stats/Chapter_28.html at main · bradduthie/stats · GitHub

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
<!DOCTYPE html>
<html lang="" xml:lang="">
<head>

  <meta charset="utf-8" />
  <meta http-equiv="X-UA-Compatible" content="IE=edge" />
  <title>Chapter 28 Practical. ANOVA and associated tests | Fundamental statistical concepts and techniques in the biological and environmental sciences: With jamovi</title>
  <meta name="description" content="This is an introductory statistics textbook for students in the biological and environmental sciences with examples using jamovi statistical software." />
  <meta name="generator" content="bookdown 0.39 and GitBook 2.6.7" />

  <meta property="og:title" content="Chapter 28 Practical. ANOVA and associated tests | Fundamental statistical concepts and techniques in the biological and environmental sciences: With jamovi" />
  <meta property="og:type" content="book" />
  <meta property="og:image" content="/img/cover.png" />
  <meta property="og:description" content="This is an introductory statistics textbook for students in the biological and environmental sciences with examples using jamovi statistical software." />
  <meta name="github-repo" content="bradduthie/stats" />

  <meta name="twitter:card" content="summary" />
  <meta name="twitter:title" content="Chapter 28 Practical. ANOVA and associated tests | Fundamental statistical concepts and techniques in the biological and environmental sciences: With jamovi" />

  <meta name="twitter:description" content="This is an introductory statistics textbook for students in the biological and environmental sciences with examples using jamovi statistical software." />
  <meta name="twitter:image" content="/img/cover.png" />

<meta name="author" content="A. Bradley Duthie" />


<meta name="date" content="2025-02-20" />

  <meta name="viewport" content="width=device-width, initial-scale=1" />
  <meta name="apple-mobile-web-app-capable" content="yes" />
  <meta name="apple-mobile-web-app-status-bar-style" content="black" />


<link rel="prev" href="Chapter_27.html"/>
<link rel="next" href="Chapter_29.html"/>
<script src="libs/jquery-3.6.0/jquery-3.6.0.min.js"></script>
<script src="https://cdn.jsdelivr.net/npm/fuse.js@6.4.6/dist/fuse.min.js"></script>
<link href="libs/gitbook-2.6.7/css/style.css" rel="stylesheet" />
<link href="libs/gitbook-2.6.7/css/plugin-table.css" rel="stylesheet" />
<link href="libs/gitbook-2.6.7/css/plugin-bookdown.css" rel="stylesheet" />
<link href="libs/gitbook-2.6.7/css/plugin-highlight.css" rel="stylesheet" />
<link href="libs/gitbook-2.6.7/css/plugin-search.css" rel="stylesheet" />
<link href="libs/gitbook-2.6.7/css/plugin-fontsettings.css" rel="stylesheet" />
<link href="libs/gitbook-2.6.7/css/plugin-clipboard.css" rel="stylesheet" />


<link href="libs/anchor-sections-1.1.0/anchor-sections.css" rel="stylesheet" />
<link href="libs/anchor-sections-1.1.0/anchor-sections-hash.css" rel="stylesheet" />
<script src="libs/anchor-sections-1.1.0/anchor-sections.js"></script>


<style type="text/css">
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; }
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;
  }
pre.numberSource { margin-left: 3em;  padding-left: 4px; }
div.sourceCode
  {   }
@media screen {
pre > code.sourceCode > span > a:first-child::before { text-decoration: underline; }
}
code span.al { font-weight: bold; } /* Alert */
code span.an { font-style: italic; } /* Annotation */
code span.cf { font-weight: bold; } /* ControlFlow */
code span.co { font-style: italic; } /* Comment */
code span.cv { font-style: italic; } /* CommentVar */
code span.do { font-style: italic; } /* Documentation */
code span.dt { text-decoration: underline; } /* DataType */
code span.er { font-weight: bold; } /* Error */
code span.in { font-style: italic; } /* Information */
code span.kw { font-weight: bold; } /* Keyword */
code span.pp { font-weight: bold; } /* Preprocessor */
code span.wa { font-style: italic; } /* Warning */
</style>

<style type="text/css">

  div.hanging-indent{margin-left: 1.5em; text-indent: -1.5em;}
</style>
<style type="text/css">
/* Used with Pandoc 2.11+ new --citeproc when CSL is used */
div.csl-bib-body { }
div.csl-entry {
  clear: both;
}
.hanging div.csl-entry {
  margin-left:2em;
  text-indent:-2em;
}
div.csl-left-margin {
  min-width:2em;
  float:left;
}
div.csl-right-inline {
  margin-left:2em;
  padding-left:1em;
}
div.csl-indent {
  margin-left: 2em;
}
</style>

<link rel="stylesheet" href="style.css" type="text/css" />
</head>

<body>


  <div class="book without-animation with-summary font-size-2 font-family-1" data-basepath=".">

    <div class="book-summary">
      <nav role="navigation">

<ul class="summary">
<li><a href="./">Statistics with jamovi</a></li>

<li class="divider"></li>
<li class="chapter" data-level="" data-path="index.html"><a href="index.html"><i class="fa fa-check"></i>Preface</a>
<ul>
<li class="chapter" data-level="" data-path="index.html"><a href="index.html#structure"><i class="fa fa-check"></i>How this book is structured</a></li>
<li class="chapter" data-level="" data-path="index.html"><a href="index.html#datasets"><i class="fa fa-check"></i>Datasets used in this book</a></li>
<li class="chapter" data-level="" data-path="index.html"><a href="index.html#acknowledgements"><i class="fa fa-check"></i>Acknowledgements</a></li>
<li class="chapter" data-level="" data-path="index.html"><a href="index.html#author"><i class="fa fa-check"></i>About the author</a></li>
</ul></li>
<li class="chapter" data-level="1" data-path="Chapter_1.html"><a href="Chapter_1.html"><i class="fa fa-check"></i><b>1</b> Background mathematics</a>
<ul>
<li class="chapter" data-level="1.1" data-path="Chapter_1.html"><a href="Chapter_1.html#numbers-and-operations"><i class="fa fa-check"></i><b>1.1</b> Numbers and operations</a></li>
<li class="chapter" data-level="1.2" data-path="Chapter_1.html"><a href="Chapter_1.html#logarithms"><i class="fa fa-check"></i><b>1.2</b> Logarithms</a></li>
<li class="chapter" data-level="1.3" data-path="Chapter_1.html"><a href="Chapter_1.html#order-of-operations"><i class="fa fa-check"></i><b>1.3</b> Order of operations</a></li>
</ul></li>
<li class="chapter" data-level="2" data-path="Chapter_2.html"><a href="Chapter_2.html"><i class="fa fa-check"></i><b>2</b> Data organisation</a>
<ul>
<li class="chapter" data-level="2.1" data-path="Chapter_2.html"><a href="Chapter_2.html#tidy-data"><i class="fa fa-check"></i><b>2.1</b> Tidy data</a></li>
<li class="chapter" data-level="2.2" data-path="Chapter_2.html"><a href="Chapter_2.html#data-files"><i class="fa fa-check"></i><b>2.2</b> Data files</a></li>
<li class="chapter" data-level="2.3" data-path="Chapter_2.html"><a href="Chapter_2.html#managing-data-files"><i class="fa fa-check"></i><b>2.3</b> Managing data files</a></li>
</ul></li>
<li class="chapter" data-level="3" data-path="Chapter_3.html"><a href="Chapter_3.html"><i class="fa fa-check"></i><b>3</b> <em>Practical</em>. Preparing data</a>
<ul>
<li class="chapter" data-level="3.1" data-path="Chapter_3.html"><a href="Chapter_3.html#transferring-data-to-a-spreadsheet"><i class="fa fa-check"></i><b>3.1</b> Transferring data to a spreadsheet</a></li>
<li class="chapter" data-level="3.2" data-path="Chapter_3.html"><a href="Chapter_3.html#making-spreadsheet-data-tidy"><i class="fa fa-check"></i><b>3.2</b> Making spreadsheet data tidy</a></li>
<li class="chapter" data-level="3.3" data-path="Chapter_3.html"><a href="Chapter_3.html#making-data-tidy-again"><i class="fa fa-check"></i><b>3.3</b> Making data tidy again</a></li>
<li class="chapter" data-level="3.4" data-path="Chapter_3.html"><a href="Chapter_3.html#tidy-data-and-spreadsheet-calculations"><i class="fa fa-check"></i><b>3.4</b> Tidy data and spreadsheet calculations</a></li>
<li class="chapter" data-level="3.5" data-path="Chapter_3.html"><a href="Chapter_3.html#summary"><i class="fa fa-check"></i><b>3.5</b> Summary</a></li>
</ul></li>
<li class="chapter" data-level="4" data-path="Chapter_4.html"><a href="Chapter_4.html"><i class="fa fa-check"></i><b>4</b> Populations and samples</a></li>
<li class="chapter" data-level="5" data-path="Chapter_5.html"><a href="Chapter_5.html"><i class="fa fa-check"></i><b>5</b> Types of variables</a></li>
<li class="chapter" data-level="6" data-path="Chapter_6.html"><a href="Chapter_6.html"><i class="fa fa-check"></i><b>6</b> Accuracy, precision, and units</a>
<ul>
<li class="chapter" data-level="6.1" data-path="Chapter_6.html"><a href="Chapter_6.html#accuracy"><i class="fa fa-check"></i><b>6.1</b> Accuracy</a></li>
<li class="chapter" data-level="6.2" data-path="Chapter_6.html"><a href="Chapter_6.html#precision"><i class="fa fa-check"></i><b>6.2</b> Precision</a></li>
<li class="chapter" data-level="6.3" data-path="Chapter_6.html"><a href="Chapter_6.html#systems-of-units"><i class="fa fa-check"></i><b>6.3</b> Systems of units</a></li>
</ul></li>
<li class="chapter" data-level="7" data-path="Chapter_7.html"><a href="Chapter_7.html"><i class="fa fa-check"></i><b>7</b> Uncertainty propagation</a>
<ul>
<li class="chapter" data-level="7.1" data-path="Chapter_7.html"><a href="Chapter_7.html#adding-or-subtracting-errors"><i class="fa fa-check"></i><b>7.1</b> Adding or subtracting errors</a></li>
<li class="chapter" data-level="7.2" data-path="Chapter_7.html"><a href="Chapter_7.html#multiplying-or-dividing-errors"><i class="fa fa-check"></i><b>7.2</b> Multiplying or dividing errors</a></li>
</ul></li>
<li class="chapter" data-level="8" data-path="Chapter_8.html"><a href="Chapter_8.html"><i class="fa fa-check"></i><b>8</b> <em>Practical</em>. Introduction to jamovi</a>
<ul>
<li class="chapter" data-level="8.1" data-path="Chapter_8.html"><a href="Chapter_8.html#summary_statistics_02"><i class="fa fa-check"></i><b>8.1</b> Summary statistics</a></li>
<li class="chapter" data-level="8.2" data-path="Chapter_8.html"><a href="Chapter_8.html#transforming_variables_02"><i class="fa fa-check"></i><b>8.2</b> Transforming variables</a></li>
<li class="chapter" data-level="8.3" data-path="Chapter_8.html"><a href="Chapter_8.html#computing_variables_02"><i class="fa fa-check"></i><b>8.3</b> Computing variables</a></li>
<li class="chapter" data-level="8.4" data-path="Chapter_8.html"><a href="Chapter_8.html#summary-1"><i class="fa fa-check"></i><b>8.4</b> Summary</a></li>
</ul></li>
<li class="chapter" data-level="9" data-path="Chapter_9.html"><a href="Chapter_9.html"><i class="fa fa-check"></i><b>9</b> Decimal places, significant figures, and rounding</a>
<ul>
<li class="chapter" data-level="9.1" data-path="Chapter_9.html"><a href="Chapter_9.html#decimal-places-and-significant-figures"><i class="fa fa-check"></i><b>9.1</b> Decimal places and significant figures</a></li>
<li class="chapter" data-level="9.2" data-path="Chapter_9.html"><a href="Chapter_9.html#rounding"><i class="fa fa-check"></i><b>9.2</b> Rounding</a></li>
</ul></li>
<li class="chapter" data-level="10" data-path="Chapter_10.html"><a href="Chapter_10.html"><i class="fa fa-check"></i><b>10</b> Graphs</a>
<ul>
<li class="chapter" data-level="10.1" data-path="Chapter_10.html"><a href="Chapter_10.html#histograms"><i class="fa fa-check"></i><b>10.1</b> Histograms</a></li>
<li class="chapter" data-level="10.2" data-path="Chapter_10.html"><a href="Chapter_10.html#barplots-and-pie-charts"><i class="fa fa-check"></i><b>10.2</b> Barplots and pie charts</a></li>
<li class="chapter" data-level="10.3" data-path="Chapter_10.html"><a href="Chapter_10.html#box-whisker-plots"><i class="fa fa-check"></i><b>10.3</b> Box-whisker plots</a></li>
</ul></li>
<li class="chapter" data-level="11" data-path="Chapter_11.html"><a href="Chapter_11.html"><i class="fa fa-check"></i><b>11</b> Measures of central tendency</a>
<ul>
<li class="chapter" data-level="11.1" data-path="Chapter_11.html"><a href="Chapter_11.html#the-mean"><i class="fa fa-check"></i><b>11.1</b> The mean</a></li>
<li class="chapter" data-level="11.2" data-path="Chapter_11.html"><a href="Chapter_11.html#the-mode"><i class="fa fa-check"></i><b>11.2</b> The mode</a></li>
<li class="chapter" data-level="11.3" data-path="Chapter_11.html"><a href="Chapter_11.html#the-median-and-quantiles"><i class="fa fa-check"></i><b>11.3</b> The median and quantiles</a></li>
</ul></li>
<li class="chapter" data-level="12" data-path="Chapter_12.html"><a href="Chapter_12.html"><i class="fa fa-check"></i><b>12</b> Measures of spread</a>
<ul>
<li class="chapter" data-level="12.1" data-path="Chapter_12.html"><a href="Chapter_12.html#the-range"><i class="fa fa-check"></i><b>12.1</b> The range</a></li>
<li class="chapter" data-level="12.2" data-path="Chapter_12.html"><a href="Chapter_12.html#the-inter-quartile-range"><i class="fa fa-check"></i><b>12.2</b> The inter-quartile range</a></li>
<li class="chapter" data-level="12.3" data-path="Chapter_12.html"><a href="Chapter_12.html#the-variance"><i class="fa fa-check"></i><b>12.3</b> The variance</a></li>
<li class="chapter" data-level="12.4" data-path="Chapter_12.html"><a href="Chapter_12.html#the-standard-deviation"><i class="fa fa-check"></i><b>12.4</b> The standard deviation</a></li>
<li class="chapter" data-level="12.5" data-path="Chapter_12.html"><a href="Chapter_12.html#the-coefficient-of-variation"><i class="fa fa-check"></i><b>12.5</b> The coefficient of variation</a></li>
<li class="chapter" data-level="12.6" data-path="Chapter_12.html"><a href="Chapter_12.html#the-standard-error"><i class="fa fa-check"></i><b>12.6</b> The standard error</a></li>
</ul></li>
<li class="chapter" data-level="13" data-path="Chapter_13.html"><a href="Chapter_13.html"><i class="fa fa-check"></i><b>13</b> Skew and kurtosis</a>
<ul>
<li class="chapter" data-level="13.1" data-path="Chapter_13.html"><a href="Chapter_13.html#skew"><i class="fa fa-check"></i><b>13.1</b> Skew</a></li>
<li class="chapter" data-level="13.2" data-path="Chapter_13.html"><a href="Chapter_13.html#kurtosis"><i class="fa fa-check"></i><b>13.2</b> Kurtosis</a></li>
<li class="chapter" data-level="13.3" data-path="Chapter_13.html"><a href="Chapter_13.html#moments"><i class="fa fa-check"></i><b>13.3</b> Moments</a></li>
</ul></li>
<li class="chapter" data-level="14" data-path="Chapter_14.html"><a href="Chapter_14.html"><i class="fa fa-check"></i><b>14</b> <em>Practical</em>. Plotting and statistical summaries in jamovi</a>
<ul>
<li class="chapter" data-level="14.1" data-path="Chapter_14.html"><a href="Chapter_14.html#reorganise-the-dataset-into-a-tidy-format"><i class="fa fa-check"></i><b>14.1</b> Reorganise the dataset into a tidy format</a></li>
<li class="chapter" data-level="14.2" data-path="Chapter_14.html"><a href="Chapter_14.html#histograms-and-box-whisker-plots"><i class="fa fa-check"></i><b>14.2</b> Histograms and box-whisker plots</a></li>
<li class="chapter" data-level="14.3" data-path="Chapter_14.html"><a href="Chapter_14.html#calculate-summary-statistics"><i class="fa fa-check"></i><b>14.3</b> Calculate summary statistics</a></li>
<li class="chapter" data-level="14.4" data-path="Chapter_14.html"><a href="Chapter_14.html#reporting-decimals-and-significant-figures"><i class="fa fa-check"></i><b>14.4</b> Reporting decimals and significant figures</a></li>
<li class="chapter" data-level="14.5" data-path="Chapter_14.html"><a href="Chapter_14.html#comparing-across-sites"><i class="fa fa-check"></i><b>14.5</b> Comparing across sites</a></li>
</ul></li>
<li class="chapter" data-level="15" data-path="Chapter_15.html"><a href="Chapter_15.html"><i class="fa fa-check"></i><b>15</b> Introduction to probability models</a>
<ul>
<li class="chapter" data-level="15.1" data-path="Chapter_15.html"><a href="Chapter_15.html#instructive-example"><i class="fa fa-check"></i><b>15.1</b> Instructive example</a></li>
<li class="chapter" data-level="15.2" data-path="Chapter_15.html"><a href="Chapter_15.html#biological-applications"><i class="fa fa-check"></i><b>15.2</b> Biological applications</a></li>
<li class="chapter" data-level="15.3" data-path="Chapter_15.html"><a href="Chapter_15.html#sampling-with-and-without-replacement"><i class="fa fa-check"></i><b>15.3</b> Sampling with and without replacement</a></li>
<li class="chapter" data-level="15.4" data-path="Chapter_15.html"><a href="Chapter_15.html#probability-distributions"><i class="fa fa-check"></i><b>15.4</b> Probability distributions</a>
<ul>
<li class="chapter" data-level="15.4.1" data-path="Chapter_15.html"><a href="Chapter_15.html#binomial-distribution"><i class="fa fa-check"></i><b>15.4.1</b> Binomial distribution</a></li>
<li class="chapter" data-level="15.4.2" data-path="Chapter_15.html"><a href="Chapter_15.html#poisson-distribution"><i class="fa fa-check"></i><b>15.4.2</b> Poisson distribution</a></li>
<li class="chapter" data-level="15.4.3" data-path="Chapter_15.html"><a href="Chapter_15.html#uniform-distribution"><i class="fa fa-check"></i><b>15.4.3</b> Uniform distribution</a></li>
<li class="chapter" data-level="15.4.4" data-path="Chapter_15.html"><a href="Chapter_15.html#normal-distribution"><i class="fa fa-check"></i><b>15.4.4</b> Normal distribution</a></li>
</ul></li>
<li class="chapter" data-level="15.5" data-path="Chapter_15.html"><a href="Chapter_15.html#summary-2"><i class="fa fa-check"></i><b>15.5</b> Summary</a></li>
</ul></li>
<li class="chapter" data-level="16" data-path="Chapter_16.html"><a href="Chapter_16.html"><i class="fa fa-check"></i><b>16</b> Central Limit Theorem</a>
<ul>
<li class="chapter" data-level="16.1" data-path="Chapter_16.html"><a href="Chapter_16.html#the-distribution-of-means-is-normal"><i class="fa fa-check"></i><b>16.1</b> The distribution of means is normal</a></li>
<li class="chapter" data-level="16.2" data-path="Chapter_16.html"><a href="Chapter_16.html#probability-and-z-scores"><i class="fa fa-check"></i><b>16.2</b> Probability and z-scores</a></li>
</ul></li>
<li class="chapter" data-level="17" data-path="Chapter_17.html"><a href="Chapter_17.html"><i class="fa fa-check"></i><b>17</b> <em>Practical</em>. Probability and simulation</a>
<ul>
<li class="chapter" data-level="17.1" data-path="Chapter_17.html"><a href="Chapter_17.html#probabilities-from-a-dataset"><i class="fa fa-check"></i><b>17.1</b> Probabilities from a dataset</a></li>
<li class="chapter" data-level="17.2" data-path="Chapter_17.html"><a href="Chapter_17.html#probabilities-from-a-normal-distribution"><i class="fa fa-check"></i><b>17.2</b> Probabilities from a normal distribution</a></li>
<li class="chapter" data-level="17.3" data-path="Chapter_17.html"><a href="Chapter_17.html#central-limit-theorem"><i class="fa fa-check"></i><b>17.3</b> Central limit theorem</a></li>
</ul></li>
<li class="chapter" data-level="18" data-path="Chapter_18.html"><a href="Chapter_18.html"><i class="fa fa-check"></i><b>18</b> Confidence intervals</a>
<ul>
<li class="chapter" data-level="18.1" data-path="Chapter_18.html"><a href="Chapter_18.html#normal-distribution-cis"><i class="fa fa-check"></i><b>18.1</b> Normal distribution CIs</a></li>
<li class="chapter" data-level="18.2" data-path="Chapter_18.html"><a href="Chapter_18.html#binomial-distribution-cis"><i class="fa fa-check"></i><b>18.2</b> Binomial distribution CIs</a></li>
</ul></li>
<li class="chapter" data-level="19" data-path="Chapter_19.html"><a href="Chapter_19.html"><i class="fa fa-check"></i><b>19</b> The t-interval</a></li>
<li class="chapter" data-level="20" data-path="Chapter_20.html"><a href="Chapter_20.html"><i class="fa fa-check"></i><b>20</b> <em>Practical</em>. z- and t-intervals</a>
<ul>
<li class="chapter" data-level="20.1" data-path="Chapter_20.html"><a href="Chapter_20.html#confidence-intervals-with-distraction"><i class="fa fa-check"></i><b>20.1</b> Confidence intervals with distrACTION</a></li>
<li class="chapter" data-level="20.2" data-path="Chapter_20.html"><a href="Chapter_20.html#confidence-intervals-from-z--and-t-scores"><i class="fa fa-check"></i><b>20.2</b> Confidence intervals from z- and t-scores</a></li>
<li class="chapter" data-level="20.3" data-path="Chapter_20.html"><a href="Chapter_20.html#confidence-intervals-for-different-sample-sizes"><i class="fa fa-check"></i><b>20.3</b> Confidence intervals for different sample sizes</a></li>
<li class="chapter" data-level="20.4" data-path="Chapter_20.html"><a href="Chapter_20.html#proportion-confidence-intervals"><i class="fa fa-check"></i><b>20.4</b> Proportion confidence intervals</a></li>
<li class="chapter" data-level="20.5" data-path="Chapter_20.html"><a href="Chapter_20.html#another-proportion-confidence-interval"><i class="fa fa-check"></i><b>20.5</b> Another proportion confidence interval</a></li>
</ul></li>
<li class="chapter" data-level="21" data-path="Chapter_21.html"><a href="Chapter_21.html"><i class="fa fa-check"></i><b>21</b> What is hypothesis testing?</a>
<ul>
<li class="chapter" data-level="21.1" data-path="Chapter_21.html"><a href="Chapter_21.html#how-ridiculous-is-our-hypothesis"><i class="fa fa-check"></i><b>21.1</b> How ridiculous is our hypothesis?</a></li>
<li class="chapter" data-level="21.2" data-path="Chapter_21.html"><a href="Chapter_21.html#statistical-hypothesis-testing"><i class="fa fa-check"></i><b>21.2</b> Statistical hypothesis testing</a></li>
<li class="chapter" data-level="21.3" data-path="Chapter_21.html"><a href="Chapter_21.html#p-values-false-positives-and-power"><i class="fa fa-check"></i><b>21.3</b> P-values, false positives, and power</a></li>
</ul></li>
<li class="chapter" data-level="22" data-path="Chapter_22.html"><a href="Chapter_22.html"><i class="fa fa-check"></i><b>22</b> The t-test</a>
<ul>
<li class="chapter" data-level="22.1" data-path="Chapter_22.html"><a href="Chapter_22.html#one-sample-t-test"><i class="fa fa-check"></i><b>22.1</b> One sample t-test</a></li>
<li class="chapter" data-level="22.2" data-path="Chapter_22.html"><a href="Chapter_22.html#independent-samples-t-test"><i class="fa fa-check"></i><b>22.2</b> Independent samples t-test</a></li>
<li class="chapter" data-level="22.3" data-path="Chapter_22.html"><a href="Chapter_22.html#paired-samples-t-test"><i class="fa fa-check"></i><b>22.3</b> Paired samples t-test</a></li>
<li class="chapter" data-level="22.4" data-path="Chapter_22.html"><a href="Chapter_22.html#assumptions-of-t-tests"><i class="fa fa-check"></i><b>22.4</b> Assumptions of t-tests</a></li>
<li class="chapter" data-level="22.5" data-path="Chapter_22.html"><a href="Chapter_22.html#non-parametric-alternatives"><i class="fa fa-check"></i><b>22.5</b> Non-parametric alternatives</a>
<ul>
<li class="chapter" data-level="22.5.1" data-path="Chapter_22.html"><a href="Chapter_22.html#wilcoxon-test"><i class="fa fa-check"></i><b>22.5.1</b> Wilcoxon test</a></li>
<li class="chapter" data-level="22.5.2" data-path="Chapter_22.html"><a href="Chapter_22.html#mann-whitney-u-test"><i class="fa fa-check"></i><b>22.5.2</b> Mann-Whitney U test</a></li>
</ul></li>
<li class="chapter" data-level="22.6" data-path="Chapter_22.html"><a href="Chapter_22.html#summary-3"><i class="fa fa-check"></i><b>22.6</b> Summary</a></li>
</ul></li>
<li class="chapter" data-level="23" data-path="Chapter_23.html"><a href="Chapter_23.html"><i class="fa fa-check"></i><b>23</b> <em>Practical</em>. Hypothesis testing and t-tests</a>
<ul>
<li class="chapter" data-level="23.1" data-path="Chapter_23.html"><a href="Chapter_23.html#one-sample-t-test-1"><i class="fa fa-check"></i><b>23.1</b> One sample t-test</a></li>
<li class="chapter" data-level="23.2" data-path="Chapter_23.html"><a href="Chapter_23.html#paired-t-test"><i class="fa fa-check"></i><b>23.2</b> Paired t-test</a></li>
<li class="chapter" data-level="23.3" data-path="Chapter_23.html"><a href="Chapter_23.html#wilcoxon-test-1"><i class="fa fa-check"></i><b>23.3</b> Wilcoxon test</a></li>
<li class="chapter" data-level="23.4" data-path="Chapter_23.html"><a href="Chapter_23.html#independent-samples-t-test-1"><i class="fa fa-check"></i><b>23.4</b> Independent samples t-test</a></li>
<li class="chapter" data-level="23.5" data-path="Chapter_23.html"><a href="Chapter_23.html#mann-whitney-u-test-1"><i class="fa fa-check"></i><b>23.5</b> Mann-Whitney U Test</a></li>
</ul></li>
<li class="chapter" data-level="24" data-path="Chapter_24.html"><a href="Chapter_24.html"><i class="fa fa-check"></i><b>24</b> Analysis of variance</a>
<ul>
<li class="chapter" data-level="24.1" data-path="Chapter_24.html"><a href="Chapter_24.html#f-distribution"><i class="fa fa-check"></i><b>24.1</b> F-distribution</a></li>
<li class="chapter" data-level="24.2" data-path="Chapter_24.html"><a href="Chapter_24.html#one-way-anova"><i class="fa fa-check"></i><b>24.2</b> One-way ANOVA</a>
<ul>
<li class="chapter" data-level="24.2.1" data-path="Chapter_24.html"><a href="Chapter_24.html#anova-mean-variance-among-groups"><i class="fa fa-check"></i><b>24.2.1</b> ANOVA mean variance among groups</a></li>
<li class="chapter" data-level="24.2.2" data-path="Chapter_24.html"><a href="Chapter_24.html#anova-mean-variance-within-groups"><i class="fa fa-check"></i><b>24.2.2</b> ANOVA mean variance within groups</a></li>
<li class="chapter" data-level="24.2.3" data-path="Chapter_24.html"><a href="Chapter_24.html#anova-f-statistic-calculation"><i class="fa fa-check"></i><b>24.2.3</b> ANOVA F-statistic calculation</a></li>
</ul></li>
<li class="chapter" data-level="24.3" data-path="Chapter_24.html"><a href="Chapter_24.html#assumptions-of-anova"><i class="fa fa-check"></i><b>24.3</b> Assumptions of ANOVA</a></li>
</ul></li>
<li class="chapter" data-level="25" data-path="Chapter_25.html"><a href="Chapter_25.html"><i class="fa fa-check"></i><b>25</b> Multiple comparisons</a></li>
<li class="chapter" data-level="26" data-path="Chapter_26.html"><a href="Chapter_26.html"><i class="fa fa-check"></i><b>26</b> Kruskal-Wallis H test</a></li>
<li class="chapter" data-level="27" data-path="Chapter_27.html"><a href="Chapter_27.html"><i class="fa fa-check"></i><b>27</b> Two-way ANOVA</a></li>
<li class="chapter" data-level="28" data-path="Chapter_28.html"><a href="Chapter_28.html"><i class="fa fa-check"></i><b>28</b> <em>Practical</em>. ANOVA and associated tests</a>
<ul>
<li class="chapter" data-level="28.1" data-path="Chapter_28.html"><a href="Chapter_28.html#one-way-anova-site"><i class="fa fa-check"></i><b>28.1</b> One-way ANOVA (site)</a></li>
<li class="chapter" data-level="28.2" data-path="Chapter_28.html"><a href="Chapter_28.html#one-way-anova-profile"><i class="fa fa-check"></i><b>28.2</b> One-way ANOVA (profile)</a></li>
<li class="chapter" data-level="28.3" data-path="Chapter_28.html"><a href="Chapter_28.html#multiple-comparisons"><i class="fa fa-check"></i><b>28.3</b> Multiple comparisons</a></li>
<li class="chapter" data-level="28.4" data-path="Chapter_28.html"><a href="Chapter_28.html#kruskal-wallis-h-test"><i class="fa fa-check"></i><b>28.4</b> Kruskal-Wallis H test</a></li>
<li class="chapter" data-level="28.5" data-path="Chapter_28.html"><a href="Chapter_28.html#two-way-anova"><i class="fa fa-check"></i><b>28.5</b> Two-way ANOVA</a></li>
</ul></li>
<li class="chapter" data-level="29" data-path="Chapter_29.html"><a href="Chapter_29.html"><i class="fa fa-check"></i><b>29</b> Frequency and count data</a>
<ul>
<li class="chapter" data-level="29.1" data-path="Chapter_29.html"><a href="Chapter_29.html#chi-square-distribution"><i class="fa fa-check"></i><b>29.1</b> Chi-square distribution</a></li>
<li class="chapter" data-level="29.2" data-path="Chapter_29.html"><a href="Chapter_29.html#chi-square-goodness-of-fit"><i class="fa fa-check"></i><b>29.2</b> Chi-square goodness of fit</a></li>
<li class="chapter" data-level="29.3" data-path="Chapter_29.html"><a href="Chapter_29.html#chi-square-test-of-association"><i class="fa fa-check"></i><b>29.3</b> Chi-square test of association</a></li>
</ul></li>
<li class="chapter" data-level="30" data-path="Chapter_30.html"><a href="Chapter_30.html"><i class="fa fa-check"></i><b>30</b> Correlation</a>
<ul>
<li class="chapter" data-level="30.1" data-path="Chapter_30.html"><a href="Chapter_30.html#scatterplots"><i class="fa fa-check"></i><b>30.1</b> Scatterplots</a></li>
<li class="chapter" data-level="30.2" data-path="Chapter_30.html"><a href="Chapter_30.html#correlation-coefficient"><i class="fa fa-check"></i><b>30.2</b> Correlation coefficient</a>
<ul>
<li class="chapter" data-level="30.2.1" data-path="Chapter_30.html"><a href="Chapter_30.html#pearson-product-moment-correlation-coefficient"><i class="fa fa-check"></i><b>30.2.1</b> Pearson product moment correlation coefficient</a></li>
<li class="chapter" data-level="30.2.2" data-path="Chapter_30.html"><a href="Chapter_30.html#spearmans-rank-correlation-coefficient"><i class="fa fa-check"></i><b>30.2.2</b> Spearman’s rank correlation coefficient</a></li>
</ul></li>
<li class="chapter" data-level="30.3" data-path="Chapter_30.html"><a href="Chapter_30.html#correlation-hypothesis-testing"><i class="fa fa-check"></i><b>30.3</b> Correlation hypothesis testing</a></li>
</ul></li>
<li class="chapter" data-level="31" data-path="Chapter_31.html"><a href="Chapter_31.html"><i class="fa fa-check"></i><b>31</b> <em>Practical</em>. Analysis of counts and correlations</a>
<ul>
<li class="chapter" data-level="31.1" data-path="Chapter_31.html"><a href="Chapter_31.html#survival-goodness-of-fit"><i class="fa fa-check"></i><b>31.1</b> Survival goodness of fit</a></li>
<li class="chapter" data-level="31.2" data-path="Chapter_31.html"><a href="Chapter_31.html#colony-goodness-of-fit"><i class="fa fa-check"></i><b>31.2</b> Colony goodness of fit</a></li>
<li class="chapter" data-level="31.3" data-path="Chapter_31.html"><a href="Chapter_31.html#chi-square-test-of-association-1"><i class="fa fa-check"></i><b>31.3</b> Chi-Square test of association</a></li>
<li class="chapter" data-level="31.4" data-path="Chapter_31.html"><a href="Chapter_31.html#pearson-product-moment-correlation-test"><i class="fa fa-check"></i><b>31.4</b> Pearson product moment correlation test</a></li>
<li class="chapter" data-level="31.5" data-path="Chapter_31.html"><a href="Chapter_31.html#spearmans-rank-correlation-test"><i class="fa fa-check"></i><b>31.5</b> Spearman’s rank correlation test</a></li>
<li class="chapter" data-level="31.6" data-path="Chapter_31.html"><a href="Chapter_31.html#untidy-goodness-of-fit"><i class="fa fa-check"></i><b>31.6</b> Untidy goodness of fit</a></li>
</ul></li>
<li class="chapter" data-level="32" data-path="Chapter_32.html"><a href="Chapter_32.html"><i class="fa fa-check"></i><b>32</b> Simple linear regression</a>
<ul>
<li class="chapter" data-level="32.1" data-path="Chapter_32.html"><a href="Chapter_32.html#visual-interpretation-of-regression"><i class="fa fa-check"></i><b>32.1</b> Visual interpretation of regression</a></li>
<li class="chapter" data-level="32.2" data-path="Chapter_32.html"><a href="Chapter_32.html#intercepts-slopes-and-residuals"><i class="fa fa-check"></i><b>32.2</b> Intercepts, slopes, and residuals</a></li>
<li class="chapter" data-level="32.3" data-path="Chapter_32.html"><a href="Chapter_32.html#regression-coefficients"><i class="fa fa-check"></i><b>32.3</b> Regression coefficients</a></li>
<li class="chapter" data-level="32.4" data-path="Chapter_32.html"><a href="Chapter_32.html#regression-line-calculation"><i class="fa fa-check"></i><b>32.4</b> Regression line calculation</a></li>
<li class="chapter" data-level="32.5" data-path="Chapter_32.html"><a href="Chapter_32.html#coefficient-of-determination"><i class="fa fa-check"></i><b>32.5</b> Coefficient of determination</a></li>
<li class="chapter" data-level="32.6" data-path="Chapter_32.html"><a href="Chapter_32.html#regression-assumptions"><i class="fa fa-check"></i><b>32.6</b> Regression assumptions</a></li>
<li class="chapter" data-level="32.7" data-path="Chapter_32.html"><a href="Chapter_32.html#regression-hypothesis-testing"><i class="fa fa-check"></i><b>32.7</b> Regression hypothesis testing</a>
<ul>
<li class="chapter" data-level="32.7.1" data-path="Chapter_32.html"><a href="Chapter_32.html#overall-model-significance"><i class="fa fa-check"></i><b>32.7.1</b> Overall model significance</a></li>
<li class="chapter" data-level="32.7.2" data-path="Chapter_32.html"><a href="Chapter_32.html#significance-of-the-intercept"><i class="fa fa-check"></i><b>32.7.2</b> Significance of the intercept</a></li>
<li class="chapter" data-level="32.7.3" data-path="Chapter_32.html"><a href="Chapter_32.html#significance-of-the-slope"><i class="fa fa-check"></i><b>32.7.3</b> Significance of the slope</a></li>
<li class="chapter" data-level="32.7.4" data-path="Chapter_32.html"><a href="Chapter_32.html#simple-regression-output"><i class="fa fa-check"></i><b>32.7.4</b> Simple regression output</a></li>
</ul></li>
<li class="chapter" data-level="32.8" data-path="Chapter_32.html"><a href="Chapter_32.html#prediction-with-linear-models"><i class="fa fa-check"></i><b>32.8</b> Prediction with linear models</a></li>
<li class="chapter" data-level="32.9" data-path="Chapter_32.html"><a href="Chapter_32.html#conclusion"><i class="fa fa-check"></i><b>32.9</b> Conclusion</a></li>
</ul></li>
<li class="chapter" data-level="33" data-path="Chapter_33.html"><a href="Chapter_33.html"><i class="fa fa-check"></i><b>33</b> Multiple regression</a>
<ul>
<li class="chapter" data-level="33.1" data-path="Chapter_33.html"><a href="Chapter_33.html#adjusted-coefficient-of-determination"><i class="fa fa-check"></i><b>33.1</b> Adjusted coefficient of determination</a></li>
</ul></li>
<li class="chapter" data-level="34" data-path="Chapter_34.html"><a href="Chapter_34.html"><i class="fa fa-check"></i><b>34</b> <em>Practical</em>. Using regression</a>
<ul>
<li class="chapter" data-level="34.1" data-path="Chapter_34.html"><a href="Chapter_34.html#predicting-pyrogenic-carbon-from-soil-depth"><i class="fa fa-check"></i><b>34.1</b> Predicting pyrogenic carbon from soil depth</a></li>
<li class="chapter" data-level="34.2" data-path="Chapter_34.html"><a href="Chapter_34.html#predicting-pyrogenic-carbon-from-fire-frequency"><i class="fa fa-check"></i><b>34.2</b> Predicting pyrogenic carbon from fire frequency</a></li>
<li class="chapter" data-level="34.3" data-path="Chapter_34.html"><a href="Chapter_34.html#multiple-regression-depth-and-fire-frequency"><i class="fa fa-check"></i><b>34.3</b> Multiple regression depth and fire frequency</a></li>
<li class="chapter" data-level="34.4" data-path="Chapter_34.html"><a href="Chapter_34.html#large-multiple-regression"><i class="fa fa-check"></i><b>34.4</b> Large multiple regression</a></li>
<li class="chapter" data-level="34.5" data-path="Chapter_34.html"><a href="Chapter_34.html#predicting-temperature-from-fire-frequency"><i class="fa fa-check"></i><b>34.5</b> Predicting temperature from fire frequency</a></li>
</ul></li>
<li class="chapter" data-level="35" data-path="Chapter_35.html"><a href="Chapter_35.html"><i class="fa fa-check"></i><b>35</b> Randomisation</a>
<ul>
<li class="chapter" data-level="35.1" data-path="Chapter_35.html"><a href="Chapter_35.html#summary-of-parametric-hypothesis-testing"><i class="fa fa-check"></i><b>35.1</b> Summary of parametric hypothesis testing</a></li>
<li class="chapter" data-level="35.2" data-path="Chapter_35.html"><a href="Chapter_35.html#randomisation-approach"><i class="fa fa-check"></i><b>35.2</b> Randomisation approach</a></li>
<li class="chapter" data-level="35.3" data-path="Chapter_35.html"><a href="Chapter_35.html#randomisation-for-hypothesis-testing"><i class="fa fa-check"></i><b>35.3</b> Randomisation for hypothesis testing</a></li>
<li class="chapter" data-level="35.4" data-path="Chapter_35.html"><a href="Chapter_35.html#randomisation-assumptions"><i class="fa fa-check"></i><b>35.4</b> Randomisation assumptions</a></li>
<li class="chapter" data-level="35.5" data-path="Chapter_35.html"><a href="Chapter_35.html#bootstrapping"><i class="fa fa-check"></i><b>35.5</b> Bootstrapping</a></li>
<li class="chapter" data-level="35.6" data-path="Chapter_35.html"><a href="Chapter_35.html#randomisation-conclusions"><i class="fa fa-check"></i><b>35.6</b> Randomisation conclusions</a></li>
</ul></li>
<li class="appendix"><span><b>Appendix</b></span></li>
<li class="chapter" data-level="A" data-path="appendexA.html"><a href="appendexA.html"><i class="fa fa-check"></i><b>A</b> Answers to chapter exercises</a>
<ul>
<li class="chapter" data-level="A.1" data-path="appendexA.html"><a href="appendexA.html#chapter-3"><i class="fa fa-check"></i><b>A.1</b> Chapter 3</a>
<ul>
<li class="chapter" data-level="A.1.1" data-path="appendexA.html"><a href="appendexA.html#exercise-3.1"><i class="fa fa-check"></i><b>A.1.1</b> Exercise 3.1:</a></li>
<li class="chapter" data-level="A.1.2" data-path="appendexA.html"><a href="appendexA.html#exercise-3.2"><i class="fa fa-check"></i><b>A.1.2</b> Exercise 3.2</a></li>
<li class="chapter" data-level="A.1.3" data-path="appendexA.html"><a href="appendexA.html#exercise-3.3"><i class="fa fa-check"></i><b>A.1.3</b> Exercise 3.3</a></li>
<li class="chapter" data-level="A.1.4" data-path="appendexA.html"><a href="appendexA.html#exercise-3.4"><i class="fa fa-check"></i><b>A.1.4</b> Exercise 3.4</a></li>
</ul></li>
<li class="chapter" data-level="A.2" data-path="appendexA.html"><a href="appendexA.html#chapter-8"><i class="fa fa-check"></i><b>A.2</b> Chapter 8</a>
<ul>
<li class="chapter" data-level="A.2.1" data-path="appendexA.html"><a href="appendexA.html#exercise-8.1"><i class="fa fa-check"></i><b>A.2.1</b> Exercise 8.1</a></li>
<li class="chapter" data-level="A.2.2" data-path="appendexA.html"><a href="appendexA.html#exercise-8.2"><i class="fa fa-check"></i><b>A.2.2</b> Exercise 8.2</a></li>
<li class="chapter" data-level="A.2.3" data-path="appendexA.html"><a href="appendexA.html#exercise-8.3"><i class="fa fa-check"></i><b>A.2.3</b> Exercise 8.3</a></li>
</ul></li>
<li class="chapter" data-level="A.3" data-path="appendexA.html"><a href="appendexA.html#chapter-14"><i class="fa fa-check"></i><b>A.3</b> Chapter 14</a>
<ul>
<li class="chapter" data-level="A.3.1" data-path="appendexA.html"><a href="appendexA.html#exercise-14.1"><i class="fa fa-check"></i><b>A.3.1</b> Exercise 14.1</a></li>
<li class="chapter" data-level="A.3.2" data-path="appendexA.html"><a href="appendexA.html#exercise-14.2"><i class="fa fa-check"></i><b>A.3.2</b> Exercise 14.2</a></li>
<li class="chapter" data-level="A.3.3" data-path="appendexA.html"><a href="appendexA.html#exercise-14.3"><i class="fa fa-check"></i><b>A.3.3</b> Exercise 14.3</a></li>
<li class="chapter" data-level="A.3.4" data-path="appendexA.html"><a href="appendexA.html#exercise-14.4"><i class="fa fa-check"></i><b>A.3.4</b> Exercise 14.4</a></li>
<li class="chapter" data-level="A.3.5" data-path="appendexA.html"><a href="appendexA.html#exercise-14.5"><i class="fa fa-check"></i><b>A.3.5</b> Exercise 14.5</a></li>
</ul></li>
<li class="chapter" data-level="A.4" data-path="appendexA.html"><a href="appendexA.html#chapter-17"><i class="fa fa-check"></i><b>A.4</b> Chapter 17</a>
<ul>
<li class="chapter" data-level="A.4.1" data-path="appendexA.html"><a href="appendexA.html#exercise-17.1"><i class="fa fa-check"></i><b>A.4.1</b> Exercise 17.1</a></li>
<li class="chapter" data-level="A.4.2" data-path="appendexA.html"><a href="appendexA.html#exercise-17.2"><i class="fa fa-check"></i><b>A.4.2</b> Exercise 17.2</a></li>
<li class="chapter" data-level="A.4.3" data-path="appendexA.html"><a href="appendexA.html#exercise-17.3"><i class="fa fa-check"></i><b>A.4.3</b> Exercise 17.3</a></li>
</ul></li>
<li class="chapter" data-level="A.5" data-path="appendexA.html"><a href="appendexA.html#chapter-20"><i class="fa fa-check"></i><b>A.5</b> Chapter 20</a>
<ul>
<li class="chapter" data-level="A.5.1" data-path="appendexA.html"><a href="appendexA.html#exercise-20.1"><i class="fa fa-check"></i><b>A.5.1</b> Exercise 20.1</a></li>
<li class="chapter" data-level="A.5.2" data-path="appendexA.html"><a href="appendexA.html#exercise-20.2"><i class="fa fa-check"></i><b>A.5.2</b> Exercise 20.2</a></li>
<li class="chapter" data-level="A.5.3" data-path="appendexA.html"><a href="appendexA.html#exercise-20.3"><i class="fa fa-check"></i><b>A.5.3</b> Exercise 20.3</a></li>
<li class="chapter" data-level="A.5.4" data-path="appendexA.html"><a href="appendexA.html#exercise-20.4"><i class="fa fa-check"></i><b>A.5.4</b> Exercise 20.4</a></li>
<li class="chapter" data-level="A.5.5" data-path="appendexA.html"><a href="appendexA.html#exercise-20.5"><i class="fa fa-check"></i><b>A.5.5</b> Exercise 20.5</a></li>
</ul></li>
<li class="chapter" data-level="A.6" data-path="appendexA.html"><a href="appendexA.html#chapter-23"><i class="fa fa-check"></i><b>A.6</b> Chapter 23</a>
<ul>
<li class="chapter" data-level="A.6.1" data-path="appendexA.html"><a href="appendexA.html#exercise-23.1"><i class="fa fa-check"></i><b>A.6.1</b> Exercise 23.1</a></li>
<li class="chapter" data-level="A.6.2" data-path="appendexA.html"><a href="appendexA.html#exercise-23.2"><i class="fa fa-check"></i><b>A.6.2</b> Exercise 23.2</a></li>
<li class="chapter" data-level="A.6.3" data-path="appendexA.html"><a href="appendexA.html#exercise-23.3"><i class="fa fa-check"></i><b>A.6.3</b> Exercise 23.3</a></li>
<li class="chapter" data-level="A.6.4" data-path="appendexA.html"><a href="appendexA.html#exercise-23.4"><i class="fa fa-check"></i><b>A.6.4</b> Exercise 23.4</a></li>
<li class="chapter" data-level="A.6.5" data-path="appendexA.html"><a href="appendexA.html#exercise-23.5"><i class="fa fa-check"></i><b>A.6.5</b> Exercise 23.5</a></li>
</ul></li>
<li class="chapter" data-level="A.7" data-path="appendexA.html"><a href="appendexA.html#chapter-28"><i class="fa fa-check"></i><b>A.7</b> Chapter 28</a>
<ul>
<li class="chapter" data-level="A.7.1" data-path="appendexA.html"><a href="appendexA.html#exercise-28.1"><i class="fa fa-check"></i><b>A.7.1</b> Exercise 28.1</a></li>
<li class="chapter" data-level="A.7.2" data-path="appendexA.html"><a href="appendexA.html#exercise-28.2"><i class="fa fa-check"></i><b>A.7.2</b> Exercise 28.2</a></li>
<li class="chapter" data-level="A.7.3" data-path="appendexA.html"><a href="appendexA.html#exercise-28.3"><i class="fa fa-check"></i><b>A.7.3</b> Exercise 28.3</a></li>
<li class="chapter" data-level="A.7.4" data-path="appendexA.html"><a href="appendexA.html#exercise-28.4"><i class="fa fa-check"></i><b>A.7.4</b> Exercise 28.4</a></li>
</ul></li>
<li class="chapter" data-level="A.8" data-path="appendexA.html"><a href="appendexA.html#chapter-31"><i class="fa fa-check"></i><b>A.8</b> Chapter 31</a>
<ul>
<li class="chapter" data-level="A.8.1" data-path="appendexA.html"><a href="appendexA.html#exercise-31.1"><i class="fa fa-check"></i><b>A.8.1</b> Exercise 31.1</a></li>
<li class="chapter" data-level="A.8.2" data-path="appendexA.html"><a href="appendexA.html#exercise-31.2"><i class="fa fa-check"></i><b>A.8.2</b> Exercise 31.2</a></li>
<li class="chapter" data-level="A.8.3" data-path="appendexA.html"><a href="appendexA.html#exercise-31.3"><i class="fa fa-check"></i><b>A.8.3</b> Exercise 31.3</a></li>
<li class="chapter" data-level="A.8.4" data-path="appendexA.html"><a href="appendexA.html#exercise-31.4"><i class="fa fa-check"></i><b>A.8.4</b> Exercise 31.4</a></li>
<li class="chapter" data-level="A.8.5" data-path="appendexA.html"><a href="appendexA.html#exercise-31.5"><i class="fa fa-check"></i><b>A.8.5</b> Exercise 31.5</a></li>
</ul></li>
<li class="chapter" data-level="A.9" data-path="appendexA.html"><a href="appendexA.html#chapter-34"><i class="fa fa-check"></i><b>A.9</b> Chapter 34</a>
<ul>
<li class="chapter" data-level="A.9.1" data-path="appendexA.html"><a href="appendexA.html#exercise-34.1"><i class="fa fa-check"></i><b>A.9.1</b> Exercise 34.1</a></li>
<li class="chapter" data-level="A.9.2" data-path="appendexA.html"><a href="appendexA.html#exercise-34.2"><i class="fa fa-check"></i><b>A.9.2</b> Exercise 34.2</a></li>
<li class="chapter" data-level="A.9.3" data-path="appendexA.html"><a href="appendexA.html#exercise-34.3"><i class="fa fa-check"></i><b>A.9.3</b> Exercise 34.3</a></li>
<li class="chapter" data-level="A.9.4" data-path="appendexA.html"><a href="appendexA.html#exercise-34.4"><i class="fa fa-check"></i><b>A.9.4</b> Exercise 34.4</a></li>
<li class="chapter" data-level="A.9.5" data-path="appendexA.html"><a href="appendexA.html#exercise-33.5"><i class="fa fa-check"></i><b>A.9.5</b> Exercise 33.5</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="B" data-path="uncertainty_derivation.html"><a href="uncertainty_derivation.html"><i class="fa fa-check"></i><b>B</b> Uncertainty derivation</a>
<ul>
<li class="chapter" data-level="B.1" data-path="uncertainty_derivation.html"><a href="uncertainty_derivation.html#propagation-of-error-for-addition-and-subtraction"><i class="fa fa-check"></i><b>B.1</b> Propagation of error for addition and subtraction</a></li>
<li class="chapter" data-level="B.2" data-path="uncertainty_derivation.html"><a href="uncertainty_derivation.html#propagation-of-error-for-multiplication-and-division"><i class="fa fa-check"></i><b>B.2</b> Propagation of error for multiplication and division</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="references.html"><a href="references.html"><i class="fa fa-check"></i>References</a></li>
<li class="divider"></li>
<li><a href="https://github.com/rstudio/bookdown" target="blank">Published with bookdown</a></li>

</ul>

      </nav>
    </div>

    <div class="book-body">
      <div class="body-inner">
        <div class="book-header" role="navigation">
          <h1>
            <i class="fa fa-circle-o-notch fa-spin"></i><a href="./">Fundamental statistical concepts and techniques in the biological and environmental sciences: With jamovi</a>
          </h1>
        </div>

        <div class="page-wrapper" tabindex="-1" role="main">
          <div class="page-inner">

            <section class="normal" id="section-">
<div id="Chapter_28" class="section level1 hasAnchor" number="28">
<h1><span class="header-section-number">Chapter 28</span> <em>Practical</em>. ANOVA and associated tests<a href="Chapter_28.html#Chapter_28" class="anchor-section" aria-label="Anchor link to header"></a></h1>
<p>This chapter focuses on applying the concepts from Chapters 24-27 in jamovi <span class="citation">(<a href="#ref-Jamovi2022" role="doc-biblioref">The jamovi project, 2024</a>)</span>.
The focus will be on ANOVA and associated tests, with five exercises in total.
The data for this chapter are inspired by the work of Dr Lidia de Sousa Teixeira at the University of Stirling <span class="citation">(<a href="#ref-Teixeira2022" role="doc-biblioref">de Sousa Teixeira, 2022</a>)</span>.
This doctoral work included a nutrient analysis of agricultural soil in different regions of Angola.
Measuring soil nutrient concentrations is essential for assessing soil quality, and these data include measurements of Nitrogen (N), Phosphorus (P), and Potassium (K) concentrations.
Here we will focus on testing whether or not the concentrations of N, P, and K differ among two different sites and three soil profiles (upper, middle, and lower).
This chapter uses the Angola soils dataset<a href="#fn63" class="footnote-ref" id="fnref63"><sup>63</sup></a>.
All concentrations of Nitrogen, Phosphorus, and Potassium are given in parts per million (ppm).</p>
<div id="one-way-anova-site" class="section level2 hasAnchor" number="28.1">
<h2><span class="header-section-number">28.1</span> One-way ANOVA (site)<a href="Chapter_28.html#one-way-anova-site" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>Suppose that we first want to test whether or not mean Nitrogen concentration is the same in different sites.
Notice that there are only two sites in the dataset: Funda and Bailundo.
We could therefore also use an Independent samples t-test.
We will do this first, then compare the t-test output to the ANOVA output.
What are the null (<span class="math inline">\(H_{0}\)</span>) and alternative (<span class="math inline">\(H_{A}\)</span>) hypotheses for the t-test?</p>
<p><span class="math inline">\(H_{0}:\)</span> ____________________</p>
<p><span class="math inline">\(H_{A}:\)</span> ____________________</p>
<p>Before running a t-test, remember that we need to check the assumptions of a t-test to see if any of them are violated (<a href="Chapter_22.html#assumptions-of-t-tests">see Section 22.4</a>).
Use the <strong>Assumption Checks</strong> in jamovi as we did in <a href="Chapter_23.html#independent-samples-t-test-1">Section 23.4</a> to test for normality and homogeneity of variances in Nitrogen concentration.
What can you conclude from these two tests?</p>
<p>Normality conclusion: ___________________________</p>
<p>Homogeneity of variances conclusion: ______________</p>
<p>Given the conclusions from the checks of normality and homogeneity of variances above, what kind of test should you use to see if the mean Nitrogen concentration is significantly different in Funda versus Bailundo?</p>
<p>Test: __________________</p>
<p>Run the test above in jamovi.
What is the p-value of the test, and what conclusion do you make about Nitrogen concentration at the two sites?</p>
<p><span class="math inline">\(P =\)</span> _________________</p>
<p>Conclusion: ____________________</p>
<p>Now we will use an ANOVA to test if the mean Nitrogen concentration differs between sites.
Remember from <a href="Chapter_24.html#Chapter_24">Chapter 24</a> that the ANOVA compares the variance among groups to the variance within groups, calculating an F-statistic and finding where it is in the null F-distribution.
To run an ANOVA, navigate to the ‘Analyses’ tab in jamovi, then select the ‘ANOVA’ button in the toolbar.
From the ANOVA pull-down, select ‘One-Way ANOVA’.
After selecting the one-way ANOVA, a familiar interface will open up.
Place ‘Nitrogen’ in the Dependent Variables box and ‘Site’ in the Grouping Variable box.
Although we have already checked the assumptions of normality and homogeneity of variances when we ran the t-test, check these boxes under <strong>Assumption Checks</strong> too (Figure 28.1).</p>
<div class="figure"><span style="display:block;" id="fig:unnamed-chunk-113"></span>
<img src="img/jamovi_one-way_ANOVA_analysis.png" alt="Jamovi interface is shown for running a one-way ANOVA, with a dependent variable of Nitrogen selected split by site. Check boxes indicate an assumption of equal variances between groups and assumption checks of homogeneity of variances and normality." width="100%" />
<p class="caption">
Figure 28.1: Jamovi interface for running a one-way ANOVA to test if Nitrogen concentration (ppm) differs among sites in the soils of Angola. Data for this test were inspired by the doctoral thesis of Dr Lidia de Sousa Teixeira.
</p>
</div>
<p>Confirm that the Shapiro-Wilk test of normality and Levene’s test of homogeneity of variances are consistent with what you concluded when testing the assumptions of the t-test above.
Since there is no reason to reject the null hypothesis that group variances are equal, we can use Fisher’s One-Way ANOVA by checking ‘Assume equal (Fisher’s)’ under <strong>Variances</strong>.
A table called ‘One-Way ANOVA’ will appear in the panel on the right.
Write down the test statistic (<span class="math inline">\(F\)</span>), degrees of freedom, and p-value from this table below.</p>
<p><span class="math inline">\(F =\)</span> _______________</p>
<p><span class="math inline">\(df1 =\)</span> _______________</p>
<p><span class="math inline">\(df2 =\)</span> _______________</p>
<p><span class="math inline">\(P =\)</span> _______________</p>
<p>Remember from <a href="Chapter_24.html#Chapter_24">Chapter 24</a> that the ANOVA calculates an F-statistic (mean variance among groups divided by mean variance withing groups).
This F-statistic is then compared to the null F-distribution with the correct degrees of freedom to calculate the p-value.
You can use the interactive app to visualise this from the above jamovi output<a href="#fn64" class="footnote-ref" id="fnref64"><sup>64</sup></a>.
To do this, move the ‘Variance 1’ slider in the app until it is approximately equal to <span class="math inline">\(F\)</span>, then change <span class="math inline">\(df1\)</span> and <span class="math inline">\(df2\)</span> to the values above.
From the interactive app, what is the approximate area under the curve (i.e., orange area) where the <span class="math inline">\(F\)</span> value on the x-axis is greater than your calculated <span class="math inline">\(F\)</span>?</p>
<p><span class="math inline">\(P =\)</span> _________________</p>
<p>Slide the ‘Variance 1’ to the left now until you find the <span class="math inline">\(F\)</span> value where the probability density in the tail (orange) is about <span class="math inline">\(P = 0.05\)</span>.
Approximately, what is this threshold <span class="math inline">\(F\)</span> value above which we will reject the null hypothesis?</p>
<p>Approximate threshold <span class="math inline">\(F\)</span>: __________________</p>
<p>What should you conclude regarding the null hypothesis that sites have the same mean?</p>
<p>Conclusion: _________________</p>
<p>Look again at the p-value from the one-way ANOVA output and the Student’s t-test output.
Are the two values the same, or different?
Why might this be?</p>
<pre><code>


</code></pre>
<p>Next, we will run a one-way ANOVA to test the null hypothesis that all profiles have the same mean Nitrogen concentration.</p>
</div>
<div id="one-way-anova-profile" class="section level2 hasAnchor" number="28.2">
<h2><span class="header-section-number">28.2</span> One-way ANOVA (profile)<a href="Chapter_28.html#one-way-anova-profile" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>We will now run an ANOVA to see if Nitrogen concentration differs among profiles.
In this dataset, there are lower, middle, and upper profiles, which refer to the location along a slope from which soil samples were obtained.
Using the same approach as the previous Exercise 28.1, we will run a one-way ANOVA with profile as the independent variable instead of site.
Again, navigate to the ‘Analyses’ tab in jamovi, then select the ‘ANOVA’ button in the toolbar.
From the ANOVA pull-down menu, select ‘One-Way ANOVA’.
First check the assumptions of normality and homogeneity of variances.
What can you conclude?</p>
<p>Normality conclusion: ___________________________</p>
<p>Homogeneity of variances conclusion: ______________</p>
<p>It appears that the assumptions of normality and homogeneity of variances are met.
We can therefore proceed with the one-way ANOVA.
Run the one-way ANOVA with the assumption of equal variances (i.e., Fisher’s test).
What are the output statistics in the One-Way ANOVA table?</p>
<p><span class="math inline">\(F =\)</span> _______________</p>
<p><span class="math inline">\(df1 =\)</span> _______________</p>
<p><span class="math inline">\(df2 =\)</span> _______________</p>
<p><span class="math inline">\(P =\)</span> _______________</p>
<p>From these statistics, what do you conclude about the difference in Nitrogen concentration among profiles?</p>
<p>Conclusion: _____________________</p>
<p>In the previous Exercise 28.1, we used an <a href="https://bradduthie.github.io/stats/app/f_distribution/">interactive app</a> to visualise the relationship between the F-statistic and the p-value.
We can do the same thing with the distrACTION module in jamovi.
To do this, go to the distrACTION option in the jamovi toolbar and select ‘F-Distribution’ from the pull-down menu.
Place the <span class="math inline">\(df1\)</span> and <span class="math inline">\(df2\)</span> from the One-Way ANOVA table into the df1 and df2 boxes under <strong>Parameters</strong> (ignore <span class="math inline">\(\lambda\)</span>).
Under <strong>Function</strong>, select ‘Compute probability’, then place the <span class="math inline">\(F\)</span> value from the One-Way ANOVA table in the box for x1.
We want the upper tail of the <span class="math inline">\(F\)</span> probability distribution, so choose <span class="math inline">\(P(X \geq x1)\)</span> from the radio buttons below.
Write down the ‘Probability’ value from the Results table in the panel to the right.</p>
<p>Probability: _________________</p>
<p>Note that this is the same value (perhaps with a rounding error) as the p-value from the One-Way ANOVA table above.
We can also find the threshold value of <span class="math inline">\(F\)</span>, above which we will reject the null hypothesis.
To do this, check the ‘Compute quantile’ box and set p = 0.95 in the box below.
From the Results table, what is the critical <span class="math inline">\(F\)</span> value (‘Quantile’), above which we would reject the null hypothesis that all groups have the same mean?</p>
<p>Critical <span class="math inline">\(F\)</span> value: ________________</p>
<p>Note that the objective of working this out in the distrACTION module (and with the interactive app) is to help explain what these different values in the One-Way ANOVA table actually mean.
To actually test the null hypothesis, the One-Way ANOVA output table is all that we really need.</p>
<p>Finally, note that in the ANOVA pull-down menu from the jamovi toolbar, the option ‘ANOVA’ is just below the ‘One-way ANOVA’ that we used in this exercise and Exercise 28.1.
This is just a more general tool for running an ANOVA, which includes the two-way ANOVA that we will use in Exercise 28.5 below.
For now, give this a try by selecting ‘ANOVA’ from the pull-down menu.
In the ANOVA interface, place ‘Nitrogen’ into the ‘Dependent Variable’ box and ‘Profile’ in the ‘Fixed Factors’ box (Figure 28.2).</p>
<div class="figure"><span style="display:block;" id="fig:unnamed-chunk-114"></span>
<img src="img/jamovi_ANOVA_input.png" alt="Jamovi interface is shown for running an ANOVA, with a dependent variable of Nitrogen and profile as a fixed factor." width="100%" />
<p class="caption">
Figure 28.2: Jamovi interface for running an ANOVA to test if Nitrogen concentration (ppm) differs among soil profiles in Angola. Data for this test were inspired by the doctoral thesis of Dr Lidia de Sousa Teixeira.
</p>
</div>
<p>The output in the right panel shows an ANOVA table.
It includes the sum of squares of the among-group (Profile) and within-group (Residuals) sum of squares and mean square.
This is often how ANOVA results are presented in the literature.
Fill in the table below (Table 28.1) with the information for degrees of freedom, <span class="math inline">\(F\)</span>, and <span class="math inline">\(p\)</span>.</p>
<table style="width:83%;">
<caption><strong>TABLE 28.1</strong> ANOVA output testing the null hypothesis that mean Nitrogen concentration is the same across three different soil profiles in Angola. Data for this test were inspired by the doctoral thesis of Dr Lidia de Sousa Teixeira.</caption>
<colgroup>
<col width="22%" />
<col width="23%" />
<col width="6%" />
<col width="19%" />
<col width="5%" />
<col width="5%" />
</colgroup>
<thead>
<tr class="header">
<th align="center"> </th>
<th align="center">Sum of Squares</th>
<th align="center">df</th>
<th align="center">Mean Square</th>
<th align="center">F</th>
<th align="center">p</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td align="center"><strong>Profile</strong></td>
<td align="center">16888.18606</td>
<td align="center"></td>
<td align="center">8444.09303</td>
<td align="center"></td>
<td align="center"></td>
</tr>
<tr class="even">
<td align="center"><strong>Residuals</strong></td>
<td align="center">118092.02927</td>
<td align="center"></td>
<td align="center">2460.25061</td>
<td align="center"></td>
<td align="center"></td>
</tr>
</tbody>
</table>
<p>Now that we have established from the one-way ANOVA that mean Nitrogen concentration is not the same across all soil profiles, we can use a test of multiple comparisons to test which profile(s) are significantly different from one another.</p>
</div>
<div id="multiple-comparisons" class="section level2 hasAnchor" number="28.3">
<h2><span class="header-section-number">28.3</span> Multiple comparisons<a href="Chapter_28.html#multiple-comparisons" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>In this exercise, we will pick up where we left off in the ANOVA of Exercise 28.2.
We have established that not all soil profiles have the same mean.
Next, we will run a post hoc multiple comparisons test to evaluate which, if any, soil profiles have different means.
In the ANOVA input panel, scroll down to the pull-down option called ‘Post Hoc Tests’.
Move ‘Profile’ to the box to the right, then select the ‘Tukey’ checkbox under <strong>Correction</strong>.
Doing this will run Tukey’s honestly significant difference (HSD) test introduced in <a href="Chapter_25.html#Chapter_25">Chapter 25</a>.
The output will appear in the panel on the right in a table called ‘Post Hoc Tests’.
Note that these post hoc tests use the t-distribution to test for significance.
Find the p-values associated with Tukey’s HSD (<span class="math inline">\(p_{\mathrm{tukey}}\)</span>) for each profile pairing.
Report these below.</p>
<p>Tukey’s HSD Lower - Middle: <span class="math inline">\(P =\)</span> _____________</p>
<p>Tukey’s HSD Lower - Upper: <span class="math inline">\(P =\)</span> _____________</p>
<p>Tukey’s HSD Middle - Upper: <span class="math inline">\(P =\)</span> _____________</p>
<p>From this output, what can we conclude about the difference among soil profiles?</p>
<pre><code>


</code></pre>
<p>Next, instead of running Tukey’s HSD test, we will use a series of t-tests with a Bonferroni correction.
Check the box for ‘Bonferroni’ in the ANOVA Post Hoc Tests input panel, then find the p-values for the Bonferroni correction (<span class="math inline">\(p_{\mathrm{bonferroni}}\)</span>).
Note that we do not need to change the <span class="math inline">\(\alpha\)</span> threshold ourselves (i.e., we do not need to see if <span class="math inline">\(P\)</span> is less than <span class="math inline">\(\alpha = 0.05/3 = 0.016667\)</span> instead of <span class="math inline">\(\alpha = 0.05\)</span>).
Jamovi modifies the p-values appropriately for the Bonferroni correction (we can see the difference by clicking the checkbox for ‘No correction’ in the Post Hoc Tests input panel).
Report the p-values for the Bonferroni correction below.</p>
<p>Bonferroni Lower - Middle: <span class="math inline">\(P =\)</span> _____________</p>
<p>Bonferroni Lower - Upper: <span class="math inline">\(P =\)</span> _____________</p>
<p>Bonferroni Middle - Upper: <span class="math inline">\(P =\)</span> _____________</p>
<p>In general, how are the p-values different between Tukey’s HSD and the Bonferroni correction?
Are they about the same, higher, or lower?</p>
<pre><code>
</code></pre>
<p>What does this difference mean in terms of making a Type I error?
In other words, based on this output, are we more likely to make a Type I error with Tukey’s HSD test or the Bonferroni test?</p>
<pre><code>

</code></pre>
<p>Note that we ran Tukey’s HSD test and the Bonferroni test separately.
This is because, when doing a post hoc test, we should choose which test to use in advance.
This will avoid biasing our results to get the conclusion that we <em>want</em> rather than the conclusion that is <em>accurate</em>.
If, for example, we first decided to use a Bonferroni correction, but then found that none of our p-values were below 0.05, it would not be okay to try a Tukey’s HSD test instead in hopes of changing this result.
This kind of practice is colloquially called ‘p-hacking’ (or ‘data dredging’), and it causes an elevated risk of Type I error and a potential for bias in scientific results.
Put more simply, trying to game the system to get results in which <span class="math inline">\(P &lt; 0.05\)</span> can lead to mistakes in science <span class="citation">(<a href="#ref-Head2015" role="doc-biblioref">Head et al., 2015</a>)</span>.
Specifically, p-hacking can lead us to believe that there are patterns in nature where none really exist, which is definitely something that we want to avoid!</p>
</div>
<div id="kruskal-wallis-h-test" class="section level2 hasAnchor" number="28.4">
<h2><span class="header-section-number">28.4</span> Kruskal-Wallis H test<a href="Chapter_28.html#kruskal-wallis-h-test" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>In this exercise, we will apply the non-parametric equivalent of the one-way ANOVA: the Kruskal-Wallis H test.
Suppose that we now want to know if Potassium concentration differs among soil profiles.
We therefore want to test the null hypothesis that the mean Potassium concentration is the same for all soil profiles.
Before opening the ANOVA input panel, have a look at a histogram of Potassium concentration.
How would you describe the distribution?
Do the data appear to be normally distributed?</p>
<pre><code>

</code></pre>
<p>We can test the assumption of normality using a Shapiro-Wilk test.
This can be done in the Descriptives panel of jamovi, or we can do it in the One-Way ANOVA panel.
To do it in the one-way ANOVA panel, first select ‘ANOVA’ from the pull-down menu as we did at the end of Exercise 28.2.
In the ANOVA interface, place ‘Potassium’ into the ‘Dependent Variable’ box and ‘Profile’ in the ‘Fixed Factors’ box.
Next, scroll down to the ‘Assumption Checks’ pull-down menu and select all three options.
From Levene’s test, the Shapiro-Wilk test, and the Q-Q plot, what assumptions of ANOVA might be violated?</p>
<pre><code>


</code></pre>
<p>Given the violation of ANOVA assumptions, we should consider a non-parametric option.
As introduced in <a href="Chapter_26.html#Chapter_26">Chapter 26</a>, the Kruskal-Wallis H test is a non-parametric alternative to a one-way ANOVA.
Like other non-parametric tests introduced in this book, the Kruskal-Wallis H test uses the ranks of a dataset instead of the actual values.
To run a Kruskal-Wallis H test, select the Analyses tab, then the ‘ANOVA’ button from the jamovi toolbar.
In the pull-down ANOVA menu, choose ‘One-Way ANOVA: Kruskal-Wallis’.</p>
<p>The Kruskal-Wallis input is basically the same as the one-way ANOVA input.
We just need to put ‘Potassium’ in the dependent variable list and ‘Profile’ in the Grouping Variable box.
The output table includes the test statistic (jamovi uses a <span class="math inline">\(\chi^{2}\)</span> value as a test statistic, which I will introduce in <a href="Chapter_29.html#Chapter_29">Chapter 29</a>), degrees of freedom, and p-value.
Report these values below.</p>
<p><span class="math inline">\(\chi^{2} =\)</span> _____________</p>
<p><span class="math inline">\(df =\)</span> _____________</p>
<p><span class="math inline">\(P =\)</span> ____________</p>
<p>From the above output, should we reject or not reject our null hypothesis?</p>
<p><span class="math inline">\(H_{0}:\)</span> _____________________</p>
<p>Note that the Kruskal-Wallis test in jamovi also includes a type of multiple comparisons test (DSCF pairwise comparisons checkbox).
We will not use the Dwass-Steel-Critchlow-Fligner pairwise comparisons, but the general idea is the same as Tukey’s HSD test for post hoc multiple comparisons in the ANOVA.</p>
</div>
<div id="two-way-anova" class="section level2 hasAnchor" number="28.5">
<h2><span class="header-section-number">28.5</span> Two-way ANOVA<a href="Chapter_28.html#two-way-anova" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>Since we have two types of categorical variables (site and profile), we might want to know if either has a significant effect on the concentration of an element, and if there is any interaction between site and profile.
The two-way ANOVA was introduced in <a href="Chapter_27.html#Chapter_27">Chapter 27</a> with an example of fig wasp wing lengths.
Here we will test the effects of site, profile, and their interaction on Nitrogen concentration.
Recall from <a href="Chapter_27.html#Chapter_27">Chapter 27</a> that a two-way ANOVA actually tests three separate null hypotheses.
Write these null hypotheses down below (the order does not matter).</p>
<p>First <span class="math inline">\(H_{0}\)</span>: ___________________________________</p>
<p>Second <span class="math inline">\(H_{0}\)</span>: ___________________________________</p>
<p>Third <span class="math inline">\(H_{0}\)</span>: ___________________________________</p>
<p>To test these null hypotheses again, select ‘ANOVA’ from the pull-down menu as we did at the end of Exercise 28.2.
In the ANOVA interface, place ‘Nitrogen’ into the ‘Dependent Variable’ box and both ‘Site’ and ‘Profile’ in the ‘Fixed Factors’ box.
Next, scroll down to the ‘Assumption Checks’ pull-down menu and select all three options.
From the assumption checks output tables, is there any reason to be concerned about using the two-way ANOVA?</p>
<pre><code>
</code></pre>
<p>In the two-way ANOVA output, we see the same ANOVA table as in Exercise 28.2 (Table 28.1).
This time, however, there are four rows in total.
The first two rows correspond with tests of the main effects of Site and Profile, and the third row tests the interaction between these two variables.
Fill in Table 28.2 with the relevant information from the two-way ANOVA output.</p>
<table style="width:90%;">
<caption><strong>TABLE 28.2</strong> Two-way ANOVA output testing the effects of two sites and three different soil profiles on soil Nitrogen concentration in Angola. Data for this test were inspired by the doctoral thesis of Dr Lidia de Sousa Teixeira.</caption>
<colgroup>
<col width="29%" />
<col width="23%" />
<col width="6%" />
<col width="19%" />
<col width="5%" />
<col width="5%" />
</colgroup>
<thead>
<tr class="header">
<th align="center"> </th>
<th align="center">Sum of Squares</th>
<th align="center">df</th>
<th align="center">Mean Square</th>
<th align="center">F</th>
<th align="center">p</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td align="center"><strong>Site</strong></td>
<td align="center">21522.18384</td>
<td align="center"></td>
<td align="center">21522.18384</td>
<td align="center"></td>
<td align="center"></td>
</tr>
<tr class="even">
<td align="center"><strong>Profile</strong></td>
<td align="center">22811.1368</td>
<td align="center"></td>
<td align="center">11405.5684</td>
<td align="center"></td>
<td align="center"></td>
</tr>
<tr class="odd">
<td align="center"><strong>Site * Profile</strong></td>
<td align="center">16209.13035</td>
<td align="center"></td>
<td align="center">8104.56517</td>
<td align="center"></td>
<td align="center"></td>
</tr>
<tr class="even">
<td align="center"><strong>Residuals</strong></td>
<td align="center">80497.68348</td>
<td align="center"></td>
<td align="center">1788.83741</td>
<td align="center"></td>
<td align="center"></td>
</tr>
</tbody>
</table>
<p>From this output table, should you reject or not reject your null hypotheses?</p>
<p>First <span class="math inline">\(H_{0}\)</span>: ___________________________________</p>
<p>Second <span class="math inline">\(H_{0}\)</span>: ___________________________________</p>
<p>Third <span class="math inline">\(H_{0}\)</span>: ___________________________________</p>
<p>In non-technical language, what should you conclude from this two-way ANOVA?</p>
<pre><code>


</code></pre>
<p>Lastly, we can look at the interaction effect between Site and Profile visually.
To do this, scroll down to the ‘Estimated Marginal Means’ pull-down option.
Move ‘Site’ and ‘Profile’ from the box on the left to the ‘Marginal Means’ box on the right (Figure 28.3).</p>
<div class="figure"><span style="display:block;" id="fig:unnamed-chunk-117"></span>
<img src="img/jamovi_marginal_means.png" alt="Jamovi interface is shown with an ANOVA test being run and a pull-down menu for Estimated Marginal Means." width="100%" />
<p class="caption">
Figure 28.3: Jamovi two-way ANOVA test with the pull-down menu for Estimated Marginal Means, which will produce a plot showing the interaction effect of the two-way ANOVA.
</p>
</div>
<p>In the panel on the right-hand side, a plot will appear under the heading ‘Estimated Marginal Means’.
Based on what you learnt in <a href="Chapter_27.html#Chapter_27">Chapter 27</a> about interaction effects, what can you say about the interaction between Site and Profile?
Does one Profile, in particular, appear to be causing the interaction to be significant?
How can you infer this from the Estimated Marginal Means plot?</p>
<pre><code>


</code></pre>
<p>Try running a two-way ANOVA to test the effects of Site and Profile on Phosphorus concentration.
Based on the ANOVA output, what can you conclude?</p>
<pre><code>


</code></pre>

</div>
</div>
<h3>References<a href="references.html#references" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<div id="refs" class="references csl-bib-body hanging-indent" line-spacing="2">
<div id="ref-Teixeira2022" class="csl-entry">
de Sousa Teixeira, L. P. (2022). <em>Geochemical, textural and micromorphological properties of <span>A</span>ngolan agroecosystem soils in relation to region, landscape position and land management</em> [PhD thesis]. University of Stirling, Stirling, UK.
</div>
<div id="ref-Head2015" class="csl-entry">
Head, M. L., Holman, L., Lanfear, R., Kahn, A. T., &amp; Jennions, M. D. (2015). <span class="nocase">The extent and consequences of p-hacking in science</span>. <em>PLoS Biology</em>, <em>13</em>(3), e1002106. <a href="https://doi.org/10.1371/journal.pbio.1002106">https://doi.org/10.1371/journal.pbio.1002106</a>
</div>
<div id="ref-Jamovi2022" class="csl-entry">
The jamovi project. (2024). <em>Jamovi (version 2.5)</em>. <a href="https://www.jamovi.org">https://www.jamovi.org</a>
</div>
</div>
<div class="footnotes">
<hr />
<ol start="63">
<li id="fn63"><p><a href="https://bradduthie.github.io/stats/data/Angola_soils.csv">https://bradduthie.github.io/stats/data/Angola_soils.csv</a><a href="Chapter_28.html#fnref63" class="footnote-back">↩︎</a></p></li>
<li id="fn64"><p><a href="https://bradduthie.github.io/stats/app/f_distribution/" class="uri">https://bradduthie.github.io/stats/app/f_distribution/</a><a href="Chapter_28.html#fnref64" class="footnote-back">↩︎</a></p></li>
</ol>
</div>
            </section>

          </div>
        </div>
      </div>
<a href="Chapter_27.html" class="navigation navigation-prev " aria-label="Previous page"><i class="fa fa-angle-left"></i></a>
<a href="Chapter_29.html" class="navigation navigation-next " aria-label="Next page"><i class="fa fa-angle-right"></i></a>
    </div>
  </div>
<script src="libs/gitbook-2.6.7/js/app.min.js"></script>
<script src="libs/gitbook-2.6.7/js/clipboard.min.js"></script>
<script src="libs/gitbook-2.6.7/js/plugin-search.js"></script>
<script src="libs/gitbook-2.6.7/js/plugin-sharing.js"></script>
<script src="libs/gitbook-2.6.7/js/plugin-fontsettings.js"></script>
<script src="libs/gitbook-2.6.7/js/plugin-bookdown.js"></script>
<script src="libs/gitbook-2.6.7/js/jquery.highlight.js"></script>
<script src="libs/gitbook-2.6.7/js/plugin-clipboard.js"></script>
<script>
gitbook.require(["gitbook"], function(gitbook) {
gitbook.start({
"sharing": {
"github": false,
"facebook": true,
"twitter": true,
"linkedin": false,
"weibo": false,
"instapaper": false,
"vk": false,
"whatsapp": false,
"all": ["facebook", "twitter", "linkedin", "weibo", "instapaper"]
},
"fontsettings": {
"theme": "white",
"family": "sans",
"size": 2
},
"edit": {
"link": "https://github.com/rstudio/bookdown-demo/edit/master/07-ANOVA.Rmd",
"text": "Edit"
},
"history": {
"link": null,
"text": null
},
"view": {
"link": null,
"text": null
},
"download": null,
"search": {
"engine": "fuse",
"options": null
},
"toc": {
"collapse": "subsection"
}
});
});
</script>

<!-- dynamically load mathjax for compatibility with self-contained -->
<script>
  (function () {
    var script = document.createElement("script");
    script.type = "text/javascript";
    var src = "true";
    if (src === "" || src === "true") src = "https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.9/latest.js?config=TeX-MML-AM_CHTML";
    if (location.protocol !== "file:")
      if (/^https?:/.test(src))
        src = src.replace(/^https?:/, '');
    script.src = src;
    document.getElementsByTagName("head")[0].appendChild(script);
  })();
</script>
</body>

</html>