本文以人對事物現象認識的感覺與模糊性作為切入點，闡述模糊性是人對事物認識的一種表徵及反應。然後，引入模糊集合的定義及刻劃模糊集合的表示函數—隸屬度，對模糊集合的各種運算、模糊矩陣、模糊差集以及宇集等內容進行較詳細的討論，並以各種事例說明一些相關概念和運算。 最後，再深入探討如何以模糊矩陣表示圖學中有向圖的問題。 / This article is to focus on the understanding of human being to the phenomenon of things as well as the fuzziness. Then, by applying the definition of the fuzzy set and explaining the membership of fuzzy set, we are going to have a detailed discussion of the operation of fuzzy set, fuzzy matrix, fuzzy subtraction and universal set. Examples are given to demonstrate some of the related concepts and expression. Next, further questions about how to display directed graph in the graph theory with fuzzy matrix will be discussed .

本研究的目的在於結合部份計分模式(partial credit model, PCM)與模糊集合論(fuzzy set theory)，提出評定量表的不同計分方式：模糊部份計分法(fuzzy partial credit scaling, FPCS)。FPCS是根據 PCM 所估計出的梯度參數(step parameters)來建構三角形模糊數，三角形模糊數代表選擇某個特定選項的受試者的能力分配情形。接著，利用中心法(center of gravity method) 將三角形模糊數解模糊化為純量。最後，利用隸屬度當作權重，計算個別受試者的模糊觀察分數，並且用模糊觀察分數當作量表的總分。 本研究採用貝克憂鬱量表(Beck Depression Inventory-II, BDI)中文版為研究工具。本研究的樣本分為憂鬱症病患與非憂鬱症的一般大學生兩大類。240位憂鬱症病患樣本是由台北市立和平醫院精神科門診募集而來；321位大學生則以便利抽樣的方式募集而來。 為了驗証FPCS的有效性，本研究進行三個子研究，來比較FPCS與傳統計分法在信度、效度、集群分析的分類正確性。 子研究一探討FPCS的信度。本研究以Cronbach alpha係數來衡量量表的內部一致性，並且以結構方程式模式(structure equation modeling)進行驗證性因素分析所估計的各試題的變異數被潛在構念解釋的比例當作信度的指標。由研究結果顯示，以量表整體而言，FPCS計分的結果得到較高的內部一致性；以各題而言，量表各試題的變異數被潛在構念解釋的百分比高於傳統的原始分數。此結果顯示FPCS的計分方式可以降低測量誤差，提升信度。 子研究二探討FPCS的效度，本研究以精神科醫師的診斷當作效標，分別以FPCS與原始分數兩種不同的計分法當作自變項，以預測效度當作效度的指標。首先，將是否罹患憂鬱症編碼為二元變數，不同計分法所得到的量表分數當作自變數，進行Logistic迴歸分析。研究結果顯示，相較於原始分數，FPCS預測罹患憂鬱症的正確率由 74.8% 提升到 77.2%。接下來，依照所有樣本的憂鬱程度，區分為一般樣本、憂鬱症且緩解、憂鬱症無緩解三類，進行區別分析。研究結果顯示，相較於原始分數，FPCS分類正確率由 71.2% 提升到 80.7%。上述的研究結果顯示，FPCS具有較高的效度，可以降低誤判憂鬱症的機率。 子研究三比較模糊集群分析(fuzzy c-means, FCM)與傳統明確邏輯的集群分析。首先利用分群效度(clustering validity)指標，決定群數為三群。並以此結果，指定模糊集群、Wald法、k-means法之群數。為了比較分類的效果，將模糊集群之樣本，指定給獲得最大隸屬度之集群。並且以醫師的診斷的憂鬱程度當作評估分類結果之標準。研究結果顯示，相較於傳統明確邏輯的集群分析(Wald法、k-means法)，模糊集群分析得到分群結果，與醫師的診斷的結果有最高的相關。結果顯示模糊集群分析更能夠忠實的反映資料結構。 整體而言，相較於原始分數，FPCS有較高的信度、效度、分類正確性。此實証性研究結果支持了模糊集合論應用於心理學研究的可行性；多值的模糊邏輯比二值明確邏輯更能夠正確反映出人類的思維。 / The aim of this study was to propose and validate the new scaling method, fuzzy partial credit scaling (FPCS), which combines fuzzy set theory with the partial credit model (PCM) to score rating scales. To achieve this goal, the Chinese version of BDI (Beck Depression Inventory-II) was administrated to a depressed sample of patients and a non-depressed sample. The depressed sample consisted of 240 outpatients who were diagnosed as depressed by a psychiatric doctor, while 321 undergraduate students were recruited for the nondepressed sample. In FPCS, triangular fuzzy numbers were generated by step parameters to characterize distributions of each alternative value. Next, the center of gravity (COG) method was applied to “de-fuzzify” the fuzzy number into a scalar. Then, the “observed fuzzy scores” defined in FPCS were calculated as the sums of fuzzy number values weighted by membership degrees for the following analysis. Three studies were performed to compare the differences in reliability, validity and clustering precision between the raw score and FPCS. In Study One, the reliability issue of FPCS was discussed. The results of confirmatory factor analysis demonstrate that the BDI reliability was higher in FCPS than in raw scoring. That is, compared with raw scoring, scoring via FPCS produced fewer measurement errors, meaning that more variances in an item of BDI were explained by depression. In Study Two, the predictive validity issue of FPCS was investigated. First, logistic regression analysis was used to predict the odds of suffering depression based on FPCS and the raw scores. The analytical results showed that, via FPCS, the probability of correct classification of depressed and non-depressed was raised from 74.8% to 77.2%. Next, discrimination analysis was performed to classify the subjects according to the severity of depression into three categories: non-depression, depression with remission and depression without remission. The analytical results exhibited that, via FPCS, the probability of correct classification of severity of depression was raised from 71.2% to 80.7%. These two statistical analyses consistently show that FPCS exhibited higher predictive validity than did the raw score. That is, BDI scoring via FPCS makes more accuracy predictions for depression than raw score. In Study Three, fuzzy c-means (FCM) clustering was applied to partition the sample according to severity of depression. To examine explore whether fuzzy-based clustering methods uncover the information inherent in the latent structure more accurately than crisp clustering, FCM, Wald’s method, and k-means method were performed. The analytical results reveal that the association between the original and classified membership generated by FCM was stronger than that of the Wald and k-means methods. Hence, FCM revealed the data structure most accurately. Overall, FPCS has been consistently shown to be superior to raw scoring in terms of reliability, validity, and clustering accuracy. This study has empirically shown that fuzzy set theory is applicable to psychological research.

在資料分析上，調查者通常需要決定，不同的樣本是否可被視為來自相同的母體。一般最常使用的統計量為Pearson’s 統計量。然而，傳統的統計方法皆是利用二元邏輯觀念來呈現。如果我們想要用模糊邏輯的概念來做樣本調查，此時，使用傳統 檢定來分析這些模糊樣本資料是否仍然適當？透過這樣的觀念，我們使用傳統統計方法，找出一個能處理這些模糊樣本資料的公式，稱之為模糊 。結果顯示，此公式可用來檢定，模糊樣本資料在不同母體下機率的一致性。 / In the analysis of research data, the investigator often needs to decide whether several independent samples may be regarded as having come from the same population. The most commonly used statistic is Pearson’s statistic. However, traditional statistics reflect the result from a two-valued logic concept. If we want to survey sampling with fuzzy logic concept, is it still appropriate to use the traditional -test for analysing those fuzzy sample data? Through this concept, we try to use a traditional statistic method to find out a formula, called fuzzy , that enables us to deal with those fuzzy sample data. The result shows that we can use the formula to test hypotheses about probabilities of various outcomes in fuzzy sample data.