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1、INTERNATIONAL STANDARD INTERNATIONAL ORGANIZATION FOR STANDARDIZATION .ME.liI, H4POLIHAR OPrAHM3AUMR n0 CTAHiIAPTM3AUMM.ORGANlSATlON INTERNATIONALE DE NORMALlSATlON Statistical interpretation of data - Comparison of two means in the case of paired observations Interpr _ UDC 519.28 - -_ Ref. No. IS0
2、3301-1975 (E) Descriptors : statistical analysis, mean, comparison. Price based on 6 pages FOREWORD IS0 (the International Organization for Standardization) is a worldwide federation of national standards institutes (IS0 Member Bodies). The work of developing International Standards is carried out t
3、hrough IS0 Technical Committees. Every Member Body interested in a subject for which a Technical Committee has been set up has the right to be represented on that Committee. International organizations, governmental and non-governmental, in liaison with ISO, also take part in the work. Draft Interna
4、tional Standards adopted by the Technical Committees are circulated to the Member Bodies for approval before their acceptance as International Standards by the IS0 Council. International Standard IS0 3301 was drawn up by Technical Committee ISOITC 69, Applications of statistical methods, and circula
5、ted to the Member Bodies in March 1974. It has been approved by the Member Bodies of the following countries : Austria . India Belgium Israel Brazil Italy Czechoslovakia Netherlands France Poland Germany Portugal Hungary Romania South Africa, Rep. of Spain Switzerland Turkey United Kingdom U.S.S.R.
6、Yugoslavia The Member Bodies of the following countries expressed disapproval of the document on technical grounds : Sweden U.S.A. 0 International Organization for Standardization, 1975 l Printed in Switzerland -,-,- INTERNATIONAL STANDARD IS0 3301-1975 (E) Statistical interpretation of data - Compa
7、rison of two means in the case of paired observations 0 INTRODUCTION The method specified in this international Standard, known as the method of paired observations, is a special case of the method described in table A of IS0 2854, Statistical interpretation of data - Techniques of estimation and te
8、sts relating to means and variances. l ) This special case is mentioned in section two of IS0 2854 immediately after the numerical illustration of table A , and a complete example of applications of the method of paired comparisons has been given in annex A of that International Standard The importa
9、nce and wide applicability of the method justify a separate International Standard being devoted to it 1 SCOPE This International Standard specifies a method for comparing the mean of a population of differences between paired observations with zero or any other preassigned value. 2 DEFINITION paire
10、d observations : Two observations xi ard yi of a certain property or characteristic are said to be paired if they are made : - on the same element i from a population of elements but under different conditions (for example, comparison of results of two methods of analysis on the same product), - on
11、two distinct e.lements considered similar in all respects except for the systematic difference which is the subject of the test (for example, comparison of the yield from adjacent plots sown with two distinct varieties of seed). However, it should be noted that in the second case the efficiency of t
12、he test depends on the validity of the hypothesis that there is no other systematic difference between the individuals in the same pair other than the systematic difference under test. 3 FIELD OF APPLICATION The method may be applied to establish a difference between two treatments. In this case, th
13、e observations xi are carried out after the first treatment and yi after the second treatment The two series of results of the observations are not independent because each resultxi of the first series (first treatment) is associated with a result yi of the second series (second treatment). The term
14、 “treatment” should be understood in a wide sense. The two treatments to be compared may, for instance, be two test methods, two measuring instruments or two laboratories, in order to detect a possible systematic error. Two treatments carried out successively on the same experimental material might
15、interact and the value obtained might depend on the order. Good experimental design should enable such biases to be eliminated. Alternatively, only one treatment may be applied and its effect may be compared to the absence of treatment; the purpose of this comparison is then to establish the effect
16、of that treatment 4 CONDITIONS FOR APPLICATION The method can be applied validly if the following two conditions are satisfied : - the series of differences di = xi- vi can be considered as a series of independent random items; - the distribution of the differences di = Xi - yi between the paired ob
17、servations is supposed to be normal or approximately normal. If the distribution of these differences deviates from the normal, the technique described remains valid, provided the sample size is sufficiently large; the greater the deviation from normality, the larger the sample size required. Even i
18、n extreme cases, however, a sample size of 100 will be sufficient for most practical applications. 1) At present W the stage of draft. iso3301-1975 (E) 5 FORMAL PRESENTATION OF CALCULATIONS roblem studied . . . _ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixperimental conditions . . .
19、 . . . . . . . . . . . . . . . . . . . . . . . . . . . itatistical data iample size : n= Sum of the observed values : ZXi = Eyi = jum of the differences : Zdi = Sum of the squares of the differences : Zdi = Siven value : do = Degrees of freedom : v=n-I= Chosen significance level : (II= Calculations
20、a =+xi - xyj) +Ji = s* = - -+di)*) = d cJ;= A, = t, -.(v)/fi u; = A* = t, - cu,*b41 0; = Results Two-sided case : The hypothesis that the population mean of the difference is equal to d,-, (null hypothesis) is rejected if : ia-d,iA* One-sided cases : a) The hypothesis that the population mean of the
21、 differences is at least equal to do (null hypothesis) is rejected if : ddo + A, NOTE t, -,(v) is the fractile of order 1 - 01 of Student s variate t with v degrees of freedom. The values of tl - /J to.95 - v , n to.99 6 1 8,985 45,013 4,465 22,501 2 2,434 5,730 1,686 4,021 3 1,591 2,920 1,177 2,270
22、 4 1,242 2,059 0,953 1.676 5 1,049 1,646 0,823 1,374 6 0,925 1,401 0,734 1,188 7 0,836 1,237 0,670 1,060 8 0,769 1,118 0,620 0,966 9 0,715 1,028 0,580 0,892 10 0,672 0,956 0,546 0,833 11 0,635 0,897 0,518 0.785 12 0,604 0,847 0,494 0,744 13 0,577 0,805 0,473 0.70% 14 0,554 0,769 0.455 0,678 15 0,533
23、 0,737 0.43% 0,651 16 0,514 0,708 0,423 0,626 17 0,497 0,683 0,410 0,605 1% 0,482 0,660 0,398 0,586 19 0,468 0,640 0,387 0,568 20 0,455 0,621 0,376 0,552 21 0,443 0,604 0,367 0.537 22 0,432 0,588 0,358 0,523 23 0,422 0.573 0,350 0,510 24 0,413 0,559 0,342 0,498 25 0,404 0,547 0,335 0,487 26 0,396 0,
24、535 0,328 0,477 27 0.38% 0,524 0,322 0,467 28 0,380 0,513 0,316 0,458 29 0,373 0,503 0,310 0,449 30 0,367 0,494 0,305 0.44 1 40 0,316 0,422 0,263 0,378 50 0,281 0.375 0,235 0,337 60 0,256 0,341 0,214 0,306 70 0,237 0,314 0,198 0,283 80 0,221 0,293 0,185 0,264 90 100 200 500 0,208 0,276 0,174 0,248 0
25、,197 0,261 0,165 0,235 0,139 0,183 0,117 0,165 0,088 0,116 0,074 0,104 0 0 0 0 r Two-sided case r One-sided case 1 IS0 3301-1975 (E) EXAMPLE : The data tabled below were collected during an investigation designed to determine whether the average rate of shaft-wear caused by various bearing metals in
26、 an internal combustion engine differed between metals. TABLE 2 - Shaft-wear after a given working time in 0.000 01 in Shaft i 1 2 3 4 5 6 7 8 9 Total Wear with Difference copper-lead white metal di = Xi - yi Xi Yi 3,5 1.5 2.0 2.0 1.3 0.7 4.7 4.5 0.2 2.8 2.5 0.3 6.5 4.5 20 2.2 1.7 0.5 2.5 1.8 0.7 5.
27、8 3.3 2.5 4.2 2.3 1.9 34.2 23.4 10.8 Technical characteristics . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . Statistical data Calculations Sample size : n=9 a=$(34.2-23.4)= 1 2 Sum of the observed values : s; = 19.22 - IO.82 EXiz34.2 Xvi= 23.4 8 - = 0.782 5 9 Sum of the differences : c
28、di= 10.8 Sum of the squares of the differences : Ed; = 19.22 Given value : do =0 Degrees of freedom : v=8 ut;=dm= 0.884 6 to 995/ the probability of an error of the second kind is therefore (3. For a given sample n and error of the first kind, these probabilities depend not only on the true mean D o
29、f the observed differences di = Xi-vi for which one can postulate different alternative hypotheses but also on the standard deviation od of these differences. This standard deviation is in general unknown and if n is small the sample will provide only a poor estimator. The result is that it is impos
30、sible to set an upper limit to the probability of an error of the second kind. Nevertheless, in the following graphs the relation is shown between the power of the test, 1 -0, and the actual population mean divided by the corresponding standard deviation, D/Od for one-sided tests of the hypothesis H
31、o ; D G 0, and for various values of n and for the significance levels 0,05 and 0,Ol respectively. From these graphs the following conclusions may be drawn : 1) The power of the test is uniquely determined by the true mean of the differences, measured in units of their standard deviation, by the sig
32、nificance level (Y and the sample size. 2) The power function is a strictly increasing function of the true mean difference. It is also strictly increasing with the sample size and the significance level (Y, provided D 0 and Q! different from 0 and from 1. 3) With a significance level of 0,05 and a
33、sample size of 50, a power of at least 0.95 is already obtained when the true mean difference exceeds one-half of the standard deviation of the differences. For n = 20, this power is obtained for Dlad = 0.78 or more. 5 ISO 3301-1975 (El 0,80 - 0,70 - 0,60 - 0,50 - 0,40 - 0,30 - 0,20 - 0.10 - FIGURE
34、1 - Power of Student s one-sample test (onesided), a= 0,Ol 1-p 0,80 0,60 I I I I , I , 0,1 02 0 3 0,4 0,s 0,6 - ad FIGURE 2 - Power of Student s one-sample test (one-sided), a = O,05 NOTE - The graphs are based on the work of D.B. OWEN, Handbook of statistical tables, Addison Weslav. D - Od 6 -,-,-