Educational researchers and practitioners are often interested in questions about how individuals grow and develop over time. For example, how rapidly does a student's understanding of various mathematical concepts changes during secondary school? (Ai, 2002). To effectively answer such questions, longitudinal data (Singer & Willett, 2003) must be collected in which the same information may be obtained from individuals at different times. For example, to study the development of mathematics achievement during secondary school, yearly mathematics achievement tests could be administered. Longitudinal data is contrasted with cross-sectional data, in which information is collected from individuals at one point in time (e.g., mathematics achievement tests of children of different ages at the same point in time). While cross-sectional data is easier to collect than longitudinal data, it cannot be used to accurately answer questions about change.
This entry provides an example of the use of longitudinal data to study change in mathematics achievement during secondary school. The research addresses the following questions:
- How much does mathematics achievement change during secondary school?
- In what ways do trajectories of mathematics achievement differ across students?
- Does mathematics achievement development differ as a function of student race?
These questions are addressed by fitting a statistical model called the multilevel model for change to five years of mathematics achievement data collected as part of the Longitudinal Study of American Youth (LSAY), a national longitudinal study of U.S. secondary school students (Miller, Kimmel, Hoffer, & Nelson, 2000). LSAY data were collected from 5,945 students over 7 years, beginning in the fall of 1987 when the students were in either 7th or 10th grade. A primary focus of the LSAY was on measuring students' mathematics achievement over time, using items from the National Assessment of Educational Progress. Here, in the example, are analyses of mathematics achievement data from a sub-sample of 1,322 White and African American students between 7th grade and 11th grade, asking about the effects of race on changes in the mathematics achievement over time.
When studying change over time, the first questions are about each person's individual change trajectory. For example, does a particular student's mathematics achievement
improve rapidly during secondary school? Does another student's achievement increase less rapidly? Might yet another student's achievement actually decrease over time? These questions are addressed with the level-1 statistical model, or individual growth model, which represents the change that, according to the hypothesis, each member of the population will experience during the time period under study.
To develop an understanding of the level-1 model, the left-hand panel of Figure 1 should be considered, in which the researchers have plotted the mathematics achievement (MATHACH) of one African American girl from the dataset against her grade, between 7th and 11th grade. For this girl, mathematics achievement improves steadily over time. This upward trend is summarized in the empirical growth record by superimposing a fitted ordinary least squares (OLS) “achievement on grade” linear regression line. Observations based on this plot suggest two important components of this level-1 statistical model. First, the model should capture systematic underlying change in mathematics achievement over time (represented by the fitted growth trajectory plotted in Figure 1). Second, the model must account for differences between observed values of mathematics achievement (represented by the datapoints plotted in Figure 1), and the predicted values from the fitted growth trajectory. These observations lead the researchers to hypothesize the following level-1 model:
This model asserts that, in the population of students from which this sample was drawn, Yij, the observed value of MATHACH for student i at time j is constituted from two important parts. The first part—in brackets in equation (1)—describes the underlying true change for this student as a linear function of his (or her) grade in school (GRADEij). This trajectory is characterized by two individual growth parameters, π0i and π1l, which determine its shape for the ith student. The intercept, π0i, represents student i's true mathematics achievement in 7th grade. (This interpretation applies because the researchers centered GRADE in the level-1 model by subtracting the constant “7” from it.) The slope, π1l, represents the yearly rate of change of student is true mathematics achievement. The second part of the level-1 model is a random error term (eij), which accounts for the difference between individual i's true and observed value of MATHACH, on occasion j. This level-1 residual represents that part of student i's value of mathach at timej not predicted by grade level.
In specifying a level-1 model, it is implicitly assumed that all students' true individual change trajectories have a common algebraic form, here represented by a straight line. But because all the students have their own value of the intercept and slope parameters, everyone does not necessarily follow exactly the same trajectory. Students' true mathematics achievement levels in seventh grade may vary, as may their rates of true change in achievement. Therefore, we may study inter-individual differences in individual growth trajectories by studying inter-individual variation in individual growth parameters. These observations form the foundation of the level-2 statistical model.
At level-2—the “between-person” or inter-individual level—questions are asked about predictors of change. Here, in the mathematics achievement example, at level-2 it was asked whether the average African American seventh grader has lower mathematics achievement than the average White seventh grader, and also whether rates of change in mathematics achievement differ as a function of race. These questions are addressed by modeling the relationship between inter-individual differences in the change trajectories (i.e., intercept and slope parameters) and student characteristics (here, race). To develop an intuition about the level-2 model, the middle panel of Figure 1 should be examined. It represents an exploratory analysis in which the researchers have plotted fitted OLS individual growth trajectories for a random subset of 10 White and 10 African American students (solid lines represent African American students and dashed lines represent White students). As noted for the single student in the left panel, mathematics achievement increases over time for most students. In addition, African American students seem to have generally lower mathematics achievement scores in seventh grade than do White students, and their rates of increase in achievement over time are not as large. But the substantial inter-individual heterogeneity in growth trajectories within groups should also be noted. Not all African American students have lower intercepts than do White students; many of them have higher mathematics achievement in seventh grade. Similarly, not all African American students have less steep slopes; some of them have very rapid increases in mathematics achievement over time. Furthermore, within both groups there are students whose mathematics achievement actually decreases over time.
The level-2 model must simultaneously account for both these general patterns (the between-group differences in intercepts and slopes) and inter-individual heterogeneity that remains within groups. This suggests that an appropriate level-2 model would have outcomes that are the level-1 individual growth parameters themselves (the π0i and π1i parameters from equation (1)). In addition, the level-2 model must specify the relationship between each individual growth parameter and predictor AFAM (0 = White, 1 = African American). Finally, the level-2 model must allow even individuals who share common predictor values to differ in their individual change trajectories, by permitting random variation in the individual growth parameters across students. These considerations lead to the following level-2 model:
Equation (2) has two main components which simultaneously treat the intercept (π0i) and the slope (π1l) of a student's growth trajectory as level-2 outcomes that are associated with predictor AFAM. The level-2 model contains four parameters known collectively as the fixed effects—γ00, γ01, γ10, and γ11—that capture systematic inter-individual differences in change trajectories. In equation (2), γ00 and γ10 are level-2 intercepts; γ01 and γ11 are level-2 slopes. γ00 represents the average true seventh grade mathematics achievement for White students in the population, while γ01 represents the hypothesized population difference in average true initial status between African American and White students. Similarly, γ10 represents the population average true annual rate of change in mathematics achievement for White students, while γ11 represents the hypothesized population difference in average true annual rate of change between African American and White students. The level-2 slopes, γ01 and γ11, jointly capture the effects of AFAM. If γ01 and γ11 are non-zero, the average population trajectories in true mathematics achievement differ between the two racial groups; on the other hand, if γ01 and γ11 are both 0, then the trajectories do not differ by race. These two level-2 slope parameters therefore address the question: What is the difference in the average trajectory of true change in mathematics achievement between White students and African American students?
An important feature of both the level-1 and level-2 models is the presence of the residuals (εij at level-1 and ζ01 and ζ1i at level-2). As is the case with most residuals, researchers are usually less interested in their specific values than in their variability. Level-1 residual variance, σ2ε, summarizes the scatter of the level-1 residuals around each person's true change trajectory, in the population. The level-2 residual variances, σ20 and σ21, summarize the population inter-individual variation in true individual intercept and slope around their averages that is left over after controlling for the effect of AFAM.
As a final question at level-2, the researchers consider a potential association between seventh grade mathematics achievement and change in achievement. For example, do students with higher seventh grade mathematics achievement also experience larger gains in achievement? The researchers permit this possibility by allowing the level-2 residuals to be correlated. Their population covariance, σ01, summarizes the association between true seventh grade math achievement and true rate of change in achievement, controlling for race.
Estimates from the fitted multilevel model for change are presented in Table 1. Substituting the ŷ's from Table 1 into the hypothesized level-2 model in equation (2), the following fitted level-2 model is obtained:
The first part of this fitted model describes the estimated effects of AFAM on true seventh grade mathematics achievement; the second part describes its estimated effects on the annual rate of true change in mathematics achievement. Beginning with the first part of the fitted model, it is estimated that true seventh grade mathematics achievement for the average White student is 53.02. For the average African American seventh grader, it is estimated that true seventh grade mathematics achievement is 5.93 points lower (47.09). In addition, the researchers reject (at the .001 level) the null hypotheses that γ00 and γ01 are 0 and conclude that the average White student had non-zero true mathematics achievement in seventh grade (hardly surprising) and that there is a statistically significant difference in the average true seventh grade mathematics achievement of White students compared with their African American peers.
In the second part of the fitted model, it is estimated that the annual rate of true change in mathematics achievement for the average White student is 2.87 points per year. For the average African American student, it is estimated to be nearly half a point lower (at 2.39). In rejecting (at the .001 level) the null hypothesis on γ10, it is concluded that the average White student experienced a statistically significant increase in true mathematics achievement over time. Because the researchers also reject (at the .05 level) the null hypothesis on γ11, they conclude that differences between African American and White students in their annual rates of true change are also statistically significant. The estimated mathematics achievement for the average White student increased 11.48 points from 7th grade to 11th grade, while the increase for African American students was two points lower (9.56). African American students begin seventh grade with lower average mathematics achievement than their White counterparts, and the achievement gap increases over time.
Another way of interpreting the estimated fixed effects is to plot fitted trajectories for prototypical individuals. For this particular model, only two prototypes are possible: an African American student (AFAM=1) and a White student (AFAM=0). Substituting these predictor values into equation (3) yields the estimated seventh grade mathematics achievement and annual growth rates for each:
When AFAM = 0:
When AFAM= 1:
These estimates are then substituted into the level-1 model in equation (1) to obtain the fitted individual change trajectories:
When AFAM = 0:
When AFAM = 1 :
These fitted trajectories are plotted in the right-hand panel of Figure 1, and reinforce the numeric conclusions articulated above. In comparison to White students, the average African American student has lower mathematics achievement in seventh grade and a slower rate of increase in mathematics achievement.
The estimated variance components assess the amount of outcome variability left after including the predictor AFAM. The level-1 residual variance, εij, summarizes the population variability in average student's outcome values around their own true change trajectory. Its estimate here is 37.17. Rejection of the associated null hypothesis test (at the .001 level) suggests the existence of additional within-person outcome variability that may be predictable in subsequent analyses by time-varying predictors other than time.
The level-2 variance components, σ20 and σ21, summarize the variability in true seventh grade achievement and rate of true change remaining after controlling for AFAM. Tests associated with these variance components evaluate whether there is any remaining outcome variation that could potentially be explained by further predictors at level-2. For these data, the researchers reject both of these null hypotheses (at the .001 level), and conclude that additional level-2 predictors may help explain some of this residual variation. Finally, the level-2 covariance component, σ01 is considered. The researchers reject the null hypothesis on the covariance and conclude that the intercepts and slopes of the individual true change trajectories are indeed correlated in the population. Controlling for student race, on average, African American and White students who have higher true mathematics achievement in seventh grade also have greater rates of increase in true mathematics achievement between 7th and 11th grade.
The mathematics achievement example presented in this entry has many features that simplify analysis and interpretation. However, the multilevel model for change is a very flexible, powerful method for analyzing longitudinal data and may be used to address quite complex longitudinal research questions. Five of these possibilities may be considered here. First, although only one predictor has been included in the analysis, it is straightforward to examine the impact of additional substantive level-2 predictors. For example, in addition to the race variable studied here, it could also be asked whether girls and boys have different mathematics achievement trajectories or whether there is an impact of various instructional methods on the development of mathematics achievement. Second, while all students in the example were assessed on exactly five occasions (a balanced design), the model may also be fitted to longitudinal datasets containing individuals with varying numbers of measurement occasions (an unbalanced design). Third, here measures of mathematics achievement that were taken in the fall of every year were analyzed, but occasions of measurement need not be equally spaced and different participants can have different data collection schedules. Fourth, individual change can be represented not only as a linear function presented here, but also curvilinear and discontinuous functions representing substantively interesting hypotheses of change in educational outcomes over time. Finally, in addition to time-invariant predictors of change, such as race and gender, the effects of predictors whose values change over time can also be estimated, such as type and level of mathematics course in which a student is enrolled each year. Readers wishing to learn more about using longitudinal data to analyze change over time should consult books devoted to the topic, including Diggle, Heagerty, Liang, and Zeger (2002); Fitzmaurice, Laird, and Ware (2004); Hedeker and Gibbons (2006); Raudenbush and Bryk (2002); Singer and Willett (2003); Snijders and Bosker (1999); Verbeke and Molenberghs (2000); Walls and Schafer (2006); and Weiss (2005).
Ai, X. (2002). Gender differences in growth in mathematics achievement: Three-level longitudinal and multilevel analyses of individual, home, and school influences. Mathematical Thinking and Learning, 4, 1–22.
Diggle, P., Heagerty, P., Liang, K-Y., & Zeger, S. (2002). Analysis of longitudinal data (2nd ed.). New York: Oxford University Press.
Fitzmaurice, G. M., Laird, N. M., & Ware, J. H. (2004). Applied longitudinal analysis. New York: Wiley.
Hedeker, D., & Gibbons, R. D. (2006). Longitudinal data Analysis. New York: Wiley.
Kreft, I. G. G., & de Leeuw, J. (1998). Introducing multilevel modeling. Thousand Oaks, CA: Sage.
Miller, J. D., Kimmel, L., Hoffer, T. B., & Nelson, Cynthia. (2000). Longitudinal study of American youth: User's manual. Chicago: International Center for the Advancement of Scientific Literacy, Northwestern University.
Raudenbush, S. W., & Bryk, A. S. (2002). Hierarchical linear models: Applications and data analysis methods (2nd ed.). Thousand Oaks, CA: Sage.
Singer, J. D., & Willett, J. B. (2003). Applied longitudinal data analysis: Modeling change and event occurrence. New York: Oxford University Press.
Snijders, T. A. B., & Bosker, R. (1999). Multilevel analysis: An introduction to basic and advanced multilevel modeling. London: Sage.
Verbeke, G., & Molenberghs, G. (2000). Linear mixed models for longitudinal data. New York: Springer.
Walls, T. A., & Schafer, J. L. (2006). Models for intensive longitudinal data. New York: Oxford University Press.
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