Optimal Experimental Design for single substance and interaction trials

This Application allows computation of D-optimal designs for interaction trials in a dose response context. Designs are computed for two singular treatments as well as up to 5 combination treatments. Furthermore, the efficiency of prespecified designs can be checked, and more
robust designs suitable for several parameter conditions can be computed.

For details see:
Holland-Letz, T and Kopp-Schneider, A (2020): An R-Shiny application to calculate optimal designs for single substance and interaction trials in dose response experiments, Toxicology Letters 337, 18-27, FEB 2021, DOI: 10.1016/j.toxlet.2020.11.018

The robustness heatmap is based on:
Holland-Letz, T and Kopp-Schneider, A (2025): Robust and near optimal designs for the estimation of log-logistic dose response functions, under review

For the (outdated) previous application from Optimal experimental designs for dose-response studies with continuous endpoints, Archives of Toxicology (2015), 89(11), 2059-68, DOI: 10.1007/s00204-014-1335-2, see https://biostatistics-dkfz.shinyapps.io/dosis/

Two different dose response functions can be considered:
$$Log-logistic: y=c+\frac{d-c}{1+{\exp}^{b(ln(x)-ln(e))}}$$ $$Weibull: y=c+(d-c) \exp^{(-\exp^{(-b (ln(x)-ln(e))})}$$

This part computes D-optimal designs for a single treatment on the specified design space. One of two available dose response functions can be chosen, and an
a priori assumption regarding the assumed slope and ED50 parameters can be made.
Lowering the value for the reduction parameter will try to find a design with fewer support points, but might reduce the efficiency.

Function Parameters

Function:

Log-logistic

Weibull

Slope (Parameter b):

ED50 (Parameter e):

Plot of function

Result

Proposed designs, values of parameters b and e, and resulting D-Efficiency of proposed designs:

Design Heatmap

Points marked red on the diagonal are potential design points. Pairs of different design points marked red when crossreferenced are interchangeable with negligible loss of efficiency.

This part allows specification of any experimental design, and calculates the D-efficiency of this design compared to the optimal one. Up to nine different log dose levels can be specified. The graph below will also show the performance of the design under parameters deviating from the assumptions

Function Parameters

Function:

Log-logistic

Weibull

Slope (Parameter b):

ED50 (Parameter e):

Log Dose levels

# of Design Points (max 9):

Log Dose 1:

Log Dose 2:

Log Dose 3:

Log Dose 4:

Log Dose 5:

Log Dose 6:

Log Dose 7:

Log Dose 8:

Log Dose 9:

Weights

Choose weights. Weights with a sum>1 will be normalized to 1

Weight 1:

Weight 2:

Weight 3:

Weight 4:

Weight 5:

Weight 6:

Weight 7:

Weight 8:

Weight 9:

Click the Compute button to check the efficiency of the entered design.

Result

D-Efficiency of proposed design at assumed parameters b and e:

Average D-Efficiency of proposed design over all values of the parameters on the Robustness Heatmap below:

Best performance at:

Robustness Heatmap, (see Holland-Letz and Kopp-Schneider 2025):

This plot shows the D-Efficiency of the given design under different parameters. This plot is not affected by settings for b and e chosen above.
(Red:100% efficiency, Green: 60% efficiency, Dark blue: 0% efficiency)

This part allows specification of several possible parameters sets, and computes a Design which is D-optimal on a (weighted) average of these parameter settings.
Different weights can be assigned to the parameter settings.

Function Parameters

Function:

Log-logistic

Weibull

# of Parameter Sets (max 5):

Parameter e 1:

Parameter e 2:

Parameter e 3:

Parameter e 4:

Parameter e 5:

Click the Compute button to calculate the Bayesian design.

Parameter b 1:

Parameter b 2:

Parameter b 3:

Parameter b 4:

Parameter b 5:

Importance of Set 1:

Importance of Set 2:

Importance of Set 3:

Importance of Set 4:

Importance of Set 5:

Result

Proposed design and D-efficiency:

Design Heatmap

Points marked red on the diagonal are potential design points. Pairs of different design points marked red when crossreferenced are interchangeable with negligible loss of efficiency.

This part computes D-optimal designs for two singular treatments and up to five combination rays on the specified design space. One of two available dose response functions can be chosen, and an
a-priori assumption regarding the assumed slope and ED50 parameters can be made for both singular treatments.
Lowering the value for the reduction parameter will try to find a design with fewer support points, but might reduce the efficiency.

Function Parameters

Function:

Log-logistic

Weibull

Slope Substance A (Parameter b):

ED50 Substance A (Parameter e):

Slope Substance B (Parameter b):

ED50 Substance B (Parameter e):

Mixtures

# mixture proportions (max 5):

Which Heatmap to plot?
(1,2:Single, 3+:Combinations)

Mixture Proportion 1:

Combination Index 1:

Mixture Proportion 2:

Combination Index 2:

Mixture Proportion 3:

Combination Index 3:

Mixture Proportion 4:

Combination Index 4:

Mixture Proportion 5:

Combination Index 5:

Click the Compute button to calculate or update the results below.

Plot of function (Parameters A)

Result

Proposed designs, expected values of parameters b and e, and resulting D-Efficiency of proposed designs:

Design Heatmap (for selected combination only)

Points marked red on the diagonal are potential design points. Pairs of different design points marked red when crossreferenced are interchangeable with negligible loss of efficiency.