R：将 PCA 分数转换为百分位数和载荷符号

Question

我使用 prcomp 在包含 24 个健康变量和其他社会经济变量的数据集上执行 PCA。目的是获得一个可用于回归分析的单一健康指数。我使用以下代码：

total_pca <- prcomp(health[,-1], scale. = FALSE, rank. = 1)

data$pca <- total_pca$x

data$PVW <- ecdf(-data$pca)(-data$pca) # Convert into percentile

从一篇文章中说：“所有载荷都是正的，这意味着 第一个主成分的值越大，表示健康状况越差。然后将第一个主成分转换为各个百分位数，以便更高的值反映更好的健康状况。因此，我们可以将估计的健康参数解释为由于健康指数增加百分位而导致的工作概率的变化。”

我的问题是我得到了正面和负面的加载，所以我仍然可以使用与文章中相同的解释吗？或者我应该颠倒负数的符号？如果是这样，我该怎么做？

另一个问题是我的数据中有不同的波，但我在某处读到，我无法根据波将我的数据子集到不同的数据集中，并为每个波做单独的 PCA - 对吗？如果是这样，那我该怎么做呢？

我的数据的可重现示例是：

health <- structure(list(wave = c(1, 2, 4, 5, 1, 5, 5, 4, 4, 1, 1, 1, 4, 2, 4, 2, 4, 6, 2, 4, 5, 1, 4, 1, 1, 2, 1, 2, 5, 2, 2, 4, 2, 1, 4, 4, 4, 1, 4, 2), fairpoor = c(1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0), adl = c(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), mental = c(0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0), heart = c(1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0), blood = c(1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0), stroke = c(1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), diabetes = c(1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), lung = c(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0), arthritis = c(1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1), cancer = c(0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), backjoint = c(1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0), doctor = c(1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1), hospital = c(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0), nursinghome = c(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), bmi = structure(c(32.1, 28.7, 24.7, 23.5, 25.1, 26.3, 22.8, 26.3, 17.2, 32.2, 21.2, 23.6, 28.3, 35.8, 28.3, 28.7, 28.1, 20.4, 23.7, 22.7, 20.4, 25.5, 29.7, 20.3, 20.8, 23.1, 23.3, 26.3, 34.2, 40.6, 24.9, 27.2, 26.4, 23.5, 32.1, 32.8, 26.0, 23.4, 23.7, 22.8), labels = structure(c(-3, -2, -1), .Names = c("Implausible/ suspected wrong", "Refusal", "Don't know")), class = "labelled"), walking = c(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), sitting = c(0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), chair = c(0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), stairs = c(0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), kneeling = c(0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), arm = c(0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0), pullpush = c(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), lifting = c(0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0), coin = c(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)), .Names = c("wave", "fairpoor", "adl", "mental", "heart", "blood", "stroke", "diabetes", "lung", "arthritis", "cancer", "backjoint", "doctor", "hospital", "nursinghome", "bmi", "walking", "sitting", "chair", "stairs", "kneeling", "arm", "pullpush", "lifting", "coin"), row.names = c(323L, 1847L, 3731L, 5973L, 244L, 5914L, 6289L, 3847L, 3804L, 75L, 247L, 212L, 3878L, 1858L, 3994L, 2046L, 3920L, 9459L, 1850L, 4000L, 6072L, 253L, 3826L, 148L, 319L, 1855L, 17L, 1849L, 5683L, 1791L, 2002L, 3744L, 2027L, 219L, 4052L, 3837L, 4008L, 127L, 3906L, 1880L), class = "data.frame")

Answer 1

biplot 可以让您更好地解释PCA 对您的健康数据的影响。

在您给定的数据中，bmi 的幅度值高于所有其他变量，因此对于相同百分比变化的数据变化贡献更大。

因此，始终建议进行缩放，以使每个变量具有同等重要性。此外，您的示例数据似乎具有固定列（所有个人均为 0），因此从分析中删除。

  non_zero_counts <- apply(health, 2, function(x) sum(x != 0))  
  all_zero_col <- which(non_zero_counts == 0)

  # removing columns with all zeroes from PCA
  total_pca <- prcomp(health[,-c(1,all_zero_col)], scale. = TRUE)                        

主成分系数让您了解主成分如何与分析中的原始变量相关。

 library(ggbiplot)
 ggbiplot(total_pca) + xlim(-2,2) + ylim(-2,2)

如您所见，几乎所有原始变量（糖尿病、肺除外）都与 PC1 负相关。如果高分条件/原始变量表明健康状况不佳，则较高的 PC1 分数与 健康状况 相关。 p>