• The Mathematical Education
• /
• v.55 no.3
• /
• pp.251-279
• /
• 2016

There are many studies on 'how' students solve mathematical problems, but few of them sufficiently explained 'why' they have to solve the problems in their own different ways. As quantitative reasoning is the basis for algebraic reasoning, to scrutinize a student's way of dealing with quantities in a problem situation is critical for understanding why the student has to solve it in such a way. From our teaching experiments with two ninth-grade students, we found that emergences of a certain level of covariational reasoning were highly consistent across different types of problems within each participating student. They conceived the given problem situations at different levels of covariation and constructed their own quantity-structures. It led them to solve the problems with the resources accessible to their structures only, and never reconciled with the other's solving strategies even after having reflection and discussion on their solutions. It indicates that their own structure of quantities constrained the whole process of problem solving and they could not discard the structures. Based on the results, we argue that teachers, in order to provide practical supports for students' problem solving, need to focus on the students' way of covariational reasoning of problem situations.

• Journal of the Korean Data and Information Science Society
• /
• v.9 no.2
• /
• pp.149-158
• /
• 1998

In this paper we suggest the joint model of death times and counts with covariates. We assume that the death times follow a Weibull distribution with rate that depends on covariates. For the counts, a Poisson process is assumed with the intensity depending on time and the covariates. We obtain the maximum likelihood estimators of model parameters. This model is applied to data set of patients with breast cancer who received a bone marrow transplant.

• Journal of the Korean Data and Information Science Society
• /
• v.7 no.1
• /
• pp.145-159
• /
• 1996

다변량 발병시간자료는 각 개개 환자에게 있어 합병증이 발생되거나 혹은 유사 환자군(집락) 내의 발병시간이 상관되어진 생의학자료에서 흔히 볼 수 있다. HMULCOX는 그런 자료를 분석하기 위한 한글 통계 패키지 가운데 하나이다. 이 프로그램은 관련된 발병시간들이 독립이 아닐때에도 COX 비례 위험 모형의 주변확률분포를 계산해 준다. 주어진 조건으로는 주변확률모형의 기본위험율은 일정한 상수, 흑은 변수라도 관계없다. 또한 치료실패율의 치료변수들(공변량)의 효과에 대해 다양한 통계적 추론이 가능하다. 기본적으로 주변확률분포접근법으로 설계되었지만 HMULCOX는 여러 가지 추론 방법을 선택하는 데 일반적으로 충분하다. 이 프로그램으로 2개의 예를 들어 실행하겠다.

• Journal of the Korean Data and Information Science Society
• /
• v.9 no.2
• /
• pp.311-322
• /
• 1998

In this paper the problem of modeling count data where the observation period is determined by the survival time of the individual under study is considered. We assume marginal frailty model in the counts. We assume that the death times follow a Weibull distribution with a rate that depends on some covariates. For the counts, given a frailty, a Poisson process is assumed with the intensity depending on time and the covariates. A gamma model is assumed for the frailty. Maximum likelihood estimators of the model parameters are obtained. The model is applied to data set of patients with breast cancer who received a bone marrow transplant. A model for the time to death and the number of supportive transfusions a patient received is constructed and consequences of the model are examined.

• The Korean Journal of Applied Statistics
• /
• v.34 no.3
• /
• pp.411-425
• /
• 2021

Accelerated failure time (AFT) model represents a linear relationship between the log-survival time and covariates. We are interested in the inference of covariate's effect affecting the variation of survival times in the AFT model. Thus, we need to model the variance as well as the mean of survival times. We call the resulting model mean and variance AFT (MV-AFT) model. In this paper, we propose a variable selection procedure of regression parameters of mean and variance in MV-AFT model using penalized likelihood function. For the variable selection, we study four penalty functions, i.e. least absolute shrinkage and selection operator (LASSO), adaptive lasso (ALASSO), smoothly clipped absolute deviation (SCAD) and hierarchical likelihood (HL). With this procedure we can select important covariates and estimate the regression parameters at the same time. The performance of the proposed method is evaluated using simulation studies. The proposed method is illustrated with a clinical example dataset.

• Journal of Educational Research in Mathematics
• /
• v.25 no.2
• /
• pp.189-206
• /
• 2015

There has been a recurring call for using visual representations in textbooks to improve the teaching and learning of proportional reasoning. However, the quantity as well as quality of visual representations used in textbooks is still very limited. In this article, we analyzed visual representations presented in a Grade 6 textbook from two perspectives of proportional reasoning, multiple-batches perspective and variable-parts perspective, and discussed the potential of the double number line and the double tape diagram to help develop the idea 'things covary while something stays the same', which is critical to reason proportionally. We also classified situations that require proportional reasoning into five categories and provided ways of using the double number line and the double tape diagram for each category.

• The Korean Journal of Applied Statistics
• /
• v.19 no.3
• /
• pp.505-519
• /
• 2006

We consider zero-inflated count data, which is discrete count data but has too many zeroes compared to the Poisson distribution. Zero-inflated data can be found in various areas. Despite its increasing importance in practice, appropriate statistical inference on zero-inflated data is limited. Classical inference based on a large number theory does not fit unless the sample size is very large. And regular Poisson model shows lack of St due to many zeroes. To handle the difficulties, a mixture of distributions are considered for the zero-inflated data. Specifically, a mixture of a point mass at zero and a Poisson distribution is employed for the data. In addition, when there exist meaningful covariates selected to the response variable, loglinear link is used between the mean of the response and the covariates in the Poisson distribution part. We propose a Bayesian inference for the zero-inflated Poisson regression model by using a Markov Chain Monte Carlo method. We applied the proposed method to a Korean oral hygienic data and compared the inference results with other models. We found that the proposed method is superior in that it gives small parameter estimation error and more accurate predictions.

• The Korean Journal of Applied Statistics
• /
• v.36 no.4
• /
• pp.323-335
• /
• 2023

In causal analysis of high dimensional data, it is important to reduce the dimension of covariates and transform them appropriately to control confounders that affect treatment and potential outcomes. The augmented inverse probability weighting (AIPW) method is mainly used for estimation of average treatment effect (ATE). AIPW estimator can be obtained by using estimated propensity score and outcome model. ATE estimator can be inconsistent or have large asymptotic variance when using estimated propensity score and outcome model obtained by parametric methods that includes all covariates, especially for high dimensional data. For this reason, an ATE estimation using an appropriate dimension reduction method and semiparametric model for high dimensional data is attracting attention. Semiparametric method or sparse sufficient dimensionality reduction method can be uesd for dimension reduction for the estimation of propensity score and outcome model. Recently, another method has been proposed that does not use propensity score and outcome regression. After reducing dimension of covariates, ATE estimation can be performed using matching. Among the studies on ATE estimation methods for high dimensional data, four recently proposed studies will be introduced, and how to interpret the estimated ATE will be discussed.

• Journal of the Korean School Mathematics Society
• /
• v.23 no.1
• /
• pp.129-158
• /
• 2020

In this study, we presented two geometric tasks to three 11th grade students to identify the characteristics of the images that the students had at the beginning of problem-solving in the problem situations and investigated how their images changed during problem-solving and effected their problem-solving behaviors. In the first task, student A had a static image (type 1) at the beginning of his problem-solving process, but later developed into a dynamic image of type 3 and recognized the invariant relationship between the quantities in the problem situation. Student B and student C were observed as type 3 students throughout their problem-solving process. No differences were found in student B's and student C's images of the problem context in the first task, but apparent differences appeared in the second task. In the second task, both student B and student C demonstrated a dynamic image of the problem context. However, student B did not recognize the invariant relationship between the related quantities. In contrast, student C constructed a robust quantitative structure, which seemed to support him to perceive the invariant relationship. The results of this study also show that the success of solving the task 1 was determined by whether the students had reached the level of theoretical generalization with a dynamic image of the related quantities in the problem situation. In the case of task 2, the level of covariational reasoning with the two varying quantities in the problem situation was brought forth differences between the two students.

• The Mathematical Education
• /
• v.54 no.4
• /
• pp.299-315
• /
• 2015

The aim of this qualitative case study is twofold: 1) to analyze how an eleventh-grader, Min-Seon, conceive and represent a pattern of change between two varying quantities in a quadratic functional situation, and 2) further to help her form a concept of 'derivative' as a tool to express the relationship with employing a concept of 'rate of change.' The result indicates that Min-Seon was able to construct graphs of piecewise functions that take average rates of change as range of the functions, and managed to conjecture the derivative of a quadratic function, $y=x^2$. In conclusion, we argue that covariational approach could not only facilitate students' construction of an initial function concept, but also support their understanding of the concept of 'derivative.'