15 Section 17. Notes on ‘Ch 19. Parametric nonlinear models’

2021-12-09

These are just notes on a single chapter of BDA3 that were not part of the course.

15.1 Chapter 19. Parametric nonlinear models

examples:
- a ratio; \(\text{E}(y = \frac{a_1 + b_1 x_1}{a_2 + b_2 x_2}\)
- sum of nonlinear functions: \(\text{E}(y) = A_1 e^{-\alpha_1x} + A_2 e^{-\alpha_2x}\)
parameters of a nonlinear model are often harder to interpret
- often requires custom visualization techniques
“Generally each new modeling problem must be tackled afresh.” (pg 471)
- these models are less systematic than linear modeling

estimate 10 unknown concentrations of an allergen based off of serial dilutions of a known standard

Notation:
- parameters of interest: concentrations of unknown samples \(\theta_1, \dots, \theta_10\)
- known concentration of the standard \(\theta_0\)
- dilution of measure \(i\) as \(x_i\) and color intensity (measurement) as \(y_i\)
Curve of expected measurements given the concentration
- use the following equation that is standard in the field
- parameters:
  - \(\beta_1\): color intensity at the limit of 0 concentration
  - \(\beta_2\): the increase to saturation
  - \(\beta_3\): concentration at which the gradient of the curve turns
  - \(\beta_4\): rate at which saturation occurs

\[ \text{E}(y | x, \beta) = g(x, \beta) = \beta_1 + \frac{\beta_2}{1 + (x / \beta_3)^{-\beta_4}} \]

Measurement error
- modeled as normally distributed with unequal variances
- parameters:
  - \(\alpha\): models the pattern that variances are higher for larger measurements
    - restricted \([0, 1]\)
  - \(A\) is a arbitrary constant to scale the data so \(\sigma\) can be interpreted as the deviation from “typical” values
  - \(\sigma\): deviation of a measure from the “typical”

\[ y_i \sim \text{N}(g(x_i, \beta), (\frac{g(x_i, \beta)}{A})^{2\alpha} \sigma_y^2) \]

Dilution errors
- two possible sources:
  1. initial dilution: the accuracy of the creation of the initial standard concentration
  2. serial dilutions: error in creation of the subsequent dilutions (low enough to ignore for this analysis)
- use a normal model on the log scale of the initial dilution error
- parameters:
  - \(\theta_0\): known concentration of the standard solution
  - \(d_0^\text{init}\): known initial dilution of the standard that is called for
    - without error, the concentration of the initial solution would be \(d_0^\text{init} \theta_0\)
  - \(x_0^\text{init}\): the actual (unknown) concentration of the initial dilution

\[ \log(x_0^\text{init}) \sim \text{N}(\log(d_0^\text{init} \cdot \theta_0), (\sigma^\text{init})^2) \]

Dilution errors (cont)
- there is no initial dilution for the unknown samples being tested
  - therefore, the unknown initial concentration for sample \(j\) is \(x^\text{init} = \theta_j\) for \(j = 1, \dots, 10\)
  - for the dilutions of the unknown samples, set \(x_i = d_i \cdot x_{j(i)}^\text{init}\)
    - \(j(i)\) is the sample \(j\) corresponding to measurement \(i\)
    - \(d_i\) is the dilution of measurement \(i\) relative to the initial concentration

priors used as described by book (are likely different than what would be recommended now):
- \(\log(\beta) \sim U(-\infty, \infty)\)
- \(\sigma_y \sim U(0, \infty)\)
- \(\alpha \sim U(0,1)\)
- \(p(\log \theta_j) \propto 1\) for each unknown \(j = 1, \dots, 10\)
cannot estimate \(\sigma^\text{init}\) because we only have a single standard
- use a fixed value of 0.02 based on a previous analysis of different plates

this is a more complex model
uses a physiological model with parameters that cannot be solely determined using the data
- requires informative priors based on previous studies