Note: In the following discussion, replace C with E when considering effect observations.
In the first field below C, type the name of the observed quantity variable (e.g., CObs).
In the second field below 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.
Additive assumes error magnitudes are constant, regardless of concentration. The error bars in the residual plots are set to CEps.
Multiplicative assumes error magnitudes are proportional to concentration. The error bars are scaled by C*CEps.
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.
Power 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 corresponds to a form like C*exp(epsilon). If there is only one error model, such as one observe statement, then the predictions and observations are log-transformed and are fit in that space. This is because the error model becomes additive in log-space, which allows for higher performance and accuracy. This affects all the plots results and residuals because they are in log-space. The simulation tables are transformed back so they are not in log-space.
If Custom is selected, type a custom error model definition in the Defn field.
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.
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
The Log-additive option corresponds to a form like C*exp(epsilon). If the Log-additive error model is specified, and if there is only one error model, such as one observe statement, then the predictions and observations are log-transformed and are fit in that space. This is because the error model becomes additive in log-space, which allows for higher performance and accuracy. This affects all the plots 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.
Since the modeling engines in NLME can only handle a single error variable (or epsilon), and some error models are best specified as having an additive component and a multiplicative component, some complexities are needed. The Mix Ratio uses the following formula:
C+eps*(1+C*mixRatio)
where mixRatio is a fixed effect and is understood to be the multiplicative sigma (i.e., standard deviation of the multiplicative error variable) divided by additive sigma (i.e., standard deviation of the additive error model).
Another way to specify a mixed error model having a fixed effect, but with the fixed effect signifying the multiplicative sigma, rather than the ratio of multiplicative to additive sigma is Add+Multi. It makes use of a built-in function called “sigma()” that can only be used in this context, and its value is the current estimate of the standard deviation of eps. The formula is:
C+eps*sqrt(1+(C*multStdev/sigma())^2)
where multStdev is the multiplicative sigma.
When this error model is used, the additive sigma is called stdev, and the multiplicative sigma is called multStdev. Since multStdev is a fixed effect, its name can be changed as desire.
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 eps1 and eps2 are drawn from a unit normal distribution. Then the formula would be:
C+eps1*sigma1+C*eps2*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+eps*sqrt(1+(C*r)2)
Then replace r with sigma2/sigma1 to obtain:
C+eps*sqrt(1+(C*sigma2/sigma1)2)
where sigma1 is represented by the sigma() function, and sigma2 is represented by multStdev.
So, by choosing the option Add+Mult, this formula will be used to estimate both stdev (the additive standard deviation) and multStdev (the multiplicative standard deviation).
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