Package {gorica}

Academic awards data

Description

Simulated dataset based on the UCLA Statistical Consulting Group's website.

Usage

data(academic_awards)

Format

A data frame with 200 rows and 4 variables.

Details

References

Introduction to SAS. UCLA: Statistical Consulting Group. from https://stats.oarc.ucla.edu/sas/modules/introduction-to-the-features-of-sas/ (accessed August 22, 2021).

Constraint to equation

Description

Formats a constraint as an equation that can be evaluated. Adds scalars to all parameters in the constraint, and adds a XXXconstant parameter to any constants. Also adds "*" and "+" operators where necessary for evaluation.

Formats a constraint as an equation that can be evaluated. Adds scalars to all parameters in the constraint, and adds a XXXconstant parameter to any constants. Also adds "*" and "+" operators where necessary for evaluation.

Usage

constraint_to_equation(hyp)

constraint_to_equation(hyp)

Arguments

Value

Character

Character

Constraint to row

Description

Evaluate a constraint, which has been formatted as an equation, with all terms moved to the left hand side, and return a single row of a (in)equality constraint matrix.

Evaluate a constraint, which has been formatted as an equation, with all terms moved to the left hand side, and return a single row of a (in)equality constraint matrix.

Usage

constraint_to_row(varnames, hyp)

constraint_to_row(varnames, hyp)

Arguments

Value

Numeric vector.

Numeric vector.

Create (in)equality constraint matrices

Description

Parses a character string describing a set of informative hypotheses, and returns (in)equality constraint matrices

Parses a character string describing a set of informative hypotheses, and returns (in)equality constraint matrices

Usage

create_matrices(varnames, hyp)

create_matrices(varnames, hyp)

Arguments

Details

Informative hypotheses specified as a character string by "hyp" should adhere to the following simple syntax:

• Competing hypotheses are separated by ";". Thus, "a=b;a>b" means that H1: a=b, and H2: a>b.

• Each individual hypothesis consists of a (series of) (in)equality constraint(s). Every single (in)equality constraint is of the form "R1*mu1 + R2*mu2+... = r", where capital Rs refer to numeric scaling constants, must refer to the names of parameters in the model, and the lower case r refers to a constant. Standard mathematical simplification rules apply; thus, "R1*mu1 = R2*mu2" is equivalent to "R1*mu1 - R2*mu2 = 0".

• Multiple unrelated constraints within one hypothesis can be chained by "&". Thus, "a=b&c=d" means that H1: a=b AND c=d.

• Multiple related constraints within one hypothesis can be chained by repeating the (in)equality operators "=", "<", or ">". Thus, "a<b<c" means that H1: a < b AND b < c.

• Parameters can be grouped by placing them in a parenthesized, comma separated list. Thus, "(a,b)>c" means that H1: a > c AND b > c. Similarly, "(a,b)>(c,d)" means that H1: a > c AND b > c AND b > c AND b > d.

Informative hypotheses specified as a character string by "hyp" should adhere to the following simple syntax:

• Competing hypotheses are separated by ";". Thus, "a=b;a>b" means that H1: a=b, and H2: a>b.

• Each individual hypothesis consists of a (series of) (in)equality constraint(s). Every single (in)equality constraint is of the form "R1*mu1 + R2*mu2+... = r", where capital Rs refer to numeric scaling constants, must refer to the names of parameters in the model, and the lower case r refers to a constant. Standard mathematical simplification rules apply; thus, "R1*mu1 = R2*mu2" is equivalent to "R1*mu1 - R2*mu2 = 0".

• Multiple unrelated constraints within one hypothesis can be chained by "&". Thus, "a=b&c=d" means that H1: a=b AND c=d.

• Multiple related constraints within one hypothesis can be chained by repeating the (in)equality operators "=", "<", or ">". Thus, "a<b<c" means that H1: a < b AND b < c.

• Parameters can be grouped by placing them in a parenthesized, comma separated list. Thus, "(a,b)>c" means that H1: a > c AND b > c. Similarly, "(a,b)>(c,d)" means that H1: a > c AND b > c AND b > c AND b > d.

Value

A pair of named matrices for every hypothesis specified in the hyp argument; one matrix named ERr, specifying equality constraints, and one matrix named IRr, specifying inequality constraints.

A pair of named matrices for every hypothesis specified in the hyp argument; one matrix named ERr, specifying equality constraints, and one matrix named IRr, specifying inequality constraints.

Author(s)

Caspar van Lissa

Expand compound constraints

Description

Takes a compound constraint, with multiple (in)equality operators, and expands it into simple constraints.

Takes a compound constraint, with multiple (in)equality operators, and expands it into simple constraints.

Usage

expand_compound_constraints(hyp)

expand_compound_constraints(hyp)

Arguments

Value

A character vector with one element for each simple constraint

A character vector with one element for each simple constraint

Expand parentheses

Description

Takes a constraint with parentheses containing a "vector" of parameter labels, and recursively expands the parenthesized "vectors".

Takes a constraint with parentheses containing a "vector" of parameter labels, and recursively expands the parenthesized "vectors".

Usage

expand_parentheses(hyp)

expand_parentheses(hyp)

Arguments

Value

A character vector with one element for each simple constraint

A character vector with one element for each simple constraint

Flip inequality

Description

Takes a constraint and flips elements on both sides of any "<" operators so that only the ">" operator is used to define inequality constraints.

Takes a constraint and flips elements on both sides of any "<" operators so that only the ">" operator is used to define inequality constraints.

Usage

flip_inequality(hyp)

flip_inequality(hyp)

Arguments

Value

Character

Character

Evaluate informative hypotheses using the GORICA

Description

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.

Usage

gorica(x, hypothesis, comparison = "unconstrained", iterations = 1e+05, ...)

## S3 method for class 'lavaan'
gorica(
  x,
  hypothesis,
  comparison = "unconstrained",
  iterations = 1e+05,
  ...,
  standardize = FALSE
)

## S3 method for class 'table'
gorica(x, hypothesis, comparison = "unconstrained", ...)

Arguments

Details

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:

1. Parameters are referred to using the names specified in names().

2. Linear combinations of parameters must be specified adhering to the following rules:

   1. Each parameter name is used at most once.

   2. Each parameter name may or may not be pre-multiplied with a number.

   3. A constant may be added or subtracted from each parameter name.

   4. A linear combination can also be a single number.

   Examples are: 3 * a + 5; a + 2 * b + 3 * c - 2; a - b; and 5.

3. (Linear combinations of) parameters can be constrained using <, >, and =. For example, a > 0 or a > b = 0 or 2 * a < b + c > 5.

4. The ampersand & can be used to combine different parts of a hypothesis. For example, a > b & b > c which is equivalent to a > b > c or a > 0 & b > 0 & c > 0.

5. Sets of (linear combinations of) parameters subjected to the same constraints can be specified using (). For example, a > (b,c) which is equivalent to a > b & a > c.

6. The specification of a hypothesis is completed by typing ; For example, hypotheses <- "a > b > c; a = b = c;", specifies two hypotheses.

7. 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:

1. 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.

2. Replace in each hypothesis the < and > by =. For example, a1 = a2 > a3 > a4 becomes a1 = a2 = a3 = a4.

3. 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 is hypotheses <- "a = 0; a > 2;" because there is no solution to the equations a=0 and a=2.

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. For example, a = b & a > 0 & b > 0 is redundant because it can also be specified as a = b & a > 0. gorica will work correctly if hypotheses specified using only < and > are redundant. gorica will return an error message if hypotheses specified using at least one = are redundant.

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 Inf's.

Value

An object of class gorica, containing the following elements:

• fit A 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 (x).

• estimates The parameters extracted from the model.

• Sigma The asymptotic covariance matrix of the estimates.

• comparison Which alternative hypothesis was used.

• hypotheses The hypotheses evaluated in fit.

• relative_weights The relative weights of each hypothesis (rows) versus each other hypothesis in the set (cols). The diagonal is equal to one, as each hypothesis is equally likely as itself. A value of, e.g., 6, means that the hypothesis in the row is 6 times more likely than the hypothesis in the column.

Contingency tables

When specifying hypotheses about contingency tables, the asymptotic covariance matrix of the model estimates is derived by means of bootstrapping. This makes it possible for users to define derived parameters: For example, a ratio between cell probabilities. For this purpose, the bain syntax has been enhanced with the command :=. Thus, the syntax "a := x[1,1]/(x[1,1]+x[1,2])" defines a new parameter a by reference to specific cells of the table x. This new parameter can now be named in hypotheses.

Author(s)

Caspar van Lissa, Yasin Altinisik, Rebecca Kuiper

References

Altinisik, Y., Van Lissa, C. J., Hoijtink, H., Oldehinkel, A. J., & Kuiper, R. M. (2021). Evaluation of inequality constrained hypotheses using a generalization of the AIC. Psychological Methods, 26(5), 599–621. doi:10.31234/osf.io/t3c8g.

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:10.31219/osf.io/ekxsn

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

Vanbrabant, L., Van Loey, N., and Kuiper, R.M. (2019). Evaluating a theory-based hypothesis against its complement using an AIC-type information criterion with an application to facial burn injury. Psychological Methods. doi:10.31234/osf.io/n6ydv

McCullagh, P. & Nelder, J. (1989). Generalized linear models (2nd ed.). Boca Raton, FL: Chapman & Hall / CRC.

McCulloch, C. E., & Searle, S. R. (2001). Generalized linear and mixed models. New York, NY: Wiley.

Examples

# 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. glm
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

Sesame Street data based on Hox (2010)

Description

Synthetic data based on Hox (2010, p. 16). In the study, the outcome variable popular represents the popularity score of pupils, ranging from 0 (very unpopular) to 10 (very popular), for pupils nested in 100 classes of varying size. The popularity scores are predicted by pupil level predictors gender (G) and pupil extroversion scores (PE) that range from 1 (introversion) to 10 (extroversion), a class-level predictor teacher experience (TE), and the cross-level interactions between G and TE as well as PE and TE. Since standardization is recommended when the model contains interactions, we standardize PS, PE and TE by means of grand mean centering. That is, we first subtract the overall means of the continuous variables PS, PE, and TE from each of their values, before dividing these values by their standard deviations.

Usage

data(hox_2010)

Format

A data frame with 2000 rows and 6 variables.

Details

References

Hox, J. J. (2010). Multilevel analysis: Techniques and applications (2nd ed.). New York, NY: Routledge.

Data based on Nederhof, Ormel, and Oldehinkel (2014)

Description

Synthetic data, (N = 310) based on Nederhof, Ormel, and Oldehinkel (2014). The 11 years old participants are divided into three groups: Sustainers, Shifters, and Comparison group, based on their performance on a sustained-attention task and on a shifting-set task. The outcome depressive episode (D: no depressive episode, versus experienced an episode) is predicted by the categorical variable early life stress (ES: Low versus High), the standardized continuous variable recent stress, RS, and the interaction between both predictors. The continuous variable recent stress, RS, is standardized to improve the interpretation of main effects when interactions exist.

Usage

data(nederhof_2014)

Format

A data frame with 310 rows and 4 variables.

Details

References

Nederhof, E., Ormel, J., & Oldehinkel, A. J. (2014). Mismatch or cumulative stress: The pathway to depression is conditional on attention style. Psychological Science, 25, 684-692. doi:10.1177/0956797613513473.

Order terms

Description

Moves all terms on the right hand side of a constraint, which has been formatted as an equation, to the left hand side.

Moves all terms on the right hand side of a constraint, which has been formatted as an equation, to the left hand side.

Usage

order_terms(hyp)

order_terms(hyp)

Arguments

Value

Character

Character

Reading achievement data

Description

Dataset based on Finch, Bolin, and Kelley (2014, p.32).

Usage

data(reading_ach)

Format

A data frame with 10320 rows and 5 variables.

Details

References

Finch, W. H., Bolin, J. E., & Kelley, K. (2014). Multilevel modeling using r. CRC Press 2014.

High School Admissions Data

Description

This dataset, provided by the UCLA Statistical Consulting Group (2021), contains information on factors that influence whether or not a high school senior is admitted into a very competitive engineering school. The dataset includes the following variables:

Usage

data(school_admissions)

Format

A data frame with 30 rows and 3 variables.

Details

The dataset is used for exact logistic regression analysis due to the binary outcome variable. It aims to identify the factors that contribute to admission decisions in a highly competitive engineering school. Since the dataset has a small sample size, specialized procedures are required for accurate estimation.

References

Introduction to SAS. UCLA: Statistical Consulting Group. from https://stats.oarc.ucla.edu/sas/modules/introduction-to-the-features-of-sas/ (accessed August 22, 2021).

Sesame Street data based on Stevens (1999)

Description

Synthetic data based Stevens (1999, p. 596). This study evaluates the effects of the first year of the Sesame Street television series in a sample of 3-5 years old children in the USA (N = 240).

Usage

data(stevens_1999)

Format

A data frame with 240 rows and 14 variables.

Details

References

Stevens, J. (1999). Applied multivariate statistics for the social sciences. (3rd ed.). New Jersey, Lawrance Erlbaum Associates, Inc.

Wechsler intelligence test data

Description

Dataset based on McArdle and Prescott (1992, p.90). This study evaluates intelligence and cognitive ability in a sample of individuals over 18 years of age (N = 1680) using the IQ test Wechsler Adult Intelligence Scale-Revised (WAIS-R) (Wechsler, 1981).

Usage

data(wechsler)

Format

A data frame with 1680 rows and 10 variables.

Details

References

McArdle, J. J., & Prescott, C. A. (1992). Age-based construct validation using structural equation modeling. Experimental Aging Research, 18, 87-115.