1、迴歸分析的例子,黃熾森 香港中文大學(xué)管理學(xué)系教授 地址: 香港新界沙田香港中文大學(xué)管理學(xué)系 電郵:.hk 2006年3月,大綱,迴歸分析及其統(tǒng)計測試原理 虛擬變項(Dummy Variable)的運(yùn)用 增加效度(Incremental Validity) 調(diào)節(jié)變項(Moderator)的測試 中介變項(Mediator)的測試 應(yīng)用迴歸分析時要注意的重點 討論幾個應(yīng)用迴歸分析的研究例子,迴歸分析的數(shù)學(xué)方程,自變項與依變項成線性關(guān)係(Linear Relationship)，假如X1及X2(即自變項；Independent Variables)是

2、Y(即依變項；Dependent Variable)的原因，那麼它們的關(guān)係可用以下方程式代表： Y = 0 + 1X1 + 2X2 + 其中： (1) 0為一常數(shù)； (2) 1代表了如果X1改變了一個單位，Y會改變的程度； (3) 2代表了如果X2改變了一個單位，Y會改變的程度； (4) 代表了隨機(jī)的誤差；,變異量的角度:最簡單的情況,最簡單的情況: 1和2是獨立的,如果A佔Y的總變異量的比重愈高，那麼1便會愈大，反映X1對Y的影響愈大； 如果B佔Y的總變異量的比重愈高，那麼2便會愈大，反映X2對Y的影響愈大。 而C則是代表了X1及X2無法影響Y的部分，也就是的變異量了。因此C佔Y的總變異量的

3、比重愈高，以X1及X2來預(yù)測或解釋Y的變異情況的能力便愈差。 A和B的部分是沒有關(guān)係的，那就是說1和2是獨立的，不會互相影響。,變異量的角度:更常見的情形,常見的情形: 1和2不是獨立的,如果我們假設(shè)這個圖中C佔Y的總變異量與之前的圖一樣，那麼，X1及X2對預(yù)測或解釋Y的變異情況的能力也會與之前的圖一樣(即A+B=a+b+D) 1和2不是獨立的，它們會互相影響，因為如果我們不考慮X2， 1便會較大；同樣地，因為如果我們不考慮X1， 2便會較大。 這一點對我們了解真實的現(xiàn)象是很重要的，因為如果在真實的現(xiàn)象中，X1及X2都同時存在而對Y有所影響，但我們的理論卻沒有考慮X1及X2都同時存在的情形，那

4、麼我們的理論便不能正確地描述這些自變項和依變項的關(guān)係了。,迴歸分析的原理,迴歸分析的原理是同時(simultaneously)考慮不同自變項對某一依變項的影響，兩點是很重要的： (1)整體而言，這些自變項對依變項的預(yù)測或解釋能力有多大，即的變異量佔Y的總變異量的大小，如果愈小，則預(yù)測或解釋能力愈高； (2)在同時考慮了所有自變項的情況下，個別自變項對依變項的影響，因此，我們可作出這樣的結(jié)論：在其他因素不變的情況下，這個自變項(例如X1)對依變項(例如Y)的影響是當(dāng)X1改變一個單位時，Y會改變1的單位(Given other things equal, Y will change by 1 un

5、it when X1 changes one unit)。,迴歸分析的統(tǒng)計數(shù)和參數(shù),R2 ， b0及bi的計算,因為我們是要以自變項來預(yù)測或解釋依變項，因此，在樣本的數(shù)據(jù)中，我們是找出一組b0及bi的數(shù)值使e的變異量最小的，然後用這一組的R2及bi來作統(tǒng)計測試(Hypothesis testing)。,R2 的統(tǒng)計測試,證整體而言，自變項對依變項的預(yù)測或解釋能力是否存在： (1)設(shè)立保守假設(shè)，即在母體中自變項對依變項沒有影響，所有自變項與依變項均無共變量，即母體的1 (的變異量)/(Y的變異量) 等於零。 (2)抽取樣本、測量各自變項及依變項，以取得數(shù)據(jù)計算R2 ； (3)計算在保守假設(shè)正確時

6、，我們會看到這個樣本的R2的機(jī)會有多大(即P值；P value)； (4)根據(jù)P值判斷是否要推翻原來保守的假設(shè)。,i 的統(tǒng)計測試,在其他因素不變的情況下，各自變頂對依變項的影響。我們以X1為例： (1)設(shè)立保守假設(shè)，即在母體中X1對Y沒有影響，所以1等於零； (2)抽取樣本、測量各自變項及依變項，以取得數(shù)據(jù)計算b1； (3)計算在保守假設(shè)正確時(即1等於零)，我們會看到這個樣本的b1的機(jī)會有多大(即P值；P value)； (4)根據(jù)P值判斷是否要推翻原來保守的假設(shè)。,虛擬變項的需要,由於我們以: Y = 0 + 1X1 + 2X2 + 這樣的方程式來代表X1、X2及Y的關(guān)係，事實上我們已經(jīng)假

7、設(shè)了X1、X2及Y最起碼是等距尺度的了，否則數(shù)學(xué)上無法運(yùn)算。,類別尺度的虛擬變項,例如X2是性別，那麼我們可創(chuàng)造一個新的虛擬變項(D)代替，當(dāng)回應(yīng)者是男性時，把D設(shè)定為1，而當(dāng)回應(yīng)者是女性時，把D設(shè)定為0，這樣一來，迴歸的方程式是： Y = 0 + 1 X1 + 2D + 。 (1) 當(dāng)D等於1時，變成：Y = 0 + 1X1 + 2 + ； (2) 當(dāng)D等於0時，變成：Y = 0 + 1X1 + 。 如果在統(tǒng)計測試中我們的結(jié)論是2等於零時，便代表男性和女性在預(yù)測或解釋Y方面沒有作用，因為無論回應(yīng)者是男性還是女性，我們接受的結(jié)論均為： Y = 0 + 1X1 + 。 所以，以虛擬變項(D)代表

8、性別後，我們便可以如常地進(jìn)行迴歸分析。,依變項的虛擬變項,假如依變項(Y)是類別尺度測量及分為兩類的，我們?nèi)钥稍O(shè)立虛擬變項，進(jìn)行特別的迴歸分析，稱為Logistic Regression。 例如:離職(Turnover)的研究。,多於兩個類別的虛擬變項,假如X2是多於兩個類別，例如是公司的種類：國營企業(yè)(SOE)、中外合資企業(yè)(JV)、外資獨資企業(yè)(WOFE) ，這樣我們便要創(chuàng)造兩個新的虛擬變項(D1及D2)來代替這變項。 例如當(dāng)企業(yè)是SOE時，把D1設(shè)定為1，而其他企業(yè)則把D1設(shè)定為0；當(dāng)企業(yè)是JV時，把D2設(shè)定為1，而其他企業(yè)則把D2設(shè)定為0。 我們的迴歸方程式便是： Y = 0 + 1

9、X1 + 2D1 + 3D2 + 如果在統(tǒng)計測試中我們的結(jié)論是2及3均等於零時，則代表企業(yè)類別對預(yù)測或解釋Y方面沒有用。 如果自變項的類別數(shù)目為n時，我們祗要設(shè)定(n-1)個虛擬變項，便可進(jìn)行迴歸分析以測試此自變項對依變項的影響。,增加效度的測試,增加效度(Incremental validity):即某一自變項在考慮了已知其他對依變項有影響的自變項後，仍對依變項有影響。 有些理論也可能描述了各自變項對依變項的影響是一個(或一類)接一個(或一類)的 我們不能單靠R2及bi的測試來驗證這些理論的正確性，而要用Hierarchical Regression的方法 。,Hierarchical Re

10、gression的測試-1,如我們要驗證X2是否在X1之上，對Y仍有預(yù)測及解釋能力，我們可比較以下兩個方程式： (1) Y = 01 + 11X1 + 1 (2) Y = 02 + 12X1 + 2X2 + 2 如果第二個方程式對Y的預(yù)測及解釋能力較第一個方程式為高，那麼我們便可以說X2是在X1之上，對Y仍有預(yù)測及解釋能力。在樣本的數(shù)據(jù)中，我們便是比較兩個方程式的R2的分別(稱為delta R-square；R2)。,Hierarchical Regression的測試-2,(1)設(shè)立保守假設(shè)，即在母體中兩個方程式對Y的預(yù)測及解釋能力沒有分別，即兩個方程式的 1 (的變異量)/(Y的變異量)

11、是一樣的。 (2)抽取樣本、測量各自變項及依變項，以取得數(shù)據(jù)計算兩個方程的R2及R2； (3)計算在保守假設(shè)正確時，我們會看到這個樣本的R2的機(jī)會有多大(即P值；P value)； (4)根據(jù)P值判斷是否要推翻原來保守的假設(shè)。,調(diào)節(jié)變項的測試 -1,在迴歸分析中我們可用交互變項(Interaction term)來驗證調(diào)節(jié)變項。所謂交互變項，就是兩個可能是調(diào)節(jié)變項相乘的積(Cross-product term)， 例如我們要驗證X2是否在X1和Y的關(guān)係中，擔(dān)當(dāng)了調(diào)節(jié)的作用，我們可先計算X1及X2相乘的積(X1*X2)。我們可用Hierarchical Regression 的測試方法，比較以下

12、兩個方程式： (1) Y = 0 + 1X1 + 2X2 + 1 (2) Y = 0 + 1X1 + 2X2 + 3(X1*X2) + 2 如果我們的結(jié)論是接受第二個方程式，即在R2的測試中我們推翻它等於零的保守假設(shè)，便等於承認(rèn)了X2在X1和Y的關(guān)係中擔(dān)當(dāng)了調(diào)節(jié)的作用。,調(diào)節(jié)變項的測試 -2,我們說當(dāng)X2不變時，而X1增加了一個單位，那麼Y的改變(Y)會是： (1) Y1 = 0 + 1X1 + 2X2 + 3(X1*X2) + (2) Y2 = 0 + 1(X1+1) + 2X2 + 3【(X1+1)*X2】+ Y = Y2 Y1 = 1 + 3X2 明顯地，由X1的改變而帶來對Y的轉(zhuǎn)變，仍

13、要視乎X2實際的數(shù)值而定。,調(diào)節(jié)變項的測試 -3,在檢定了調(diào)節(jié)變項後，如有需要，我們應(yīng)以圖示其實際的調(diào)節(jié)形態(tài)，由迴歸分析的結(jié)果如何繪圖來表示調(diào)節(jié)的形態(tài)，可參看Aiken and West (1991) 。 如果我們要測試更高層次的交互作用，例如三個自變項(X1、X2及X3)的交互作用，也是以層級迴歸咎於Hierarchical Regression) 的測試方法，檢定加入了調(diào)節(jié)變項相乘的積(即X1*X2*X3)後的R2。 唯一要注意的是，在最後加入X1*X2*X3之前，我們需先把所有較低層次的交互作用(即X1*X2、X1*X3、X2*X3)包括在迴歸分析中，可參看Aiken and West

14、(1991)。,中介變項,中介變項(Mediator)的意思，就是說自變項對依變項的影響是透過中介變項的，如果M真的是X和Y的中介變項，那麼，它們的關(guān)係應(yīng)該是：XMY。這裡有三個因果關(guān)係的條件： X是M的原因之一； X是Y的原因之一； X對Y的影響是透過M的。,中介變項的證據(jù)-1,在對樣本的迴歸的分析中，我們應(yīng)該看到以下的結(jié)果(Judd & Kenny, 1981；Baron and Kenny, 1986)： (1) X = b01 + b11M + e1 (2) Y = b02 + b21X + e2 (3) Y = b03 + b31X + b32M + e3 在第一個方程式中，以b11

15、來測試M和X的關(guān)係，結(jié)論應(yīng)是：11不等於零。在第二個方程式中，以b21來測試X和Y的關(guān)係，結(jié)論應(yīng)是：21不等於零。在第三個方程式中，是以b31和b32來測試當(dāng)M被同時考慮時，X對Y的影響，最理想的結(jié)論是：31等於零，但32不等於零。 如果這三個條件都符合，我們的結(jié)論便會是：M是X和Y的中介變項。,中介變項的證據(jù)-2,有些時候，雖然第一個和第二個方程式的結(jié)論都得到支持，但在第三個方程式中我們的結(jié)論是31和32都不等於零，這樣我們便要看b21和b31的分別，或者是第二和第三個方程式的R2分別了?；旧希绻鸐是X和Y的中介變項，那麼這些分別應(yīng)該是頗大的。 (MacKinnon, Lockwood,

16、 Hoffman, West, & Sheets (2002)有很詳細(xì)的總結(jié)。),應(yīng)用迴歸分析時要注意的重點,(1) 自變項與依變項的因果關(guān)係。 (2) 自變項與依變項的測量尺度。 (3) 控制變項。 (4) 線性關(guān)係的設(shè)定。 (5) 測量的誤差。 (6) 數(shù)據(jù)方面的要求：例如應(yīng)該是隨機(jī)和常態(tài)分佈的；當(dāng)各自變項互相的共變量很大，bi便會很不穩(wěn)定，使我們難以判斷最終對依變項的影響到底是來自那一個自變項，這問題稱為多線性問題(multicollinearity) 。,例子一：Law, Wong and Wang (2004)(1) 研究問題(Research Question)-1,這個研究要探討

17、的是在中國，跨國企業(yè)(Transnational Corporations；TNC)要本土化(localization)其中高層員工(即以本地員工取代從國外派駐的員工)，其成功的因素是否有一個層次，順序為：(1)企業(yè)視本土化為重要目標(biāo)；(2)本土化的計劃週詳程度；(3)與本土化相關(guān)的人力資源管理措施的落實程度。,(1) 研究問題(Research Question)-2,H1: The extent to which localization is regarded as an important goal of the TNC is positively related to localiz

18、ation success. H2: Localization planning efforts such as top management commitment to localization and selection of appropriate expatriates are positively related to localization success. H3: Specific human resource practices favoring the implementation of localization plans (training opportunities

19、for local managers, performance evaluation and rewards for expatriates and local managers, and repatriation arrangements) are positively related to localization success. H4: The TNCs localization planning efforts would explain variation in localization success over and beyond that of setting localiz

20、ation as an important objective. H5: The TNCs localizarion-related human resources management practices would explain variation in localization success over and beyond that of setting localization as an important objective and localization planning efforts.,測量變項的方法-1,Final participants in our valida

21、tion sample were 139 human resources managers of TNCs operating in Fujian Province in the PRC. We chose TNCs from one single province in order to control for the differences in governmental regulations. With the help of this professor in Xiamen, we sent out 180 questionnaires to current and graduate

22、d MBA students who are top or middle-level managers in TNCs in Fujian Province. These managers were asked to fill out the questionnaires themselves if they were the human resources manager of the company. They were asked to refer to their human resources manager for necessary information if they wer

23、e top executives of the company. After distributing the questionnaires and one round of telephone follow-up, a total of 139 responses were received,測量變項的方法-2,Subjective, multiple item measures with a development sample Objective indicator of localization success. In addition to the four subjective q

24、uestions, we added an objective indicator of localization success. Specifically, we used a ratio of the “Number of local managers occupying positions originally occupied by expatriates” to the “Total number of positions occupied by expatriates when the PRC operations started.” The numerator is a mea

25、sure of actual localization success, while the denominator is a comparison base of the starting number of expatriate positions. This variable is very important because it allows us to double check the validity of the subjective indicator of localization success. Also, unless the respondents delibera

26、tely lied to us, this objective indicator can be a good dependent variable that has little respondents biases with the independent variables. Control Variables, e.g., dummy coded organizational type,(3) 迴歸分析-1,由於要檢定各組自變項的順序?qū)哟危赞挌w的方式是Hierarchical Regression： In order to have a more rigorous test of

27、the five hypotheses in this study, we used hierarchical regression analyses to identify the important determinants of localization success. Results of these analyses are shown in Table 2. Table 2 shows that the inclusion of the three controlling variables (i.e., YEAR, MANUFACTURING, and JV) is neces

28、sary because they have significant effects on the localization success. Changes in R2 are .16 (p.05) and .13 (p.01) respectively for the objective and subjective indicator of localization success.,(3) 迴歸分析-2,As expected, localization objectives (i.e., GOAL) explained a significant portion of varianc

29、e in the localization result. The increase in model R2 for in predicting the objective and subjective success of localization were .20 (p.01) and .18 (p.01) respectively. Planning efforts for localization explained an additional significant portion of the variance of localization results on top of l

30、ocalization objectives. The changes in R2 for the objective and subjective success of localization measures were .13 (p.01) and .10 (p.01) respectively. Thus, H4 is supported. Human resources practices related to localization further explained a significant portion of the variances of the dependent

31、variables. The change in model R2 for the objective and subjective success of localization measures as dependent variables were .07 (p.10) and .08 (p.05) respectively. Thus, H5 is supported.,例子二：Wong, Wong and Law (2005) (1) 研究問題(Research Question),這個研究要探討的問題是：(1)工作對情緒表現(xiàn)的要求(Emotional Labor；EL)是否會成為情

32、緒智能(Emotional Intelligence；EI)與工作滿足感(Job Satisfaction)的調(diào)節(jié)變項；(2)傳統(tǒng)廣為接受的職業(yè)分類模型(Hollands Model)是否可代表工作對情緒表現(xiàn)的要求： Hypothesis 1. EI is positively related to life satisfaction. Hypothesis 2. EI is positively related to job satisfaction. Hypothesis 3. The effect of EI on job satisfaction is dependent on the

33、 EL of the job. Specifically, the higher the EL of the job, the stronger would be the effects of EI on job satisfaction. Hypothesis 4. Following Hollands model of vocational choice, the effects of EI on job satisfaction would be highest for social types of jobs. The effect sizes of the EI-job satisf

34、action relationship for different types of jobs follow Hollands calculus assumption. Hypothesis 5. The effects of EI on life satisfaction are independent of the EL of the job.,(2) 測量變項的方法-1,The sample of this study came from two sources. The first source is union members of five types of job. The fi

35、ve jobs included bus driver (realistic), computer programmer (investigative), art designer of advertising companies (artistic), shop manager of retailing shops (enterprising), and clerks (conventional). A total of 300 questionnaires were given to the union and 218 valid responses were returned, repr

36、esenting a response rate of 72.7%. However, since there are no social jobs in the union, our second sample source was teachers of two secondary schools. One hundred and ten questionnaires were sent to all the teachers of two schools and 89 valid responses were returned, representing a response rate

37、of 80.9%. Thus, the final sample consisted of 307 respondents (46 bus drivers, 103 clerks, 17 computer programmers, 9 art designers, 43 shop managers, and 89 secondary school teachers).,(2) 測量變項的方法-2,Proxy of emotional labor by Hollands occupational model. To test the importance of EI in various occ

38、upational types, we created a second proxy measure of emotional labor according to Hollands (RIASEC) model. As argued before, social type of jobs would probably have the highest level of emotional labor because these jobs have the greatest requirement of social interaction. Following the calculus as

39、sumption of Hollands (RIASEC) model, the order of emotional labor will thus be social, its adjacent types (i.e., artistic and enterprising), its alternative types (i.e., investigative and conventional), and its opposite type (i.e., realistic). Thus, this proxy measure of emotional labor was coded as

40、 follows: the social type (i.e., secondary school teachers) was coded as 4, its adjacent types (i.e., art designers and shop mangers) were coded as 3, the alternate types (i.e., computer programmers and clerks) were coded as 2, and the opposite type (i.e., bus drivers) was coded as 1.,(3) 迴歸分析,Hierarchical regression was conducted to test the main effect of EI and the interaction effect between EI and emotional labor on job satisfact