Note: In the following discussion, replace C with E when considering effect observations.
In the first field next to C, type the name of the observed quantity variable (e.g., CObs).
In the second field next to C, type the name of the epsilon variable (e.g., CEps).
The epsilon variable represents a normal error with standard deviation as specified in the stdev field.
Select the type of error model from the Type pulldown (see descriptions below).
If AdditiveMultiplicative is selected, type a name for the parameter in the Mult Stdev field.
If MixRatio is selected, type a name for the parameter in the mix Ratio field.
If Power is selected, enter the value to use in the Power field.
If Custom is selected, type a custom error model definition in the Defn field.
In the StDev field, type a value for the standard deviation.
Check the BQL box if the dataset contains BQL values for the observation data.
When checked, the engine automatically reverts to the Laplacian method.
To use a worksheet with a censored column as input for a model with BQL selected, map the censored column to the CObsBQL context. This column can contain two categories of values: non-zero number (censored) or zero/blank (non-censored).
A concentration value marked as censored (CObsBQL<>0 and it is not empty) means that the true value of the observation is unknown, but it is not greater than the observed value (e.g., LLOQ, which is provided in the CObs cell for that row) and then the cumulative distribution function for the normally distributed error is used to calculate the likelihood. (The likelihood is the probability of falling into the interval between minus infinity and LLOQ, where LLOQ is given the value of the CObs or EObs column on that row.)
If a concentration value is flagged as non-censored, then the probability density function is used to calculate the likelihood.
The context name CObsBQL changes based on what is typed in the observed quantity variable field. For example, instead of CObsBQL, it could be ConcBQL.
Note: Maximum Likelihood Models with censored data (BQL? option) use the log of the probabilities between 0 and the censored number in the log likelihoods. If the censoring numbers are small, the loglikelihood might overflow, resulting in a Fortran error. This is more often the case when using multiplicative error models. If the error occurs, try increasing the BQL value if possible or change error types.
Note: For a straightforward way to create an observation column and its associated BQL flag column, use the Phoenix BQL object.
When BQL is checked, the Static LLOQ switch becomes available. Turn the switch on to enter a numeric value of LLOQ (>0) in the displayed field.
If CObsBQL is not mapped to a column in the dataset, then the static value of LLOQ will be used, so any observed value less than or equal to that LLOQ value is treated as censored.
If a value is specified and the CObsBQL column (or other column with a BQL flag) is mapped, the value in the observation column will be used as the LLOQ and will override the static LLOQ.
Check the Freeze box to freeze the standard deviation to the value shown in the StDev field and prevent estimation of this part of the model.
Additional residual error model details
Note: For PK models, Multiplicative, then Power are the preferred error models over Additive. This is because PK model types usually have concentrations spanning several orders of magnitude and, on a log scale, Additive has large errors at low concentrations.
Additive
The Addition option assumes error magnitudes are constant, regardless of concentration: CObs = C + CEps, where C denotes the predicted concentration and CEps represents the error.
Multiplicative
The Multiplicative option assumes error magnitudes are proportional to concentration: CObs = C*(1 + CEps).
Additive + Multiplicative
This Additive + Multiplicative option assumes a combined additive and multiplicative error.
C+CEps*sqrt(1+(CMultStdev/sigma())^2)
where CMultStdev is the standard deviation of multiplicative error and “sigma()” a built-in function whose value is the current estimate of the standard deviation of CEps.
To justify the above formula, look at the variance. Suppose the additive standard deviation is called sigma1, the multiplicative standard deviation is called sigma2, and suppose the corresponding epsilons CEps1 and CEps2 are drawn from a unit normal distribution. Then the formula would be:
C+CEps1*sigma1+C*CEps2*sigma2
The variance of this is the sum of the variances from each term, or:
sigma12+(C*sigma2)2
Now let r be the ratio: r=sigma2/sigma1, then the variance is:
sigma12+(C*r*sigma1)2
or sigma12 *(1+(C*r)2)
which is the variance of C+CEps*sqrt(1+(C*r)2)
Then replace r with sigma2/sigma1 to obtain:
C+CEps*sqrt(1+(C*sigma2/sigma1)2)
where sigma1 is represented by the sigma() function, and sigma2 is represented by CMultStdev.
MixRatio
The MixRatio option provides another way to specify a combined additive and multiplicative error. It uses the following formula:
C+CEps*(1+C*CMixRatio)
where CMixRatio is a fixed effect and is understood to be the ratio of the multiplicative sigma (i.e., standard deviation of the multiplicative error variable) to the additive sigma (i.e., standard deviation of the additive error).
Power
The Power option assumes the error magnitude is proportional to the concentration raised to the given power (i.e., CObs = C + CEps * C^p, where p is the value entered in the Power field).
LogAdditive
The LogAdditive option appears only when models contain one continuous observed variable (as this option behaves the same as the Multiplicative option when there are more than one continuous observed variable). It uses the formula:
CObs=C*exp(CEps)
For this option, both the predictions and observations are log-transformed and the fitting is done in the log-transformed space by default. This is because the error model becomes additive in log-space, allowing higher performance and accuracy. This affects all plot results and residuals because they are in log-space. The simulation tables are transformed back so they are not in log-space.
Because the logs of zero or negative numbers are not allowed, they are truncated to a value which is ¼ (0.25) of the smallest positive observation value. If the model is log-additive, but the conditions have not been met for log-transforming to take place, the model behaves the same as multiplicative.
The residual model that is displayed in the model text is:
observe(CObs=C*exp(CEps))
or
observe(CObs=exp(log(C)+CEps))
The engines implement the model by log-transforming both sides and seeing if the derivative of the right-hand side with respect to CEps is 1. If so, and if there is only one observe statement, then it does log-transformation.
observe(log(CObs)=log(C)+CEps)
This check is accomplished by examining the text of the model, so it can be applied to models other than built-in models.
The main advantages of log-additive are that the engines, particularly the Lindstrom-Bates FOCE engine, can run faster when a simple additive error model is used, and the FOCE approximation can be more accurate.
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