inst/help/goric/restriktorSyntax.md

In jasp-stats/jaspAnova: ANOVA Module for JASP

Syntax for Creating Order Restrictions

Order Restrictions

A special syntax is necessary to specify order restrictions in the General Linear Model. Order restrictions mean that effects in a model are required to be equal or unequal. For example, when performing an ANOVA with three groups, the following order constraint could be applied:

μLow \< μMed \< μHigh.

This is an example for an inequality constraint and restricts the mean of the first group to be smaller than the mean of the second group, which must be smaller than the mean of the third group. The null-model for the ANOVA could also be expressed as an equality constraint:

μLow = μMed = μHigh,

which means that all three groups must have the same mean.

Basic Syntax

To specify an order constraint, the (in-) equality between two coefficients must be expressed using =/==, <, or > operators. Coefficients can be accessed by their name in the data set. For instance:

variable1 == variable2,

indicates that the coefficient (effect) for variable1 is equal to the coefficient for variable2. Coefficients can also be constrained to a constant, e.g., variable1 == 0.

The constraints are set up as relations between two coefficients and each relation is written on a new line. Alternatively, they can also be put on the same line and separated with a semicolon (i.e., ;). It is possible to specify multiple relations simultaneously, e.g. variable1 < variable2 = variable3 is the same as writing variable1 < variable2; variable2 = variable3.

Available coefficients

Coefficients that are available in the syntax depend on the model and the data. Tick "Show available coefficients" to display the list of coefficients that are available.

Define custom coefficients

It is possible to define a custom parameter using := operator and then subsequently use it further in the model. For example newPar := variable1 + variable2; newPar = 0 defines a new parameter called newPar, which is the sum of coefficients associated with predictors variable1 and variable2. The second statement sets newPar to zero, which effectively implements a sum-to-zero constraint of the two coefficients.

Factors

Factor predictors are represented by multiple coefficients, depending on the number of levels in the factor. They can be accessed by the factor variable name immediately followed by the respective level:

factorLow == factorHigh.

Factors are dummy coded.

Intercept

When the intercept is included in the model, it can be accessed using the .Intercept. keyword. The reference level of the first factor in the model would then correspond to this .Intercept. coefficient.

Example

Consider a factor called factor with three levels: Low, Med, High. We will construct the inequality constraint μLow \< μMed \< μHigh.

If the intercept is included in the model, the reference category (Low) is estimated as the .Intercept.. The factors are dummy coded, so the mean of the Med category is equal to .Intercept. + factorMed, and the mean of the High category is equal to .Intercept. + factorHigh. That is, factorMed and factorHigh here refer to the difference between the respective category and the reference category. The restriction could be implemented as follows:

.Intercept. < .Intercept. + factorMed

.Intercept. + factorMed < .Intercept. + factorHigh.

The syntax could be simplified as:

factorMed > 0

factorHigh > factorMed.

Alternatively, we could define new parameters that correspond to the means of the three categories, and then specify the constraint in terms of those:

low := .Intercept.

med := .Intercept. + factorMed

high := .Intercept. + factorHigh

low < med < high.

In case that the intercept is not included in the model, there are three coefficients factorLow, factorMed, factorHigh, which already correspond to the means of the three categories. The following syntax would yield the same results:

factorLow < factorMed

factorMed < factorHigh.

Advanced Syntax

Interactions

Constraints on interaction effects can be specified using variable1.variable2 to refer to an interaction term between variable1 and variable2.

In case of interaction between factors, one needs to take into account that the factors are dummy coded, and whether or not intercept is included in the model. For example, if we have the first predictor Group with levels Control and Experimental, and another predictor Fitness with level Low and High. If we include the intercept in the model, the .Intercept coefficient then correspond to the first level of each factor (in this case Control Group and Low Fitness). We can define the means of the four cells in the design as:

ControlLow := .Intercept.

ControlHigh := .Intercept. + FitnessHigh

ExperimentalLow := .Intercept. + GroupExperimental

ExperimentalHigh := .Intercept. + FithessHigh + GroupExperimental + FitnessHigh.GroupExperimental,

Which makes it easier to make complex predictions in terms of the means of the cells in the design instead of the coefficients in the model.

When intercept is not included, defining the corresponding group means would be:

ControlLow := GroupControl

ControlHigh := GroupControl + FitnessHigh

ExperimentalLow := GroupExperimental

ExperimentalHigh := GroupExperimental + FitnessHigh + GroupExperimenal.FitnessHigh.

Repeated Measures

Constraints on coefficients of within-subject factors can be specified similar to between-subject factors. To access the coefficient associates with a repeated measures cell, use the variable name that is included in the "Repeated Measures Cells" field.

If both within- and between-subjects terms are in the model, their coefficients can also only be accessed through the dot operator .; the within-subject terms preceed those that are between-subjects, e.g.

withinSubjectsCell.betweenFactorLevel1.

Including or excluding the intercept has the same implications for the between-subject terms (and covariates) as in AN(C)OVA. There is no intercept for the within-subject terms.

Summary

The syntax for specifying order constraints contains the following elements: - =/== for equality constraints - < and > for inequality constraints - .Intercept. to access the intercept - variable1.variable2 to access interaction terms - := to define new parameters - Interaction syntax if both within- and between-subject terms are included

Examples

For further information and examples, see: https://restriktor.org/tutorial/syntax.html

References

Kuiper, R. M., Hoijtink, H. and Silvapulle, M.J. (2011). An Akaike type information criterion for model selection under inequality constraints. Biometrika, 98, 495-501.

Kuiper, R. M., Hoijtink, H. and Silvapulle, M.J. (2012). Generalization of the order restricted information criterion for multivariate normal linear models. Journal of Statistical Planning and Inference, 142, 2454-2463.

Kuiper, R. M., & Hoijtink, H. (2013). A Fortran 90 program for the generalization of the order restricted information criterion. Journal of Statistical Software, 54(8), 1-19.

Vanbrabant, L., Van Loey, N., & Kuiper, R. M. (2020). Evaluating a Theory-Based Hypothesis Against Its Complement Using an AIC-Type Information Criterion With an Application to Facial Burn Injury. Psychological Methods, 25(2), 129-142.

jasp-stats/jaspAnova documentation built on June 14, 2024, 6:48 p.m.