20:
775:
1224:
It seems surprising that the warnings of three such eminent statistician-scientists as
Pearson, Galton and Weldon should have largely gone unheeded for so long: even today uncritical applications of inappropriate statistical methods to compositional data with consequent dubious inferences are
474:
1151:
1478:
Lovell, David; MĂĽller, Warren; Taylor, Jen; Zwart, Alec; Helliwell, Chris (2011). "Chapter 14: Proportions, Percentages, PPM: Do the
Molecular Biosciences Treat Compositional Data Right?". In Pawlowsky-Glahn, Vera; Buccianti, Antonella (eds.).
245:
1208:
measurements by dividing them by a particular variable or total. The danger he saw was that conclusions would be drawn from correlations that are artifacts of the analysis method, rather than actual “organic” relationships.
770:{\displaystyle \rho ={\frac {r_{12}v_{1}v_{2}-r_{14}v_{1}v_{4}-r_{23}v_{2}v_{3}+r_{34}v_{3}v_{4}}{{\sqrt {v_{1}^{2}+v_{3}^{2}-2r_{13}v_{1}v_{3}}}{\sqrt {v_{2}^{2}+v_{4}^{2}-2r_{24}v_{2}v_{4}}}}}}
164:
1251:
Pearson, Karl (1896). "Mathematical
Contributions to the Theory of Evolution – On a Form of Spurious Correlation Which May Arise When Indices Are Used in the Measurement of Organs".
466:
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78:, which deals with the analysis of variables that carry only relative information, such as proportions, percentages and parts-per-million.
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are drawn from normal distributions with means 10, 10, and 30, respectively, and standard deviations 1, 1, and 3 respectively, i.e.,
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263:
1212:
However, it would appear that spurious correlation (and its potential to mislead) is not yet widely understood. In 1986
1201:
1156:
For the special case in which all coefficients of variation are equal (as is the case in the illustrations at right),
1205:
1204:
in cautioning scientists to be wary of spurious correlation, especially in biology where it is common to scale or
51:
are statistically independent of each other (i.e., the pairwise correlations between each of them are zero). The
808:
405:
1512:
Lovell, David; Pawlowsky-Glahn, Vera; Egozcue, Juan José; Marguerat, Samuel; Bähler, Jürg (16 March 2015).
1146:{\displaystyle \rho _{0}={\frac {v_{3}^{2}}{{\sqrt {v_{1}^{2}+v_{3}^{2}}}{\sqrt {v_{2}^{2}+v_{3}^{2}}}}}.}
1384:
Galton, Francis (1896). "Note to the memoir by
Professor Karl Pearson, F.R.S., on spurious correlation".
1229:
More recent publications suggest that this lack of awareness prevails, at least in molecular bioscience.
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Pearson derived an approximation of the correlation that would be observed between two indices (
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The phenomenon of spurious correlation of ratios is one of the main motives for the field of
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This expression can be simplified for situations where there is a common divisor by setting
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240:{\displaystyle {\begin{aligned}x,y&\sim N(10,1)\\z&\sim N(30,3)\\\end{aligned}}}
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71:
that arises between ratios of absolute measurements which themselves are uncorrelated.
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1405:
1464:
306:
values tend to appear in the bottom left of the plot; trios with relatively small
286:) and can be better understood if we colour the points in the scatter plot by the
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1287:
869:
23:
An illustration of spurious correlation, this figure shows 500 observations of
60:
1304:
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1397:
1264:
1488:
1360:
129:
The scatter plot above illustrates this example using 500 observations of
1419:
Jackson, DA; Somers, KM (1991). "The
Spectre of 'Spurious' Correlation".
282:
have a correlation of 0.53. This is because of the common divisor (
1514:"Proportionality: A Valid Alternative to Correlation for Relative Data"
1448:
1440:
1272:
266:
and therefore uncorrelated, in the depicted typical sample the ratios
125:
for each triplet, and correlation will be found between these indices.
109:, these will be pair and pair uncorrelated. Form the proper fractions
1197:
18:
81:
Spurious correlation is distinct from misconceptions about
97:
Select three numbers within certain ranges at random, say
93:
Pearson states a simple example of spurious correlation:
1288:"Correlations Genuine and Spurious in Pearson and Yule"
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1031:
975:
935:
905:
878:
844:
817:
786:
477:
408:
366:
324:
162:
1481:
Compositional Data
Analysis: Theory and Applications
1353:
Compositional Data
Analysis: Theory and Applications
1022:are uncorrelated, giving the spurious correlation:
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1324:The statistical analysis of compositional data
39:. The sample correlation is 0.53, even though
402:), i.e., ratios of the absolute measurements
8:
1317:
1315:
1386:Proceedings of the Royal Society of London
1253:Proceedings of the Royal Society of London
1216:, who pioneered the log-ratio approach to
314:Approximate amount of spurious correlation
55:-values are highlighted on a colour scale.
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310:values tend to appear in the top right.
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1192:Relevance to biology and other sciences
461:{\displaystyle x_{1},x_{2},x_{3},x_{4}}
7:
89:Illustration of spurious correlation
14:
1015:{\displaystyle x_{1},x_{2},x_{3}}
230:
218:
198:
186:
65:spurious correlation of ratios
1:
1182:{\displaystyle \rho _{0}=0.5}
1539:10.1371/journal.pcbi.1004075
1218:compositional data analysis
1202:Walter Frank Raphael Weldon
962:{\displaystyle x_{3}=x_{4}}
395:{\displaystyle x_{2}/x_{4}}
353:{\displaystyle x_{1}/x_{3}}
76:compositional data analysis
1594:
1578:Covariance and correlation
1518:PLOS Computational Biology
264:statistically independent
83:correlation and causality
1322:Aitchison, John (1986).
809:coefficient of variation
302:) with relatively large
1347:Pawlowsky-Glahn, Vera;
1398:10.1098/rspl.1896.0077
1326:. Chapman & Hall.
1286:Aldrich, John (1995).
1265:10.1098/rspl.1896.0076
1227:
1196:Pearson was joined by
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1147:
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861:{\displaystyle r_{ij}}
832:
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1489:10.1002/9781119976462
1361:10.1002/9781119976462
1305:10.1214/ss/1177009870
1184:
1148:
1017:
964:
921:
919:{\displaystyle x_{j}}
894:
892:{\displaystyle x_{i}}
863:
833:
831:{\displaystyle x_{i}}
802:
800:{\displaystyle v_{i}}
772:
463:
397:
355:
242:
22:
16:Concept in statistics
1392:(359–367): 498–502.
1349:Buccianti, Antonella
1259:(359–367): 489–498.
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1029:
973:
933:
903:
876:
842:
815:
784:
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364:
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69:spurious correlation
1530:2015PLSCB..11E4075L
1433:1991Oecol..86..147J
1292:Statistical Science
1225:regularly reported.
1134:
1116:
1097:
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870:Pearson correlation
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707:
652:
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1441:10.1007/bf00317404
1198:Sir Francis Galton
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290:-value. Trios of (
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1333:978-0-412-28060-3
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31:plotted against
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1524:(3): e1004075.
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1351:, eds. (2011).
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1427:(1): 147–151.
1411:
1376:
1370:978-0470711354
1369:
1339:
1332:
1311:
1298:(4): 364–376.
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1214:John Aitchison
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67:is a form of
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34:
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26:
21:
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317:
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259:
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251:
250:Even though
249:
150:
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141:. Variables
138:
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98:
96:
92:
80:
73:
64:
58:
52:
48:
44:
40:
36:
32:
28:
24:
1233:References
61:statistics
1483:. Wiley.
1421:Oecologia
1406:170846631
1355:. Wiley.
1206:normalize
1165:ρ
1034:ρ
727:−
654:−
551:−
518:−
479:ρ
213:∼
181:∼
1572:Category
1558:25775355
1457:28313173
872:between
1549:4361748
1526:Bibcode
1465:1116627
1449:4219582
1429:Bibcode
1220:wrote:
807:is the
298:,
294:,
1556:
1546:
1495:
1463:
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1367:
1330:
1273:115879
1271:
969:, and
838:, and
780:where
258:, and
137:, and
47:, and
1461:S2CID
1445:JSTOR
1402:S2CID
1269:JSTOR
1554:PMID
1493:ISBN
1453:PMID
1365:ISBN
1328:ISBN
1200:and
899:and
868:the
360:and
274:and
262:are
149:and
117:and
1544:PMC
1534:doi
1485:doi
1437:doi
1394:doi
1357:doi
1300:doi
1261:doi
1177:0.5
811:of
59:In
1574::
1552:.
1542:.
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1275:.
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1174:=
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216:N
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41:x
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35:/
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27:/
25:x
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