="cb6-25"><a href="#cb6-25" tabindex="-1"></a><span class="do">## ABUND ~ LOGDIST + LOGLDIST + YR.ISOL + ALT + LOGAREA + FGRAZE + </span></span>
<span id="cb6-26"><a href="#cb6-26" tabindex="-1"></a><span class="do">## FGRAZE:LOGAREA</span></span>
<span id="cb6-27"><a href="#cb6-27" tabindex="-1"></a><span class="do">## Df Sum of Sq RSS AIC F value Pr(>F) </span></span>
<span id="cb6-28"><a href="#cb6-28" tabindex="-1"></a><span class="do">## <none> 1329.8 228.20 </span></span>
<span id="cb6-29"><a href="#cb6-29" tabindex="-1"></a><span class="do">## LOGDIST 1 4.02 1333.8 226.40 0.1602 0.690605 </span></span>
<span id="cb6-30"><a href="#cb6-30" tabindex="-1"></a><span class="do">## LOGLDIST 1 0.55 1330.3 226.23 0.0220 0.882644 </span></span>
<span id="cb6-31"><a href="#cb6-31" tabindex="-1"></a><span class="do">## YR.ISOL 1 0.77 1330.6 226.24 0.0306 0.861862 </span></span>
<span id="cb6-32"><a href="#cb6-32" tabindex="-1"></a><span class="do">## ALT 1 20.84 1350.6 227.24 0.8306 0.366220 </span></span>
<span id="cb6-33"><a href="#cb6-33" tabindex="-1"></a><span class="do">## LOGAREA:FGRAZE 4 405.04 1734.8 238.02 4.0358 0.006259 **</span></span>
<span id="cb6-34"><a href="#cb6-34" tabindex="-1"></a><span class="do">## ---</span></span>
<span id="cb6-35"><a href="#cb6-35" tabindex="-1"></a><span class="do">## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1</span></span>
<span id="cb6-36"><a href="#cb6-36" tabindex="-1"></a></span>
<span id="cb6-37"><a href="#cb6-37" tabindex="-1"></a><span class="co"># LOGLDIST is the least significant (p = 0.88), and therefore makes the least </span></span>
<span id="cb6-38"><a href="#cb6-38" tabindex="-1"></a><span class="co"># contribution to the variability explained by the model, with respect to </span></span>
<span id="cb6-39"><a href="#cb6-39" tabindex="-1"></a><span class="co"># the number of degrees of freedom it uses (1). This variable is a good candidate </span></span>
<span id="cb6-40"><a href="#cb6-40" tabindex="-1"></a><span class="co"># to remove from the model </span></span></code></pre></div>
<p> </p>
<p>6. Update and refit your model and remove the least significant
explanatory variable (from above). Repeat single term deletions with
<code>drop1()</code> again using this updated model. You can update the
model by just fitting a new model without the appropriate explanatory
variable and assign it to a new name (<code>M2</code>). Alternatively
you can use the <code>update()</code> function instead (I show you how
to do this in the solutions).</p>
<div class="sourceCode" id="cb7"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb7-1"><a href="#cb7-1" tabindex="-1"></a><span class="co"># new model removing LOGLDIST </span></span>
<span id="cb7-2"><a href="#cb7-2" tabindex="-1"></a>M2 <span class="ot"><-</span> <span class="fu">lm</span>(ABUND <span class="sc">~</span> LOGDIST <span class="sc">+</span> YR.ISOL <span class="sc">+</span> ALT <span class="sc">+</span> LOGAREA <span class="sc">+</span> FGRAZE <span class="sc">+</span></span>
<span id="cb7-3"><a href="#cb7-3" tabindex="-1"></a> LOGAREA<span class="sc">:</span>FGRAZE, <span class="at">data =</span> loyn) </span>
<span id="cb7-4"><a href="#cb7-4" tabindex="-1"></a></span>
<span id="cb7-5"><a href="#cb7-5" tabindex="-1"></a><span class="co"># or use a shortcut with the update() function:</span></span>
<span id="cb7-6"><a href="#cb7-6" tabindex="-1"></a>M2 <span class="ot"><-</span> <span class="fu">update</span>(M1, <span class="at">formula =</span> . <span class="sc">~</span> . <span class="sc">-</span> LOGLDIST) <span class="co"># "." means all previous variables</span></span>
<span id="cb7-7"><a href="#cb7-7" tabindex="-1"></a></span>
<span id="cb7-8"><a href="#cb7-8" tabindex="-1"></a><span class="co"># now redo drop1() on the new model</span></span>
<span id="cb7-9"><a href="#cb7-9" tabindex="-1"></a><span class="fu">drop1</span>(M2, <span class="at">test =</span> <span class="st">"F"</span>)</span>
<span id="cb7-10"><a href="#cb7-10" tabindex="-1"></a><span class="do">## Single term deletions</span></span>
<span id="cb7-11"><a href="#cb7-11" tabindex="-1"></a><span class="do">## </span></span>
<span id="cb7-12"><a href="#cb7-12" tabindex="-1"></a><span class="do">## Model:</span></span>
<span id="cb7-13"><a href="#cb7-13" tabindex="-1"></a><span class="do">## ABUND ~ LOGDIST + YR.ISOL + ALT + LOGAREA + FGRAZE + LOGAREA:FGRAZE</span></span>
<span id="cb7-14"><a href="#cb7-14" tabindex="-1"></a><span class="do">## Df Sum of Sq RSS AIC F value Pr(>F) </span></span>
<span id="cb7-15"><a href="#cb7-15" tabindex="-1"></a><span class="do">## <none> 1330.3 226.23 </span></span>
<span id="cb7-16"><a href="#cb7-16" tabindex="-1"></a><span class="do">## LOGDIST 1 3.64 1334.0 224.41 0.1478 0.702134 </span></span>
<span id="cb7-17"><a href="#cb7-17" tabindex="-1"></a><span class="do">## YR.ISOL 1 0.78 1331.1 224.27 0.0317 0.859332 </span></span>
<span id="cb7-18"><a href="#cb7-18" tabindex="-1"></a><span class="do">## ALT 1 22.24 1352.6 225.34 0.9029 0.346233 </span></span>
<span id="cb7-19"><a href="#cb7-19" tabindex="-1"></a><span class="do">## LOGAREA:FGRAZE 4 406.74 1737.1 236.10 4.1275 0.005451 **</span></span>
<span id="cb7-20"><a href="#cb7-20" tabindex="-1"></a><span class="do">## ---</span></span>
<span id="cb7-21"><a href="#cb7-21" tabindex="-1"></a><span class="do">## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1</span></span>
<span id="cb7-22"><a href="#cb7-22" tabindex="-1"></a></span>
<span id="cb7-23"><a href="#cb7-23" tabindex="-1"></a><span class="co"># YR.ISOL is now the least significant (p = 0.859), hence makes the least </span></span>
<span id="cb7-24"><a href="#cb7-24" tabindex="-1"></a><span class="co"># contribution to the variability explained by the model, </span></span>
<span id="cb7-25"><a href="#cb7-25" tabindex="-1"></a><span class="co"># with respect to the number of degrees of freedom it uses (1)</span></span></code></pre></div>
<p> </p>
<p>7. Again, update the model to remove the least significant
explanatory variable (from above) and repeat single term deletions with
<code>drop1()</code>.</p>
<div class="sourceCode" id="cb8"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb8-1"><a href="#cb8-1" tabindex="-1"></a>M3 <span class="ot"><-</span> <span class="fu">update</span>(M2, <span class="at">formula =</span> . <span class="sc">~</span> . <span class="sc">-</span> YR.ISOL)</span>
<span id="cb8-2"><a href="#cb8-2" tabindex="-1"></a></span>
<span id="cb8-3"><a href="#cb8-3" tabindex="-1"></a><span class="fu">drop1</span>(M3, <span class="at">test =</span> <span class="st">"F"</span>)</span>
<span id="cb8-4"><a href="#cb8-4" tabindex="-1"></a><span class="do">## Single term deletions</span></span>
<span id="cb8-5"><a href="#cb8-5" tabindex="-1"></a><span class="do">## </span></span>
<span id="cb8-6"><a href="#cb8-6" tabindex="-1"></a><span class="do">## Model:</span></span>
<span id="cb8-7"><a href="#cb8-7" tabindex="-1"></a><span class="do">## ABUND ~ LOGDIST + ALT + LOGAREA + FGRAZE + LOGAREA:FGRAZE</span></span>
<span id="cb8-8"><a href="#cb8-8" tabindex="-1"></a><span class="do">## Df Sum of Sq RSS AIC F value Pr(>F) </span></span>
<span id="cb8-9"><a href="#cb8-9" tabindex="-1"></a><span class="do">## <none> 1331.1 224.27 </span></span>
<span id="cb8-10"><a href="#cb8-10" tabindex="-1"></a><span class="do">## LOGDIST 1 3.28 1334.4 222.43 0.1355 0.714237 </span></span>
<span id="cb8-11"><a href="#cb8-11" tabindex="-1"></a><span class="do">## ALT 1 25.33 1356.5 223.53 1.0468 0.310729 </span></span>
<span id="cb8-12"><a href="#cb8-12" tabindex="-1"></a><span class="do">## LOGAREA:FGRAZE 4 405.99 1737.1 234.10 4.1936 0.004916 **</span></span>
<span id="cb8-13"><a href="#cb8-13" tabindex="-1"></a><span class="do">## ---</span></span>
<span id="cb8-14"><a href="#cb8-14" tabindex="-1"></a><span class="do">## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1</span></span>
<span id="cb8-15"><a href="#cb8-15" tabindex="-1"></a></span>
<span id="cb8-16"><a href="#cb8-16" tabindex="-1"></a><span class="co"># LOGDIST now the least significant (p = 0.714) and should be removed from </span></span>
<span id="cb8-17"><a href="#cb8-17" tabindex="-1"></a><span class="co"># the next model.</span></span></code></pre></div>
<p> </p>
<p>8. Once again, update the model to remove the least significant
explanatory variable (from above) and repeat single term deletions with
<code>drop1()</code>.</p>
<div class="sourceCode" id="cb9"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb9-1"><a href="#cb9-1" tabindex="-1"></a>M4 <span class="ot"><-</span> <span class="fu">update</span>(M3, <span class="at">formula =</span> . <span class="sc">~</span> . <span class="sc">-</span> LOGDIST)</span>
<span id="cb9-2"><a href="#cb9-2" tabindex="-1"></a><span class="fu">drop1</span>(M4, <span class="at">test =</span> <span class="st">"F"</span>)</span>
<span id="cb9-3"><a href="#cb9-3" tabindex="-1"></a><span class="do">## Single term deletions</span></span>
<span id="cb9-4"><a href="#cb9-4" tabindex="-1"></a><span class="do">## </span></span>
<span id="cb9-5"><a href="#cb9-5" tabindex="-1"></a><span class="do">## Model:</span></span>
<span id="cb9-6"><a href="#cb9-6" tabindex="-1"></a><span class="do">## ABUND ~ ALT + LOGAREA + FGRAZE + LOGAREA:FGRAZE</span></span>
<span id="cb9-7"><a href="#cb9-7" tabindex="-1"></a><span class="do">## Df Sum of Sq RSS AIC F value Pr(>F) </span></span>
<span id="cb9-8"><a href="#cb9-8" tabindex="-1"></a><span class="do">## <none> 1334.4 222.43 </span></span>
<span id="cb9-9"><a href="#cb9-9" tabindex="-1"></a><span class="do">## ALT 1 22.84 1357.2 221.57 0.9584 0.331805 </span></span>
<span id="cb9-10"><a href="#cb9-10" tabindex="-1"></a><span class="do">## LOGAREA:FGRAZE 4 408.56 1743.0 232.33 4.2864 0.004273 **</span></span>
<span id="cb9-11"><a href="#cb9-11" tabindex="-1"></a><span class="do">## ---</span></span>
<span id="cb9-12"><a href="#cb9-12" tabindex="-1"></a><span class="do">## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1</span></span>
<span id="cb9-13"><a href="#cb9-13" tabindex="-1"></a></span>
<span id="cb9-14"><a href="#cb9-14" tabindex="-1"></a><span class="co"># ALT is not significant (p = 0.331)</span></span></code></pre></div>
<p> </p>
<p>9. And finally, update the model to remove the least significant
explanatory variable (from above) and repeat single term deletions with
<code>drop1()</code>.</p>
<div class="sourceCode" id="cb10"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb10-1"><a href="#cb10-1" tabindex="-1"></a><span class="co"># and finally drop ALT from the model</span></span>
<span id="cb10-2"><a href="#cb10-2" tabindex="-1"></a>M5 <span class="ot"><-</span> <span class="fu">update</span>(M4, <span class="at">formula =</span> . <span class="sc">~</span> . <span class="sc">-</span> ALT)</span>
<span id="cb10-3"><a href="#cb10-3" tabindex="-1"></a><span class="fu">drop1</span>(M5, <span class="at">test =</span> <span class="st">"F"</span>)</span>
<span id="cb10-4"><a href="#cb10-4" tabindex="-1"></a><span class="do">## Single term deletions</span></span>
<span id="cb10-5"><a href="#cb10-5" tabindex="-1"></a><span class="do">## </span></span>
<span id="cb10-6"><a href="#cb10-6" tabindex="-1"></a><span class="do">## Model:</span></span>
<span id="cb10-7"><a href="#cb10-7" tabindex="-1"></a><span class="do">## ABUND ~ LOGAREA + FGRAZE + LOGAREA:FGRAZE</span></span>
<span id="cb10-8"><a href="#cb10-8" tabindex="-1"></a><span class="do">## Df Sum of Sq RSS AIC F value Pr(>F) </span></span>
<span id="cb10-9"><a href="#cb10-9" tabindex="-1"></a><span class="do">## <none> 1357.2 221.57 </span></span>
<span id="cb10-10"><a href="#cb10-10" tabindex="-1"></a><span class="do">## LOGAREA:FGRAZE 4 389.3 1746.6 230.47 4.0874 0.00556 **</span></span>
<span id="cb10-11"><a href="#cb10-11" tabindex="-1"></a><span class="do">## ---</span></span>
<span id="cb10-12"><a href="#cb10-12" tabindex="-1"></a><span class="do">## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1</span></span>
<span id="cb10-13"><a href="#cb10-13" tabindex="-1"></a></span>
<span id="cb10-14"><a href="#cb10-14" tabindex="-1"></a><span class="co"># the LOGAREA:FGRAZE term represents the interaction between LOGAREA and</span></span>
<span id="cb10-15"><a href="#cb10-15" tabindex="-1"></a><span class="co"># FGRAZE. This is significant (p = 0.005) and so our model selection</span></span>
<span id="cb10-16"><a href="#cb10-16" tabindex="-1"></a><span class="co"># process comes to an end.</span></span></code></pre></div>
<p> </p>
<p>10. If all goes well, your final model should be
<code>lm(ABUND ~ LOGAREA + FGRAZE + LOGAREA:FGRAZE)</code> which you
encountered in the previous exercise. Also, you may have noticed that
the output from the <code>drop1()</code> function does not include the
main effects of <code>LOGAREA</code> or <code>FRGRAZE</code>. Can you
think why this might be the case?</p>
<div class="sourceCode" id="cb11"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb11-1"><a href="#cb11-1" tabindex="-1"></a><span class="co"># As the interaction between LOGAREA and FGRAZE was significant at each step of</span></span>
<span id="cb11-2"><a href="#cb11-2" tabindex="-1"></a><span class="co"># model selection process the main effects should be left in our model,</span></span>
<span id="cb11-3"><a href="#cb11-3" tabindex="-1"></a><span class="co"># irrespective of significance. This is because it is quite difficult to </span></span>
<span id="cb11-4"><a href="#cb11-4" tabindex="-1"></a><span class="co"># interpret an interaction without the main effects. The drop1 </span></span>
<span id="cb11-5"><a href="#cb11-5" tabindex="-1"></a><span class="co"># function is clever enough that it doesn't let you see the P values for the </span></span>
<span id="cb11-6"><a href="#cb11-6" tabindex="-1"></a><span class="co"># main effects, in the presence of their significant interaction.</span></span>
<span id="cb11-7"><a href="#cb11-7" tabindex="-1"></a></span>
<span id="cb11-8"><a href="#cb11-8" tabindex="-1"></a><span class="co"># Also note, because R always includes interactions *after* their main effects</span></span>
<span id="cb11-9"><a href="#cb11-9" tabindex="-1"></a><span class="co"># the P value of the interaction term (p = 0.005) from the model selection </span></span>
<span id="cb11-10"><a href="#cb11-10" tabindex="-1"></a><span class="co"># is the same as P value if we use the anova() function on our final model</span></span>
<span id="cb11-11"><a href="#cb11-11" tabindex="-1"></a></span>
<span id="cb11-12"><a href="#cb11-12" tabindex="-1"></a><span class="co"># Check this:</span></span>
<span id="cb11-13"><a href="#cb11-13" tabindex="-1"></a><span class="fu">anova</span>(M5)</span>
<span id="cb11-14"><a href="#cb11-14" tabindex="-1"></a><span class="do">## Analysis of Variance Table</span></span>
<span id="cb11-15"><a href="#cb11-15" tabindex="-1"></a><span class="do">## </span></span>
<span id="cb11-16"><a href="#cb11-16" tabindex="-1"></a><span class="do">## Response: ABUND</span></span>
<span id="cb11-17"><a href="#cb11-17" tabindex="-1"></a><span class="do">## Df Sum Sq Mean Sq F value Pr(>F) </span></span>
<span id="cb11-18"><a href="#cb11-18" tabindex="-1"></a><span class="do">## LOGAREA 1 3978.1 3978.1 167.0669 < 2.2e-16 ***</span></span>
<span id="cb11-19"><a href="#cb11-19" tabindex="-1"></a><span class="do">## FGRAZE 4 1038.2 259.5 10.9000 1.241e-06 ***</span></span>
<span id="cb11-20"><a href="#cb11-20" tabindex="-1"></a><span class="do">## LOGAREA:FGRAZE 4 389.3 97.3 4.0874 0.00556 ** </span></span>
<span id="cb11-21"><a href="#cb11-21" tabindex="-1"></a><span class="do">## Residuals 57 1357.3 23.8 </span></span>
<span id="cb11-22"><a href="#cb11-22" tabindex="-1"></a><span class="do">## ---</span></span>
<span id="cb11-23"><a href="#cb11-23" tabindex="-1"></a><span class="do">## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1</span></span>
<span id="cb11-24"><a href="#cb11-24" tabindex="-1"></a><span class="fu">drop1</span>(M5, <span class="at">test=</span> <span class="st">"F"</span>) </span>
<span id="cb11-25"><a href="#cb11-25" tabindex="-1"></a><span class="do">## Single term deletions</span></span>
<span id="cb11-26"><a href="#cb11-26" tabindex="-1"></a><span class="do">## </span></span>
<span id="cb11-27"><a href="#cb11-27" tabindex="-1"></a><span class="do">## Model:</span></span>
<span id="cb11-28"><a href="#cb11-28" tabindex="-1"></a><span class="do">## ABUND ~ LOGAREA + FGRAZE + LOGAREA:FGRAZE</span></span>
<span id="cb11-29"><a href="#cb11-29" tabindex="-1"></a><span class="do">## Df Sum of Sq RSS AIC F value Pr(>F) </span></span>
<span id="cb11-30"><a href="#cb11-30" tabindex="-1"></a><span class="do">## <none> 1357.2 221.57 </span></span>
<span id="cb11-31"><a href="#cb11-31" tabindex="-1"></a><span class="do">## LOGAREA:FGRAZE 4 389.3 1746.6 230.47 4.0874 0.00556 **</span></span>
<span id="cb11-32"><a href="#cb11-32" tabindex="-1"></a><span class="do">## ---</span></span>
<span id="cb11-33"><a href="#cb11-33" tabindex="-1"></a><span class="do">## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1</span></span></code></pre></div>
<p> </p>
<p>11. Now that you have your final model, you should go through your
model validation and model interpretation as usual. As we have already
completed this in the previous exercise I’ll leave it up to you to
decide whether you include it here (you should be able to just copy and
paste the code).</p>
<p>Please make sure you understand the biological interpretation of each
of the parameter estimates and the interpretation of the hypotheses you
are testing.</p>
<div class="sourceCode" id="cb12"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb12-1"><a href="#cb12-1" tabindex="-1"></a><span class="co"># Biologically: confirming what we already found out in the previous exercise:</span></span>
<span id="cb12-2"><a href="#cb12-2" tabindex="-1"></a><span class="co"># There is a significant interaction between the area of the patch and the level </span></span>
<span id="cb12-3"><a href="#cb12-3" tabindex="-1"></a><span class="co"># of grazing </span></span>
<span id="cb12-4"><a href="#cb12-4" tabindex="-1"></a></span>
<span id="cb12-5"><a href="#cb12-5" tabindex="-1"></a><span class="co"># However, some observations are poorly predicted (fitted) using the set of</span></span>
<span id="cb12-6"><a href="#cb12-6" tabindex="-1"></a><span class="co"># available explanatory variables (i.e. the two very large forest patches)</span></span>
<span id="cb12-7"><a href="#cb12-7" tabindex="-1"></a></span>
<span id="cb12-8"><a href="#cb12-8" tabindex="-1"></a><span class="co"># Interpretation: </span></span>
<span id="cb12-9"><a href="#cb12-9" tabindex="-1"></a><span class="co"># Bird abundance might increase with patch area due to populations being more</span></span>
<span id="cb12-10"><a href="#cb12-10" tabindex="-1"></a><span class="co"># viable in large patches (e.g. less prone to extinction), </span></span>
<span id="cb12-11"><a href="#cb12-11" tabindex="-1"></a><span class="co"># or perhaps because there is proportionally less edge effect in larger</span></span>
<span id="cb12-12"><a href="#cb12-12" tabindex="-1"></a><span class="co"># patches, and this in turn provides more high quality habitat for species </span></span>
<span id="cb12-13"><a href="#cb12-13" tabindex="-1"></a><span class="co"># associated with these habitat patches</span></span>
<span id="cb12-14"><a href="#cb12-14" tabindex="-1"></a></span>
<span id="cb12-15"><a href="#cb12-15" tabindex="-1"></a><span class="co"># The negative effect of grazing may be due to grazing decreasing resource</span></span>
<span id="cb12-16"><a href="#cb12-16" tabindex="-1"></a><span class="co"># availability for birds, for example plants or seeds directly, or insects</span></span>
<span id="cb12-17"><a href="#cb12-17" tabindex="-1"></a><span class="co"># associated with the grazed plants. There may also be more disturbance of birds</span></span>
<span id="cb12-18"><a href="#cb12-18" tabindex="-1"></a><span class="co"># in highly grazed forest patches resulting in fewer foraging opportunities</span></span>
<span id="cb12-19"><a href="#cb12-19" tabindex="-1"></a><span class="co"># or chances to mate (this is all speculation mind you!).</span></span>
<span id="cb12-20"><a href="#cb12-20" tabindex="-1"></a></span>
<span id="cb12-21"><a href="#cb12-21" tabindex="-1"></a><span class="co"># Methodologically:</span></span>
<span id="cb12-22"><a href="#cb12-22" tabindex="-1"></a><span class="co"># Doing model selection is difficult without intrinsic / expert knowledge</span></span>
<span id="cb12-23"><a href="#cb12-23" tabindex="-1"></a><span class="co"># of the system, to guide what variables to include.</span></span>
<span id="cb12-24"><a href="#cb12-24" tabindex="-1"></a><span class="co"># Even with this data set, many more models could have been formulated.</span></span>
<span id="cb12-25"><a href="#cb12-25" tabindex="-1"></a><span class="co"># For example, for me, theory would have suggested to test an interaction </span></span>
<span id="cb12-26"><a href="#cb12-26" tabindex="-1"></a><span class="co"># between YR.ISOL and LOGDIST (or LOGLDIST?), </span></span>
<span id="cb12-27"><a href="#cb12-27" tabindex="-1"></a><span class="co"># because LOGDIST will affect the dispersal of birds between patches </span></span>
<span id="cb12-28"><a href="#cb12-28" tabindex="-1"></a><span class="co"># (hence the colonisation rate), and the time since isolation of the patch may </span></span>
<span id="cb12-29"><a href="#cb12-29" tabindex="-1"></a><span class="co"># affect how important dispersal has been to maintain or rescue populations </span></span>
<span id="cb12-30"><a href="#cb12-30" tabindex="-1"></a><span class="co"># (for recently isolated patches, dispersal, and hence distance to nearest</span></span>
<span id="cb12-31"><a href="#cb12-31" tabindex="-1"></a><span class="co"># patches may have a less important effect)</span></span></code></pre></div>
<p> </p>
<p><strong>OPTIONAL questions</strong> if you have time / energy /
inclination!</p>
<p> </p>
<p>A1. If we weren’t aiming to directly test the effect of the
<code>LOGAREA:FGRAZE</code> interaction statistically (i.e. test this
specific hypothesis), we could use AIC to perform model selection.
Repeat the model selection you did above, but this time use the
<code>drop1()</code> function and perform model selection using AIC
instead. Don’t forget, if we want to perform model selection based on
AIC with the <code>drop1()</code> function we need to omit the
<code>test = "F"</code> argument)</p>
<div class="sourceCode" id="cb13"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb13-1"><a href="#cb13-1" tabindex="-1"></a><span class="co"># This time, we are not doing any specific hypothesis testing, rather we are </span></span>
<span id="cb13-2"><a href="#cb13-2" tabindex="-1"></a><span class="co"># attempting to select a model with the 'best' goodness of fit with the minimal</span></span>
<span id="cb13-3"><a href="#cb13-3" tabindex="-1"></a><span class="co"># number of estimated parameters. </span></span>
<span id="cb13-4"><a href="#cb13-4" tabindex="-1"></a></span>
<span id="cb13-5"><a href="#cb13-5" tabindex="-1"></a><span class="co"># We will start with a reasonably complex but PLAUSABLE model (this is the same </span></span>
<span id="cb13-6"><a href="#cb13-6" tabindex="-1"></a><span class="co"># model we started with using F test based model selection above.</span></span>
<span id="cb13-7"><a href="#cb13-7" tabindex="-1"></a></span>
<span id="cb13-8"><a href="#cb13-8" tabindex="-1"></a>M.start.AIC<span class="ot"><-</span> <span class="fu">lm</span>(ABUND <span class="sc">~</span> LOGLDIST <span class="sc">+</span> LOGDIST <span class="sc">+</span> YR.ISOL <span class="sc">+</span> ALT <span class="sc">+</span> LOGAREA <span class="sc">+</span> FGRAZE <span class="sc">+</span></span>
<span id="cb13-9"><a href="#cb13-9" tabindex="-1"></a> LOGAREA<span class="sc">:</span>FGRAZE, <span class="at">data =</span> loyn)</span>
<span id="cb13-10"><a href="#cb13-10" tabindex="-1"></a></span>
<span id="cb13-11"><a href="#cb13-11" tabindex="-1"></a><span class="fu">drop1</span>(M.start.AIC)</span>
<span id="cb13-12"><a href="#cb13-12" tabindex="-1"></a><span class="do">## Single term deletions</span></span>
<span id="cb13-13"><a href="#cb13-13" tabindex="-1"></a><span class="do">## </span></span>
<span id="cb13-14"><a href="#cb13-14" tabindex="-1"></a><span class="do">## Model:</span></span>
<span id="cb13-15"><a href="#cb13-15" tabindex="-1"></a><span class="do">## ABUND ~ LOGLDIST + LOGDIST + YR.ISOL + ALT + LOGAREA + FGRAZE + </span></span>
<span id="cb13-16"><a href="#cb13-16" tabindex="-1"></a><span class="do">## LOGAREA:FGRAZE</span></span>
<span id="cb13-17"><a href="#cb13-17" tabindex="-1"></a><span class="do">## Df Sum of Sq RSS AIC</span></span>
<span id="cb13-18"><a href="#cb13-18" tabindex="-1"></a><span class="do">## <none> 1329.8 228.20</span></span>
<span id="cb13-19"><a href="#cb13-19" tabindex="-1"></a><span class="do">## LOGLDIST 1 0.55 1330.3 226.23</span></span>
<span id="cb13-20"><a href="#cb13-20" tabindex="-1"></a><span class="do">## LOGDIST 1 4.02 1333.8 226.40</span></span>
<span id="cb13-21"><a href="#cb13-21" tabindex="-1"></a><span class="do">## YR.ISOL 1 0.77 1330.6 226.24</span></span>
<span id="cb13-22"><a href="#cb13-22" tabindex="-1"></a><span class="do">## ALT 1 20.84 1350.6 227.24</span></span>
<span id="cb13-23"><a href="#cb13-23" tabindex="-1"></a><span class="do">## LOGAREA:FGRAZE 4 405.04 1734.8 238.02</span></span></code></pre></div>
<p> </p>
<p>A2. Refit your model with the variable associated with the lowest AIC
value removed. Run <code>drop1()</code> again on your updated model.
Perhaps call this new model <code>M2.AIC</code>.</p>
<div class="sourceCode" id="cb14"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb14-1"><a href="#cb14-1" tabindex="-1"></a><span class="co"># So, our starting model with no variables removed has an AIC of 228.20. If we </span></span>
<span id="cb14-2"><a href="#cb14-2" tabindex="-1"></a><span class="co"># remove the interaction term `LOGAREA:FGRAZE` from the model then this results </span></span>
<span id="cb14-3"><a href="#cb14-3" tabindex="-1"></a><span class="co"># in a big increase in AIC (238.02 - 228.20 = 9.82) so this suggests that there </span></span>
<span id="cb14-4"><a href="#cb14-4" tabindex="-1"></a><span class="co"># is substantial evidence that the interaction should remain in the model. The </span></span>
<span id="cb14-5"><a href="#cb14-5" tabindex="-1"></a><span class="co"># models without `LOGLDIST`, `LOGDIST`, `YR.ISOL` all have pretty much the </span></span>
<span id="cb14-6"><a href="#cb14-6" tabindex="-1"></a><span class="co"># same AIC value (around 226) so in practice we could remove any of them. Let's </span></span>
<span id="cb14-7"><a href="#cb14-7" tabindex="-1"></a><span class="co"># remove the term that results in the model with the lowest AIC which is the </span></span>
<span id="cb14-8"><a href="#cb14-8" tabindex="-1"></a><span class="co"># `LOGLDIST` variable (AIC 226.23).</span></span>
<span id="cb14-9"><a href="#cb14-9" tabindex="-1"></a></span>
<span id="cb14-10"><a href="#cb14-10" tabindex="-1"></a>M2.AIC <span class="ot"><-</span> <span class="fu">update</span>(M.start.AIC, <span class="at">formula =</span> . <span class="sc">~</span> . <span class="sc">-</span> LOGLDIST)</span>
<span id="cb14-11"><a href="#cb14-11" tabindex="-1"></a><span class="fu">drop1</span>(M2.AIC)</span>
<span id="cb14-12"><a href="#cb14-12" tabindex="-1"></a><span class="do">## Single term deletions</span></span>
<span id="cb14-13"><a href="#cb14-13" tabindex="-1"></a><span class="do">## </span></span>
<span id="cb14-14"><a href="#cb14-14" tabindex="-1"></a><span class="do">## Model:</span></span>
<span id="cb14-15"><a href="#cb14-15" tabindex="-1"></a><span class="do">## ABUND ~ LOGDIST + YR.ISOL + ALT + LOGAREA + FGRAZE + LOGAREA:FGRAZE</span></span>
<span id="cb14-16"><a href="#cb14-16" tabindex="-1"></a><span class="do">## Df Sum of Sq RSS AIC</span></span>
<span id="cb14-17"><a href="#cb14-17" tabindex="-1"></a><span class="do">## <none> 1330.3 226.23</span></span>
<span id="cb14-18"><a href="#cb14-18" tabindex="-1"></a><span class="do">## LOGDIST 1 3.64 1334.0 224.41</span></span>
<span id="cb14-19"><a href="#cb14-19" tabindex="-1"></a><span class="do">## YR.ISOL 1 0.78 1331.1 224.27</span></span>
<span id="cb14-20"><a href="#cb14-20" tabindex="-1"></a><span class="do">## ALT 1 22.24 1352.6 225.34</span></span>
<span id="cb14-21"><a href="#cb14-21" tabindex="-1"></a><span class="do">## LOGAREA:FGRAZE 4 406.74 1737.1 236.10</span></span></code></pre></div>
<p> </p>
<p>A3. Refit your model with the variable associated with the lowest AIC
value removed and run <code>drop1()</code> again on your new model
(<code>M3.AIC</code>).</p>
<div class="sourceCode" id="cb15"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb15-1"><a href="#cb15-1" tabindex="-1"></a><span class="co"># Ok, as the model without the variable `YR.ISOL` has the lowest AIC (224.27) </span></span>
<span id="cb15-2"><a href="#cb15-2" tabindex="-1"></a><span class="co"># let's update our model and remove this variable.</span></span>
<span id="cb15-3"><a href="#cb15-3" tabindex="-1"></a></span>
<span id="cb15-4"><a href="#cb15-4" tabindex="-1"></a>M3.AIC <span class="ot"><-</span> <span class="fu">update</span>(M2.AIC, <span class="at">formula =</span> . <span class="sc">~</span> . <span class="sc">-</span> YR.ISOL)</span>
<span id="cb15-5"><a href="#cb15-5" tabindex="-1"></a><span class="fu">drop1</span>(M3.AIC)</span>
<span id="cb15-6"><a href="#cb15-6" tabindex="-1"></a><span class="do">## Single term deletions</span></span>
<span id="cb15-7"><a href="#cb15-7" tabindex="-1"></a><span class="do">## </span></span>
<span id="cb15-8"><a href="#cb15-8" tabindex="-1"></a><span class="do">## Model:</span></span>
<span id="cb15-9"><a href="#cb15-9" tabindex="-1"></a><span class="do">## ABUND ~ LOGDIST + ALT + LOGAREA + FGRAZE + LOGAREA:FGRAZE</span></span>
<span id="cb15-10"><a href="#cb15-10" tabindex="-1"></a><span class="do">## Df Sum of Sq RSS AIC</span></span>
<span id="cb15-11"><a href="#cb15-11" tabindex="-1"></a><span class="do">## <none> 1331.1 224.27</span></span>
<span id="cb15-12"><a href="#cb15-12" tabindex="-1"></a><span class="do">## LOGDIST 1 3.28 1334.4 222.43</span></span>
<span id="cb15-13"><a href="#cb15-13" tabindex="-1"></a><span class="do">## ALT 1 25.33 1356.5 223.53</span></span>
<span id="cb15-14"><a href="#cb15-14" tabindex="-1"></a><span class="do">## LOGAREA:FGRAZE 4 405.99 1737.1 234.10</span></span></code></pre></div>
<p> </p>
<p>A4. Repeat your model selection by removing the variable indicated by
the model with the lowest AIC.</p>
<div class="sourceCode" id="cb16"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb16-1"><a href="#cb16-1" tabindex="-1"></a><span class="co"># So, now the model without `LOGDIST` has the lowest AIC (222.43) so we should </span></span>
<span id="cb16-2"><a href="#cb16-2" tabindex="-1"></a><span class="co"># refit the model without this variable and run `drop1()` again.</span></span>
<span id="cb16-3"><a href="#cb16-3" tabindex="-1"></a></span>
<span id="cb16-4"><a href="#cb16-4" tabindex="-1"></a>M4.AIC <span class="ot"><-</span> <span class="fu">update</span>(M3.AIC, <span class="at">formula =</span> . <span class="sc">~</span> . <span class="sc">-</span> LOGDIST)</span>
<span id="cb16-5"><a href="#cb16-5" tabindex="-1"></a><span class="fu">drop1</span>(M4.AIC)</span>
<span id="cb16-6"><a href="#cb16-6" tabindex="-1"></a><span class="do">## Single term deletions</span></span>
<span id="cb16-7"><a href="#cb16-7" tabindex="-1"></a><span class="do">## </span></span>
<span id="cb16-8"><a href="#cb16-8" tabindex="-1"></a><span class="do">## Model:</span></span>
<span id="cb16-9"><a href="#cb16-9" tabindex="-1"></a><span class="do">## ABUND ~ ALT + LOGAREA + FGRAZE + LOGAREA:FGRAZE</span></span>
<span id="cb16-10"><a href="#cb16-10" tabindex="-1"></a><span class="do">## Df Sum of Sq RSS AIC</span></span>
<span id="cb16-11"><a href="#cb16-11" tabindex="-1"></a><span class="do">## <none> 1334.4 222.43</span></span>
<span id="cb16-12"><a href="#cb16-12" tabindex="-1"></a><span class="do">## ALT 1 22.84 1357.2 221.57</span></span>
<span id="cb16-13"><a href="#cb16-13" tabindex="-1"></a><span class="do">## LOGAREA:FGRAZE 4 408.56 1743.0 232.33</span></span></code></pre></div>
<p> </p>
<p>A5. Rinse and repeat as above.</p>
<div class="sourceCode" id="cb17"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb17-1"><a href="#cb17-1" tabindex="-1"></a><span class="co"># And the model without the variable `ALT` has an AIC of 221.57 which is about </span></span>
<span id="cb17-2"><a href="#cb17-2" tabindex="-1"></a><span class="co"># the same as the model with `ALT` (AIC 222.43), so let's remove this variable </span></span>
<span id="cb17-3"><a href="#cb17-3" tabindex="-1"></a><span class="co"># from the model as this suggests that the simpler model fits our data just as </span></span>
<span id="cb17-4"><a href="#cb17-4" tabindex="-1"></a><span class="co"># well as the more complicated model.</span></span>