-
Notifications
You must be signed in to change notification settings - Fork 1
/
Copy pathChapter_29.html
993 lines (955 loc) · 87.1 KB
/
Chapter_29.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
<!DOCTYPE html>
<html lang="" xml:lang="">
<head>
<meta charset="utf-8" />
<meta http-equiv="X-UA-Compatible" content="IE=edge" />
<title>Chapter 29 Frequency and count data | 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 29 Frequency and count data | 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 29 Frequency and count data | 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="2024-08-06" />
<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_28.html"/>
<link rel="next" href="Chapter_30.html"/>
<script src="libs/jquery-3.6.0/jquery-3.6.0.min.js"></script>
<script src="https://cdn.jsdelivr.net/npm/[email protected]/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_29" class="section level1 hasAnchor" number="29">
<h1><span class="header-section-number">Chapter 29</span> Frequency and count data<a href="Chapter_29.html#Chapter_29" class="anchor-section" aria-label="Anchor link to header"></a></h1>
<p>In this book, we have introduced hypothesis testing as a tool to determine if variables were sampled from a population with a specific mean (one sample t-test in <a href="Chapter_22.html#one-sample-t-test">Section 22.1</a>), or if different groups of variables were sampled from a population with the same mean (the independent samples t-test in <a href="Chapter_22.html#independent-samples-t-test">Section 22.2</a> and ANOVA in <a href="Chapter_24.html#Chapter_24">Chapter 24</a>).
In these tests, the variables for which we calculated the means were always continuous (e.g., fig wasp wing lengths, nitrogen concentration in parts per million).
That is, the dependent variables of the t-test and ANOVA could always, at least in theory, take any real value (i.e., any decimal).
And the comparison was always between the means of categorical groups (e.g., fig wasp species or study sites).
But not every variable that we measure will be continuous.
For example, in <a href="Chapter_5.html#Chapter_5">Chapter 5</a>, we also introduced discrete variables, which can only take discrete counts (1, 2, 3, 4, and so forth).
Examples of such <strong>count data</strong> might include the number of species of birds in a forest or the number of days in the year for which an extreme temperature is recorded.
<a href="Chapter_15.html#Chapter_15">Chapter 15</a> included some examples of count data when introducing probability distributions (e.g., counts of heads or tails in coin flips, or the number of people testing positive for COVID-19).
Count data are discrete because they can only take integer values.
For example, there cannot be 14.24 bird species in a forest; it needs to be a whole number.</p>
<p>In the biological and environmental sciences, we often want to test whether or not observed counts are significantly different from some expectation.
For example, we might hypothesise that the probability of flowers being red versus blue in a particular field is the same.
In other words, <span class="math inline">\(P(flower = red) = 0.5\)</span> and <span class="math inline">\(P(flower = Blue) = 0.5\)</span>.
By this logic, if we were to collect 100 flowers at random from the field, we would expect 50 to be red and 50 to be blue.
If we actually went out and collected 100 flowers at random, but found 46 to be red and 54 to be blue, would this be sufficiently different from our expectation to reject the null hypothesis that the probability of sampling a red versus blue flower is the same?
We could test this null hypothesis using a Chi-square goodness of fit test (<a href="#the-chi-square-distribution">Section 29.1</a>).
Similarly, we might want to test if two different count variables (e.g., flower colour and flower species) are associated with one another (e.g., if blue flowers are more common in one species than another species).
We could test this kind of hypothesis using a Chi-square test of association.</p>
<p>Before introducing the Chi-square goodness of fit test or the Chi-square test of association, it makes sense to first introduce the Chi-square (<span class="math inline">\(\chi^{2}\)</span>) distribution.
The general motivation for introducing the Chi-square distribution is the same as it was for the t-distribution (<a href="Chapter_19.html#Chapter_19">Chapter 19</a>) or F-distribution (<a href="#the-f-distribution">Section 24.1</a>).
We need some probability density distribution that is our null distribution, which is what we predict if our null hypothesis is true.
We then compare this null distribution to our test statistic to find the probability of sampling a test statistic as or more extreme if the null hypothesis is really true (i.e., a p-value).</p>
<div id="chi-square-distribution" class="section level2 hasAnchor" number="29.1">
<h2><span class="header-section-number">29.1</span> Chi-square distribution<a href="Chapter_29.html#chi-square-distribution" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>The Chi-square (<span class="math inline">\(\chi^{2}\)</span>) distribution is a continuous distribution in which values of <span class="math inline">\(\chi^{2}\)</span> can be any real number greater than or equal to 0. We can generate a <span class="math inline">\(\chi^{2}\)</span> distribution by adding up squared values that are sampled from a standard normal distribution <span class="citation">(<a href="#ref-Sokal1995" role="doc-biblioref">Sokal & Rohlf, 1995</a>)</span>, hence the ‘square’ in ‘Chi-square’.
There is a lot to unpack in the previous sentence, so we can go through it step by step.
First, we can take another look at the standard normal distribution from <a href="Chapter_15.html#normal-distribution">Section 15.4.4</a> (Figure 15.9).
Suppose that we randomly sampled four values from the standard normal distribution.</p>
<ul>
<li><span class="math inline">\(x_{1} = -1.244\)</span></li>
<li><span class="math inline">\(x_{2} = 0.162\)</span></li>
<li><span class="math inline">\(x_{3} = -2.214\)</span></li>
<li><span class="math inline">\(x_{4} = 2.071\)</span></li>
</ul>
<p>We can square all of these values, then add up the squares,</p>
<p><span class="math display">\[\chi^{2} = (-1.244)^{2} + (0.162)^{2} + (-2.214)^{2} + (2.071)^{2}.\]</span></p>
<p>Note that <span class="math inline">\(\chi^{2}\)</span> cannot be negative because when we square a number that is either positive or negative, we always end up with a positive value (e.g., <span class="math inline">\(-2^{2} = 4\)</span>, see <a href="Chapter_1.html#numbers-and-operations">Section 1.1</a>).
The final value is <span class="math inline">\(\chi^{2} = 10.76462\)</span>.
Of course, this <span class="math inline">\(\chi^{2}\)</span> value would have been different if our <span class="math inline">\(x_{i}\)</span> values (<span class="math inline">\(x_{1}\)</span>, <span class="math inline">\(x_{2}\)</span>, <span class="math inline">\(x_{3}\)</span>, and <span class="math inline">\(x_{4}\)</span>) had been different.
And if we are sampling randomly from the normal distribution, we should not expect to get the same <span class="math inline">\(\chi^{2}\)</span> value from four new random standard normal deviates.
We can therefore ask: if we were to keep sampling four standard normal deviates and calculating new <span class="math inline">\(\chi^{2}\)</span> values, what would be the distribution of these <span class="math inline">\(\chi^{2}\)</span> values?
The answer is shown in Figure 29.1.</p>
<div class="figure"><span style="display:block;" id="fig:unnamed-chunk-118"></span>
<img src="bookdown-demo_files/figure-html/unnamed-chunk-118-1.png" alt="Plot of a curved distribution that rapidly increases to a maximum at around 2 then slowly decreases." width="672" />
<p class="caption">
Figure 29.1: A Chi-square distribution that is the expected sum of four squared standard normal deviates, i.e., the sum of four values sampled from a standard normal distribution and squared.
</p>
</div>
<p>Looking at the shape of Figure 29.1, we can see that most of the time, the sum of deviations from the mean of <span class="math inline">\(\mu = 0\)</span> will be about 2.
But sometimes we will get a much lower or higher value of <span class="math inline">\(\chi^{2}\)</span> by chance, if we sample particularly low or high values of <span class="math inline">\(x_{i}\)</span>.</p>
<p>If we summed a different number of squared <span class="math inline">\(x_{i}\)</span> values, then we would expect the distribution of <span class="math inline">\(\chi^{2}\)</span> to change.
Had we sampled fewer than four <span class="math inline">\(x_{i}\)</span> values, the expected <span class="math inline">\(\chi^{2}\)</span> would be lower just because we are adding up fewer numbers.
Similarly, had we sampled more than four <span class="math inline">\(x_{i}\)</span> values, the expected <span class="math inline">\(\chi^{2}\)</span> would be higher just because we are adding up more numbers.
The shape of the <span class="math inline">\(\chi^{2}\)</span> distribution<a href="#fn65" class="footnote-ref" id="fnref65"><sup>65</sup></a> is therefore determined by the number of values sampled (<span class="math inline">\(n\)</span>), or more specifically the degrees of freedom (<span class="math inline">\(df\)</span>, or sometimes <span class="math inline">\(v\)</span>), which in a sample is <span class="math inline">\(df = n - 1\)</span>.
This is the same idea as the t-distribution from <a href="Chapter_19.html#Chapter_19">Chapter 19</a>, which also changed shape depending on the degrees of freedom.
Figure 29.2 shows the different <span class="math inline">\(\chi^{2}\)</span> probability density distributions for different degrees of freedom.</p>
<div class="figure"><span style="display:block;" id="fig:unnamed-chunk-119"></span>
<img src="bookdown-demo_files/figure-html/unnamed-chunk-119-1.png" alt="A plot is shown with three different curve lines, which show three different Chi-square distributions with different degrees of freedom." width="672" />
<p class="caption">
Figure 29.2: Probability density functions for three different Chi-square distributions, each of which has different degrees of freedom (<span class="math inline">\(df\)</span>).
</p>
</div>
<p>As with the F-distribution from <a href="#the-f-distribution">Section 24.1</a>, visualising the <span class="math inline">\(\chi^{2}\)</span> distribution is easier using an interactive application<a href="#fn66" class="footnote-ref" id="fnref66"><sup>66</sup></a>.
And as with the F-distribution, it is not necessary to memorise how the <span class="math inline">\(\chi^{2}\)</span> distribution changes with different degrees of freedom.
The important point is that the distribution changes with different degrees of freedom, and we can map probabilities to the <span class="math inline">\(\chi^{2}\)</span> value on the x-axis in the same way as any other distribution.</p>
<p>What does any of this have to do with count data? It actually is a bit messy.
The <span class="math inline">\(\chi^{2}\)</span> distribution is not a perfect tool for comparing observed and expected counts <span class="citation">(<a href="#ref-Sokal1995" role="doc-biblioref">Sokal & Rohlf, 1995</a>)</span>.
After all, counts are integer values, and the <span class="math inline">\(\chi^{2}\)</span> distribution is clearly continuous (unlike, e.g., the binomial or Poisson distributions from <a href="##probability-distributions">Section 15.4</a>).
The <span class="math inline">\(\chi^{2}\)</span> distribution is in fact a useful approximation for testing counts, and one that becomes less accurate when sample size <span class="citation">(<a href="#ref-Slakter1968" role="doc-biblioref">Slakter, 1968</a>)</span> or expected count size <span class="citation">(<a href="#ref-Tate1973" role="doc-biblioref">Tate & Hyer, 1973</a>)</span> is small.
Nevertheless, we can use the <span class="math inline">\(\chi^{2}\)</span> distribution as a tool for testing whether observed counts are significantly different from expected counts.
The first test that we will look at is the goodness of fit test.</p>
</div>
<div id="chi-square-goodness-of-fit" class="section level2 hasAnchor" number="29.2">
<h2><span class="header-section-number">29.2</span> Chi-square goodness of fit<a href="Chapter_29.html#chi-square-goodness-of-fit" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>The first kind of test that we will consider for count data is the goodness of fit test.
In this test, we have some number of counts that we expect to observe (e.g., expected counts of red versus blue flowers), then compare this expectation to the counts that we actually observe.
If the expected and observed counts differ by a lot, then we will get a large test statistic and reject the null hypothesis.
A simple concrete example will make this a bit more clear.</p>
<p>Recall the practical in <a href="Chapter_17.html#Chapter_17">Chapter 17</a>, in which players of the mobile app game ‘Power Up!’ chose a small, medium, or large dam at the start of the game.
Suppose that we are interested in the size of dam that policy-makers choose to build when playing the game, so we find 60 such people in Scotland and ask them to play the game.
Perhaps we do not think that the policy-makers will have any preference for a particular dam size (and therefore just pick one of the three dam sizes at random).
We would therefore expect an equal number of small, medium, and large dams to be selected among the 60 players.
That is, for our expected counts of each dam size (<span class="math inline">\(E_{\mathrm{size}}\)</span>), we expect 20 small (<span class="math inline">\(E_{\mathrm{small}} = 20\)</span>), 20 medium (<span class="math inline">\(E_{\mathrm{medium}} = 20\)</span>), and 20 large (<span class="math inline">\(E_{\mathrm{large}} = 20\)</span>) dams in total (because <span class="math inline">\(60/3 = 20\)</span>).</p>
<p>Of course, even if our players have no preference for a particular dam size, the number of small, medium, and large dams will not always be <em>exactly</em> the same.
The expected counts might still be a bit different from the observed counts of each dam size (<span class="math inline">\(O_{\mathrm{size}}\)</span>).
Suppose, for example, we find that out of our total 60 policy-makers, we observe 18 small (<span class="math inline">\(O_{\mathrm{small}} = 18\)</span>), 24 medium (<span class="math inline">\(O_{\mathrm{medium}} = 24\)</span>), and 18 large (<span class="math inline">\(O_{\mathrm{large}} = 18\)</span>), dams were actually chosen by game players.
What we want to test is the null hypothesis that there is no significant difference between expected and observed counts.</p>
<ul>
<li><span class="math inline">\(H_{0}\)</span>: There is no significant difference between expected and observed counts.</li>
<li><span class="math inline">\(H_{A}\)</span>: There is a significant difference between expected and observed counts.</li>
</ul>
<p>To get our test statistic<a href="#fn67" class="footnote-ref" id="fnref67"><sup>67</sup></a>, we now just need to take each observed count, subtract the expected count, square this difference, divide by the expected count, then add everything up,</p>
<p><span class="math display">\[\chi^{2} = \frac{(18 - 20)^{2}}{20} + \frac{(24 - 20)^{2}}{20} + \frac{(18 - 20)^{2}}{20}.\]</span></p>
<p>We can calculate the values in the numerator.
Note that all of these numbers must be positive (e.g., <span class="math inline">\(18 - 20 = -2\)</span>, but <span class="math inline">\(-2^{2} = 4\)</span>),</p>
<p><span class="math display">\[\chi^{2} = \frac{4}{20} + \frac{16}{20} + \frac{4}{20}.\]</span></p>
<p>When we sum the three terms, we get a value of <span class="math inline">\(\chi^{2} = 1.2\)</span>.
Note that if all of our observed values had been the same as the expected values (i.e., 20 small, medium, and large dams actually chosen), then we would get a <span class="math inline">\(\chi^{2}\)</span> value of 0.
The more the observed values differ from the expectation of 20, the higher the <span class="math inline">\(\chi^{2}\)</span> will be.
We can now check to see if the test statistic <span class="math inline">\(\chi^{2} = 1.2\)</span> is sufficiently large to reject the null hypothesis that our policy-makers have no preference for small, medium, or large dams.
There are <span class="math inline">\(n = 3\)</span> categories of counts (small, medium, and large), meaning that there are <span class="math inline">\(df = 3 - 1 = 2\)</span> degrees of freedom.
As it turns out, if the null hypothesis is true, then the probability of observing a value of <span class="math inline">\(\chi^{2} = 1.2\)</span> or higher (i.e., the p-value) is <span class="math inline">\(P = 0.5488\)</span>.
Figure 29.3 shows the appropriate <span class="math inline">\(\chi^{2}\)</span> distribution plotted, with the area above the test statistic <span class="math inline">\(\chi^{2} = 1.2\)</span> shaded in grey.</p>
<div class="figure"><span style="display:block;" id="fig:unnamed-chunk-120"></span>
<img src="bookdown-demo_files/figure-html/unnamed-chunk-120-1.png" alt="Plot of curved distribution that starts around 0.5 and decreases, asymptoting toward 0." width="672" />
<p class="caption">
Figure 29.3: A Chi-square distribution with 2 degrees of freedom, used in a goodness of fit test to determine if all players of the mobile game ‘Power Up!’ have the same preference for initial dam size. An arrow indicates a <span class="math inline">\(\chi^{2}\)</span> value of 1.2 on the x-axis, and grey shading indicates the probability of a more extreme value under the null hypothesis.
</p>
</div>
<p>Because <span class="math inline">\(P > 0.05\)</span>, we do not reject the null hypothesis that there is no significant difference between expected and observed counts of chosen dam sizes.</p>
<p>Note that this was a simple example.
For a goodness of fit test, we can have any number of different count categories (at least, any number greater than 1).
The expectations also do not need to be integers.
For example, if we only managed to find 59 policy makers instead of 60, then our expected counts would have been <span class="math inline">\(59/3 = 19.33\)</span> instead of <span class="math inline">\(60/3 = 20\)</span>.
The expectations also do not <em>need</em> to be the same.
For example, we could have tested the null hypothesis that twice as many policy-makers would choose large dams (i.e., <span class="math inline">\(E_{\mathrm{large}} = 40\)</span>, <span class="math inline">\(E_{\mathrm{medium}} = 10\)</span>, and <span class="math inline">\(E_{\mathrm{small}} = 10\)</span>).
For <span class="math inline">\(n\)</span> categories, the more general equation for the <span class="math inline">\(\chi^{2}\)</span> statistic is,</p>
<p><span class="math display">\[\chi^{2} = \sum_{i = 1}^{n} \frac{\left(O_{i} - E_{i}\right)^{2}}{E_{i}}.\]</span></p>
<p>We can therefore use this general equation to calculate a <span class="math inline">\(\chi^{2}\)</span> for any number of categories (<span class="math inline">\(n\)</span>).
Next, we will look at testing associations between counts in different types of categories.</p>
</div>
<div id="chi-square-test-of-association" class="section level2 hasAnchor" number="29.3">
<h2><span class="header-section-number">29.3</span> Chi-square test of association<a href="Chapter_29.html#chi-square-test-of-association" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>The second kind of test that we will consider for count data is the Chi-square test of association.
While the goodness of fit test focused on a single categorical variable (dam sizes in the example above), the Chi-square test of association focuses on two different categorical variables.
What we want to know is whether or not the two categorical variables are independent of one another <span class="citation">(<a href="#ref-Box1978" role="doc-biblioref">Box et al., 1978</a>)</span>.
In other words, does knowing something about one variable tell us anything about the other variable?
A concrete example will make it easier to explain.
We can again make use of the <a href="Chapter_17.html#Chapter_17">Chapter 17</a> game ‘Power Up!’.
As mentioned in the <a href="#chi-squared-goodness-of-fit">previous section</a>, game players choose a small, medium, or large dam at the start of the game.
Players can play the game on either an Android or macOS mobile device.
We therefore have two categorical variables: dam size and OS type.
We might want to know, do Android users choose the same dam sizes as macOS users?
In other words, are dam size and OS type associated?
We can state this as a null and alternative hypothesis.</p>
<ul>
<li><span class="math inline">\(H_{0}\)</span>: There is no association between OS and dam size choice.</li>
<li><span class="math inline">\(H_{A}\)</span>: There is an association between OS and dam size choice.</li>
</ul>
<p>Consider the data in Table 29.1, which show counts of Android versus macOS users and their dam choices.</p>
<table style="width:56%;">
<caption><strong>TABLE 29.1</strong> Counts (<span class="math inline">\(N = 60\)</span>) from a mobile game called ‘Power Up!’, in which players are confronted with trade-offs between energy output, energy justice, and biodiversity. Players can use one of two categories of Operating System (Android or macOS) and build one of three types of dam in the game (Small, Medium, or Large).</caption>
<colgroup>
<col width="19%" />
<col width="11%" />
<col width="12%" />
<col width="12%" />
</colgroup>
<thead>
<tr class="header">
<th align="center"> </th>
<th align="center">Small</th>
<th align="center">Medium</th>
<th align="center">Large</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td align="center"><strong>Android</strong></td>
<td align="center">8</td>
<td align="center">16</td>
<td align="center">6</td>
</tr>
<tr class="even">
<td align="center"><strong>macOS</strong></td>
<td align="center">10</td>
<td align="center">8</td>
<td align="center">12</td>
</tr>
</tbody>
</table>
<p>Just looking at the counts in Table 29.1, it appears that there might be an association between the two variables.
For example, Android users appear to be more likely to choose a medium dam than macOS users.
Medium dams are the most popular choice for Android users, but they are the least popular choice for macOS users.
Nevertheless, could this just be due to chance?
If it were due to chance, then how unlikely are the counts in Table 29.1?
In other words, if Android and macOS users in the whole population really do choose dam sizes at the same frequencies, then what is the probability of getting a sample of 60 players in which the choices are as or more unbalanced as this?
This is what we want to answer with our Chi-square test of association.</p>
<p>The general idea is the same as with the Chi-square goodness of fit test.
We have our observed values (Table 29.1).
We now need to find the expected values to calculate a <span class="math inline">\(\chi^{2}\)</span> value.
But the expected values are now a bit more complicated.
With the goodness of fit test in <a href="#chi-squared-goodness-of-fit">Section 29.2</a>, we just assumed that all categories were equally likely (i.e., the probability of choosing each size dam was the same).
There were 60 players and three dam sizes, so the expected frequency of each dam choice was 60/3 = 20.
Now it is different.
We are not testing if dam sizes or OS choices are the same.
We want to know if they are <em>associated</em> with one another.
That is, regardless of the popularity of Android versus macOS, or the frequency with which small, medium and large dams are selected, do Android users choose different dam sizes than macOS users?
If dam size is not associated with OS, then we would predict that the relative frequency of small, medium, and large dams would be the same for both Android and macOS.</p>
<p>To find the expected counts of each variable combination (e.g., Android and Small, or macOS and Large), we need to get the probability that each category is selected independently.
For example, what is the probability of a player selecting a large dam, regardless of the OS that they are using?
Table 29.2 shows these probabilities as additional rows and columns added onto Table 29.1</p>
<table style="width:79%;">
<caption><strong>TABLE 29.2</strong> Counts (<span class="math inline">\(N = 60\)</span>) from a mobile game called ‘Power Up!’, in which players are confronted with trade-offs between energy output, energy justice, and biodiversity. Players can use one of two categories of Operating System (Android or macOS) and build one of three categories of dam in the game (Small, Medium, or Large). Outer rows and columns show the probabilities of categories being selected.</caption>
<colgroup>
<col width="25%" />
<col width="11%" />
<col width="12%" />
<col width="11%" />
<col width="19%" />
</colgroup>
<thead>
<tr class="header">
<th align="center"> </th>
<th align="center">Small</th>
<th align="center">Medium</th>
<th align="center">Large</th>
<th align="center">Probability</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td align="center"><strong>Android</strong></td>
<td align="center">8</td>
<td align="center">16</td>
<td align="center">6</td>
<td align="center">0.5</td>
</tr>
<tr class="even">
<td align="center"><strong>macOS</strong></td>
<td align="center">10</td>
<td align="center">8</td>
<td align="center">12</td>
<td align="center">0.5</td>
</tr>
<tr class="odd">
<td align="center"><strong>Probability</strong></td>
<td align="center">0.3</td>
<td align="center">0.4</td>
<td align="center">0.3</td>
<td align="center">–</td>
</tr>
</tbody>
</table>
<p>Since there are 30 total Android users (<span class="math inline">\(8 + 16 + 6 = 30\)</span>) and 30 total macOS users (<span class="math inline">\(10 + 8 + 12 = 30\)</span>), the probability of a player having an Android OS is <span class="math inline">\(30/60 = 0.5\)</span>, and the probability of a player having a macOS is also <span class="math inline">\(30 / 60 = 0.5\)</span>.
Similarly, there are 18 small, 24 medium, and 18 large dam choices in total.
Hence, the probability of a player choosing a small dam is <span class="math inline">\(18/60 = 0.3\)</span>, medium is <span class="math inline">\(24/60 = 0.4\)</span>, and large is <span class="math inline">\(18/60 = 0.3\)</span>.
If these probabilities combine independently<a href="#fn68" class="footnote-ref" id="fnref68"><sup>68</sup></a>, then we can multiply them to find the probability of a particular combination of categories.
For example, the probability of a player using Android is 0.5 and choosing a small dam is 0.3, so the probability of a player having both Android <strong>and</strong> a small dam is <span class="math inline">\(0.5 \times 0.3 = 0.15\)</span> (see <a href="Chapter_16.html#Chapter_16">Chapter 16</a> for an introduction to probability models).
The probability of a player using Android <strong>and</strong> choosing a medium dam is <span class="math inline">\(0.5 \times 0.4 = 0.2\)</span>.
We can fill in all of these joint probabilities in a new Table 29.3.</p>
<table style="width:79%;">
<caption><strong>TABLE 29.3</strong> Probabilities for each combination of categorical variables from a dataset in which players on either an Android or macOS choose a dam size in the mobile app game ‘Power Up!’, assuming that variables are independent of one another.</caption>
<colgroup>
<col width="25%" />
<col width="11%" />
<col width="12%" />
<col width="11%" />
<col width="19%" />
</colgroup>
<thead>
<tr class="header">
<th align="center"> </th>
<th align="center">Small</th>
<th align="center">Medium</th>
<th align="center">Large</th>
<th align="center">Probability</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td align="center"><strong>Android</strong></td>
<td align="center">0.15</td>
<td align="center">0.2</td>
<td align="center">0.15</td>
<td align="center">0.5</td>
</tr>
<tr class="even">
<td align="center"><strong>macOS</strong></td>
<td align="center">0.15</td>
<td align="center">0.2</td>
<td align="center">0.15</td>
<td align="center">0.5</td>
</tr>
<tr class="odd">
<td align="center"><strong>Probability</strong></td>
<td align="center">0.3</td>
<td align="center">0.4</td>
<td align="center">0.3</td>
<td align="center">–</td>
</tr>
</tbody>
</table>
<p>From Table 29.3, we now have the probability of each combination of variables.
Note that all of these probabilities sum to 1.</p>
<p><span class="math display">\[0.15 + 0.2 + 0.15 + 0.15 + 0.2 + 0.15 = 1.\]</span></p>
<p>To get the expected count of each combination, we just need to multiply the probability by the sample size, i.e., the total number of players (<span class="math inline">\(N = 60\)</span>).
For example, the expected count of players who use Android and choose a small dam will be <span class="math inline">\(0.15 \times 60 = 9\)</span>.
Table 29.4 fills in all of the expected counts.
Note that the sum of all the counts equals our sample size of 60.</p>
<table style="width:65%;">
<caption><strong>TABLE 29.4</strong> Expected counts for each combination of categorical variables from a dataset in which players on either an Android or macOS choose a dam size in the mobile app game ‘Power Up!’, assuming that variables are independent of one another.</caption>
<colgroup>
<col width="19%" />
<col width="11%" />
<col width="12%" />
<col width="11%" />
<col width="11%" />
</colgroup>
<thead>
<tr class="header">
<th align="center"> </th>
<th align="center">Small</th>
<th align="center">Medium</th>
<th align="center">Large</th>
<th align="center">Sum</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td align="center"><strong>Android</strong></td>
<td align="center">9</td>
<td align="center">12</td>
<td align="center">9</td>
<td align="center">30</td>
</tr>
<tr class="even">
<td align="center"><strong>macOS</strong></td>
<td align="center">9</td>
<td align="center">12</td>
<td align="center">9</td>
<td align="center">30</td>
</tr>
<tr class="odd">
<td align="center"><strong>Sum</strong></td>
<td align="center">18</td>
<td align="center">24</td>
<td align="center">18</td>
<td align="center">–</td>
</tr>
</tbody>
</table>
<p>We now have both the observed (Table 29.2) and expected (Table 29.4) counts (remember that the expected counts do not <em>need</em> to be integers).
To get our <span class="math inline">\(\chi^{2}\)</span> test statistic, we use the same formula as in <a href="#chi-squared-goodness-of-fit">Section 29.2</a>,</p>
<p><span class="math display">\[\chi^{2} = \sum_{i = 1}^{n} \frac{\left(O_{i} - E_{i}\right)^{2}}{E_{i}}.\]</span></p>
<p>There are six total combinations of OS and dam size, so there are <span class="math inline">\(n = 6\)</span> values to sum up,</p>
<p><span class="math display">\[\chi^{2} = \frac{(8-9)^2}{9} + \frac{(16 - 12)^{2}}{12} + ... + \frac{(16 - 12)^{2}}{12} + \frac{(8-9)^2}{9}.\]</span></p>
<p>If we sum all of the six terms, we get a value of <span class="math inline">\(\chi^{2} = 4.889\)</span>.
We can compare this to the null <span class="math inline">\(\chi^{2}\)</span> distribution as we did in the <a href="#chi-squared-goodness-of-fit">Section 29.2</a> goodness of fit test, but we need to know the correct degrees of freedom.
The correct degrees of freedom<a href="#fn69" class="footnote-ref" id="fnref69"><sup>69</sup></a> is the number of categories in variable 1 (<span class="math inline">\(n_{1}\)</span>) minus 1, times the number of categories in variable 2 (<span class="math inline">\(n_{2}\)</span>) minus 1,</p>
<p><span class="math display">\[df = (n_{1} - 1) \times (n_{2} - 1).\]</span></p>
<p>In our example, the degrees of freedom equals the number of dam types minus 1 (<span class="math inline">\(n_{\mathrm{dam}} = 3 - 1)\)</span> times the number of operating systems minus 1 (<span class="math inline">\(n_{\mathrm{OS}} = 2 - 1\)</span>).
The correct degrees of freedom is therefore <span class="math inline">\(df = 2 \times 1 = 2\)</span>.
We now just need to find the p-value for a Chi-square distribution with two degrees of freedom and a test statistic of <span class="math inline">\(\chi^{2} = 4.889\)</span>.
We get a value of about <span class="math inline">\(P = 0.0868\)</span>.
In other words, if <span class="math inline">\(H_{0}\)</span> is true, then the probability of getting a <span class="math inline">\(\chi^{2}\)</span> of 4.889 or higher is <span class="math inline">\(P = 0.0868\)</span>.
Consequently, because <span class="math inline">\(P > 0.05\)</span>, we would not reject the null hypothesis.
We should therefore conclude that there is no evidence for an association between OS and dam size choice.</p>
<p>Jamovi will calculate the <span class="math inline">\(\chi^{2}\)</span> value and get the p-value for the appropriate degrees of freedom <span class="citation">(<a href="#ref-Jamovi2022" role="doc-biblioref">The jamovi project, 2024</a>)</span>.
To do this, it is necessary to input the categorical data (e.g., Android, macOS) in a tidy format, which will be a focus of <a href="Chapter_31.html#Chapter_31">Chapter 31</a>.</p>
<p>There is one final point regarding expected and observed values of the Chi-square test of association.
There is another way of getting these expected values that is a bit faster (and more widely taught) but does not demonstrate the logic of expected counts as clearly.
If we wanted to, we could sum the rows and columns of our original observations.
Table 29.5 shows the original observations with the sum of each row and column.</p>
<table style="width:65%;">
<caption><strong>TABLE 29.5</strong> Observed counts for each combination of categorical variables from a dataset in which players on either an Android or macOS choose a dam size in the mobile app game ‘Power Up!’. The last row and column shows the sum of observed dam sizes and OS users, respectively.</caption>
<colgroup>
<col width="19%" />
<col width="11%" />
<col width="12%" />
<col width="11%" />
<col width="11%" />
</colgroup>
<thead>
<tr class="header">
<th align="center"> </th>
<th align="center">Small</th>
<th align="center">Medium</th>
<th align="center">Large</th>
<th align="center">Sum</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td align="center"><strong>Android</strong></td>
<td align="center">8</td>
<td align="center">16</td>
<td align="center">6</td>
<td align="center">30</td>
</tr>
<tr class="even">
<td align="center"><strong>macOS</strong></td>
<td align="center">10</td>
<td align="center">8</td>
<td align="center">12</td>
<td align="center">30</td>
</tr>
<tr class="odd">
<td align="center"><strong>Sum</strong></td>
<td align="center">18</td>
<td align="center">24</td>
<td align="center">18</td>
<td align="center">–</td>
</tr>
</tbody>
</table>
<p>We can get the expected counts from Table 29.5 if we multiply each row sum by each column sum, then divide by the total sample size (<span class="math inline">\(N = 60\)</span>).
For example, to get the expected counts of Android users who choose a small dam, we can multiply <span class="math inline">\((18 \times 30)/60 = 9\)</span>.
To get the expected counts of macOS users who choose a medium dam, we can multiply <span class="math inline">\((30 \times 24)/60 = 12\)</span>.
This works for all of combinations of rows and columns, so we could do it to find all of the expected counts from Table 29.4.</p>
</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-Box1978" class="csl-entry">
Box, G. E. P., Hunter, W. G., & Hunter, S. J. (1978). <em><span class="nocase">Statistics for Experimenters: An Introduction to Design, Data Analysis, and Model Building</span></em>. John Wiley & Sons, New York, USA.
</div>
<div id="ref-Miller2004" class="csl-entry">
Miller, I., & Miller, M. (2004). <em><span class="nocase">John E. Freund’s mathematical statistics</span></em> (7th ed., p. 614). Pearson Prentice Hall, Upper Saddle River, New Jersey, USA.
</div>
<div id="ref-Slakter1968" class="csl-entry">
Slakter, M. J. (1968). Accuracy of an approximation to the power of the chi-square goodness of fit test with small but equal expected frequencies. <em>Journal of the American Statistical Association</em>, <em>63</em>(323), 912–918. <a href="https://doi.org/10.1080/01621459.1968.11009319">https://doi.org/10.1080/01621459.1968.11009319</a>
</div>
<div id="ref-Sokal1995" class="csl-entry">
Sokal, R. R., & Rohlf, F. J. (1995). <em><span>Biometry</span></em> (3rd ed., p. 887). W. H. Freeman & Company, New York, USA.
</div>
<div id="ref-Tate1973" class="csl-entry">
Tate, M. W., & Hyer, L. A. (1973). Inaccuracy of the <span class="math inline">\(X^{2}\)</span> test of goodness of fit when expected frequencies are small. <em>Journal of the American Statistical Association</em>, <em>68</em>(344), 836–841. <a href="https://doi.org/10.1080/01621459.1973.10481433">https://doi.org/10.1080/01621459.1973.10481433</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="65">
<li id="fn65"><p>A random variable <span class="math inline">\(X\)</span> has a <span class="math inline">\(\chi^{2}\)</span> distribution if and only if its probability density function is defined by <span class="citation">(<a href="#ref-Miller2004" role="doc-biblioref">Miller & Miller, 2004</a>)</span>, <span class="math display">\[f(x) = \left\{\begin{array}{ll}\frac{1}{2^{\frac{2}{v}}\Gamma\left(\frac{v}{2}\right)}x^{\frac{v-2}{2}}e^{-\frac{x}{2}} & \quad for\:x > 0 \\ 0 & \quad elsewhere \end{array}\right.\]</span> In this equation, <span class="math inline">\(v\)</span> is the degrees of freedom of the distribution.<a href="Chapter_29.html#fnref65" class="footnote-back">↩︎</a></p></li>
<li id="fn66"><p><a href="https://bradduthie.github.io/stats/app/chi-square/" class="uri">https://bradduthie.github.io/stats/app/chi-square/</a><a href="Chapter_29.html#fnref66" class="footnote-back">↩︎</a></p></li>
<li id="fn67"><p>A lot of statisticians will use <span class="math inline">\(X^{2}\)</span> to represent the test statistic here instead of <span class="math inline">\(\chi^{2}\)</span> <span class="citation">(<a href="#ref-Sokal1995" role="doc-biblioref">Sokal & Rohlf, 1995</a>)</span>. The difference is the upper case ‘X’ versus the Greek letter Chi, ‘<span class="math inline">\(\chi\)</span>’. The X is used since the test statistic we calculate here is not <em>technically</em> from the <span class="math inline">\(\chi^{2}\)</span> distribution, just an approximation. We will not worry about the distinction here, and to avoid confusion, we will just go with <span class="math inline">\(\chi^{2}\)</span>.<a href="Chapter_29.html#fnref67" class="footnote-back">↩︎</a></p></li>
<li id="fn68"><p>We can call these the ‘marginal probabilities’.<a href="Chapter_29.html#fnref68" class="footnote-back">↩︎</a></p></li>
<li id="fn69"><p>This formula works due to a bit of a mathematical trick <span class="citation">(<a href="#ref-Sokal1995" role="doc-biblioref">Sokal & Rohlf, 1995</a>)</span>. The actual logic of the degrees of freedom is a bit more involved. From our total of <span class="math inline">\(k = 6\)</span> different combinations, we actually need to subtract one degree of freedom for the total sample size (<span class="math inline">\(N = 60\)</span>), then a degree of freedom for each variable probability estimated (i.e., subtract <span class="math inline">\(n_{1} - 1\)</span> and <span class="math inline">\(n_{2} - 1\)</span> because we need this many degrees of freedom to get the <span class="math inline">\(n_{1}\)</span> and <span class="math inline">\(n_{2}\)</span> probabilities, respectively; if we have all but one probability, then we know the last probability because the probabilities must sum to 1). Since we lose <span class="math inline">\(n_{1} - 1\)</span> and <span class="math inline">\(n_{2} - 1\)</span> degrees of freedom, and 1 for the sample size, this results in <span class="math inline">\(df = k - (n_{1} - 1) - (n_{2} - 1) - 1\)</span>. In the case of the ‘Power Up!’ example, we get <span class="math inline">\(df = 6 - (3 - 1) - (2 - 1) - 1 = 2\)</span>. The <span class="math inline">\(df = (n_{1} - 1) \times (n_{2} - 1)\)</span> formulation is possible because <span class="math inline">\(k = n_{1} \times n_{2}\)</span> <span class="citation">(<a href="#ref-Sokal1995" role="doc-biblioref">Sokal & Rohlf, 1995</a>)</span>.<a href="Chapter_29.html#fnref69" class="footnote-back">↩︎</a></p></li>
</ol>
</div>
</section>
</div>
</div>
</div>
<a href="Chapter_28.html" class="navigation navigation-prev " aria-label="Previous page"><i class="fa fa-angle-left"></i></a>
<a href="Chapter_30.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/08-Correlation.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>