# Creating a design

The syntax of spdesign is made to be intuitive and easy, while at the same time very flexible. To achieve this, we use a combination of regular expressions and R’s powerful expression parser “under the hood” to translate the user-specified utility functions into objects that can be manipulated by the package.

It is impossible to write checks for all possible edge cases of inputs. Therefore, you are strongly encouraged to follow the recommendations in this manual carefully when defining your utility functions.

## Specifying a prior

### A fixed prior

In spdesign, all parameters must have a prior. This is a deliberate design choice to make it clear that using a 0 prior is also an assumption in our design. The name of the prior is followed by a square bracket, which contains the assumed value of the prior. All priors have to start with b_. This is to separate them from attributes and to easily allow for non-linear utility functions.

If no prior is specified for a given parameter, an error is returned. If the parameter is generic, i.e. it enters multiple utility expressions, then you only need to specify the prior once. It is strongly recommended that priors for generic parameters are specified the first time they are used (this is a good tip for attributes as well!).

For example, to specify a parameter b_x1 with a prior of 0, we would write:

b_x1[0]

### A Bayesian prior

We can specify a Bayesian prior by specifying the distribution of the prior. For example, uniform_p(-1, 1) means that we allow the prior to follow a uniform distribution between -1 and 1.

"b_x1[uniform_p(-1, 1)]"

We can use the following distributions:

• Normal - normal_p(mean, sd)
• Log-normal - lognormal_p(mean, sd)
• Triangular - triangular_p(location, spread)
• Uniform - uniform_p(min, max)

All distributions are specified with a mean (location) or standard deviation (spread). In the case of the log-normal distribution we are specifying the mean and standard deviation of the underlying normal distribution.

## Specifying a parameter with levels

### Linear attributes

The syntax for specifying attributes and levels is similar to how we specify parameters and priors. The attribute name is specified to the left and levels are specified inside []. Below, the attribute x_1 can take on the levels 1, 2, 3, 4 and 5.

"x1[1:5]"

When we specify the levels, we can make use of R’s built in functions for generating sequences and vectors. For example, the following three specifications are identical:

# Specification 1
"x1[1:5]"

# Specification 2
"x1[seq(1, 5, 1)]"

# Specification 3
"x1[c(1, 2, 3, 4, 5)]"

However,

"x1[1, 2, 3, 4, 5]"

is not a valid specification and will throw an error. The error will most likely be of the form:

Error in parse(text = x) : <text>:1:2: unexpected ','
1: 1,
^ 

While we have not written a new intuitive error message for this misspecification, it will be caught by R’s parser and we feel that this is sufficient. As a compromise, we state the most likely error here to help users catch why their specifications fail.

Important: Make sure that all attribute levels are numeric. Even in the cases where you use the _dummy coding syntax it is always safer to use numeric values for the levels.

## Combining parameters, priors, attributes and levels to form a utility function

We combine the parameter and attribute as in the example below to start building our utility expression. We recommend that people use the naming convention b_x1 being the parameter for the attribute x1. It has no practical implication for how the code works or interprets the utility function, but it makes it much easier for the user to see which parameter goes with which attribute. It is also important that the parameter b_x1 precedes the attribute x1. There is currently no checks written to ensure this, but failing to do so may cause unintended errors in certain instances. For example, if you are using the dummy-coding shorthand detailed below.

Below is an example:

"b_x1[0] * x1[1:5]"

It is strongly recommended to use identical names for priors and attributes, with the exception of “b_” for priors. This makes reading of the code easier.

## Building a set of utility functions

The utility functions take the form of named lists. Let’s say that we would like to create a design comprising two alternatives with three attributes each and a status quo. We want our first attribute to take on the five levels 1, 2, 3, 4, and 5; our second attribute to take on the levels 0 and 1; and our third attribute to take on the values betwween 0 and 1 in .25 increments. One way to specify this would be the following:

utility <- list(
alt1 = "b_x1[0.1]  * x1[1:5] + b_x2[0.4] * x2[c(0, 1)] + b_x3[-0.2] * x3[seq(0, 1, 0.25)]",
alt2 = "b_x1       * x1      + b_x2      * x2          + b_x3       * x3",
alt3 = "b_sq[0.15] * sq[1]"
)

Where the status quo or opt out alternative has a single constant. It is important that when you add constants the level is 1. You cannot currently specify specific levels of the attributes for the SQ. Since these do not vary, this is equivalent to a constant (which can be calculated beforehand, see more details below) because only differences in utility matter for the model.

## Generating a design

To generate a design, we need to pass the utility function along with other options to the generate_design function. Currently, we can only generate designs for the MNL model. To see the full set of options for the function: ?generate_design.

design <- generate_design(utility, rows = 20,
model = "mnl", efficiency_criteria = "d-error",
algorithm = "rsc", draws = "scrambled-sobol",
control = list(
max_iter = 21000
))

The list control contains options that can be passed along to the function to change default values. The default values and options are:

  default_control <- list(
cores = 1,
max_iter = 10000,
max_relabel = 10000,
max_swap = 10000,
efficiency_threshold = 0.1,
sample_with_replacement = FALSE
)

### Algorithms and attribute level balance

No default algorithm for generating a design is implemented because they have different properties with respect to how easy it is to achieve attribute level balance (see more details below) and how easy it is to include restrictions/exclusions in the design (see more details below). Three algorithms are implemented: rsc, federov, and random.

The rsc algorithm creates an initial design candidate based on the provided attributes, levels and number of rows. This design candidate is attribute level balanced by default (if the number of attribute levels for all attributes are a multiple of the number of rows) or near attribute level balanced. It will not deviate from this. It finds new candidates by relabling or swapping attribute levels. The default is to cycle between relabling and swapping each 10000 candidates. The cycling part of the algorithm is not included because it is rarely used in practice. You cannot include restrictions or exclusions in your design when using the rsc algorithm. For details: ?rsc

The federov and random algorithms will systematically or randomly select potential design candidates from the candidate set. The candidate set can be either supplied or assumed to be the full factorial. These algorithms completely lets go of attribute level balance and you do run the risk of some attribute levels not showing up in the final design. We show syntax below to include level occurrence restrictions below. You can easily include restrictions in your design using these to algorithms by excluding attribute level combinations from the candidate set. That way, they will never be part of the design.

Note that in general, both restrictions and attribute level balance may lead to less efficient designs and that if your restrictions are too tight and you are trying to force attribute level balance, no designs may be found. Few checks are made to notify the user of this.

## Blocking the design

Blocking the design is done after the design has been generate and will add a blocking column to the design. The following will block the design into 4 trying to minimize the average squared correlation between the blocking column and each of the design columns. More information: ?block

design <- block(design, 4)

## Summary of the design.

The design object is stored as a list of class spdesign. Individual parts can be accessed using the \$ operator. Functions such as summary, coef, print, vcov and cor all work as expected.

## Attribute level balance

As stated above, the rsc algorithm wil ensure attribute level balance, whereas the federov and random algorithms will not. However, there is some flexibility in how this is specified. The following will only work for the federov and random algorithms and will be ignored without warning if the rsc algorithm is used.

The implicit restriction in the package is that each attribute level can occur between 0 and rows number of times. For example, if you have a design with 20 rows, then the attribute level can occur between 0 and 20 times. In practice, anywhere between never and always. The package does provide the user with the ability to specify a narrower range.

To do this, we would specify a parenthesis behind the attribute levels where we give the range that the attribute levels can take. Assume that our design has 18 rows. The following specification would force attribute level balance by saying that each level has to occur exactly 6 times.

"b_x1[0] * x1[c(0, 1, 2)](6)"

As stated, it is not always possible to get attribute level balance and indeed it can be quite hard with the federov and random algorithms. Specifying a range may be better. For example, we could specify that all attribute levels should occur between 5 and 7 times.

"b_x1[0] * x1[c(0, 1, 2)](5:7)"

In very special cases, it may be desirable to let some attribute levels occur more frequently than others and we could specify:

"b_x1[0] * x1[c(0, 1, 2)](3:5, 5:7, 6:9)"

which would mean that 0 occurs between 3 and 5 times, 1 between 5 and 7, and 2 between 6 and 9 times. Indeed, in the case where we only specify a single range, it is expanded to be equal to the number of attribute levels such that:

"b_x1[0] * x1[c(0, 1, 2)](5:7)"

is equivalent to

"b_x1[0] * x1[c(0, 1, 2)](5:7, 5:7, 5:7)"

A few important things to note:

1. The sum of minimum level occurrences must be less than rows and the sum of maximum level occurrences must be larger than rows. Otherwise, no design can be found. No checks are implemented for this.
2. You can only specify either a single number or range, or number or ranges equal to the number of levels. Deviating from this may have unintended consequences and no checks are implemented.
3. You can only specify the number of times an attribute should occur in the design at the same time as you specify the number of levels,

## Interactions

If you believe that there are interaction effects between attributes in your model, then these will have to be considered at the design stage. Any design that is not a full factorial assumes that some higher order interaction effects are insignificant. To ensure that you are able to identify the effect of an interaction in your data, this will have to be included in the design. To include an interaction term in the design, we can use the following syntax.

"b_x1[0] * x1[c(0, 1)] + b_x2 * x2[c(0, 1)] + b_x1x2 * I(x1 * x2)"

Two things to note about the specification of the utility function with interactions. First, the levels of the attributes are specified when the attribute is first introduced and not in the interaction term itself, and two, for the parser to correctly interpret an interaction term, it needs to be wrapped in R’s internal function for handling interactions I().

## Dummy-coded attributes

Often we have attributes in mind that are categorical in nature, e.g., comfort or environmental quality. In the design, it is often desirable to let these be dummy-coded. We can easily specify an attribute to dummy-coded using the _dummy syntax. Assume that we have a single categorical attribute that can take on three levels. We would then specify the utility functions as follows:

V = list(
alt1 = "b_x1_dummy[c(0.2, 0.5)] * x1[c(1, 2, 3)]",
alt2 = "b_x1_dummy * x1"
)

Here we have specified 2 priors and 3 levels. The dummy coding expands the attribute levels by transforming the attribute into a factor and expanding it using a formula and the model.matrix. The first level is chosen as the baseline and dropped. Similarly, the priors are expanded into separate priors for each level. This also works with Bayesian priors.

It is recommended to use 1, 2, 3, 4 for the levels of a dummy-coded attribute. Note that the levels don’t matter, but this naming convention secures that there is a correspondence between the names for the priors and attributes.

A few things to note about using the _dummy syntax

1. The priors and levels must be specified simultaneously.
2. You must specify one more level for the attribute than priors for the parameter.
3. You cannot use _dummy as an extension for your attribute name.
4. If you are using specified levels for the status-quo or opt out alternative, make sure that this corresponds to the base level and is coded as a 0. See more details below.

## Constrained designs

When using the federov or random algorithms, we can pass along a list of exclusions to generate_design. The exclusions will exclude the specified combinations from the full factorial prior to generating design candidates. The exclusions take the form of an unamed list of strings. You must ensure that the names of the list corresponds to the names of the expanded attribute levels (see ?expand_attribute_levels).

For example:

exclusions = list(
"alt1_x1 == 2 & alt1_x2 == 0 & alt1_x3 == 0",
"alt2_x2 == 1 & alt2_x3 == 1"
)


## Specifying constants and levels of the SQ alternative

You cannot currently specify fixed SQ levels. Specifying new attributes that can only take on a single level for the SQ leads to a computationally singular Hessian matrix (because there is no variation). To get around this, you can calculate what the size of the SQ constant would be given the defined attribute levels for the SQ and the assumed prior and include this as a single constant for the SQ alternative. This will ensure that the correct utility differences are used in the design.

A fix with additional level restrictions will be included in a future version of the package.

Remember, that when you specify fixed levels for the SQ, you cannot also specify alternative specific constants for the non-SQ alternatives. This is a violation of the J-1 ‘rule’ and will result in a singular Fisher information matrix.

There is no point specifying SQ levels that only take on the value of 0. These have no bearing on the design and may be problematic when inverting the Fisher information matrix.

Related to the above point, in the case where your attribute is dummy-coded the easiest is to define it such that the base level of your dummy-coded attribute corresponds to the level of the SQ. Remember, that in your design, these only take on the values of 1 and 0 with the base level being zero for all non-base-level levels. Defining the base level to be equal to the SQ obviates the need to explicitly specify the SQ beyond a constant.

# Examples

Many examples are provided along with the package documentation and on Github to showcase different syntaxes.