r - fit$loadings 和 fit$Vaccounted 之间的差异在因素分析中占方差？

Question

无论是使用 fit$loadings 还是使用 fit$Vaccounted 检查它们，我都得到不同的方差值，这些值由因子分析中的因子解释。我正在使用带有 fa() 函数的 psych 包。如果它们应该是完全相同的东西(我猜它们不是，或者它们的计算方式不同)，为什么会出现这种情况？

总差异不是很大，但仍然不是微不足道的(累计大约 0.7)。我在下面有一个代表。

(对于大型数据集，我很抱歉，我无法用不同的数据集或子集复制这个问题，所以它可能与数据中的一些奇怪的东西有关。)

data <- structure(list(X1 = c(5, 5, 5, 7, 2, 2, 2, 2, 7, 5, 4, 9, 8, 
8, 6, 9, 9, 2, 2, 2, 2, 3, 2, 2, 9, 7, 8, 4, 3, 4, 6, 6, 3, 4, 
4, 4, 8, 7, 6, 7, 5, 6, 6, 4, 8, 8, 8, 3, 9, 9, 6, 4, 8, 7, 8, 
7, 8, 8, 8, 8), X2 = c(6, 4, 4, 6, 2, 2, 2, 2, 6, 5, 4, 8, 7, 
9, 6, 9, 4, 2, 2, 2, 6, 4, 6, 7, 9, 6, 8, 4, 3, 3, 5, 5, 2, 3, 
4, 7, 7, 5, 5, 6, 7, 7, 7, 3, 8, 5, 3, 2, 9, 9, 4, 4, 4, 6, 4, 
4, 8, 8, 8, 8), X3 = c(7, 5, 4, 7, 2, 2, 2, 2, 7, 5, 3, 7, 8, 
9, 7, 9, 2, 2, 2, 2, 4, 2, 5, 4, 9, 6, 8, 4, 3, 2, 4, 5, 3, 2, 
2, 7, 7, 6, 6, 5, 7, 7, 7, 4, 8, 7, 3, 2, 9, 9, 4, 3, 4, 4, 5, 
5, 8, 7, 7, 7), X5 = c(7, 6, 4, 6, 2, 2, 2, 2, 6, 4, 3, 7, 7, 
9, 6, 9, 2, 2, 2, 2, 2, 2, 4, 4, 9, 8, 6, 5, 2, 2, 4, 3, 2, 2, 
4, 7, 7, 6, 5, 6, 7, 7, 7, 3, 4, 5, 3, 2, 9, 9, 4, 2, 4, 4, 4, 
5, 8, 4, 6, 5), X6 = c(8, 4, 3, 8, 3, 2, 2, 2, 6, 5, 3, 7, 9, 
9, 7, 9, 2, 2, 2, 2, 6, 4, 6, 5, 8, 7, 6, 3, 2, 2, 2, 2, 4, 5, 
8, 8, 8, 2, 3, 4, 8, 8, 5, 3, 2, 2, 2, 2, 9, 9, 4, 4, 4, 4, 4, 
4, 5, 3, 4, 5), X7 = c(6, 6, 4, 4, 2, 2, 2, 2, 7, 4, 3, 7, 6, 
7, 4, 6, 2, 2, 2, 2, 2, 2, 4, 2, 7, 4, 8, 2, 2, 2, 4, 3, 3, 3, 
2, 5, 8, 4, 6, 7, 6, 6, 4, 2, 4, 8, 7, 2, 8, 8, 3, 3, 5, 5, 6, 
6, 5, 8, 8, 8), X8 = c(6, 6, 4, 4, 2, 2, 2, 2, 7, 4, 3, 7, 5, 
7, 6, 6, 2, 2, 2, 2, 2, 2, 2, 2, 6, 3, 7, 3, 2, 2, 4, 2, 2, 2, 
2, 4, 7, 4, 4, 6, 6, 6, 5, 2, 2, 7, 3, 2, 8, 7, 3, 3, 4, 5, 5, 
5, 4, 6, 8, 8), X10 = c(9, 9, 9, 8, 9, 9, 9, 9, 4, 6, 8, 3, 6, 
5, 6, 4, 9, 9, 9, 9, 8, 7, 8, 8, 2, 8, 3, 9, 9, 9, 9, 7, 7, 8, 
7, 7, 4, 3, 7, 6, 9, 6, 9, 9, 9, 9, 9, 9, 4, 4, 8, 9, 9, 6, 8, 
8, 9, 9, 9, 9), X11 = c(5, 6, 4, 7, 2, 3, 2, 3, 7, 6, 2, 3, 8, 
7, 6, 7, 2, 2, 2, 2, 3, 2, 2, 3, 9, 4, 8, 2, 2, 2, 6, 5, 3, 2, 
2, 2, 5, 7, 4, 6, 8, 5, 8, 2, 7, 7, 2, 2, 8, 8, 4, 4, 5, 4, 5, 
4, 5, 3, 5, 3), X12 = c(8, 6, 4, 6, 2, 2, 2, 2, 2, 5, 2, 2, 3, 
3, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 9, 4, 4, 2, 2, 3, 6, 2, 3, 
3, 3, 4, 4, 8, 7, 5, 8, 6, 4, 5, 8, 2, 2, 2, 4, 4, 3, 5, 5, 4, 
4, 7, 4, 6, 6), X13 = c(9, 8, 8, 8, 2, 2, 2, 2, 3, 5, 3, 2, 7, 
5, 8, 5, 2, 2, 2, 2, 2, 2, 2, 2, 2, 8, 3, 3, 2, 2, 5, 6, 7, 7, 
8, 6, 3, 4, 8, 6, 4, 6, 6, 6, 9, 9, 9, 4, 3, 5, 6, 8, 8, 8, 8, 
9, 7, 8, 9, 9), X14 = c(7, 5, 6, 8, 2, 2, 2, 2, 7, 5, 3, 9, 8, 
8, 6, 9, 2, 2, 2, 2, 5, 2, 3, 3, 9, 6, 8, 2, 5, 4, 6, 4, 4, 5, 
5, 6, 6, 8, 3, 5, 9, 7, 6, 8, 9, 9, 4, 3, 9, 9, 4, 4, 6, 7, 6, 
7, 8, 8, 8, 9), X15 = c(7, 6, 4, 6, 2, 2, 2, 2, 6, 5, 3, 8, 9, 
7, 6, 5, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 3, 3, 4, 4, 5, 3, 
4, 7, 2, 3, 5, 2, 6, 5, 6, 3, 4, 7, 5, 3, 8, 8, 3, 4, 5, 5, 6, 
6, 8, 7, 6, 7), X16 = c(7, 6, 4, 6, 2, 3, 2, 2, 7, 5, 3, 8, 9, 
9, 7, 9, 2, 2, 2, 2, 2, 2, 7, 5, 9, 7, 8, 2, 2, 2, 4, 4, 5, 4, 
4, 6, 9, 8, 6, 6, 6, 5, 6, 3, 8, 7, 3, 3, 8, 8, 4, 4, 4, 5, 5, 
5, 8, 7, 5, 7), X17 = c(9, 4, 3, 7, 3, 3, 2, 2, 2, 2, 2, 2, 9, 
8, 7, 4, 2, 2, 2, 2, 2, 2, 2, 2, 9, 5, 8, 3, 2, 2, 7, 6, 4, 2, 
3, 3, 4, 7, 6, 6, 8, 7, 7, 3, 2, 2, 3, 3, 2, 7, 5, 4, 4, 4, 4, 
4, 4, 4, 4, 3), X18 = c(8, 5, 7, 7, 2, 2, 2, 2, 2, 5, 3, 7, 9, 
8, 9, 9, 2, 2, 2, 2, 4, 4, 5, 3, 9, 8, 9, 3, 3, 2, 5, 4, 3, 4, 
6, 5, 6, 8, 8, 8, 4, 5, 3, 2, 9, 8, 7, 3, 6, 8, 4, 2, 2, 4, 4, 
3, 6, 4, 3, 6), X19 = c(4, 5, 7, 8, 2, 2, 2, 2, 7, 4, 3, 8, 9, 
8, 7, 9, 2, 2, 2, 2, 2, 2, 4, 2, 9, 6, 8, 2, 2, 2, 5, 4, 3, 2, 
2, 2, 8, 9, 3, 7, 6, 6, 2, 2, 8, 5, 2, 3, 7, 9, 3, 3, 5, 3, 4, 
2, 7, 5, 4, 5), X20 = c(8, 7, 7, 7, 5, 6, 6, 6, 4, 3, 4, 4, 8, 
5, 6, 7, 6, 6, 6, 6, 4, 2, 4, 4, 9, 4, 7, 6, 5, 5, 5, 5, 6, 6, 
6, 6, 8, 5, 6, 5, 5, 3, 2, 2, 8, 9, 9, 9, 9, 9, 6, 7, 8, 8, 8, 
9, 9, 8, 9, 8), X21 = c(9, 8, 7, 7, 4, 4, 5, 5, 9, 3, 8, 9, 9, 
9, 9, 9, 4, 4, 4, 4, 8, 7, 7, 4, 9, 8, 9, 9, 4, 5, 5, 5, 5, 6, 
5, 6, 9, 7, 7, 7, 6, 6, 6, 6, 9, 9, 9, 9, 9, 9, 6, 8, 8, 8, 8, 
9, 9, 8, 9, 9), X23 = c(4, 4, 3, 6, 3, 2, 2, 2, 2, 2, 2, 2, 6, 
7, 4, 7, 3, 3, 3, 3, 3, 2, 2, 2, 7, 5, 7, 4, 2, 2, 2, 2, 4, 6, 
8, 7, 4, 2, 5, 4, 2, 2, 2, 2, 2, 2, 2, 2, 8, 9, 5, 5, 4, 6, 5, 
5, 5, 3, 5, 8), X24 = c(4, 3, 6, 3, 2, 2, 2, 4, 2, 2, 2, 2, 8, 
8, 7, 7, 2, 2, 2, 2, 7, 8, 5, 5, 3, 2, 3, 2, 2, 2, 2, 2, 2, 2, 
2, 2, 7, 5, 6, 5, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 8, 2, 2, 2, 
2, 2, 2, 2, 2), X25 = c(6, 6, 6, 7, 3, 5, 3, 3, 7, 5, 3, 5, 8, 
8, 9, 9, 2, 2, 2, 2, 6, 7, 6, 5, 7, 2, 3, 2, 2, 2, 2, 2, 2, 3, 
3, 4, 5, 4, 6, 6, 7, 9, 7, 4, 2, 2, 2, 2, 5, 6, 2, 9, 2, 5, 4, 
3, 4, 3, 3, 6), X26 = c(8, 7, 5, 7, 3, 5, 3, 4, 4, 5, 3, 6, 7, 
6, 7, 4, 2, 2, 2, 2, 2, 6, 5, 4, 2, 9, 9, 3, 2, 2, 2, 2, 4, 6, 
7, 4, 5, 6, 8, 6, 6, 6, 7, 3, 3, 7, 5, 4, 4, 5, 3, 5, 4, 5, 5, 
4, 4, 4, 5, 6), X28 = c(6, 4, 5, 6, 2, 2, 2, 2, 7, 4, 2, 5, 8, 
6, 7, 5, 3, 3, 3, 3, 2, 2, 2, 2, 7, 4, 6, 2, 2, 2, 2, 2, 3, 3, 
2, 4, 5, 7, 7, 6, 5, 3, 6, 5, 2, 8, 2, 2, 5, 5, 7, 7, 4, 4, 4, 
5, 4, 3, 4, 7), X29 = c(5, 8, 6, 6, 9, 9, 9, 9, 5, 6, 9, 5, 3, 
4, 4, 6, 8, 8, 8, 8, 9, 8, 9, 8, 5, 8, 8, 8, 8, 8, 6, 7, 6, 7, 
7, 5, 4, 3, 4, 4, 6, 4, 6, 5, 8, 5, 8, 8, 7, 7, 4, 5, 7, 7, 6, 
7, 8, 8, 9, 8), X30 = c(3, 3, 4, 5, 2, 2, 2, 2, 5, 4, 2, 5, 8, 
7, 7, 6, 2, 2, 2, 2, 2, 2, 2, 2, 6, 5, 6, 3, 3, 2, 2, 2, 2, 2, 
4, 3, 7, 8, 7, 6, 2, 2, 2, 2, 2, 9, 3, 2, 4, 3, 6, 5, 3, 2, 4, 
3, 2, 2, 2, 4), X32 = c(2, 3, 3, 3, 2, 4, 2, 3, 3, 2, 2, 6, 8, 
7, 8, 8, 2, 2, 2, 2, 2, 2, 2, 2, 8, 5, 8, 2, 2, 2, 2, 2, 3, 2, 
2, 3, 2, 6, 4, 6, 9, 9, 9, 5, 2, 9, 2, 2, 5, 4, 6, 7, 2, 2, 2, 
2, 5, 6, 5, 6), X34 = c(2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 4, 
3, 4, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 
6, 6, 2, 2, 3, 2, 6, 8, 7, 2, 2, 2, 3, 2, 6, 4, 3, 3, 3, 4, 3, 
3, 4, 3, 4, 2)), class = "data.frame", row.names = c(NA, 60L))

现在我们已经定义了数据集，让我们开始编写代码吧。

library(psych)
fit <- fa(data, nfactors = 4)
#> Loading required namespace: GPArotation
print(fit$loadings)
#> 
#> [Loadings truncated for brevity]
#> 
#>                  MR1   MR2   MR3   MR4
#> SS loadings    9.464 3.571 2.171 1.682
#> Proportion Var 0.338 0.128 0.078 0.060
#> Cumulative Var 0.338 0.466 0.543 0.603

print(fit$Vaccounted, digits = 3)
#>                  MR1   MR2   MR3    MR4
#> SS loadings    10.392 4.328 2.324 1.8283
#> Proportion Var 0.371 0.155 0.083 0.0653
#> Cumulative Var 0.371 0.526 0.609 0.6740

^{由 reprex package 创建于 2022-02-10 (v2.0.1)}

我们可以看到值不同。有什么想法吗？

Answer 1

https://www.researchgate.net/post/How_can_of_Variance_of_factors_in_exploratory_factor_analysis_be_calculated_when_factors_are_correlated

我不熟悉因子分析，但是如此处所示，由于使用倾斜旋转时因子间相关性，SS 加载似乎无法计算为平方和。也许，fit$Vaccounted 考虑了这个问题，但 fit$loadings 只是平方和。我认为这种差异出现了。

注意fa包中默认的rotation是oblimin也就是obliqu rotation，所以我觉得会出现这个区别。