523:"Whilst the total effect of birth weight on BP is not affected by the numbers of intermediate body size variables in the model, the estimation of 'direct' effect differs when different intermediate variables are adjusted for. Unless there is experimental evidence to support the notion that there are indeed different paths of direct and indirect effects from birth weight to BP, we are cautious of using such terminology to label the results from multiple regression, as with model 3. In other words, to determine whether the unconditional or conditional relationship reflects the true physiological relationship between birth weight and blood pressure, experiments in which birth weight and current weight can be manipulated are required in order to estimate the impact of birth weight on blood pressure." (pg8)
133:) statement that for males and females of equal September weight, the males gain more than the females. In contrast, if the statisticians turned these descriptive statements into causal statements, neither would be correct or incorrect because untestable assumptions determine the correctness of causal statements... Statistician 1 is wrong because he makes a causal statement without specifying the assumption needed to make it true. Statistician 2 is more cautious, since he makes only a descriptive statement. However, unless he too makes further assumptions, his descriptive statement is completely irrelevant to the campus dietician's interest in the effect of the dining hall diet." (pg. 19)
266:
both through a direct effect and an indirect effect (by influencing initial weight, which then influences final weight). Note that none of these variables are confounders, so controls are not strictly necessary in this model. However, the choice of whether to control for initial weight dictates which effect the researcher is measuring: the first statistician does not control and measures a total effect, while the second does control and measures a direct effect.
163:
270:"Cases where total and direct effects differ in sign are commonplace. For example, we are not at all surprised when smallpox inoculation carries risks of fatal reaction, yet reduces overall mortality by eradicating smallpox. The direct effect (fatal reaction) in this case is negative for every stratum of the population, yet the total effect (on mortality) is positive for the population as a whole." (pg 4)
71:
evidence of any interesting effect of diet (or of anything else) on student weights. In particular, there is no evidence of any differential effect on the two sexes, since neither group shows any systematic change." (pg. 305) Visually, the first statistician sees that neither group mean ('A' and 'B') has changed, and concludes that the new diet had no causal impact.
45:
58:“A large university is interested in investigating the effects on the students of the diet provided in the university dining halls and any sex differences in these effects. Various types of data are gathered. In particular, the weight of each student at the time of his arrival in September and his weight the following June are recorded.” (Lord 1967, p. 304)
129:“In summary, we believe that the following views resolve Lord's Paradox. If both statisticians made only descriptive statements, they would both be correct. Statistician 1 makes their unconditional descriptive statements that the average weight gains for males and females are equal; Statistician 2 makes the conditional (on
78:(ANCOVA), and compares (adjusted) final weights as the outcome. He finds a significant difference between the two dining halls. Visually, the second statistician fits a regression model (green dotted lines), finds that the intercept differs for boys vs girls, and concludes that the new diet had a larger impact for males.
155:
between groups?" Moreover, since the answer depends on the causal model assumed, we should explain: (1) Why people find Lord’s story to be "Paradoxical" rather than "In need of more information" and, (2) How to properly utilize causal models to answer Lord’s question, regardless of whether they are testable or not.
124:
Holland & Rubin (1983) argue that both statisticians have captured accurate descriptive features of the data: Statistician 1 accurately finds no difference in relative weight changes across the two genders, while
Statistician 2 accurately finds a larger average weight gain for boys conditional on
106:
Bock responded to the paradox by positing that both statisticians in the scenario are correct once the question being asked is clarified. The first statistician (who compares group means and distributions) is asking "are there differences in average weight gain?", whereas the second is asking "what
265:
Going back to Lord's original problem of comparing boys and girls, Pearl (2016) posits another causal model where sex and initial weight both influence the final weight. Moreover, since sex also influences the initial weight, Initial Weight becomes a mediating variable: sex influences final weight
233:
One intuition claims that, to get the needed effect, we must make “proper allowances” for uncontrolled preexisting differences between groups” (i.e. initial weights). The second claims that the overall effect (of Diet on Gain) is just the average of the stratum-specific effects. The two intuitions
70:
One statistician does not adjust for initial weight, instead using t-test and comparing gain scores (individuals' average final weight − average initial weight) as the outcome. The first statistician claims no significant difference between genders: "s far as these data are concerned, there is no
170:
The two ellipses represent two dining halls, each serving a different diet, and each point represents a student's initial and final weights. Note that students who weigh more in the beginning tend to eat in dining hall B, while the ones who weigh less eat in dining hall A. The first statistician
154:
Pearl (2016) agrees with Lord’s conclusion that the answer cannot be found in the data, but he finds
Holland and Rubin’s account to be incomplete. In his views, a complete resolution of the Paradox should provide an answer to Lord’s essential question: "How to allow for preexisting differences
145:
happened absent an intervention. The latter is unobservable in the real world, a fact that
Holland & Rubin term "the fundamental problem of causal inference" (pg. 10). This is explains why researchers often turn to experiments: while we still never observe both counterfactuals for a single
115:
Cox and McCullagh interpret the problem by constructing a model of what could have happened had the students not dined in the dining hall, where they assume that a student's weight would have stayed constant. They conclude that in fact the first statistician was right when asking about group
31:
gave examples when statisticians could reach different conclusions depending on whether they adjust for pre-existing differences. Holland & Rubin (1983) use these examples to illustrate how there may be multiple valid descriptive comparisons in the data, but causal conclusions require an
125:
a boy and girl have the same starting weight. However, when turning these descriptions into causal statements, they implicitly assert that weight would have otherwise stayed constant (Statistician 1) or that it would have followed the posited linear model (Statistician 2).
146:
subject, experiments let us make statistical claims about these differences in the population under minimal assumptions. Absent an experiment, modelers should carefully describe the model they use to make causal statements and justify those models as strongly as possible.
199:. He concludes therefore that the students on Diet B gain more than those on Diet A. As before, the data can’t tell us whom to believe, and a causal model must be assumed to settle the issue. One plausible model is shown in Figure 2(b). In this model,
158:
To this end, Pearl used a simplified version of Lord’s
Paradox, proposed by Wainer and Brown, in which gender differences are not considered. Instead, the quantity of interest is the effect of diet on weight gain, as shown in Figure 2(a).
310:
More directly, Lord's
Paradox may have implications for both education and health policies that attempt to reward educators or hospitals for the improvements that their children/patients made under their care, which is the basis for
578:
DB Rubin, EA Stuart, EL Zanutto, A potential outcomes view of value-added assessment in education, Journal of educational and behavioral
Statistics, Vol. 29, No. 1, Value-Added Assessment Special Issue (Spring, 2004), pp. 103–116
274:
Tu, Gunnell, and
Gilthorpe (2008) use a similar causal framework, but counter that the conceptualization of direct and total effect is not the best framework in many cases because there are many different variables that
507:
Wainer, H.; Brown, L. (2007) "Three statistical paradoxes in the interpretation of group differences: Illustrated with medical school admission and licensing data" In Rao, C.; Sinharay, S. (Eds.)
555:
Yu-Kang Tu, David
Gunnell, Mark S Gilthorpe. Simpson's Paradox, Lord's Paradox, and Suppression Effects are the same phenomenon – the reversal paradox. Emerg Themes Epidemiol. 2008; 5: 2.
234:
are valid, but seem to clash when we interpret the first statistician’s finding as zero effect when, in fact, his finding merely entails equality of distributions, and says nothing about
137:
Moreover, the underlying assumptions necessary to turn descriptive statements into causal statements are untestable. Unlike descriptive statements (e.g. "the average height in the US is
230:
According to Pearl, the data triggers a clash between two strong intuitions, both are valid in causal thinking, but not in the non-causal thinking invoked by the first statistician.
64:
In both
September and June, the overall distribution of male weights is the same, although individuals' weights have changed, and likewise for the distribution of female weights.
366:
Lord, E. M. (1975). Lord's paradox. In S. B. Anderson, S. Ball, R. T. Murphy, & Associates, Encyclopedia of
Educational Evaluation (pp. 232–236). San Francisco, CA: Jossey-Bass.
185:
has the same distribution in both ellipses. The second statistician compares the final weights under Diet A to those of Diet B for a group of students with same initial weight
81:
Lord concluded: "there simply is no logical or statistical procedure that can be counted on to make proper allowance for uncontrolled preexisting differences between groups."
249:
This resolution of Lord’s Paradox answers both questions: (1) How to allow for preexisting differences between groups and (2) Why the data appear paradoxical. Pearl's
307:
play major roles in applied statistics. Lord's Paradox and associated analyses provide a powerful teaching tool to understand these fundamental statistical concepts.
531:, Mark S Gilthorpe. Simpson's Paradox, Lord's Paradox, and Suppression Effects are the same phenomenon – the reversal paradox. Emerg Themes Epidemiol. 2008; 5: 2.
291:. Those authors state that Lord's Paradox, Simpson's Paradox, and the suppression of covariates by uncorrelated predictor variables are all the same thing, namely a
224:
is essential for deconfounding the causal effect needed. Assuming this model, the second statistician would be correct and the first statistician would be wrong.
634:
may fail in non-causal relations. In other words, statistical associations may disappear or reverse upon aggregation when strata are of different sizes.
93:. While initially framed as a paradox, later authors have used the example to clarify the importance of untestable assumptions in causal inference.
24:
394:
227:
This analysis also unveils why Lord’s story appears paradoxical, and why generations of statisticians have found it perplexing.
650:
90:
350:
Lord, F. M. (1969). Statistical adjustments when comparing preexisting groups. Psychological Bulletin, 72, 336–337.
334:
Lord, E. M. (1967). A paradox in the interpretation of group comparisons. Psychological Bulletin, 68, 304–305.
67:
Lord imagines two statisticians who use different common statistical methods but reach opposite conclusions.
75:
89:
There have been many attempts and interpretations of the paradox, along with its relationship to other
631:
627:
312:
304:
288:
442:
390:
600:
580:
563:
556:
539:
532:
432:
382:
351:
335:
32:
underlying (untestable) causal model. Pearl used these examples to illustrate how graphical
28:
171:
claims that switching from Diet A to B would have no effect on weight gain, since the gain
567:
543:
303:
Broadly, the "fundamental problem of causal inference" and related aggregation concepts
116:
differences, while the second was right when asking about the effect on an individual.
246:
while, simultaneously, be statistically independent of it (due to path cancelations).
162:
644:
528:
287:
According to Tu, Gunnell, and Gilthorpe, Lord's paradox is the continuous version of
33:
446:
279:
controlled for, without an experimental basis that these are separate causal paths.
254:
604:
315:. It has also been suspected to be at work in studies linking IQ to eye defects.
595:
Refractive state, intelligence, education, and Lord's paradox Sorjonen, K et al
584:
44:
560:
536:
437:
420:
141:"), causal statements involve a comparison between what happened and what
52:
The most famous formulation of Lord's paradox comes from his 1967 paper:
36:
resolve the issue of when control for baseline status is appropriate.
630:
when we give causal interpretations to statistical associations; the
355:
339:
192:
and finds that latter is larger than the former in every level of
161:
43:
257:
assumed, including models with multiple unobserved confounders.
166:
Revised version of Lord’s paradox and its causal diagram (from )
483:
74:
The second statistician adjusts for initial weight, using
23:
raises the issue of when it is appropriate to control for
238:. This can also be seen from Figure 2(b), which allows
469:
The Book of Why: The New Science of Cause and Effect
511:Vol. 26 North Holland: Elsevier B.V., pp. 893-918.
421:"Lord's Paradox Revisited – (Oh Lord! Kumbaya!)"
414:
412:
410:
408:
406:
107:are the differences in individual weight gain?"
253:-calculus further answers question (1) for any
484:"Lord's Paradox: The Power of Causal Thinking"
387:Principals of modern psychological measurement
8:
381:Holland, Paul W.; Rubin, Donald B. (1983).
436:
509:Handbook of Statistics 26: Psychometrics
619:
324:
488:Causal Analysis in Theory and Practice
467:Pearl, Judea; Mackenzie, Dana (2018).
462:
460:
458:
456:
7:
376:
374:
372:
330:
328:
97:Importance of modeling assumptions
14:
626:An identical clash surfaces in
599:61, March–April 2017, 115–119
1:
605:10.1016/j.intell.2017.01.011
471:. New York, NY: Basic Books.
389:. Routledge. pp. 3–25.
261:Initial weight as a mediator
425:Journal of Causal Inference
283:Relation to other paradoxes
39:
667:
206:is the only confounder of
27:status. In three papers,
585:10.3102/10769986029001103
48:Sketch of Lord's paradox.
120:Holland and Rubin (1983)
111:Cox and McCullagh (1982)
525:
272:
167:
135:
76:analysis of covariance
49:
651:Statistical paradoxes
561:10.1186/1742-7622-5-2
537:10.1186/1742-7622-5-2
521:
438:10.1515/jci-2016-0021
419:Pearl, Judea (2016).
268:
217:, so controlling for
165:
127:
91:statistical paradoxes
47:
632:sure-thing principle
313:No Child Left Behind
383:"On Lord's paradox"
242:to causally affect
16:Statistical paradox
168:
50:
40:Lord's formulation
628:Simpson's paradox
305:Simpson's paradox
289:Simpson's paradox
658:
635:
624:
607:
593:
587:
576:
570:
553:
547:
518:
512:
505:
499:
498:
496:
494:
479:
473:
472:
464:
451:
450:
440:
416:
401:
400:
378:
367:
364:
358:
356:10.1037/h0028108
348:
342:
340:10.1037/h0025105
332:
293:reversal paradox
29:Frederic M. Lord
666:
665:
661:
660:
659:
657:
656:
655:
641:
640:
639:
638:
625:
621:
616:
611:
610:
594:
590:
577:
573:
554:
550:
520:From the text:
519:
515:
506:
502:
492:
490:
481:
480:
476:
466:
465:
454:
431:(2): 20160021.
418:
417:
404:
397:
380:
379:
370:
365:
361:
349:
345:
333:
326:
321:
301:
285:
263:
222:
215:
204:
197:
190:
183:
176:
152:
122:
113:
104:
99:
87:
42:
19:In statistics,
17:
12:
11:
5:
664:
662:
654:
653:
643:
642:
637:
636:
618:
617:
615:
612:
609:
608:
588:
571:
548:
513:
500:
482:Pearl, Judea.
474:
452:
402:
395:
368:
359:
343:
323:
322:
320:
317:
300:
297:
284:
281:
262:
259:
220:
213:
202:
195:
188:
181:
174:
151:
148:
121:
118:
112:
109:
103:
100:
98:
95:
86:
83:
62:
61:
60:
59:
41:
38:
21:Lord's paradox
15:
13:
10:
9:
6:
4:
3:
2:
663:
652:
649:
648:
646:
633:
629:
623:
620:
613:
606:
602:
598:
592:
589:
586:
582:
575:
572:
569:
565:
562:
558:
552:
549:
546:
545:
541:
538:
534:
530:
529:David Gunnell
524:
517:
514:
510:
504:
501:
489:
485:
478:
475:
470:
463:
461:
459:
457:
453:
448:
444:
439:
434:
430:
426:
422:
415:
413:
411:
409:
407:
403:
398:
396:9780898592771
392:
388:
384:
377:
375:
373:
369:
363:
360:
357:
353:
347:
344:
341:
337:
331:
329:
325:
318:
316:
314:
308:
306:
298:
296:
294:
290:
282:
280:
278:
271:
267:
260:
258:
256:
252:
247:
245:
241:
237:
231:
228:
225:
223:
216:
209:
205:
198:
191:
184:
177:
164:
160:
156:
149:
147:
144:
140:
134:
132:
126:
119:
117:
110:
108:
101:
96:
94:
92:
84:
82:
79:
77:
72:
68:
65:
57:
56:
55:
54:
53:
46:
37:
35:
34:causal models
30:
26:
22:
622:
597:Intelligence
596:
591:
574:
551:
527:Yu-Kang Tu,
526:
522:
516:
508:
503:
491:. Retrieved
487:
477:
468:
428:
424:
386:
362:
346:
309:
302:
292:
286:
276:
273:
269:
264:
255:causal model
250:
248:
243:
239:
235:
232:
229:
226:
218:
211:
207:
200:
193:
186:
179:
172:
169:
157:
153:
150:Pearl (2016)
142:
138:
136:
130:
128:
123:
114:
105:
88:
80:
73:
69:
66:
63:
51:
20:
18:
102:Bock (1975)
493:August 13,
319:References
299:Importance
143:would have
85:Responses
645:Category
447:17506239
277:could be
25:baseline
568:2254615
544:2254615
236:effects
566:
542:
445:
393:
614:Notes
443:S2CID
495:2019
391:ISBN
210:and
601:doi
581:doi
564:PMC
557:doi
540:PMC
533:doi
433:doi
352:doi
336:doi
295:.
647::
486:.
455:^
441:.
427:.
423:.
405:^
385:.
371:^
327:^
251:do
178:–
603::
583::
559::
535::
497:.
449:.
435::
429:4
399:.
354::
338::
244:Y
240:D
221:I
219:W
214:F
212:W
208:D
203:I
201:W
196:0
194:W
189:0
187:W
182:I
180:W
175:F
173:W
139:X
131:X
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
