# Half-life calculation with Tobit regression

Half-life calculation with Tobit regression allows inclusion of concentrations that are below the lower limit of quantification in the half-life estimate.

## Comparison to semi-log regression

Typical half-life calculation uses curve-stripping semi-log regression of the natural logarithm of the concentration by time. Based on the use of the logarithm of the concentration, concentrations below the lower limit of quantification (LLOQ) which are set to zero are ignored.

Tobit regression allows inclusion of the concentrations below the LLOQ in the half-life calculation. Tobit regression for half-life is equivalent to using Bealâ€™s M3 method in population pharmacokinetic (PK) models.

With Tobit regression, a line is fit using maximum likelihood. For points above the LLOQ, the likelihood is based on the probability density at the observed concentration. For points below the LLOQ, the likelihood is based on the cumulative probability distribution function from negative infinity to the limit of quantification.

## Automatic point selection with semi-log regression

With semi-log regression, the typical method used to automatically select concentrations for inclusion in the half-life estimate is to:

1. Omit all concentrations that are missing.
2. Omit all concentrations that are below the LLOQ.
3. Estimate the half-life for each set of points from the first concentration measure after Tmax to the third measure before Tlast.
4. Select the best half-life with the following criteria, in order:
1. The adjusted r-squared must be within a tolerance factor (typically 0.0001) of the largest adjusted r-squared.
2. The $$\lambda_z$$ value (slope for the half-life line) must be positive; in other words, the half-life slope must be decreasing.
3. If multiple choices of points fit the above criteria, choose the one with the most concentration measurements included.

For comparison with PKNCA, note that Phoenix WinNonlin switches the order for selection of 4.1 and 4.2 above. So, if the best adjusted r-squared is for an increasing slope but there is another adjusted r-squared with a decreasing slope, Phoenix will report the half-life.

## Automatic point selection with Tobit regression

With Tobit regression, the method is generally similar to the semi-log regression with two changes. The first change is that concentrations below the LLOQ are retained in the estimate. The second change is that the adjusted r-squared is not possible to calculate when including points below the LLOQ, so the minimum standard deviation estimate is used.

The selection method below results in effectively the same estimates for half-life when all points are above the LLOQ and improved estimates for half-life when some points are below the LLOQ. Future research may investigate optimization of this method.

The steps for Tobit regression are:

1. Omit all concentrations that are missing.
2. Estimate the half-life for each set of points from the first concentration measure after Tmax to the third measure before Tlast while including all points below the LLOQ after Tlast.
3. Select the best half-life with the following criteria, in order:
1. The estimated standard deviation of the slope is minimized.
2. The $$\lambda_z$$ value (slope for the half-life line) must be positive; in other words, the half-life slope must be decreasing.

# Comparison of Tobit and semi-log regression

In almost all scenarios, Tobit regression using the algorithm above improves the half-life estimate compared to semi-log regression. In the figure below, concentration-time profiles were simulated with 1-, 2-, and 3-compartment linear PK models with intravenous or extravascular administration and a variety of compartmental model parameters. The true half-life was calculated based on the compartmental model parameters. Then, the ratio of the estimated to true half-life was calculated. Values closer to 1 indicate a better fit and values farther from 1 indicate a poorer fit.

The empirical cumulative distribution function for ratio of estimated to theoretical half-lives for Tobit regression (aqua line) and semi-log regression (red line) are shown.

Tobit regression performs universally better than least-squares up to the estimated theoretical half-life, and better at >2-fold above the theoretical half-life while least-squares performs slightly better between the theoretical and 2-fold above. The fact that the Tobit regression cumulative distribution function is closer to 1 across the range of simulations indicates that Tobit regression provides better half-life estimate across a broad range of data.