1 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 (Riegel et al. 2006; Kantarjian et al. 2007). 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 et al. 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.
2 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.
3 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
\[\log\left(\mu_{ab(ij)}\right)=\gamma_{ij}+\eta_{a(ij)}^{W}+\eta_{b(ij)}^{Y} \label{eq:1} \tag{1}\]
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 (1) leads to different types of association structures between the MRCVs. For example, adding \(\lambda_{ab}\) to Equation (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
(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.
4 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 et al. 2000). The help file for
MI.test() shows how to use functions in the
geepack package
(Yan et al. 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 et al. (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.
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