분할 선형 성장모형을 사용한 불연속 성장궤적의 분석에서 도약의 추정 = Estimating jumps in disjointed trajectories using piecewise linear growth model

국문 초록 (Abstract)

분할 선형 성장모형(piecewise linear growth model)은 비선형 성장궤적을 가지는 종단자료의 분석에 유용하다. 분할 선형 성장모형에서는 변곡점을 기준으로 측정기간을 복수의 구간으로 나누고 각 구간에 개별적인 선형모형을 적용함으로써 비선형 성장궤적을 모형화할 수 있을 뿐만 아니라 성장요인에 대해 직관적이고 이론적으로 의미 있는 해석이 용이하다. 이러한 장점에 더해서, 변곡점에서 발생하는 변화로 인해 성장궤적이 연결되지 않는 불연속 성장궤적(이하 ‘도약’) 또한 모형화가 가능하다. 그러나 이러한 장점은 분할성장모형의 적용에서 자주 간과된다. 본 연구에서는 도약을 포함한 분할 선형 성장모형의 모수화 방법을 개괄하고 분할 선형 성장모형의 적용에서 자료에 실재하는 도약을 생략할 때 발생할 수 있는 모수 추정의 편향과 적합도의 변화를 시뮬레이션을 통해 확인하였다. 다양한 크기의 도약이 존재하는 불연속 성장궤적을 기반으로 자료를 생성하고 이를 Harring 등(2006)이 제안한 분할 선형 성장모형으로 분석하였다. Harring 등(2006)의 모형은 이론에 기반 하여 선험적으로 변곡점을 설정하는 대신, 자료로부터 변곡점의 위치를 추정한다. 따라서 본 연구의 결과를 통해 도약의 생략으로 인해 발생할 수 있는 변곡점 추정의 편향 또한 확인하였다. 분석 결과, 도약의 크기가 커질수록 이를 포함하지 않는 분할 선형 성장모형의 결과에서 모수 추정의 편향이 대체로 커지고 모형의 적합도가 낮아졌다. 모수 추정의 편향은 성장요인의 종류에 따라 그 양상이 달랐다. 본 연구의 결과는 분할 선형 성장모형의 적용에서 도약을 추정하거나 실증적으로 검정하는 것이 대부분의 경우 더 적절함을 시사한다. 마지막으로 본 연구의 결과를 토대로 분할 선형 성장모형의 적용을 위한 전략을 제안하였다.

다국어 초록 (Multilingual Abstract)

Piecewise linear growth model(PLGM) is useful to model for non-linear trajectories. In PLGM, entire assessment period is split into multiple phases at points called “knots’, and separate linear growth model is applied to each phase. Because linear growth model is used at each phase, the interpretation of growth factors is more straightforward and theoretically meaningful compared to other methods for modeling non-linear growths. In addition, radical changes at the knot can lead to disjointed trajectories (referred to as ”jump“ in the follwing) at knots, and PLGM can model the jump. However, such advantage of PLGM　is often overlooked in applications of PLGM. In this study, we reviewed parameterizations of PLGM that allow the estimation of the jump in disjointed trajectories, and examined consequences, in terms of estimation bias and model fit, of model misspecification by omitting the jump. For this purpose, we generated datasets with trajectories with various degrees of jumps and analyzed the datasets using the PLGM proposed by Harring et al. (2006), which estimates the location of the knot, instead of setting it at an a priori point. Thus, we were also able to examine the estimation of the knot locations in the presence of the model misspecification. In our results, with increasing degrees of the jump, in general, the bias of parameter estimates increased and the model fit declined. The results showed that, in most situations, it is a good idea to include the jump in the applications of PLGM, unless there is a strong theoretical background to omit the jump. We also provided practical strategies in the applications of PLGM based on our results.

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