# Selection of data range changes coefficients too much in lmer (inverse regression)

Background: I have the lmer model

lmer(log(log(Amplification)) ~ poly(Voltage, 3) +
(poly(Voltage, 3) | Serial_number), data = APD)


for a data frame APD. This contains lot of devices, separated by Serial_number. Each device's amplification was measured in the voltage range from 0 V to about 400 V. In total I have about 1500 devices (and nearly 80,000 observables).

Motivation: My goal is to get the corresponding voltage of a certain amplification for each device; Voltage(Amplification=150)?. With the help of extracting the coefficients of the polynomials I'm able to do an inverse regression and to calculate this voltage.

Problem: When I select the data range in different ways I calculate different voltages. This is more or less reasonable because the fits vary. How can I ensure which voltage is correct? When I take the whole data range the calculated (Amplification=150, Voltage(A=150))-point is far beside the measurement data. When I limit the data range then the point moves closer and closer to the measurement data. So obviously I can influence that strongly by selecting the data range.

Bottom line: My very goal is to find the very most reliable voltage for each device having an amplification of Amplification=150. A model which is often used in literature is

$$\text{Amplification}=\frac{1}{1-\left(\frac{\text{Voltage}}{\theta_0}\right)^{\theta_1}}+\theta_2+\epsilon$$

however I would have to invert the model in my case because I want to predict the Voltage given that Amplification=150.

The plots of the calculated data points:

Whole data range for fitting with lmer:

Limited data range with ignoring all measurement data below Amplification<100 and all higher than Amplification>200.

The origin curve of one single device:

Minimal data set and script: https://files.fm/u/5yy22kkm