# Propensity Score Estimates in Multilevel Models for Causal Inference

##### Abstract

Background: Teenage obesity is a national epidemic that requires school- and community-based initiatives to support healthy behaviors of students regarding exercise and nutrition to decrease the prevalence.

Objectives: The aim of this study was to demonstrate a methodology for an estimation of causal effects of the adoption of healthy behaviors with a potential outcomes approach within a multilevel treatment setting of school program adoption of a socially supportive environment.

Methods: Propensity score estimates within a multilevel model provided causal estimates of the impact of the adoption of health habits by students within supportive school environments (SSEs) and non-SSEs. A potential outcomes approach to causal modeling was shown with a secondary analysis ofthe National Longitudinal Study of Adolescent Health study. The student participants consisted of 13,854 adolescent students, with an accompanying sample of 164 school administrators.

Results: The effect of healthy eating habits in an SSE was a statistically nonsignificant decrease in body mass index (BMI). The effect of healthy eating habits in a non-SSE was a statistically nonsignificant increase in BMI. The difference between the healthy habit practices for students in supportive and nonsupportive schools was a resultant difference in BMI of 0.3484.

Discussion: The results demonstrate a difference in causal effects of eating habits in different school settings. Further research regarding causal effects of student habits and school programs is indicated.

##### Author Information

**Patricia Eckardt, PhD,** is Assistant Professor, School of Nursing, Stony Brook University, New York.

Accepted for publication February 28, 2012.

The author thanks the Educational Psychology Department at the Graduate Center CUNY for its approval of this study and Dr. David Rindskopf, PhD, a distinguished professor at the Graduate Center of the City University of New York, for his suggestions in the design of this study and in the analysis of the data.

The author has no funding or conflicts of interest to disclose.

Corresponding author: Patricia Eckardt, PhD, School of Nursing, Health Sciences Center, Level 2, Room 247, Stony Brook University, Stony Brook, NY 11794-8240 (e-mail: patricia.eckardt@stonybrook.edu).

Teenage obesity is a national epidemic (^{Ben-Sefer, Ben-Natan, & Ehrenfeld, 2009}). Approximately 17% (12.5 million) of children and teens in the United States are obese (^{Ogden, Carroll, Curtin, Lamb, & Flegal, 2010}). It has been hypothesized that students who follow healthy habits regarding exercise and nutrition decrease their chances of becoming obese in adulthood (^{Liou, Liou, & Chang, 2010}). Some school policymakers and nurses believe that the adoption of a supportive school environment (SSE) program, one that includes increased fruits and vegetables available at the cafeteria and a variety of available forms of exercise and recreation, increase opportunities for students to practice healthy habits and decrease their chances of developing obesity (^{Cassady, Vogt, Oto-Kent, Mosley, & Lincoln, 2006}; ^{Vitale, 2010}). Body mass index (BMI) is the current standard measure of obesity in adolescents (^{Lin & Lam, 2011}; ^{Liou et al., 2010}). However, a standardized BMI *z* score often is considered a more accurate measure of an individual teen’s obesity (^{Boylan et al., 2010}; ^{Rausch, Perito, & Hametz, 2011}). Multiple observational studies have examined these hypotheses with conflicting results (^{Liou et al., 2010}).

An observational study, like an experimental study, is an empirical study conducted to explain a cause-and-effect relationship between two variables (^{Cochran, 1965}; ^{Little & Rubin, 2000}: ^{Rosenbaum, 2002}; ^{Rubin, 1997}). In nursing literature, the term “quasi-experimental study” is used more commonly than *observational study* to describe a study that lacks the critical causal element of random assignment (^{Guo & Fraser, 2010}). In ^{Clayton, Chin, Blackburn, and Echeverria’s (2010)} prospective quasi-experimental study, a difference was found in obesity rates between groups after a healthy habits program was offered at school. ^{Fulkerson et al. (2010)} and ^{Hollar et al. (2010)} did not find a significant difference in means of postintervention BMI *z* score between groups with healthy habits at home and school intervention. Randomized and multilevel randomized control trials to study the effect of healthy habit interventions on adolescents’ BMI (^{de Heer, Koehly, Pederson, & Morera, 2011}) and BMI *z* score (^{DeBar et al., 2011}; ^{Dzewaltowski et al., 2010}) were equivocal in findings.

To estimate the causal effect of students’ healthy habits on their development of obesity in young adulthood and the effect of an SSE on students’ practice of healthy habits, a multilevel potential outcomes model is required to account for the natural nesting of the teenage students within the schools they attend. Multilevel models are also known as hierarchical models, mixed effect models, and random coefficient models (^{Albright, 2007}; ^{Albright & Marinova, 2010}; ^{Laird & Ware, 1982}; ^{Muthen & Muthen, 2012}; ^{Rasbash, Steele, Browne, & Goldstein, 2009}; ^{Raudenbush & Bryk, 2002}; ^{Singer, 1998}; ^{Singer & Willett, 2003}). Multilevel modeling of students within school settings frames the research within the socioecological theory by considering social and environmental factors in which the students are nested naturally (^{Maclean et al., 2010}). Multilevel modeling also allows for variance component estimates to be examined explicitly at each level of the proposed model (^{Albright, 2007}; ^{Albright & Marinova, 2010}; ^{Raudenbush & Bryk, 2002}; ^{Singer, 1998}; ^{Singer & Willett, 2003}).

A potential outcomes framework at each level of the multilevel model allows for causal inference regarding these factors and not simply comparison across groups (^{Pearl, 2003, 2010}). This framework accounts for the probabilistic mechanism of treatment assignment that observational studies and most random experiments usually do not. Assignment to treatment in an observational study is not random, although sampling from a population of interest often is; whereas assignment to treatment in random experiments is assigned randomly, yet sampling from the population of interest is not a random process (^{Little & Rubin, 2000}). Propensity score estimates allow for potential outcomes inference by accounting for manipulation of treatment assignment. There can be no causal inference without manipulation of treatment (^{Rubin, 1986}).

^{Rubin’s (1974)} counterfactual causal model approaches causal modeling with a fundamental assumption of potential outcomes (^{Angrist & Pischke, 2008}; ^{Rosenbaum, 2002}; ^{Rubin, 1974}). This theory asserts that there is one potential outcome associated with each treatment a subject may receive. For the following example, *Y* denotes an individual subject’s potential outcome and *Z* denotes treatment assignment. If there are two alternate treatments (a binary treatment condition), then each subject has two potential outcomes represented commonly in the potential outcomes literature as *Y*(0) and *Y*(1). Specifically, *Y*(0) is the outcome for a subject who does not receive the treatment (i.e., *Z* = 0), whereas *Y*(1) is the outcome if the subject does receive the treatment (i.e., *Z* = 1). However, one never gets to observe both of these potential outcomes, as the subject can receive only one treatment at a time (^{Rubin, 2004}). Therefore, the causal effect *Y*(1) − *Y*(0) cannot be estimated for an individual, but only an average effect for a group of people.

In a randomized experiment, the observed outcomes of the subjects in the sample can be used to estimate an average treatment effect conditioned on treatment assigned. An unbiased estimate of the average treatment effect *E*[*Yi*(1) − *Yi*(0)] = *E*[*Yi*(1)] − *E*[*Yi*(0)] within an observed population sample is the difference in observed outcomes in the two distinct treatment groups. Therefore, in a randomized experiment, if *Z* stands for treatment assignment, then the average treatment effect *E*[*Yi*(1)] − *E*[*Yi*(0)] can be estimated as *E*[*Yi*(1)] = *E*[*Yi*(1)|*Zi* = 1] and *E*[*Yi*(0)] = *E*[*Yi*(0)|*Zi* = 0]. One can then obtain an unbiased estimate of the average treatment effect by estimating *E*[*Yi*(1)] with

and *E*[*Yi*(0)] with

.

The assumptions of the potential outcomes model are ignorability of treatment assignment and stable unit treatment value. Treatment assignment is ignorable when the potential outcomes are independent of the treatment variable, as is the case with randomized assignment to treatment (^{Rubin, 2004}). The stable unit treatment value (SUTVA) is the assumption that there is only one potential outcome for each subject associated with the policy or treatment adopted, with no effect on an individual subject’s potential outcome due to the policy or treatment given another subject (^{Rubin, 1986}). The assumption of SUTVA at the school level is plausible because schools in the proposed nationally sampled data set are located in distinct communities with different geographic conditions; have various combinations of socioeconomic levels, governmental resources, and political affiliations in the community profiles; and span the continental United States. Here, an assumption that the identities and treatment assignments of other schools are uninformative about student *i*’s outcomes holds (SUTVA assumption at the school level).

However, ^{Rubin (1990)} cautioned that SUTVA becomes problematic when treatments are given to children who interact with one another, as in an observational study wherein student-level treatment is nested within school-level program adoption. To account for the possible violation of a strict interpretation of SUTVA at the student level within a multilevel setting, the ^{Hong and Raudenbush (2006)} relaxed form of SUTVA can be applied. For a binary treatment, let *zi* = 1 if student *i* practices healthy habits and *zi =* 0 if the student does not. Then, given *N* units overall in a school, there is a 1 × *N* vector of possible treatment assignments **z** = (*z*_{1}*, z*_{2}*,* …, *zN*) = (*zi,* **z_***i*), wherein **z**_*i* is the 1 × (*N* − 1) vector of treatment assignments with *zi* removed, for

. Under these set conditions, subject *i* has 2*N* potential outcomes, *Yi*(**z**), corresponding to all possible treatment assignments of the *N* subjects.

A contrast between any two of the 2*N *potential outcomes for a given subject is a causal effect. The SUTVA is a distinctive case wherein *Yi*(**z**) ≡ *Yi* (*zi*, **z**_*i*) = *Yi* (*zi*), demonstrating that an individual student’s potential outcome, given all other student treatments, remains the same under SUTVA. To summarize the school program adoption information in a way that will be useful for average causal estimates, the impact of *z* (treatment assignment) on a single subject’s potential outcome can be modeled as operating through *zi* (individual’s treatment assignment) as well as through a simple function of **z**_*i*, which will be denoted as *v*(**z**), the treatment assignments of all other students within an individual’s school, except the individual’s assignment. This is represented as

*E*{*Y*[*z,v*(**z**)] − *Y*[*z*′,*v*(**z′**)]} is a generic causal estimand, wherein *z* and *z*′ are alternative treatment assignments for an individual and **z** and **z′** are alternative treatment assignments for all individuals within that group (^{Hong & Raudenbush, 2006}). For the purposes of the SSE policy, let *v*(**z**) = 1 if a school is an SSE policy school and *v*(**z**) = 0 if this is not the case. Now, *Y*(*z,v*(**z**)) can take on four possible values: *Y*(1,1), *Y*(0,1), *Y*(1,0), and *Y*(0,0).

Therefore, the potential outcomes for an individual student *i* attending a school *j*, denoted by *Yij*(*zij*, *vj*), could take on four values: *Yij*(1,1), *Yij*(0,1), *Yij*(1,0), and *Yij*(0,0). For example, *Yij*(1,1) represents the potential outcome of a student *i* who practices healthy habits from an SSE school, and *Yij* (1,0) is the potential outcome of a student *i* who practices healthy habits from a non-SSE policy school.

Through the use of the relaxed form of SUTVA outlined above and the propensity score estimates for a potential outcomes framework of causal estimates (^{Rosenbaum & Rubin, 1984}) outlined in the next section, causal effects of treatment on a student-level outcome variable of interest can be estimated. This causal modeling methodology allows school assignment and peer treatments to affect potential outcomes within a framework that accounted for the natural nesting of the data and did not violate assumptions required for these estimates.

Supportive school environment policies are not assigned at random to schools, and students within schools are not assigned at random to be practice of healthy habits, so the ignorability of treatment assignment assumption must be considered. A propensity score is the conditioning on covariates of an observed probability for receiving the treatment. As such, if there is no hidden bias, then the conditional distribution of treatment assignment is uniform and treatment assignment is ignorable (^{Rosenbaum, 2002}).

Let **X** be a vector of student-level covariates and **W** be a vector of school-level covariates. **W** includes the school-level aggregates of student-level covariates such as the demographic makeup of students attending a school. Causal inferences are possible if treatment assignments are ignorable within levels of covariates (wherein **X** = **x** and **W** = **w**) for schools such that

In this case, the conditional average causal effect *E*[*Y(z, v)*|*X* = *x*,**W** = **w**] − *E*[*Y*(*z*′*, v*′)|**X** = **x**, **W** = **w**] is equivalent to the observed data estimand, *E*[*Y*(*z*,*v*)|*Z* = *z, V* = *v,* **X** = **x***,* **W** = **w**] − *E*[*Y*(*z*′*,v*′)|*Z* = *z*′*, V* = *v*′*,* **X** = **x**, **W** = **w**]. The conditional average causal effect, being estimable from observed data, allows causal inference regarding the difference between students attending an SSE school versus a non-SSE school (^{Hong & Raudenbush, 2006}).

*v*(*Z*) = *V* was denoted as the random variable that takes on values *v*(*z*) = *v* = 0 for non-SSE policy schools or *v*(*z*) = *v* = 1 for SSE policy schools; **X** was assigned to be a vector of child-level covariates and **W** as a vector of school-level covariates.

Let *Q* be the probability of a school having a supportive environment. Conditioning on covariates, schools adopting a policy (*V* = 1 or *V* = 0) is influenced by *W* (school-level covariates) with a probability *Q* = *Q* (**W**); thus,

If the assignment of schools to a zero or non-SSE policy is ignorable given the observed school-level pretreatment covariates **W**, then an unbiased estimate of school *j*’s propensity of selecting an SSE policy, *Qj,* can be made as a function of **W***j*.

Let *q* be the probability that a student practices healthy habits according to the school SSE policy within the school that student attends. This could be written as

If student practice or nonpractice of healthy habits, under an SSE or a non-SSE policy, is ignorable given the observed student-level pretreatment covariates **X** and the school-level covariates **W**, then student *i*’s propensity of practicing healthy habits in school *j*, denoted by *q*_{1}*ij* if the school has adopted an SSE policy and by *q*_{0}*ij* otherwise, can be estimated as functions of **X***i* and **W***j*. Hence *q*_{1} = *q*_{1}(**X**, **W**) and *q*_{0} = *q*_{0}(**X**, **W**), wherein

A student’s probability of practicing a specific treatment (practice of healthy habits or nonpractice of healthy habits) under a specific school program (SSE or not) can be expressed as *P*(*Z = z, V = v*|**X**, **W**) = *P*(*Z = z|V = v*, **X**,**W**)*P*(*V = v*|**W**).

If these potential outcomes are considered carefully, it is clear that not all of the four potential outcomes are defined for some of the students. A student who may practice healthy habits in an SSE policy school may not have such a propensity in a non-SSE, and another student may never practice healthy habits (even within a school with an SSE policy). Monotonicity, described by ^{Angrist (1990)}, is applied here as the assumption that the probability of practicing healthy habits in an SSE policy school is always greater than or equal to that within a non-SSE policy school. To address this natural distinction into subpopulations of students, the monotonicity assumption regarding the student level propensity score estimates (*q*_{1} and *q*_{0}), wherein *q*_{1} denotes a student’s probability of practicing healthy habits under an SSE policy and *q*_{0} denotes the student’s probability of practicing healthy habits under a non-SSE policy was applied. As monotonicity is plausible in this case, three subpopulations of students of interest can be identified within the data set, noted as Subpopulations A, B, and C. The potential outcomes and the causal effects of interest for each of these subpopulations are listed in Table 1.

This results in the following causal estimands: The average causal effect of practice of healthy habits relative to nonpractice of healthy habits, conditioned on the covariates (**X** and **W**), under a non-SSE policy for students in Subpopulation A (students with propensity of practicing healthy habits in a non-SSE school), may be expressed as

Also, the average conditional practice of healthy habits effect under an SSE policy for students in Subpopulation A and those in Subpopulation B. Combining these subpopulations (A and B) produces a group of students who have a propensity for practicing healthy habits. Denoting this group of students as Subpopulation AB (A and B), this estimand is expressed as

For students in Subpopulation A, the difference between Equations (6) and (7) can be considered an estimation of the extent to which the causal effect of the individual-level practice of healthy habits depends on the school-level practice of healthy habits policy. Only one of the two propensity scores for every student can be estimated from observed data. This will affect the ability to estimate the causal effects defined in Equations (6) and (7).

After decomposition of Equation (6) as

it is clear to see that only the second term of this expression can be estimated for this population:

wherein δ_{Z0} is the conditional effect of healthy habits practice under a non-SSE policy for students attending non-SSE policy schools.

Following this logic with the estimates of conditional average practice of healthy habits effects for students with propensity to practice healthy habits subpopulation (AB), if Expression (7) was decomposed now as

then the second term, *E*[*Y*AB(1, 1) − *Y*AB(0,1)|*V* = 0, **X, W**], cannot be estimated directly from the observational adolescent health data, but the first term,

can be estimated. δ*Z*1 is the conditional effect of following healthy habits under the SSE policy for students attending SSE policy schools.

The research questions of interest were as follows: (a) What are the causal effects of a student’s practice of healthy habits in an SSE on the student’s BMI? (b) What are the causal effects of a student’s practice of healthy habits in a non-SSE on the student’s BMI? The multilevel approach to answer these questions accounts for the socioecological influence of students nested within schools, and propensity score estimation allows for a cause-and-effect estimation if assumptions are met.

## Methods

##### Sample

The National Longitudinal Study of Adolescent Health (Add Health; http://www.cpc.unc.edu/projects/addhealth/data) is a longitudinal study of a nationally representative sample of adolescents in Grades 7–12 in the United States that began during the 1994–1995 school year (^{Harris & Udry, 2009}). Data were used on a subsample of subjects from Wave I (1994–1995 school year) and Wave III (2001 year) and consisted of students sampled from Wave I (*n* = 13,854) and school administrators sampled from Wave I (*n* = 164). Protection of human subject rights was insured with institutional review board approval and a Sensitive Data Security Plan that was developed according to the strict guidelines of the Add Health Data Administrator at the University of North Carolina at Chapel Hill for the National Longitudinal Study of Adolescent Health data when stored on an external hard drive.

##### Variables

The treatment variables of interest in this research study occurred at the student and school levels. At the school level, the adoption of an SSE for healthy habits was the treatment of interest. This variable was constructed from responses by school administrators regarding school programs of nutrition and exercise. At the student level, the practice of healthy habits by a student was the treatment level variable of interest and was constructed from student responses regarding daily eating and exercise habits. The Wave I covariates that were hypothesized to affect either of the treatment variables at their corresponding level or the outcome variable of interest were included in the propensity score estimates for the assumption of ignorability of treatment assignment to hold (^{Gelman & Hill, 2006}). These variables at the school level included measures of parents’ average socioeconomic status, school district available resources in dollars and space, teacher and administrator demographics, geographic location of school, and type of neighborhood (rural, urban, suburban; Table 2). At the student level, the covariates used in the propensity score estimates included student age, gender, ethnicity, number of people in household, club affiliation, measures of self-worth, body image, behaviors regarding alcohol and tobacco, friends’ behaviors regarding alcohol, grades, and parental influence regarding teen’s choices and daily habits (Table 3). Wave III data collected on a subset of students from Wave I provided the outcome variable of interest calculated BMI from measured height and weight of student participants.

##### Analysis

The causal estimates of interest were obtained using logistic regression equations to compute the estimates of

in Excel 2010. To demonstrate the diagnostic properties of the propensity score estimates, common axis bar graphs were generated with STATA SE 11.

is each school’s conditional propensity of adopting an SSE policy given the observed school-level covariates (*W*_{1}, …, *Wn*). Seventeen covariates considered predictive of treatment or outcome (^{Drews et al., 2009}) were identified and included in the propensity score estimate. On the basis of the logit of

, the sample of schools were divided into five strata. Five subclasses are sufficient to remove at least 90% of the bias for many continuous distributions (^{Cochran, 1968}).

The estimate of *q*_{1} (the probability of practicing healthy habits within an SSE policy school) was estimated using a logistic regression model. A total of 65 student-level covariates considered predictive of treatment or outcome (^{Liou et al., 2010}; ^{Power, Bindler, Goetz, & Daratha, 2010}) were included in the propensity estimate of practice of healthy habits,

. In parallel to the estimation of *q*_{1}, *q*_{0} was estimated through a logistic regression model for students attending non-SSE policy schools using the same student-level covariates. Five strata for the logit of

also were constructed.

The initial propensity score estimates of

were examined for balance on covariates within each stratum. The goal of balancing is to verify that subclassification on the estimated propensity score removes any initial bias on confounders. Balance on covariates was checked using multiple two-way analyses of variance, wherein treatment (SSE school assignment or practice of healthy habits, for *Q* and *q*, respectively) was one factor, the propensity score strata to which the individual was assigned was a second factor (coded as a categorical variable with four levels), and each of the covariates (or confounders) was a dependent variable. These balancing steps were repeated until balance was achieved or until no further improvement in balance could be made (^{Yanovitzky, Zanutto, & Hornik, 2005}). However, reliance on significance testing to check for covariate balance is sensitive to sample size (^{Shadish, Clark, & Steiner, 2008}).

The minimum value of the logit of *q*_{1} (constructed from the data set) for the practice of healthy habits students attending SSE schools was then used as the cutoff point to identify students from Subpopulation C (students unlikely to practice healthy habits even in an SSE). These students were omitted from the causal analysis, because they did not have the potential outcomes related to the causal question of interest. Students whose logit of *q*_{1} was above the cutoff value were considered to belong to Subpopulations A or B (students who have a propensity to practice healthy habits) and were combined into a subpopulation of AB.

Similarly, students from non-SSE policy schools whose *q*_{0} logit was below the minimum cutoff for any practice of healthy habits students were excluded from the sample used in the causal estimates, leaving only students from Subpopulation A (students with a propensity to practice healthy habits even in non-SSE policy schools).

After the initial estimate of causal effects by propensity score stratification, the clustering of students within schools was modeled with two hierarchical linear models using HLM 6 software. The resulting models were estimates of δ_{z0}, a causal estimand of the effect of healthy habit practice relative to non-healthy habit practice on measured BMI scores for the students in Subpopulation A, and δ_{z1}, a causal estimand of effect of student non-healthy habits relative to healthy habits on measured BMI scores for the students in Subpopulation AB in an SSE. These models included an estimate of the variation of the practice of healthy habits effect across the students and schools by using a hierarchical two-level regression model, with student at Level 1 and school at Level 2.

## Results

As shown in a bar graph in Figure 1 representing the distribution of schools according to the logit of the *Q* (propensity for adopting SSE), most schools in the sample were non-SSE schools (*n* = 146) and the remaining were SSE schools (*n* = 18). Balance was achieved on over 95% of the school-level pretreatment covariates. On the basis of the logit of the *Q* estimate, the sample of schools (*n* = 164) was divided into five strata. The results of the analysis are displayed in Table 4. Schools with supportive environments had a predicted propensity of a supportive environment, with over 40% of the SSE schools in the highest predicted stratum. An overall higher logit of the *Q* estimate within stratum for the SSE schools was noted in contrast to the non-SSE schools. Schools with a propensity for SSE programs had, on average, more English-as-a-primary-language households and higher family household incomes. The non-SSE schools also had a lower percentage of schools with immunization programs and tended to be located in rural areas.

Student propensity for practice of healthy habits within a non-SSE school, *q*_{0}, was estimated through a logistic regression model for children attending non-SSE schools only. In Figure 2, students practicing healthy habits (*n* = 6,775) are represented across the upper half of graph and the nonpractice of healthy habits students (*n* = 5,581) is represented across the lower half of the graph. On the basis of the logit of the *q*_{0} estimate, the sample of students (*n* = 12,356) was divided into five strata. The students who practiced healthy habits and the students who did not were dispersed across all strata in this estimate, with no clear concentration of students in either area of the propensity strata. Balance was achieved on more than 95% of the student-level pretreatment covariates. Students who had lower propensity estimates to practice healthy habits were more likely to be students born outside of the United States, watched more hours of television, ate less meals per week with parents, had more friends who drank alcohol regularly, and made their own choices regarding time to sleep on school nights.

Student propensity for practice of healthy habits within an SSE school, *q*_{1}, was estimated through a logistic regression model for children attending SSE schools only. Balance was achieved on 95% of the student-level pretreatment covariates believed to effect the treatment or outcome variable. A graph representing the distribution of students according to the propensity score estimates of practicing healthy habits from an SSE school is presented in Figure 3. On the basis of the logit of the *q*_{1} estimate, the sample of students (*n* = 1,495) was divided into five strata. There was a higher propensity for students who practiced healthy habits to have more frequent medical and dental exams, to eat dinner with their parents, to get more hours of sleep, to feel safe in their schools, and to plan on attending college. The students who did not practice healthy habits had an increased hours of television watched, had an increased amount of video games played, reported more asthmatic conditions, ate nothing for breakfast often, and had a higher report of being left back a grade in school.

Empirical identification of subpopulations was accomplished by constructing mutually exclusive groups based on the estimated value of the logits of students to practice healthy habits. The value of −3.341 was used as the cutoff point to identify the 20 non-healthy habits students in SSE schools belonging to Subpopulation C, because there were no practice of healthy habits students with a logit less than or equal to −3.341. The minimum value of the logit of

was used as the cutoff point to identify the 21 students attending non-SSE schools belonging to Subpopulation C, bringing the Subpopulation C total sample size to 42. The remaining sample size for students in Subpopulation AB in SSEs was 1,475, and the sample size for students identified as belonging to Subpopulation A in nonsupportive schools was 12,335.

The raw difference in the average BMI score outcome was observed across strata for students in Subpopulation AB in non-SSE schools, and an estimate of the overall differences in BMI was calculated (Table 5). The mean differences in BMI were not significant within the strata. The raw difference in the average BMI score outcome was observed across strata for students in Subpopulation A in SSE schools, and the average mean difference was not significant (Table 6).

The causal estimands of interest δ*z*_{0} and δ*z*_{1} were then obtained with a hierarchical two-level regression model, students at Level 1 within school at Level 2 (Tables 7 and 8). The raw difference results in BMI outcomes were supported with the causal estimates of the effect of practicing healthy habits obtained with the hierarchical linear model estimates. The causal effect of practicing healthy habits within a non-SSE school, δ*z*_{0}, for students in Subpopulation A was 0.16 (*SE* = 0.14). The causal effect on BMI scores for students from Subpopulation AB, practicing healthy habits in SSE schools, δ_{z1}, was −0.18 (*SE* = 0.39).

## Discussion

The purpose of this research was to obtain causal estimates of the impact of healthy habits on individual students’ obesity outcomes with a nationally representative observational data set to demonstrate a methodology to estimate causal effects in a multilevel setting. To obtain these causal estimates, the distribution of covariates between the units of comparison at both school and student levels and the natural clustering of students within schools was considered in the potential outcome models with a relaxed form of SUTVA.

At the school level of this data, the estimated distribution of the logit for adopting an SSE policy allowed for comparison between types of schools. Most schools were non-SSE; the SSE-identified schools appear to have a dichotomous distribution, with approximately half of the schools profiling as private schools with high SES and resources and the other half profiling as larger public schools with lower income and resources. The private schools have tuition resources, and the federally and state funded programs may be funding the larger, economically less-advantaged public schools, resulting in the emergence of these two profiles of SSE schools.

At the student level of the sample, there were two propensity score estimates obtained empirically, the first being *q*_{0} from children attending non-SSE schools. The students within the non-SSEs who practiced healthy habits did differ from their non-healthy habits practice peers on many covariates of interest, but a strong profile did not emerge. The second student-level estimate, *q*_{1}, was obtained empirically from students who attended SSE schools. The students within these schools also differed when compared between students who practiced healthy habits and those who did not.

The results, although not statistically significant, support the finding that SSE has a positive effect on students with a propensity for practicing healthy habits in SSE and non-SSE schools. Random assignment to treatment allows for causal inference if the sample size is sufficient (^{Pearl, 2003}). This study had a larger sample size, in both levels of the treatment, than previous multilevel randomized control trials that studied the effect of exercise and eating habits on adolescents’ BMI and BMI *z* scores and unlike the randomized studies accounted for ignorability of treatment assignment at both levels of the model (^{DeBar et al., 2011}; ^{de Heer et al., 2011}; ^{Dzewaltowski et al., 2010}). The nonsignificant findings may be due to low power as the Level 2 sample size was only *n* = 18 for SSE schools. However, one strategic advantage of propensity score estimates for causal inference, over simple matching and regression techniques, is that it provides a one-factor covariate comprised of all other hypothesized predictor covariates, allowing for stratification on one identified covariate instead of 65 at the student level. More importantly, the nonsignificant findings may indicate that the causal relationship between the adoption of healthy habit behaviors at a student level, whether in an SSE or not, may not address the multidimensional influences on obesity outcomes fully (^{Maclean et al., 2010}). A dependent measure more proximal in time than the 6-year lag BMI used in this analysis would add to causal inference estimates. Body mass index is not considered by some experts to be the best measure for adolescent obesity, so a standardized BMI *z* score may be a more accurate measure of an individual teen’s obesity (^{Rausch et al., 2011}). However, although BMI and BMI *z* score are the most current measures of childhood obesity, these are not direct measures of body fat. Other measures such as percentage of body fat and bioelectrical impedance analysis are being proposed by some obesity researchers as more valid obesity assessment tools (^{Boylan et al., 2010}). Alternative or multiple obesity measures should be considered for further research. A continuous measurement level of SSE as opposed to a binary assignment may be helpful in adding information regarding the causal effect of varying levels of SSE on students’ BMIs. The results obtained with this research cannot be generalized to students in Subpopulation C. Subpopulation C will require a different causal model. Although small (*n* = 41), this subpopulation needs to be considered in healthy habits policy and program adoption research to decrease adulthood obesity. A further limitation of this research was that most variables at both levels were from self-reported survey data, with the exception of measured heights and weights for the outcome BMI variable.

Multilevel modeling can be achieved with multiple types of software, such as HLM, Mplus, MLwiN, SAS PROC MIXED, STATA, R, and IBM SPSS 19 and 20. Versions of HLM, like MLwiN and R, are available for free, an appealing consideration when allocating research funding. HLM allows for specification of the model at each level separately, such as at the student and school levels, and does not require the researcher to derive a single equation specification (^{Singer, 1998}). This can be quite helpful to the nonexpert quantitative researcher by providing explicit notation of the theorized variables at each level and avoiding the common omission of cross-level interactions. HLM does derive and display the singular multiple-level equation, in addition to the individual-level equations. HLM, like MLwiN, does not require command input in syntax, making it appealing to many nonstatistician researchers. There are manuscripts and books available that provide a thorough discussion of the various software products available to model multilevel research (^{Albright, 2007}; ^{Albright & Marinova, 2010}; ^{Hong & Raudenbush, 2006}; ^{Muthen & Muthen, 2012}; ^{Rasbash et al., 2009}; ^{Singer, 1998}).

Multilevel modeling accounts for the natural clustering of students within schools and can be used to estimate the variance components at each level of the model, allowing researchers to examine distinct phenomena and socioecological factors. However, multilevel modeling without a causal framework will not produce cause-and-effect relationship answers. The inclusion of propensity score estimates at both levels of the model, with a relaxed form of SUTVA at the student level, can meet the assumptions required to establish a cause and effect relationship.

Propensity score estimates for causal inference need to be used in consideration of potential limitations. Ignorability of treatment assignment is an untestable assumption for all causal inference modeling and may be violated if an unobserved relevant covariate is omitted from the model (^{Pearl, 2003, 2010}). Sensitivity analysis can provide information regarding plausibility of assumptions being tenable, but confidence in causal conclusions must be viewed by their consistency with findings of other evidence in addition to how sensitive the conclusions are to reasonable deviations. Propensity score estimates, like experimental studies, work better with larger samples, as a covariate in a small sample may be unbalanced enough between two groups to make the samples nonequivalent (^{Rubin, 1997}). Balancing can be observed with a variety of methods on observed covariates only (^{Yanovitzky et al., 2005} ). The SUTVA assumption may not be tenable in multilevel settings. This violation can be addressed by a relaxation of SUTVA as done in this study. The primary limitation with propensity score estimates, as with all causal inference, remains the inability to test if a relevant covariate was eliminated from the model (^{Pearl, 2003, 2010}; ^{Rubin, 1974, 1997}). Convergence of strong evidence over multiple studies supports causal conclusions.

Cause-and-effect relationships in a multilevel setting are often the relationships nurse researchers want to test. Various constraints of resources, ethics, and political opposition often preclude the design of treatment effect studies that meet the assumptions for causal inference. Propensity scores can provide these estimates in observational settings and can be produced with the use of spreadsheet software that has basic statistical functions, making a causal modeling approach of analysis readily available to most researchers.

## References

*Estimating multilevel models Using SPSS, Stata, and SAS*. Retrieved from http://www.indiana.edu/˜statmath/stat/all/hlm/hlm.pdf

*National Longitudinal Study of Adolescent Health (Add Health), 1994–2002*[Computer file]. ICPSR21600-v2. Ann Arbor, MI: Inter-university Consortium for Political and Social Research [distributor], 2009-03-11. doi:10.3886/ICPSR21600.

*Mplus user’s guide (v. 6)*. Los Angeles, CA: Author. Retrieved from http://www.statmodel.com/ugexcerpts.shtml

*Test*,

*12,*281–345. Retrieved from http://www.springer.com/statistics/journal/11749

**Keywords:**

adolescent; causal effects estimates; counterfactual; hierarchical linear modeling; nutrition; pediatric obesity prevention; potential outcomes