MRCV: A Package for Analyzing Categorical Variables with Multiple Response Options
Natalie A. Koziol
Department of Statistics; Department of Educational Psychology, University of Nebraska-LincolnChristopher R. Bilder
Department of Statistics, University of Nebraska-Lincoln2014-04-19
Abstract
Multiple response categorical variables (MRCVs), also known as “pick any” or “choose all that apply” variables, summarize survey questions for which respondents are allowed to select more than one category response option. Traditional methods for analyzing the association between categorical variables are not appropriate with MRCVs due to the within-subject dependence among responses. We have developed the MRCV package as the first R package available to correctly analyze MRCV data. Statistical methods offered by our package include counterparts to traditional Pearson chi-square tests for independence and loglinear models, where bootstrap methods and Rao-Scott adjustments are relied on to obtain valid inferences. We demonstrate the primary functions within the package by analyzing data from a survey assessing the swine waste management practices of Kansas farmers.Introduction
Survey questions often instruct respondents to “choose all that apply” from a list of response categories. For example, surveys instituted by U.S. government agencies are mandated to ask race and ethnicity questions in this format ((Office of Management and Budget 1997), p. 58781). In medical applications, “choose all that apply” questions have been used for a variety of purposes, including gathering information about treatment and monitoring strategies (Kantarjian et al. 2007; Riegel et al. 2006). Outside of surveys, this format can appear in unexpected applications. For example, wildlife management researchers are often interested in the food habits of animal species. Traces of prey in scats provide these researchers with a “choose all that apply” type of response because multiple prey types may be present ((Lemons et al. 2010; Riemer, Wright, and Brown 2011)).
Variables that summarize data arising from a “choose all that apply” format are referred to as multiple response categorical variables (MRCVs), and the response categories within each MRCV are referred to as items (Bilder and Loughin 2004). Because individual subjects are allowed to choose multiple items, the responses are likely dependent, and therefore traditional methods for analyzing categorical variables (e.g., Pearson chi-square tests for independence, loglinear models) are not appropriate. Unfortunately, numerous examples exist where these traditional methods are still used (see (Wright 2010) for a review), which can lead to erroneous results (Loughin and Scherer 1998).
While MRCVs have been identified since at least Coombs (1964), methods for correctly analyzing MRCVs in the context of common categorical data analysis interests, such as examining associations between variables, have only been available for approximately 15 years (e.g., see (Agresti and Liu 1999)). Our MRCV package (Koziol and Bilder 2014) is the first R package available to implement valid inference procedures for this type of data. The functions within the package can be used by researchers who want to examine the relationship among items from up to three MRCVs.
We begin this paper by first illustrating functions within the package for summarizing MRCV data and testing for independence. Then, we illustrate functions for fitting a generalized loglinear model to MRCV data and for performing follow-up analyses using method functions. Our examples focus on only two MRCVs for brevity reasons, but we discuss extensions in the conclusion.
Test for independence
We begin with an example from Bilder and Loughin (2007) involving a simple random sample of Kansas swine farmers. There are two MRCVs to be examined here, and we denote them generically as \(W\) and \(Y\). The first MRCV (\(W\)) corresponds to a survey question that asked farmers to state which contaminants they tested for from the items “nitrogen”, “phosphorous”, and “salt” (\(W_{1}\), \(W_{2}\), \(W_{3}\), respectively). The second MRCV (\(Y\)) corresponds to a survey question that asked farmers to identify their swine waste storage methods from the items “lagoon”, “pit”, “natural drainage”, and “holding tank” (\(Y_{1}\), \(Y_{2}\), \(Y_{3}\), \(Y_{4}\), respectively). Farmers were instructed to “choose all that apply” from each of these predefined lists. By using a \(0\) to denote an item not chosen (negative response) and a \(1\) to denote an item chosen (positive response), each observation consists of a set of correlated binary responses, as shown below:
> head(farmer2, n = 3)
w1 w2 w3 y1 y2 y3 y4
1 0 0 0 0 0 0 0
2 0 0 0 0 0 0 1
3 0 0 0 0 0 0 1
> tail(farmer2, n = 3)
w1 w2 w3 y1 y2 y3 y4
277 1 1 1 1 1 0 0
278 1 1 1 1 1 0 0
279 1 1 1 1 1 1 0We see, for example, that the third farmer does not test for any contaminants and uses only a holding tank for waste storage.
Contingency table-like summaries of MRCV data are often given in papers. In particular, marginal counts for all pairwise positive responses between items in \(W\) and \(Y\) are shown in Table 1. This display format can lead researchers to want to apply Pearson chi-square tests (or other simple categorical measures) to the table of counts in order to understand associations between the MRCVs. However, this approach is not correct because it does not take into account the fact that an individual subject can contribute to multiple counts in the table, which violates any type of multinomial distribution underlying assumption for these specific counts. Furthermore, three other tables summarizing the pairwise positive/negative responses (e.g., summarizing responses for items “not” chosen) of this type could also be constructed. Agresti and Liu (1999) and Bilder and Loughin (2001) show that testing procedures are not invariant to whether positive or negative responses are summarized and that different conclusions about the data can be reached depending on the types of responses summarized.
| Waste storage method | |||||
| Lagoon | Pit | Natural Drainage | Holding tank | ||
| Nitrogen | 27 | 16 | 2 | 2 | |
| Contaminant | Phosphorous | 22 | 12 | 1 | 1 |
| Salt | 19 | 6 | 1 | 0 |
Examining all possible combinations of the positive/negative item
responses between MRCVs is the preferred way to display and subsequently
analyze MRCV data. The item.response.table() function
provides this summary for each (\(W_{i}\), \(Y_{j}\)) pair:
> item.response.table(data = farmer2, I = 3, J = 4)
y1 y2 y3 y4
0 1 0 1 0 1 0 1
w1 0 123 116 175 64 156 83 228 11
1 13 27 24 16 38 2 38 2
w2 0 128 121 181 68 165 84 237 12
1 8 22 18 12 29 1 29 1
w3 0 134 124 184 74 174 84 245 13
1 2 19 15 6 20 1 21 0 where I is the number of items for \(W\) and J is the number of
items for \(Y\). The pairwise
item-response table indicates, for example, that \(27\) farmers tested for nitrogen and used
lagoon as a waste storage method (i.e., \(W_{1}=1\), \(Y_{1}=1\)). Furthermore, \(123\) farmers did not test for nitrogen and
did not use a lagoon, \(13\) farmers
tested for nitrogen without using a lagoon, and \(116\) farmers did not test for nitrogen
while using a lagoon. In total, \(27+123+13+116=279\) farmers participated in
the survey (there are no missing responses to any item).
Agresti and Liu (1999) provided the
MRCV extension to testing for independence between single response
categorical variables (SRCVs). This test, known as a test for
simultaneous pairwise marginal independence (SPMI), involves determining
whether each \(W_{1}\), …, \(W_{I}\) is pairwise independent of each
\(Y_{1}\), …, \(Y_{J}\). Our MI.test()
function calculates their modified Pearson statistic as \(X_{S}^{2}=\sum\nolimits_{i=1}^{I}\sum\nolimits_{j=1}^{J}X_{S,i,j}^{2}\)
where \(X_{S,i,j}^{2}\) is the usual
Pearson chi-square statistic used in this situation to test for
independence in the 2\(\times\)2 tables formed by each
(\(W_{i}\), \(Y_{j}\)) response combination. In our
example, \(X_{S}^{2}\) is the sum of 12
pairwise marginal tests for independence. In general, \(X_{S}^{2}\) does not have an asymptotic
\(\chi_{IJ}^{2}\) distribution due to
dependency among the \(X_{S,i,j}^{2}\).
Rather, the asymptotic distribution is a linear combination of
independent \(\chi_{1}^{2}\) random
variables (Bilder and Loughin 2004).
The MI.test() function offers three methods, available
through its type argument, that can be used with \(X_{S}^{2}\) or the \(X_{S,i,j}^{2}\) individual statistics to
perform valid tests for SPMI. The type = "boot" argument
value specifies the use of the nonparametric bootstrap to estimate the
sampling distribution of \(X_{S}^{2}\)
under SPMI and to calculate an appropriate p-value using B
resamples. In addition, two p-value combination methods—the product and
minimum of p-values—are implemented to combine the p-values obtained
from \(X_{S,i,j}^{2}\) and a \(\chi_{1}^{2}\) approximation. Details on
these bootstrap approaches are given in Bilder
and Loughin (2004). The type = "rs2" argument value
applies a Rao-Scott second-order adjustment to \(X_{S}^{2}\) and its sampling distribution.
This procedure adjusts \(X_{S}^{2}\) to
match the first two moments of a chi-square random variable,
asymptotically. Details on this approach are provided in Bilder and Loughin (2004) and Thomas and Decady (2004). Finally, the
type = "bon" argument value simply applies a Bonferroni
adjustment with each \(X_{S,i,j}^{2}\)
and a \(\chi_{1}^{2}\) approximation.
To implement all three methods, we can use the type = "all"
argument value:
> set.seed(102211) # Set seed to replicate bootstrap results
> MI.test(data = farmer2, I = 3, J = 4, type = "all", B = 1999, plot.hist = TRUE)
Test for Simultaneous Pairwise Marginal Independence (SPMI)
Unadjusted Pearson Chi-Square Tests for Independence:
X^2_S = 64.03
X^2_S.ij =
y1 y2 y3 y4
w1 4.93 2.93 14.29 0.01
w2 6.56 2.11 11.68 0.13
w3 13.98 0.00 7.08 0.32
Bootstrap Results:
Final results based on 1999 resamples
p.boot = 0.0005
p.combo.prod = 0.0005
p.combo.min = 0.001
Second-Order Rao-Scott Adjusted Results:
X^2_S.adj = 36.17
df.adj = 6.78
p.adj < 0.0001
Bonferroni Adjusted Results:
p.adj = 0.0019
p.ij.adj =
y1 y2 y3 y4
w1 0.3163 1.0000 0.0019 1.0000
w2 0.1253 1.0000 0.0076 1.0000
w3 0.0022 1.0000 0.0934 1.0000Figure 1 shows histograms from the bootstrap implementations. All of the methods provide strong evidence for rejecting SPMI. The \(X_{S,i,j}^{2}\) and corresponding Bonferroni adjusted p-values indicate a significant association for the (\(W_{1}\), \(Y_{3}\)), (\(W_{2}\), \(Y_{3}\)), and (\(W_{3}\), \(Y_{1}\)) combinations.
Loglinear modeling
SPMI is only one possible association structure between MRCVs. Bilder and Loughin (2007) introduced a flexible loglinear modeling approach that allows researchers to consider alternative association structures somewhere between SPMI and complete dependence. Within this framework, a model under SPMI is given as
\[\log\left(\mu_{ab(ij)}\right)=\gamma_{ij}+\eta_{a(ij)}^{W}+\eta_{b(ij)}^{Y} \label{eq:1} (\#eq:1)\]
where \(\mu_{ab(ij)}\) is the expected number of subjects who responded (\(W_{i}=a\), \(Y_{j}=b\)) for \(a\), \(b\) \(\in \{0,1\}\). The terms on the right side of the model are the same as for a loglinear model under independence between two SRCVs, where we have added a subscript (\(ij\)) to indicate a particular 2\(\times\)2 table for (\(W_{i}\), \(Y_{j}\)) within the pairwise item-response table. The usual constraints are placed on the model parameters to maintain identifiability.
Adding additional terms to Equation @ref(eq:1) leads to different types of association structures between the MRCVs. For example, adding \(\lambda_{ab}\) to Equation @ref(eq:1) produces a homogeneous association model (i.e., a model that implies equal odds ratios, not necessarily equal to \(1\), for each 2\(\times\)2 table), adding \(\lambda_{ab(i)}^{W}\) or \(\lambda_{ab(j)}^{Y}\) to the homogeneous association model produces a W- or Y-main effects model, respectively, and adding both of these terms to the homogeneous association model produces a W- and Y-main effects model. The addition of a WY interaction term, \(\lambda_{ab(ij)}^{WY}\), produces the saturated model.
The genloglin() function estimates the above models
through a marginal estimation approach. Within genloglin(),
a new data frame is created by converting the raw data into the pairwise
item-response counts:
> item.response.table(data = farmer2, I = 3, J = 4, create.dataframe = TRUE)
W Y wi yj count
1 w1 y1 0 0 123
2 w1 y1 0 1 116
3 w1 y1 1 0 13
4 w1 y1 1 1 27
5 w1 y2 0 0 175
< output omitted >
48 w3 y4 1 1 0The glm() function is subsequently called from within
genloglin() to estimate a loglinear model to these counts.
Rao-Scott adjustments are then applied to obtain valid large-sample
standard error estimates. The model argument of
genloglin() can take the names of "spmi",
"homogeneous", "w.main",
"y.main", "wy.main", and
"saturated" to specify a particular model. Alternatively, a
user-supplied formula allows for more flexibility by specifying the
model in terms of W, Y, wi,
yj, count, W1, …,
WI, and Y1, …, YJ, which we
illustrate shortly. The boot = TRUE (the default) value for
genloglin() specifies that resamples should be taken under
the fitted model. We use the method of Gange
(1995) for generating correlated binary data to perform
semi-parametric bootstrap resampling in this case. These resamples are
subsequently used for hypothesis tests, confidence intervals, and/or
standardized residuals with our related method functions for objects
returned by genloglin().
We demonstrate the genloglin() function by estimating
the Y-main effects model to the farmer2 data, and then
summarize the results using our summary() method
function:
> set.seed(499077) # Set seed to replicate bootstrap results
> mod.fit <- genloglin(data = farmer2, I = 3, J = 4, model = "y.main", B = 1999,
+ print.status = FALSE)
> summary(mod.fit)
Call:
glm(formula = count ~ -1 + W:Y + wi %in% W:Y + yj %in% W:Y + wi:yj + wi:yj %in% Y,
family = poisson(link = log), data = model.data)
Deviance Residuals:
Min 1Q Median 3Q Max
-1.58007 -0.13272 0.00043 0.10282 0.79587
Coefficients:
Estimate RS SE z value Pr(>|z|)
Ww1:Yy1 4.83360 0.06535 73.969 < 2e-16 ***
Ww2:Yy1 4.85571 0.06387 76.023 < 2e-16 ***
Ww3:Yy1 4.87418 0.06314 77.199 < 2e-16 ***
< output omitted >
Null deviance: 25401.0663 Residual deviance: 5.8825
Number of Fisher Scoring iterations: 4The print.status argument can be changed to
TRUE (default) in order to print model fitting information
while the function is running. Information typically provided by the
glm() function can be extracted from
mod.fit.
The formula argument within the Call:
portion of the output displays an alternative way that the Y-main
effects model could have been specified using variable names. For a
model under SPMI (Equation @ref(eq:1)), the syntax
-1 + W:Y + wi %in% W:Y + yj %in% W:Y specifies an ordinary
loglinear model under independence within each 2\(\times\)2 table formed by the
(\(W_{i}\), \(Y_{j}\)) pairs; i.e., the intercept
(W:Y), “row effect” (wi %in% W:Y), and “column
effect” (yj %in% W:Y) terms. Note that the addition of
wi:yj %in% W:Y would then lead to a saturated loglinear
model within the 2\(\times\)2
tables. Instead, the addition of wi:yj + wi:yj %in% Y
allows for the associations to vary across the items in \(Y\) (waste storage) but to be the same
across items in \(W\)
(contaminant).
The deviance values in the output should not be used with chi-square
distributional approximations to construct traditional model comparison
tests. Instead, our anova() method function offers
bootstrap and Rao-Scott second-order adjustments
(type = "boot" and type = "rs2", respectively,
or type = "all" for both methods) to obtain appropriate
tests for comparing the model specified in genloglin() to
an alternative model given by its model.HA argument.
Comparing the Y-main effects model to the saturated model shows moderate
evidence of lack-of-fit:
> anova(object = mod.fit, model.HA = "saturated", type = "all")
Model comparison statistics for
H0 = y.main
HA = saturated
Pearson chi-square statistic = 5.34
LRT statistic = 5.88
Second-Order Rao-Scott Adjusted Results:
Rao-Scott Pearson chi-square statistic = 10.85, df = 5.23, p = 0.0624
Rao-Scott LRT statistic = 11.96, df = 5.23, p = 0.0409
Bootstrap Results:
Final results based on 1999 resamples
Pearson chi-square p-value = 0.0385
LRT p-value = 0.0255Our residuals() method function provides Pearson
standardized residuals, where bootstrap or asymptotic standard errors
can be used in their formation. For the Y-main effects model, we find
that lack-of-fit occurs for the (\(W_{3}\), \(Y_{1}\)) association. This suggests the
need to estimate a new model that explicitly accounts for the
heterogeneity:
mod.fit.w3y1 <- genloglin(data = farmer2, I = 3, J = 4, model = count ~ -1 + W:Y +
wi %in% W:Y + yj %in% W:Y + wi:yj + wi:yj %in% Y +
wi:yj %in% W3:Y1, B = 1999)where the wi:yj %in% W3:Y1 term forces a perfect fit to
the (\(W_{3}\), \(Y_{1}\)) association while still
maintaining a Y-main effects model elsewhere.
Once an appropriate model has been identified, our
predict() method function can be used to obtain observed
and model-estimated odds ratios with corresponding asymptotic and
bootstrap BC\(_{\text{a}}\) (Efron 1987) confidence intervals. These odds
ratios help to facilitate interpretation of the association among items
between the two MRCVs.
Summary
The equivalents of many traditional categorical data analysis methods
are implemented within our package in the context of MRCVs. We
demonstrated a few of the package’s primary functions for analyzing the
association between two MRCVs. While not shown here, these functions can
be used to analyze MRCVs in other settings. For instance, tests for
multiple marginal independence (MMI; (Agresti and Liu 1999)) between an MRCV and an
SRCV can be performed by MI.test(), where the
I argument is set to a value of \(1\). An example is given within the help
file for this function. Additionally, the MRCV package can be used to
analyze the association between three MRCVs. For example, Bilder and Loughin (2007) discuss a third
“choose all that apply” question asked of the swine farmers that relates
to the farmers’ sources of veterinary information. We show in the help
file for genloglin() how to estimate a generalized
loglinear model for this setting.
Agresti and Liu (1999, 2001) show how
to take advantage of many commonly used modeling methods (e.g.,
generalized linear mixed models) for MRCV data. Most of these methods
have disadvantages to their use—for example, standard generalized linear
mixed models induce a positive correlation between binary responses
within subjects, but a negative correlation can occur with MRCV data.
Their recommended modeling method, a generalized loglinear model fit
through generalized estimating equation (GEE) methodology, can work
reasonably well in very large sample sizes (Bilder, Loughin, and Nettleton 2000). The help
file for MI.test() shows how to use functions in the geepack
package (Yan, Hojsgaard, and Halekoh 2012)
to estimate this model and then subsequently test for MMI via a Wald
test.
We envision future additions to the package that will allow for extensions to other situations. For example, “choose all that apply” questions are often asked in complex survey sampling settings. Bilder and Loughin (2009) propose using the same generalized loglinear models, where now different Rao-Scott adjustments are needed to take into account the sampling design. Also, MRCV data can arise over a longitudinal setting, and Suesse and Liu (2013) propose the use of GEE methodology to fit models for this situation. Finally, Nandram, Toto, and Katzoff (2009) offer a Bayesian perspective to the MMI testing problem. Due to the ubiquitous nature of “choose all that apply” type data formats, we expect there to be many other unique settings where new statistical methods need to be developed. We encourage readers to contact us about their novel methods and/or interest in collaboration.