Bivariate Regression Analysis - KOCWcontents.kocw.net/document/11_Simple Regression Analysis.pdf ·...
Transcript of Bivariate Regression Analysis - KOCWcontents.kocw.net/document/11_Simple Regression Analysis.pdf ·...
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11th Week
Bivariate Regression Analysis
이변량 회귀분석
Jongseok Lee
Business Administration
Hallym University
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주식 가격 = f (?)
Source: http://finance.naver.com
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변수들 관계의 모형화: 확정적 모형과 확률적 모형
확률적 모형 Probabilistic Model
확정적 모형 Deterministic Model
□ 직사각형의 면적 = 가로 × 세로
□ 원의 면적 = 3.14 × 반지름의 제곱
□ 몸무게 = 체지방 + 제지방
□ 몸무게 = f (키) + ε
e : 입실론(epsilon) = 오차항(error term) 혹은 무작위항(random term)
☞ 실제 몸무게와 키를 근거로 추정된 몸무게의 차이
□ 몸무게 = f (키, 허리둘레, … ) + ε
오차항이 없다!
오차항이 있다!
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1.284 g/cm2
BMD, bond mineral density
골밀도 = f (?) + ε
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Act 1
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골밀도 알아 맞추기
1.284 g/cm2
= BMC / Area = 2,557 (g) / 1,991(cm2)
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젂체의 골밀도 평균으로 추정
상대에 대한 정보가 하나도 없는데
골밀도를 어떻게 맞추지?
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정규 분포
Normal Distribution
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49.0 kg 1.284 g/cm2
X
Body Weight
Y
Bone Mineral Density ~
Weight and BMD
BMD is associated with body weight !
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체중이 49 kg인 사람들의 골밀도 평균으로 추정
체중이 49 kg이어요.
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체중에 관한 정보만 있는 경우
= 0.006 × 49 + 0.719
= 1.013
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체중뿐 아니라 여자라는 것을 안 경우
= 0.006 × 49 + 0.795
= 1.089
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49.0 kg 1.284 g/cm2
X
Body Weight
Y
Bone Mineral Density
yi = b0 + b1xi + ei
선형모형 (linear model)
y = b0 + b1x
y = 0.795 + 0.006 x
= 0.795 + 0.006 × 49
= 1.089
e = 1.284 – 1.089
= 0.195
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체중이 112 kg인 여자의 골밀도 추정치는?
= 0.006 × 112 + 0.795
= 1.467
(112, 1.591)
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= 1.119
= 1.467
= 1.591
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Act 2
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Relationship between Weight and BMD
Normal
Osteopaenia
Osteoporosis
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SPSS를 이용한 회귀 분석
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SPSS 회귀 분석 결과
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분석결과의 제시
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체중과 골밀도 Weight and BMD
저체중은 골다공증의 원인으로 알려져 있다. 체중은 골밀도의 좋은 설명변수인가? 골밀도
(T-Score)의 체중(kg)에 대한 이변량 회귀분석의 결과는 다음과 같다.
• 적합된 회귀직선의 골밀도(T-Score)의 변동에 대한 설명력은 6.1% 정도 이었다. 회귀모형의
유의성에 대한 분산분석의 결과, 유의수준 0.01에서 적합된 회귀모형이 유의하지 않다는 귀
무가설을 기각할 수 있었다.
• 체중과 관련된 회귀계수는 0.033이 되어 체중이 1(kg) 증가하면 골밀도 T값이 0.033 개선되는
것으로 추정되었다. 회귀계수에 대한 유의성 검정 결과, p값이 0.000이 되어 추정된 회귀계수
가 유의하지 않다는 귀무가설을 유의수준 0.001에서 기각할 수 있었다.
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Act 3
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Is BMD associated with body weight ?
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Body Weight
50 kg 65 kg
Group 2 Group 4 Group 1
저체중 과체중 보통 미만
Group 3
57.5 kg
보통 이상
체중 집단들의 적합된 회귀식의 기울기에
통계적으로 유의한 차이가 있는가?
가설 : 체중증가가 골밀도를 향상시킨다면,
과체중 집단의 기울기는 저체중 집단의 기울기 보다
완만할 것이다.
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Relationship between Age and BMD
Table. Simple Regression Results
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Table. Results of Bivariate Linear Regression Analysis with Age, as an
Independent Variable, by Each Weight Group
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Figure: Overweight vs. Underweight Group