Multiple simulations of a system under different conditions - paired data?

Question

I am currently generating data by simulating a model of chemical system under different conditions (temperature) over time. In each simulation, the starting structure being modeled is exactly the same - only the temperature is different. The system is allowed to propagate over time and the length and number of observations in each simulation is identical. I would like to compare mean values e.g. distances between two atom under the different conditions. I have two questions:

1. Should I regard the two simulations I have (high and low temperature) as paired data. How would an analogous human study be treated (e.g. comparing the behaviour of a single human participant under during a 1 hour period under condition 1 and another 1 hour period after an extensive washout period - under condition 2)?

2. Since I effectively have two time series, what are the implications of the distance I want to measure being autocorrelated in some way?

Answer 1

Stochastic model

Consider the following scenario:

• You measure the blood pressure of the same man every day for two weeks;
• You measure the blood pressure of the same woman every day for two weeks;

What can you say about gender and blood pressure. Answer: not very much. This example is very similar to what you have. Until your sample size is bigger than one, you can't say very much.

As I commented above, you know that your two simulations are different. If you simulated the process enough times, you would get a "small p-value". What I suspect you want is to estimate the magnitude of these differences. So,

• You need more simulations to estimate the variance. There's no way around this.
• At each time-point store $x$ and $x^2$. This will allow you estimate the mean and variance of $x$.
• Once you have the mean and variance you could plot:
  • $x$ with a 95% confidence interval over time;
  • $x_H - x_L$ with a 95% confidence interval over time.

Deterministic model

Consider the following deterministic model for a simple death process:

\begin{equation} \frac{dX(t)}{dt} = -\mu X(t) \end{equation}

We can solve this equation to get $X(0) e^{-\mu t}$. Now suppose:

• For a high temperature, X(0) = 50
• For a low temperature, X(0) = 30.

Then for all $t$, the population of the high temperature is always greater than the low temperature. As we have a deterministic model, we have no uncertainty. If you did a t-test, your variance would be zero.

This is your scenario. Your model is deterministic. So the simulations are either the same or they are different. You have no uncertainty in the result.