Quadratic fitting for grouped data in R

Question

although I've found plenty of help regarding fitting models in general, I keep running into a specific issue with my data because of the way it's organized. It's from an intro stats book and is supposed to represent sample data of errors as a function of milligrams of some drug.

|-----|-------|-------|-------|
| 0mg | 100mg | 200mg | 300mg |
|-----|-------|-------|-------|
| 25  |  16   |   6   |   8   |
| 19  |  15   |  14   |  18   |
| 22  |  19   |   9   |   9   |
| 15  |  11   |   5   |  10   |
| 16  |  14   |   9   |  12   |
| 20  |  23   |  11   |  13   |

The data looks like it dips around group C, then goes up a bit for D, hence looking for a quadratic fit.

I've tried the following:

scores = c(25, 19, 22, 15, 16, 20,
           16, 15, 19, 11, 14, 23,
            6, 14,  9,  5,  9, 11,
            8, 18,  9, 10, 12, 13)

x_groups = rep(c(0,100, 200, 300), each = 6)
scores.quadratic = lm(scores ~ poly(x_groups, 2, raw = TRUE))

I can then use the summary() function to view the results. I'm confused about the lm() function and how it's supposed to fit a quadratic function. My understanding is that it will take each index in x_groups and square that, then use a linear fit with that new vector, but that doesn't seem correct to me.

Can someone provide feedback on how this is supposed to fit a quadratic to my data, or if it's not doing that, please help me understand where I'm going wrong.

Thank you.

Answer 1

Let's go through your way of thinking step by step. First, you spot this dip via your numbers for group C. The best way to visualise this is

library(ggplot2)
library(dplyr)

scores = c(25, 19, 22, 15, 16, 20,
           16, 15, 19, 11, 14, 23,
           6, 14,  9,  5,  9, 11,
           8, 18,  9, 10, 12, 13)

x_groups = rep(c(0,100, 200, 300), each = 6)

# create dataset
d1 = data.frame(scores, x_groups) 

# calcuate average scores for each group
d2 = d1 %>% group_by(x_groups) %>% summarise(Avg = mean(scores))

# plot them
ggplot() + 
  geom_point(data = d1, aes(x_groups, scores)) +
  geom_line(data = d2, aes(x_groups, Avg), col="blue")

Now you can actually see the dip and that's the pattern you want to model.

Then, you want to fit your quadratic model. Keep in mind that quadratic is a specific case of a polynomial formula, but it has order = 2. A polynomial fit with order = n of a variable x will fit intercept + x + x^2 + x^3 + ... + x^n. Therefore, the quadratic will fit intercept + x + x^2 and that's exactly the coefficients you get in your model's output:

scores.quadratic = lm(scores ~ poly(x_groups, 2, raw = TRUE))
summary(scores.quadratic)

# Call:
#   lm(formula = scores ~ poly(x_groups, 2, raw = TRUE))
# 
# Residuals:
#   Min      1Q  Median      3Q     Max 
# -6.1250 -2.3333 -0.2083  1.8542  8.7917 
# 
# Coefficients:
#                                    Estimate Std. Error t value Pr(>|t|)    
#   (Intercept)                    20.2083333  1.5925328  12.689 2.58e-11 ***
#   poly(x_groups, 2, raw = TRUE)1 -0.0745833  0.0255747  -2.916  0.00825 ** 
#   poly(x_groups, 2, raw = TRUE)2  0.0001458  0.0000817   1.785  0.08870 .  
# ---
#   Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
# 
# Residual standard error: 4.002 on 21 degrees of freedom
# Multiple R-squared:  0.4999,  Adjusted R-squared:  0.4523 
# F-statistic:  10.5 on 2 and 21 DF,  p-value: 0.0006919

The coefficient of the quadratic term is 0.0001458 ,close to zero , but statistically significantly different than zero at a 0.1 level (p value = 0.08870). Therefore, the model kind of feels that there's a dip.

You can plot the fit like this:

# plot the model
ggplot(d1, aes(x_groups, scores)) + 
  geom_point() +
  geom_smooth(formula = y ~ poly(x, 2, raw = TRUE),
              method = "lm")

You can see this as a smoothed version of the real pattern (1st plot).