Mice Types Tumorgenesis Propensity

Table of Contents
1. Overview	4. Data and Simulation
2. Results	5. Findings
3. Analysis Methodology	6. Model Evaluation

Overview

A cancer lab is investigating the rate of tumorigenesis in two types of mice. Type A mice have been extensively studied, and existing data suggests that their tumor counts follow a Poisson distribution with a mean of 12. The parameter $\theta_A$ follows a gamma distribution defined as gamma(120, 12). In contrast, the distribution of tumor counts for Type B mice is unknown.

Results

The results from our analysis provide compelling evidence of a significant difference between Type A and Type B mice in terms of their tumorigenesis rates.

• Type B Mice exhibit a higher affinity for tumorigenesis when compared to Type A mice. This was confirmed through extensive Monte Carlo simulations, where Type B mice were shown to have higher predicted tumor counts in 70% of the simulations.

• Poisson Model Fit: The Poisson model, which is effective for Type A mice, underpredicts the standardized tumor counts for Type B mice. This discrepancy suggests that the Poisson model is not suitable for Type B, leading to the use of the model as a lower boundary for the tumorigenesis propensity of Type B mice. Even when treating the Poisson model as a conservative estimate, the simulations show a stronger propensity for tumorigenesis in Type B mice compared to Type A.

• Implication: This result underscores that Type B mice have a significantly higher propensity for developing tumors, a distinction that becomes clearer as we treat the Poisson model as a lower bound for their tumor rates.

Analysis Methodology

The study employed Bayesian analysis in combination with Monte Carlo simulations to estimate and compare the tumorigenesis rates between the two mouse types. The goal was to generate posterior distributions for each type of mouse based on the observed tumor count data.

Steps Involved:

1. Monte Carlo Simulation: We used the results from prior distributions and conducted simulations to generate a range of tumor counts for both mouse types.
2. Comparison of Posterior Distributions: We compared the tumor count distributions between Type A and Type B mice, providing a statistical foundation to determine which group has a higher tumorigenesis propensity.

Data and Simulation

The data for this study includes tumor counts from both mouse types, obtained through experimental observations.

• Type A Tumor Counts: 10 samples with counts
• Type B Tumor Counts: 13 samples with counts

Approach:

• Prior Distributions: Prior distributions were derived based on existing research data for Type A mice, while Type B data were used to derive posterior distributions.
• Monte Carlo Simulations: These were conducted to estimate the tumor counts and generate relevant posterior distributions for both mouse types.

# Theta_a_mont and theta_b_mont from above
# select 1000 random samples from a poisson distribution with the thetas calculated from previous monte carlo simulation

y_a_mont <- rpois(1000, theta_a_mont)
y_b_mont <- rpois(1000, theta_b_mont)


# Print probability
cat("P(theta_b < theta_a | y_a, y_b) =",mean(y_a_mont > y_b_mont))

Findings

Probability of Tumorigenesis Affinity

The Monte Carlo simulations revealed a strong probability that the tumorigenesis affinity of Type A mice, $\theta_A$, is higher than that of Type B mice, $\theta_B$. This was based on the comparison of the posterior distributions derived from the simulation.

• The simulation results suggest that, in general, Type B mice show a higher tendency for tumor development, even with the conservative estimate provided by the Poisson model for Type B.

Effect of Sample Size on Tumorigenesis Probability

In further analysis, we explored the effect of varying sample sizes, $n_0$, on the probability of Type B mice having a higher tumor count than Type A mice.

• Increasing Sample Size: As $n_0$ increases, the probability that $\theta_A$ exceeds $\theta_B$ decreases, making the conclusions more stable and less sensitive to prior assumptions.

Comparison of Posterior Distributions

By comparing the posterior distributions of tumor counts for Type A and Type B, we assessed the probability that samples from Type B have fewer tumors than samples from Type A.

• As expected, larger sample sizes lead to a stronger differentiation between the two types, reinforcing the conclusion that Type B mice have a significantly higher tumorigenesis rate.

Model Evaluation

Type A Mice

The Poisson model fits the tumor count data for Type A mice well. The observed tumor count statistics closely match the predictions of the Poisson model, suggesting that the model accurately represents the tumor distribution for Type A.

Type B Mice

For Type B mice, the Poisson model does not provide a good fit. The observed tumor count statistics deviate significantly from the model's predictions. This discrepancy suggests that Type B mice require a more complex model or a different distribution to accurately represent their tumor count data.