-
Notifications
You must be signed in to change notification settings - Fork 0
/
correlation.html
970 lines (931 loc) · 76.2 KB
/
correlation.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
<!DOCTYPE html>
<html lang="" xml:lang="">
<head>
<meta charset="utf-8" />
<meta http-equiv="X-UA-Compatible" content="IE=edge" />
<title>Chapter 29 Correlation | Statistical Techniques for Biological and Environmental Sciences</title>
<meta name="description" content="This is a lab book for the University of Stirling second year undergraduate Biological and Environmental Sciences statistics module." />
<meta name="generator" content="bookdown 0.27 and GitBook 2.6.7" />
<meta property="og:title" content="Chapter 29 Correlation | Statistical Techniques for Biological and Environmental Sciences" />
<meta property="og:type" content="book" />
<meta property="og:description" content="This is a lab book for the University of Stirling second year undergraduate Biological and Environmental Sciences statistics module." />
<meta name="github-repo" content="bradduthie/statistical_techniques" />
<meta name="twitter:card" content="summary" />
<meta name="twitter:title" content="Chapter 29 Correlation | Statistical Techniques for Biological and Environmental Sciences" />
<meta name="twitter:description" content="This is a lab book for the University of Stirling second year undergraduate Biological and Environmental Sciences statistics module." />
<meta name="author" content="Brad Duthie" />
<meta name="date" content="2023-03-19" />
<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="practical.-analysis-of-counts-and-correlations.html"/>
<script src="libs/jquery-3.6.0/jquery-3.6.0.min.js"></script>
<script src="https://cdn.jsdelivr.net/npm/fuse.js@6.4.6/dist/fuse.min.js"></script>
<link href="libs/gitbook-2.6.7/css/style.css" rel="stylesheet" />
<link href="libs/gitbook-2.6.7/css/plugin-table.css" rel="stylesheet" />
<link href="libs/gitbook-2.6.7/css/plugin-bookdown.css" rel="stylesheet" />
<link href="libs/gitbook-2.6.7/css/plugin-highlight.css" rel="stylesheet" />
<link href="libs/gitbook-2.6.7/css/plugin-search.css" rel="stylesheet" />
<link href="libs/gitbook-2.6.7/css/plugin-fontsettings.css" rel="stylesheet" />
<link href="libs/gitbook-2.6.7/css/plugin-clipboard.css" rel="stylesheet" />
<link href="libs/anchor-sections-1.1.0/anchor-sections.css" rel="stylesheet" />
<link href="libs/anchor-sections-1.1.0/anchor-sections-hash.css" rel="stylesheet" />
<script src="libs/anchor-sections-1.1.0/anchor-sections.js"></script>
<style type="text/css">
pre > code.sourceCode { white-space: pre; position: relative; }
pre > code.sourceCode > span { display: inline-block; line-height: 1.25; }
pre > code.sourceCode > span:empty { height: 1.2em; }
.sourceCode { overflow: visible; }
code.sourceCode > span { color: inherit; text-decoration: inherit; }
pre.sourceCode { margin: 0; }
@media screen {
div.sourceCode { overflow: auto; }
}
@media print {
pre > code.sourceCode { white-space: pre-wrap; }
pre > code.sourceCode > span { text-indent: -5em; padding-left: 5em; }
}
pre.numberSource code
{ counter-reset: source-line 0; }
pre.numberSource code > span
{ position: relative; left: -4em; counter-increment: source-line; }
pre.numberSource code > span > a:first-child::before
{ content: counter(source-line);
position: relative; left: -1em; text-align: right; vertical-align: baseline;
border: none; display: inline-block;
-webkit-touch-callout: none; -webkit-user-select: none;
-khtml-user-select: none; -moz-user-select: none;
-ms-user-select: none; user-select: none;
padding: 0 4px; width: 4em;
}
pre.numberSource { margin-left: 3em; padding-left: 4px; }
div.sourceCode
{ }
@media screen {
pre > code.sourceCode > span > a:first-child::before { text-decoration: underline; }
}
code span.al { font-weight: bold; } /* Alert */
code span.an { font-style: italic; } /* Annotation */
code span.cf { font-weight: bold; } /* ControlFlow */
code span.co { font-style: italic; } /* Comment */
code span.cv { font-style: italic; } /* CommentVar */
code span.do { font-style: italic; } /* Documentation */
code span.dt { text-decoration: underline; } /* DataType */
code span.er { font-weight: bold; } /* Error */
code span.in { font-style: italic; } /* Information */
code span.kw { font-weight: bold; } /* Keyword */
code span.pp { font-weight: bold; } /* Preprocessor */
code span.wa { font-style: italic; } /* Warning */
</style>
<style type="text/css">
/* 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="./">Statistical Techniques</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#why-this-module-is-important"><i class="fa fa-check"></i>Why this module is important</a></li>
<li class="chapter" data-level="" data-path="index.html"><a href="index.html#ILOs"><i class="fa fa-check"></i>Intended learning outcomes (ILOs)</a></li>
<li class="chapter" data-level="" data-path="index.html"><a href="index.html#accessibility"><i class="fa fa-check"></i>Accessibility</a></li>
<li class="chapter" data-level="" data-path="index.html"><a href="index.html#teaching_overview"><i class="fa fa-check"></i>Teaching overview</a>
<ul>
<li class="chapter" data-level="" data-path="index.html"><a href="index.html#book_chapters"><i class="fa fa-check"></i>Book chapters</a></li>
<li class="chapter" data-level="" data-path="index.html"><a href="index.html#additional_readings"><i class="fa fa-check"></i>Additional readings</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="index.html"><a href="index.html#Canvas"><i class="fa fa-check"></i>Canvas</a></li>
<li class="chapter" data-level="" data-path="index.html"><a href="index.html#assessment-overview"><i class="fa fa-check"></i>Assessment overview</a>
<ul>
<li class="chapter" data-level="" data-path="index.html"><a href="index.html#tests"><i class="fa fa-check"></i>Tests</a></li>
<li class="chapter" data-level="" data-path="index.html"><a href="index.html#exams"><i class="fa fa-check"></i>Exams</a></li>
<li class="chapter" data-level="" data-path="index.html"><a href="index.html#feedback"><i class="fa fa-check"></i>Feedback</a></li>
<li class="chapter" data-level="" data-path="index.html"><a href="index.html#extenuating_circumstances"><i class="fa fa-check"></i>Extenuating circumstances</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="index.html"><a href="index.html#practicals"><i class="fa fa-check"></i>Practicals</a></li>
<li class="chapter" data-level="" data-path="index.html"><a href="index.html#help"><i class="fa fa-check"></i>Optional help hours</a></li>
<li class="chapter" data-level="" data-path="index.html"><a href="index.html#jamovi"><i class="fa fa-check"></i>Jamovi statistical software</a></li>
<li class="chapter" data-level="" data-path="index.html"><a href="index.html#timetable"><i class="fa fa-check"></i>Timetable</a></li>
<li class="chapter" data-level="" data-path="index.html"><a href="index.html#license"><i class="fa fa-check"></i>License</a></li>
</ul></li>
<li class="part"><span><b>I Background mathematics and data organisation</b></span></li>
<li class="chapter" data-level="" data-path="Week1.html"><a href="Week1.html"><i class="fa fa-check"></i>Week 1 Overview</a></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> Practical: Preparing data</a>
<ul>
<li class="chapter" data-level="3.1" data-path="Chapter_3.html"><a href="Chapter_3.html#exercise-1-transferring-data-to-a-spreadsheet"><i class="fa fa-check"></i><b>3.1</b> Exercise 1: Transferring data to a spreadsheet</a></li>
<li class="chapter" data-level="3.2" data-path="Chapter_3.html"><a href="Chapter_3.html#exercise-2-making-spreadsheet-data-tidy"><i class="fa fa-check"></i><b>3.2</b> Exercise 2: Making spreadsheet data tidy</a></li>
<li class="chapter" data-level="3.3" data-path="Chapter_3.html"><a href="Chapter_3.html#exercise-3-making-data-tidy-again"><i class="fa fa-check"></i><b>3.3</b> Exercise 3: Making data tidy again</a></li>
<li class="chapter" data-level="3.4" data-path="Chapter_3.html"><a href="Chapter_3.html#exercise-4-tidy-data-and-spreadsheet-calculations"><i class="fa fa-check"></i><b>3.4</b> Exercise 4: 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="part"><span><b>II Statistical concepts</b></span></li>
<li class="chapter" data-level="" data-path="Week2.html"><a href="Week2.html"><i class="fa fa-check"></i>Week 2 Overview</a></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>
<li class="chapter" data-level="6.4" data-path="Chapter_6.html"><a href="Chapter_6.html#other-examples-of-units"><i class="fa fa-check"></i><b>6.4</b> Other examples of units</a>
<ul>
<li class="chapter" data-level="6.4.1" data-path="Chapter_6.html"><a href="Chapter_6.html#units-of-density"><i class="fa fa-check"></i><b>6.4.1</b> Units of density</a></li>
<li class="chapter" data-level="6.4.2" data-path="Chapter_6.html"><a href="Chapter_6.html#mass-of-metal-discharged-from-a-catchment"><i class="fa fa-check"></i><b>6.4.2</b> Mass of metal discharged from a catchment</a></li>
<li class="chapter" data-level="6.4.3" data-path="Chapter_6.html"><a href="Chapter_6.html#soil-carbon-inventories"><i class="fa fa-check"></i><b>6.4.3</b> Soil carbon inventories</a></li>
</ul></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 propogation</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>
<li class="chapter" data-level="7.3" data-path="Chapter_7.html"><a href="Chapter_7.html#applying-formulas-for-combining-errors"><i class="fa fa-check"></i><b>7.3</b> Applying formulas for combining 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> Practical. 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> Exercise for 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> Exercise on 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> Exercise on 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="part"><span><b>III Summary statistics</b></span></li>
<li class="chapter" data-level="" data-path="Week3.html"><a href="Week3.html"><i class="fa fa-check"></i>Week 3 Overview</a></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> <em>Practical</em>. Plotting and statistical summaries in Jamovi</a>
<ul>
<li class="chapter" data-level="13.1" data-path="Chapter_13.html"><a href="Chapter_13.html#reorganise-the-dataset-into-a-tidy-format"><i class="fa fa-check"></i><b>13.1</b> Reorganise the dataset into a tidy format</a></li>
<li class="chapter" data-level="13.2" data-path="Chapter_13.html"><a href="Chapter_13.html#histograms-and-box-whisker-plots"><i class="fa fa-check"></i><b>13.2</b> Histograms and box-whisker plots</a></li>
<li class="chapter" data-level="13.3" data-path="Chapter_13.html"><a href="Chapter_13.html#calculate-summary-statistics"><i class="fa fa-check"></i><b>13.3</b> Calculate summary statistics</a></li>
<li class="chapter" data-level="13.4" data-path="Chapter_13.html"><a href="Chapter_13.html#reporting-decimals-and-significant-figures"><i class="fa fa-check"></i><b>13.4</b> Reporting decimals and significant figures</a></li>
<li class="chapter" data-level="13.5" data-path="Chapter_13.html"><a href="Chapter_13.html#comparing-across-sites"><i class="fa fa-check"></i><b>13.5</b> Comparing across sites</a></li>
</ul></li>
<li class="part"><span><b>IV Probability models and the Central Limit Theorem</b></span></li>
<li class="chapter" data-level="" data-path="Week4.html"><a href="Week4.html"><i class="fa fa-check"></i>Week 4 Overview</a></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> Introduction to probability models</a>
<ul>
<li class="chapter" data-level="14.1" data-path="Chapter_14.html"><a href="Chapter_14.html#an-instructive-example"><i class="fa fa-check"></i><b>14.1</b> An instructive example</a></li>
<li class="chapter" data-level="14.2" data-path="Chapter_14.html"><a href="Chapter_14.html#biological-applications"><i class="fa fa-check"></i><b>14.2</b> Biological applications</a></li>
<li class="chapter" data-level="14.3" data-path="Chapter_14.html"><a href="Chapter_14.html#sampling-with-and-without-replacement"><i class="fa fa-check"></i><b>14.3</b> Sampling with and without replacement</a></li>
<li class="chapter" data-level="14.4" data-path="Chapter_14.html"><a href="Chapter_14.html#probability-distributions"><i class="fa fa-check"></i><b>14.4</b> Probability distributions</a>
<ul>
<li class="chapter" data-level="14.4.1" data-path="Chapter_14.html"><a href="Chapter_14.html#binomial-distribution"><i class="fa fa-check"></i><b>14.4.1</b> Binomial distribution</a></li>
<li class="chapter" data-level="14.4.2" data-path="Chapter_14.html"><a href="Chapter_14.html#poisson-distribution"><i class="fa fa-check"></i><b>14.4.2</b> Poisson distribution</a></li>
<li class="chapter" data-level="14.4.3" data-path="Chapter_14.html"><a href="Chapter_14.html#uniform-distribution"><i class="fa fa-check"></i><b>14.4.3</b> Uniform distribution</a></li>
<li class="chapter" data-level="14.4.4" data-path="Chapter_14.html"><a href="Chapter_14.html#normal-distribution"><i class="fa fa-check"></i><b>14.4.4</b> Normal distribution</a></li>
</ul></li>
<li class="chapter" data-level="14.5" data-path="Chapter_14.html"><a href="Chapter_14.html#summary-2"><i class="fa fa-check"></i><b>14.5</b> Summary</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> The Central Limit Theorem (CLT)</a>
<ul>
<li class="chapter" data-level="15.1" data-path="Chapter_15.html"><a href="Chapter_15.html#the-distribution-of-means-is-normal"><i class="fa fa-check"></i><b>15.1</b> The distribution of means is normal</a></li>
<li class="chapter" data-level="15.2" data-path="Chapter_15.html"><a href="Chapter_15.html#probability-and-z-scores"><i class="fa fa-check"></i><b>15.2</b> Probability and z-scores</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> <em>Practical</em>. Probability and simulation</a>
<ul>
<li class="chapter" data-level="16.1" data-path="Chapter_16.html"><a href="Chapter_16.html#probabilities-from-a-dataset"><i class="fa fa-check"></i><b>16.1</b> Probabilities from a dataset</a></li>
<li class="chapter" data-level="16.2" data-path="Chapter_16.html"><a href="Chapter_16.html#probabilities-from-a-normal-distribution"><i class="fa fa-check"></i><b>16.2</b> Probabilities from a normal distribution</a></li>
<li class="chapter" data-level="16.3" data-path="Chapter_16.html"><a href="Chapter_16.html#central-limit-theorem"><i class="fa fa-check"></i><b>16.3</b> Central limit theorem</a></li>
</ul></li>
<li class="part"><span><b>V Statistical inference</b></span></li>
<li class="chapter" data-level="" data-path="Week5.html"><a href="Week5.html"><i class="fa fa-check"></i>Week 5 Overview</a></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> Confidence intervals (CIs)</a>
<ul>
<li class="chapter" data-level="17.1" data-path="Chapter_17.html"><a href="Chapter_17.html#normal-distribution-cis"><i class="fa fa-check"></i><b>17.1</b> Normal distribution CIs</a></li>
<li class="chapter" data-level="17.2" data-path="Chapter_17.html"><a href="Chapter_17.html#binomial-distribution-cis"><i class="fa fa-check"></i><b>17.2</b> Binomial distribution CIs</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> The t-interval</a></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> <em>Practical</em>. z- and t- intervals</a>
<ul>
<li class="chapter" data-level="19.1" data-path="Chapter_19.html"><a href="Chapter_19.html#confidence-intervals-with-distraction"><i class="fa fa-check"></i><b>19.1</b> Confidence intervals with distrACTION</a></li>
<li class="chapter" data-level="19.2" data-path="Chapter_19.html"><a href="Chapter_19.html#confidence-intervals-from-z--and-t-scores"><i class="fa fa-check"></i><b>19.2</b> Confidence intervals from z- and t-scores</a></li>
<li class="chapter" data-level="19.3" data-path="Chapter_19.html"><a href="Chapter_19.html#confidence-intervals-for-different-sample-sizes-t--and-z-"><i class="fa fa-check"></i><b>19.3</b> Confidence intervals for different sample sizes (t- and z-)</a></li>
<li class="chapter" data-level="19.4" data-path="Chapter_19.html"><a href="Chapter_19.html#proportion-confidence-intervals"><i class="fa fa-check"></i><b>19.4</b> Proportion confidence intervals</a></li>
<li class="chapter" data-level="19.5" data-path="Chapter_19.html"><a href="Chapter_19.html#another-proportion-confidence-interval"><i class="fa fa-check"></i><b>19.5</b> Another proportion confidence interval</a></li>
</ul></li>
<li class="part"><span><b>VI Hypothesis testing</b></span></li>
<li class="chapter" data-level="" data-path="Week6.html"><a href="Week6.html"><i class="fa fa-check"></i>Week 6 Overview</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> What is hypothesis testing?</a>
<ul>
<li class="chapter" data-level="20.1" data-path="Chapter_20.html"><a href="Chapter_20.html#how-ridiculous-is-our-hypothesis"><i class="fa fa-check"></i><b>20.1</b> How ridiculous is our hypothesis?</a></li>
<li class="chapter" data-level="20.2" data-path="Chapter_20.html"><a href="Chapter_20.html#statistical-hypothesis-testing"><i class="fa fa-check"></i><b>20.2</b> Statistical hypothesis testing</a></li>
<li class="chapter" data-level="20.3" data-path="Chapter_20.html"><a href="Chapter_20.html#p-values-false-positives-and-power"><i class="fa fa-check"></i><b>20.3</b> P-values, false positives, and power</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> The t-test</a>
<ul>
<li class="chapter" data-level="21.1" data-path="Chapter_21.html"><a href="Chapter_21.html#one-sample-t-test"><i class="fa fa-check"></i><b>21.1</b> One sample t-test</a></li>
<li class="chapter" data-level="21.2" data-path="Chapter_21.html"><a href="Chapter_21.html#independent-samples-t-test"><i class="fa fa-check"></i><b>21.2</b> Independent samples t-test</a></li>
<li class="chapter" data-level="21.3" data-path="Chapter_21.html"><a href="Chapter_21.html#paired-sample-t-test"><i class="fa fa-check"></i><b>21.3</b> Paired sample t-test</a></li>
<li class="chapter" data-level="21.4" data-path="Chapter_21.html"><a href="Chapter_21.html#assumptions-of-t-tests"><i class="fa fa-check"></i><b>21.4</b> Assumptions of t-tests</a></li>
<li class="chapter" data-level="21.5" data-path="Chapter_21.html"><a href="Chapter_21.html#non-parametric-alternatives"><i class="fa fa-check"></i><b>21.5</b> Non-parametric alternatives</a>
<ul>
<li class="chapter" data-level="21.5.1" data-path="Chapter_21.html"><a href="Chapter_21.html#wilcoxon-test"><i class="fa fa-check"></i><b>21.5.1</b> Wilcoxon test</a></li>
<li class="chapter" data-level="21.5.2" data-path="Chapter_21.html"><a href="Chapter_21.html#mann-whitney-u-test"><i class="fa fa-check"></i><b>21.5.2</b> Mann-Whitney U test</a></li>
</ul></li>
<li class="chapter" data-level="21.6" data-path="Chapter_21.html"><a href="Chapter_21.html#summary-3"><i class="fa fa-check"></i><b>21.6</b> Summary</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> <em>Practical</em>. Hypothesis testing and t-tests</a>
<ul>
<li class="chapter" data-level="22.1" data-path="Chapter_22.html"><a href="Chapter_22.html#exercise-on-a-simple-one-sample-t-test"><i class="fa fa-check"></i><b>22.1</b> Exercise on a simple one sample t-test</a></li>
<li class="chapter" data-level="22.2" data-path="Chapter_22.html"><a href="Chapter_22.html#exercise-on-a-paired-t-test"><i class="fa fa-check"></i><b>22.2</b> Exercise on a paired t-test</a></li>
<li class="chapter" data-level="22.3" data-path="Chapter_22.html"><a href="Chapter_22.html#wilcoxon-test-1"><i class="fa fa-check"></i><b>22.3</b> Wilcoxon test</a></li>
<li class="chapter" data-level="22.4" data-path="Chapter_22.html"><a href="Chapter_22.html#independent-samples-t-test-1"><i class="fa fa-check"></i><b>22.4</b> Independent samples t-test</a></li>
<li class="chapter" data-level="22.5" data-path="Chapter_22.html"><a href="Chapter_22.html#mann-whitney-u-test-1"><i class="fa fa-check"></i><b>22.5</b> Mann-Whitney U Test</a></li>
</ul></li>
<li class="part"><span><b>VII Review of parts I-V</b></span></li>
<li class="chapter" data-level="" data-path="Week7.html"><a href="Week7.html"><i class="fa fa-check"></i>Week 7 Overview (Reading week)</a></li>
<li class="part"><span><b>VIII Analysis of Variance (ANOVA)</b></span></li>
<li class="chapter" data-level="" data-path="Week8.html"><a href="Week8.html"><i class="fa fa-check"></i>Week 8 Overview</a></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> Analysis of variance</a>
<ul>
<li class="chapter" data-level="23.1" data-path="Chapter_23.html"><a href="Chapter_23.html#the-f-distribution"><i class="fa fa-check"></i><b>23.1</b> The F-distribution</a></li>
<li class="chapter" data-level="23.2" data-path="Chapter_23.html"><a href="Chapter_23.html#one-way-anova"><i class="fa fa-check"></i><b>23.2</b> One-way ANOVA</a>
<ul>
<li class="chapter" data-level="23.2.1" data-path="Chapter_23.html"><a href="Chapter_23.html#anova-mean-variance-among-groups"><i class="fa fa-check"></i><b>23.2.1</b> ANOVA mean variance among groups</a></li>
<li class="chapter" data-level="23.2.2" data-path="Chapter_23.html"><a href="Chapter_23.html#anova-mean-variance-within-groups"><i class="fa fa-check"></i><b>23.2.2</b> ANOVA mean variance within groups</a></li>
<li class="chapter" data-level="23.2.3" data-path="Chapter_23.html"><a href="Chapter_23.html#anova-f-statistic-calculation"><i class="fa fa-check"></i><b>23.2.3</b> ANOVA F statistic calculation</a></li>
</ul></li>
<li class="chapter" data-level="23.3" data-path="Chapter_23.html"><a href="Chapter_23.html#assumptions-of-anova"><i class="fa fa-check"></i><b>23.3</b> Assumptions of ANOVA</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> Multiple comparisons</a></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> Kruskall-Wallis H test</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> Two-way ANOVA</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> <em>Practical</em>. ANOVA and associated tests</a>
<ul>
<li class="chapter" data-level="27.1" data-path="Chapter_27.html"><a href="Chapter_27.html#one-way-anova-site"><i class="fa fa-check"></i><b>27.1</b> One-way ANOVA (site)</a></li>
<li class="chapter" data-level="27.2" data-path="Chapter_27.html"><a href="Chapter_27.html#one-way-anova-profile"><i class="fa fa-check"></i><b>27.2</b> One-way ANOVA (profile)</a></li>
<li class="chapter" data-level="27.3" data-path="Chapter_27.html"><a href="Chapter_27.html#multiple-comparisons"><i class="fa fa-check"></i><b>27.3</b> Multiple comparisons</a></li>
<li class="chapter" data-level="27.4" data-path="Chapter_27.html"><a href="Chapter_27.html#kruskall-wallis-h-test"><i class="fa fa-check"></i><b>27.4</b> Kruskall-Wallis H test</a></li>
<li class="chapter" data-level="27.5" data-path="Chapter_27.html"><a href="Chapter_27.html#two-way-anova"><i class="fa fa-check"></i><b>27.5</b> Two-way ANOVA</a></li>
</ul></li>
<li class="part"><span><b>IX Counts and Correlation</b></span></li>
<li class="chapter" data-level="" data-path="Week9.html"><a href="Week9.html"><i class="fa fa-check"></i>Week 9 Overview</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> Frequency and count data</a>
<ul>
<li class="chapter" data-level="28.1" data-path="Chapter_28.html"><a href="Chapter_28.html#the-chi-square-distribution"><i class="fa fa-check"></i><b>28.1</b> The Chi-square distribution</a></li>
<li class="chapter" data-level="28.2" data-path="Chapter_28.html"><a href="Chapter_28.html#chi-squared-goodness-of-fit"><i class="fa fa-check"></i><b>28.2</b> Chi-squared goodness of fit</a></li>
<li class="chapter" data-level="28.3" data-path="Chapter_28.html"><a href="Chapter_28.html#chi-squared-test-of-association"><i class="fa fa-check"></i><b>28.3</b> Chi-squared test of association</a></li>
</ul></li>
<li class="chapter" data-level="29" data-path="correlation.html"><a href="correlation.html"><i class="fa fa-check"></i><b>29</b> Correlation</a>
<ul>
<li class="chapter" data-level="29.1" data-path="correlation.html"><a href="correlation.html#scatterplots"><i class="fa fa-check"></i><b>29.1</b> Scatterplots</a></li>
<li class="chapter" data-level="29.2" data-path="correlation.html"><a href="correlation.html#the-correlation-coefficient"><i class="fa fa-check"></i><b>29.2</b> The correlation coefficient</a>
<ul>
<li class="chapter" data-level="29.2.1" data-path="correlation.html"><a href="correlation.html#pearson-product-moment-correlation-coefficient"><i class="fa fa-check"></i><b>29.2.1</b> Pearson product moment correlation coefficient</a></li>
<li class="chapter" data-level="29.2.2" data-path="correlation.html"><a href="correlation.html#spearman-rank-correlation-coefficient"><i class="fa fa-check"></i><b>29.2.2</b> Spearman rank correlation coefficient</a></li>
</ul></li>
<li class="chapter" data-level="29.3" data-path="correlation.html"><a href="correlation.html#correlation-hypothesis-testing"><i class="fa fa-check"></i><b>29.3</b> Correlation hypothesis testing</a></li>
</ul></li>
<li class="chapter" data-level="30" data-path="practical.-analysis-of-counts-and-correlations.html"><a href="practical.-analysis-of-counts-and-correlations.html"><i class="fa fa-check"></i><b>30</b> <em>Practical</em>. Analysis of counts and correlations</a>
<ul>
<li class="chapter" data-level="30.1" data-path="practical.-analysis-of-counts-and-correlations.html"><a href="practical.-analysis-of-counts-and-correlations.html#survival-goodness-of-fit"><i class="fa fa-check"></i><b>30.1</b> Survival goodness of fit</a></li>
<li class="chapter" data-level="30.2" data-path="practical.-analysis-of-counts-and-correlations.html"><a href="practical.-analysis-of-counts-and-correlations.html#colony-goodness-of-fit"><i class="fa fa-check"></i><b>30.2</b> Colony goodness of fit</a></li>
<li class="chapter" data-level="30.3" data-path="practical.-analysis-of-counts-and-correlations.html"><a href="practical.-analysis-of-counts-and-correlations.html#chi-square-test-of-association"><i class="fa fa-check"></i><b>30.3</b> Chi-Square test of association</a></li>
<li class="chapter" data-level="30.4" data-path="practical.-analysis-of-counts-and-correlations.html"><a href="practical.-analysis-of-counts-and-correlations.html#pearson-product-moment-correlation-test"><i class="fa fa-check"></i><b>30.4</b> Pearson product moment correlation test</a></li>
<li class="chapter" data-level="30.5" data-path="practical.-analysis-of-counts-and-correlations.html"><a href="practical.-analysis-of-counts-and-correlations.html#spearman-rank-correlation-test"><i class="fa fa-check"></i><b>30.5</b> Spearman rank correlation test</a></li>
<li class="chapter" data-level="30.6" data-path="practical.-analysis-of-counts-and-correlations.html"><a href="practical.-analysis-of-counts-and-correlations.html#untidy-goodness-of-fit"><i class="fa fa-check"></i><b>30.6</b> Untidy goodness of fit</a></li>
</ul></li>
<li class="part"><span><b>X Linear Regression</b></span></li>
<li class="chapter" data-level="" data-path="Week10.html"><a href="Week10.html"><i class="fa fa-check"></i>Week 10 Overview</a></li>
<li class="chapter" data-level="31" data-path="regression-key-concepts.html"><a href="regression-key-concepts.html"><i class="fa fa-check"></i><b>31</b> Regression key concepts</a></li>
<li class="chapter" data-level="32" data-path="regression-validity.html"><a href="regression-validity.html"><i class="fa fa-check"></i><b>32</b> Regression validity</a></li>
<li class="chapter" data-level="33" data-path="introduction-to-multiple-regression.html"><a href="introduction-to-multiple-regression.html"><i class="fa fa-check"></i><b>33</b> Introduction to multiple regression</a></li>
<li class="chapter" data-level="34" data-path="model-selection-maybe-remove-this.html"><a href="model-selection-maybe-remove-this.html"><i class="fa fa-check"></i><b>34</b> Model selection (maybe remove this?)</a></li>
<li class="chapter" data-level="35" data-path="practical.-using-regression.html"><a href="practical.-using-regression.html"><i class="fa fa-check"></i><b>35</b> <em>Practical</em>. Using regression</a>
<ul>
<li class="chapter" data-level="35.1" data-path="practical.-using-regression.html"><a href="practical.-using-regression.html#regression-exercise-1"><i class="fa fa-check"></i><b>35.1</b> Regression Exercise 1</a></li>
<li class="chapter" data-level="35.2" data-path="practical.-using-regression.html"><a href="practical.-using-regression.html#regression-exercise-2"><i class="fa fa-check"></i><b>35.2</b> Regression Exercise 2</a></li>
<li class="chapter" data-level="35.3" data-path="practical.-using-regression.html"><a href="practical.-using-regression.html#regression-exercise-3"><i class="fa fa-check"></i><b>35.3</b> Regression Exercise 3</a></li>
<li class="chapter" data-level="35.4" data-path="practical.-using-regression.html"><a href="practical.-using-regression.html#regression-exercise-4"><i class="fa fa-check"></i><b>35.4</b> Regression Exercise 4</a></li>
</ul></li>
<li class="part"><span><b>XI Randomisation approaches</b></span></li>
<li class="chapter" data-level="" data-path="Week11.html"><a href="Week11.html"><i class="fa fa-check"></i>Week 11 Overview</a></li>
<li class="chapter" data-level="36" data-path="introduction-to-randomisation.html"><a href="introduction-to-randomisation.html"><i class="fa fa-check"></i><b>36</b> Introduction to randomisation</a></li>
<li class="chapter" data-level="37" data-path="assumptions-of-randomisation.html"><a href="assumptions-of-randomisation.html"><i class="fa fa-check"></i><b>37</b> Assumptions of randomisation</a></li>
<li class="chapter" data-level="38" data-path="bootstrapping.html"><a href="bootstrapping.html"><i class="fa fa-check"></i><b>38</b> Bootstrapping</a></li>
<li class="chapter" data-level="39" data-path="monte-carlo.html"><a href="monte-carlo.html"><i class="fa fa-check"></i><b>39</b> Monte Carlo</a></li>
<li class="chapter" data-level="40" data-path="practical.-using-r.html"><a href="practical.-using-r.html"><i class="fa fa-check"></i><b>40</b> <em>Practical</em>. Using R</a>
<ul>
<li class="chapter" data-level="40.1" data-path="practical.-using-r.html"><a href="practical.-using-r.html#r-exercise-1"><i class="fa fa-check"></i><b>40.1</b> R Exercise 1</a></li>
<li class="chapter" data-level="40.2" data-path="practical.-using-r.html"><a href="practical.-using-r.html#r-exercise-2"><i class="fa fa-check"></i><b>40.2</b> R Exercise 2</a></li>
<li class="chapter" data-level="40.3" data-path="practical.-using-r.html"><a href="practical.-using-r.html#r-exercise-3"><i class="fa fa-check"></i><b>40.3</b> R Exercise 3</a></li>
</ul></li>
<li class="part"><span><b>XII Statistical Reporting</b></span></li>
<li class="chapter" data-level="" data-path="Week12.html"><a href="Week12.html"><i class="fa fa-check"></i>Week 12 Overview</a></li>
<li class="chapter" data-level="41" data-path="reporting-statistics.html"><a href="reporting-statistics.html"><i class="fa fa-check"></i><b>41</b> Reporting statistics</a></li>
<li class="chapter" data-level="42" data-path="more-introduction-to-r.html"><a href="more-introduction-to-r.html"><i class="fa fa-check"></i><b>42</b> More introduction to R</a></li>
<li class="chapter" data-level="43" data-path="more-getting-started-with-r.html"><a href="more-getting-started-with-r.html"><i class="fa fa-check"></i><b>43</b> More getting started with R</a></li>
<li class="chapter" data-level="44" data-path="practical.-using-r-1.html"><a href="practical.-using-r-1.html"><i class="fa fa-check"></i><b>44</b> <em>Practical</em>. Using R</a>
<ul>
<li class="chapter" data-level="44.1" data-path="practical.-using-r-1.html"><a href="practical.-using-r-1.html#r-exercise-1-1"><i class="fa fa-check"></i><b>44.1</b> R Exercise 1</a></li>
<li class="chapter" data-level="44.2" data-path="practical.-using-r-1.html"><a href="practical.-using-r-1.html#r-exercise-2-1"><i class="fa fa-check"></i><b>44.2</b> R Exercise 2</a></li>
<li class="chapter" data-level="44.3" data-path="practical.-using-r-1.html"><a href="practical.-using-r-1.html#r-exercise-3-1"><i class="fa fa-check"></i><b>44.3</b> R Exercise 3</a></li>
</ul></li>
<li class="part"><span><b>XIII Review of parts (VII-XII)</b></span></li>
<li class="chapter" data-level="" data-path="Week13.html"><a href="Week13.html"><i class="fa fa-check"></i>Module summary</a></li>
<li class="appendix"><span><b>Appendix</b></span></li>
<li class="chapter" data-level="A" data-path="appendexA_CMS.html"><a href="appendexA_CMS.html"><i class="fa fa-check"></i><b>A</b> Common Marking Scheme</a></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></li>
<li class="chapter" data-level="C" data-path="appendixC_tables.html"><a href="appendixC_tables.html"><i class="fa fa-check"></i><b>C</b> Statistical tables</a>
<ul>
<li class="chapter" data-level="C.1" data-path="appendixC_tables.html"><a href="appendixC_tables.html#wilcoxon-signed-rank-critical-values"><i class="fa fa-check"></i><b>C.1</b> Wilcoxon signed rank critical values</a></li>
<li class="chapter" data-level="C.2" data-path="appendixC_tables.html"><a href="appendixC_tables.html#mann-whitney-u-critical-values"><i class="fa fa-check"></i><b>C.2</b> Mann-Whitney U critical values</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="./">Statistical Techniques for Biological and Environmental Sciences</a>
</h1>
</div>
<div class="page-wrapper" tabindex="-1" role="main">
<div class="page-inner">
<section class="normal" id="section-">
<div id="correlation" class="section level1 hasAnchor" number="29">
<h1><span class="header-section-number">Chapter 29</span> Correlation<a href="correlation.html#correlation" class="anchor-section" aria-label="Anchor link to header"></a></h1>
<p>This chapter focuses on the association between <a href="Chapter_5.html#Chapter_5">types of variables</a> that are quantitative (i.e., represented by numbers).
It is similar to the <a href="Chapter_28.html#chi-squared-test-of-association">Chi-squared test of association</a> from <a href="Chapter_28.html#Chapter_28">Chapter 28</a> in the sense that it is about how variables are associated.
The focus of the <a href="Chapter_28.html#chi-squared-test-of-association">Chi-squared test of association</a> was on the association when data were categorical (e.g., ‘Android’ or ‘MacOS’ operating system).
Here we focus instead of the association when data are numeric.
But the concept is generally the same; are variables independent, or does knowing something about one variable tell us something about the other variable?
For example, does knowing something about the latitude of a location tell us something about its average yearly temperature?</p>
<div id="scatterplots" class="section level2 hasAnchor" number="29.1">
<h2><span class="header-section-number">29.1</span> Scatterplots<a href="correlation.html#scatterplots" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>The easiest way to visualise the concept of a correlation is by using a scatter-plot.
Scatterplots are useful for visualising the association between two quantitative variables.
In a scatterplot the values of one variable are plotted on the x-axis, and the values of a second variable are plotted on the y-axis.
Consider two fig wasp species of the genus <em>Heterandrium</em> (Figure 29.1).</p>
<div class="figure"><span style="display:block;" id="fig:unnamed-chunk-143"></span>
<img src="img/heterandrium.png" alt="Two small black iridescent wasps are shown side by side with wings facing upwards." width="100%" />
<p class="caption">
Figure 29.1: Fig wasps from 2 different species are shown. Wasps were collected from Baja, Mexico.
</p>
</div>
<p>Both fig wasp species in Figure 29.1 are unnamed.
We can call the species in Figure 29.1A ‘Het1’ and the species in Figure 29.1B ‘Het2’.
We might want to collect morphological measurements of fig wasp head, thorax, and abdomen lengths in these 2 species <span class="citation">(<a href="#ref-Duthie2015b" role="doc-biblioref">Duthie, Abbott, and Nason 2015</a>)</span>.
Table 29.1 shows these measurements for 11 wasps.</p>
<table>
<caption><span id="tab:unnamed-chunk-144">Table 29.1: </span>Body segment length measurements (mm) from fig wasps of 2 species. Data were collected from Baja, Mexico.</caption>
<thead>
<tr class="header">
<th align="left">Species</th>
<th align="right">Head</th>
<th align="right">Thorax</th>
<th align="right">Abdomen</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td align="left">Het1</td>
<td align="right">0.566</td>
<td align="right">0.767</td>
<td align="right">1.288</td>
</tr>
<tr class="even">
<td align="left">Het1</td>
<td align="right">0.505</td>
<td align="right">0.784</td>
<td align="right">1.059</td>
</tr>
<tr class="odd">
<td align="left">Het1</td>
<td align="right">0.511</td>
<td align="right">0.769</td>
<td align="right">1.107</td>
</tr>
<tr class="even">
<td align="left">Het1</td>
<td align="right">0.479</td>
<td align="right">0.766</td>
<td align="right">1.242</td>
</tr>
<tr class="odd">
<td align="left">Het1</td>
<td align="right">0.545</td>
<td align="right">0.828</td>
<td align="right">1.367</td>
</tr>
<tr class="even">
<td align="left">Het1</td>
<td align="right">0.525</td>
<td align="right">0.852</td>
<td align="right">1.408</td>
</tr>
<tr class="odd">
<td align="left">Het2</td>
<td align="right">0.497</td>
<td align="right">0.781</td>
<td align="right">1.248</td>
</tr>
<tr class="even">
<td align="left">Het2</td>
<td align="right">0.450</td>
<td align="right">0.696</td>
<td align="right">1.092</td>
</tr>
<tr class="odd">
<td align="left">Het2</td>
<td align="right">0.557</td>
<td align="right">0.792</td>
<td align="right">1.240</td>
</tr>
<tr class="even">
<td align="left">Het2</td>
<td align="right">0.519</td>
<td align="right">0.814</td>
<td align="right">1.221</td>
</tr>
<tr class="odd">
<td align="left">Het2</td>
<td align="right">0.430</td>
<td align="right">0.621</td>
<td align="right">1.034</td>
</tr>
</tbody>
</table>
<p>Intuitively, we might expect most of these measurements to be associated with one another.
For example, if a fig wasp has a relatively long thorax, then it probably also has a relatively long abdomen (i.e., it could just be a big wasp).
We can check this visually by plotting one variable on the x-axis and the other on the y-axis.
Figure 29.2 does this for wasp thorax length (x-axis) and abdomen length (y-axis).</p>
<div class="figure"><span style="display:block;" id="fig:unnamed-chunk-145"></span>
<img src="bookdown-demo_files/figure-html/unnamed-chunk-145-1.png" alt="A plot is shown with 11 points, such tha tpoints that are higher on the x-axis are generally higher on the y-axis." width="672" />
<p class="caption">
Figure 29.2: An example of a scatterplot in which fig wasp thorax length (x-axis) is plotted against fig wasp abdomen length (y-axis). Each point is a different fig wasp. Wasps were collected in 2010 in Baja, Mexico.
</p>
</div>
<p>In Figure 29.2, each point is a different wasp from Table 29.1.
For example, in the last row of Table 29.1, there is a wasp with a particularly low thorax length (0.621 mm) and abdomen length (1.034 mm).
In the scatterplot, we can see this wasp represented by the point that is lowest and furthest to the left (Figure 29.2).</p>
<p>There is a clear association between thorax length and abdomen length in Figure 29.2.
Fig wasps that have low thorax lengths also tend to have low abdomen lengths, and wasps that high thorax lengths also tend to have high abdomen lengths.
In this sense, the 2 variables are associated.
More specifically, they are positively correlated.
As thorax length increases, so does abdomen length.</p>
</div>
<div id="the-correlation-coefficient" class="section level2 hasAnchor" number="29.2">
<h2><span class="header-section-number">29.2</span> The correlation coefficient<a href="correlation.html#the-correlation-coefficient" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>The <strong>correlation coefficient</strong> formalises the association described in the previous section.
It gives us a single number that defines how 2 variables are correlated.
We represent this number with the letter ‘<span class="math inline">\(r\)</span>’, which can range from values of -1 to 1<a href="#fn53" class="footnote-ref" id="fnref53"><sup>53</sup></a>.
Positive values indicate that two variables are positively correlated, such that a higher value of one variable is associated with higher values of the other variable (as was the case with fig wasp thorax and abdomen measurements).
Negative values indicate that two variables are negatively correlated, such that a higher values of one variable are associated with lower values of the other variable.
Values of zero (or not significantly different from zero, more on this later) indicate that 2 variables are uncorrelated (i.e., independent of one another).
Figure 29.3 shows scatterplots for 8 different correlation coefficients between values of <span class="math inline">\(x\)</span> and <span class="math inline">\(y\)</span>.</p>
<div class="figure"><span style="display:block;" id="fig:unnamed-chunk-146"></span>
<img src="bookdown-demo_files/figure-html/unnamed-chunk-146-1.png" alt="Four scatter plots are shown in a 2 by 2 grid with different correlations between x and y variables." width="480" />
<p class="caption">
Figure 29.3: Examples of scatterplots with different correlation coefficients (r) between 2 variables (x and y).
</p>
</div>
<p>We will look at 2 types of correlation coefficient, the Pearson product moment correlation coefficient and the Spearman rank correlation coefficient.
The 2 are basically the same; the Spearman rank correlation coefficient is just a correlation of the ranks of values instead of the actual values.</p>
<div id="pearson-product-moment-correlation-coefficient" class="section level3 hasAnchor" number="29.2.1">
<h3><span class="header-section-number">29.2.1</span> Pearson product moment correlation coefficient<a href="correlation.html#pearson-product-moment-correlation-coefficient" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>To understand the correlation coefficient, we need to first understand covariance.
<a href="Chapter_12.html#the-variance">Chapter 23.3</a> introduced the variance (<span class="math inline">\(s^{2}\)</span>) as a measure of spread in some variable <span class="math inline">\(x\)</span>,</p>
<p><span class="math display">\[s^{2} = \frac{1}{N - 1}\sum_{i = 1}^{N}\left(x_{i} - \bar{x} \right)^{2}.\]</span></p>
<p>The variance is actually just a special case of a covariance.
The variance describes how a variable <span class="math inline">\(x\)</span> covaries with itself.
The covariance (<span class="math inline">\(cov_{x,y}\)</span>) describes how a variable <span class="math inline">\(x\)</span> covaries with another variable <span class="math inline">\(y\)</span>,</p>
<p><span class="math display">\[cov_{x, y} = \frac{1}{N - 1} \sum_{i = 1}^{N}\left(x_{i} - \bar{x} \right) \left(y_{i} - \bar{y} \right).\]</span></p>
<p>The <span class="math inline">\(\bar{x}\)</span> and <span class="math inline">\(\bar{y}\)</span> are the means of <span class="math inline">\(x\)</span> and <span class="math inline">\(y\)</span>, respectively.
Note that if <span class="math inline">\(x_{i} = y_{i}\)</span>, then the equation for <span class="math inline">\(cov_{x,y}\)</span> is identical to the equation for <span class="math inline">\(s^{2}\)</span> because <span class="math inline">\(\left(x_{i} - \bar{x} \right) \left(x_{i} - \bar{x} \right) = \left(x_{i} - \bar{x} \right)^{2}\)</span>.</p>
<p>What the equation for <span class="math inline">\(cov_{x,y}\)</span> is describing is how the variation in <span class="math inline">\(x\)</span> is relates to variation in <span class="math inline">\(y\)</span>.
If a value of <span class="math inline">\(x_{i}\)</span> is much higher than the mean <span class="math inline">\(\bar{x}\)</span>, and a value of <span class="math inline">\(y_{i}\)</span> is much higher than the mean <span class="math inline">\(\bar{y}\)</span>, then the product of <span class="math inline">\(\left(x_{i} - \bar{x} \right)\)</span> and <span class="math inline">\(\left(y_{i} - \bar{y} \right)\)</span> will be especially high because we will be multiplying 2 large positive number together.
If a value of <span class="math inline">\(x_{i}\)</span> is much higher than the mean <span class="math inline">\(\bar{x}\)</span>, but the corresponding <span class="math inline">\(y_{i}\)</span> is much lower than the mean <span class="math inline">\(\bar{y}\)</span>, then the product of <span class="math inline">\(\left(x_{i} - \bar{x} \right)\)</span> and <span class="math inline">\(\left(y_{i} - \bar{y} \right)\)</span> will be especially low because we will be multiplying a large positive number and a large negative number.
Consequently, when <span class="math inline">\(x_{i}\)</span> and <span class="math inline">\(y_{i}\)</span> tend to deviate from their means <span class="math inline">\(\bar{x}\)</span> and <span class="math inline">\(\bar{y}\)</span> in a consistent way, we get either high or low values of <span class="math inline">\(cov_{x,y}\)</span>.
If there is no such relationship between <span class="math inline">\(x\)</span> and <span class="math inline">\(y\)</span>, then we will get <span class="math inline">\(cov_{x,y}\)</span> values closer to zero.</p>
<p>The covariance can, at least in theory, be any real number.
How low or high it is will depend on the magnitudes of <span class="math inline">\(x\)</span> and <span class="math inline">\(y\)</span>, just like <a href="Chapter_12.html#the-variance">the variance</a>.
To get the Pearson product moment correlation coefficient<a href="#fn54" class="footnote-ref" id="fnref54"><sup>54</sup></a>: (<span class="math inline">\(r\)</span>), we need to standardise the covariance so that the minimum possible value of <span class="math inline">\(r\)</span> is -1 and the maximum possible value of <span class="math inline">\(r\)</span> is 1.
We can do this by dividing <span class="math inline">\(cov_{x,y}\)</span> by the product of the standard deviation of <span class="math inline">\(x\)</span> (<span class="math inline">\(s_{x}\)</span>) and the standard deviation of <span class="math inline">\(y\)</span> (<span class="math inline">\(s_{x}\)</span>),</p>
<p><span class="math display">\[r = \frac{cov_{x,y}}{s_{x} \times s_{y}}.\]</span></p>
<p>This works because <span class="math inline">\(s_{x} \times s_{y}\)</span> describes the total variation between the two variables, so the absolute value of <span class="math inline">\(cov_{x,y}\)</span> cannot be larger <span class="math inline">\(s_{x} \times s_{y}\)</span>.
We can again think about the special case in which <span class="math inline">\(x = y\)</span>.
Since the covariance between <span class="math inline">\(x\)</span> and itself is just the variance of <span class="math inline">\(x\)</span> (<span class="math inline">\(s_{x}^{2}\)</span>), and <span class="math inline">\(s_{x} \times s_{x} = s^{2}_{x}\)</span>, we end up with the same value on the top and bottom and <span class="math inline">\(r = 1\)</span>.</p>
<p>We can expand <span class="math inline">\(cov_{x,y}\)</span>, <span class="math inline">\(s_{x}\)</span>, and <span class="math inline">\(s_{y}\)</span> to see the details of the equation for <span class="math inline">\(r\)</span>,</p>
<p><span class="math display">\[r = \frac{\frac{1}{N - 1} \sum_{i = 1}^{N}\left(x_{i} - \bar{x} \right) \left(y_{i} - \bar{y} \right)}{\sqrt{\frac{1}{N - 1}\sum_{i = 1}^{N}\left(x_{i} - \bar{x} \right)^{2}} \sqrt{\frac{1}{N - 1}\sum_{i = 1}^{N}\left(y_{i} - \bar{y} \right)^{2}}}.\]</span></p>
<p>This looks like a lot, but we can clean the equation up a bit because the <span class="math inline">\(1 / (N-1)\)</span> expressions cancel on the top and bottom of the equation,</p>
<p><span class="math display">\[r = \frac{\sum_{i = 1}^{N}\left(x_{i} - \bar{x} \right) \left(y_{i} - \bar{y} \right)}{\sqrt{\sum_{i = 1}^{N}\left(x_{i} - \bar{x} \right)^{2}} \sqrt{\sum_{i = 1}^{N}\left(y_{i} - \bar{y} \right)^{2}}}.\]</span></p>
<p>As with other statistics defined in this book, it is almost never necessary to calculate <span class="math inline">\(r\)</span> by hand.
Statistical programs such as Jamovi and R will make these calculations for us <span class="citation">(<a href="#ref-Jamovi2022" role="doc-biblioref">The Jamovi Project 2022</a>; <a href="#ref-Rproject" role="doc-biblioref">R Core Team 2022</a>)</span>.
The reason for working through all of these equations is to help make the conceptual link between <span class="math inline">\(r\)</span> and the variance of the variables of interest.
To make this link a bit more clear, we can calculate the correlation coefficient of thorax and abdomen length from Table 29.1.
We can set thorax to be the <span class="math inline">\(x\)</span> variable and abdomen to be the <span class="math inline">\(y\)</span> variable.
Mean thorax length is <span class="math inline">\(\bar{x} = 0.770\)</span>, and mean abdomen length is <span class="math inline">\(\bar{y} = 1.210\)</span>.
The standard deviation of thorax length is <span class="math inline">\(s_{x} = 0.064\)</span>, and the standard deviation of abdomen length is <span class="math inline">\(s_{y} = 0.123\)</span>.
This gives us the numbers that we need to calculate the bottom of the fraction for <span class="math inline">\(r\)</span>, which is <span class="math inline">\(s_{x} \times s_{y} = 0.007872\)</span>.
We now need to calculate the covariance on the top.
To get the covariance, we first need to calculate <span class="math inline">\(\left(x_{i} - \bar{x} \right) \left(y_{i} - \bar{y} \right)\)</span> for each row (<span class="math inline">\(i\)</span>) in Table 29.1.
For example, for row 1, <span class="math inline">\(\left(0.767 - 0.770\right) \left(1.288 - 1.210\right) = -0.000234.\)</span>
For row 2, <span class="math inline">\(\left(0.784 - 0.770\right) \left(1.059 - 1.210\right) = -0.002114.\)</span>
We continue this for all rows.
Table 29.2 shows the thorax length (<span class="math inline">\(x_{i}\)</span>), abdomen length (<span class="math inline">\(y_{i}\)</span>), and <span class="math inline">\(\left(x_{i} - \bar{x} \right) \left(y_{i} - \bar{y} \right)\)</span> for rows <span class="math inline">\(i = 1\)</span> to <span class="math inline">\(i = 11\)</span>.</p>
<table>
<caption><span id="tab:unnamed-chunk-147">Table 29.2: </span>Measurements of 11 fig wasp thorax and abdomen lengths (mm). The fourth column shows the product of the deviations of each measurement from the mean, where mean thorax length is 0.770 and mean abdomen length is 1.210.</caption>
<thead>
<tr class="header">
<th align="right">Row (i)</th>
<th align="right">Thorax</th>
<th align="right">Abdomen</th>
<th align="right">Squared Deviation</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td align="right">1</td>
<td align="right">0.767</td>
<td align="right">1.288</td>
<td align="right">-0.000234</td>
</tr>
<tr class="even">
<td align="right">2</td>
<td align="right">0.784</td>
<td align="right">1.059</td>
<td align="right">-0.002114</td>
</tr>
<tr class="odd">
<td align="right">3</td>
<td align="right">0.769</td>
<td align="right">1.107</td>
<td align="right">0.000103</td>
</tr>
<tr class="even">
<td align="right">4</td>
<td align="right">0.766</td>
<td align="right">1.242</td>
<td align="right">-0.000128</td>
</tr>
<tr class="odd">
<td align="right">5</td>
<td align="right">0.828</td>
<td align="right">1.367</td>
<td align="right">0.009106</td>
</tr>
<tr class="even">
<td align="right">6</td>
<td align="right">0.852</td>
<td align="right">1.408</td>
<td align="right">0.016236</td>
</tr>
<tr class="odd">
<td align="right">7</td>
<td align="right">0.781</td>
<td align="right">1.248</td>
<td align="right">0.000418</td>
</tr>
<tr class="even">
<td align="right">8</td>
<td align="right">0.696</td>
<td align="right">1.092</td>
<td align="right">0.008732</td>
</tr>
<tr class="odd">
<td align="right">9</td>
<td align="right">0.792</td>
<td align="right">1.240</td>
<td align="right">0.000660</td>
</tr>
<tr class="even">
<td align="right">10</td>
<td align="right">0.814</td>
<td align="right">1.221</td>
<td align="right">0.000484</td>
</tr>
<tr class="odd">
<td align="right">11</td>
<td align="right">0.621</td>
<td align="right">1.034</td>
<td align="right">0.026224</td>
</tr>
</tbody>
</table>
<p>If we sum up all of the values in the column “Squared Deviation” from Table 29.2, we get a value of 0.059487.
We can multiply this value by <span class="math inline">\(1 / (N - 1)\)</span> to get the top of the equation for <span class="math inline">\(r\)</span>, <span class="math inline">\((1 / (11-1)) \times 0.059487 = 0.0059487\)</span>.
We now have all of the values we need to calculate <span class="math inline">\(r\)</span> between fig wasp thorax and abdomen length,</p>
<p><span class="math display">\[r_{x,y} = \frac{0.0059487}{0.064 \times 0.123}.\]</span></p>
<p>Our final value is <span class="math inline">\(r_{x, y} = 0.7556784\)</span>.
As suggested by the scatterplot in Figure 29.2, thorax and abdomen lengths are highly correlated.
We will test whether or not this value of <span class="math inline">\(r\)</span> is statistically significant in <a href="correlation.html#correlation-hypothesis-testing">Chapter 23.3</a> below, but first we will introduce the Spearman’s rank correlation coefficient.</p>
</div>
<div id="spearman-rank-correlation-coefficient" class="section level3 hasAnchor" number="29.2.2">
<h3><span class="header-section-number">29.2.2</span> Spearman rank correlation coefficient<a href="correlation.html#spearman-rank-correlation-coefficient" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>Throughout this book, we have seen how the <em>ranks</em> of data can be substituted in place of the actual values.
This has been useful when data violate the assumptions of a statistical test, and we need a nonparametric test instead (e.g., the <a href="Chapter_22.html#wilcoxon-test-1">Wilcoxon signed rank test</a>, the <a href="Chapter_22.html#mann-whitney-u-test-1">Mann-Whitney U test</a>, or the <a href="Chapter_25.html#Chapter_25">Kruskall-Wallis H test</a>).
We can use the same trick for the correlation coefficient.
The Spearman rank correlation coefficient is calculated the exact same way as the Pearson product moment correlation coefficient, except on the ranks of values.
To calculate the Spearman rank correlation coefficient for the fig wasp example in <a href="correlation.html#pearson-product-moment-correlation-coefficient">the previous section</a>, we just need to rank the thorax and abdomen lengths from 1 to 11, then calculate <span class="math inline">\(r\)</span> using the rank values instead of the actual measurements of length.
Figure 29.3 shows the same 11 fig wasp measurements as Figures 29.1 and 29.2, but with columns added to show the ranks of thorax and abdomen lengths.</p>
<table>
<caption><span id="tab:unnamed-chunk-148">Table 29.3: </span>Measurements of 11 fig wasp thorax and abdomen lengths (mm) and their ranks.</caption>
<thead>
<tr class="header">
<th align="right">Wasp (i)</th>
<th align="right">Thorax</th>
<th align="right">Thorax rank</th>
<th align="right">Abdomen</th>
<th align="right">Abdomen rank</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td align="right">1</td>
<td align="right">0.767</td>
<td align="right">4</td>
<td align="right">1.288</td>
<td align="right">9</td>
</tr>
<tr class="even">
<td align="right">2</td>
<td align="right">0.784</td>
<td align="right">7</td>
<td align="right">1.059</td>
<td align="right">2</td>
</tr>
<tr class="odd">
<td align="right">3</td>
<td align="right">0.769</td>
<td align="right">5</td>
<td align="right">1.107</td>
<td align="right">4</td>
</tr>
<tr class="even">
<td align="right">4</td>
<td align="right">0.766</td>
<td align="right">3</td>
<td align="right">1.242</td>
<td align="right">7</td>
</tr>
<tr class="odd">
<td align="right">5</td>
<td align="right">0.828</td>
<td align="right">10</td>
<td align="right">1.367</td>
<td align="right">10</td>
</tr>
<tr class="even">
<td align="right">6</td>
<td align="right">0.852</td>
<td align="right">11</td>
<td align="right">1.408</td>
<td align="right">11</td>
</tr>
<tr class="odd">
<td align="right">7</td>
<td align="right">0.781</td>
<td align="right">6</td>
<td align="right">1.248</td>
<td align="right">8</td>
</tr>
<tr class="even">
<td align="right">8</td>
<td align="right">0.696</td>
<td align="right">2</td>
<td align="right">1.092</td>
<td align="right">3</td>
</tr>
<tr class="odd">
<td align="right">9</td>
<td align="right">0.792</td>
<td align="right">8</td>
<td align="right">1.240</td>
<td align="right">6</td>
</tr>
<tr class="even">
<td align="right">10</td>
<td align="right">0.814</td>
<td align="right">9</td>
<td align="right">1.221</td>
<td align="right">5</td>
</tr>
<tr class="odd">
<td align="right">11</td>
<td align="right">0.621</td>
<td align="right">1</td>
<td align="right">1.034</td>
<td align="right">1</td>
</tr>
</tbody>
</table>
<p>Note from Table 29.3 that the lowest value of Thorax is 0.621, so it gets a rank of 1.
The highest value of Thorax is 0.852, so it gets a rank of 11.
We do the same for abdomen ranks.
To get the Spearman rank correlation coefficient, we just calculate <span class="math inline">\(r\)</span> using the ranks.
The ranks number from 1 to 11 for both variables, so the mean rank is 6 and the standard deviation is 3.317 for both thorax and abdomen ranks.
We can then go through each row and calculate <span class="math inline">\(\left(x_{i} - \bar{x} \right) \times \left(y_{i} - \bar{y} \right)\)</span> using the ranks.
For the first row, this gives us <span class="math inline">\(\left(4 - 6 \right) \left(9 - 6 \right) = -6\)</span>.
If we do this same calculation for each row and sum them up, then multiply by 1/(N-1), we get a value of 6.4.
To calculate <span class="math inline">\(r\)</span>,</p>
<p><span class="math display">\[r_{rank(x),rank(y)} = \frac{6.4}{3.317 \times 3.317}\]</span></p>
<p>Our Spearman rank correlation coefficient is therefore <span class="math inline">\(r = 0.582\)</span>, which is a bit lower than our Pearson product moment correlation was.
The key point here is that the definition of the correlation coefficient has not changed; we are just using the ranks of our measurements instead of the measurements themselves.
The reason why we might want to use the Spearman rank correlation coefficient instead of the Pearson product moment correlation coefficient is explained in the next section.</p>
</div>
</div>
<div id="correlation-hypothesis-testing" class="section level2 hasAnchor" number="29.3">
<h2><span class="header-section-number">29.3</span> Correlation hypothesis testing<a href="correlation.html#correlation-hypothesis-testing" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>We often want to test if 2 variables are correlated.
In other words, is <span class="math inline">\(r\)</span> significantly different from zero?
We therefore want to test the null hypothesis that <span class="math inline">\(r\)</span> is not significantly different from zero.</p>
<ul>
<li><span class="math inline">\(H_{0}:\)</span> The population correlation coefficient is zero.</li>
<li><span class="math inline">\(H_{A}:\)</span> The correlation coefficient is significantly different from zero.</li>
</ul>
<p>Note that <span class="math inline">\(H_{A}\)</span> above is for a two-tailed test, in which we do not care about direction.
We could also use a one-tailed test if our <span class="math inline">\(H_{A}\)</span> is that the correlation coefficient is greater than or less than zero.</p>
<p>How do we actually test the null hypothesis?
As it turns out, the sample correlation coefficient (<span class="math inline">\(r\)</span>) will be approximately t-distributed around a true mean (<span class="math inline">\(\rho\)</span>) with a t-score defined by <span class="math inline">\(r - \rho\)</span> divided by its standard error (<span class="math inline">\(SE(r)\)</span>)<a href="#fn55" class="footnote-ref" id="fnref55"><sup>55</sup></a>,</p>
<p><span class="math display">\[t = \frac{r - \rho}{SE(r)}.\]</span></p>
<p>Since our null hypothesis is that variables are uncorrelated, <span class="math inline">\(\rho = 0\)</span>.
Statistical programs such as Jamovi or R will use this equation to test whether or not the correlation coefficient is significantly different from zero <span class="citation">(<a href="#ref-Jamovi2022" role="doc-biblioref">The Jamovi Project 2022</a>; <a href="#ref-Rproject" role="doc-biblioref">R Core Team 2022</a>)</span>.
The reason for presenting it here is to show the conceptual link to other hypothesis tests in earlier chapters.
In <a href="Chapter_21.html#one-sample-t-test">Chapter 21.1</a>, we saw that the one sample t-test defined <span class="math inline">\(t\)</span> as the deviation of the sample mean from the true mean, divided by the standard error.
Here we are doing the same for the correlation coefficient.
One consequence of this is that, like the one sample t-test, the test of the correlation coefficient assumes that <span class="math inline">\(r\)</span> will be normally distributed around the true mean <span class="math inline">\(\rho\)</span>.
If this is not the case, and the assumption of normality is violated, then the test might have a misleading Type I error rate.</p>
<p>To be cautious, we should check whether or not the variables that we are correlating are normally distributed (especially if the sample size is small).
If they are normally distributed, then we can use the Pearson’s product moment correlation to test the null hypothesis.
If the assumption of normality is violated, then we might consider using the non-parametric Spearman rank correlation coefficient instead.
The fig wasp thorax and abdomen lengths from Table 29.1 are normally distributed, so we can use the Pearson product moment correlation coefficient to test whether or not the correlation between these two variables is significant.
In Jamovi, the t-score is not even reported as output when using a correlation test.
We only see <span class="math inline">\(r\)</span> and the p-value (Figure 23.4).</p>
<div class="figure"><span style="display:block;" id="fig:unnamed-chunk-149"></span>
<img src="img/Jamovi_correlation_output.png" alt="A table called 'Correlation Matrix' is shown, which is mostly empty except for two values in the lower left of a 2 by 2 matrix." width="100%" />
<p class="caption">
Figure 29.4: Jamovi output for a test of the null hypothesis that thorax length and abdomen length are not significantly correlated in a sample of fig wasps collected in 2010 from Baja, Mexico.
</p>
</div>
<p>From Figure 29.4, we can see that the sample <span class="math inline">\(r = 0.75858\)</span>, and the p-value is <span class="math inline">\(P = 0.00680\)</span>.
Since our p-value is less than 0.05, we can reject the null hypothesis that fig was thorax and abdomen lengths are not correlated.
To get a more intuitive sense of how the correlation coefficient works, we can use an <a href="https://bradduthie.shinyapps.io/corr_click/">interactive application</a>.</p>
<blockquote>
<p><a href="https://bradduthie.shinyapps.io/corr_click/">Click here</a> for an interactive application showing the relationship between a scatterplot, a correlation coefficient, and statistical significant.</p>
</blockquote>
<p>Points can be added to the <a href="https://bradduthie.shinyapps.io/corr_click/">interactive application</a> by clicking anywhere within the plot.
A table of <span class="math inline">\(x\)</span> and <span class="math inline">\(y\)</span> points is shown to the right, and the correlation coefficient and p-value can be shown or hidden using the buttons on the top.</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">
<div id="ref-Duthie2015b" class="csl-entry">
Duthie, A Bradley, Karen C Abbott, and John D Nason. 2015. <span>“<span class="nocase">Trade-offs and coexistence in fluctuating environments: evidence for a key dispersal-fecundity trade-off in five nonpollinating fig wasps</span>.”</span> <em>American Naturalist</em> 186 (1): 151–58. <a href="https://doi.org/10.1086/681621">https://doi.org/10.1086/681621</a>.
</div>
<div id="ref-Rproject" class="csl-entry">
R Core Team. 2022. <em>R: A Language and Environment for Statistical Computing</em>. Vienna, Austria: R Foundation for Statistical Computing. <a href="https://www.R-project.org/">https://www.R-project.org/</a>.
</div>
<div id="ref-Rahman1968" class="csl-entry">
Rahman, N A. 1968. <em><span class="nocase">A Course in Theoretical Statistics</span></em>. Charles Griffin; Company.
</div>
<div id="ref-Jamovi2022" class="csl-entry">
The Jamovi Project. 2022. <span>“Jamovi.”</span> Sydney, Australia. <a href="https://www.jamovi.org">https://www.jamovi.org</a>.
</div>
</div>
<div class="footnotes">
<hr />
<ol start="53">
<li id="fn53"><p>Note that <span class="math inline">\(r\)</span> is the sample correlation coefficient, which is an estimate of the population correlation coefficient. The population correlation coefficient is represented by the Greek letter ‘<span class="math inline">\(\rho\)</span>’ (‘rho’), and sometimes the sample correlation coefficient is represented as <span class="math inline">\(\hat{\rho}\)</span>.<a href="correlation.html#fnref53" class="footnote-back">↩︎</a></p></li>
<li id="fn54"><p>We can usually just call this the ‘correlation coefficient’.<a href="correlation.html#fnref54" class="footnote-back">↩︎</a></p></li>
<li id="fn55"><p>We can calculate the standard error of r as <span class="citation">(<a href="#ref-Rahman1968" role="doc-biblioref">Rahman 1968</a>)</span>, <span class="math display">\[SE(r) = \sqrt{\frac{1 - r^{2}}{N - 2}}.\]</span> But this is not necessary to ever do by hand. Note that we lose 2 degrees of freedom (N - 2), one for calculating each variable mean.<a href="correlation.html#fnref55" 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="practical.-analysis-of-counts-and-correlations.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/09-Correlation.Rmd",
"text": "Edit"
},
"history": {
"link": null,
"text": null
},
"view": {
"link": null,
"text": null
},
"download": ["bookdown-demo.pdf", "bookdown-demo.epub"],
"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://mathjax.rstudio.com/latest/MathJax.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>