Question

The predicted values for the four phenotypes form a
ratio. The simulated values
match this predicted ratio.

Answer 1

That sounds good — if your simulated phenotype counts follow the same relative proportions as the predicted ratio, that supports the model. To make that conclusion quantitative and repeatable, you can:

1. Convert the predicted ratio to expected counts
- If the predicted ratio is r1:r2:r3:r4 and your total simulated sample size is N, then expected count for phenotype i is Ei = N * ri / (r1+r2+r3+r4).

2. Compute a goodness-of-fit test
- Pearson chi-square: X2 = sum((Oi − Ei)^2 / Ei) over the four phenotypes. Degrees of freedom = k − 1 = 3 (unless you estimated parameters from the data, in which case df reduces).
- Get the p-value from the chi-square distribution. A large p-value (e.g. p > 0.05) means you do not reject the null that observed = expected (i.e., they match).

3. Check assumptions and alternatives
- If any Ei < 5, chi-square may be unreliable: use exact/Monte Carlo methods or combine categories.
- You can also report effect-size metrics (e.g., root-mean-square deviation, percent deviation for each category) and plot observed vs expected proportions.

4. Optional: use simulation to assess variability
- Repeat the simulation many times and examine the distribution of chi-square statistics or of the deviations from expected ratios.

Example formulas and quick code

- Expected counts:
Ei = N * ri / sum(r)

- Chi-square:
X2 = sum((Oi - Ei)**2 / Ei)

Python (SciPy):
from scipy.stats import chi2
import numpy as np

obs = np.array([O1, O2, O3, O4])
ratio = np.array([r1, r2, r3, r4])
N = obs.sum()
exp = N * ratio / ratio.sum()
X2 = ((obs - exp)**2 / exp).sum()
p = 1 - chi2.cdf(X2, df=3)

R:
obs <- c(O1, O2, O3, O4)
ratio <- c(r1, r2, r3, r4)
N <- sum(obs)
exp <- N * ratio / sum(ratio)
chisq.test(x = obs, p = ratio/sum(ratio), rescale.p = TRUE, correct = FALSE)

If you want, tell me the predicted ratio and your observed counts (or paste your simulation results) and I’ll compute the test and/or make a plot.