How to calculate RMSE and standard deviation between a coordinate point and a non-linear regression under the same abscissa value?

Question

The original data is as below:

ISIDOR <- structure(list(Pos_heliaphen = c("W30", "X41", "Y27", "Z24", 
                                       "Y27", "W30", "W30", "X41", "Y27", "W30", "X41", "Z40", "Z99"
), traitement = c("WW", "WW", "WW", "WW", "WW", "WW", "WW", "WW", 
                  "WW", "WW", "WW", "WW", "WW"), Variete = c("Isidor", "Isidor", 
                                                             "Isidor", "Isidor", "Isidor", "Isidor", "Isidor", "Isidor", "Isidor", 
                                                             "Isidor", "Isidor", "Isidor", "Cali"), FTSW_apres_arros = c(0.462837958498518, 
                                                                                                                         0.400045032939416, 0.352560790392534, 0.377856799586057, 0.170933345859364, 
                                                                                                                         0.315689846065931, 0.116825600914318, 0.0332444780173884, 0.00966070114456602, 
                                                                                                                         0.0871102539376406, 0.0107280083093036, 0.195548432729584, 1), 
NLE = c(0.903498791068124, 0.954670066942938, 0.970762905436272, 
        0.873838605282389, 0.647875257025359, 0.53056603773585, 0.0384548155916796, 
        0.0470924009989314, 0.00403163281128882, 0.193696514297641, 
        0.0718450645564359, 0.295346695941639, 1)), class = c("tbl_df", 
                                                              "tbl", "data.frame"), row.names = c(NA, -13L))

Here is the code:

pred_df <- data.frame(FTSW_apres_arros = seq(min(ISIDOR$FTSW_apres_arros), 
                                             max(ISIDOR$FTSW_apres_arros),
                                             length.out = 100))

pred_df$NLE <- predict(mod, newdata = pred_df)

mod = nls(NLE ~ 2/(1+exp(a*FTSW_apres_arros))-1,start = list(a=1),data = ISIDOR)
ISIDOR$pred = predict(mod,ISIDOR)
a = coef(mod)
RMSE = rmse(ISIDOR$NLE, ISIDOR$pred)
MSE = mse(ISIDOR$NLE, ISIDOR$pred)
Rsquared = summary(lm(ISIDOR$NLE~ ISIDOR$pred))$r.squared

ggplot(ISIDOR, aes(FTSW_apres_arros, NLE)) + 
  geom_point(aes(color = Variete), pch = 19, cex = 3) +
  geom_line(data = pred_df) +
  scale_color_manual(values = c("red3","blue3"))+
  scale_y_continuous(limits = c(0, 1.0)) +
  scale_x_continuous(limits = c(0, 1)) +
  labs(title = "Isidor", 
       y = "Expansion folliaire totale relative",
       x = "FTSW",
       subtitle = paste0("y = 2/(1 + exp(", round(a, 3), "* x)) -1)","\n",
                         "R^2 = ", round(Rsquared, 3),"  RMSE = ",
                         round(RMSE, 3), "   MSE = ", round(MSE, 3)))+
  theme(plot.title = element_text(hjust = 0, size = 14, face = "bold", 
                                  colour = "black"),
        plot.subtitle = element_text(hjust = 0,size=10, face = "italic", 
                                     colour = "black"),
        legend.position = "none")

Then I get the figure below:

Now I want to calculate RMSE and standard deviation between the coordinate point circled by orange (0.352560790, 0.970762905) and the non-linear regression y = 2/(1 + exp(-5.674* x))-1 (the coordinate point in the regression line with the same abscissa 0.352560790). Could anyone please give me suggestions?

Thank you so much!

Answer 1

You didn't show us how you made your model, so I'm assuming it's something like this:

mod <- nls(NLE ~ 2/(1 + exp(a * FTSW_apres_arros)) - 1, start = list(a = -6),
           data = ISIDOR)

We can use our model to insert the expected value of NLE for each row in our original data frame like this:

ISIDOR$pred  <- predict(mod)

And we can get the residuals (or error) of each point (i.e. its distance from the regression line like this:

ISIDOR$error <- summary(mod)$residuals 

This results in the following data frame:

head(ISIDOR)
#> # A tibble: 6 x 7
#>   Pos_heliaphen traitement Variete FTSW_apres_arros   NLE  pred   error
#>   <chr>         <chr>      <chr>              <dbl> <dbl> <dbl>   <dbl>
#> 1 W30           WW         Isidor             0.463 0.903 0.865  0.0385
#> 2 X41           WW         Isidor             0.400 0.955 0.813  0.142 
#> 3 Y27           WW         Isidor             0.353 0.971 0.762  0.209 
#> 4 Z24           WW         Isidor             0.378 0.874 0.790  0.0837
#> 5 Y27           WW         Isidor             0.171 0.648 0.450  0.198 
#> 6 W30           WW         Isidor             0.316 0.531 0.714 -0.184

For our plot, we calculate the various metrics and use geom_smooth to plot our regression line (this means we don't need a prediction data frame)

library(geomtextpath)

Rsquared <- 1 - var(ISIDOR$error) / var(ISIDOR$NLE)
MSE      <- mean(ISIDOR$error^2)
RMSE     <- sqrt(MSE)

ggplot(ISIDOR, aes(FTSW_apres_arros, NLE)) + 
  geom_segment(aes(xend = FTSW_apres_arros, yend = pred), color = 'gray') +
  geom_textsegment(aes(xend = FTSW_apres_arros, yend = pred, 
                       label = round(error, 3)),
               color = 'red', data = ISIDOR[3,], vjust = 1.1) +
  geom_point(aes(color = Variete), shape = 19, cex = 3) +
  geom_smooth(method = nls, formula = y ~ 2/(1 + exp(a * x)) - 1,
              method.args = list(start = list(a = -6)), se = FALSE,
              color = 'black', size = 0.75) +
  scale_color_manual(values = c("red3", "blue3"))+
  scale_y_continuous(limits = c(0, 1.0)) +
  scale_x_continuous(limits = c(0, 1)) +
  labs(title = "Isidor", 
       y = "Expansion folliaire totale relative",
       x = "FTSW",
       subtitle = paste0("y = 2/(1 + exp(", round(coef(mod), 3), "* x)) - 1",
                         "\n", "R^2 = ", round(Rsquared, 3),"  RMSE = ",
                         round(RMSE, 3), "   MSE = ", round(MSE, 3)))+
  theme(plot.title = element_text(hjust = 0, size = 14, face = "bold", 
                                  colour = "black"),
        plot.subtitle = element_text(hjust = 0,size=10, face = "italic", 
                                     colour = "black"),
        legend.position = "none")