Example of an A-efficient RUM design with ChoiceDesign
This notebook illustrates how to use ChoiceDesign to generate a simple A-efficient experimental design for a Random Utility Maximisation (RUM) model. Instead of minimising the D-error (the determinant of the inverse Fisher information matrix), the A-criterion minimises the A-error:
where \(K\) is the number of non-ASC parameters. The A-error is the average variance across all parameter estimates, making it more sensitive to individual parameters that are poorly estimated.
Step 1: Load modules, define design parameters and set attributes
The following lines load:
EffDesign: the class of efficient designs,AttributeandParameter: the classes of attributes and parameters, respectively.
[1]:
from choicedesign.design import EffDesign
from choicedesign.expressions import Attribute, Parameter
The following lines define 2 alternatives, named alt1 and alt2, and 4 attributes named from \(A\) to \(D\):
[2]:
alt1_A = Attribute('alt1_A',[1,2,3])
alt1_B = Attribute('alt1_B',[10,15,15.5])
alt1_C = Attribute('alt1_C',[0,3,5])
alt1_D = Attribute('alt1_D',[0,1,2])
alt2_A = Attribute('alt2_A',[1,2,3])
alt2_B = Attribute('alt2_B',[10,15,15.5])
alt2_C = Attribute('alt2_C',[0,3,5])
alt2_D = Attribute('alt2_D',[0,1,2])
Step 2: Construct efficient design object and generate initial design matrix
The design object is constructed with EffDesign, passing the list of attributes and the number of choice situations:
[3]:
design = EffDesign(
X = [alt1_A,alt1_B,alt1_C,alt1_D,
alt2_A,alt2_B,alt2_C,alt2_D],
ncs=18)
[4]:
init_design = design.gen_initdesign()
init_design
[4]:
| alt1_A | alt1_B | alt1_C | alt1_D | alt2_A | alt2_B | alt2_C | alt2_D | |
|---|---|---|---|---|---|---|---|---|
| 0 | 3.0 | 15.0 | 5.0 | 1.0 | 2.0 | 15.5 | 0.0 | 0.0 |
| 1 | 2.0 | 15.0 | 5.0 | 1.0 | 1.0 | 15.5 | 0.0 | 2.0 |
| 2 | 2.0 | 15.5 | 3.0 | 0.0 | 1.0 | 10.0 | 0.0 | 0.0 |
| 3 | 2.0 | 15.0 | 0.0 | 2.0 | 2.0 | 15.0 | 5.0 | 2.0 |
| 4 | 3.0 | 15.5 | 3.0 | 0.0 | 2.0 | 10.0 | 3.0 | 1.0 |
| 5 | 2.0 | 10.0 | 3.0 | 1.0 | 3.0 | 15.0 | 3.0 | 0.0 |
| 6 | 1.0 | 15.0 | 0.0 | 2.0 | 3.0 | 15.5 | 0.0 | 0.0 |
| 7 | 1.0 | 10.0 | 0.0 | 1.0 | 3.0 | 15.5 | 5.0 | 0.0 |
| 8 | 1.0 | 10.0 | 5.0 | 1.0 | 1.0 | 15.0 | 3.0 | 2.0 |
| 9 | 2.0 | 15.5 | 3.0 | 0.0 | 2.0 | 15.0 | 3.0 | 2.0 |
| 10 | 1.0 | 15.5 | 0.0 | 1.0 | 1.0 | 15.5 | 5.0 | 1.0 |
| 11 | 2.0 | 10.0 | 3.0 | 2.0 | 1.0 | 10.0 | 5.0 | 2.0 |
| 12 | 3.0 | 15.0 | 0.0 | 0.0 | 3.0 | 10.0 | 5.0 | 2.0 |
| 13 | 3.0 | 10.0 | 3.0 | 0.0 | 1.0 | 10.0 | 3.0 | 1.0 |
| 14 | 1.0 | 15.5 | 0.0 | 2.0 | 2.0 | 15.5 | 5.0 | 1.0 |
| 15 | 3.0 | 15.5 | 5.0 | 2.0 | 3.0 | 15.0 | 0.0 | 0.0 |
| 16 | 3.0 | 10.0 | 5.0 | 2.0 | 2.0 | 15.0 | 3.0 | 1.0 |
| 17 | 1.0 | 15.0 | 5.0 | 0.0 | 3.0 | 10.0 | 0.0 | 1.0 |
Step 3: Set the utility functions
Parameters are defined with the Parameter class. The arguments are:
name: The parameter nameprior: The prior value
The following lines define four parameters:
[5]:
beta_A = Parameter('beta_A',-0.1)
beta_B = Parameter('beta_B',-0.02)
beta_C = Parameter('beta_C',0.1)
beta_D = Parameter('beta_D',0.15)
[6]:
V1 = beta_A * alt1_A + beta_B * alt1_B + beta_C * alt1_C + beta_D * alt1_D
V2 = beta_A * alt2_A + beta_B * alt2_B + beta_C * alt2_C + beta_D * alt2_D
V = {1: V1, 2: V2}
Step 4: Optimise the design using the A-error criterion
The optimise() method accepts a criterion argument that selects the optimality criterion:
'd'(default): minimise the D-error — \(\det(I^{-1})^{1/K}\)'a': minimise the A-error — \(\operatorname{trace}(I^{-1})/K\)'c': minimise the C-error (WTP variance sum)
Setting criterion='a' yields an A-efficient design. All other arguments remain the same as in the D-efficient case:
[7]:
optimal_design, init_perf, final_perf, final_iter, ubalance_ratio = design.optimise(
init_design=init_design,
V=V,
model='mnl',
criterion='a',
time_lim=1,
verbose=True
)
Evaluating initial design
Optimization complete 0:00:59 / A-error: 0.050462
Elapsed time: 0:01:00
A-error of initial design: 0.118812
A-error of last stored design: 0.050462
Utility Balance ratio: 94.56 %
Algorithm iterations: 183454
Blocking the design
The optimal design can be blocked using the method gen_blocks(). The following line creates 3 blocks:
[10]:
optimal_design_blocked, corr_history = design.gen_blocks(optimal_design, n_blocks=3)
optimal_design_blocked
[10]:
| CS | alt1_A | alt1_B | alt1_C | alt1_D | alt2_A | alt2_B | alt2_C | alt2_D | Block | |
|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 1.0 | 3.0 | 15.5 | 0.0 | 2.0 | 1.0 | 10.0 | 3.0 | 0.0 | 1 |
| 1 | 2.0 | 2.0 | 15.0 | 5.0 | 1.0 | 2.0 | 15.0 | 0.0 | 1.0 | 2 |
| 2 | 3.0 | 2.0 | 15.0 | 5.0 | 2.0 | 2.0 | 15.0 | 0.0 | 0.0 | 1 |
| 3 | 4.0 | 2.0 | 15.0 | 0.0 | 0.0 | 2.0 | 15.0 | 5.0 | 2.0 | 3 |
| 4 | 5.0 | 3.0 | 10.0 | 3.0 | 0.0 | 1.0 | 15.5 | 3.0 | 2.0 | 3 |
| 5 | 6.0 | 1.0 | 10.0 | 0.0 | 2.0 | 3.0 | 15.5 | 5.0 | 0.0 | 1 |
| 6 | 7.0 | 1.0 | 15.5 | 5.0 | 1.0 | 3.0 | 10.0 | 0.0 | 1.0 | 3 |
| 7 | 8.0 | 1.0 | 10.0 | 0.0 | 2.0 | 3.0 | 15.5 | 3.0 | 0.0 | 3 |
| 8 | 9.0 | 1.0 | 15.0 | 3.0 | 0.0 | 3.0 | 15.0 | 5.0 | 2.0 | 2 |
| 9 | 10.0 | 3.0 | 15.0 | 5.0 | 0.0 | 1.0 | 15.0 | 0.0 | 2.0 | 1 |
| 10 | 11.0 | 2.0 | 15.0 | 0.0 | 1.0 | 2.0 | 15.0 | 5.0 | 1.0 | 2 |
| 11 | 12.0 | 2.0 | 15.5 | 3.0 | 2.0 | 2.0 | 10.0 | 3.0 | 0.0 | 3 |
| 12 | 13.0 | 2.0 | 15.5 | 3.0 | 0.0 | 2.0 | 10.0 | 5.0 | 2.0 | 1 |
| 13 | 14.0 | 3.0 | 10.0 | 5.0 | 1.0 | 1.0 | 15.5 | 0.0 | 1.0 | 3 |
| 14 | 15.0 | 1.0 | 15.5 | 3.0 | 0.0 | 3.0 | 10.0 | 3.0 | 2.0 | 2 |
| 15 | 16.0 | 3.0 | 15.5 | 3.0 | 2.0 | 1.0 | 10.0 | 3.0 | 0.0 | 2 |
| 16 | 17.0 | 3.0 | 10.0 | 0.0 | 1.0 | 1.0 | 15.5 | 5.0 | 1.0 | 2 |
| 17 | 18.0 | 1.0 | 10.0 | 5.0 | 1.0 | 3.0 | 15.5 | 0.0 | 1.0 | 1 |
(optional) Evaluate the design
Pass criterion='a' to evaluate() to compute the A-error of a stored design:
[11]:
perf, ubalance = design.evaluate(optimal_design, V, model='mnl', criterion='a')
print('A-error:', perf)
print('Utility balance:', ubalance)
A-error: 0.050462107185962744
Utility balance: 94.56297363503425
References
[1] Quan, W., Rose, J. M., Collins, A. T., & Bliemer, M. C. (2011). A comparison of algorithms for generating efficient choice experiments.