Effex app documentation – Design detail

Docs: Summary

Thu, 05 Jan 2017 00:00:00 +0000

Selection of entries in the other tabs. Importantly, this tab informs us about the price of the design.

When you place the pointer of your mouse over the text on the Properties column, you will get an explanation.

The price appears in the last row and it is expressed in credits. The credits are the currency of our software and the way of purchasing designs.

Docs: Morphology

Thu, 05 Jan 2017 00:00:00 +0000

The content on this tab refer to qualities of the design matrix. For example, there is information on the factors, the number of runs, whether the design is foldover, etc.

As in the previous tab, when you place the pointer of your mouse over the text on the Properties column, you will get an explanation.

Docs: Aliasing

Thu, 05 Jan 2017 00:00:00 +0000

This tab contains a colormap on correlations with individual absolute correlations between the effects in a full second-order effects model, together with some averages and maximum values that appear on it.

As in the previous tab, when you place the pointer of your mouse over the text on the Properties column, you will get an explanation.

Docs: Efficiencies

Thu, 05 Jan 2017 00:00:00 +0000

This tab displays the efficiencies for the models that are estimable.

As in the previous tab, when you place the pointer of your mouse over the text on the Properties column, you will get an explanation. You can change the model using the radio buttons on top.

The D-, A-, G- and I-efficiency are standard attributes of an experimental design which inform us about how well the model effects are estimated and how precise the predictions will be. All efficiencies are, in one way or another, based on the Fisher information matrix.

Design matrix and model matrix

The design matrix, denoted as $\mathbf{D}$, has $n$ rows (one per test), and $m$ columns (one per factor). Schematically, it is represented as $\mathbf{D} = \begin{pmatrix} d_{11} & d_{12} & \cdots & d_{1m} \\ d_{21} & d_{22} & \cdots & d_{2m} \\ \vdots & \vdots & \cdots & \vdots \\ d_{n1} & d_{n2} & \cdots & d_{nm} \end{pmatrix}$, where an entry $d_{ij}$ indicates the level of the $j$-th factor at the $i$-th run.

The expansion vector function, denoted as $\mathbf{f}(\mathbf{d})$ takes a vector of factor settings (this is, a row of $\mathbf{D}$), and expands that vector to its correponding model terms, which can include intercept, main effects, two-factor interaction effects, quadratic effects, etc.

The model matrix, denoted as $\mathbf{X}$, has $n$ rows (one per test), and $p$ columns (one per effect). It is built by applying the expansion vector function to each row of the design matrix.

D-efficiency and A-efficiency

Notation:

D-efficiency for a specific model: D(model)
A-efficiency for a specific model: A(Model)

Both measure how precisely the effects in a model can be estimated.

The inverse of the information matrix, $(\mathbf{X}^T\mathbf{X})^{-1}$, indicates how well the model effects can be estimated. In an orthogonal design, where the model effects can be estimated independently from each other, both the information matrix and its inverse are diagonal matrices.

For example, the diagonal entries quantify the estimation error of each one of the model effects, and its determinant gives an overall measure on how good all model effects can be estimated compared to an orthogonal design.

The D-efficiency is a number that lies between $0$ and $100$ and gives us information on the confidence ellipsoid of the estimation error of the model effects. The lower this value, the better. It is calculated from the determinant of the information matrix as follows: $D-efficiency = 100 \cdot\frac{|\mathbf{X}^T\mathbf{X}|}{n}^{1/p}$, where $|\cdot|$ denotes the determinant of a matrix, $p$ is the number of effects in the model and $n$ is the number of runs. The D-efficiency takes into account the correlation between the estimates of the model effects.

The A-efficiency is a number that lies between $0$ and $100$ and, similarly to the D-efficiency, indicates how well the model effects can be estimated overall. It is calculated using the trace of the information matrix as follows: $A-efficiency = 100 \cdot \frac{p}{n \cdot tr((\mathbf{X}^T\mathbf{X})^{-1})}$, where $tr(\cdot)$ is a matrix operator result of the sum of the diagonal entries of the matrix (matrix trace).

G-efficiency and average prediction variance

Notation:

G-efficiency for a specific model: G(model)
Average prediction variance for a specific model: Avg pred var(Model)

The variance of the predicted value at a point $\mathbf{x}$ is given by: $pv_{\mathbf{x}}= \mathbf{f}^T(\mathbf{x}) ( \mathbf{X}^T\mathbf{X} )^{-1} \mathbf{f}(\mathbf{x})\sigma_{\varepsilon}^2$, where $\mathbf{f}(\mathbf{x})$ is the model expansion of $\mathbf{x}$ over the experimental region $\chi$, and $\sigma_{\varepsilon}^2$ is the error variance.

The maximum prediction variance then equals $\max pv_{\mathbf{x}}, \mathbf{x} \in \chi$, and the average prediction variance $avp_{\chi} = \int_{\chi} pv_{\mathbf{x}} d\mathbf{x}$.

Finally, $G-efficiency = 100 \cdot \frac{p}{n \cdot avp_{\chi}}$.

Docs: Powers

Thu, 05 Jan 2017 00:00:00 +0000

We look at the designs from different angles with respect to the power calculation. Some of the powers we present here are not present in any other commercial software. Here you will find detailed information on how we calculate them.

The information on this interactive table helps assess how good a design is regarding the capacity to estimate each effect type. The table has three columns:

Model (1st column): underlying set of effects considered for the power calculation. Only models that are estimable within the design are displayed. The models may be one of the following:
- Model with a constant term (intercept)
- Model with a constant term and main effects (me)
- Model with a constant term, main effects, and two-factor interaction effects (meie)
- Model with a constant term, main effects, two-factor interaction effects, and quadratic effects (mesoe)
- Model with a constant term, main effects, and quadratic effects (meqe)
Effects (2nd column): the effect type considered. It can be one of the following:
- Main effect (main)
- Two-factor interaction effect (interaction)
- Quadratic effect (quadratic)
Value (3rd column): minimum value of the power to detect one effect of the type mentioned in the second column when the underlying model has the effects mentioned in the first column.

Try to change the values of $\alpha$ and the signal-to-noise ratio (SNR) to see how the values of the power change.

Docs: Projection properties

Thu, 05 Jan 2017 00:00:00 +0000

While a design may not allow the estimation of a full second-order effects model, it may allow the estimation of a large subset of the effects, so that it is suitable for a screening + optimization experiment at the same time. Second-order models for subsets of factors.

This page offers details on the estimation quality of second-order models based on subsets of the factors. For any subset, these models include an intercept, all the main effects, all the two-factor interactions, and all the quadratic effects of the numerical three-level factors in the subset.

The numbers of numerical three-level factors in the subsets are specified in the first column of the table.
The numbers of two-level factors are specified in the table’s second column.
The third column displays the so-called projection estimation capacity for up to three designs that are to be compared. This is the fraction of estimable models for the numbers of numerical three-level factors and two-level factors specified in the first and second columns, respectively.
The last column contains clickable buttons that produce, for each subset with the numbers of two-level and three-level factors specified in the first two columns, the average D-efficiencies and A-efficiencies. Further characteristics are the averages of (1) the maximum prediction variance and (2) the average prediction variance over 500 randomly chosen combinations of two-level and three-level factors.
The combinations of the two-level factors are randomly chosen from the full factorial design in the specified number of two-level factors. The combinations of the three-level factors are randomly chosen from between the low and high levels.
As the maximum prediction variance is often located at the edges and corners, a few of such points are added to the 500 random points.

Docs: Model effects

Thu, 05 Jan 2017 00:00:00 +0000

When a certain model is estimable, we can obtain statistical quality indicators that gives us information on the power to estimate each effect in the model, the variance inflation factor of each estimate in the model, or the relative error of estimation of each individual effect.

The power to detect an active effect.

Calculated for different values of Signal-to-noise ratio and significance level ($\alpha$). For detailed information on how the power is calculated, please click here.

Variance inflation factor

Variance inflation factor (VIF) is a measure of how much the variance of the estimate of an individual effect is inflated due to aliasing or multicollinearity with other effects. The expression for VIF is given as:

$VIF = \frac{1}{1-R^2}$

where $R^2$ is the coefficient of determination of a model where the dependent variable is the selected effect and the independent variables are all the other terms in the model selected. A VIF value of 1 ($R^2 = 0$) indicates that there is no aliasing between the selected effect and all other effects included in the model. Ideally this number should be small ($<10$).

Relative error of estimation

The Relative error of estimation (REE) is the standard error of the estimate of an effect relative to the standard error of the model. This is given by the expression:

$REE = \sqrt{(\mathbf{X’X})^{-1}_{ii}}$ ,

where $\mathbf{X}$ is the selected model matrix and $(\mathbf{X’X})^{-1}_{ii}$ is the $i^{th}$ diagonal element of the variance covariance matrix $(\mathbf{X’X})^{-1}$ . The actual standard error of an estimate of an effect can be calculated by multiplying this expression with the standard error from the model.

All the three characterizations (POWER, VIF, REE) can be calculated for the following models:

ME – All main effects model
MEIE – All main effects + all two-factor interactions model
MESOE – All main effects + all second order effects (two-factor interactions and quadratic effects) model.
MEQE – Main effects + all quadratic effects model.

Additional notes:

Some model options may not appear for the chosen design. This happens when the said model cannot be fit with the design selected.

The powers and REE for the blocking effect is only displayed when blocking option is selected (Under – ‘Define your experiment’) and the intrablock correlation coefficient is set to Extreme (For more information, refer to ‘power page’).

The VIF and REE calculations are independent from the specified value of Signal-to-noise ratio and significance level $\alpha$.