CN103198208B - A kind of Weight Determination being applicable to System in Small Sample Situation situation - Google Patents

Abstract

The invention provides a kind of Weight Determination being applicable to System in Small Sample Situation situation, the method is according to the appraisal result of several expert to factors importance, first carry out Kruskal-Wo Lisi to check, between the appraisal result seeing each factor, whether there were significant differences, if there were significant differences, then carry out graceful-Whitney-Wilcoxon-test, by calculating the weight ratio between the sum of ranks determination Different factor of each factor scores result in all score value composition sequences, if assay can not illustrate that between factors appraisal result, there were significant differences, then collect data, carry out regretional analysis to determine the weight ratio between factor.The present invention utilizes the concept of nonparametric statistics method of inspection, regression analysis and sum of ranks, the Weight Determination that this subjective weighting method and objective weighted model combine both had combined subjective method and had calculated advantage that is easy and objective approach objective reality, and part overcomes the shortcoming that subjectivity is random and objective approach is higher to sample size requirements of subjective method again.

Description

Weight determination method suitable for small subsample situation

Technical Field

The invention relates to a weight determination method, in particular to a weight determination method suitable for a small subsample situation, and belongs to the technical field of product quality.

Background

The weights are relative values representing the importance of the factors. The weight can be defined in the composite evaluation as the relative importance of the element to the overall contribution. The determination of the weights falls into a decision theory category.

The determination method of the weight mainly has two main types: subjective weighting method and objective weighting method. The subjective weighting method is a method in which a decision maker weights according to his own experience and subjective weighting degree of each attribute, and mainly includes a point estimation method (statistical averaging method), a judgment matrix method (pairwise comparison method, priority matrix method), an analytic hierarchy method, a delphire method, and the like. The objective method is a method for determining a weight by simply using objective information of attributes, and includes an entropy method (entropy weight information method), a variation coefficient method, a principal component analysis method, and the like.

The subjective weighting method is given by experts according to own experience and actual judgment, and the weights obtained by different selected experts are different. The method has the advantages that the expert can reasonably determine the importance of each component according to practical problems, the operation is simple and convenient, the method is not limited by the sample size of each factor, and the method is advantageous when the data size is insufficient. However, the method has obvious defects that the expert scoring statistical average method (Delphi method) and the causal matrix method do not apply scientific statistical analysis and hypothesis test methods, have large subjective randomness and are not fundamentally improved by increasing the number of experts and carefully selecting the experts, so that the subjective weighting adopted under individual conditions possibly has larger difference from the actual condition, while the analytic hierarchy process carries out consistency test on the expert scoring results of a plurality of factors but needs to carry out importance comparison between every two factors, when the factors needing weighting are more, the scoring method has higher difficulty and poorer credibility, and is not directly graded by the experts according to the scale of 1 to 9, but the Delphi method and the causal matrix method which determine importance weight according to the scale of 1 to 9 at present directly add and average the expert scoring results without carrying out statistical test on the scoring results, is greatly influenced by subjective factors of experts.

The original data of the objective weighting method is derived from the actual data of each index, and has absolute objectivity. However, in the field of aerospace, sometimes, because the sampled samples are not large enough or sufficient enough, the most important components may not have the largest weight, the unimportant components may have larger weights, and when the factors to be considered are more, the workload of collecting data by the objective weighting method is too large to be practical, and the factors needing to be collected are preferably selected first, so as to improve the working efficiency.

In the process of actually determining the index weight, the advantages of a subjective weighting method and an objective weighting method can be combined together, so that the method is called as a combined weighting method, and the combined weighting method is not strong in operability in engineering.

Disclosure of Invention

In view of this, the invention provides a weight determination method suitable for a small sample situation, which is used for determining importance weights of factors influencing product quality by applying statistical methods such as non-parametric statistics and regression analysis, and can be used for determining key characteristics influencing the quality of aerospace products in the aerospace field.

A weight determination method suitable for a small sub-sample situation comprises the following specific steps:

determining all factors influencing the product quality, and assuming that there are m, respectively A₁，…,A_mN experts respectively score according to the influence degree of each factor on the product quality according to a scale from 1 to 9, wherein 1 represents unimportant, 9 represents particularly important, and the importance of 1 to 9 is increased in sequence to obtain a scoring matrix C ═ (C)_ij)_m×nWherein c is_ijIs the scoring result of the importance of the jth expert to the ith factor;

the first step is as follows: performing a Kruskal-Wallace statistical test on the collected scoring matrix to determine whether the scoring values of each factor are significantly different, judging whether the scoring values of each factor are significantly different by the Kruskal-Wallace test for the case that m is more than or equal to 3, and calculating test statistics according to a formula (1):

$W=\[\frac{12}{n_T\left(n_T+1\right)}{\Σ}_{i=1}^m\frac{R_i^2}{n_i}\]-3\left(n_T+1\right)---\left(1\right)$

wherein n is_TIs the total number of all scores, here n × m_iA score that is a factor i; r_iIs the rank sum of each score value in the factor i;

when the W value is larger than a critical value of chi-square distribution alpha with the degree of freedom m-1 being 0.05, the factors are considered to have significant difference, and the second step is executed; otherwise, considering that the factors have no obvious difference, and executing a third step;

the second step is that: and (3) sequentially carrying out a Mann-Whitney-Wilcoxon test on the scoring results of all factors and the scoring result of the rank sum minimum factor to determine whether the two factors participating in the test have obvious difference, wherein the test method comprises the following steps:

① setting the rank and minimum factor to A_iSelecting a factor set as A_jOrdering 2n data obtained by scoring the two factors by n experts from low to high to obtain a vector d, and respectively calculating the rank sum of the n scoring results of the two factors in d;

② according to factor A_iSample size n of the score value₁N and factor A_jSample of the value of creditQuantity n₂Determining the upper limit of the rejection range T of the rank sum by querying a Wilcoxon rank sum test table₁And a lower limit T₂；

③ when step ① indicates that the smaller of the two factors in the rank sum is less than T₁Or greater than T₂If not, the importance weights of the two factors are not considered to be significantly different, then executing a third step, and determining a weight ratio by using a correlation coefficient in regression analysis;

④ minimum factor A of rank sum_iThe sum of the n scoring results in c is a_iFactor A_jThe corresponding sum of ranks is a_jThen, the weight ratio of the two factors is obtained as a_i：a_j；

⑤ loop ① - ④ sequentially combines the scoring results of other factors with rank and the minimum factor A_iThe scoring result is tested until all factors are tested;

the third step: two factors that do not significantly differ in importance weight after the second step of the test are set as A_OAnd A_PDetermining a weight ratio by utilizing a correlation coefficient between two factors and the dependent variable Y, wherein the weight ratio is specifically as follows:

let us gather the factor A_OS data pairs (x) with dependent variable Y_i，y_i) Then the correlation coefficient r_OThe solution of (a) is shown in formula (2),

$r_O=\[{\Σ}_{O=1}^s(x_O-\stackrel{\‾}{x})(y_O-\stackrel{\‾}{y})\]/\sqrt{{\mathrm{\Σ}}_{O=1}^s{(x_O-\stackrel{\‾}{x})}^2{\mathrm{\Σ}}_{O=1}^s{(y_O-\stackrel{\‾}{y})}^2}---\left(2\right)$

wherein,andfor s data pairs (x)_i，y_i) Average of x and y in (1);

the factor A is obtained in the same way_PAnd the dependent variable Yr_PThe ratio of the absolute values of the two correlation coefficients is used as the weight ratio of the two factors;

the fourth step: since the sum of the weights of all the factors is 1, the obtained weight ratios of all the factors are normalized to obtain the weight of each factor.

Has the advantages that:

(1) due to the subjectivity of expert scoring, if the existing Delphi method or statistical averaging method is used, and only the average value of the expert scoring results is taken as the weight, the final results have great difference due to personal factors such as the knowledge background of the expert. At this time, firstly, statistical test is carried out on the scoring results of all experts, only when the test conclusion is that 'significant differences exist among factors', the scoring results of the experts can be used for weighting, and if 'the conclusion that' significant differences exist among the factors 'cannot be obtained', the scoring results of the experts are considered to be incapable of explaining the significant differences among the factors, and only other objective weighting methods can be adopted. This reduces the degree to which expert scoring is affected by human factors.

(2) The robustness of the obtained weight result (namely the degree of fluctuation of the weight result according to the grading result) can be increased by adopting the ratio of the rank sum to replace the ratio of the score in the conventional grading method as the importance weight ratio of different factors, the importance score of an actual factor can be slightly different from the importance score of an expert due to the subjectivity of the expert grading, for example, if a certain factor A is particularly important and corresponds to a score 9, and another factor B is particularly important and corresponds to a score 6, if the ratio of the scores is 3:2, the factor B can be slightly viewed, and the weighting is determined by the method of the rank sum only according to the ranking (namely, the relative size) of the scores and does not view the absolute size of the scores, so the robustness is better, and the influence of the subjective factors of the expert is smaller.

(3) When the conclusion that the importance of the factors is obviously different cannot be obtained from the inspection result, the correlation coefficient between the objective data and the response variable is obtained by adopting a regression analysis method based on the objective data, and the importance of the factors is reflected by the size of the correlation coefficient, so that the test data can be collected only for a small part of the factors, and large sample amount data required by factor analysis and principal component analysis is not required, and the method is suitable for the engineering field.

Drawings

FIG. 1 is a flow chart of a weight determination method for small subsamples according to the present invention.

Detailed Description

The invention is described in detail below by way of example with reference to the accompanying drawings.

As shown in fig. 1, the present invention provides a weight determination method suitable for a small sub-sample situation, which comprises the following specific steps:

determining all factors influencing dependent variables, and assuming that there are m factors, respectively A₁，…，A_mN experts respectively score the dependent variable according to the influence degree of each factor on a scale from 1 to 9, wherein 1 represents unimportant, 9 represents particularly important, and the importance of 1 to 9 is increased in sequence to obtain a scoring matrix C ═ (C)_ij)_m×nWherein c is_ijIs the scoring result of the importance of the jth expert to the ith factor;

the first step is as follows: performing a Kruskal-Wallace statistical test on the collected scoring matrix to determine whether the scoring values of each factor are significantly different, judging whether the scoring values of each factor are significantly different by the Kruskal-Wallace test for the case that m is more than or equal to 3, and calculating test statistics according to a formula (1):

$W=\[\frac{12}{n_T\left(n_T+1\right)}{\Σ}_{i=1}^m\frac{R_i^2}{n_i}\]-3\left(n_T+1\right)---\left(1\right)$

wherein n is_TIs the total number of all scores, here n × m_iA score that is a factor i; r_iRank of each score value in factor i, so-called rank sum of factor i, ranks all n × m experts from small to large to obtain vector c ═ c (c is₍₁₎,……,c_(m×n)) Calculating the rank sum of n scores of the ith factor, wherein the scored rank is the position serial number of the score in c, and if the score x has a plurality of x in c, the average value of the position serial numbers of all x is taken as the rank of x, the minimum score rank in all the score values is 1, the maximum score rank is n × m, and under the original assumption that the importance of each factor is the same, the statistic W approximately follows chi-square distribution with the degree of freedom of m-1;

when the W value is larger than a critical value of chi-square distribution alpha with the degree of freedom m-1 being 0.05, the factors are considered to have significant difference, and the second step is executed; otherwise, considering that the factors have no obvious difference, and executing a third step;

the second step is that: and (3) sequentially carrying out a Man-Whitney-Wilcoxon test (MWW) test on the scoring result of each factor and the scoring result of the rank sum minimum factor to determine whether the two factors participating in the test have obvious difference, wherein the test method comprises the following steps:

① setting the rank and minimum factor to A_iSelecting a factor set as A_jOrdering 2n data obtained by scoring the two factors by n experts from low to high to obtain a vector d, and respectively calculating the rank sum of the n scoring results of the two factors in d;

② according to factor A_iSample size n of the score value₁N and factor A_jSample size n of the score value₂Determining the upper limit of the rejection range T of the rank sum by querying a Wilcoxon rank sum test table₁And a lower limit T₂；

③ when step ① indicates that the smaller of the two factors in the rank sum is less than T₁Or greater than T₂If not, the importance weights of the two factors are not considered to be significantly different, then executing a third step, and determining a weight ratio by using a correlation coefficient in regression analysis;

④ minimum factor A of rank sum_iThe sum of the n scoring results in c is a_iFactor A_jThe corresponding sum of ranks is a_jThen, the weight ratio of the two factors is obtained as a_i：a_j；

⑤ loop ① - ④ sequentially combines the scoring results of other factors with rank and the minimum factor A_iUntil all factors are tested.

In the above description, the MWW test is performed on the factor with the smallest sum of the scores in the ranking vector c composed of all the m × n scoring results, and the other factors are sequentially subjected to MWW test, and for the factor with significantly different importance weight from the factor with the smallest sum of the scores, in order to increase the robustness of the weighting result, the ratio of the sum of the ranks of the two factors is used as the weight ratio without using the statistical averaging method, and the sum of the scores of n of a certain factor is the sum of the scores in the mn-dimensional vector c composed of all the scores sorted from small to large.

The third step: two factors that do not significantly differ in importance weight after the second step of the test are set as A_OAnd A_PDetermining a weight ratio by utilizing a correlation coefficient between two factors and the dependent variable Y, wherein the weight ratio is specifically as follows:

let us gather the factor A_OS data pairs (x) with dependent variable Y_i，y_i) Then the correlation coefficient r_OThe solution of (a) is shown in formula (2),

$r_O=\[{\Σ}_{O=1}^s(x_O-\stackrel{\‾}{x})(y_O-\stackrel{\‾}{y})\]/\sqrt{{\mathrm{\Σ}}_{O=1}^s{(x_O-\stackrel{\‾}{x})}^2{\mathrm{\Σ}}_{O=1}^s{(y_O-\stackrel{\‾}{y})}^2}---\left(2\right)$

wherein,andfor s data pairs (x)_i，y_i) Average of x and y in (1);

the factor A is obtained in the same way_PAnd the dependent variable Y_PThe ratio of the absolute values of the two correlation coefficients is used as the weight ratio of the two factors.

r_OThe larger the absolute value of (a) is, the more closely the factor is related to the dependent variable Y, i.e., the factor should be given a larger importance weight.

The fourth step: since the sum of the weights of all the factors is 1, the obtained weight ratios of all the factors are normalized to obtain the weight of each factor.

The method according to the invention is described below as an example.

Four factors are assumed: the environmental temperature, the humidity, the vibration frequency and the pressure are factors influencing the service life of the product, four experts of A, B, C and D score the four factors according to the scale from 1 to 9 from the least important to the particularly important, and a scoring matrix is obtained as shown in Table 1:

TABLE 1 expert Scoring matrix

The first step is as follows: in this example, the number of factors m is 4, the number of experts n is 4, and the total number of scores n_T16, 16 scoring values in order from small to large constituting a vector:

c＝(3,3,3,4,4,5,5,5,6,7,7,7,8,8,9,9)

temperature factor 4 score values in c and R₁Position numbers of 9 in c are 15 and 16, i.e. rank of 9 is 55.5 ═ 15.5+15.5+13.5+11 ═ 55.5 ═ 55The position numbers of 8 in c are 13 and 14, i.e. the rank of 8 isThe position numbers of 7 in c are 10, 11 and 12, i.e. the rank of 7 isIn the same way, R can be obtained₂＝31.5，R₃＝33.5，R₄＝15.5，n_iSubstituting the above values into equation (1) to obtain W of 8.945;

left side α of chi-square distribution with degree of freedom 4-1-3-0.05 quantileBecause of 8.94>7.81, namely, the importance weight of each factor is considered to beA significant difference.

The second step is that: as can be seen from table 1, the smallest rank sum factor is a stress factor, and therefore, the MWW test was performed on the basis of the stress with three other factors in turn.

(1) When the temperature and pressure factors are tested, the ranking vector composed of all 8 scoring values is (3,3,4,5,7,8,9,9), wherein the sum of the four scoring values corresponding to the pressure factor is 1.5 × 2+3+ 4-10, the sum of the temperature factors is 26, the smaller sum of the four scoring values is T-10, and when n is equal to 10₁＝n₂Looking up the Wilcoxon rank sum table when the number is 4, the rejection range T is less than or equal to 12, the rejection range 10 falls into the rejection range, so that the two factors are considered to have a significant difference, and the weight ratio of the two factors is:

R₁：R₄＝55.5:15.5(3)

(2) the vibration frequency and the pressure factor are tested, the ranking vector of the scores is (3,3,4,4,5,5,6,7), the sum of the scores corresponding to the pressure factor is 1.5+1.5+3.5+5.5 ═ 12, the sum of the vibration frequency factors is 24, the smaller sum of the scores T ═ 12 falls in the rejection domain, the two factors are considered to have significant difference, and the weight ratio of the two factors is:

R₂：R₄＝31.5:15.5(4)

(3) and (3) inspecting the humidity and the pressure factors, wherein the ranking vector of the score values is (3,3,3,4,5,5,7 and 8), the rank sum of the four score values corresponding to the pressure factors is 2+2+4+5.5 which is 13.5, the rank sum of the four score values corresponding to the humidity factors is 22.5, the smaller rank sum of the four score values is T13.5, and the smaller rank sum does not fall into a rejection region T which is less than or equal to 12, so that the two score values cannot be considered to have a significant difference, and turning to a third step, collecting data, and determining the weight ratio by using the correlation coefficient.

The weight ratio of the temperature, the vibration frequency and the pressure factors is known through the first step and the second step, and the ratio of the rank and the sum is used as the weight ratio after statistical test, so that the defect that the evaluation method is greatly influenced by the subjective factors of experts is overcome, and the number of the factors needing to collect test data for correlation analysis is greatly reduced.

The third step: humidity data and pressure data and corresponding product life data are collected, and correlation coefficients of humidity and product life and correlation coefficients of pressure and product life are calculated respectively, as shown in tables 1 and 2.

TABLE 2 humidity-Life datasheet

TABLE 3 pressure-Life datasheet

Calculating the correlation coefficients of humidity, pressure and life according to the formula (2), which are respectively: r is₁＝0.941，r₂＝-0.913；

Taking the ratio of the absolute values of the correlation coefficients as 0.941: 0.913 is the ratio of the importance weights of humidity to pressure. And (3) integrating the two formulas (4) and (3) obtained in the second step to obtain the weight ratio of the importance of the four factors of temperature, vibration frequency, humidity and pressure to the service life of the product, wherein the weight ratio is 55.5: 31.5: 16: and 15.5, performing normalization to obtain weight vectors of the four factors as (0.47, 0.26, 0.14 and 0.13).

The method combines the expert scoring, statistical inference and data analysis, integrates the advantages of a subjective weighting method and an objective weighting method, and is suitable for the engineering field.

In summary, the above description is only a preferred embodiment of the present invention, and is not intended to limit the scope of the present invention. Any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims (1)

1. A weight determination method suitable for a small sub-sample situation is characterized by comprising the following specific steps:

determining all factors influencing the product quality, and assuming that there are m, respectively A₁，…,A_mN experts respectively score according to the influence degree of each factor on the product quality according to a scale from 1 to 9, wherein 1 represents unimportant, 9 represents particularly important, and the importance of 1 to 9 is increased in sequence to obtain a scoring matrix C ═ (C)_ij)_m×nWherein c is_ijIs the evaluation of the importance of the jth expert on the ith factorDividing the result;

the first step is as follows: performing a Kruskal-Wallace statistical test on the collected scoring matrix to determine whether the scoring values of each factor are significantly different, judging whether the scoring values of each factor are significantly different by the Kruskal-Wallace test for the case that m is more than or equal to 3, and calculating test statistics according to a formula (1):

$W=\[\frac{12}{n_T(n_T+1)}{\Σ}_{i=1}^m\frac{R_i^2}{n_i}\]-3(n_T+1)---\left(1\right)$

wherein n is_TIs the total number of all scores, here n × m_iA score that is a factor i; r_iIs the rank sum of each score value in the factor i;

when the W value is larger than a critical value of chi-square distribution alpha with the degree of freedom m-1 being 0.05, the factors are considered to have significant difference, and the second step is executed; otherwise, considering that the factors have no obvious difference, and executing a third step;

the second step is that: and (3) sequentially carrying out a Mann-Whitney-Wilcoxon test on the scoring results of all factors and the scoring result of the rank sum minimum factor to determine whether the two factors participating in the test have obvious difference, wherein the test method comprises the following steps:

① setting the rank and minimum factor to A_iSelecting a factor set as A_jOrdering 2n data obtained by scoring the two factors by n experts from low to high to obtain a vector d, and respectively calculating the rank sum of the n scoring results of the two factors in d;

② according to factor A_iSample size n of the score value₁N and factor A_jSample size n of the score value₂Determining the upper limit of the rejection range T of the rank sum by querying a Wilcoxon rank sum test table₁And a lower limit T₂；

③ when step ① indicates that the smaller of the two factors in the rank sum is less than T₁Or greater than T₂If not, the importance weights of the two factors are not considered to be significantly different, then executing a third step, and determining a weight ratio by using a correlation coefficient in regression analysis;

④ minimum factor A of rank sum_iThe sum of the n scoring results in c is a_iFactor A_jThe corresponding sum of ranks is a_jThen, the weight ratio of the two factors is obtained as a_i：a_j；

⑤ loop ① - ④ sequentially combines the scoring results of other factors with rank and the minimum factor A_iThe scoring result is tested until all factors are tested;

the third step: two factors that do not significantly differ in importance weight after the second step of the test are set as A_OAnd A_PDetermining a weight ratio by utilizing a correlation coefficient between two factors and the dependent variable Y, wherein the weight ratio is specifically as follows:

let us gather the factor A_OS data pairs (x) with dependent variable Y_i，y_i) Then the correlation coefficient r_OThe solution of (a) is shown in formula (2),

$r_O=\[{\mathrm{\Σ}}_{O=1}^s(x_O-\stackrel{\‾}{x})(y_O-\stackrel{\‾}{y})\]/\sqrt{{\mathrm{\Σ}}_{O=1}^s{(x_O-\stackrel{\‾}{x})}^2{\mathrm{\Σ}}_{O=1}^s{(y_O-\stackrel{\‾}{y})}^2}---\left(2\right)$

wherein,andfor s data pairs (x)_i，y_i) Average of x and y in (1);

the factor A is obtained in the same way_PAnd the dependent variable Y_PThe ratio of the absolute values of the two correlation coefficients is used as the weight ratio of the two factors;

the fourth step: since the sum of the weights of all the factors is 1, the obtained weight ratios of all the factors are normalized to obtain the weight of each factor.