使用lm()与ggplot2绘制y轴取100-y时的凹形对数回归方法咨询

我需要将y轴数据从百分比y调整为100-y时修改对数回归模型，但暂时找不到正确的实现方法，以下是我使用的测试示例：

m=c(3,1,3,2,4,7,10,2,1,44,11,24,1,3,5,11,7,17,8,10,17,34,31,10,7,3,1,6,4,3,4,7,11,14,6,8,34,80,23,54,2,3,2,1,13,8,10,15,2,15,8,11,8,12,9,18,85)
q=c(91,78,72,84,74,76,81,80,70,130,131,136,53,56,57,111,60,80,78,76,84,107,85,76,57,57,45,50,68,61,52,59,60,60,51,63,95,93,85,97,52,39,34,40,70,124,90,68,39,81,50,60,74,70,59,53,98)
Age=c(rep("Young",24),rep("Old",33))
df=data.frame(Age,m,q)

原始回归尝试 #

结合散点图，我分别尝试了线性回归与对数回归：

model_lin_df <- df %>% group_by(Age) %>% do(model = glance(lm(m ~ q,data = .))) %>% unnest(model)
> model_lin_df
# A tibble: 2 x 13
  Age   r.squared adj.r.squared sigma statistic  p.value    df logLik   AIC   BIC deviance df.residual  nobs
  <chr>     <dbl>         <dbl> <dbl>     <dbl>    <dbl> <dbl>  <dbl> <dbl> <dbl>    <dbl>       <int> <int>
1 Old       0.370         0.350 16.4       18.2 0.000172     1 -138.   282.  287.    8339.          31    33
2 Young     0.418         0.391  8.97      15.8 0.000646     1  -85.7  177.  181.    1771.          22    24

model_exp_df <- df %>% group_by(Age) %>% do(model = glance(lm(log(m) ~ q,data = .))) %>% unnest(model)
model_exp_df
# A tibble: 2 x 13
  Age   r.squared adj.r.squared sigma statistic    p.value    df logLik   AIC   BIC deviance df.residual  nobs
  <chr>     <dbl>         <dbl> <dbl>     <dbl>      <dbl> <dbl>  <dbl> <dbl> <dbl>    <dbl>       <int> <int>
1 Old       0.517         0.502 0.776      33.2 0.00000242     1  -37.4  80.8  85.3     18.7          31    33
2 Young     0.362         0.333 0.916      12.5 0.00187        1  -30.9  67.8  71.4     18.5          22    24

并绘制了对应的拟合线：

model_lin <- lm(m ~ Age/q + 0, df,na.action=na.exclude)
model_exp <- lm(log(m) ~ Age/q + 0, df,na.action=na.exclude)

df_plot <- ggplot(df, aes(x = q, y = m, shape=Age, color=Age)) + geom_point() +
                geom_line(aes(y = ifelse(Age == "Young", fitted(model_lin), NA))) +
                geom_line(aes(y = ifelse(Age == "Old", exp(fitted(model_exp)), NA))) +
                theme_bw() + theme(panel.grid = element_blank())

原始拟合结果图：

y轴调整后的问题 #

但实际业务中我需要将y轴设置为100-y，对数据表意更合理，我目前无法绘制对数回归的「反转」凹形版本：

s=100-m
df=cbind(df,s)

拟合模型的结果显示我需要调整公式：

> model_lin_df2 <- df %>% group_by(Age) %>% do(model = glance(lm(s ~ q,data = .))) %>% unnest(model)
> model_lin_df2
# A tibble: 2 x 13
  Age   r.squared adj.r.squared sigma statistic  p.value    df logLik   AIC   BIC deviance df.residual  nobs
  <chr>     <dbl>         <dbl> <dbl>     <dbl>    <dbl> <dbl>  <dbl> <dbl> <dbl>    <dbl>       <int> <int>
1 Old       0.370         0.350 16.4       18.2 0.000172     1 -138.   282.  287.    8339.          31    33
2 Young     0.418         0.391  8.97      15.8 0.000646     1  -85.7  177.  181.    1771.          22    24

> model_exp_df2 <- df %>% group_by(Age) %>% do(model = glance(lm(log(s) ~ q,data = .))) %>% unnest(model)
> model_exp_df2
    # A tibble: 2 x 13
      Age   r.squared adj.r.squared sigma statistic  p.value    df logLik   AIC   BIC deviance df.residual  nobs
      <chr>     <dbl>         <dbl> <dbl>     <dbl>    <dbl> <dbl>  <dbl> <dbl> <dbl>    <dbl>       <int> <int>
    1 Old       0.286         0.263 0.363      12.4 0.00134      1  -12.3  30.7  35.2    4.08           31    33
    2 Young     0.408         0.381 0.115      15.1 0.000788     1   18.9 -31.8 -28.3    0.291          22    24

线性模型的统计指标和预期一致没有变化（仅曲线为负向），但对数模型的指标出现了差异。我尝试过拟合指数模型lm(exp(s) ~ q，也试过添加负号的"-exp()"写法，但都无法让R²等指标与原log(m) ~ q模型匹配，还试过1/log等其他写法，也找不到适合「Old」分组的模型写法与曲线绘制方法。下图是我当前用log(s)拟合得到的结果：

model_lin2 <- lm(s ~ Age/q + 0, df,na.action=na.exclude)
model_exp2 <- lm(log(s) ~ Age/q + 0, df,na.action=na.exclude)

df_plot2 <- ggplot(df, aes(x = q, y = s, shape=Age, color=Age)) + geom_point() +
              geom_line(aes(y = ifelse(Age == "Young", fitted(model_lin2), NA))) +
              geom_line(aes(y = ifelse(Age == "Old", exp(fitted(model_exp2)), NA))) +
              theme_bw() + theme(panel.grid = element_blank()) 

修改y轴后的拟合结果图：

需求 #

希望能得到可行的解决方法，我认为和线性回归的逻辑一致，100-y版本的对数回归的r²、p值等统计指标应该和原始模型保持一致。

内容的提问来源于stack exchange，提问作者Mata