Ch03FactorAnalysis

– The factor matrix contains factor loadings for each variable on each factor
– Factor loadings are the correlation of each variable and the factor.
– With higher loadings means the variable is representative of the factor.
• 析出的因子占总方差的百分比 • 自然科学95% • 社会科学可能60%都可以了
4) Scree test criterion
• 一般而言，scree test比latent root的结果要多 出2个或以上的因子。
Stage 5: Interpreting the
factors
1. The initial unrotated factor matrix
– More important
1. If visual inspection reveals no substantial number of correlations greater than .30 then FA is probably inappropriate.
2. If “true” factors exist in the data, the partial correlation should be small, because the variable can be explained by the factors.
大于等于0.7: middling;
大于等于0.6: mediocre;
大于等于0.5: miserable;
小于0.5: unacceptable.
要检查每个变量的MSA，把小于.5的去掉，再做 FA。若有多个变量小于.5，则先去掉最小的一 个，再做一次，看MSA和overall MSA.
• Conceptual assumptions:
– 如果研究者想把较大数量的变量减少为较少 数量的不相关变量，并为后来的回归或其他 prediction techniques，则正交转换是最好的；
– 如果研究目的是得到若干个具有理论意义的 因子或constructs，则斜交转换合适。
3. Criteria for the significance of factor loadings
• 已知items对性别是差异的，而对男女样本的数 据进行FA是不合适的。
• 样本中存在差异的组，应该分组做FA，然后比 较结果，再去比较合并组的FA的结果。看差异 的性质。
Stage 4: Deriving factors and
assessing overall fit
1. The method of extracting the factors: common FA versus component analysis
• 如果研究者用PCA，那么common FA也 应该再使用，来比较一下。
2. Criteria for the number of factors to extract
– Most representative
– Parsimony
1) Latent root criterion:
• Only the factors having latent roots or eigenvalues greater than 1 are considered significant.
Ch02 Factor Analysis
Jia Liangding
Stage 1: Objectives of FA
1. Identifying structure through data summarization
2. Data reduction
• Exploratory
– R FA – Q FA
– In some cases, dummy variables can be used
• Boolean FA
– The strength of FA lies in finding patterns among groups of variables, and it is of little use in identifying factors composed of only a single variable.
• Confirmatory
– SEM (Chapter 11)
Stage 2: Designing a FA
1. Calculation of the input data (a correlation matrix) to meet the specified objectives of grouping variables or respondents
• 因子间相关系数为0
– An oblique rotation
• 因子间相关系数不为0
– 大多数人认为，rotation是必需的，关键是选择正 交还是斜交转换。
– 正交转换常用些，因为大多数软件都提供此，并且 斜交转换还存在一些争议。
– 目的是得到简约而有代表性的因子结构。 – Simplifying the rows:使行中的尽可能多的值接近0 ，
– Preferably the sample size should be 100 or larger. – The minimum: cases-per-variable ratio is at least 5. – More acceptable size: a 10-to-1 ratio. – The researcher should always try to obtain the
If the partial correlations are high, then there are no “true” underlying factors, and the FA is inappropriate.
3. Larger partial or anti-image correlations are indicative of a data matrix perhaps not suited to FA.
highest cases-per-variable ratio to minimize the chances of overfitting the data.
Stage 3: Assumptions in FA
• Statistical assumptions • Conceptual assumptions
2. The design of the study in terms of number of variables, measurement properties of variables, and the types of allowable variables
– Metric measurement
– Principal component analysis:
• Consider the total variance (unity)对角线值是总 方差
• Derive factors that contain small proportions of unique variance
– Common FA:
• 当变量数在20－50间时，此法最可靠。 • 当变量数小于20，此法析出因子过少； • 当变量数多于50，此法析出因子过多。
2) A priori criterion:
• 研究者已经知道有多少个因子 • 当检验有关因子数的理论或假设时，此法有用 • 重复别人的工作时
3) Percentage of variance criterion
1) Ensuring practical significance
±0.3以上，满足最低水平 ±0.4以上，more important ±0.5以上，practically significant
• Common FA合适：
– 目的：找到潜维度或因子
– 事先知识：对特殊的和误方差只占总方差的部分不 了解
• In most applications, both PCA and Common FA arrive at essentially identical results if the number of variables exceeds 30, or the communalities exceed .60 for most variables.
即使变量在单一因子中的负荷最大化
– Simplifying the columns:使列中的尽可能多的值接 近0，即使高负荷的数目尽可能少
三种正交转换方法：
1) QUARTIMAX
• Simplify the rows：使行中的尽可能多的值接 近0 ，即使变量在单一因子中的负荷Baidu Nhomakorabea大化
• 转换以使变量在一因子上负荷高，而在其他所 以因子上尽可能低
• Focuses on the common variance.对角线值是公 因子方差communality
• 选择标准：
– FA的目的 – 有关变量的事先知识
• PCA合适：
– 目的：有关prediction或找到代表原变量集的方差 最大部分的最少因子
– 事先知识：特殊的和误方差只占总方差中的相对小 部分
• 往往会：在factor matrix 的每列中，一些高负 荷，如接近1或－1，或一些低负荷，如接近0。
• 这让人们更加容易解释。 • 被证明是成功、有效的方法
3) EQUIMAX
• 上两种方法的折衷 • 未被广泛接受
• 选择转换方法：看研究问题的需要
– 如果研究目的是减少一些就是，而不管所得 因子的意思，则正交转换合适；
• 往往大多数变量（可能所有）在第一因子中高 负荷
• 往往会：许多变量在相同因子中高负荷或接近 高负荷
• 在生成更简单结构中，此法并没有被证明很成 功
2) VARIMAX
• Simplify the columns of the factor matrix：使 列中的尽可能多的值接近0，即使高负荷的数 目尽可能少
– The unrotated factor solution may not provide a meaningful pattern of variable loadings
– 在factor matrix中，列代表因子，行代表变量的负 荷
2. Rotation of factors
– An orthogonal rotation
3. The sample size necessary, both in absolute terms and as a function of the number of variables in the analysis
– Do not factor analyze a sample of fewer than 50 observations
– Some underlying structure does exist in the set of selected variables.
• 把自变量和因变量放在一起做FA是不合适的。
– The sample is homogeneous with respect to the underlying factor structure.
– Q-type FA: the intercorrelation between respondents;
– Cluster Analysis: a distance-based similarity measure
– R-type FA: the intercorrelation between variables;
4. The Bartlett test of sphericity: 变量间相关关 系的检验
5. MSA (the measurement of sampling adequacy). 0-1. 1表示每个变量都可以由其他变量完美无 误地predicted.
大于等于0.8: meritorious;