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Longitudinal Dynamic Analyses of Depression andAcademic Achievement in the Hawaiian High SchoolsHealth Survey Using Contemporary Latent VariableChange ModelsJohn J. McArdlea, Fumiaki Hamagamib, Janice Y. Changc & Earl S. Hishinumac

a University of Southern Californiab Hawai’i State Hospitalc University of Hawai’i, MānoaPublished online: 08 Aug 2014.

To cite this article: John J. McArdle, Fumiaki Hamagami, Janice Y. Chang & Earl S. Hishinuma (2014) Longitudinal DynamicAnalyses of Depression and Academic Achievement in the Hawaiian High Schools Health Survey Using ContemporaryLatent Variable Change Models, Structural Equation Modeling: A Multidisciplinary Journal, 21:4, 608-629, DOI:10.1080/10705511.2014.919824

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Structural Equation Modeling: A Multidisciplinary Journal, 21: 608–629, 2014Copyright © Taylor & Francis Group, LLCISSN: 1070-5511 print / 1532-8007 onlineDOI: 10.1080/10705511.2014.919824

Longitudinal Dynamic Analyses of Depression andAcademic Achievement in the Hawaiian High Schools

Health Survey Using Contemporary Latent VariableChange Models

John J. McArdle,1 Fumiaki Hamagami,2 Janice Y. Chang,3 and Earl S. Hishinuma3

1University of Southern California2Hawai’i State Hospital

3University of Hawai’i, Manoa

The scientific literature consistently supports a negative relationship between adolescentdepression and educational achievement, but we are certainly less sure on the causal deter-minants for this robust association. In this article we present multivariate data from alongitudinal cohort-sequential study of high school students in Hawai‘i (following McArdle,2008; McArdle, Johnson, Hishinuma, Miyamoto, & Andrade, 2001). We first describe the fullset of data on academic achievements and self-reported depression. We then carry out andpresent a progression of analyses in an effort to determine the accuracy, size, and directionof the dynamic relationships among depression and academic achievement, including genderand ethnic group differences. We apply 3 recently available forms of longitudinal data analy-sis: (a) Dealing with incomplete data—We apply these methods to cohort-sequential data withrelatively large blocks of data that are incomplete for a variety of reasons (Little & Rubin,1987; McArdle & Hamagami, 1992). (b) Ordinal measurement models (Muthén & Muthén,2006)—We use a variety of statistical and psychometric measurement models, including ordi-nal measurement models, to help clarify the strongest patterns of influence. (c) Dynamicstructural equation models (DSEMs; McArdle, 2008). We found the DSEM approach takenhere was viable for a large amount of data, the assumption of an invariant metric over timewas reasonable for ordinal estimates, and there were very few group differences in dynamicsystems. We conclude that our dynamic evidence suggests that depression affects academicachievement, and not the other way around. We further discuss the methodological implicationsof the study.

Keywords: academic achievements (GPA), cohort sequential design, depression (C-ESD), dynamic structural equations (DSEM), incomplete data, Mplus Software, ordinalmeasurements

Many important studies in psychology deal with the rela-tionships among some forms of psychopathology connectedwith real-life outcomes. Questions about the likely sequence

Color versions of one or more of the figures in the article can be foundonline at www.tandfonline.com/hsem.

Correspondence should be addressed to John J. McArdle, University ofSouthern California, Department of Psychology, Los Angeles, CA 90089,USA. E-mail: jmcardle@usc.edu

of effects are often a fundamental aspect of developmentalresearch (e.g., Ferrer & McArdle, 2010; Flaherty, 2008;McArdle, 2008). In cross-cultural research, questions raisedabout cultural differences in beliefs about normal and clinicalpsychological behaviors have led some researchers to con-clude that both measurement and treatment should be cul-ture specific (Higgenbothman & Marsella, 1988; Werner& Smith, 2001). However, other researchers using differ-ent analytic approaches seem to have come to notably

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DYNAMIC ANALYSES OF DEPRESSION AND ACHIEVEMENT 609

different conclusions (e.g., regarding measurement, seeBeals, Manson, Keane, & Dick, 1991; Johnson, Danko,Andrade, & Markoff, 1997). A key question raised from suchanalyses is how much of the resulting differences can beattributed to the people themselves, and how much of thesedifferences can be attributed to the methods of data analysisused. We consider whether different results can come fromdifferent forms of analysis.

In this article we present multivariate data from a lon-gitudinal cohort-sequential study of high school students inHawai‘i (see McArdle, 2008; McArdle, Johnson, Hishinuma,Miyamoto, & Andrade, 2001). We first describe the full setof data on academic achievements and self-reported depres-sion. We then carry out and present a progression of analysesin an effort to determine the accuracy, size, and directionof the gender and ethnic group differences in the dynamicrelationships among depression and academic achievements.We apply three recently available forms of longitudinal dataanalysis:

1. Dealing with incomplete data. We apply these methodsto cohort-sequential data with relatively large blocks ofdata that are incomplete for a variety of reasons (Little& Rubin, 1987; McArdle & Hamagami, 1992).

2. Ordinal measurement models (Hamagami, 1998;Muthén & Muthén, 2006). We use a variety of statis-tical and psychometric measurement models, includ-ing ordinal measurement models, to help clarify thestrongest patterns of influence.

3. Dynamic structural equation models (DSEMs;McArdle, 2008. The DSEMs are used here to providea formal assessment of the possible directionality ofeffects from observational data, and we include groupdifferences in dynamic systems.

After the full set of dynamic analyses is presented we con-clude that our dynamic evidence suggests that depressionaffects academic achievement, and not the other way around.

PRIOR RESEARCH ON DEPRESSION ANDACADEMIC ACHIEVEMENT

Despite the enormous amount of research that has beenconducted on educational achievement, only a small propor-tion of this research body has investigated the associationbetween academic achievement in the form of academicachievements in school and individual psychological adjust-ment such as depressive symptoms. A parallel publication tothis study is by Hishinuma, Chang, McArdle, and Hamagami(2012). Their Table 1 is an overview of previously publishedresearch, and this summary suggests prior research has foundsignificant but small, negative associations between indica-tors of academic achievement, such as high school grade-point average (GPA), and depressive symptoms. It seems

that when the GPA range was not restricted and the depres-sive symptom measure was not combined with anxiety, thecorrelation between GPA and depressive symptoms rangedfrom –.16 to –.36 (see their Table 1), with only a few excep-tions. This negative relationship between GPA and depres-sive symptoms was further confirmed by other studies. Forexample, Juvonen, Nishina, and Graham (2000), using struc-tural equation modeling (SEM) and a middle school sample,found evidence for perceived peer harassment leading topsychological distress (i.e., depressive symptoms as mea-sured by the Children’s Depression Inventory, self-worth),which in turn led to poor school adjustment (i.e., actualGPA, number of classroom hours missed during semesterof data collection). This suggested that depressive symptomsoccurred before lowered GPAs, although this study was notlongitudinal in nature.

In attempting to determine any causal relationshipsbetween GPA (school achievement) and depressive symp-toms (psychological adjustment), it is likely that lon-gitudinal studies will be needed (see Duncan, 1975;Flaherty, 2008; McArdle, 2008). For example, Lehtinen,Raikkonen, Heinonen, Raitakari, and Keltikangas-Jarvinen(2006) assessed parents of Finnish students when the stu-dents were 9, 12, and 15 years of age, and then assessedthe students when they were 21 to 36 years of age. Theresearchers found an overall negative association betweenGPA and later depressive symptoms, but no consistent rela-tionship across age and gender. In particular, 12- and 15-year-old girls’ lower GPA predicted depressive symptoms12 and 17 years later. Although GPA was thought to be thepredictor of the depressive symptoms, the following conclu-sion was drawn: “The current findings, thus, suggest thatdepressive outcomes have a detrimental effect on success atschool rather than vice versa: presumably, failure to copewith challenges at school could lead to different kinds ofvicious circles, in which case deterioration of school per-formance would serve as a risk factor for adult depressiveoutcomes” (Lehtinen et al., 2006, p. 289).

A variety of other relevant studies are reviewed byHishinuma et al. (2012). In general, only two studies couldbe identified that included both GPA and depressive symp-toms in a longitudinal design that involved examining trajec-tories. Repetto, Caldwell, and Zimmerman (2004) surveyedAfrican American high school students from the 9th gradewith GPAs of 3.0 or lower, at risk for high school dropout,and not diagnosed with emotional impairment or devel-opmental disability. Depressive symptoms (six items fromthe Brief Symptom Inventory) and GPA (8th–10th grades)were utilized; however, the analyses did not include incom-plete data. The students were clustered in four groups basedon their longitudinal depressive symptoms: (a) consistentlyhigh, (b) consistently low, (c) decreasing, and (d) increas-ing. Using multivariate analysis of variance (MANOVA),students who were consistently high in depressive symp-toms were more likely to be female, have anxiety and stress,

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610 MCARDLE, HAMAGAMI, CHANG, HISHINUMA

TABLE 1Summary of Available Data in the Hawaiian High Schools Health Survey (HHSHS) Collection Waves by Year, High School, and Grade Level

School Year

1 2 3 4 5

High School Grade Level 1991−92 1992−93 1993−94 1994−95 1995−96 Total

1 9 103 70 100 27310 88 82 76 24611 75 70 84 22912 75 47 66 71 259Incompletea 0 0 1 1

2 9 119 110 118 34710 94 69 100 26311 85 56 79 22012 71 37 54 23 185Incompletea 0 0 3 3

3 9 354 364 290 1, 00810 315 335 326 97611 293 319 297 90912 318 304 298 296 1, 216Incompletea 0 2 3 1 6

4 9 417 385 80210 388 371 75911 329 312 327 96812 229 275 294 310 1, 108Incompletea 1 4 4 9

5 9 265 288 55310 243 244 48711 251 220 221 69212 172 185 194 205 756Incompletea 4 3 2 9

Total 1, 990 4, 164 4, 182 1, 433 515 12, 284

Note. N = 7,317; D = 12,284 surveys completed.aStudents who did not provide their grade level within the survey.

and have lower self-esteem and GPA. In independent work,W. Johnson, McGue, and Iacono (2006) examined academicachievement trajectories in adolescents in the MinnesotaTwin Family Study. Although data on a composite mea-sure of “grades” were collected across multiple years, thedepression indicator was collected only in Year 1 and wasnot significantly associated with the measure of academicachievement.

The issues surrounding the scaling of variables havegenerated a great deal of prior research on these kinds ofmeasures. Scales of achievement have been developed thathave interval properties (e.g., the Woodcock–Johnson scales,the American College Testing [ACT], etc.; see McArdle &Woodcock, 1998), but easy-to-obtain scales such as GPAhave engendered lots of criticism. “In college, for example,instructors grade students on an ordinal scale, A, B, C, D,F, and the registrars assign to these grades the numbers 4,3, 2, 1, 0 respectively, in order to compute the ‘grade pointaverage.’ Clearly such assignments are arbitrary” (Duncan,1975, p. 159).

Similarly, the use of the 20-item scale of the Center forEpidemiologic Studies–Depression (CES–D; Radloff, 1977)

was critiqued soon after it started to be used (Devins &Orme, 1985). Although group comparisons on the CES–D scale have been around for a long time (e.g., Gatz &Hurwicz, 1990), the more recent work of Cole, Kawachi,Maller, and Berkman (2000) pointed out significant item biasbetween African American and White American responsesto the CES–D items (also see Yang & Jones, 2007). In ourown prior work (McArdle et al., 2001) we summarize theprevious factor analytic work on the depressive symptomscales and find evidence for at least three or four measurablefactors in the CES–D (also see Beals et al., 1991). In thisprior work, we also show how a simple form of rescalingof the CES–D can lead to a simple and direct interpreta-tion of resulting score as “the number of days of depressionsymptoms reported per week” (WCES–D).

On a theoretical note, we are aware that the GPA andCES–D scores have substantially different norms of refer-ence. That is, the CES–D is based on the same set of itemsmeasured on the same rating scale, so it is possible to exam-ine both systematic growth and rapid changes in a commonscale. However, GPA is not an ability score with a freerange; instead, GPA is a score that is relative to a grade

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DYNAMIC ANALYSES OF DEPRESSION AND ACHIEVEMENT 611

norm—that is, even if an individual presents a constant GPAacross Grades 9 through 12, they are likely to be growingin the ability underlying the academic performance. In thisway, GPA is much like an IQ (i.e., mental age/chronologicalage) score rather than a raw test score (i.e., mental age), andthis limits our interpretations of growth and change that canemerge from the analysis of GPA trajectories.

Based on the overall literature, the past research on thistopic seems limited in several respects:

1. The sample sizes are relatively small.2. The samples have not involved large samples of Asian

American and Pacific Islander adolescents.3. The literature shows little consideration of longitudinal

data on GPA and depressive symptoms.4. The literature shows little consideration of the issues of

incomplete data in analyses (e.g., Repetto et al., 2004)and in SEM (Juvonen et al., 2000).

5. There is little consideration given to the measurementor scaling of the key outcome variables as part ofdata analysis, even though it is known that scalingof measurements is critical to understanding lead-lagrelationships.

6. More recent SEM analyses have been presented usingmore contemporary growth and trajectory modeling(e.g., Hong, Veach, & Lawrenz, 2005; Johnson et al.,2006; Juvonen et al., 2000; Lehtinen et al., 2006;Repetto et al., 2004; Shahar et al., 2006).

The approach taken here attempts to combine and over-come problems of all six types. The models used here are allbased on incomplete longitudinal data, and the DSEM anal-ysis approach has been used in several recent studies (seeFerrer & McArdle, 2010; Gerstorf et al., 2009; McArdle,2008). In examples related to this specific problem, McArdleand Hamagami (2001) examined longitudinal data from theNational Longitudinal Study of Youth (NLSY) and con-cluded that reading ability was a precursor to antisocialbehavior. In a more extensive study using the NLSY, Grimm(2007) studied academic achievement and depression, andhe concluded that academic achievement preceded the devel-opment of depression. These researchers all emphasize thatpeople using the same data could come to different dynamicconclusions using different multivariate models. Therefore,we examine a few alternatives based on different forms ofmeasurement models here.

THE HAWAIIAN HIGH SCHOOLSHEALTH SURVEY STUDY

A wide variety of psychosocial problems are being studiedby the National Center on Indigenous Hawaiian BehavioralHealth (NCIHBH), including the relationships between psy-chological health and culture. Data from a variety of sources

indicate that persons of aboriginal Hawaiian ancestry aremore “at risk for psychological problems” than residents ofHawai‘i of other ethnic backgrounds (e.g., Andrade et al.,2006; Danko et al., 1988; Werner & Smith, 2001; Yuen,Nahulu, Hishinuma, & Miyamoto, 2000). The aim of theHawaiian High Schools Health Survey Study (HHSHS)was to assess the adjustment of a large community-basedsample of adolescents of Hawaiian ancestry as well asa comparison sample of adolescents from other ancestralgroups. The longitudinal approach used in prior work in theHawaiian islands (Werner & Smith, 2001) partly inspired thiscohort-sequential data collection design. The study furtherinvestigated the importance of demographic variation andmeasures of identification with the Native Hawaiian culturewith indices of adjustment.

The NCIHBH has already conducted studies involvingself-reported GPA and adjustment, using a primarily AsianAmerican and Pacific Islander high school sample. A sta-tistically significant and negative relationship of –.18 (p <.05) was found between self-reported GPA and the CES–D(Hishinuma et al., 2006; Hishinuma, Foster, et al., 2001).This negative association remained significant even whencontrolling for other demographic variables (i.e., ethnicity,gender, grade level, main wage earners’ educational attain-ment, main wage earners’ employment status) and academicmeasures (i.e., actual cumulative GPA, absolute differencebetween self-reported GPA and cumulative GPA).

This negative relationship found among GPA and CES–D scores was particularly salient for children and adoles-cents of ethnic minority ancestry (Hishinuma et al., 2006;Hishinuma, Foster, et al., 2001). Given the general find-ing of lower educational achievement being associated withpoorer psychological adjustment (e.g., Hishinuma, Foster,et al., 2001), and given the differential academic achievementamong racial and ethnic groups (e.g., African Americanswith lower high school GPAs than Whites; Whites withlower high school GPAs than Asian-Pacific Islanders), thisarea of study is critical in understanding the well-beingof ethnic minorities. In the context of this study, NativeHawaiian adolescents tend to have lower levels of achieve-ment than non-Hawaiians in Hawai‘i (e.g., Hishinuma et al.,2006; Hishinuma, Foster, et al., 2001; Hishinuma, Johnson,et al., 2001).

DYNAMIC MODELINGWITH INCOMPLETE DATA

The previous HHSHS analyses described the use of a mul-tiple imputation (MI) procedure for handling incompletedata (McArdle et al., 2001). This approach is reasonablefor data missing within or over time, and it can be usedwhen the incompleteness is due to attrition and other fac-tors and the lack of data are considered missing at random(MAR; after Little & Rubin, 1987). The same assumptions

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underlie analyses based on any latent variable structuralequation model that included “all the data”—not simplythe complete cases (e.g., Horn & McArdle, 1980; McArdle,1996). We expect nonrandom attrition, but our goal is toinclude all the longitudinal and cross-sectional data to pro-vide the best estimate of the parameters of change as ifeveryone had continued to participate (Diggle, Liang, &Zeger, 1994; Little, 1995; McArdle & Anderson, 1990;McArdle & Bell, 2000; McArdle & Hamagami, 1991;McArdle, Prescott, Hamagami, & Horn, 1998). In compu-tational terms, the available information for any participanton any data point (i.e., any variable at any occasion) is usedto build up maximum likelihood estimates (MLEs) that opti-mize the model parameters with respect to any available data.

Another aspect of the methods used here is the investiga-tion of a measurement model for each key outcome variable.In the case of GPAs, the scores are first treated in the typ-ical way, as an interval scale, but due to its more uniformdistribution an effort is made to consider this more realisti-cally as an ordinal scale. Likewise, the CES–D measurementof depressive symptoms is considered to be multifactorial,so only some items are selected, and then due to the result-ing skewness, models with both interval scaling and ordinalscaling are considered here. In addition, the requirement ofinvariance of measurement over high school grades is con-sidered for each outcome separately in latent curve models,and then together in the dynamic systems models. Thesefinal dynamic models are compared over groups defined bygender and ethnicity.

METHOD

Participants

In previous analyses, we selected data from any student whoparticipated in the HHSHS in 1993 and 1994 (N > 5,000;e.g., McArdle et al., 2001). The analyses presented here arebased on survey data obtained from high school students inHawai‘i who participated in at least one of five waves of mea-surement (1992–1996; see Andrade et al., 2006; Andradeet al., 1994). In this longitudinal-cohort study, a large sam-ple (N = 7,317) of high school students were surveyed ona variety of well-known psychometric instruments designedto indicate different aspects of psychological health. Thesedata reflect a large but generally nonrepresentative epidemi-ological sample of high school students; namely, in this studywe intentionally selected primarily Hawaiian communities inGrades 9 through 12 from three of Hawai‘i’s major islands.

For the purposes of those analyses, we classified allstudents into one of four broad groups. Hawaiian versusnon-Hawaiian ancestry was based on questions about theparents’ ethnic background. Students whose parents hadany Hawaiian ancestry were classified as Hawaiian and allothers were classified as non-native Hawaiians. A second

grouping was based on the students’ self-reported gender(male or female). These two groupings led to four stu-dent groups: (a) Native Hawaiian females (n = 1,733), (b)Native Hawaiian males (n = 1,456), (c) non-native Hawaiianfemales (n = 973), and (d) non-native Hawaiian males (n =881). Additional information about these four groups canbe found in other more detailed reports (see Andrade et al.,2006). The same HHSHS data collection is used here, butnow we use all the longitudinal data available at any wave ofmeasurement.

Procedures

Parents and students were given written notification of thenature and purpose of the research study prior to admin-istration, with the opportunity for parents to decline theiryouths’ participation. Students who provided their assentwere administered the survey in their homerooms by theirteachers. The majority of the surveys were completed bythe students within 30 to 45 min. Based on the existingenrollments, approximately 60% of the students were sur-veyed. Separate analyses indicated that a higher proportionof males were not surveyed. In addition, those who werenot surveyed had more absences, suspensions, and conductinfractions, and had lower actual GPAs (Andrade et al.,2006). All procedures were approved by the University ofHawai‘i’s Institutional Review Board.

Patterns of Available Longitudinal Data

This investigation entailed a cross-sequential (i.e., cross-sectional and longitudinal) design whereby students fromtwo to five high schools (located on three of the HawaiianIslands) were surveyed across a 5-year period (school years1991–1992 to 1995–1996). The initial plan was to over-sample Native Hawaiian adolescents by surveying studentsfrom three high schools (High Schools 1–3) that had a largeproportion of youths of Native Hawaiian ancestry. Table 1provides a description of the patterns of responses at the indi-vidual level (see also Andrade et al., 2006) and Figure 1 is aplot of a small sample of trajectories from this larger longi-tudinal database. Figure 1a is a plot of the trajectories of theGPAs of 30 randomly selected individuals, and Figure 1b is aplot of the same 30 people on their CES–D scores. Complexpatterns of fluctuation are evident and the connection amongthese trajectories is not obvious.

The HHSHS data used here are based on a complexsampling strategy highlighted in Table 1. This survey wasconducted for the first three years of the study (1991–1992 to1993–1994 school years) and for all high school grade lev-els (9–12). During Year 1 (1991–1992), the decision wasmade to also survey in Year 2 (1992–1993) students fromtwo other high schools (High Schools 4–5) that would allowfor more meaningful comparisons with non-Hawaiian ado-lescent cohort groups. As with High Schools 1 through 3,

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DYNAMIC ANALYSES OF DEPRESSION AND ACHIEVEMENT 613

9 10 11 121

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2

2.5

3

3.5

4(a)

(b)

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GPA

9 10 11 121

2

3

4

5

Grade Level

CE

SD

FIGURE 1 Observed longitudinal trajectories on the two key indicatorsvariable for a random subset of n = 30 persons. (a) Observed longitudi-nal trajectories for grade-point average (GPA). (b) Observed longitudinaltrajectories for Weekly Center for Epidemiologic Studies–Depression Scale(WCES–D).

students from High Schools 4 and 5 were surveyed for allhigh school grade levels (9–12) in Years 2 and 3 (1992–1993& 1993–1994). To obtain complete longitudinal data acrossall four grade levels for the 9th graders who were surveyedin Year 1 (1991–1992) for High Schools 1 through 3, and forthe 9th graders who were surveyed in Year 2 (1992–1993) forHigh Schools 4 and 5, the decision was made to: (a) in Year4 (1994–1995) survey the 12th graders from High Schools1 through 3, and (b) in Year 4 (1994–1995), survey 11thand 12th graders from High Schools 4 and 5, and in Year 5(1995–1996) survey 12th graders from High Schools 4 and 5.

Among students who were in the 9th grade, 3,644 (50.3%)never had an opportunity to take the survey, 2,938 (40.6%)took the survey, and 660 (9.1%) had the opportunity to takethe survey, but did not for whatever reason (e.g., parentdeclined, student declined, student moved to another school).Seventy-five participants were not included in the previouscount due to anomalies such as repeating a grade level. Forthe 10th grade, 3,286 (45.4% of students that had no oppor-tunity to take the survey), 2,707 (37.4% of students that tookthe survey), 1,249 (17.3% of students that had an opportunity

but did not take the survey), respectively; for the 11th grade,2,685 (37.1%), 2,984 (41.2%), 1,573 (21.7%), respectively;and for 12th grade, 1,822 (25.2%), 3,498 (48.3%), 1,922(26.5%), respectively.

This series of data collection decisions and methodsresulted in the project starting during the 1991–1992 schoolyear, ending during the 1995–1996 school year, and variablysurveying 9th through 12th graders across five high schoolsat different points in time. A total of 7,317 students were sur-veyed resulting in 12,284 completed questionnaires. We notethat the potential for a lack of convergence of these data(McArdle & Hamagami, 1991) is considered in all analyses.

Measures

Self-reported GPA. This variable was operationallydefined by a single survey question: “On the average, whatwere your grades on your last report card?” with 10 responsechoices offered. This measurement of academic achievementdemonstrated high concurrent validity with actual cumu-lative GPA in a substudy of the same persons (r = .76,Hishinuma, Johnson, et al., 2001) and construct validity withadjustment indicators, arrests or serious trouble with thelaw, and substance use (Hishinuma et al., 2006; Hishinuma,Foster, et al., 2001).

In subsequent calculations, we reconstructed this GPAvariable using numerical values (in parentheses): A (4.0), A–(3.7), B+ (3.3), B (3.0), B– (2.7), C+ (2.3), C (2.0), C– (1.7),D or less (1.0), or Don’t know (converted to a missing score).Table 2 gives the percentage of persons responding to each of

TABLE 2Summary of Estimated Categorical Data Percentages

Category Interpretation Grade 9 Grade 10 Grade 11 Grade 12

Observed GPA category percentages1 A (4.0) 5.9 5.1 3.4 2.22 A– (3.7) 6.0 5.3 4.9 3.43 B+ (3.3) 16.0 14.0 12.0 9.04 B (3.0) 12.2 12.9 12.0 10.15 B– (2.7) 10.4 10.7 11.9 11.96 C+ (2.3) 20.4 21.8 20.7 20.37 C (2.0) 11.0 14.7 16.2 17.48 C– (1.7) 9.4 9.6 11.8 15.09 D or less (1.0) 8.7 5.9 7.1 10.8

Observed weekly CES–D category percentages (rounded weekly ratings of13 items)1 0–1 days 64.2 67.2 66.9 68.62 1–2 days 21.3 18.7 20.3 19.93 2–3 days 8.1 8.0 7.5 6.94 3–4 days 4.1 4.1 3.6 3.35 4–5 days 1.8 1.8 1.4 1.16 5–7 days 0.4 0.2 0.3 0.2

Note. GPA = grade-point average; CES–D = Center for EpidemiologicStudies–Depression scale. For sample sizes, see Table 1.

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614 MCARDLE, HAMAGAMI, CHANG, HISHINUMA

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FIGURE 2 Observed within-grade-level distributions of grade-point average (GPA) for all persons (N > 7,000).

these categories, and here we find a wide spread of responsesat each year, with the highest proportion (> 20%) of studentssuggesting they had a C+ average each year. As can be seenin Table 2, we also found lower proportions of A averagesand greater proportion of D averages over successive grades.We note that any selection due to flunking out of high schooldue to low GPA is not specifically indicated in these data(except perhaps by the earlier GPAs). Figure 2 includeshistograms describing these GPA data at each grade.

Self-reported depression. In one section of the 30- to45-min survey, all students were asked to rate their depres-sive symptoms using the well-known CES–D inventory (seeRadloff, 1977; Zimmerman & Coryell, 1994). In the stan-dard scoring system, the 20 CES–D items are scored on ascale ranging from 0 to 3 and these scores are summed overthe 20 items to form a CES–D total score ranging from 0 to60 (see Santor & Coyne, 1997). As mentioned earlier, a prac-tical scoring system for the CES–D was created and used inour earlier work (in McArdle et al., 2001), and we expandon this process here. The typical CES–D item responses arebased on “the number of days per week,” with four responsecategories: 1 (rarely or none of the time [< 1 day]), 2 (someor a little of the time [1–2 days]), 3 (occasionally or a mod-erate amount of the time [3–4 days]), and 4 (most or all ofthe time [5–7 days]). It is possible to simply treat this asa 1-to-4 scale. As an alternative, it is also possible to alter

the scale points to more directly reflect the average numberof days per week for each symptom (so 1 = 0.5 days, 2 =1.5 days, 3 = 2.5 days, and 4 = 6 days).

The unidimensionality of the CES–D items has beenquestioned in prior research (Beals et al., 1991; Dick, Beals,Keane, & Manson, 1994; Radloff, 1977). In our own workon this topic (McArdle et al., 2001), we found more thanone common factor was needed to account for all inter-relationships among the CES–D items. In these analyses,the first common factor was indicated by the most items(13) and these items reflected the basic construct of depres-sion. Therefore, we considered only these 13 items here.To simplify the overall scaling, but retain the individual dif-ferences across time in these scores, we first (a) altered allitems to be in the metric of days per week, then (b) aver-aged over the responses to the unidimensional items (13 outof 20), and finally (c) rounded the resulting averages into sixreasonably sized categories of days per week (average daysper week = 0–1, 1–2, 2–3, 3–4, 4–5, and 5–7). Incompletedata were not placed in any specific category.

Table 2 gives the percentage of persons responding toeach of these rounded six categories, and it is clear thatthe scale is highly skewed. The first category (0–1 days)includes over 64% of the responses, and the last category(5–7 days) includes less than 0.5% of responses withinany year. Figure 3 includes histograms describing theseWCES–D data at each grade level. This skew seems rather

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DYNAMIC ANALYSES OF DEPRESSION AND ACHIEVEMENT 615

2 4 60

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FIGURE 3 Observed within-grade-level distributions for Weekly Center for Epidemiologic Studies–Depression Scale (WCES–D) for all persons (N >

7,000).

severe, but a likely reflection of the construct of depressivesymptoms (i.e., with no positive scores); therefore, we try toexamine this variable with this statistical issue in mind.

Modeling With Incomplete Data

Incomplete data techniques are now available in manycurrent computer programs (e.g., SAS PROC MIXED;Littell, Milliken, Stroup, Wolfinger, & Schabenberger, 2006;Verbeke & Molenberghs, 2000), and we use Mplus 5.0(Muthén & Muthén, 2006; see Ferrer, Hamagami, &McArdle, 2004). The Mplus program seems advantageousbecause it also allows us to deal with (a) survey samplingweights, (b) categorical measurement models, (c) multilevelmodels, and (d) a random-slopes approach to latent curvemodels (but see Ghisletta & McArdle, 2012). The goodnessof fit of each model presented here will be assessed usingclassical statistical principles relying on the model likelihood(f MLE) and change in fit (χ2). In most models to follow, weuse the MAR convergence assumption to deal with incom-plete longitudinal records, and we discuss these assumptionslater (e.g., Cnaan, Laird, & Slasor, 1997; Little, 1995).

These statistical models are also used to address groupdifferences about the CES–D and GPA at several differentlevels, and comparative statistical results are presented inthe next four sections. Due to our relatively large sample

size (N = 7,317), only estimated coefficients with a 99%confidence boundary that does not include zero are consid-ered accurate estimates of effects. Although this is preciselyequivalent to stating an effect is not significant (at the α <.01 test level), this emphasis on the accuracy of estimatesis more consistent with our overall modeling strategy. Theoverall goodness of fit of these models can also be exam-ined in several ways (see Bollen & Long, 1993). However,most current SEM analyses rely on comparing observed andexpected values based on a noncentral chi-square (χ2) distri-bution (paralleling the F-distribution of analysis of variance).Due to the relatively large sample size (N > 7,000), wealso examine statistical differences between models usingthe Browne and Cudeck (1993) root mean square error ofapproximation (RMSEA, εa) and we calculate the “proba-bility of close fit” to indicate a model with a discrepancyεa < .05. Our main goal is to use these empirical analy-ses to separate (a) models that seem consistent with the dataand should be useful in future research, from (b) models thatseem inconsistent with the data and should be dropped fromfurther work.

Dealing With Nonnormal Outcomes

The evaluation of models for change over time is conceptu-ally based on longitudinal analyses of multiple trajectories

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616 MCARDLE, HAMAGAMI, CHANG, HISHINUMA

(e.g., McArdle, 2008). There is a long history of embed-ding categorical concepts within repeated measures lon-gitudinal data (e.g., Hamagami, 1998; Koch, Landis,Freeman, Freeman, & Lehnen, 1977; Muthén, 1984). Theseapproaches can clarify the results if we have either (a) cat-egorical outcomes that are not ordinal, or (b) if the ordercategories are not equal in the interval between scale points.There are many techniques that can be used to deal with non-normally distributed outcomes and we will not be exhaustivehere.

One classic way to deal with nonnormal outcomes is touse score transformations, but these will not help here due tothe extreme limits of some of our outcomes (e.g., Table 2).Instead, we first use standard SEM approaches for estimatinggrowth and change under the assumption of an interval scal-ing of the outcomes (as in McArdle et al., 2001). This is notoften based on MLE–MAR assumptions, but it is possible.In a parallel set of analyses we consider the same longitudi-nal data but we expand the model to include a set of ordinalthresholds (τ ; see Hamagami, 1998) and we highlight anydifferences in the substantive results for the same data.

To carry out calculations for the ordinal approach, werely on the approach created by Muthén (for LISCOMP soft-ware; see Muthén & Satorra, 1995). In this approach, the firststep is to use MLE–MAR to estimate the empirical distancebetween each entry of a categorical variable. Because wehave nine possible ratings of GPA at each occasion, we canestimate eight thresholds describing the difference betweenthese categories. Because we have only six possible cat-egories of weekly CES–D, we estimate five thresholds todescribe the difference among these categories. In the secondstep, we assume the underlying latent variables are normallydistributed (with mean 0 and variance 1) and we estimatethe correlations among latent variables using polychoric pro-cedures. Finally, in a third step, a consistent estimator ofthe asymptotic covariance matrix of the latent correlationsis based on a weighted least squares mean and variance(WLSMV) adjusted estimator. Additional model assump-tions based on the structure of the thresholds (i.e., invariantover time) or the model correlations (following a latent curvehypothesis) are added and the comparative goodness of fit isobtained. If needed, formal tests of the difference betweenmodels are made more accurate by using kurtosis adjust-ments (scaling correction factors; see Browne, 1984; Satorra& Bentler, 1994). In any case, each of these steps is nowfairly easy to implement with contemporary software suchas Mplus (Muthén & Muthén, 2006).

Methods for Univariate Dynamic SEM Analyses

All analyses presented will be based on longitudinal struc-tural equation models (e.g., McArdle & Nesselroade, 2014,in press; Singer & Willett, 2003). To be clear, we willemphasize the representation of trajectories over time and thedynamic interpretations of model parameters (see McArdle,

X[1]

τ

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αα α

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x [4]x [3]x [2]x [1]

x* [1] x* [2] x* [3] x* [4]

g1

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1σg0

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FIGURE 4 A univariate dual change score model to examine trajecto-ries on grade-level changes for each indicator (X). The model includes anestimate for intercept (g0), mean intercept (μ0), and mean slope (μ1). Theα represents constant changes and defines the form of the slope factor g1

(i.e., a = 1 is linear change). The β represents the size of the proportionalautoregressive changes. A correlation between the intercept and slope (ρ01)is allowed, and the error variance (ψ2) is assumed to be constant at eachgrade level.

2008; McArdle & Hamagami, 2004). In recent research, wehave found it most flexible to represent the classical growthand change models based on multiple latent change scores(see Figure 4; McArdle, 2008). These models are practicallyuseful because they encompass the popular latent growthmodels and autoregressive time series models. In addition,because they are formed in a different way, they easily extendto multivariate forms (to follow).

More formally, we first assume we have observed scoresX[t] and X[t – 1] measured over a defined interval of time(�t), but we assume the latent variables (g[t]) are definedover an equal interval of time (�t = 1). This definition ofan equal interval latent time scale is nontrivial because itallows us to eliminate �t from the rest of the equations.This allows us to use latent scores g[t] and g[t – 1], resid-ual scores u[t] representing measurement error, and latentchange scores�g[t]. Even though this change�g[t] is a the-oretical score, and not simply a fixed linear combination, wecan now write a structural model for any latent change tra-jectory. This simple algebraic device allows us to generallydefine the trajectory equation based on a summation (

∑i=1, t)

or accumulation of the latent changes (�g[t]) up to time t.These latent change score models allow a family of fairly

complex nonlinear trajectory equations (e.g., nonhomoge-neous equations). These trajectories can be described by

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DYNAMIC ANALYSES OF DEPRESSION AND ACHIEVEMENT 617

writing the implied basis coefficients (Aj[t]) as the linearaccumulation of first differences for each variable (

∑�g[j],

j = 0 to t). Of course, this accumulation can be created fromany well-defined change model, and therefore, the procedureis quite flexible (e.g., see Ghisletta & McArdle, 2012).

On a practical note, these latent change score structuralexpectations are automatically generated using any standardSEM software (e.g., R, LISREL, Mplus, Mx, etc.). That is,we do not directly define the basis (A[t]) coefficients of atrajectory equation, but we directly define changes as anaccumulation of the first differences among latent variables.These dynamic models permit new features of the factor

analytic assessment of the previous models (e.g., Ferrer& McArdle, 2004; McArdle & Hamagami, 2003, 2004).

Methods for Bivariate Dynamic SEM Analyses

A bivariate dynamic model can be used to relate the latentscores from one variable to another over time (see Figure 5).More formally, we first assume we have observed both setsof scores Y[t] and X[t] measured over a defined interval oftime (�t), but we assume the latent variables are defined overan equal interval of time (�t = 1). This is most useful whenwe start to examine time-dependent interrelationships among

FIGURE 5 A bivariate biometric dual-change score model trajectories over grade-level changes in two measured variables (Y and X). Error variance (ψ2) isassumed to be constant at each grade level within each factor; αy and αx represent constant change related to the slope factors ys and xs; βy and βx representproportional change in Y and X; and cross-trait coupling is indicated by γ yx and γ xy. The model includes estimates for intercepts (y0 and x0), mean intercepts(μy0 and μx0), and mean slopes (μy1 and μx1). Other parameters are used to generate the decomposition of the correlation between the intercept and slope forX and Y .

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multiple growth processes. In general, we write an expansionof our previous latent change scores logic as the bivariatedynamic change score model. In this model, the g1 and h1

are the latent slope score that is constant over time, and thechanges are based on additive parameters (αy and αx), mul-tiplicative parameters (βy and βx), and coupling parameters(γ yx and γ xy). The coupling parameter (γ yx) represents thetime-dependent effect of latent x[t] on y[t], and the other cou-pling parameter (γ xy) represents the time-dependent effect oflatent y[t] on x[t]. When there are multiple measured vari-ables within each occasion, additional unique covarianceswithin occasions (uy[t] ux[t]) are identifiable as well.

RESULTS

Initial Statistical Description Including Thresholds

An initial description of some relevant summary statisticsappears in Table 3 and Table 4. Here we include estimatedthresholds, means, deviations, and correlations estimated

using the most typical incomplete data algorithms (MLE–MAR in Mplus 5.0; Muthén & Muthén, 2006; programscripts available on our Web site). One thing to note isthat the interval score estimates presented here do not haveany correction for nonnormality, although the ordinal scoreestimates have been corrected.

Table 3 presents the estimated thresholds for GPA (nowtermed GPA∗) and the WCES–D (now termed WCES–D∗). There were obtained using the SEM software Mplus5.0 under the constraint that these thresholds representeddifferences among the response categories, and that theywere the same (i.e., invariant MLE) at each of the fourgrades. We note that this simplification of parameters is nota necessary feature of the data, and the scale could changefrom one grade to the next, but this is a prerequisite for allmodels to follow, and therefore, we present it first.

Table 3a shows the estimated threshold for GPAs fromcategory D to C– (i.e., 1.0–1.7) has a value of τ 1 = –2.14.Because this was estimated in a normal probability, or probitmetric, this value indicates the location on a normal curve for

TABLE 3aEstimated (Maximum Likelihood Estimate [MLE]) of Ordinal Thresholds for Grade-Point Average

(Invariant Over All Four Grade Levels) From the Hawaiian High Schools Health Survey

Estimated Thresholds τ 1 τ 2 τ 3 τ 4 τ 5 τ 6 τ 7 τ 8

GPA units 1.0−1.7 1.7−2.0 2.0−2.3 2.3−2.7 2.7−3.0 3.0−3.3 3.3−3.7 3.7−4.0MLE GPAa −2.14 −1.60 −0.86 −.039 0.01 0.71 1.31 2.03Ratio of distances =0 1.80 2.47 2.05 0.13 2.33 1.50 3.43

Note. GPA = grade-point average.aIndicates ordinal version of interval scale; see sample sizes in Table 1.

TABLE 3bEstimated (Maximum Likelihood Estimate [MLE]) Ordinal Thresholds for CES–D

(Invariant Over All Four Grade Levels) From the Hawaiian High Schools Health Survey

Estimated Thresholds τ 1 τ 2 τ 3 τ 4 τ 5

WCES–D (days) 0–1 to 1–2 1–2 to 2–3 2–3 to 3–4 3–4 to 4–5 4–5 to 5–7MLE WCES–Da 0.53 1.48 2.14 2.85 3.78Ratio of distances =0 0.95 0.66 0.71 0.46

Note. CES–D = Center for Epidemiological Studies–Depression Scale; WCES–D = Weekly Center for Epidemiological Studies–Depression Scale.aIndicates ordinal version of interval scale; see sample sizes in Table 1.

TABLE 3cEstimated Means (MLE) and Standard Deviations for Four Grade Levels From the Hawaiian High Schools Health Survey

9th Grade 10th Grade 11th Grade 12th Grade

Parameters and Fit M SD M SD M SD M SD

Estimated MGPA (GPAa) 2.69 =0 2.72 0.31 2.78 0.19 2.95 0.47WCES–D (WCES–Da) 1.48 =0 1.44 −0.09 1.44 −0.08 1.38 −0.13

Estimated SDGPA (GPAa) .792 =1 .749 .670 .732 .625 .712 .642WCES–D (WCES–Da) 1.02 =1 1.01 1.060 .976 .919 .929 .800

Note. GPA = grade-point average; WCES–D = Weekly Center for Epidemiological Studies–Depression Scale.aIndicates ordinal version of interval scale; see sample sizes in Table 1.

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DYNAMIC ANALYSES OF DEPRESSION AND ACHIEVEMENT 619

people above and below this GPA point (i.e., approximately2% below and 98% above). The next estimated value of τ 2 =–1.60 suggests a slightly larger number of people are likely torespond between 1.7 and 2.0. The vector of eight thresholdsT = [–2.1, –1.6, –0.9, –.00, +0.0, +0.7, +1.3, +2.0] is seento increase ordinally, because this was restricted (it was notsimply nominal). The nonlinear nature of these changes inresponses can be seen from the difficulty of shifting betweenresponses formed here as a ratio of the estimated differencesto the observed differences (as in McArdle & Epstein, 1987).These differences would be constant if the scaling was at anequal interval, but these are � =[= 0, 1.8, 2.5, 2.1, 0.1, 2.3,1.5, 3.4], and this indicates the apparent difficulty of mov-ing between the response categories not being equal to thestated distance. For this reason, linear relations within andamong variables are likely to be much better represented byan ordinal scale of GPA∗.

Table 3b shows the estimated thresholds for the WCES–D for the category based on 0 to 1 days with a value ofτ 1 = 0.53. Again, because this was estimated in a probitmetric, this value indicates the location on a normal curve forpeople above and below this WCES-D point (i.e., approx-imately 52% below and 48% above). The next estimatedvalue of τ 2 = 1.48 suggests a slightly smaller number ofpeople are likely to respond between 1 and 2 days. The vec-tor of five thresholds T = [+0.5, +1.5, +2.1, +2.9, +3.8] isseen to increase in an ordered progression, because this wasrestricted (it was not simply nominal). The simple ratio of theestimated differences to the observed differences would beconstant if the scaling was at an equal interval, but these are� =[=0, 1.0, 0.7, 0.7, 0.5]. Therefore, these findings reflectthe difficulty of moving between the response categories notbeing equal. Thus, we expect that linear relations within andamong variables are likely to be much better represented bythis kind of an ordinal scale of WCES–D∗.

Table 3c is a listing of the means and deviations over fourgrade levels that would result from an interval or ordinal scal-ing of the two variables. These estimates are all based onMLE–MAR, so they use all available longitudinal data (seeTable 1). GPA in an interval scale has easy-to-understandmeans, which seem to increase over Grades 9, 10, 11, and12 from 2.69, to 2.72, to 2.78, to 2.95. Once again, highschool dropouts have been included in these calculations, butonly to the degree that we had other measured information

TABLE 4aEstimated Pearson Product–Moment (or Tetrachoric) CorrelationsFrom Hawaiian High Schools Health Survey of Interval GPA Over

Time (and Ordinal GPAa)

Correlations GPA[9] GPA[10] GPA[11] GPA[12]

GPA[9] 1.00 (1.00)GPA[10] .590 (.610) 1.00 (1.00)GPA[11] .561 (.568) .601 (.618) 1.00 (1.00)GPA[12] .494 (.524) .476 (.513) .587 (.609) 1.00 (1.00)

Note. GPA = grade-point average.aIndicates ordinal version of interval scale. See sample sizes in Table 1.

TABLE 4bEstimated Pearson Product–Moment (or Tetrachoric) Correlations of

From Hawaiian High Schools Health Survey of Interval WCES–DOver Time (and Ordinal WCES–Da)

Correlations WCES–D[9] WCES–D[10] WCES–D[11] WCES–D[12]

WCES–D[9] 1.00 (1.00)WCES–D[10] .472 (.550) 1.00 (1.00)WCES–D[11] .452 (.482) .503 (.539) 1.00 (1.00)WCES–D[12] .458 (.509) .453 (.511) .520 (.563) 1.00 (1.00)

Note. WCES–D = Weekly Centers for Epidemiological Studies–Depression Scale.

aIndicates ordinal version of interval scale. See sample sizes in Table 1.

(i.e., 9th-grade GPA). The estimates for the ordinal scalingof the GPA are slightly different, suggesting the GPA∗ goesfrom 0 (fixed for identification purposes) to 0.3 to 0.2 to 0.5,but the gain is neither linear nor as large. The estimated stan-dard deviations are reduced in the GPA∗ metrics, and thechanges are more complex for the WCES–D∗ metrics.

Table 4 is a listing of the estimated Pearson correlationsover four grade levels that would result from an intervalscale (using invariant means and deviations). The correla-tions in parentheses are polychoric estimates of relationshipsbased on ordinal scaling of the variables (using invariantthresholds). Each of these estimates is based on MLE–MAR,and therefore, they use all available longitudinal data (seeTable 1). When comparing Pearson correlations and poly-choric correlations, we find attenuation of correlations ifPearson correlations were opted for analyses due to the factthat measurements are nonnormal (as in Hamagami, 1998).Table 4a is a list of the correlations of GPA over grade levels,and these are strong and positive (∼r = .5–.6). However,

TABLE 4cEstimated Pearson Product–Moment (or Tetrachoric) Correlations From Hawaiian High Schools

Health Survey of Interval GPA With Interval WCES–D (and Ordinal GPAa with WCES–Da)

Correlations GPA[9] GPA[10] GPA[11] GPA[12]

WCES–D[9] −.130 (−.154) −.086 (−.085) −.063 (−.037) −.122 (−.082)WCES–D[10] −.095 (−.077) −.148 (−.162) −.082 (−.152) −.059 (−.076)WCES–D[11] −.098 (−.109) −.120 (−.149) −.135 (−.141) −.080 (−.117)WCES–D[12] −.096 (−.223) −.094 (−.174) −.083 (−.148) −.122 (−.156)

Note. WCES–D = Weekly Centers for Epidemiological Studies–Depression Scale.aIndicates ordinal version of interval scale. See sample sizes in Table 1.

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the use of ordinal scoring in GPA∗ yields a slightly highercorrelation in every case. Table 4b is a list of the correla-tions of the WCES–D over grade levels, and these are almostas strong and positive (∼r = .4–.5). Similar to GPA, how-ever, the use of ordinal scores in WCES–D∗ yields a slightlyhigher correlation in every case, suggesting that using theWCES–D∗ is more accurate. The critical cross-correlationsover time among these measures are listed in Table 4c forselected pairs of variables: (a) GPA[t] and WCES–D[t] and(b) GPA[t]∗ and WCES–D[t]∗. The GPA[t] and WCES–D[t]correlations are largely low and negative (∼r = –.06 to –.15),and the use of the ordinal scaling does not seem to change thesimple relationships very much (∼r = –.04 to –.22). Thesedifferences in the patterns of correlations between Pearsonand polychoric approaches indicate that dynamic character-istics derived from two different approaches might not beexactly the same.

Results From Univariate Dynamic Models of GradePoint Averages Over Four Grade Levels

The first set of longitudinal results of Table 5 is basedon alternative change models for the GPA trajectories (inFigure 1a). The first three models (5a, 5b, 5c) assumeinterval scaled data and models are fit in increasing order

of complexity. First, a baseline model with intercepts onlyis fitted (not shown) and this yields χ2(11) = 398. The firstmodel listed in Table 5 is the linear change model with oneslope (labeled Model 5a) and fits much better with χ2(8) =48 (so ε = 0.027), so the change in fit is large as well (�χ2

= 350 on �df = 3). This is a remarkably small misfit forsuch a large sample size (N > 7,000). The mean interceptfor the 9th grade is reasonable (μ0 = 2.65) and the meanslope is positive but small (μ1 = 0.09), indicating only smallaverage changes over grade levels. The variability in thesecomponents shows somewhat large initial differences (σ 0

2 =0.42) with small systematic changes (σ 1

2 = 0.02) and largerrandom changes (ψ2 = 0.22). The second model (Model5b) forces the systematic slope components to be zero, butallows an autoregressive component (β = 0.037) to allowchanges that accumulate. This autoregression AR(1) modeldoes not seem to fit the GPA data so well, χ2(8) = 114(so ε = 0.044). The third model (Model 5c), termed a dualchange, allows both linear and autoregressive changes tooccur, and this leads to the best fit, χ2(7) = 24 (so ε =0.019). This model suggests more complex changes dueto the negative mean slope (μ1 = –1.79) combined withpositive autoregressive changes (β = 0.69), implying thatthe GPA would go down over grade levels except for theimpact of the prior grade’s GPA.

TABLE 5Alternative Latent Curve Results From Fitting Univariate Dual-Change ScoreModel to Different Scalings of Grade-Point Averages (GPAs) (Ordinal GPA∗)

Model 5a Model 5b Model 5c Model 5d Model 5e Model 5fInterval Interval Interval Ordinal Ordinal Ordinal

GPA GPA GPA GPA∗ GPA∗ GPA∗Parameters and Fit Linear Change AR(1) Change Dual Change Linear Change AR(1) Change Dual Change

Fixed effectsMean intercept, μ0 2.65 (211) 2.69 (201) 2.68 (210) =0 =0 =0Mean slope, μ1 0.09 (17) =0 −1.79 (3.3) 0.14 (14) =0 0.07 (4.6)Constant change, α =1 =0 =1 =1 =0 =1Proportion change, β =0 0.037 (15) 0.69 (3.4) =0 = 0.69 (3.2) 0.84 (3.3)Scaling coefficients, T =1 =1 =1 −1.8, –1.3 −2.0, –1.5 −1.9, –1.4

−.70, –.30 −.77, –.31 −.80, –.30 .03,.06, .66 .07, .75 .671.2, 1.8 1.3, 2.0 1.2, 1.9

Variable intercepts, μ =0 −0.07 (3.6) =0 =0 0.16 (15) =0Random effects

Intercept variance, σ 02 0.42 (25) 0.42 (25) 0.40 (25) =1 = 1 =1

Slope variance, σ 12 0.02 (5.1) =0 1.22 (2.1) 0.04 (6.3) =0 0.78 (1.81)

Common slope-level, σ 01 −0.05 (7) =0 −0.29 (4.1) −0.11 (8.7) =0 −.88 (−3.6)Unique variance, ψ2 .22 (35) .027 (38) .22 (35) 0.48 (18) 0.70 (27) 0.53 (18)Residual variance, σ e

2 =0 −0.03 (5.7) =0 =0 −0.05 (2.9) =0Goodness of fit

Misfit index χ2/df 48/8 114/8 24/7 174/26 246/26 94/25Change in fit �χ2/�df 350/3 284/3 24/1; 90/1 985/3 913/3 80/1; 152/1WSLMV scaling factor κ 1 1 1 0.723 0.739 0.651

RMSEA ε .027 .044 .019 .029 .034 .020

Note. Weighted least squares mean and variance (WLSMV) estimator based on weighted least squares plus mean adjusted variances. Sample size n =6,896, 15 patterns of incomplete data. Level-only baseline for the two scalings are GPA = 398/11 with ε = 0.071, and GPA∗ = 1,159/29, κ = 0.871, ε =0.075. Obtain Technical Appendix (from authors) for Mplus input and output. GPA = grade-point average; AR = autoregression; RMSEA = root mean squareerror of approximation.

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DYNAMIC ANALYSES OF DEPRESSION AND ACHIEVEMENT 621

The second set of three models (Models 5d, 5e, and 5f)assumes some ordinal thresholds are needed and then reex-amines the same change models. Here we tried to placeseveral additional constraints on the latent variables to assureidentification (i.e., μ0 = 0, σ 0

2 = 1) and estimate the eightthresholds (t1–t8). Although we could allow the thresholds tochange over grade levels, we do not consider this possibil-ity here, and this assumption of scale invariance over gradelevels leads to additional misfit (but see McArdle, 2007).The scaling coefficients (described earlier) range about –2 to+2 but suggest an unequal distance between GPA points.The baseline model with intercepts only is fitted (not shown)and this yields χ2(29) = 1,159, so the ordinal linear modelfitted to these is a great improvement in fit; that is, χ2(26)= 174, so �χ2 = 985 on �df = 3; with parameters repre-senting some positive changes (μ1 = 0.14, σ 1

2 = 0.04) andlarger random changes (ψ2 = 0.48). (We note that the kurto-sis scaling coefficient for the WLSMV estimator is reported(ω4 = 0.7230, but, because our overall results will remainthe same with or without this correction, this was not appliedto adjust the chi-square tests.)

The next model (Model 5e) is the same, but allows pro-portional autoregressive changes without slopes, and it doesnot fit as well. The final model (Model 5f) allows both typesof changes to the ordinal scales, and it fits much better; thatis, χ2(25) = 94, so �χ2 = 80 on �df = 1. Assuming this isour final GPA model, we have (a) unequal distances in GPAscaling, (b) small positive changes in slopes over grade lev-els with large variations (μ1 = 0.07, σ 1

2 = 0.78), (c) largepositive proportional changes from one GPA level to another(β = 0.84), and (d) even larger random changes (ψ2 = 0.70).In this comparison, the first three models for GPA suggest thedual change is needed—all models for ordinal GPA∗ suggestunequal but ordered intervals, and the dual change model fitsbest. A plot of the expected values from this GPA∗ model isdrawn in Figure 6a and this shows a small but significantlypositive shift in the mean over grade levels, with increasingindividual differences in trajectories.

Results for Univariate Dynamic Models of DepressiveSymptoms Over Four Grade Levels

The second set of longitudinal results of Table 6 is based onalternative change models for the WCES–D trajectories (inFigure 1b). The first three models (Models 6a, 6b, and 6c)assume interval scaled data and models are fit in increasingorder of complexity. First, a baseline model with interceptsonly is fitted (not shown) and this yields χ2(11) = 74. Thefirst model listed in Table 6 is the linear change model withone slope (Model 6a) and fits much better with χ2(8) = 31(so ε = 0.027), so the change in fit is large, too (�χ2 = 43 on�df = 3). This is also a remarkably small misfit for such alarge sample size (N > 7,000). The mean intercept for the9th grade is reasonable (μ0 = 1.48 days) and the mean slopeis negative but small (μ1 = –0.03) indicating only small

9 9.5 10 10.5 11 11.5 12–6

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(a)

Grade Level

GPA

*

9 9.5 10 10.5 11 11.5 12–2

0

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8

10

Grade Level

WC

ESD

*

FIGURE 6 Expected longitudinal trajectories on the two key variableindicators for a random subset of n = 30 persons. (a) Expected longitudinaltrajectories for GPA∗ (from Table 5, Model 5d); (b) expected longitudinaltrajectories for WCES–D∗ (from Table 6, Model 6d).

Note. GPA = grade-point average; WCES–D = Weekly Center forEpidemiologic Studies–Depression Scale.

average changes over grade levels. The variability in thesecomponents shows somewhat large initial differences (σ 0

2 =0.57) with small systematic changes (σ 1

2 = 0.01) and largerrandom changes (ψ2 = 0.49). The second model (Model 6b)forces the systematic slope components to be zero, but allowsan autoregressive component (β = –0.13) to allow changesto accumulate and level off. This AR(1) model seems to fitthe WCES–D data very well, χ2(8) = 21, so ε = 0.02. Thethird model, termed a dual change, allows both linear andautoregressive changes to occur, and this leads to the best fit,χ2(7) = 29, so ε = 0.02. This model suggests more com-plex changes due to the negative mean slope (μ1 = −1.48)combined with positive autoregressive changes (β = 0.03),implying more complex grade-level patterns in the WCES–D; however, this model is not an improvement in fit. Indeed,the interval WCES–D seems to fluctuate from one time tothe next without a systematic linear change.

The second set of three models (Models 6d, 6e, and6f) assumes some ordinal thresholds are needed and thenreexamines the same change models. Here we tried to placeseveral additional constraints on the latent variables to assure

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TABLE 6Alternative Latent Curve Results From Fitting Univariate Dual-Change Score Model to Interval and Ordinal Scalings of the Weekly Center for

Epidemiologic Studies–Depression Scale (WCES–D; Rescaled Version of 13 items of Factor 1)

Model 6a Model 6b Model 6c Model 6d Model 6e Model 6fInterval Interval Interval Ordinal Ordinal Ordinal

WCES–D WCES–D WCES–D WCES–D∗ WCES–D∗ WCES–D∗Parameters and Fit No AR Change AR(1) Change Dual Change Linear Change AR(1) Change Dual Change

Fixed effectsMean intercept, μ0 1.48 (91) 1.60 (100) 1.47 (103) =0 =0 =0Mean slope, μ1 −0.03 (4.3) =0 −1.48 (−9.1) −0.46 (2.5) =0 −0.04 (1.5)Constant change, α =1 =0 =1 =1 =0 =1Proportion change, β =0 −0.13 (4.0) 0.03 (0.8) =0 −0.07 (2.0) 0.03 (0.4)Scaling coefficients, T =1 =1 =1 0.5, 1.4 0.5, 1.4 0.5, 1.4

2.0, 2.0 2.02.7, 3.6 2.6, 3.4 2.7, 3.6

Variable intercepts, μ =0 0.18 (3.4) =0 =0 −0.05 (2.6) =0Random effects

Intercept variance, σ 02 0.57 (20) 0.51 (15) 0.53 (22) =1 =1 =1

Slope variance, σ 12 0.01 (1.2) =0 0.56 (17) 0.011 (0.6) =0 0.10 (0.3)

Common slope-level, σ 01 −0.04 (20) =0 −0.54 (20) −0.04 (1.1) =0 −0.30 (0.5)Unique variance, ψ2 0.49 (38) 0.44 (19) 0.50 (37) 0.78 (8.8) 0.67 (7.0) 0.77 (9.7)Residual variance, σ e

2 =0 0.06 (2.0) =0 =0 0.11 (1.7) =0Goodness of fit

Misfit index χ2/df 31.3/8 21.9/8 29.2/7 13.7/17 10.7/17 13.8/16Change in fit �χ2/�df 42.7/3 51.1/3 1.9/1; –8.3/1 47.3/3 50.3/3 −0.1/1; –2.9/1WLSM scaling factor κ 1 1 1 0.772 0.796 0.734

RMSEA ε 0.02 0.02 0.02 0.00 0.00 0.00

Note. Weighted least squares mean and variance (WLSMV) estimator based on weighted least squares plus mean adjusted variances. Sample size n =6,942, 15 patterns of incomplete data. Level-only baseline for the two scalings are WCES–D = 74/11 with ε = 0.03, and WCES–D∗ = 61/20, μ = 0.900, ε

= 0.02. AR = autoregression; RMSEA = root mean square error of approximation.

identification (i.e., μ0 = 0, σ 02 = 1) and estimate the five

thresholds (τ 1–τ 5). Although we could allow the thresholdsto change over grade levels, we do not consider this pos-sibility here, and this assumption of scale invariance overgrade levels leads to very little misfit. The estimated scalingcoefficients (described earlier) range from about 0 to +4 butsuggest an unequal distance between WCES–D points. Thebaseline model with intercepts only is fitted (not shown) andthis yields χ2(29) = 61, and therefore, the ordinal linearmodel fitted to WCES–D∗ is a minor improvement in fit; thatis, χ2(17) = 14, so �χ2 = 47 on �df = 12, with parame-ters representing some negative changes (μ1 = –0.46, σ 1

2 =0.01) and larger random changes (ψ2 = 0.78). (Again, wenote that the kurtosis scaling coefficient for the WLSM esti-mator is reported [ω4 = 0.772], because this has an effect onthe resulting chi-square tests.)

The next model (Model 6e) is the same, but allows propor-tional autoregressive changes without slopes, and it fits onlyslightly better; that is, χ2(17) = 11.). The final model (Model6f) allows both types of changes to the ordinal scales, and itfits about the same; that is, χ2(16) = 14. Assuming this isour final WCES–D∗ model, we find (a) unequal distances inWCES–D scaling, (b) small negative changes in slopes overgrade levels with large variations (μ1 = –0.04, σ 1

2 = 0.10),(c) small positive proportional changes from one WCES–D level to another (β = 0.03), and (d) even larger random

changes (ψ2 = 0.77). In this comparison, the first three mod-els for WCES–D suggest that only the autoregressive part ofthe dual change is needed. All models for ordinal WCES–D∗suggest unequal but ordered intervals, and the dual changemodel is possible. A plot of the expected values from thisWCES–D∗ model is drawn in Figure 6b and this shows asmall but significantly negative shift in the mean over grades,with increasing individual differences in trajectories.

Results From Bivariate Dynamic Latent ChangeModels

The results of Table 7 give the SEM parameter estimates fora dynamic system where GPA and WCES–D are allowed toimpact the trajectory of each other over 9th to 12th gradesbased on the full bivariate dynamic path diagram of Figure 5.The first model (Model 7a) is based on interval scaling ofboth variables, GPA and WCES–D as observed, and thesecond model (Model 7b) is based on estimation using theordinal concepts developed up to this point. (The authorsprovide an Mplus computer program for these models in aTechnical Appendix on their Web site). Perhaps it is obvious,but the dynamic results differ depending on how the variablesare scaled.

In the interval measurement model (Model 7a), we usethe scale of measurement as it is calculated from the scores

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DYNAMIC ANALYSES OF DEPRESSION AND ACHIEVEMENT 623

TABLE 7Dynamic Structural Equation Model Results From a Bivariate Dual-Change Score Model for Interval and Ordinal Scalings of Grade-Point

Averages (GPAs) and Weekly Center for Epidemiologic Studies–Depression Scale (WCES–D), Including Alternative Misfits

Interval Scaling Ordinal∗ Scaling

Parameters GPA WCES–D GPA∗ WCES–D∗

Fixed effectsMean intercept, μ0 2.68 (202) 1.48 (81) = 0 ( = 0) = 0 ( = 0)Mean slope, μ1 −0.66 ( − .58) 0.45 (.35) 0.026 (1.00) −0.052 ( − 2.34)Constant change, α =1 =1 =1 =1Proportional change, β 0.607 (2.57) 0.140 (.22) −0.109 (0.27) −0.010 ( − 0.02)Coupling, γ GPA → WCES–D −0.25 (−1.05) −0.074 (−0.37)Coupling, γ WCES–D → GPA −0.63 (−1.19) −2.170 (−2.57)Scaling coefficients, T −1.8, –1.4, –0.7, 0.5, 1.4,

−0.3, 0.01, 0.6, 2.0,1.1, 1.7 2.7, 3.6

Random effectsIntercept variance, σ 0

2 0.40 (24) 0.56 (15) = 1 ( = 0) = 1 ( = 0)Slope variance, σ 1

2 0.44 (1.2) 0.05 (.38) 4.58 (1.3) 0.01 (0.71)Covariance slope & level, σ 01 −0.31 (3.7) −0.12 (.39) −0.45 ( − 1.5) −0.03 ( − 0.07)Unique variance, ψ2 0.22 (35) 0.49 (35) 0.43 (14.7) 0.81 (13.1)

Goodness of fitDual coupling misfit 55.9/23/0.014 82.6/50/0.009No GPA → WCES–D 56.7/24/0.014 82/51/0.009No WCES–D → GPA 57.7/24/0.014 113/51/0.013No couplings misfit 58.0/25/0.014 111/52/0.013No cross-lags misfit 83.0/27/0.017 191/54/0.019No slopes misfit 168/32/0.024 1172/59/0.060

Note. Dual model sample size is N = 7,258 with 130 patterns of incomplete data. Ordinal ω4 = .725. Not all model parameters are listed here. ObtainTechnical Appendix (from authors) for Mplus input and output.

in the data, and this assumes an equal interval between scorepoints for both GPA and WCES–D. Of course, our prior plotsand models fitted have suggested this is largely an incorrectassumption for both variables. Nevertheless, because inter-val scaling is typically assumed, we start from this point.As a result, we observe strong dynamic parameters withGPA, little or no dynamic action of the WCES–D, and lit-tle or no systematic coupling across variables. The impact ofGPA on the changes in WCES–D is now negligible (γ = –0.25, t ∼ 1), and the impact of WCES–D on the changes inGPA, although seeming a bit larger, is also negligible (χ =–0.63, t ∼ 1). The overall model fits the data well, χ2(23)= 56 (ε = 0.014), but a model with either or both cou-plings set to zero, χ2(24) = 57, or χ2(25) = 58, fit about thesame (ε = 0.014). In contrast, a model where we assume nocross-lagged impacts at all (i.e., a “parallel process” model)does lead to a substantial loss of fit, χ2(27) = 83, and amodel assuming no slopes at all (i.e., a standard cross-laggedregression) is even worse, χ2(32) = 168. Therefore, althoughthere are some dynamic influences operating in the intervalscaling, the direction of impacts over grade levels is not veryclear—GPA and WCES–D appear to be correlated by notdependent processes.

In the ordinal measurement model (Table 7), we do notassume the scale points have equal distance among them,but, as defined earlier, we estimate the differences using the

thresholds (τ ). This use of a latent variable measurementmodel for the GPA∗ and the WCES–D∗ requires some kindsof identification constraints and we employ the ones usedearlier (μ0 = 0, σ 0

2 = 1), but now for both variables. Whenthe dynamic model is estimated for these ordinal latent vari-ables, a result emerges: The impact of GPA∗ on the changesin WCES–D∗ is clearly negligible (γ = –0.07, t < 1), butthe impact of WCES–D∗ on the changes in GPA∗ is nownoteworthy and large (γ = –2.17, t > 2.5). The fits listedshow models where (a) the overall model fits the data well,χ2(50) = 83 (ε = 0.009); (b) a GPA∗→ WCES–D∗ cou-pling set to zero fits just as well, χ2(51) = 82; (c) theWCES–D∗→ GPA∗ coupling set to zero does not fit as well,χ2(51) = 113; and (d) both couplings set to zero does not fiteither, χ2(52) = 111. In addition, a model where we assumeno cross-lagged impacts at all (i.e., a “parallel process”model) leads to a substantial loss of fit, χ2(54) = 191, and amodel assuming no slopes at all (i.e., a standard cross-laggedregression) is even worse, χ2(59) = 1172. Therefore, in thisordinal scaling there seem to be directional dynamic influ-ences operating—the direction of impacts accumulated overall grade levels suggests that WCES–D∗ → GPA∗ and notthe other way around.

The use of a vector field plot was introduced to assist inthe interpretation of the size of the impacts of the dynamicresults (see Boker & McArdle, 2005). In Figure 7, these

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624 MCARDLE, HAMAGAMI, CHANG, HISHINUMA

–2 –1 0 1 2–1

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0 1 2 3 4 50

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Interval Scale BLCS Model

Ordinal Scale BLCS Model(b)

(a)

FIGURE 7 Vector field plots of expected longitudinal trajectories fromthe interval and ordinal bivariate latent change score (BLCS) models (seeTable 7). (a) interval scale; (b) ordinal scale.

Note. WCES–D = Weekly Center for Epidemiologic Studies–DepressionScale; GPA = grade-point average.

kinds of vector field plots are displayed for both the inter-val model (Table 7, Figure 7a) and for the ordinal model(Table 7, Figure 7b). In both plots we derive the expecteddirection of the trajectory for all points across a reason-able span of scale points. We then display these expectationsas small arrows (i.e., directional derivatives) and we inter-pret the overall picture in terms of the overall flow of thearrows. We include a 95% confidence ellipsoid around thestarting points based on the available data, but we recog-nize that expectation outside of these boundaries might beamong the most interesting flows in the diagram. The firstdisplay (Figure 7a) shows the interval model with very littlecross-variable movement, especially inside the 95% boundsof the observed data. The second display (Figure 7b) showsthe ordinal model has a little trace of cross-variable move-ment, where the GPA∗ is moving down for increasing levels

of WCES–D∗. In both cases there is exaggerated movementoutside the 95% bounds of the observed data, but thesearrows reflect expectations for scores that do not appear inthe current data. There does not appear to be any dynamicaction in the interval model, but there is some importantdynamic action in the ordinal model where the WCES–D∗effects changes in the GPA∗ and not the other way around.

Results for Group Differences in Linear DynamicModels

To add to our understanding of the dynamics, we cancompare demographic groups, such as Hawaiians and non-Hawaiians, and females versus males. There are several waysto consider group differences in the bivariate latent differ-ence score models, and we present a few of these analysesnext (for details, see McArdle, 2008; McArdle & Hamagami,1996). In one form of a multilevel DSEM, we simply use thegroup information to describe the distribution of the latentlevel and latent slopes. This is equivalent to suggesting thatthe dynamic process based on the coupling coefficients isinvariant over groups, but that the position of the groupsmight differ in the vector field (i.e., different 95% confidenceellipsoids). A more flexible way to deal with these dynamicmodels is to examine the possibility of qualitative differencesin dynamic systems among the WCES–D∗ and GPA∗ lon-gitudinal processes over the groups—that is, we ask, “Dothe dynamics operate the same way in each subgroup?”One practical way to examine this form of invariance is tosplit up the data into groups, allow the dynamics to dif-fer, and see if these relaxed assumptions improve the modelfit substantially. This simple strategy requires a multiple-group approach and, to retain power within each group, manyparticipants are needed within in each group (McArdle &Hamagami, 1996). By splitting up the data into only fourgroups based only on ethnicity and gender we can meetthe sample size requirements in the HHSHS. Results forthis multiple-group form of group dynamics is presented inTable 8 and Figure 8.

The resultant bivariate dynamics for the four adolescentgroups are first listed separately in Table 8—that is, resultsare listed for the male-Hawaiian adolescent group (Table 8and Figure 8a), the male non-Hawaiian group (Table 8 andFigure 8b), and so on. In all groups, the status of WCES–D∗dictates how both WCES–D∗ and GPA∗ change. WCES–D∗ alone seems to decline itself, whereas GPA∗ is nega-tively influenced by the status of depression. The significantnegative coupling of WCES–D∗ on a change score of GPA∗intimates that a change in GPA∗ is solely based on the previ-ous state of WCES–D∗. The higher the depressive state atthe current state, the larger the negative change in GPA∗.Although the plots in Figure 8 are not exactly the same, theyare all very similar.

We started with the separate models of Table 8 and addedinvariance constraints to the four-group model, and we found

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DYNAMIC ANALYSES OF DEPRESSION AND ACHIEVEMENT 625

TABLE 8Maximum Likelihood Estimate–Missing at Random Estimates of Dynamical Parameters (and z Values) Based on Four Separate Bivariate

Latent Change Score Models With Fixed Ordinal Measurement Model (Threshold Parameters Based on Table 7)

MaleHawaiian

MaleNon-Hawaiian

FemaleHawaiian

FemalevNon-Hawaiian

n 2,531 1,192 2,286 1,184μWCES–D −.16 −.10 −.09 −.11

(2.37) (1.20) (3.32) (1.07)μGPA .04 .08 −.02 .002

(1.50) (1.91) (.40) (.04)βWCES–D −.98 −1.02 −.86 −.65

(11.1) (9.61) (3.79) (3.78)βGPA .42 .26 −.07 .25

(1.09) (.73) (.18) (.67)γGPA →WCES–D −.45 −.22 −.34 .15

(1.76) (.77) (.69) (.33)γWCES–D →GPA −.72 −.92 −.99 −.65

(2.72) (6.22) (7.24) (2.78)χ2 81 74 42 63df 52 51a 52 50b

εa .015 .014 .000 .015

Note. Variances of all slope components are fixed at 1.0 for simplicity. WCES–D = Weekly Center for Epidemiological Studies–Depressions Scale; GPA =grade-point average; εa = root mean square error of approximation.

aThere is no response in one category in WCES–D, thus one less df . bThere is no response in the last category in WCES–D in Grades 9 and 10, with thehighest threshold parameter rendered un-estimable.

–2 0 2

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CE

SD*

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SD*

Male Hawaiian Male Non–Hawaiian

Female Hawaiian Female Non–Hawaiian(b) (d)

(b)(a)

FIGURE 8 Vector field plots of expected longitudinal trajectories from the multiple group bivariate latent change score (BLCS) models (see Table 8).

Note. GPA = grade-point average; CES–D = Center for Epidemiologic Studies–Depression Scale.

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626 MCARDLE, HAMAGAMI, CHANG, HISHINUMA

(a) a model that is invariant over ethnicity yields a relativelysmall loss of fit of �χ2 =50 on �df = 13; and (b) a modelthat is invariant over gender yields a relatively large lossof fit of �χ2 = 321 on �df = 13. As a result, the ideathat these subgroups should be treated as one large grouphaving equivalent dynamic systems is a reasonable one,except for some notable gender differences. That is, althoughthere are slightly different parameters for Hawaiians andnon-Hawaiians, there is little evidence for differences indynamic systems for different ethnic groups. However, itappears males and females are operating on slightly differentdynamic systems and should probably not be combined.

Finally, to check our within-time equality constraints (onβs and γ s) we examined the equality of dynamic constraintsover time. In the previous models listed earlier we requiredthe same dynamic result to appear over every time interval—from 9th to 10th, 10th to 11th, and 11th to 12th. In these finalanalyses, we relaxed these dynamic assumptions, looked atthe groups as a whole, and within each separate subgroup,and we basically found no evidence for differential dynamicsover grade level or time.

DISCUSSION

The primary issue raised in this article is not a new one—Does depression lead to poor academic achievement ORdoes poor academic achievement lead to depression? Ofcourse, it would not be ethical to deal with this kind of ques-tion on a randomized experimental trial design (see McArdle& Prindle, 2008). As a practical alternative, we dealt withthis substantive question on a statistical basis by using con-temporary models with three features: (a) a large set ofrelevant data was collected on adolescents using a cohort-sequential design with intentionally incomplete pieces, (b)an accounting was made of scale differences between GPAand CES–D, (c) a linear dynamic model was fitted, andexamined in several ways, including some accounting forgroup differences. The overall result differs from the previ-ous models in at least three important ways—dealing withincomplete data, scaling issues, and dynamic inferences.First, we used all the data at all occasions of measurementin an attempt to provide the best estimates available for anymodel fitted.

Second, we demonstrated that that the scaling of the vari-ables can make a difference in the dynamic interpretationbetween academic achievement (GPA) and depressive symp-toms (CES–D). From our results, we learned that the scalingof the WCES–D is highly skewed (see Figure 3) and weknow this limits the inferences that can be made using stan-dard interval score models. In addition, the categories ofthe GPA scale used here were also not equal, and this ledto additional limits. It is not so surprising that when weexamined alternative latent change models for either vari-able, we found that both the form and size of the changes

were related to the measurement or scaling of the variablesused in the change model. But the biggest differences arosewhen we tried to link the repeated measures data from onegrade level to the next over both variables. Here we firstestablished no dynamic linkage in the interval scaled data.However, when we used an ordinal scaling of both vari-ables, we found out that prior levels of WCES–D∗ lead tosubsequent GPA∗ changes in the next grade level. Any truedynamic expression, no matter how large, could have beenmasked by impoverished measurement. The typical GPA andCES–D scaling could be considered in this respect. That is,it could be that the constructs of achievement and depressionare clearly linked over time, but the typical measurementwith GPA and the CES–D might not be the best way tounderstand these processes. Perhaps the fact that we can-not fit a good dynamic expression of change without goodmeasurement of the construct should not be a great surpriseeither.

Third, in general, affective states of youth in our samplefrom Hawai‘i seemed to manifest in how they focused onschool performances. This has many implications from clin-ical and prevention standpoints (see also Hishinuma et al.,2012). For example, screening for depressive symptoms,major depression, and dysthymia, in conjunction with appro-priate identification and treatment, might prevent future pooracademic performance and perhaps prevent academic failureand other comorbid delinquency behaviors (e.g., droppingout of school, substance use, suicide). In addition, giventhat depression is typically internalized and often not iden-tified by schools aside from the outward manifestations ofdepression (e.g., changes in patterns in eating and sleeping),decreases in academic achievement might be a marker forschools that depressive symptoms might have preceded orare currently occurring and causing the lowered academicperformance.

In our final analyses, we learned more about gender dif-ferences and ethnicity differences. In these analyses, wefound the ethnic differences were not apparent, but thegender differences were large from a number of points ofview. That is, the females seemed to be experiencing bothmore of, and a potentially more virulent form of an impactof increasing depressive symptoms, leading to loweringof changes in academic achievement. These findings sug-gest that screening for prevention and treatment could beparticularly important for girls.

There are many limitations of the modeling approachused here, some based on the current analyses and somebased on the available data. The statistical model used isfairly flexible, but it still presumes invariance at several lev-els largely because parameter identification becomes muchmore difficult when more complexity is added. Our initialassumptions about interval scaling were relaxed and madeordinal, but the results presented here did not fully evalu-ate the assumptions of an invariant over-grade measurementmodel, and these assumptions are worth examining in more

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DYNAMIC ANALYSES OF DEPRESSION AND ACHIEVEMENT 627

detail. In addition, for example, in all the DSEMs fitted here,we assumed a dynamic process that takes the same amountof time no matter what pair of grade levels we were con-sidering. We recognize that it is possible that more changesoccur between some grade levels (i.e., Grades 9 and 10) thanbetween others (i.e., Grades 10 and 11), or that the changesbetween longer times are not a simple accumulation (i.e.,Grades 9 to 11 is a function of the other times).

The data we used were somewhat limited as well. Forexample, the assumption of invariance over different patternsof incomplete data is likely to be incorrect to some degreeand we know that specific forms of self-selection effects canlead to parameter bias (see McArdle & Hamgami, 1992).There were also no other measured variables defined inthis system, although surely there are other sources to thechanges in both GPA and CES–D (Hishinuma, Foster, et al.,2001). For example, it is likely that sources of individual dif-ferences outside the school setting are responsible for theadditional influences, and we have not tried to take intoaccount other sources of family support (but see Andradeet al., 2006). Finally, the group differences models onlyattempted to account for observed group differences in eth-nicity and gender, and it is likely that latent subgroups orclasses of similar groups of adolescents were involved (seeFlaherty, 2008; McArdle & Prindle, 2008; Van Horn et al.,2009). It is possible, for example, to isolate groups of ado-lescents for whom GPA → �Depression, and to find othersfor whom Depression →�GPA. For all of these reasons, thesubstantive results presented might not replicate in furtheranalyses of key subgroups.

The dynamic analyses presented here clearly pointed outa few problems that can be overcome by using contempo-rary modeling procedures. This DSEM approach can be usedto turn important developmental questions about temporalsequences into statistically powerful hypotheses. The factthat this can be done in the presence of ordinal-level mea-surement and large amounts of incomplete data was alsodemonstrated, and this is a necessity for most real-life sit-uations. We hope this approach can be useful for many otherstudies in which multivariate longitudinal data have been col-lected to gain some insight into an ongoing developmentalprocess. Such findings, in turn, should have both scientificand applied value.

ACKNOWLEDGMENTS

Thanks must be given to our mentors, Dr. Ronald C.Johnson (deceased) and Dr. Naleen Andrade, for their con-stant support in data collection and data analysis. The initialpresentation of this work was done by the first authorat the 2003 Statistical Modeling Seminars, Department ofPsychiatry, John A. Burns School of Medicine, Universityof Hawai‘i at Manoa, Honolulu, Hawai‘i, January 2003.The same longitudinal data and latent change analyses are

described and presented in a more substantive context byHishinuma, Chang, McArdle, and Hamagami (2012). Theauthors would also like to express their appreciation to theresearchers and administrators of the Asian/Pacific IslanderYouth Violence Prevention Center and National Center onIndigenous Hawaiian Behavioral Health.

FUNDING

This methodologically focused article was supported bythe Research Centers in Minority Institutions (RCMI)Supplement, National Institutes of Health (NIH; RR0361-06S1), National Institute of Mental Health (NIMH;R24 MH5015-01, R24 MH57079-A1), and Centers forDisease Control and Prevention (CDC; R49/CCR918619-05; Cooperative Agreement #1 U49/CE000749-01). Thecontents of this article are solely the responsibility of theauthors and do not necessarily represent the official viewsof the funding agencies.

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