GORICA is an acronym for "generalized order-restricted information criterion approximation". It can be utilized to evaluate informative hypotheses, which specify directional relationships between model parameters in terms of (in)equality constraints.
An R object containing the outcome of a statistical analysis. Currently, the following objects can be processed:
A character string containing the informative hypotheses to evaluate (see Details).
A character string indicating what the
Additional arguments passed to the internal function
The GORICA is applicable to not only normal linear models, but also applicable to generalized linear models (GLMs) (McCullagh & Nelder, 1989), generalized linear mixed models (GLMMs) (McCullogh & Searle, 2001), and structural equation models (SEMs) (Bollen, 1989). In addition, the GORICA can be utilized in the context of contingency tables for which (in)equality constrained hypotheses do not necessarily contain linear restrictions on cell probabilities, but instead often contain non-linear restrictions on cell probabilities.
hypotheses is a character string that specifies which informative
hypotheses have to be evaluated. A simple example is
hypotheses <- "a >
b > c; a = b = c;" which specifies two hypotheses using three estimates with
names "a", "b", and "c", respectively.
The hypotheses specified have to adhere to the following rules:
Parameters are referred to using the names specified in
Linear combinations of parameters must be specified adhering to the following rules:
Each parameter name is used at most once.
Each parameter name may or may not be pre-multiplied with a number.
A constant may be added or subtracted from each parameter name.
A linear combination can also be a single number.
3 * a + 5;
a + 2 * b + 3 * c - 2;
a - b;
(Linear combinations of) parameters can be constrained using <, >, and
=. For example,
a > 0 or
a > b = 0 or
2 * a < b + c > 5.
The ampersand & can be used to combine different parts of a hypothesis.
a > b & b > c which is equivalent to
a > b > c or
a > 0 & b > 0 & c > 0.
Sets of (linear combinations of) parameters subjected to the same
constraints can be specified using (). For
a > (b,c) which is equivalent to
a > b & a > c.
The specification of a hypothesis is completed by typing ; For example,
hypotheses <- "a > b > c; a = b = c;", specifies two hypotheses.
Hypotheses have to be compatible, non-redundant and possible. What these terms mean will be elaborated below.
The set of hypotheses has to be compatible. For the statistical
background of this requirement see Gu, Mulder, Hoijtink (2018). Usually the
sets of hypotheses specified by researchers are compatible, and if not,
gorica will return an error message. The following steps can be used to
determine if a set of hypotheses is compatible:
Replace a range constraint, e.g.,
1 < a1 < 3, by an equality
constraint in which the parameter involved is equated to the midpoint of the
range, that is,
a1 = 2.
Replace in each hypothesis the < and > by =. For example,
a1 = a2
> a3 > a4 becomes
a1 = a2 = a3 = a4.
The hypotheses are compatible if there is at least one solution to the
resulting set of equations. For the two hypotheses considered under 1. and
2., the solution is a1 = a2 = a3 = a4 = 2. An example of two non-compatible
hypotheses <- "a = 0; a > 2;" because there is no
solution to the equations
Each hypothesis in a set of hypotheses has to be non-redundant. A
hypothesis is redundant if it can also be specified with fewer constraints.
a = b & a > 0 & b > 0 is redundant because it can also be
a = b & a > 0.
gorica will work correctly if
hypotheses specified using only < and > are redundant.
return an error message if hypotheses specified using at least one = are
Each hypothesis in a set of hypotheses has to be possible. An
hypothesis is impossible if estimates in agreement with the hypothesis do not
exist. For example: values for
a in agreement with
a = 0 &
a > 2 do not exist. It is the responsibility of the user to ensure that the
hypotheses specified are possible. If not,
gorica will either return an
error message or render an output table containing
An object of class
gorica, containing the following elements:
data.frame containing the loglikelihood, penalty
(for complexity), the GORICA value, and the GORICA weights. The GORICA
weights are calculated by taking into account the misfits and complexities of
the hypotheses under evaluation. These weights are used to quantify the
support in the data for each hypothesis under evaluation. By looking at the
pairwise ratios between the GORICA weights, one can determine the relative
importance of one hypothesis over another hypothesis.
call The original function call.
model The original model object (
estimates The parameters extracted from the
Sigma The asymptotic covariance matrix of the
comparison Which alternative hypothesis was used.
hypotheses The hypotheses evaluated in
Caspar van Lissa, Yasin Altinisik, Rebecca Kuiper
Altinisik, Y. (2018). Evaluation of Inequality Constrained Hypotheses Using an Akaike-Type Information Criterion (Doctoral dissertation, Utrecht University). ISBN: 978-90-393-6918-0. https://dspace.library.uu.nl/handle/1874/360604
Bollen, K. (1989). Structural equations with latent variables. New York, NY: John Wiley and Sons.
Kuiper, R. M., Hoijtink, H., & Silvapulle, M. J. (2011). An Akaike-type information criterion for model selection under inequality constraints. Biometrika, 98, 495-501. doi:10.1093/biomet/asr002
Kuiper, R. M., Hoijtink, H., & Silvapulle, M. J. (2012). Generalization of the order-restricted information criterion for multivariate normal linear models. Journal of statistical planning and inference, 142(8), 2454-2463. doi:10.1016/j.jspi.2012.03.007
McCullagh, P. & Nelder, J. (1989). Generalized linear models (2nd ed.). Boca Raton, FL: Chapman & Hall / CRC.
McCullogh, C. E., & Searle, S. R. (2001). Generalized linear and mixed models. New York, NY: Wiley.
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# EXAMPLE 1. One-sample t test ttest1 <- t_test(iris$Sepal.Length,mu=5) gorica(ttest1,"x<5.8") # EXAMPLE 2. ANOVA aov1 <- aov(yield ~ block-1 + N * P + K, npk) gorica(aov1,hypothesis="block1=block5; K1<0") # EXAMPLE 3. gml counts <- c(18,17,15,20,10,20,25,13,12) outcome <- gl(3,1,9) treatment <- gl(3,3) fit <- glm(counts ~ outcome-1 + treatment, family = poisson()) gorica(fit, "outcome1 > (outcome2, outcome3)") # EXAMPLE 4. ANOVA res <- lm(Sepal.Length ~ Species-1, iris) est <- get_estimates(res) est gor <- gorica(res, "Speciessetosa < (Speciesversicolor, Speciesvirginica)", comparison = "complement") gor
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