Determine the correct approach for calculating AIC correction (AICc) when comparing models with different variance structures across compartments | Step-by-Step Solution

Understanding AICc Parameter Counting in Complex Biological Models

1. What We're Solving:

We need to figure out how to properly count parameters when calculating the corrected Akaike Information Criterion (AICc) for biological models that have multiple compartments with different variance structures. This is tricky because we're comparing models across different datasets where the variance might scale differently between compartments.

2. The Approach:

When comparing different cooking methods (models) for a multi-course meal (multi-compartment system), you need to account for ALL the ingredients and techniques used (parameters), including how much each course varies in preparation time (variance structures). The AICc helps us pick the best "recipe" while penalizing overly complicated approaches.

3. Step-by-Step Solution:

Step 1: Identify ALL Parameters

• Count the biological parameters (rate constants, initial conditions, etc.)
• Count variance parameters for EACH compartment separately
• Don't forget any scaling parameters between compartments
• Why this matters: Missing parameters leads to unfair model comparisons

Step 2: Handle Heteroscedastic Variance Structure

• If compartments have different variance patterns (heteroscedastic), each gets its own variance parameter
• For example: if Compartment A has constant variance but Compartment B has variance proportional to the mean, that's 2 separate variance parameters
• Key insight: Each unique variance structure adds to your parameter count

Step 3: Account for Cross-Dataset Scaling

• When comparing across datasets with different scales, determine if you're estimating separate scaling factors
• If models share parameters across datasets, count shared parameters only once
• If datasets require separate parameter estimates, multiply accordingly
• Think of it as: Are you using one recipe for all occasions, or adapting it for each kitchen?

Step 4: Calculate AICc Properly

• Use the formula: AICc = AIC + 2k(k+1)/(n-k-1)
• Where k = total parameter count from steps 1-3
• Where n = total number of observations across ALL compartments and datasets
• Critical point: The correction term becomes more important with higher parameter counts relative to sample size

Step 5: Ensure Fair Comparison

• All models being compared must use the same dataset(s)
• Parameter counting rules must be consistent across all candidate models
• Consider whether variance parameters are estimated or assumed known

4. The Framework:

For your specific analysis:

• Parameter inventory: Create a detailed list for each model including biological + variance + scaling parameters
• Variance structure map: Document which compartments share variance structures vs. have unique ones
• Dataset accounting: Clarify whether parameters are shared or separate across datasets
• Consistency check: Verify all competing models use identical parameter counting rules

5. Memory Tip:

Remember "BVSS" - Biological parameters + Variance parameters + Scaling parameters = Smart AICc calculation. Every unique random component in your model needs its own parameter in the count!

Great job tackling such a sophisticated statistical concept! Multi-compartment models with heteroscedastic variance are challenging even for experienced analysts. The key is being systematic and consistent in your parameter accounting - it's better to be overly detailed in your documentation than to miss something important.