Some standard content:
National Standard of the People's Republic of China
Multiple comparison for test results
Multiple comparison for test results1 Subject content and scope of application
1.1 Subject content
UDC 620. 1. 001. 36
GB10092.88
This standard specifies the basic principles and methods for multiple comparison tests and statistical analysis of multiple treatments and single indicators, in order to obtain comparative conclusions.
Note: 1) The various objects involved in the comparison are called treatments. 2) The method of comparing whether there are significant differences between multiple treatments at the same time is called multiple comparisons. 1.2 Scope of application
This standard is applicable to the comparison of any single indicator (mean) in industrial and agricultural production and scientific experiments. For example, the comparison of the quality indicators of several products, the results of several process conditions or several test methods. This standard assumes that the test results of the same treatment are from the same normal population, and the variances of the different treatments involved in the comparative test are basically the same.
2 Reference standards
GB3358 Statistical terms and symbols
GB4882 Statistical processing and interpretation of data Normality test GB4883 Statistical processing and interpretation of data Judgment and processing of abnormal values in normal samples 3 Experimental leadership group and its qualifications
Multiple comparison tests should be carried out in an organized and planned manner. The unit or department responsible for the test shall organize the experimental leadership group, and at least one member of the group shall have data analysis knowledge of mathematical statistics and understand the application of multiple comparison methods. The leadership group shall discuss and determine:
a. The specific meaning of the treatment in this test, b. The single index compared in this test; which statistical method should be used for comparison. Once determined, it cannot be changed at will; c.
How many times n should the test be repeated for each treatment (see Appendix B); d.
How to arrange the test to ensure that the square meters of repetitions of the same treatment are carried out under the conditions specified in the definition of repeatability; how to ensure the randomness of sampling;
If the meaning of the treatment in this test is related to the uniformity of the sample, that is, when the uniformity of the sample will affect the conclusion of the comparison, then the uniformity of the sample must be ensured in all aspects of sample preparation, distribution, transportation, storage and testing. The leading group should fully study this,
conduct statistical analysis on the test results, discuss the report on the statistical analysis, and make comparative conclusions. h.
Approved by the State Administration of Technical Supervision on December 10, 1988 298
Implemented on October 1, 1989
4 Arrangement, calculation and verification of original test results 4.1
Arrangement of original test results
Arrange the original test results in the form of Table 1. Processing
Reference processing 1)
Number of test repetitions n:
GB 10092-88
321+9221
901302
Note: 1) In many practical problems, there is often a special processing that plays a benchmark role in the comparison of various processing, which is called reference processing. In the table:
the,th observation value of the "th treatment; the number of experimental repetitions of the "th treatment. When designing an experiment, n is generally as similar as possible. 9t
4.2 Calculate the mean value of each treatment and the sample variance within the treatment. According to formula (1), the mean value of each treatment is calculated:
According to formula (2), the sample variance within each treatment is calculated: S?
The calculation results are organized into the form of Table 2.
Reference treatment
Number of experimental repetitions n
4.3 Consistency test of variance
1,2,,h
():
1,2,
The mean value of each treatment stands at
........+.( 2)
Sample variance S within each treatment
4.3.1 In order to ensure that the variances within different treatments involved in the comparative test are basically consistent, a variance consistency test must be performed. Before the test, it should be required that there are no abnormal values in the repeated test results of each treatment. For this purpose, it is necessary to use the Grubbs test specified in 5.2.3 of GB4883 and the Dixon test specified in 6.3 to find abnormal values. For ease of use, this standard lists the relevant content in Appendix C (supplement).
4.3.2 This standard stipulates that the Cochran test method should be used to test the consistency of the variance within each treatment. According to formula (3 ) Calculate the value of the Cochran test statistic.
Where: mx—the maximum value among all s.
The critical value table of the Cochran test is shown in Table A1. In the table, n is the number of test repetitions. In general, n should be the same for each treatment. If some treatments are not repeated, n is the number of repetitions of most treatments. If the value of the Cochran test statistic C is greater than the critical value of 0.05 (or 0.01), then after removing S%x, continue to test the maximum variance among the remaining k-1 variances until it Until all the remaining variances meet the variance consistency requirements. The treatments that are eliminated due to inconsistent variances cannot be included in the comparison, and all their data should be eliminated. Note: 1) The Cochran test specified in this standard is a one-sided variance consistency test. It only detects those that do not meet the consistency requirements in the largest variance. In fact, some treatments may have small variances and do not meet the consistency requirements. This standard does not consider the test of small variances. First, it is inappropriate to eliminate the data of some treatments because of their high precision; second, let the high-precision treatments be calculated together with other treatments, and their common variance estimates and simultaneous confidence intervals are slightly smaller. Therefore, it is possible to improve the significance level. 4.4 Final calculation of the mean value and common variance estimate of each treatment After the outlier test, variance consistency test and correction of the eliminated data, the number of treatments, the number of experimental repetitions of each treatment, the mean value and the sample variance within the treatment may change. Therefore, the mean value and common variance estimate of each treatment should be finally calculated. The final calculation results are organized in the form of Table 3. For simplicity, the values of symbols, ni9, and S in Table 3 may change, but the signs remain unchanged.
Reference treatment
In the table:
Number of experimental repetitions ni
Average value of each treatment
Sample variance within each treatment S
Estimated value of common variance 62
5~——The average value of the ith treatment calculated after outlier test and correction or elimination of data: S-—The sample variance within the ith treatment calculated after outlier test and correction or elimination of data; Estimated value of common variance;
When there is no reference treatment result,
When there is a reference treatment result,
GB 10092 --
Degrees of freedom When there is no reference treatment result,, =—h; When there is a reference treatment result, = (+1);
t———The number of treatments finally retained after Cochran's test; n-The number of experimental repetitions of the ith treatment after eliminating outliers. 5 Three statistical methods for multiple comparisons
5.1 Comparison between the results of different treatments and the results of reference treatments 88
5.1.1 The number of experimental repetitions of the treatments involved in the comparison and the reference treatment is equal (no=comparison implementation steps:
n) (D method)).
After implementing the various items in Chapter 4, the mean values of the reference treatment and the other k treatments in Table 3 are finally calculated, and the estimated values of the variance a.
are obtained.
b. Given the significance level α (usually α=0.05 or 0.01), calculate the degree of freedom α=-(h+1), and from α, look up Table A2 to obtain the value of d(,zi).
Calculate t by the following formula. The value of:
ta == da(h,了)
（6）
d, Conclusion: In 92,., all treatments falling within the interval (9.一la,o+1) are judged to have no significant difference with the reference treatment at level α, and all treatments falling outside the interval (9.一l,3.十t) are judged to have significant difference with the reference treatment at level α. 5.1.2 The number of test repetitions of the reference treatment is greater than the number of test repetitions of other treatments, but the number of test repetitions of other treatments is equal (n.nl
Comparison implementation steps:
5.1.1a;
nx = n)(D method).
b. Given the significance level α (usually α=0.05 or 0.01), calculate the degrees of freedom at =N(k+1), and from α, look up Table A2. The small number in the upper right corner of the table is the correction value of d(,) when no≥n. d:(h)[1+g(1 )/100]da(s,For example: =2, no=21, lm2=11, y=40, α=0.05. Look up Table A2: do.(2,40)= 2.291.1, that is: 9=1.4, do.05(2,40)- 2.29)100× 2.29 = 2.31
de.u5(2,40) = [1 + 1.4(1 -
c. Calculate the value of by the following formula:
la - da (h
d. Same as 5.1.1 d.
5.1.3 The result of the reference treatment is a known value. Comparison implementation steps:
(8)
a. After implementing the various provisions of Chapter 1, the average value of the treatments in Table 3 is finally calculated. 92.. and the estimated value of common variance b.
Given the significance level α, calculate the value of the white degree "N-301
GB 10092-88
: α = 1 (1 α),
By looking up the distribution quantile table in Table A3, the value of t (f) is obtained. c. Calculate the value of 1(i) by the following formula:
t.() t, (f)
（9）
d． Conclusion: Compare the mean values of the treatments. 92, # and the known values of the reference treatment. All treatments with 1-y1≤() are judged to be significantly different from the reference treatment at the level α, and all treatments with ) are judged to be significantly different from the reference treatment at the level α.
5.2 Pairwise comparison of treatment results (T method) In some practical problems, there is no reference treatment as a comparison benchmark, and it is necessary to conduct any pairwise comparisons between the treatments. There are a total of h(h-1)/2 groups for this comparison.
Pairwise comparisons are divided into two situations:
5.2.1 The number of experimental repetitions of the treatments involved in the comparison is equal (n) Comparison implementation steps:
Same as 5.1.3a;
Given the significance level a, calculate the degree of freedom N, and look up Table A4 by, to obtain the value of qa (,); The calculated value is obtained by the following formula
ta ga(hf)
:（10)
dConclusion: If the relationship between the ith treatment and the rest of the treatments is to be compared, then all treatments falling within the interval of (%-la, +ta) are judged to have no significant difference with the ith treatment at level α. All treatments falling outside the interval of (-+t) are judged to have significant difference with the ith treatment at level α.
5.2.2 The number of experimental repetitions of each treatment involved in the comparison is not equal. Comparison implementation steps:
Same as 5.1.3a;
Same as 5.2.1b;
Value calculated by the following formula:
t,(i,3) m 9,(k,f)
- 1,2,*+ 1,2,.,k,i→3
Where: n, n; are the number of repetitions of any two treatments compared. ...()
d. Conclusion: If you want to compare the ith treatment with the ith treatment, find the value of l-. If l-≤(i,j), then it is judged that there is no significant difference between the ith treatment and the ith treatment at level α. If l-≥(i), then it is judged that there is a significant difference between the ith treatment and the ith treatment at level α.
5.3 If there are some practical problems in the comparison between the mean values of the treatment results (S method), it is necessary to compare the mean values of several treatment results. For example, the comparison between the mean value of a certain treatment and the total mean value, the comparison between the mean value of the results of a treatment in a treatment and the mean value of the results of 6 treatments, etc. This is collectively referred to as the method of arbitrary linear comparison. This standard only stipulates the following two special cases for arbitrary linear comparison: 5.3.1 Comparison between the mean value of one treatment result and the total mean value of 100 treatment results. Comparison implementation steps:
After implementing according to 1.【~1.3, calculate the mean value of the 9th treatment and the total mean value of the 9th treatment and the estimated value of the common variance, where: Township
GB 10092-88
Given the significance level α, by α, 1, look up Table A5, get the value of (, J), is the value of)o:
Calculate (yuan
a. Calculate the value of ta(i) by the following formula: ta(i) Sa(h, f)
（12）
e. Conclusion: Whenever 1-1≤（i), it is judged that the mean value of the 9th treatment is not significantly different from the total mean value at level α, and whenever [9l>li), it is judged that the mean value of the ith treatment is significantly different from the total mean value at level α. 5.3.2 Comparison between the mean value of any a treatment among the π treatments and the mean value of any other treatment. Comparison implementation steps:
a. After implementing each of 4.1 to 4.3, calculate the average value of any α treatments (assuming it is the first a treatments) in the treatments and the estimated value of the average value and common variance of any other treatments (assuming it is the next treatment). a
b. Same as 5.3.1 b;
c. Calculate the value of,
);
d. Calculate the value of t. a,b) by the following formula: +b≤
taa,b)=Sa(.f)/
·(13)
Conclusion: If. 一≤t（a,), then it is judged that the average value of u treatments is not significantly different from the average value of other treatments at level α; if 1一l（a,), then it is judged that the average value of α treatments is significantly different from the average value of other treatments at level α.
6 Application Examples
The color characteristics of cotton are one of the main indicators for evaluating the grade of cotton. When using a cotton color meter to measure the color characteristics of cotton, two indicators, reflectance Ra and yellow depth +, are determined at the same time. At present, my country's cotton standards are divided into 7 grades. From grade 1 to grade 7, the R. value gradually decreases, while the +h value gradually increases. People believe that there is little difference between grades 1, 2 and 3. Now 15 provinces and cities have selected 1 to 2 sets of cotton samples from grades 1 to 7, and the results obtained by using a color meter are as follows (Table 4, Table 5). This batch of data is used to compare whether there are significant differences between grades. Analysis: (1) This is a comparison problem of two indicators (R. and +5). The two indicators are calculated and compared separately, and then transformed into a comparison of two single indicators. In this experiment, the meaning of treatment is different grades of cotton. (2) According to experience, the values of H. and +b follow a normal distribution, so there is no need to perform a normality test. (3) The color characteristics of cotton grades 1 to 7 vary greatly, so it is possible that the sample variances within different grades (i.e., within different treatments) are not the same. Comparison implementation steps:
8 Arrange the test results of 15 provinces (cities) in the form of Tables 4 and 5. Calculate the mean value of each treatment and the sample variance within the treatment, respectively, and list them in the form of Tables 6 and 7. 303
1(level)
2(level)
3(level)
A(level)
5(level)
6(level)
7(level)
(level)
2(level)
3(level)
4(level)
5(level)
6(level)
7(level)
Number of test repetitions
Number of test repetitions
GB 10092-—88
Observed value
Observed value
Reflectivity Ra
Yellow depth++
Number of test repetitions
Number of test repetitions
GB10092---88
Average value of each treatment
UR = 78. 48
t2 77. 91
3R3 77.26
3R= 75.46
UR5 73.33
YRG— 69.96
YR = 66.47
Average value of each treatment
3b1 = 8. 66
96 = 8.81
64 - 9. 22
yos = 9.58
9h6 10. 29
Sample variance within each treatment
— 5. 24
Ss = 9. 57
S系— 16.33
S%— 11.98
Sample variance within each treatment
$ = 0. 202
S2 - 0. 181
S: = 0. 194
$ = 0. 212
S6s= 0. 616
s+ = 1.741
The repeated test results of each treatment were checked and the data of treatment 12.5 were detected in Table 5.0 is a highly abnormal value, and the reason is not found, so it is eliminated, and the number of test repetitions, treatment averages and sample variances within treatments are modified accordingly in Table 9. No abnormal values were found in other treatments. The sample variances within each treatment were tested, and the variances of the 5th, 6th and 7th treatments in Table 6 and the 6th and 7th treatments in Table 7 were all abnormal values. The data in Tables 6 and 7 are divided into two categories, one for the 1st to 4th treatments and the other for the 5th to 7th treatments. The data in Tables 6 and 7 are divided into two categories, one for the 1st to 4th treatments and the other for the 5th to 7th treatments. The data are tested separately according to the first and second categories, and all of them are normal values. This shows that the variances of the first category are basically consistent, and the variances of the second category are basically consistent, but the variances of the first category and the second category are inconsistent. The estimated values of the common variances of the first and second categories are calculated and listed in Tables 8 and 9. Table 8
Number of test repetitions
Average value of each treatment
#21 78.48
R2 - 77.91
9R3 — 77. 26
R4- 75.46
9R5 — 73. 33
39n6 69. 96
3R7 -- 66. 47
Sample variance within each treatment
1 — 3. 88
m2 2. 82
Quantity3 — 2. 95
SR- 11.98
Common variance estimate
Number of test repetitions
GB 10092--88
Average of each treatment
301 = 8. 66
352— 8.77
9b3 = 8. 81
30 = 9. 22
307=10.75
Sample variance within each treatment
Common variance estimate
=- 1.220
According to the provisions of 5.2.1b, given the significance level α=0.05, the degree of freedom is calculated as -N-. For the pairwise comparison of the first and second categories, 71, for the pairwise comparison of the second category, F=47, from α, look up the table in Table A4, and get the 9a(,) values for the first and second category comparisons: Class: 90.05(4,71)3.73
Class: 90.05(3,47)α3.42
According to different pairwise comparisons, different ta(i) values are calculated respectively, and the comparison results are calculated. The calculation and comparison results are listed in Table 10. Table 10
Processing of the comparison
ga (h,f)
Comparison results
No significant difference
No significant difference
Significant difference
No significant difference
Significant difference
Significant difference
Significant difference
No significant difference
Especially significant difference
Significant difference
No significant difference
Significant difference
Significant difference
Significant difference
Significant difference
No significant difference
GB 10092--88
d. Conclusion: From the reflectance R. Comparison with the two indicators of yellow depth + b shows that there is no significant difference between cotton grades 1-3, and they can be combined into grade 1. There is a significant difference between grade 4 and grades 1-3, and they are classified as grade 2. Grades 1-4 belong to the same category, called first grade. Compared with grade 6, there is a significant difference in reflectivity, but no significant difference in yellow depth, while there is a significant difference between grade 5 and grade 7 in both reflectivity and yellow depth, so the original grades 5-7 can be roughly maintained, belonging to the second grade. From the color characteristics, it can be concluded that my country's cotton can be divided into the following two grades and five grades of color listed in the table
· Grade cotton
Second grade cotton
Original grades 1 to 3 merged)
(Original grade 5)
(Original grade 6)
(Drying grade 4)
(Original grade 7)
GB10092--88
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８９mConclusion: From the comparison of the two indicators of reflectivity R and yellow depth + b, it can be seen that there is no significant difference in cotton grades 1-3, and they can be combined into grade 1. There is a significant difference between grade 4 and grades 1-3, and they are classified as grade 2. Grades 1-4 belong to the same category, called first grade. Compared with grade 6, there is a significant difference in reflectivity, but no significant difference in yellow depth. There are significant differences between grade 5 and grade 7 in both reflectivity and yellow depth, so the original grades 5-7 can be roughly maintained, belonging to the second grade. From the color characteristics, it can be concluded that my country's cotton can be divided into the following two grades and five grades of color listed in the table
· Grade cotton
Second grade cotton
Original grades 1 to 3 merged)
(Original grade 5)
(Original grade 6)
(Drying grade 4)
(Original grade 7)
GB10092--88
## # Meat## # #
## Adjustment## ## ### ### # Adjustment###
# Engraving### # Prevent### # Period Engraving# # Engraving## Please net## ### Net# ## Device####### Net#
Adjustment period T
Demei# Medicine
GB 10092 ---88
Guomei Medicine
Can#reach
Driving Room
National Limited Service
Island Strong Management
Courage to wait for medicine
Medicine Medicine Medicine
Gome Final Education
And Two and
Medicine Guoyao
Medicine Taomei
Sea Adjustment
Xi Jiji||t t||Leaving the medicine Yong District
Gao Ying wrote the road
National Body Beauty Service
National Medicine Grape
National Medicine National Medicine
Demei Period
National Driving Medicine
# Driving Chicken
National Sea Inspection
National National Reality
Ming Measurement Research
Test Complex
Do Replay Generation
８９mConclusion: From the comparison of the two indicators of reflectivity R and yellow depth + b, it can be seen that there is no significant difference in cotton grades 1-3, and they can be combined into grade 1. There is a significant difference between grade 4 and grades 1-3, and they are classified as grade 2. Grades 1-4 belong to the same category, called first grade. Compared with grade 6, there is a significant difference in reflectivity, but no significant difference in yellow depth. There are significant differences between grade 5 and grade 7 in both reflectivity and yellow depth, so the original grades 5-7 can be roughly maintained, belonging to the second grade. From the color characteristics, it can be concluded that my country's cotton can be divided into the following two grades and five grades of color listed in the table
· Grade cotton
Second grade cotton
Original grades 1 to 3 merged)
(Original grade 5)
(Original grade 6)
(Drying grade 4)
(Original grade 7)
GB10092--88
## # Meat## # #
## Adjustment## ## ### ### # Adjustment###
# Engraving### # Prevent### # Period Engraving# # Engraving## Please net## ### Net# ## Device####### Net#
Adjustment period T
Demei# Medicine
GB 10092 ---88
Guomei Medicine
Can#reach
Driving Room
National Limited Service
Island Strong Management
Courage to wait for medicine
Medicine Medicine Medicine
Gome Final Education
And Two and
Medicine Guoyao
Medicine Taomei
Sea Adjustment
Xi Jiji||t t||Leaving the medicine Yong District
Gao Ying wrote the road
National Body Beauty Service
National Medicine Grape
National Medicine National Medicine
Demei Period
National Driving Medicine
# Driving Chicken
National Sea Inspection
National National Reality
Ming Measurement Research
Test Complex
Do Replay Generation
８９m
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