15.8 MultiNormal Distributions and Correlated Inputs

Model inputs can be correlated such that changes in one input are related to changes in other input(s). If you sample these inputs from independent distributions, this correlation is ignored. If correlations are ignored within the context of probabilistic sensitivity analysis, it can lead to more variance in your results than is realistic.

Input correlation can range from -1 to 1.

  • 1 represents 100% correlation between inputs

  • 0 represents no correlation between inputs

  • -1 represents 100% correlation in the opposite direction between inputs

To incorporate correlations when sampling values, you can use a MultiNormal distribution. MultiNormal distributions include multiple correlated variates and generate correlated samples for each of those variates. For example, if you have three correlated inputs, you would create a single MultiNormal distribution with three correlated variates, generating three sets of correlated samples.

The rest of this section describes how to create and use MultiNormal distributions for parameters in the context of Probabilistic Sensitivity Analysis (PSA). But you could also use MultiNormal distributions in the context of Microsimulation.

Correlation Example Model

The Legal tutorial example model BreachOfContract-PSA-Multinormal.trex is a decision tree comparing two strategies - Litigate versus Settle at current demand.

In the Litigate strategy, the consequentials are all dependent through a MultiNormal distribution: dist_vConsDamages_All. The consequentials in the three outcomes are different – as represented by the variables vConsDamages_1Low, vConsDamages_2Med and vConsDamages_3High.

For Probabilistic Sensitivity Analysis (PSA), those variables reference distributions. The current model we are using will show the impact of correlation between the three inputs. We can reference back to the model BreachOfContract-PSA.trex to see the impact without correlation. That way we can examine the impact of correlation on the PSA results. We will examine the results from PSA with and without correlation later in this section.

Creating a MultiNormal Distribution

Although the example model BreachOfContract-PSA-Multinormal.trex used in this section already has the MultiNormal distribution created, this is how you would create one.

  • Open the Distributions View.

  • Click the “+” button to create a new distribution.

  • Enter the name of the distribution.

  • Scroll down the list of distribution types to select MultiNormal.

  • Click the Edit Matrix & Options button (See figure Screenshot #1). A second dialog “Edit Matrix & Options” will open (See figure Screenshot #2)

  • Enter the number of variates (minimum 2). In this case we have 3 because there are 3 types of damages.

  • Select the appropriate Matrix type based on the form of your correlation data. This likely depends on the software used to generate the correlations.

  • Click on the column headings above the matrix and enter names for each variate within the MultiNormal distribution (optional) (See figure Screenshot #2 where Damages_VALUE have been entered as the column headings).

  • Enter the correlation data into the matrix grid.

  • Enter mean values for the individual variates.

  • Enter standard deviation (or scaling) values for the individual variates. Note that matrix types have variance built-in, so you would not enter the variate standard deviations.

  • If you have the matrix, mean and standard deviation data in Excel, you can copy and paste the data directly into the Matrix dialogue. Use the copy and paste buttons above the matrix.

  • Click OK to return to the Add/Change Distribution dialog.

  • In this example, the resampling rate is “Resample per EV” for use in PSA, which is what we want in this context.

  • Click OK to save the distribution.

Above is Screen shot #1.

Above is Screenshot #2.

For the top two matrix types shown in Screenshot #2 above – Variance/Covariance and Correlation – TreeAge performs the Cholesky decomposition prior to sampling. Use the "Show Cholesky" button to show the calculated Cholesky Matrix as per TreeAge.

For the second two matrix types – Cholesky (Var/Covar) and Cholesky (Correlation) – the Cholesky decomposition has already been incorporated into the matrix, so TreeAge Pro does not apply it again. Depending on the software you have used to generate the correlations, you can enter data in any of these 4 formats.

Once your MultiNormal distribution is created, select the distribution in the Distributions View and click the "Graph It" icon within the Distributions View toolbar to see the samples generated for each variate of the MultiNormal distribution.

Referencing the MultiNormal Distribution in the Model

Although the Legal example model BreachOfContract-PSA-Multinormal.trex used in this section already has the MultiNormal distribution incorporated, this section describes how it is referenced.

As with any distribution, it must be referenced within the model to have any impact on results. Many distributions just return a single value each time they are sampled, and these would be referenced directly, that is pVariable = dist_for_pVariable. Distributions such as MultiNormal and Dirichlet that have variates (that is they can return several values) must be referenced differently, so you can specify the variate you want. The two forms are presented below.

  • By name: Dist(“DistName”; "VariateName")

  • By index: Dist(DistIndex; VariateIndex)

You can also reference the distribution by name, and the variate by index or vice versa. Note that name entries are always placed in quotes, while index entries are not.

In our example model, the MultiNormal distribution is referenced with the first form above, but it is used only when correlationSw = 1. Therefore we see the following definitions:

  • vConsDamages_1Low = Dist("dist_vConsDamages_All"; 1)

  • vConsDamages_2Med = Dist("dist_vConsDamages_All"; 2)

  • vConsDamages_3High = Dist("dist_vConsDamages_All"; 3)

When running PSA, samples for the probabilities are assigned from three variates of the MultiNormal distribution.

Impact of Correlation on PSA Results