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# Statistical tests

Several statistical tests may be automatically performed to test the different components of the model. These tests use individual parameters drawn from the conditional distribution, which means that you need to run the task “Conditional distribution” in order to get these results. The tests for the residuals require to have first plot residuals (scatter plot or distribution).

Results of the tests are available in the tab “Results”  and selecting “Tests” in the left menu

## The model for the individual parameters

Consider a PK example with the following model for the individual PK parameters (ka, V, Cl):

In this example, the different assumptions we make about the model are:

• The 3 parameters are lognormally distributed
• ka is function of age only
• V is function of sex and weight. More precisely, the log-volume log(V) is a linear function of the log-weight $${\rm lw70 }= \log({\rm wt}/70)$$.
• Cl is not function of any of the covariates.
• The random effects $\eta_V$ and $\eta_{Cl}$ are linearly correlated
• $\eta_{ka}$ is not correlated with $\eta_V$ and $\eta_{Cl}$

Let’s see how each of these assumptions are tested:

### The covariate model

#### Testing if covariates should be removed from the model

If an individual parameter is function of a continuous covariate, the linear correlation between the transformed parameter and the covariate is not 0 and the associated $\beta$ coefficient is not 0 either. Then, Pearson’s correlation tests and Wald tests are used to test whether continuous covariates should be removed from the model. ANOVA and Wald tests are performed for categorical covariates in a same way.
In our example, we may want to test if the absorption rate constant ka is function of sex and if the volume V is function of sex and weight. The individual model looks like the figure below

If we look at the tests results for the covariate, the ANOVA for ka clearly shows that our hypothesis that ka is function of sex should be rejected:

Remark: The two covariates weight and sex are strongly dependent. Then, the fact that both lw70 and sex are significant on the parameter V does not mean that these two covariates should be kept in the model.

The Wald test using the standard errors estimated either by linearization or by stochastic approximation lead to the same conclusion:

#### Testing if covariates should be added to the model

Pearson’s correlation tests and ANOVA are performed to test if some relationships between random effects and covariates have not been taken into account in the model. In our example, only a relationship between weight and clearance could possibly be worthy of investigation.

### The model for the random effects

#### Testing the normality of the random effects

Shapiro-Wilk tests are performed to test if the random effects are normally distributed

#### Testing the correlation between random effects

Pearson’s correlation tests are performed to test if the random effects are linearly correlated. In our example, the assumption that $\eta_V$ and $\eta_{Cl}$ are correlated should be rejected.

### The distribution of the individual parameters

#### Individual parameters not dependent on covariates

When an individual parameter doesn’t depend on covariates, its distribution is a transformation of the normal distribution. Then, a Shapiro-Wilk test can be used for testing the normality of the transformed parameter. In our example, Cl does not depend on any covariate the hypothesis of lognormality should not be rejected:

Remark:  testing the normality of a transformed individual parameter that does not depend on covariates is equivalent to testing the normality of the associated random effect. We can check in our example that the  Shapiro-Wilk tests for $\log(Cl)$ and $\eta_{Cl}$ are equivalent.

#### Individual parameters dependent on covariates

Individual parameters that depend on covariates are not anymore identically distributed. Each transformed individual parameter is normally distributed, with its own mean that depends on the value of the individual covariate. In other words, the distribution of an individual parameter is a mixture of (transformed) normal distributions. A Kolmogorov-Smirnov test is used for testing the distributional adequacy of these individual parameters

## The model for the observations

A combined error model is assumed in our example with normal residual errors.

### The distribution of the residuals

Different tests are performed for the individual residuals, the npde’s and for the population parameters.

#### Testing the symmetry of the residual distribution

A Van Der Waerden test is used for testing the symmetry of the residuals. Indeed, symmetry of the residuals around 0 is an important property that deserve to be tested, in order to decide, for instance, if some transformation of the observations should be done

#### Testing the normality of the residuals

A Shapiro Wilk test is used for testing the normality of the residuals.

Remark:  the Shapiro Wilk test is known to be very powerful. Then, a small deviation of the empirical distribution from the normal distribution may lead to a very significant test (i.e. a very small p-value), which does not necessarily means that the model should be rejected!