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1、SEM的方法基础,回归分析,CFA模型,路径分析,EFA模型,2,路径分析的基本方法,路径分析的局限,验证性因子分析的基本方法,本讲内容,关于验证性因子分析的建议,3,假定自变量线性无关 可用来检验变量之间的直接关系,经典回归分析,X,Z,Y,4,对于下面的关系，怎么办？,X,Z,Y,5,路径分析,Path analysis is a straightforward extension of multiple regression. Its aim is to provide estimates of the magnitude and significance of hypothesized
2、 causal connections between sets of variables. path diagram input path diagram ：is drawn beforehand to help plan the analysis and represents the causal connections that are predicted by the hypothesis. output path diagram: represents the results of a statistical analysis, and shows what was actually
3、 found.,6,路径分析示例,对锻炼的态度,锻炼数量,食物摄取量,体重下降数量,+,+,+,-,对锻炼的态度,锻炼数量,食物摄取量,体重下降数量,0.31,0.30,0.15,-0.44,如果因果关系不明确， 怎么办？,双向箭头 或曲线,0.20,相关系数,7,路径分析的基本类型：递归模型,路径模型中变量之间只有单向的因果关系，没有直接或间接的反馈，而且所有的误差项不相关 路径图中没有环，误差项之间没有双向（弧线）箭头 假定： 变量间为线性、可加的因果关系 每一内生变量的误差项与外生变量无关，与其他外生变量的误差项无关 因果关系不包括反馈作用 变量为定量变量 变量自身的测量不存在误差,8,
4、路径分析的基本类型：非递归模型,至少符合以下条件之一 模型中任意两个变量之间存在直接或间接的反馈作用 某变量存在自身反馈作用（自相关） 误差项相关 一个结果变量的误差项与其原因变量相关 不同变量的误差项相关 路径图中有环，误差项之间有双向（弧线）箭头,9,路径模型的协方差阵,用样本协方差阵S代替 可建立协方差方差 从而估计模型参数,10,路径模型整体的识别,设模型参数的个数为t 内生变量数：p 外生变量数：q S中的元素个数 t规则（必要条件）：tv 零B规则（充分条件）：B=0 递归规则（充分条件）：递归模型可识别,11,非递归路径模型单个方程的识别,阶条件（必要条件）：若第i个方程未包括的
5、内生变量和外生变量数之和必须大于或等于p-1 秩条件（充分条件）：将矩阵G中第i行的非零元素所在的列划掉，剩余阵Gi的秩Rank(Gi)= p-1 如果每个方程都可识别，则非递归路径模型可识别,12,例：判断可识别性,X1,Y1,X2,Y2,21,11,22,12,21,1,2,X1,Y1,X3,Y2,13,11,23,12,21,X2,1,2,12,23,12,21,13,路径分析的步骤,模型设定 参数估计 递归模型：OLS 非递归效应：ML/LS/GLS 模型检验与评价 效应分解 因果效应：变量之间由于存在因果关系而产生的影响作用 直接效应/间接效应 虚假效应：两个内生变量的相关系数中，由
6、于共同的起因产生影响作用的部分 未析效应：一个外生变量与一个内生变量的相关系数中，除去直接效应和间接效应外剩余的部分,14,例：效应分解,X1,Y1,X2,Y2,21,11,22,12,21,2,1,15,Blau和Duncan（1967）的社会分层研究,According to Marxist scholar of the time, the US is a highly stratified society. Status is determined by family background and transmitted through the school system. Questi
7、ons: How and to what degree do the circumstances of birth condition subsequent status? How does status attained (whether by ascription or achievement) at one stage of the life cycle affect the prospects for a subsequent stage?,16,Path model. Stratification, US, 1962,Duncans prestige scale (096) 考虑了收
8、入、教育和工作声望,解释一下af的含义 有没有问题？,17,路径分析的假定,X,X,Y,Z,Y=a+b +,Z=c+d +e +,因果机制,连接,You seem to be free to use your own xs and ys, rather than the ones generated by Nature, as inputs.,Potential outcome,18,Selection vs intervention,路径分析连接了两种很不相同的关于条件期望的思想 selecting the subjects with X=x intervening to set X=x,通
9、常进行回归分析的目的是不做实际试验而完成因果推断。 但是，没有实际试验，模型背后的假定是可疑的，这样得 到的推断是忽略了假定的可疑性而做的，这就是由回归做 因果推断的自相矛盾之处。 路径模型并不从关联推断出因果，实际上，路径模型假定 了因果关系，并利用附加的统计假定从观察数据来估计因 果关联。,19,Duncan(1984)的一席话,Coupled with downright incompetence in statistics, paradoxically, we often find the syndrome that I have come to call statisticism:
10、the notion that computing is synonymous with dong research, the nave faith that statistics is a complete or sufficient basis for scientific methodology, the superstition that statistical formulas exist for evaluating such things as the relative merits of different substantive theories or the importa
11、nce of the causes of a dependent variable; and the delusion that decomposing the covariations of some arbitrary and haphazardly assembled collection of variables can somehow justify not only a causal model but also, praise the mark, a measurement model. There should be no point in deploring such car
12、icatures of the scientific enterprise if there were a clearly identifiable sector of social science research wherein such fallacies were clearly recognized and emphatically out of bounds.,20,因子,Factors are influences that are not directly measured but account for commonality among a set of measureme
13、nts.,X1,X2,X3,F,X1,X2,X3,u1,u2,u3,F,21,探索性因子分析,将可测随机向量与潜在因子连接起来的线性模型 目的：寻找少数几个因子，以解释观察变量之间的相关性 没有如下先验知识 公因子数 因子载荷 因子间关系 假定误差项无关,提出假设： 公因子数 因子载荷 放松假定： 和为对称阵,验证性因子分析,22,EFA与CFA的关系,相同点： 模型相同 CFA类似于EFA的简单结构 不同点： 用途不同 EFA在于探索（归纳） CFA在于检验（演绎） 假定不同 估计方法不同 EFA常采用谱分解 CFA常采用MLE、GLS等,1,X1,X2,1,X3,X4,2,3,4,12,2
14、,1,1,42,21,31,32,12,11,41,22,EFA,CFA,23,CFA的可识别性,参数个数：t= qn+ n(n+1)/2+q(q+1)/2 x中元素：qn 中元素：n(n+1)/2 中元素：q(q+1)/2 方程个数： v=q(q+1)/2 必须对()中的参数施加约束，CFA方可识别 参考变量法（reference variable solution）：设x中每一列至少有一个ij=1 标准化法（standardization solution ）：设潜变量的方差等于1 其他：设某些ij=0；设某些ij相等；设某些参数等于给定的数,经验 对于参考变量法： x中每一列至少有一个i
15、j=1 其他行有且只有一个非零元素 每个因子至少有三个指标 为对角阵 对不做假定 对于标准化法： 的对角元素为1 x的每一个元素都不为1,24,例：三因子验证性因子模型,1,X1,X2,1,X3,X4,X5,2,3,4,5,12,X6,X7,X8,6,7,8,2,3,23,13,1,1,42,1,21,63,73,83,标准化法如何设模型？,25,CFA的分析步骤,模型设定 参数估计 选择拟合函数（fit function）F(S,() 拟合函数最小化 模型的假设检验 模型拟合优度评价 模型修正 最好本着简约的原则，移除而非添加路径 模型选择,26,CFA例：Performance Asses
16、sment Program,研究目标：检验PAP的效果 五个维度 Teachers support for the PAP Teachers emphasis on outcome/change in instruction and assessment Teachers familiarity with PAP PAPs impact on instruction/assessment PAPs impact on professional development 八个指标（4级Likert量表） 265个教师接受调查,27,EFA输出结果：四因子,Call: factanal(factor
17、s = 4, covmat = cov) Uniquenesses: V1 V2 V3 V4 V5 V6 V7 V8 0.537 0.138 0.492 0.411 0.378 0.388 0.616 0.135 Loadings: Factor1 Factor2 Factor3 Factor4 1, 0.210 0.638 2, 0.897 0.207 3, 0.232 0.654 0.157 4, 0.198 0.737 5, 0.747 0.120 0.184 0.128 6, 0.612 0.313 0.340 0.157 7, 0.543 0.107 0.202 0.192 8, 0
18、.299 0.137 0.869 Factor1 Factor2 Factor3 Factor4 SS loadings 1.460 1.348 1.190 0.909 Proportion Var 0.182 0.168 0.149 0.114 Cumulative Var 0.182 0.351 0.500 0.613 The degrees of freedom for the model is 2 and the fit was 0.0034,28,EFA输出结果：三因子,Call: factanal(factors = 3, covmat = cov, rotation = “var
19、imax“) Uniquenesses: V1 V2 V3 V4 V5 V6 V7 V8 0.005 0.613 0.634 0.005 0.492 0.378 0.600 0.669 Loadings: Factor1 Factor2 Factor3 1, 0.996 2, 0.251 0.564 3, 0.366 0.476 4, 0.169 0.980 5, 0.660 0.207 0.173 6, 0.680 0.298 0.267 7, 0.598 0.122 0.165 8, 0.569 Factor1 Factor2 Factor3 SS loadings 1.805 1.467
20、 1.331 Proportion Var 0.226 0.183 0.166 Cumulative Var 0.226 0.409 0.575 The degrees of freedom for the model is 7 and the fit was 0.0901,29,CFA输出结果：三因子（pp.23）,Estimate Std Error z value Pr(|z|) lam21 1.30574 0.22995 5.6783 1.3607e-08 X2 f1 phi13 8.90354 1.99314 4.4671 7.9288e-06 f3 f1 phi23 9.52657
21、 1.58466 6.0118 1.8352e-09 f3 f2 delta1 21.22005 3.53395 6.0046 1.9177e-09 X1 X1 delta2 12.76266 5.25386 2.4292 1.5132e-02 X2 X2 delta3 9.06085 1.77062 5.1173 3.0989e-07 X3 X3 delta4 12.33760 1.88532 6.5440 5.9882e-11 X4 X4 delta5 18.25313 2.09161 8.7268 0.0000e+00 X5 X5 delta6 12.03134 2.00074 6.01
22、35 1.8161e-09 X6 X6 delta7 30.21710 3.09192 9.7729 0.0000e+00 X7 X7 delta8 47.34513 4.49991 10.5214 0.0000e+00 X8 X8 phi11 18.37994 4.19164 4.3849 1.1604e-05 f1 f1 phi22 12.33890 2.31616 5.3273 9.9677e-08 f2 f2 phi33 17.74676 3.02662 5.8636 4.5307e-09 f3 f3,Model Chisquare = 42.394 Df = 17 Pr(Chisq)
23、 = 0.00058827 Chisquare (null model) = 598.95 Df = 28 Goodness-of-fit index = 0.96173 Adjusted goodness-of-fit index = 0.91895 RMSEA index = 0.07522 90% CI: (0.047104, 0.10396) Bentler-Bonnett NFI = 0.92922 Tucker-Lewis NNFI = 0.92675 Bentler CFI = 0.95552 SRMR = 0.040583 BIC = -52.462 Normalized Re
24、siduals Min. 1st Qu. Median Mean 3rd Qu. Max. -2.36e+00 -3.43e-01 5.37e-05 -6.59e-02 2.33e-01 9.25e-01,30,CFA输出结果：三因子,Estimate Std Error z value Pr(|z|) lam11 4.28726 0.488333 8.7794 0.0000e+00 X1 f1 phi13 0.49298 0.068025 7.2471 4.2566e-13 f3 f1 phi23 0.64378 0.060265 10.6825 0.0000e+00 f3 f2 delta
25、1 21.21941 3.532160 6.0075 1.8842e-09 X1 X1 delta2 12.76251 5.251298 2.4304 1.5084e-02 X2 X2 delta3 9.06100 1.770171 5.1187 3.0763e-07 X3 X3 delta4 12.33743 1.885110 6.5447 5.9625e-11 X4 X4 delta5 18.25358 2.091432 8.7278 0.0000e+00 X5 X5 delta6 12.03125 2.000661 6.0136 1.8140e-09 X6 X6 delta7 30.21
26、706 3.091902 9.7730 0.0000e+00 X7 X7 delta8 47.34513 4.499884 10.5214 0.0000e+00 X8 X8,Model Chisquare = 42.394 Df = 17 Pr(Chisq) = 0.00058827 Chisquare (null model) = 598.95 Df = 28 Goodness-of-fit index = 0.96173 Adjusted goodness-of-fit index = 0.91895 RMSEA index = 0.07522 90% CI: (0.047104, 0.1
27、0396) Bentler-Bonnett NFI = 0.92922 Tucker-Lewis NNFI = 0.92675 Bentler CFI = 0.95552 SRMR = 0.040584 BIC = -52.462 Normalized Residuals Min. 1st Qu. Median Mean 3rd Qu. Max. -2.36e+00 -3.43e-01 -2.05e-06 -6.60e-02 2.33e-01 9.25e-01,31,CFA对数据的要求,样本容量 应考虑模型复杂性、估计方法、分布特征、测量尺度 仅考虑模型复杂性：每个待估参数至少需要4个样本点
28、通常建议，CFA的样本容量至少为200 显现出渐进性质：400 利用修正指数来改进模型：800 分布特征：联合正态分布 如果模型正确、样本容量足够大，MLE稳健性较好 如果极端非正态，需要用渐进分布无关（asymptotic distribution free）方法或Satorra-Bentler稳健统计量 测量尺度：连续尺度 指标类型 结果（effect）/反映性（reflective） 原因（cause）/形成性（formative）,32,7 Recommendations for CFA,Users should aim for samples of at least 200, and
29、, preferably, 400 cases. If more than minimal respecification of an hypothesized model is anticipated, then a sample of at least 800 cases is necessary. The distributional properties of indicators should be well understood and corrective measures taken (e.g., transformations, parceling, scaled stati
30、stics) when distributions depart markedly from normality. At last three and, preferably, four indicators of factors should be obtained.,33,7 Recommendations for CFA(cont.),Simple structure should not be assumed in all models. With sufficient indicators per factor, cross-loadings are permissible and
31、may be an important feature of a model. The multifaceted nature of fit evaluation should e acknowledged by consulting two or more indicators of fit that rely on different computational logic. Whenever possible, multiple, nested models should be posited in order to rule out parsimonious or substantiv
32、ely interesting alternatives to the hypothesized model.,34,7 Recommendations for CFA(cont.),Respecification is not to be eschewed, but it should be undertaken in a disciplined manner with due attention to the possibility of Type I errors. Substantially respecified models should be cross-validated in an independent sample.,35,课后任务,阅读文献Sewall Wright: The theory of path coefficients reply to Niless criticism, Genetics, 1923(8): 239-255 在R中下载sem，练习数据导入和sem的基本操作 对于PAP一例，设定其他模型形式，进行CFA分析,