The scientific literature consistently supports a negative relationship between adolescent depression and educational achievement, but we are certainly less sure on the causal determinants for this robust association. In this paper we present multivariate data from a longitudinal cohort-sequential study of high school students in Hawai‘i (following

Many important studies in psychology deal with the relationships among some forms of psychopathology connected with real-life outcomes. Questions about the likely sequence of effects are often a fundamental aspect of developmental research (e.g.,

In this paper we present multivariate data from a longitudinal cohort-sequential study of high school students in Hawai’i (see

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

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

A variety of other relevant studies are reviewed by ^{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

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

Similarly, the use of the 20-item scale of the

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 samples have not involved large samples of Asian American and Pacific Islander adolescents.

The literature shows little consideration of longitudinal data on GPA and depressive symptoms.

The literature shows little consideration of the issues of incomplete data in analyses (e.g.,

There is little consideration given to the measurement or scaling of the key outcome variables as part of data analysis, even though it is known that scaling of measurements is critical to understanding lead-lag relationships.

More recent SEM analyses have been presented using more contemporary growth and trajectory modeling (e.g.,

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

A wide variety of psychosocial problems are being studied by the

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 (

This negative relationship found among GPA and CESD scores was particularly salient for children and adolescents of ethnic minority ancestry (

The previous HHSHS analyses described the use of a

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 (

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 (

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 (

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.

The HHSHS data used here are based on a complex sampling strategy highlighted in ^{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 (

This variable was operationally defined by a single survey question, “On the average, what were your grades on your last report card?” with 10 response choices offered. This measurement of academic achievement demonstrated high concurrent validity with actual cumulative GPA in a sub-study of the same persons (

In subsequent calculations, we reconstructed this GPA variable 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).

In one section of the 30–45 minute survey all students were asked to rate their depressive symptoms using the well-known

The unidimensionality of the CESD items has been questioned in prior research (

Incomplete data techniques are now available in many current computer programs (e.g., SAS PROC MIXED; _{MLE}) and change in fit (χ^{2}). 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.,

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 (^{2}) distribution (paralleling the _{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.,

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.,

To carry out calculations for the ordinal approach, we rely on the approach created by Muthén (for LISCOMP software; see

All analyses presented will be based on longitudinal structural equation models (e.g.,

More formally, we first assume we have observed scores _{i=1, t}) or accumulation of the latent changes (Δ

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 (_{j}[t]) as the linear accumulation of first differences for each variable (ΣΔ

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 bivariate dynamic model can be used to relate the latent scores from one variable to another over time (see _{1} and _{1} are the latent slope score which is constant over time, and the changes are based on additive parameters (_{y}_{x}_{y}_{x}_{yx}_{xy}_{yx}_{xy}_{.} When there are multiple measured variables within each occasion, additional unique covariances within occasions (_{y}_{x}

An initial description of some relevant summary statistics appears in

_{1}=−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 τ_{1}=−1.60 suggests a slightly larger number of people are likely to respond between 1.7 and 2.0. The vector of eight thresholds

_{1}=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 τ_{1}=+1.48 suggests a slightly smaller number of people are likely to respond between 1 and 2 days. The vector of five thresholds

^{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.

The first set of longitudinal results of ^{2}=398 on ^{2}=48 on ^{2}=350 on Δ^{th} grade is reasonable (µ_{0}=2.65) and the mean slope is positive but small (µ_{1}=0.09) indicating only small average changes over grade levels. The variability in these components shows somewhat large initial differences (σ_{0}^{2}=0.42) with small systematic changes (σ_{1}^{2}=0.02) and larger random changes (ψ^{2}=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 (χ^{2}=114 on ^{2}=24 on _{1}=−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 (_{0}=0, σ_{0}^{2}=1) and estimate the 8 thresholds (τ_{1}−τ_{8}). 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 χ^{2}=1159 on ^{2}=174 on ^{2}=985 on Δ_{1}=0.14, σ_{1}^{2}=0.04) and larger random changes (ψ^{2}=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., χ^{2}=94 on ^{2}=80 on Δ_{1}=0.07, σ_{1}^{2}=0.78), (3) large positive proportional changes from one GPA level to another (β=0.84), and (4) even larger random changes (ψ^{2}=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

The second set of longitudinal results of ^{2}=74 on ^{2}=31 on ^{2}=43 on Δ^{th} grade is reasonable (µ_{0}=1.48 days) and the mean slope is negative but small (µ_{1}=−0.03) indicating only small average changes over grade levels. The variability in these components shows somewhat large initial differences (σ_{0}^{2}=0.57) with small systematic changes (σ_{1}^{2}=0.01) and larger random changes (ψ^{2}=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 (χ^{2}=21 on ^{2}=29 on _{1}=−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., µ_{0}=0, σ_{0}^{2}=1) and estimate the 5 thresholds (τ_{1}−τ_{5}). 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 χ^{2}=61 on ^{2}=14 on ^{2}=47 on Δ_{1}=−0.46, σ_{1}^{2}=0.01) and larger random changes (ψ^{2}=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., χ^{2}=11 on ^{2}=14 on _{1}=−0.04, σ_{1}^{2}=0.10), (3) small positive proportional changes from one WCESD level to another (β=0.03), and (4) even larger random changes (ψ^{2}=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

The results of ^{th} to 12^{th} grades based on the full bivariate dynamic path diagram of

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, ^{2}=56 on ^{2}=57 on ^{2}=58 on ^{2}=83 on ^{2}=168 on

In the Ordinal measurement model (_{0}=0, σ_{0}^{2}=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, ^{2}=83 on ^{2}=82 on ^{2}=113 on ^{2}=111 on ^{2}=191, on ^{2}=1172 on

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

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

The resultant bivariate dynamics for the four adolescent groups are first listed separately in

We started with the separate models of ^{2} =50 on Δ^{2} =321 on Δ

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.

The primary issue raised in this paper is not a new one – “Does depression lead to poor academic achievement OR does poor academic achievement lead to depression?” Of course, it would not be ethical to deal with this kind of question on a randomized experimental trial design (see

Second, we demonstrated that that the scaling of the variables can make a difference in the dynamic interpretation between academic achievement (GPA) and depressive symptoms (CESD). From our results, we learned that the scaling of the WCESD is highly skewed (see

And third, in general, affective states of youth in our sample from Hawai‘i seemed to manifest in how they focused on school performances. This has many implications from clinical and prevention standpoints (see also

In our final analyses, we learned more about gender differences and ethnicity differences. In these analyses, we found the ethnic differences were not apparent, but the gender differences were large from a number of points of view. That is, the females seemed to be experiencing both more of, and a potentially more virulent form of an impact of increasing depressive symptoms leading to lowering of changes in academic achievement. These finding suggest that screening for prevention and treatment may be particularly important for girls.

There are many limitations of the modeling approach used here, some of these based on the current analyses and some based on the available data. The statistical model used is fairly flexible, but it still presumes invariance at several levels largely because parameter identification becomes much more difficult when more complexity is added. Our initial assumptions about interval scaling were relaxed and made ordinal, but the results presented here did not fully evaluate the assumptions of an invariant over-grade measurement model, and these assumptions are worthwhile examining in more detail. In addition, for example, in all the DSEMs fitted here, we assumed a dynamic process that takes the same amount of time no matter what pair of grade levels we were considering. We recognize that it is possible that more changes occur between some grade levels (i.e., Grades 9 and 10) than between others (i.e., Grades 10 and 11), or that the changes between longer times is 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. For example, the assumption of invariance over different patterns of incomplete data is likely to be incorrect to some degree and we know that specific forms of self-selection effects can lead to parameter bias (see

The dynamic analyses presented here clearly pointed out a few problems that can be overcome by using contemporary modeling procedures. This DSEM approach can be used to turn important developmental questions about temporal sequences into statistically powerful hypotheses. The fact that this can be done in the presence of ordinal level measurement and large amounts of incomplete data was also demonstrated, and this is a necessity for most real life-situations. We hope this approach can be useful for many other studies where multivariate longitudinal data have been collected to gain some insight into an ongoing developmental process. Such findings, in turn, should have both scientific and applied value.

Thanks must be given to our mentors, Dr. Ronald C Johnson (deceased) and Dr. Naleen Andrade, for their constant support in data collection and data analysis. The initial presentation of this work was done by the first author at the 2003 Statistical Modeling Seminars, Department of Psychiatry, John A. Burns School of Medicine, University of Hawai‘i at Mānoa, Honolulu, Hawai‘i, January 2003. The same longitudinal data and latent change analyses are described and presented in a more substantive context by

Observed longitudinal trajectories on the two key indicators variable for a random subset of

Observed within-grade-level distributions of Grade Point Average (GPA) for all persons (

Observed within-grade-level distributions for Weighted Center for Epidemiologic Studies-Depression (WCESD) for all persons (

A univariate dual change score model to examine trajectories on grade level changes for each indicator (_{0}), mean intercept (µ_{0}), and mean slope (µ_{1}). Notes: The α represents constant changes and defines the form of the slope factor _{1} (i.e., a=1 is linear change). The β represents the size of the proportional auto-regressive changes. A correlation between the intercept and slope (ρ_{01}) is allowed, and the error variance (ψ^{2}) is assumed to be constant at each grade level.

A bivariate biometric dual change score model trajectories over grade level changes in two measured variables (^{2}) is assumed to be constant at each grade level within each factor; α_{y} and α_{x} represent constant change related to the slope factors _{s} and _{s}; β_{y} and β_{x} represent proportional change in _{yx} and γ_{xy}. The model includes estimates for intercepts (_{0} and _{0}), mean intercepts (µ_{y0} and µ_{x0}), and mean slopes (µ_{y1} and µ_{x1}). Other parameters are used to generate the decomposition of the correlation between the intercept and slope for

Expected longitudinal trajectories on the two key indicators variable for a random subset of

Vector field plots of expected longitudinal trajectories from the Interval and Ordinal bivariate latent change score (BLCS) models (see

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

Summary of Available Data in the Hawaiian High Schools Health Survey (HHSHS) Collection Waves by Year, High School, and Grade Level (N = 7,317; D=12,284 Surveys Completed)

School Year | |||||||
---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | |||

High | Grade | 1991-92 | 1992-93 | 1993-94 | 1994-95 | 1995-96 | Total |

1 | 9^{th} | 103 | 70 | 100 | 273 | ||

10^{th} | 88 | 82 | 76 | 246 | |||

11^{th} | 75 | 70 | 84 | 229 | |||

12^{th} | 75 | 47 | 66 | 71 | 259 | ||

Incomplete | 0 | 0 | 1 | 1 | |||

2 | 9^{th} | 119 | 110 | 118 | 347 | ||

10^{th} | 94 | 69 | 100 | 263 | |||

11^{th} | 85 | 56 | 79 | 220 | |||

12^{th} | 71 | 37 | 54 | 23 | 185 | ||

Incomplete | 0 | 0 | 3 | 3 | |||

3 | 9^{th} | 354 | 364 | 290 | 1,008 | ||

10^{th} | 315 | 335 | 326 | 976 | |||

11^{th} | 293 | 319 | 297 | 909 | |||

12^{th} | 318 | 304 | 298 | 296 | 1,216 | ||

Incomplete | 0 | 2 | 3 | 1 | 6 | ||

4 | 9^{th} | 417 | 385 | 802 | |||

10^{th} | 388 | 371 | 759 | ||||

11^{th} | 329 | 312 | 327 | 968 | |||

12^{th} | 229 | 275 | 294 | 310 | 1,108 | ||

Incomplete | 1 | 4 | 4 | 9 | |||

5 | 9^{th} | 265 | 288 | 553 | |||

10^{th} | 243 | 244 | 487 | ||||

11^{th} | 251 | 220 | 221 | 692 | |||

12^{th} | 172 | 185 | 194 | 205 | 756 | ||

Incomplete | 4 | 3 | 2 | 9 | |||

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

Students who did not provide their grade level within the survey.

Summary of Estimated Categorical Data Percentages (for sample sizes, see

2a: Observed Grade Point Average Category Percentages | |||||
---|---|---|---|---|---|

Category | Interpretation | Grade 9 | Grade 10 | Grade 11 | Grade 12 |

1 | A (4.0) | 5.9 | 5.1 | 3.4 | 2.2 |

2 | A- (3.7) | 6.0 | 5.3 | 4.9 | 3.4 |

3 | B+ (3.3) | 16.0 | 14.0 | 12.0 | 9.0 |

4 | B (3.0) | 12.2 | 12.9 | 12.0 | 10.1 |

5 | B- (2.7) | 10.4 | 10.7 | 11.9 | 11.9 |

6. | C+ (2.3) | 20.4 | 21.8 | 20.7 | 20.3 |

7 | C (2.0) | 11.0 | 14.7 | 16.2 | 17.4 |

8 | C- (1.7) | 9.4 | 9.6 | 11.8 | 15.0 |

9 | D or less (1.0) | 8.7 | 5.9 | 7.1 | 10.8 |

b: Observed Weekly CESD Category Percentages (Rounded Weekly Ratings of 13 Items) | |||||
---|---|---|---|---|---|

Category | Interpretation | Grade 9 | Grade 10 | Grade 11 | Grade 12 |

1 | 0–1 Days | 64.2 | 67.2 | 66.9 | 68.6 |

2 | 1–2 Days | 21.3 | 18.7 | 20.3 | 19.9 |

3 | 2–3 Days | 8.1 | 8.0 | 7.5 | 6.9 |

4 | 3–4 Days | 4.1 | 4.1 | 3.6 | 3.3 |

5 | 4–5 Days | 1.8 | 1.8 | 1.4 | 1.1 |

6 | 5–7 Days | 0.4 | 0.2 | 0.3 | 0.2 |

Estimated Thresholds, Means, and Standard Deviations from the Hawaiian High Schools Health Survey for Grade Point Averages (GPAs) and the Center for Epidemiologic Studies-Depression (CESD) Scale

a Estimated (maximum likelihood estimate [MLE]) of ordinal thresholds for GPA (invariant over all for four grade levels) | ||||||||
---|---|---|---|---|---|---|---|---|

τ_{1} | τ_{2} | τ_{3} | τ_{4} | τ_{5} | τ_{6} | τ_{7} | τ_{8} | |

GPA units | 1.0 to 1.7 | 1.7 to 2.0 | 2.0 to 2.3 | 2.3 to 2.7 | 2.7 to 3.0 | 3.0 to 3.3 | 3.3 to 3.7 | 3.7 to 4.0 |

MLE GPA | −2.14 | −1.60 | −0.86 | −.039 | 0.01 | 0.71 | 1.31 | 2.03 |

Ratio of Distances | =0 | 1.80 | 2.47 | 2.05 | 0.13 | 2.33 | 1.50 | 3.43 |

b Estimated (MLE) ordinal thresholds for CESD (invariant over all for four grade levels) | |||||
---|---|---|---|---|---|

τ_{1} | τ_{2} | τ_{3} | τ_{4} | τ_{5} | |

WCESD (days) | 0–1 to 1–2 | 1–2 to 2–3 | 2–3 to 3–4 | 3–4 to 4–5 | 4–5 to 5–7 |

MLE WCESD | 0.53 | 1.48 | 2.14 | 2.85 | 3.78 |

Ratio of Distances | =0 | 0.95 | 0.66 | 0.71 | 0.46 |

c Estimated means (MLE) and standard deviations (SDs) for four grade levels | ||||
---|---|---|---|---|

Parameters & Fit | 9^{th} Grade | 10^{th} Grade | 11^{th} Grade | 12^{th} Grade |

GPA (GPA | 2.69 (=0) | 2.72 (0.31) | 2.78 (0.19) | 2.95 (0.47) |

WCESD (WCESD | 1.48 (=0) | 1.44 (−0.09) | 1.44 (−0.08) | 1.38 (−0.13) |

GPA (GPA | .792 (=1) | .749 (.670) | .732 (.625) | .712 (.642) |

WCESD (WCESD | 1.02 (=1) | 1.01 (1.060) | .976 (.919) | .929 (.800) |

Note:

indicates ordinal version of interval scale; see sample sizes in

Estimated Pearson Product Moment (or Tetrachoric) Correlations from Hawaiian High Schools Health Survey for Grade Point Averages (GPAs) and the Center for Epidemiologic Studies-Depression (CESD) Scale

a Correlations of Interval GPA Over Time (and Ordinal GPA | ||||
---|---|---|---|---|

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) |

b Correlations of Interval WCESD Over Time (and Ordinal WCESD | ||||
---|---|---|---|---|

Correlations | WCESD[9] | WCESD[10] | WCESD[11] | WCESD[12] |

WCESD[9] | 1.00 (1.00) | |||

WCESD[10] | .472 (.550) | 1.00 (1.00) | ||

WCESD[11] | .452 (.482) | .503 (.539) | 1.00 (1.00) | |

WCESD[12] | .458 (.509) | .453 (.511) | .520 (.563) | 1.00 (1.00) |

c Correlations of Interval GPA with Interval WCESD (and Ordinal GPA | ||||
---|---|---|---|---|

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

WCESD[9] | −.130 (−.154) | −.086 (−.085) | −.063 (−.037) | −.122 (−.082) |

WCESD[10] | −.095 (−.077) | −.148 (−.162) | −.082 (−.152) | −.059 (−.076) |

WCESD[11] | −.098 (−.109) | −.120 (−.149) | −.135 (−.141) | −.080 (−.117) |

WCESD[12] | −.096 (−.223) | −.094 (−.174) | −.083 (−.148) | −.122 (−.156) |

Note:

indicates ordinal version of interval scale; see sample sizes in

Alternative Latent Curve Results from Fitting Univariate Dual Change Score Model to Different Scalings of Grade Point Averages (GPAs) (Ordinal GPA

Parameters & Fit | (a) | (b) | (c) | (d) | (e) | (f) |
---|---|---|---|---|---|---|

Mean intercept, µ_{0} | 2.65 (211) | 2.69 (201) | 2.68 (210) | =0 | =0 | =0 |

Mean slope, µ_{1} | 0.09 (17.) | =0 | −1.79 (3.3) | 0.14 (14) | =0 | 0.07 (4.6) |

Constant change, α | =1 | =0 | =1 | =1 | =0 | =1 |

Proportion 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 |

Variable Intercepts, µ | =0 | −0.07 (3.6) | =0 | =0 | 0.16 (15) | =0 |

Intercept Variance, σ_{0}^{2} | 0.42 (25) | 0.42 (25) | 0.40 (25) | =1 | = 1 | =1 |

Slope variance, σ_{1}^{2} | 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, ψ^{2} | .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) | =0 |

Misfit index χ^{2} / | 48./8 | 114./8 | 24./7 | 174./26 | 246./26 | 94./25 |

Change in fit Δχ^{2} / Δ | 350./3 | 284./3 | 24./1; 90./1 | 985./3 | 913./3 | 80/1; 152./1 |

WLSM 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 (WLSM) estimator based on weighted least squares plus mean adjusted variances. Sample size

=1159/29, κ=0.871, ε=0.075. Obtain Technical Appendix (from authors) for Mplus input and output. AR = autoregression. Root Mean Square Error of Approximation = RMSEA.

Alternative Latent Curve Results from Fitting Univariate Dual Change Score Model to Interval and Ordinal Scalings of the Weighted Center for Epidemiologic Studies-Depression (WCESD) Scale (rescaled version of 13 items of Factor 1)

Parameters & Fit | (a) | (b) | (c) | (d) | (e) | (f) |
---|---|---|---|---|---|---|

Mean intercept, µ_{0} | 1.48 (91) | 1.60 (100) | 1.47 (103) | =0 | =0 | =0 |

Mean slope, µ_{1} | −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 | =1 |

Proportion 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 |

Variable Intercepts, µ | =0 | 0.18 (3.4) | =0 | =0 | −0.05 (2.6) | =0 |

Intercept Variance, σ_{0}^{2} | 0.57 (20) | 0.51 (15) | 0.53 (22) | =1 | =1 | =1 |

Slope Variance, σ_{1}^{2} | 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, ψ^{2} | 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) | =0 |

Misfit index χ^{2} / | 31.3 / 8 | 21.9 / 8 | 29.2 / 7 | 13.7 / 17 | 10.7 / 17 | 13.8 / 16 |

Change in fit Δχ^{2} / Δ | 42.7 / 3 | 51.1 / 3 | 1.9/1; −8.3/1 | 47.3 / 3 | 50.3 / 3 | −0.1/1;−2.9/1 |

WLSM 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 (WLSM) estimator based on weighted least squares plus mean adjusted variances. Sample size

=61./20, m=0.900, ε=0.02; AR = auto-regression; Root Mean Square Error of Approximation = RMSEA.

Dynamic Structural Equation Model (SEM) Results from a Bivariate Dual Change Score Model for Interval and Ordinal Scalings of Grade Point Averages (GPAs) and Weighted Center for Epidemiologic Studies-Depression (WCESD) Scale, including Alternative Misfits

(a) Interval Scaling | (b) Ordinal* Scaling | |||
---|---|---|---|---|

Parameters | GPA | WCESD | GPA* | WCESD* |

Mean intercept, µ_{0} | 2.68 (202) | 1.48 (81) | =0 (=0) | =0 (=0) |

Mean slope, µ_{1} | −0.66 (−.58) | 0.45 (.35) | 0.026 (1.00) | −0.052 (−2.34) |

Constant change, α | =1 | =1 | =1 | =1 |

Proportional change, β | − | − | ||

Coupling, γ GPA → WCESD | ||||

Coupling, γ WCESD → GPA | ||||

Scaling coefficients, T | −1.8, −1.4, −0.7, | 0.5, 1.4, | ||

Intercept 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) |

Cova. Slope & Level, σ_{01} | −0.31 (3.7) | −0.12 (.39) | −0.45 (−1.5) | −0.03 (−0.07) |

Unique variance, ψ^{2} | 0.22 (35) | 0.49 (35) | 0.43 (14.7) | 0.81 (13.1) |

“Dual Coupling” Misfit | 55.9 / 23 / 0.014 | 82.6 / 50 / 0.009 | ||

“No GPA→ WCESD” | 56.7 / 24 / 0.014 | 82.0 / 51 / 0.009 | ||

“No WCESD→ GPA” | 57.7 / 24 / 0.014 | 113. / 51 / 0.013 | ||

“No Couplings” Misfit | 58.0 / 25 / 0.014 | 111. / 52 / 0.013 | ||

“No Cross-Lags” Misfit | 83.0 / 27 / 0.017 | 191. / 54 / 0.019 | ||

“No Slopes” Misfit | 168. / 32 / 0.024 | 1172. / 59 / 0.060 |

Notes: “Dual” model sample size is

Maximum Likelihood Estimate-Missing at Random (MLE-MAR) 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

(a) Male | (b) Male | (c) Female | (d) Female | |
---|---|---|---|---|

µ_{WCESD} | −.16 | −.10 | −.09 | −.11 |

(2.37) | (1.20) | (3.32) | (1.07) | |

µ_{GPA} | .04 | .08 | −.02 | .002 |

(1.50) | (1.91) | (.40) | (.04) | |

β_{WCESD} | −.98 | −1.02 | −.86 | −.65 |

(11.1) | (9.61) | (3.79) | (3.78) | |

β_{GPA} | .42 | .26 | −.07 | .25 |

(1.09) | (.73) | (.18) | (.67) | |

γ_{GPA} →_{WCESD} | −. | −. | −. | . |

(1.76) | ( .77) | (.69) | (.33) | |

γ_{WCESD} →_{GPA} | −. | −. | −. | −. |

(2.72) | (6.22) | (7.24) | (2.78) | |

χ^{2} | 81 | 74 | 42 | 63 |

52 | 51 | 52 | 50 | |

ε_{a} | .015 | .014 | .000 | .015 |

Notes:

indicates that there is no response in one category in WCESD, thus one less _{a} = Root Mean Square Error of Approximation

indicates that there is no response in the last category in WCESD in Grades 9 and 10, with the highest threshold parameter rendered un-estimable;

Variances of all slope components are fixed at 1.0 for simplicity; εa = Root Mean Square Error of Approximation