# Calculating half-life

## Problem description

Half-life of drug in SCID, pigs that have no immune system. A total of 6 SCID pigs and 8 SCID pigs in 3 litters where the first litter was bottle fed and the latter two were nursed. The first litter has 2 SCID pigs and 2 non-SCID pigs.

Main scientific question is whether there is a difference in half-life between SCID and non-SCID pigs.

Approach 1) Currently modeling the log of the amount of drug and calculating the half-life. Then fitting another model for the calculated half-life using SCID and litter as fixed effects.

Approach 2) Separately fitting a mixed effect model on the log of the amount of drug and including SCID, time, and SCID x time as fixed effects and a random effect for pig.

Client question is “How are these models the same/different?”

## Consulting response

Approach 2) has a common slope for all pigs. If a random coefficient for time by pig is included in Approach 2), it gets closer to the first approach.

We would suggest modeling the logarithm of the drug amount using litter, SCID, time, and SCID x time as fixed affects and (time|pig) as a random effect.

The main quantity of interest to answer the scientific question of interest is the coefficient for the SCID x time interaction which determines if there is a difference in degradation, on average, between the SCID and non-SCID pigs. It may be of interest to calculate the slope (coefficient for time) for both SCID and non-SCID pigs along with their confidence intervals. Then to interpret the half-life, use the equation halflife = ln(2)/slope to calculate the point estimate and confidence interval for the halflife.

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