澄清回归线之间的比较

Question

我有这个数据框：

Help_df<- data.frame(
  Variety=c('Sirio CL', 'Sirio CL', 'Sirio CL', 'Sirio CL', 
            'Sirio CL', 'Sirio CL', 'Sirio CL', 'Sirio CL',
            'Sirio CL', 'Sirio CL', 'Sirio CL', 'Sirio CL',
            'Sirio CL', 'Sirio CL', 'Sirio CL', 'Augusto', 
            'Augusto', 'Augusto', 'Augusto', 'Augusto', 'Augusto', 
            'Augusto', 'Augusto', 'Augusto', 'Augusto', 'Augusto',
            'Augusto', 'Augusto', 'Augusto', 'Augusto', 'Mare CL', 
            'Mare CL', 'Mare CL', 'Mare CL', 'Mare CL', 'Mare CL',
            'Mare CL', 'Mare CL', 'Mare CL', 'Mare CL', 'Mare CL', 
            'Mare CL', 'Mare CL', 'Mare CL', 'Mare CL'),
  Yield=c(6.98, 6.41, 6.73, 7.15, 7.32, 6.55, 6.92, 7.12, 
          6.77, 6.38, 6.4, 6.1, 5.9, 5.5, 5.6, 6.66, 6.51,
          6.15, 6.03, 6.21, 6.8, 5.98, 6.52, 6.25, 5.56,
          5.9, 5.39, 4.9, 5.6, 5, 4.25, 4.65, 4.89, 4.656,
          5.32, 5.69, 5.89, 6.02, 6.32, 6.54, 6.65, 6.54,
          6.87, 7.2, 6.21 ),
  Index=c(333, 328, 271, 265, 281, 272, 337, 389, 276, 296, 250, 251, 200, 
          200, 190, 317, 371, 351, 313, 367, 338, 356, 
          351, 335, 295, 250, 250, 200, 175, 150, 317,
          371, 351, 313, 289, 265, 298, 145, 278, 295,
          250, 250, 200, 125, 198)
)

我想知道 Yield 和 Index 之间的相关性是否会随着 Variety 的不同而变化。这里是数据图：

ggplot(Help_df, aes(x=Yield, y=Index, color=Variety)) +
 geom_point(shape=16, size=3) +
 geom_smooth(method=lm,   # Add linear regression lines
              se=FALSE)    # Don't add shaded confidence region

阅读此文help 我测试了这两个Anova:

   Anova(lm(Yield~Index*Variety,data=Help_df))
   Anova(lm(formula = Yield ~ Variety + Index + Variety:Index, data = Help_df))

据我所知，术语 Index:Variety 表示相关性对于不同的 Variety 具有不同的斜率。我想知道这两个模型有什么区别，因为两个 Anova 输出非常相似，如果有一种 "post-hoc test" 表明哪个 Variety 不同与其他因素（在这种情况下很明显 Mare 是完全不同的，但识别哪个因素不同并不总是那么容易）。

此外，我已经尝试使用@PAC 提出的解决方案来使用 "Chow test"，正如您在上面发布的 link 中看到的那样。这个测试可能是最好的，因为它比较了斜率+截距。但是 p 值为 1 表示 Yield 和 Index 之间的相关性对于不同的 Variety 没有差异，这与我在数据中观察到的不一致。

mc  <- lm(formula = Index ~ YIELD, data = Help)
m1  <-  lm(formula = Index ~ YIELD, data = subset(Help, Variety == "'Augusto'"))
m2  <-  lm(formula = Index ~ YIELD, data = subset(Help, Variety == "'Sirio CL'"))
m3  <-  lm(formula = Index ~ YIELD, data = subset(Help, Variety == "'Mare CL'"))

sc  <- sum(mc$residuals^2)
s1  <- sum(m1$residuals^2)
s2  <- sum(m2$residuals^2)
s3  <- sum(m3$residuals^2)
k  <- 3
# Test statistic
fstat  <- (sc - (s1 + s2 + s3)) / k / (s1 + s2 + s3) * (length(mc$residuals) - 2*k)  
fstat

# Rejection region
qf(.95,k, length(mc$residuals) - 2*k)

# Pvalue
pf(fstat,k, length(mc$residuals) - 2*k)

Answer 1

两个方差分析不仅相似，而且完全相同。阅读help("formula")。

对于成对比较，我会这样做：

m12  <-  lm(formula = Yield ~ Index, 
            data = subset(Help_df, Variety %in% c('Augusto', 'Sirio CL')))
m12v  <- lm(formula = Yield ~ Index * Variety, 
            data = subset(Help_df, Variety %in% c('Augusto', 'Sirio CL')))

m13 <- lm(formula = Yield ~ Index, 
          data = subset(Help_df, Variety %in% c('Augusto', 'Mare CL')))
m13v  <-  lm(formula = Yield ~ Index * Variety, 
             data = subset(Help_df, Variety %in% c('Augusto', 'Mare CL')))

m23  <-  lm(formula = Yield ~ Index, 
            data = subset(Help_df, Variety %in% c('Sirio CL', 'Mare CL')))
m23v  <-  lm(formula = Yield ~ Index * Variety, 
             data = subset(Help_df, Variety %in% c('Sirio CL', 'Mare CL')))

p.adjust(
  c(anova(m12, m12v)$"Pr(>F)"[2],
    anova(m13, m13v)$"Pr(>F)"[2],
    anova(m23, m23v)$"Pr(>F)"[2]),
  method = "holm")
#[1] 1.290727e-04 2.845623e-05 1.340764e-05

澄清回归线之间的比较

Clarification on the comparison between regression lines

r

regression

anova