尺度 [1]の組み合わせ別の類似度分析の手法をまとめてみました。MIC, HSIC など比較的 新しい相関係数については触れていません。
また, 調査不足による抜けもあるかと思います…
| 手法 | 正規性 | 尺度の組み合わせ |
|---|---|---|
| Pearson’s Correlation | 要 | 量的変数 x 量的変数 |
| Cramer’s V | 不要 | 名義尺度 x 名義尺度 (2×2より大きい) |
| Goodman-Kruskal Gamma | 不要 | 順序尺度 x 順序尺度 (2×2より大きい) |
| Spearman’s Rank-Order Correlation | 不要 | 順序尺度 x 順序尺度 (順位の差) |
| Kendall Rank Correlation | 不要 | 順序尺度 x 順序尺度 (順序関係の一致率) |
| Polychoric Correlation | 要 | 順序尺度 x 順序尺度 (心理尺度で有効, 背景に連続性を仮定) |
| Polyserial Correlation | 要 | 順序尺度 x 量的変数 |
| Tetrachoric Correlation | 不要 | 順序尺度 x 順序尺度 (2値なので名義尺度?) |
これらを R で実践してみます。用いるデータは架空のデータセットを含みます。
Pearson’s Correlation
d <- read.csv("data/accident.csv")
# city post accident population
# 1 1 160 58 85
# 2 2 175 68 91
# 3 3 158 55 79
# 4 4 165 63 88
# 5 5 177 66 95
# 6 6 166 67 89
# 7 7 170 59 87
# 8 8 171 62 91
# 9 9 173 65 93
# 10 10 168 61 90
cor(d$accident, d$population, method = "p")
# [1] 0.8132599
cor(d$accident, d$post, method="p")
# [1] 0.7440411
偏相関係数を求める場合は psych::partial.r() を用いる方法がある。
require(psych)
partial.r(d, c(2, 4), 3)
# partial correlations
# post population
# post 1.00 0.76
# population 0.76 1.00
partial.r(d, c(2, 3), 4)
# partial correlations
# post accident
# post 1.00 0.04
# accident 0.04 1.00
Cramer's V
require(lsr)
d <- read.csv("data/conditions.csv")
# choice condition1 condition2
# 1 a 30 35
# 2 b 20 30
# 3 c 50 35
x <- cbind(d$condition1, d$condition2)
cramersV(x)
# [1] 0.1586139
Goodman-Kruskal Gamma
require(vcdExtra)
# See https://www.inside-r.org/packages/cran/vcdExtra/docs/JobSat
# contingency table of job satisfaction (1996 General Social Survey)
data(JobSat)
# satisfaction
# income VeryD LittleD ModerateS VeryS
# < 15k 1 3 10 6
# 15-25k 2 3 10 7
# 25-40k 1 6 14 12
# > 40k 0 1 9 11
GKgamma(JobSat, level=0.95)
# gamma : 0.221
# std. error : 0.117
# CI : -0.009 0.451
Spearman's Rank-Order Correlation
d <- read.csv("data/satisfaction.csv")
# id satisfaction price gender
# 1 1 5 4 male
# 2 2 2 1 female
# 3 3 3 2 female
# 4 4 1 2 female
# 5 5 1 0 male
# 6 6 2 3 female
# 7 7 3 1 male
# 8 8 3 3 female
# 9 9 4 5 male
# 10 10 0 1 female
# Spearman's rank correlation rho
cor.test(d$satisfaction, d$price, method="s")
# data: d$satisfaction and d$price
# S = 48.774, p-value = 0.02295
# alternative hypothesis: true rho is not equal to 0
# sample estimates:
# rho
# 0.7044025
Kendall Rank Correlation
require(Kendall)
d <- read.csv("data/satisfaction.csv")
# id satisfaction price gender
# 1 1 5 4 male
# 2 2 2 1 female
# 3 3 3 2 female
# 4 4 1 2 female
# 5 5 1 0 male
# 6 6 2 3 female
# 7 7 3 1 male
# 8 8 3 3 female
# 9 9 4 5 male
# 10 10 0 1 female
# tau Kendall’s tau statistic
Kendall(d$satisfaction, d$price)
# tau = 0.575, 2-sided pvalue =0.039549
# or cor.test()
cor.test(d$satisfaction, d$price, method="k")
# Kendall's rank correlation tau
#
# data: d$satisfaction and d$price
# z = 2.152, p-value = 0.0314
# alternative hypothesis: true tau is not equal to 0
# sample estimates:
# tau
# 0.575
Polychoric Correlation
require(polycor)
d <- read.csv("data/satisfaction.csv")
# id satisfaction price gender
# 1 1 5 4 male
# 2 2 2 1 female
# 3 3 3 2 female
# 4 4 1 2 female
# 5 5 1 0 male
# 6 6 2 3 female
# 7 7 3 1 male
# 8 8 3 3 female
# 9 9 4 5 male
# 10 10 0 1 female
# rho the polychoric correlation.
polychor(d$satisfaction, d$price)
# [1] 0.7377534
polychor(d$satisfaction, d$price, ML = TRUE, std.err = TRUE)
# Polychoric Correlation, ML est. = 0.7384 (0.1673)
# Test of bivariate normality: Chisquare = 15.39, df = 24, p = 0.9089
#
# Row Thresholds
# Threshold Std.Err.
# 1 -1.25300 0.5145
# 2 -0.53780 0.4094
# 3 -0.02593 0.3898
# 4 0.92270 0.4775
# 5 1.43200 0.5435
#
#
# Column Thresholds
# Threshold Std.Err.
# 1 -1.2430 0.5176
# 2 -0.2821 0.3923
# 3 0.2130 0.4006
# 4 0.9278 0.4769
# 5 1.4350 0.5431
Polyserial Correlation
require(polycor)
d <- read.csv("data/quality.csv")
# id quality price
# 1 1 1 980
# 2 2 2 2980
# 3 3 3 2980
# 4 4 4 4980
# 5 5 3 4480
# 6 6 2 1980
# 7 7 3 1480
# 8 8 3 1850
# 9 9 1 1200
# 10 10 5 9800
polyserial(d$price, d$quality, std.err = TRUE)
# Polyserial Correlation, 2-step est. = 0.9015 (0.06796)
# Test of bivariate normality: Chisquare = 10.83, df = 14, p = 0.6991
Tetrachoric Correlation
require(psych)
d <- read.csv("data/gender.csv")
# bought male female
# 1 true 350 50
# 2 false 210 400
x <- cbind(d$male, d$female)
# tetrachoric correlation
tetrachoric(x, na.rm = TRUE)
# [1] 0.76
#
# with tau of
# [1] -0.26 0.14
Codeは GitHub に置いた。
[1] 尺度には 名義尺度, 順序尺度, 間隔尺度, 比尺度 (比例尺度)があり, 名義尺度は定性的変数で残りが定量的変数となる。ただし, 順序尺度は離散値しか取らず, 比尺度は連続値しか取らない。
[2] 順序尺度の相関係数（ポリコリック相関係数）について
[3] 研究と検定