hopit: Generalized hierarchical ordered threshold models.

Description

The ordered response data classify a measure of interest into ordered categories collected during a survey. For example, if the dependent variable is a happiness rating, a respondent typically answers a question such as: <U+201C>Taking all things together, would you say you are ... ?" and then selects from response options along the lines of: "very happy", "pretty happy", "not too happy", and "very unhappy" Liao2005hopit. Similarly, if interviewees are asked to evaluate their health in general (e.g., <U+201C>Would you say your health is ... ?<U+201D>) they, can typically choose among several categories, such as "very good", "good", "fair", "bad", and "very bad" King2004,Jurges2007,Rebelo2014,OKSUZYAN2019hopit. In political science, a respondent may be asked for an opinion about recent legislation (e.g. <U+201C>Rate your feelings about the proposed legislation.") and asked to choose among categories like: "strongly oppose", "mildly oppose", "indifferent", "mildly support", and "strongly support" GreeneHensher2010hopit. It is easy to imagine other multi-level ordinal variables that might be used during a survey and to which the methodology described below could be applied.

In practice, it is assumed that when responding to a survey question about their general happiness, health, feelings, attitudes or other status, participants are assessing their true value of this unobserved continuous variable, and project it onto the discrete scale provided. The thresholds that individuals use to categorize their true status by selecting a specific response option may be affected by the reference group chosen, their earlier life experiences, and cross-cultural differences in using scales. Thus, the responses of individuals may differ depending on their gender, age, cultural background, education, and personality traits; among other factors King2004,Jurges2007,OKSUZYAN2019hopit. From the perspective of reporting behavior modeling, one of the main tasks researchers face is to compute this continuous estimate of the underlying, latent measures of individuals based on several specific characteristics of the responses considered (e.g., health variables or happiness variables), and to account for variations in reporting across socio-demographic and cultural groups. More specifically, to build a latent, underlying measure, a generalized hierarchical ordered threshold model is fitted that regresses the reported status/attitude/feeling on two sets of independent variables Boes2006,Green2014hopit. When the dependent reported ordered variable is self-rated health status, then the first set of variables <U+2013> i.e., health variables <U+2013> assess specific aspects of individuals<U+2019> health, such as measures of chronic conditions, mobility, difficulties with a range of daily activities, grip strength, anthropometric characteristics, and lifestyle behaviors. Using the second set of independent variables (threshold variables), the model also adjusts for differences across socio-demographic and cultural groups, such as differences in cultural background, gender, age, and education King2004,Jurges2007,OKSUZYAN2019hopit.

Ordered threshold models are used to fit ordered categorical dependent variables. The generalized ordered threshold models Terza1985,Boes2006,Green2014hopit are an extension of the ordered threshold models McKelvey1975hopit. Whereas in the latter models, the thresholds are constant, in the generalized models the thresholds are allowed to be dependent on covariates. GreeneHensher2010,Green2014;textualhopit pointed out that for a model to make sense, the thresholds must also be ordered. This observation motivated Greene and coauthors to call these models HOPIT, which stands for hierarchical ordered probit models.

The fitted hopit model is used to analyze heterogeneity in reporting behavior. See standardizeCoef, latentIndex, getCutPoints, getLevels, and boot_hopit.

Usage

hopit(latent.formula, thresh.formula = ~1, data, decreasing.levels,
  start = NULL, fit.sigma = FALSE, design = list(), weights = NULL,
  link = c("probit", "logit"), control = list(), na.action = na.fail)

Arguments

latent.formula

a formula used to model the latent variable. It should not contain any threshold variable. To specify the interactions between the latent and the threshold variables, see details.

thresh.formula

a formula used to model the threshold variable. It should not contain any latent variable. To specify interactions between the latent and the threshold variables, see details. Any dependent variable (left side of "~" in the formula) will be ignored.

data

a data frame that includes all modeled variables.

decreasing.levels

a logical indicating whether self-reported health classes are ordered in decreasing order.

start

a vector with starting coefficient values in the form c(latent_parameters, threshold_lambdas, threshold_gammas) or c(latent_parameters, threshold_lambdas, threshold_gammas, logSigma) if the fit.sigma == TRUE.

fit.sigma

a logical indicating whether to fit an additional parameter sigma, which models a standard deviation of the error term (e.g., the standard deviation of the cumulative normal distribution in the probit model).

design

an optional survey design. Use the svydesign function to specify the design. The design cannot be specified together with parameter weights.

weights

optional model weights. Use the design to construct survey weights.

link

a link function. The possible values are "probit" (default) and "logit".

control

a list with control parameters. See hopit.control.

na.action

a function that indicates what should happen when the data contain NAs. The default is na.fail, which generates an error if any missing value is found. The alternative is na.omit (or na.exclude equivalently), which removes rows with missing values from the data. Using na.pass will lead to an error.

Value

a hopit object used by other functions and methods. The object is a list with the following components:

control

a list with control parameters. See hopit.control.

link

a link function used.

hasdisp

a logical indicating whether fit.sigma was modeled.

use.weights

a logical indicating whether any weights were used.

weights

a vector with model weights.

latent.formula

a latent formula used to fit the model.

latent.mm

a latent model matrix.

latent.terms

latent variables used, and their interactions.

cross.inter.latent

a part of the latent formula used for modeling cross-interactions in the latent model

thresh.formula

a threshold formula used to fit the model.

thresh.mm

a threshold model matrix.

thresh.extd

an extended threshold model matrix.

thresh.terms

threshold variables used, and their interactions.

cross.inter.thresh

a part of the threshold formula used for modeling cross-interactions in the threshold model

thresh.no.cov

a logical indicating whether gamma parameters are present.

parcount

a 3-element vector with a number of parameters for the latent variables (beta), the threshold intercepts (lambda), and the threshold covariates (gamma).

coef

a vector with model coefficients.

coef.ls

model coefficients as a list.

start

a vector with the starting values of the coefficients.

alpha

estimated individual-specific thresholds.

y_i

a vector with individual responses - the response variable.

y_latent_i

a vector with predicted latent measures for each individual.

Ey_i

a vector with predicted categorical responses for each individual.

a number of response levels.

a number of observations.

deviance

a deviance.

a log likelihood.

AIC

an AIC for models without a survey design.

vcov

a variance-covariance matrix.

vcov.basic

a variance-covariance matrix that ignores the survey design.

hessian

a Hessian matrix.

estfun

a gradient (a vector of partial derivatives) of the log likelihood function at the estimated coefficient values.

YYY1,YYY2,YYY3

an internal objects used for the calculation of gradient and Hessian functions.

Details

The function fits generalized hierarchical ordered threshold models.

latent.formula models the latent variable. If the response variable is self-rated health, then the latent measure can depend on different health conditions and diseases (latent variables are called health variables). Latent variables are modeled with the parallel regression assumption. According to this assumption, the coefficients that describe the relationship between the lowest response category and all of the higher response categories, are the same as the coefficients that describe the relationship between another (e.g., adjacent) lowest response category and the remaining higher response categories. The predicted latent variable is modeled as a linear function of the health variables and the corresponding coefficients.

thresh.formula models the threshold variable. The thresholds (cut-points, alpha) are modeled by the threshold variables gamma and the intercepts lambda. It is assumed that they model the contextual characteristics of the respondent (e.g., country, gender, and age). The threshold variables are modeled without the parallel regression assumption; thus, each threshold is modeled by a variable independently Boes2006,Green2014hopit. The hopit() function uses the parameterization of thresholds proposed by Jurges2007;textualhopit.

decreasing.levels it is the logical that determines the ordering of the levels of the categorical response variable. It is always advisable to first check the ordering of the levels before starting (see example 1)

It is possible to model the interactions, including interactions between the latent and the threshold variables. The interactions added to the latent formula only model the latent measure, and the interactions modeled in the threshold formula only model the thresholds. The general rule for modeling any kind of interaction is to use "*" to specify interactions within a latent (or threshold) formula and to use ':' to specify interactions between the latent and the threshold variables. In the latter case, the main effects of an interaction must also be specified; i.e., the main latent effects must be specified in the latent formula, and the main threshold effect must be speciffied in the threshold formula. See also Example 3 below.

For more details, please see the package vignette: "introduction_to_hopit", which is also available under this link: introduction_to_hopit.pdf

References

Examples

Run this code

# NOT RUN {
# DATA
data(healthsurvey)

# first determine the order of the levels of the dependent variable
levels(healthsurvey$health)

# the order of response levels decreases from the best health to
# the worst health; hence the hopit() parameter decreasing.levels
# is set to TRUE

# Example 1 ---------------------

# fitting the model:
model1 <- hopit(latent.formula = health ~ hypertension + high_cholesterol +
                heart_attack_or_stroke + poor_mobility + very_poor_grip +
                depression + respiratory_problems +
                IADL_problems + obese + diabetes + other_diseases,
              thresh.formula = ~ sex + ageclass + country,
              decreasing.levels = TRUE,
              control = list(trace = FALSE),
              data = healthsurvey)

# summarize the fit:
summary(model1)

# extract parameters in the form of a list
cm1 <- coef(model1, aslist = TRUE)

# names of the returned coefficients
names(cm1)

# extract the latent health coefficients
cm1$latent.params

# check the fit
# }
# NOT RUN {
profile(model1)
# }
# NOT RUN {
# Example 2 ---------------------

# }
# NOT RUN {
# incorporate the survey design
design <- svydesign(ids = ~ country + psu, weights = healthsurvey$csw,
data = healthsurvey)

model2 <- hopit(latent.formula = health ~ hypertension + high_cholesterol +
                  heart_attack_or_stroke + poor_mobility +
                  very_poor_grip + depression + respiratory_problems +
                  IADL_problems + obese + diabetes + other_diseases,
                thresh.formula = ~ sex + ageclass + country,
                decreasing.levels = TRUE,
                design = design,
                control = list(trace = FALSE),
                data = healthsurvey)

# compare the latent variables
cbind('No survey design' = coef(model1, aslist = TRUE)$latent.par,
'Has survey design' = coef(model2, aslist = TRUE)$latent.par)
# }
# NOT RUN {
# Example 3 ---------------------

# defining the interactions between the threshold and the latent variables

# correctly defined interactions:
model3 <- hopit(latent.formula = health ~ hypertension + high_cholesterol +
                heart_attack_or_stroke + poor_mobility * very_poor_grip +
                depression + respiratory_problems +
                IADL_problems + obese + diabetes + other_diseases +
                sex : depression + sex : diabetes + ageclass:obese,
              thresh.formula = ~ sex * ageclass + country + sex : obese,
              decreasing.levels = TRUE,
              control = list(trace = FALSE),
              data = healthsurvey)
# }
# NOT RUN {
# badly defined interactions:

# 1) lack of a main effect of "other_diseases" in any formula
# it can be solved by adding " + other_diseases" to the latent formula
model3a <- hopit(latent.formula = health ~ hypertension + high_cholesterol +
                heart_attack_or_stroke + poor_mobility + very_poor_grip +
                depression + respiratory_problems +
                IADL_problems + obese + diabetes + other_diseases : sex,
              thresh.formula = ~ sex + ageclass + country,
              decreasing.levels = TRUE,
              control = list(trace = FALSE),
              data = healthsurvey)

# 2) the main effect of sex is present in both formulas.
# it can be solved by replacing "*" with ":" in "other_diseases * sex"
model3b <- hopit(latent.formula = health ~ hypertension + high_cholesterol +
                heart_attack_or_stroke + poor_mobility + very_poor_grip +
                depression + respiratory_problems +
                IADL_problems + obese + diabetes + other_diseases * sex,
              thresh.formula = ~ sex + ageclass + country,
              decreasing.levels = TRUE,
              control = list(trace = FALSE),
              data = healthsurvey)

# }
# NOT RUN {
# Example 4 ---------------------

# }
# NOT RUN {
# construct a naive continuous variable:
hs <- healthsurvey
hs$cont_var <- sample(5000:5020,nrow(hs),replace=TRUE)

latent.formula = health ~ hypertension + high_cholesterol +
  heart_attack_or_stroke + poor_mobility + very_poor_grip +
  depression + respiratory_problems +
  IADL_problems + obese + diabetes + other_diseases

# in some cases, when continuous variables are used, the hopit:::get.hopit.start() function
# do not find starting parameters (R version 3.4.4 (2018-03-15)):
model4 <- hopit(latent.formula = latent.formula,
                thresh.formula = ~ sex + cont_var,
                decreasing.levels = TRUE,
                data = hs)

# one of the solutions is to transform one or more continuous variables:
hs$cont_var_t <- hs$cont_var-min(hs$cont_var)

model4b <- hopit(latent.formula = latent.formula,
                 thresh.formula = ~ sex + cont_var_t,
                 decreasing.levels = TRUE,
                 data = hs)

# this can also be done automatically using the the control parameter
model4c <- hopit(latent.formula = latent.formula,
                 thresh.formula = ~ sex + cont_var,
                 decreasing.levels = TRUE,
                 control = list(transform.thresh = 'min',
                                transform.latent = 'none'),
                 data = hs)

model4d <- hopit(latent.formula = latent.formula,
                 thresh.formula = ~ sex + cont_var,
                 decreasing.levels = TRUE,
                 control = list(transform.thresh = 'scale_01',
                                transform.latent = 'none'),
                 data = hs)

model4e <- hopit(latent.formula = latent.formula,
                 thresh.formula = ~ sex + cont_var,
                 decreasing.levels = TRUE,
                 control = list(transform.thresh = 'standardize',
                                transform.latent = 'none'),
                 data = hs)

model4f <- hopit(latent.formula = latent.formula,
                 thresh.formula = ~ sex + cont_var,
                 decreasing.levels = TRUE,
                 control = list(transform.thresh = 'standardize_trunc',
                                transform.latent = 'none'),
                 data = hs)

round(t(rbind(coef(model4b),
              coef(model4c),
              coef(model4d),
              coef(model4e),
              coef(model4f))),4)

# }

Run the code above in your browser using DataLab