Despite the enormous amount of research that has been conducted on educational achievement, only a small proportion of this research body has investigated the association between academic achievement in the form of academic achievements in school and individual psychological adjustment such as depressive symptoms. A parallel publication to the present study is by Hishinuma, Chang, McArdle and Hamagami (in press). Their Table 1 is an overview of previously published research, and this summary suggests prior research has found significant but small, negative associations between indicators 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 depressive symptom measure was not combined with anxiety, the correlation between GPA and depressive symptoms ranged from −.16 to −.36 (see their Table 1), with only a few exceptions. This negative relationship between GPA and depressive symptoms was further confirmed by other studies. For example, Juvonen, Nishina, and Graham (2000), utilizing structural equation modeling (SEM) and a middle school sample, found evidence for perceived peer harassment leading to psychological distress (i.e., depressive symptoms as measured by the Children’s Depression Inventory, self-worth), which in turn leading to poor school adjustment (i.e., actual GPA, number of classroom hours missed during semester of data collection). This suggested that depressive symptoms occurred before lowered GPAs, although this study was not longitudinal in nature.

In attempting to determine any causal relationships between GPA (school achievement) and depressive symptoms (psychological adjustment), it is likely that longitudinal studies will be needed (see Duncan, 1975; Flaherty, 2008; McArdle, 2009). For example, Lehtinen, Raikkonen, Heinonen, Raitakari, and Keltikangas-Jarvinen (2006) assessed parents of Finnish students when the students were 9, 12, and 15 years of age, and then assessed the students when they were 21–36 years of age. The researchers found an overall negative association between GPA and later depressive symptoms, but no consistent relationship across age and gender. In particular, 12- and 15-year old girls’ lower GPA predicted depressive symptoms 12 and 17 years later. Despite GPA was thought to be the predictor of the depressive symptoms, the following conclusion was drawn: “The current findings, thus, suggest that depressive outcomes have a detrimental effect on success at school rather than vice versa: presumably, failure to cope with challenges at school could lead to different kinds of vicious circles, in which case deterioration of school performance would serve as a risk factor for adult depressive outcomes …” (p. 289).

A variety of other relevant studies are reviewed by Hishinuma et al. (in press). In general, only two studies could be identified that included both GPA and depressive symptoms in a longitudinal design that involved examining trajectories. Repetto and colleagues (2004) surveyed African American high school students from the 9^th grade with GPAs of 3.0 or lower, at-risk for high school dropout, and not diagnosed with emotional impairment or developmental disability. Depressive symptoms (six items from the Brief Symptom Inventory) and GPA (8^th to 10^th grades) were utilized; however, the analyses did not include incomplete data. The students were clustered in four groups based on their longitudinal depressive symptoms: (1) consistently high; (2) consistently low; (3) decreasing; and (4) increasing. Using multivariate analysis of variance (MANOVA), students who were consistently high in depressive symptoms were more likely to be female, have anxiety and stress, and have lower self-esteem and GPA. In independent work, Johnson and colleagues (2006) examined academic achievement trajectories in adolescences in the Minnesota Twin Family Study. Although data on a composite measure of “grades” were collected across multiple years, the depression indicator was collected only in Year 1 and was not significantly associated with the measure of academic achievement.

The issues surrounding the scaling of variables have generated a great deal of prior research on these kinds of measures. Scales of achievement have been developed which have 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 GPA, have 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 point average.’ Clearly such assignments are arbitrary …” (Duncan, 1975, p. 159).

Similarly, the use of the 20-item scale of the Center for Epidemiologic Studies-Depression (CESD, Radloff, 1977) was critiqued soon after it started to be used (Devins & Orme, 1985). Although group comparisons on the CESD scale have been around for a long time (e.g., Gatz & Hurwitz, 1990), the more recent work of Cole, Kawachi, Maller, and Berkman (2000) pointed out significant item bias between African American and White American responses to the CESD items (also see Yang & Jones, 2007). In our own prior work (McArdle, Johnson et al., 2001) we summarize the previous factor analytic work on the depressive symptom scales and find evidence for at least 3 or 4 measurable factors in the CESD (also see Beals et al., 1991). In this prior work, we also show how a simple form of rescaling of the CESD can lead to a simple and direct interpretation of resulting score as “the number of days of depression symptoms reported per week” (WCESD).

On a theoretical note, we are aware that the GPA and CESD scores have substantially different norms of reference. That is, the CESD is based on the same set of items measured on the same rating scale, so it is possible to examine both systematic growth and rapid changes in a common scale. However, GPA is not an ability score with a free range; instead, GPA is a score which is relative to a grade norm—that is, even if an individual presents a constant GPA across Grades 9–12, they are likely to be growing in the ability underlying the academic performance. In this way, GPA is much like an IQ (i.e., mental age / chronological age) score rather than a raw test score (i.e., mental age), and this limits our interpretations of growth and change that can emerge from the analysis of GPA trajectories.

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

The sample sizes are relatively small.

The approach taken here attempts to combine and overcome problems of all six types. The models used here are all based on incomplete longitudinal data, and the DSEM analysis approach has been used in several recent studies (see Ferrer & McArdle, 2010; Gerstorf et al., 2009; McArdle, 2009). In examples related to this specific problem, McArdle and Hamagami (2001) examined longitudinal data from the National Longitudinal Study of Youth (NLSY) and concluded that reading ability was a precursor to anti-social behavior. In a more extensive study using the NLSY, Grimm (2007) studied academic achievement and depression, and he concluded that academic achievement preceded the development of depression. These researchers all emphasize that people using the same data could come to different dynamic conclusions using different multivariate models. Therefore, we examine a few alternatives based on different forms of measurement models here.

A wide variety of psychosocial problems are being studied by the National Center on Indigenous Hawaiian Behavioral Health (NCIHBH), including the relationships between psychological health and culture. Data from a variety of sources indicate that persons of aboriginal Hawaiian ancestry are more “at risk for psychological problems” than residents of Hawai‘i of other ethnic backgrounds (e.g., Andrade et al., 2006; Danko, Johnson, Nagoshi, Yuen, Gidley, & Ahn, 1988; Werner & Smith, 2001; Yuen, Nahulu, Hishinuma, & Miyamoto, 2000). The aim of the HHSHS study was to assess the adjustment of a large community-based sample of adolescents of Hawaiian ancestry as well as a comparison sample of adolescents from other ancestral groups. The longitudinal approach used in prior work in the Hawaiian islands (Werner & Smith, 2001) partly inspired this cohort-sequential data collection design. The study further investigated the importance of demographic variation and measures of identification with the Native Hawaiian culture with indices of adjustment.

The NCIHBH has already conducted studies involving self-reported GPA and adjustment, utilizing a primarily Asian American and Pacific Islander high school sample. A statistically significant and negative relationship of −.18 (p < .05) was found between self-reported GPA and of the CESD (Hishinuma, Foster et al., 2001; Hishinuma et al., 2006). This negative association remained significant even when controlling for other demographic variables (i.e., ethnicity, gender, grade level, main wage earners’ educational attainment, main wage earners’ employment status) and academic measures (i.e., actual cumulative GPA, absolute difference between self-reported GPA and cumulative GPA).

This negative relationship found among GPA and CESD scores was particularly salient for children and adolescents of ethnic minority ancestry (Hishinuma, Foster et al., 2001; Hishinuma et al., 2006). Given the general finding of lower educational achievement being associated with poorer psychological adjustment (e.g., Hishinuma, Foster et al., 2001), and given the differential academic achievement among racial and ethnic groups (e.g., African Americans with lower high school GPAs than Caucasians; Caucasians with lower high school GPAs than Asian-Pacific Islanders), this area of study is critical in understanding the well-being of ethnic minorities. In the context of the present study, Native Hawaiian adolescents tend to have lower levels of achievement than non-Hawaiians in Hawai‘i (e.g., Hishinuma, Foster et al., 2001; Hishinuma, Johnson et al., 2001; Hishinuma et al., 2006).

The previous HHSHS analyses described the use of a multiple imputation (MI) procedure for handling incomplete data (McArdle, Johnson et al., 2001). This approach is reasonable for data missing within or over time, and it can be used when the incompleteness is due to attrition and other factors and the lack of data are considered missing at random (MAR; after Little & Rubin, 1987). The same assumptions underlie analyses based on any latent variable SEM which included “all the data”—not simply the complete cases (e.g., Horn & McArdle, 1980; McArdle, 1996). We expect non-random attrition, but our goal is to include all the longitudinal and cross-sectional data to provide the best estimate of the parameters of change as if everyone 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 computational terms, the available information for any participant on any data point (i.e., any variable at any occasion) is used to build up maximum likelihood estimates (MLEs) that optimize the model parameters with respect to any available data.

Another aspect of the methods used here is the investigation of a measurement model for each key outcome variable. In the case of GPAs, the scores are first treated in the typical way, as an interval scale, but due to its more uniform distribution an effort is made to consider this more realistically as an ordinal scale. Likewise, the CESD measurement of depressive symptoms is considered to be multi-factorial, so only some items are selected, and then due to the resulting skewness, models with both interval scaling and ordinal scaling are considered here. In addition, the requirement of invariance of measurement over high school grades is considered for each outcome separately in latent curve models, and then together in the dynamic systems models. These final dynamic models are compared over groups defined by gender and ethnicity.

In previous analyses, we selected data from any student who participated in the HHSHS in 1993 and 1994 (N>5,000; e.g., McArdle, Johnson et al., 2001). The analyses presented here are based on survey data obtained from high school students in Hawai‘i who participated in at least one of five waves of measurement (1992–1996; see Andrade et al., 1994, 2006). In this longitudinal-cohort study, a large sample (N=7,317) of high school students were surveyed on a variety of well-known psychometric instruments designed to indicate different aspects of psychological health. These data reflect a large but generally non-representative epidemiological sample of high school students; namely, in this study we intentionally selected primarily Hawaiian communities in Grades 9–12 from three of Hawai‘i’s major islands.

For the purposes of those analyses, we classified all students into one of four broad groups. Hawaiian versus non-Hawaiian ancestry was based on questions about the parents’ ethnic background. Students whose parents had any Hawaiian ancestry were classified as “Hawaiian” and all others were classified as “non-native Hawaiians.” A second grouping was based on the students’ self-reported gender (as Male or Female). These two groupings led to four student groups: (1) Native Hawaiian Females (N=1,733), (2) Native Hawaiian Males (N=1,456), (3) Non-native Hawaiian Females (N=973), and (4) Non-native Hawaiian Males (N=881). Additional information about these four groups can be found in other more detailed reports (see Andrade et al., 1994, 2006). The same HHSHS data collection is used here, but now we use all the longitudinal data available at any wave of measurement.

Parents and students were given written notification of the nature and purpose of the research study prior to administration, with the opportunity for parents to decline their youths’ participation. Students who provided their assent were administered the survey in their homerooms by their teachers. The majority of the surveys were completed by the students within 30–45 minutes. Based on the existing enrollments, approximately 60% of the students were surveyed. Separate analyses indicated that a higher proportion of males were not surveyed. In addition, those who were not surveyed had more absences, suspensions, and conduct infractions, and had lower actual GPAs (Andrade et al., 2006). All procedures were approved by the University of Hawai‘i’s Institutional Review Board.

This investigation entailed a cross-sequential (i.e., cross-sectional and longitudinal) design whereby students from two to five high schools (located on three of the Hawaiian Islands) were surveyed across a five-year period (school years 1991-92 to 1995-96). The initial plan was to over-sample Native Hawaiian adolescents by surveying students from three high schools (High Schools 1–3) that had a large proportion of youths of Native Hawaiian ancestry. Table 1 provides a description of the patterns of responses at the individual level (see also Andrade et al., 2006) and Figure 1 is a plot of a small sample of trajectories from this larger longitudinal database. Figure 1a is a plot of the trajectories of the GPAs of 30 randomly selected individuals, and Figure 1b is a plot of the same 30 people on their CESD scores. Complex patterns of fluctuation are evident and the connection among these trajectories is not obvious.

The HHSHS data used here are based on a complex sampling strategy highlighted in Table 1. This survey was conducted for the first three years of the study (1991-92 to 1993-94 school years) and for all high school grade levels (9^th−12^th). During Year 1 (1991-92), the decision was made to also survey in Year 2 (1992-93) students from two other high schools (High Schools 4–5) that would allow for more meaningful comparisons with non-Hawaiian adolescent cohort groups. As with High Schools 1–3, students from High Schools 4–5 were surveyed for all high school grade levels (9^th−12^th) in Years 2–3 (1992-93 & 1993-94). In order to obtain complete longitudinal data across all four grade levels for the 9^th graders who were surveyed in Year 1 (1991-92) for High Schools 1–3, and for the 9^th graders who were surveyed in Year 2 (1992-93) for High Schools 4–5, the decision was made to: (a) in Year 4 (1994-95) survey the 12^th graders from High Schools 1–3, and (b) in Year 4 (1994-95), survey 11^th and 12^th graders from High Schools 4–5, and in Year 5 (1995-96) survey 12^th graders from High Schools 4–5.

Among students who were in the 9^th 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 take the survey, but did not for whatever reason (e.g., parent declined, student declined, student moved to another school). Seventy-five participants were not included in the previous count due to anomalies such as repeating a grade level. For the 10^th grade: 3,286 (45.4%), 2,707 (37.4%), 1,249 (17.3%), respectively; for the 11^th grade: 2,685 (37.1%), 2,984 (41.2%), 1,573 (21.7%), respectively; and for 12^th grade: 1,822 (25.2%), 3,498 (48.3%), 1,922 (26.5%) respectively.

This series of data collection decisions and methods resulted in the project starting during the 1991-92 school year, ending during the 1995-96 school year, and variably surveying 9^th−12^th graders across five high schools at different points in time. A total of 7,317 students were surveyed resulting in 12,284 completed questionnaires. We note that the potential for a lack of convergence of these data (McArdle & Hamagami, 1991) is considered in all analyses.

Incomplete data techniques are now available in many current computer programs (e.g., SAS PROC MIXED; Littell, Milliken, Stroup, Wolfinger, & Schabenberger, 2006; Verbeke, Molenberghs, Krickeberg, & Fienberg, 2000), and we use Mplus 5.0 (Muthén & Muthén, 2006; see Ferrer, Hamagami, & McArdle, 2004). The Mplus program seems advantageous because it also allows us to deal with (a) survey sampling weights, (b) categorical measurement models, (c) multilevel models, and (d) a random-slopes approach to latent curve models (but see Ghisletta & McArdle, in press). The goodness-of-fit of each model presented here will be assessed using classical statistical principles relying on the model likelihood (f_MLE) and change in fit (χ²). In most models to follow, we use the MAR convergence assumption to deal with incomplete longitudinal records, and we discuss these assumptions later (e.g., Cnaan, Laird, & Slasor, 1997; Little, 1995).

These statistical models are also used to address group differences about the CESD and GPA at several different levels, and comparative statistical results are presented in the next four sections. Due to our relatively large sample size (N=7,317), only estimated coefficients whose 99% confidence boundary does not include zero are considered accurate estimates of effects. While this is precisely equivalent to stating an effect is not significant (at the α <.01 test level), this emphasis on the accuracy of estimates is more consistent with our overall modeling strategy. The overall goodness-of-fit of these models can also be examined in several ways (see Bollen & Long, 1993). However, most current SEM analyses rely on comparing observed and expected values based on a non-central chi-square (χ²) distribution (paralleling the F-distribution of ANOVA). Due to the relatively large sample size (N>7,000), we also examine statistical differences between models using the Browne and Cudeck (1993) Root Mean Square Error of Approximation (RMSEA, ε_a) and we calculate the “probability of close fit” to indicate a model with a discrepancy ε_a < .05. Our main goal is to use these empirical analyses to separate (a) which models seem consistent with the data and should be useful in future research, from (b) models which seem inconsistent with the data and should be dropped from further work.

The evaluation of models for change over time is conceptually based on longitudinal analyses of multiple trajectories (e.g., McArdle, 2009). There is a long history of embedding categorical concepts within repeated measures longitudinal data (e.g., Hamagami, 1998; Koch, Landis, Freeman, Freeman, & Lehnen, 1977; Muthén, 1984). These approaches can clarify the results if we have either (a) categorical outcomes which are not ordinal, or (b) if the order categories 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 exhaustive here.

One classic way to deal with non-normal outcomes is to use score transformations, but these will not help here due to the extreme limits of some of our outcomes (e.g., Table 2b). Instead, we first use standard SEM approaches for estimating growth and change under the assumption of an interval scaling of the outcomes (as in McArdle, Johnson et al., 2001). This is not often based on MLE-MAR assumptions, but it is possible. In a parallel set of analyses we consider the same longitudinal data but we expand the model to include a set of ordinal thresholds (τsee Hamagami, 1998) and we highlight any differences in the substantive results for the same data.

To carry out calculations for the ordinal approach, we rely on the approach created by Muthén (for LISCOMP software; see Muthén & Satorra, 1995). In this approach, the first step is to use MLE-MAR to estimate the empirical distance between each entry of a categorical variable. Since we have 9 possible ratings of GPA at each occasion, we can estimate 8 thresholds describing the difference between these categories. Since we have only 6 possible categories of Weekly CESD, we estimate 5 thresholds to describe the difference among these categories. In the second step, we assume the underlying latent variables are normally distributed (with mean 0 and variance 1) and we estimate the correlations among latent variables using polychoric procedures. Finally, in a third step, a consistent estimator of the asymptotic covariance matrix of the latent correlations is based on a Weighted Least Squares Mean (WLSM) Adjusted estimator. Additional model assumptions based on the structure of the thresholds (i.e., invariant over time) or the model correlations (following a latent curve hypothesis) are added and the comparative goodness of fit is obtained. If needed, formal tests of the difference between models are made more accurate by using kurtosis adjustments (scaling correction factors; see Browne, 1984; Satorra & Bentler, 1994). In any case, each of these steps is now fairly easy to implement with contemporary software such as Mplus (Muthén & Muthén, 2006).

All analyses presented will be based on longitudinal structural equation models (e.g., McArdle & Nesselroade, 2003, in press; Singer & Willett, 2003). To be clear, we will emphasize the representation of trajectories over time and the dynamic interpretations of model parameters (see McArdle, 2009; McArdle & Hamagami, 2004). In recent research, we have found it most flexible to represent the classical growth and change models based on multiple latent change scores (see Figure 4; McArdle, 2009). These models are practically useful because they encompass the popular latent growth models and autoregressive time series models. In addition, because they are formed in a different way, they easily extend to multivariate forms (to follow).

More formally, we first assume we have observed scores X[t] and X[t-1] measured over a defined interval of time (Δt), but we assume the latent variables (g[t]) are defined over an equal interval of time (Δt=1). This definition of an equal interval latent time scale is non-trivial because it allows us to eliminate Δt from the rest of the equations. This allows us to use latent scores g[t] and g[t-1], residual scores u[t] representing measurement error, and latent change scores Δg[t]. Even though this change Δg[t] is a theoretical score, and not simply a fixed linear combination, we can now write a structural model for any latent change trajectory. This simple algebraic device allows us to generally define the trajectory equation based on a summation (Σ _{i=1, t}) or accumulation of the latent changes (Δg[t]) up to time t.

These latent change score models allow a family of fairly complex nonlinear trajectory equations (e.g., non-homogeneous equations). These trajectories can be described by writing the implied basis coefficients (A_j[t]) as the linear accumulation of first differences for each variable (ΣΔg[j], j=0 to t). Of course, this accumulation can be created from any well-defined change model, and therefore, the procedure is quite flexible (e.g., see Ghisletta & McArdle, in press).

On a practical note, these latent change score structural expectations are automatically generated using any standard SEM software (e.g., R, LISREL, Mplus, Mx, etc.). That is, we do not directly define the basis (A[t]) coefficients of a trajectory equation, but we directly define changes as an accumulation 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).

A bivariate dynamic model can be used to relate the latent scores from one variable to another over time (see Figure 5). More formally, we first assume we have observed both sets of scores Y[t] and X[t] measured over a defined interval of time (Δt), but we assume the latent variables are defined over an equal interval of time (Δt=1). This is most useful when we start to examine time-dependent inter-relationships among multiple growth processes. In general, we write an expansion of our previous latent difference scores logic as the bivariate dynamic change score model. In this model, the g₁ and h₁ are the latent slope score which is constant over time, and the changes are based on additive parameters (α_y and α_x), multiplicative parameters (β_y and β_x), and coupling parameters (γ_yx and γ_xy). The coupling parameter (γ_yx) represents the time-dependent effect of latent x[t] on y[t], and the other coupling parameter (γ_xy) represents the time-dependent effect of latent y[t] on x[t]_. When there are multiple measured variables within each occasion, additional unique covariances within occasions (u_y[t] u_x[t]) are identifiable as well.

An initial description of some relevant summary statistics appears in Table 3 and Table 4. Here we include estimated thresholds, means, deviations, and correlations estimated using the most typical incomplete data algorithms (MLE-MAR in Mplus 5.0; Muthén & Muthén, 2006; program scripts available on our website). One thing to note is that the interval score estimates presented here do not have any correction for non-normality, while the ordinal score estimates have been corrected.

Table 3 presents the estimated thresholds for GPA (now termed GPA*) and the WCESD (now termed WCESD*). There were obtained using the SEM software Mplus 5.0 under the constraint that these thresholds represented differences among the response categories, and that they were the same (i.e., invariant MLE) at each of the four grades. We note that this simplification of parameters is not a necessary feature of the data, and the scale could change from one grade to the next, but this is a prerequisite for all models to follow, and therefore, we present it first.

Table 3a shows the estimated thresholds for GPAs from category D to C- (i.e., 1.0 to 1.7) has a value of τ₁=−2.14. Because this was estimated in a normal probability, or probit metric, this value indicates the location on a normal curve for people above and below this GPA point (i.e., approximately 2% below, and 98% above). The next estimated value of τ₁=−1.60 suggests a slightly larger number of people are likely to respond between 1.7 and 2.0. The vector of eight thresholds T=[ −2.1, −1.6, −0.9, −.00, +0.0, +0.7, +1.3, +2.0] is seen to increase ordinally, even though this was not restricted (i.e., it was simply categorical). The nonlinear nature of these changes in responses can be seen from the difficulty of shifting between responses formed here as a ratio of the estimated differences to the observed differences (as in McArdle & Epstein, 1987). These differences would be constant if the scaling was equal interval, but these are Δ=[=0, 1.8, 2.5, 2.1, 0.1, 2.3, 1.5, 3.4], and this indicates the apparent difficulty of moving between the response categories not being equal to the stated distance. For this reason, linear relations within and among variables are likely to be much better represented by an ordinal scale of GPA*.

Table 3b shows the estimated thresholds for the WCESD for the category based on 0 to 1 days with a value of τ₁=0.53. Again, because this was estimated in a probit metric, this value indicates the location on a normal curve for people above and below this GPA point (i.e., approximately 52% below and 48% above). The next estimated value of τ₁=+1.48 suggests a slightly smaller number of people are likely to respond between 1 and 2 days. The vector of five thresholds T= [ +0.5, +1.5, +2.1, +2.9, +3.8] is seen to increase in an ordered progression, even though this was not restricted (i.e., it was simply categorical). The simple ratio of the estimated differences to the observed differences would be constant if the scaling was equal interval, but these are Δ=[=0, 1.0, 0.7, 0.7, 0.5]. Therefore, these findings reflect the difficulty of moving between the response categories not being equal. Thus, we expect that linear relations within and among variables are likely to be much better represented by this kind of an ordinal scale of WCESD*.

Table 3c is a listing of the means and deviations over four grade levels that would result from an interval or ordinal scaling of the two variables. These estimates are all based on MLE-MAR, so they utilize all available longitudinal data (see Table 1). GPA in an interval scale has easy-to-understand means, which seem to increase over grade levels 9^th, 10^th, 11^th and 12^th from 2.69, 2.72, 2.78, to 2.95. Once again, high school dropouts have been included in these calculations, but only to the degree that we had other measured information (i.e., 9^th grade GPA). The estimates for the ordinal scaling of the GPA are slightly different suggesting the GPA* goes from 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 standard deviations are reduced in the GPA* metrics, and the changes are more complex for the WCESD* metrics.

Table 4 is a listing of the estimated Pearson correlations over four grade levels that would result from an interval scale (using invariant means and deviations). The correlations in parentheses are polychoric estimates of relationships based on ordinal scaling of the variables (using invariant thresholds). Each of these estimates is based on MLE-MAR, and therefore, they utilize all available longitudinal data (see Table 1). When comparing Pearson correlations and polychoric correlations, we find attenuation of correlations if Pearson correlations were opted for analyses due to the fact that measurements are non-normal (as in Hamagami, 1998). Table 4a is a list of the correlations of GPA over grade levels, and these are strong and positive (~r=+0.5 to +0.6). However, the use of ordinal scoring in GPA* yields a slightly higher correlation in every case. Table 4b is a list of the correlations of the WCESD over grade levels, and these are almost as strong and positive (~r=+0.4 to +0.5). Similar to GPA, however, the use of ordinal scores in WCESD* yields a slightly higher correlation in every case, suggesting that using the WCESD* is more accurate. The critical cross-correlations over time among these measures is listed in Table 4c for selected pairs of variables: (1) GPA[t] and WCESD[t] and (2) GPA[t]* and WCESD[t]*. The GPA[t] and WCESD[t] correlations are largely low and negative (~r=−0.06 to −0.15), and the use of the ordinal scaling does not seem to change the simple relationships very much (~r=−.04 to −.22). These differences in the patterns of correlations between Pearson and polychoric approaches indicate that dynamic characteristics derived from two different approaches may not be exactly the same.

The first set of longitudinal results of Table 5 is based on alternative change models for the GPA trajectories (in Figure 1a). The first three models (5a, 5b, 5c) assume interval scaled data and models are fit in increasing order of complexity. First, a baseline model with intercepts only is fitted (not shown) and this yields χ²=398 on df=11. The first model listed in Table 5a is the linear change model with one slope (labeled Model 5a) and fits much better with χ²=48 on df=8 (so ε=0.027), so the change in fit is large as well (Δχ²=350 on Δdf=3). This is remarkably small misfit for such a large sample size (N>7,000). The mean intercept for the 9^th grade is reasonable (µ₀=2.65) and the mean slope is positive but small (µ₁=0.09) indicating only small average changes over grade levels. The variability in these components shows somewhat large initial differences (σ₀²=0.42) with small systematic changes (σ₁²=0.02) and larger random changes (ψ²=0.22). The second model (5b) forces the systematic slope components to be zero, but allows an autoregressive component (β=0.037) to allow changes which accumulate. This auto-regression AR(1) model does not seem to fit the GPA data so well (χ²=114 on df=8 so ε=0.044). The third model (5c), termed a Dual change, allows both linear and autoregressive changes to occur, and this leads to the best fit (χ²=24 on df=7, so ε=0.019). This model suggests more complex changes due to the negative mean slope (µ₁=−1.79) combined with positive autoregressive changes (β=0.69), implying that the GPA would go down over grade levels except for the impact of the prior grades GPA.

The second set of three models (Table 5d, 5e, 5f) assume some ordinal thresholds are needed and then re-examines the same change models. Here we need to place several additional constraints on the latent variables to assure identification (i.e., µ₀=0, σ₀²=1) and estimate the 8 thresholds (τ₁−τ₈). While we could allow the thresholds to change over grade levels, we do not consider this possibility here, and this assumption of scale invariance over grade levels 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=1159 on df=29, so the ordinal linear model fitted to these is a great improvement in fit (i.e., χ²=174 on df=26, so Δχ²=985 on Δdf=3), with parameters representing some positive changes (µ₁=0.14, σ₁²=0.04) and larger random changes (ψ²=0.48). (We note that the kurtosis scaling coefficient for the WLSM estimator is reported (ω4=0.723, but, since our overall results will remain the same with or without this correction, this was not applied to adjust the chi-square tests.)

The next model (5e) is the same, but allows proportional auto-regressive changes without slopes, and it does not fit as well. The final model (5f) allows both types of changes to the ordinal scales, and it fits much better (i.e., χ²=94 on df=25, so Δχ²=80 on Δdf=1). Assuming this is our final GPA model, we have: (1) unequal distances in GPA scaling, (2) small positive changes in slopes over grade levels with large variations (µ₁=0.07, σ₁²=0.78), (3) large positive proportional changes from one GPA level to another (β=0.84), and (4) even larger random changes (ψ²=0.70). In this comparison, the first three models for GPA suggest the dual change is needed – all models for ordinal GPA* suggest unequal but ordered intervals, and the dual change model fits best. A plot of the expected values from this GPA* model is drawn in Figure 6a and this shows a small but significantly positive shift in the mean over grade levels, with increasing individual differences in trajectories.

The second set of longitudinal results of Table 6 is based on alternative change models for the WCESD trajectories (in Figure 1b). The first three models (6a, 6b, 6c) assume interval scaled data and models are fit in increasing order of complexity. First, a baseline model with intercepts only is fitted (not shown) and this yields χ²=74 on df=11. The first model listed in Table 6a is the linear change model with one slope (Model 6a) and fits much better with χ²=31 on df=8 (so ε=0.027), so the change in fit is large too (Δχ²=43 on Δdf=3). This is also a remarkably small misfit for such a large sample size (N>7,000). The mean intercept for the 9^th grade is reasonable (µ₀=1.48 days) and the mean slope is negative but small (µ₁=−0.03) indicating only small average changes over grade levels. The variability in these components shows somewhat large initial differences (σ₀²=0.57) with small systematic changes (σ₁²=0.01) and larger random changes (ψ²=0.49). The second model (6b) forces the systematic slope components to be zero, but allows an autoregressive component (β=−0.13) to allow changes to accumulate and level off. This AR(1) model seems to fit the WCESD data very well (χ²=21 on df=8 so ε=0.02). The third model, termed a Dual change, allows both linear and autoregressive changes to occur, and this leads to the best fit (χ²=29 on df=7 so ε=0.02). This model suggests more complex 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 WCESD; however, this model is not an improvement in fit. Indeed, the interval WCESD seems to fluctuate from one time to the next without a systematic linear change.

The second set of three models (6d, 6e, 6f) assumes some ordinal thresholds are needed and then re-examines the same change models. Here we need to place several additional constraints on the latent variables to assure identification (i.e., µ₀=0, σ₀²=1) and estimate the 5 thresholds (τ₁−τ₅). While we could allow the thresholds to change over grade levels, we do not consider this possibility here, and this assumption of scale invariance over grade levels leads to very little misfit. The estimated scaling coefficients (described earlier) range about 0 to +4 but suggest an unequal distance between WCESD points. The baseline model with intercepts only is fitted (not shown) and this yields χ²=61 on df=29, and therefore, the ordinal linear model fitted to WCESD* is a minor improvement in fit (i.e., χ²=14 on df=17, so Δχ²=47 on Δdf=12), with parameters representing some negative changes (µ₁=−0.46, σ₁²=0.01) and larger random changes (ψ²=0.78). (Again, we note that the kurtosis scaling coefficient for the WLSM estimator is reported [ω4=0.772], because this has an effect on the resulting chi-square tests.)

The next model (6e) is the same, but allows proportional auto-regressive changes without slopes, and it fits only slightly better (i.e., χ²=11 on df=17). The final model (6f) allows both types of changes to the ordinal scales, and it fits about the same (i.e., χ²=14 on df=16). Assuming this is our final WCESD* model, we find: (1) unequal distances in WCESD scaling, (2) small negative changes in slopes over grade levels with large variations (µ₁=−0.04, σ₁²=0.10), (3) small positive proportional changes from one WCESD level to another (β=0.03), and (4) even larger random changes (ψ²=0.77). In this comparison, the first three models for WCESD suggest that only the autoregressive part of the dual change is needed. All models for ordinal WCESD* suggest unequal but ordered intervals, and the dual change model is possible. A plot of the expected values from this WCESD* model is drawn in Figure 6b and this shows a small but significantly negative shift in the mean over grades, with increasing individual differences in trajectories.

The results of Table 7 give the SEM parameter estimates for a dynamic system where GPA and WCESD are allowed to impact the trajectory of each another over 9^th to 12^th grades based on the full bivariate dynamic path diagram of Figure 5. The first model (Table 7a) is based on interval scaling of both variables, GPA and WCESD as observed, and the second model (7b) is based on estimation using the ordinal concepts developed up to this point. (The authors provide an Mplus computer program for these models in a Technical Appendix on their website). Perhaps it is obvious, but the dynamic results differ depending upon how the variables are scaled.

In the Interval measurement model (7a), we use the scale of measurement as it is calculated from the scores in the data, and this assumes an equal interval between score points for both GPA and WCESD. Of course, our prior plots and models fitted have suggested this is largely an incorrect assumption for both variables. Nevertheless, since interval scaling is typically assumed, we start from this point. As a result, we observe strong dynamic parameters with GPA, little or no dynamic action of the WCESD, and little or no systematic coupling across variables. The impact of GPA on the changes in WCESD is now negligible (γ=−0.25, t~1), and the impact of WCESD on the changes in GPA, while seeming a bit larger, is also negligible (γ=−0.63, t~1). The overall model fits the data well (χ²=56 on df=23, ε=0.014), but a model with either or both couplings set to zero (χ²=57 on df=24, or χ²=58 on df=25) fit about the same (ε=0.014). In contrast, a model where we assume no cross-lagged impacts at all (i.e., a “parallel process” model) does lead to a substantial loss of fit (χ²=83 on df=27) and a model assuming no slopes at all (i.e., a standard cross-lagged regression) is even worse (χ²=168 on df=32). Therefore, while there are some dynamic influences operating in the interval scaling, the direction of impacts over grade levels in not very clear – GPA and WCESD appear to be correlated by not dependent processes.

In the Ordinal measurement model (Table 7b), we do not assume 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 measurement model for the GPA* and the WCESD* requires some kinds of identification constraints and we employ the ones used earlier (µ₀=0, σ₀²=1), but now for both variables. When the dynamic model is estimated for these ordinal latent variables, a result emerges: the impact of GPA* on the changes in WCESD* is clearly negligible (γ=−0.07, t<1), but the impact of WCESD* on the changes in GPA* is now significant and large (γ=−2.17, t>2.5). The fits listed show models where: (a) the overall model fits the data well (χ²=83 on df=50, ε=0.009), (b) a GPA*→ WCESD* coupling set to zero fits just as well (χ²=82 on df=51), (c) the WCESD*→ GPA* coupling set to zero does not fit as well (χ²=113 on df=51), and (d) both couplings set to zero does not fit either (χ²=111 on df=52). In addition, a model where we assume no cross-lagged impacts at all (i.e., a “parallel process” model) leads to a substantial loss of fit (χ²=191, on df=54) and a model assuming no slopes at all (i.e., a standard cross-lagged regression) is even worse (χ²=1172 on df=59). Therefore, in this ordinal scaling there seems to be directional dynamic influences operating – the direction of impacts accumulated over all grade levels suggests that WCESD*→ GPA* and not the other way around.

The use of a vector field plot was introduced to assist in the interpretation of the size of the impacts of the dynamic results (see Boker & McArdle, 2005). In Figure 7, these kinds of vector field plots are displayed for both the Interval model (Table 7a, Figure 7a) and for the Ordinal model (Table 7b, Figure 7b). In both plots we derive the expected direction of the trajectory for all points across a reasonable span of scale points. We then display these expectations as small arrows (i.e., directional derivatives) and we interpret the overall picture as in terms of the overall flow of the arrows. We include a 95% confidence ellipsoid around the starting points based on the available data, but we recognize that expectation outside of these boundaries may be among the most interesting flows in the diagram. The first display (Figure 7a) shows the Interval model with very little cross-variable movement, especially inside the 95% bounds of the observed data. The second display (Figure 7b) shows the Ordinal model has a little trace of cross-variable movement, where the GPA* is moving down for increasing levels of WCESD*. In both cases there is exaggerated movement outside the 95% bounds of the observed data, but these arrows reflect expectations for scores that do not appear in the current data. There does not appear to be any dynamic action in the Interval model, but there is some important dynamic action in the Ordinal model where the WCESD* effects changes in the GPA* and not the other way around.

To add to our understanding of the dynamics, we can compare demographic groups, such as Hawaiians and Non-Hawaiians, and Females versus Males. There are several ways to consider group differences in the bivariate latent difference scores models, and we present a few of these analyses next (for details, see McArdle, 2009; McArdle & Hamagami, 1996). In one form of a multi-level DSEM, we simply use the group information to describe the distribution of the latent level and latent slopes. This is equivalent to suggesting that the dynamic process based on the coupling coefficients is invariant over groups, but that the position of the groups may differ in the vector field (i.e., different 95% confidence ellipsoids). A more flexible way to deal with these dynamic models is to examine the possibility of qualitative differences in dynamic systems among the WCESD* and GPA* longitudinal processes over the groups – that is, we ask, “Do the dynamics operate the same way in each sub-group?” One practical way to examine this form of invariance is to split up the data into groups, allow the dynamics to differ, and see if these relaxed assumptions improve the model fit substantially. This simple strategy requires a multiple group approach and, to retain power within each group, many participants are needed within in each group (McArdle & Hamagami, 1996). By splitting up the data into only four groups based only on Ethnicity and Gender we can meet the sample size requirements in the HHSHS. Results for this multiple group form of group dynamics is presented in Table 8 and Figure 8.

The resultant bivariate dynamics for the four adolescent groups are first listed separately in Table 8 – i.e., results are listed for the Male-Hawaiian adolescent group (Table 8a and Figure 8a), the Male Non-Hawaiian group (Table 8b and Figure 8b), and so on. In all groups, the status of WCESD* dictates how both WCESD* and GPA* change. WCESD* alone seems to decline itself, while GPA* is negatively influenced by the status of depression. The significant negative coupling of WCESD* on a change score of GPA* intimates that a change in GPA* is solely based on the previous state of WCESD*. The higher the depressive state at the current state, the larger the negative change in GPA*. While the plots in Figure 8 are not exactly the same, they are all very similar.

We started with the separate models of Table 8 and added invariance constraints to the four-group model, and we found: (1) A model which is invariant over Ethnicity yields a relatively small loss of fit of Δχ² =50 on Δdf = 13; and (2) A model which is invariant over Gender yields a relatively large loss of fit of Δχ² =321 on Δdf = 13. As a result, the idea that these subgroups should be treated as one large group having equivalent dynamic systems is a reasonable one, except for some notable Gender differences. That is, while there are slightly different parameters for Hawaiian and Non-Hawaiians, there is little evidence for differences in dynamic systems for different Ethnic groups. However, it appears Males and Females are operating on slightly different dynamic systems and should probably not be combined.

Finally, to check our within-time equality constraints (on βs and γs) we examined the equality of dynamic constraints over time. In the previous models listed above we required the same dynamic result to appear over every time interval – from 9^th to 10^th, 10^th to 11^th, and 11^th to 12^th. In these final analyses, we relaxed these dynamic assumptions, looked at the groups as a whole, and within each separate sub-group, and we basically found no evidence for differential dynamics over grade level or time.