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Practical Assessment, Research & Evaluation, 8(12). Retrieved August 18, 2006 from http://edresearch.org/pare/getvn.asp?v=8&n=12 . This paper has been viewed 485 times since 5/27/03.
Studying change processes in education and the behavioral sciences has been of interest for a long time. Researchers and practitioners in the behavioral sciences are concerned with questions about how individuals change over time (Willett & Sayer, 1994; Williamson, Appelbaum, & Epanchin, 1991). That is, we are interested in the question of how individual change in certain attributes (for example, change of masculinity and femininity among adolescents) is related to selected characteristics of a person's background, family and peer environment, or training. In recent years, the number of models utilized to address questions of this kind has increased substantially (e.g., Collins & Horn, 1991; Aber & McArdle, 1991). The focus of this paper is not to compare different methods in modeling growth curve. Rather, the current study is to propose a longitudinal profile analysis via Multidimensional Scaling (PAMS), called LPAMS model. This is an exploratory approach to identify growth patterns in the data using Multidimensional Scaling, an alternative to more commonly used theory-based structural equations model approach (e.g., McArdle & Epstein, 1987). The longitudinal multidimensional scaling analysis described here is an exploratory technique. There are several reasons for using the LPAMS model rather than more traditional approaches such as structural equations modeling. First, the LPAMS model is most suitable for the situation in which the growth patterns are to be derived from the data, rather than specified by theory. It requires no a priori hypotheses about the nature of the underlying growth patterns. While a priori specification of a unidimensional growth pattern model is often feasible, the difficulty of specifying the growth patterns a priori increases as the number of growth curves increases. Second, it provides estimates of latent growth pattern as well as estimates of individual growth parameters in the model (that is, growth rate estimates for each individuals), not simply summary statistics such as means, variances, and covariances. If one wishes to study inter-individual differences in growth profiles, one may take the individual growth parameter estimates as a description of his or her entire growth pattern and subject those parameter estimates to multivariate analysis of variance or regression analysis. In a sense, LPAMS is similar to multilevel or hierarchical modeling in that it allows one to estimate both group level growth parameters and individual growth rate, where within-subject is considered level 1 model and between-subjects is considered level 2 model (Willett & Sayer, 1994). After obtaining level 1 individual growth rate estimate and initial status estimate, one can examine between-group differences in these parameter estimates. The differences between the LPAMS model and the commonly-used hierarchical linear model (HLM) are that: First, in HLM there is clustering of individuals within group and variables are measured at all available levels, thus combining variables from different levels in one statistical model, whereas in the LPAMS model one estimates within-individual change with respect to the latent change curves using MDS method and estimates between-individual variations via conventional ANOVA approach. Second, they differ on some of the assumptions, which are discussed below. Third, the assumptions are minimal. The more traditional models suffer from complaints about the theoretical restrictions and lack of applicability of many standard linear models that assume the normal distribution (Nesselroade & Ford, 1987; Nesselroad & Cattell, 1989). The LPAMS model allows for simultaneous estimation intra- and inter-individual growth processes without requirement of the multivariate normality of tests frequently employed. It can be applied to the data that are not amenable to the common methods of analysis due to limitations imposed by the specific design of experiment, or theoretical assumptions about the nature of the phenomenon under study. Fourth, the LPAMS model is based on distance model (Borg & Groenen, 1997; Davison, 1983) rather than linear model. Thus, it can be used to model data that are nonlinear in nature. In LPAMS modeling, the growth patterns are examined, not just mean score change over time, which can usually be analyzed using a standard univariate or multivariate repeated-measures design. This is especially the case when there may be more than one growth curves in the data and the LPAMS model can simultaneously identify these curves. As with any methodology, there are several limitations associated with this exploratory longitudinal MDS approach. First, as with any exploratory approach, it is designed to identify patterns of growth underlying a set of data, not to test a priori hypotheses about patterns. Therefore, it will often be most useful in the early stages of a research program in which little is known about the underlying growth patterns. Second, while it requires no distributional assumptions about the observed variables, it does require equal error variances at each time period, which may not be realistic. In rest of the paper, the focus will be on discussing the multidimensional scaling (MDS) model underlying longitudinal profile analysis and on how the LPAMS model can be used for growth curve analysis. Next, parameter estimation is considered. The parameter estimation scheme involves obtaining an initial estimate of growth curve parameters, re-scaling them to facilitate interpretation in the context of growth curve analysis, and then estimating the individual growth parameters.
Longitudinal Profile Analysis via MDS Model A multidimensional scaling (MDS) model produces a geometric or spatial representation of relationships between stimuli or individuals. The goal is to produce a geometric representation of underlying individual or stimuli dimensions that fit the data based on spatial proximity between pairs of objects as a measure of their relatedness. A classical MDS example used to identify latent stimuli dimensions is the research on the children’s cognitive reasoning of body parts (Jacobowitz, 1975). In this study, children were presented with pairs of body parts. They were asked to sort the body parts according to the functions of these parts. MDS was used to model the nature of a child’s conception in the geometric space in which subjects locate their perceptions of the different body parts. This geometric representation of
underlying stimuli can be extended to examining how a set of variables is
patterned or configured in a space. The research focus is on what typical
patterns or configurations of variables actually exist in the population and
how individuals differ with respect to these patterns. This is what MDS
profile analysis attempts to answer (e.g., Davison, Gasser, & Ding, 1996;
Davison, Kuang, & Kim, 1999). Figure 1 is an example of such a latent
profile analysis of vocational interest profiles based on well-known hexagonal
theory of vocational interest (Campbell & Hansen, 1985). As can be seen
from the figure, each interest variable is located along the two dimensions
according to their scale values
The above examples serve to
demonstrate two applications of the MDS model. One can build on these concepts
and extend the model for longitudinal studies. When the variable under the
study is repeated at different occasions, however, the repeated variable is
expected to be related to one other in the form of a particular developmental
shape along the certain dimensions. Therefore, one may conceptualize the
magnitude of that variable as a function of characteristics of that person as
well as time of measurement. The MDS model used in such longitudinal data
analysis is called longitudinal profile analysis via multidimensional scaling
(LPAMS) model. In this model, each growth curve dimension Longitudinal profile analysis via MDS model starts with the equation:
where e_{p(t)} is an error term. The point of the foregoing discussion is that a set of variables or a set of scores of a repeated variable can be configured in a multidimensional space according to their locations in k-spaces; a particular arrangement of scores based on their scale value estimates may form a latent pattern or growth curve. This pattern does not indicate homogeneity of construct of a set of variables, as in factor analysis; rather it represents a psychological characteristic shape and level of the variable under inquiry. Graphically the concepts of
The intercept or level parameter,
The LPAMS model analysis begins by
using MDS to obtain initial estimates of the scale values
t,
t’) is a squared Euclidean distance measure, d_{tt'}^{2}, computed from the raw data as
follows:
Table 1 shows the square root of squared Euclidean distance matrix thus computed based on reading achievement. The distance coefficients indicates the degree to which the reading scores differ from each other over time. The proximity module in many standard statistical packages (e.g. SAS, SPSS, SYSTAT) includes an option for the computation of squared Euclidean distance or Euclidean distance proximity measures defined over all possible pairs of variables.
When proximity measures Equation 2 are submitted
to an appropriate multidimensional scaling algorithm, the analysis should yield
one dimension for each latent growth curve. Since the objective of the LPAMS
model analysis is to see if there are particular trend shapes in the data,
whether they are linear curve, nonlinear curve, or time-series periodic curves,
MDS estimation method would identify one or more such trend shapes when such
curves exist in the data. Thus, LPAMS model configures repeated data points in
k-dimensions, and the scale value for time
Thus, LPAMS model analysis proceeds in three
steps. First, proximity measures are computed over all possible pairs of times
according to Equation 2. These proximity measures are analyzed using nonmetric
MDS algorithms to yield estimates of the scale values A concrete example may elucidate the LPAMS model analysis. The SAS codes are provided in the appendix so that reader can carry out the analysis. The data were from student reading achievement over a four-year span. Like many in the literature, it includes only one latent growth pattern, i.e., only one dimension. It should be noted that this was a coincident in that there were only four repeated measures available over four years for the current study. In fact, more data points are desirable in such applications so that different growth curves can be studied.
To illustrate the use of the LPAMS
model for exploratory growth profile analysis, a data set containing four waves
of data was used. The data was obtained from a sample of 705 elementary and
middle school students at a school district in a Southwest state. These
students consisted of 2 cohorts. Grade 3 cohort students were in 3 Table 2 shows the means and standard deviations of the reading scores for these two cohorts. These statistics are based on 705 students who had complete data on all four occasions of measurement. Inspection of the table indicated that students did seem to progress in their reading achievement over the years, with grade 3 cohort students showing decreased variations over time, indicating on average students were less scattered in their reading scores.
Exploratory longitudinal profile analysis via multidimensional scaling (LPAMS) started with estimating scale values presented in Equation 1. As mentioned above, time is specified as the latent dimension along which individuals vary with regard to the growth/decline curve. In applying this model, the reading score is considered a repeatedly measured variable on a time dimension along which individual change patterns are of interest. Since the analysis is exploratory, no specification of a particular growth/decline profile is necessary. The growth patterns are to be derived from the observed data.
Next, initial scale values, The estimates of LPAMS growth curve values (i.e., scale values of the repeated variable) are presented in Table 3. The scale values in Table 3 are the final estimates obtained by re-scaling the initial estimates in such a way that the zero point corresponds to the scale value of time 1 so that intercept or level estimate indicate the initial level. To facilitate the presentation of the scale values, they were scaled to have a mean of 5 and a standard deviation of 2. This was a simple linear transformation that did not affect the interpretation of the scale values. Growth rates in percentage for each of the three time intervals were also shown in the table. These growth rates remained the same regardless of the translation of the scale values.
Figure 3 shows the latent growth
pattern based on the final growth scale values. The dimension scale values
depict the latent growth pattern in terms of three line segments. Each segment
covers one time interval: year 97 to 98, year 98 to 99, and year 99 to 2000.
Differences in growth rate over the several time intervals are represented by
the slopes of the line segments for those intervals. As can be seen in Figure
3, for grade 3 cohort, the pattern showed the greatest growth rate from 3
In the current
data, the average of the growth profile index was 19.85, with a standard
deviation of 9.01 for grade 3 cohort, and was 16.50, with a standard deviation
of 8.08 for grade 5 cohort. This indicated that, on average, students had made
gains in reading scores over the years, with certain individual variation in
growth profiles. The correlation between the intercept
In recent years, latent growth curve modeling has been widely used in longitudinal research. Some authors (e.g., McArdle & Epstein, 1987; Meredith & Tisak, 1990; Muthen, 1991) have demonstrated how concepts of individual growth modeling can be accommodated within the framework of covariance structure analysis. In this paper, an exploratory growth profile analysis was proposed. The approach uses MDS model based on squared Euclidean distance measures of proximity defined over pairs of time periods to model growth curve in the data. Such longitudinal profile analysis by means of MDS can be used to study individual and group patterns of longitudinal change in an exploratory fashion. Latent growth rates are reflected in the estimates of scale values from MDS geometric solutions, which could simultaneously accommodate one or more growth curves in the data. Individual growth profile index is also estimated from the model so that one could directly study the inter-individual differences with respect to growth. One drawback of the current study is that there are only 4 longitudinal data points available due to the difficult of obtaining the longitudinal data with more time points. This limits the demonstration of using the LPAMS model to identify different growth curves in the data, which is one of the major insights that can be made from the analysis. Readers are encouraged to try out this technique using their own data, preferably with eight or more time points.
Aber, M. S.,
& McArdle, J. J. (1991). Latent growth curve approach to modeling the
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& Hansen, J. C. (1985). Collins, L. M.,
& Horn, J. L. (1991). Davison, M. L.,
Gasser, M., & Ding, S. (1996). Identifying major profile patterns in a
population: An exploratory study of WAIS and GATB patterns. Davison, M. L.,
Kuang, H., & Kim, S. (1999). The structure of ability profile patterns: A
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Ackerman, P. C. Kyllonen, & R.D. Roberts (Eds.). Kruskal, J. B. (1964).
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hypothesis. Jacobowitz, D.
(1975). McArdle, J. J.,
& Epstein, D. (1987). Latent growth curves within developmental
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(1991). Analysis of longitudinal data using latent variable models with
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& Cattell, R. B. (1989). Nesselroade, J.
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academic achievement.
%MACRO GROWTH (data =, var =, dim =, dimname= , pout= , subid = );
DATA trsf1 (Keep=&var); set &data;
PROC TRANSPOSE DATA=trsf1 OUT=out; var &var; RUN; %INCLUDE 'C:\program files\SAS institute\sas\v8\STAT\SAMPLE\xmacro.SAS'; %INCLUDE 'C:\program files\SAS institute\sas\v8\STAT\SAMPLE\DISTnew.SAS'; %INCLUDE 'C:\program files\SAS institute\sas\v8\STAT\SAMPLE\stdize.SAS'; %DISTANCE ( DATA=out, METHOD=EUCLID, id=_name_, OUT=dis ); RUN; PROC PRINT data=dis; RUN;
/*MDS analysis*/ PROC MDS DATA=dis CONDITION=matrix SHAPE=Triangle LEVEL=ordinal COEF=i FORMULA=1 FIT=1 DIMENSION= &dim PFINAL OUT=scale_value OUTRES=cordres ; TITLE "Nonmetric MDS Growth Scale Values"; TITLE ‘ ‘; RUN;
/*Re-scale and Estimate C and W*/
DATA profile (KEEP= &dimname); SET scale_value (FIRSTOBS=2 ); PROC PRINT data=profile; run;
PROC IML; USE &data VAR{&var}; /*use original raw score variables*/ READ all INTO M ; USE profile VAR{&dimname}; /*Read in initial scale values*/ READ ALL INTO X;
X = (X - X[1,1])* -1; /*Re-set the origin of the initial scale value and make them all positive */ Print X; Print "This is final re-scaled estimates of growth scale values";
/*computes the weights and level parameters */ START w; R=NROW(x); COL=J(r,1,1); X1=X||COL; M1=inv(X1`*X1); M2=M*X1; TW=M2*M1;
FINISH W; RUN w;
/*computes the fit measure for each subject*/ START fit; M1 = tw * x1`; K = ncol(m); R = nrow(m); COL = j(1,k,1); M1RSUM = m1-((m1[,+]*col)/k ); PVAR = (m1rsum##2)*col`; MRSUM = m - ((m[,+] * col)/k); VAR = (mrsum##2)*col`; COL= pvar/var; FINISH fit; run fit; /*Merge weights and level parameters with the fit measures */ WL=tw||col; /*Get subject id from original data for merging dataset*/ use &data Var{&subid}; read all into id; WL = id||WL; MATTRIB WL COLNAME = ({&subid &DIMNAME LEVEL FIT});
CREATE &pout var{&subid &dimname level fit}; APPEND from WL; SHOW DATASETS; SHOW CONTENTS; QUIT; %MEND;
% (data = work.g3, var = read97 read98 read99 read00, dim = pout = wldata, subid = id );
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Descriptors: Mathematical Models; Multidimensional Scaling; Classification; Hypothesis Testing |

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