지난 수업에서 LM test를 하는 practical computation을 배웠는데,
R2와 uncentered R2가 같은지 확인해볼 수 있는 데이터와 구하는 방법을 올립니다.
(절편항이 들어있는 회귀분석의 경우 uncentered R2와 R2가 다른 값인 것이 일반적입니다.)
컴퓨터 패키지를 단 한번이라도 사용해보신 분들에게는 극히 쉬운 일입니다만,
그저 노파심에 적어봅니다.
<Matlab-최근버전-의 경우>만 적겠습니다.
1. File/Import data 클릭
2. 저장한 txt 파일을 열기(*아래의 데이터를 쓰는 경우)
3. Import Wizard가 뜨는데, 이때 한 번은 그냥 next 클릭/
'create vectors from each column using column names.'선택 후 Finish 클릭
4. 다음을 입력 (맨 윗줄 '%%%%..'에서부터 맨 아랫줄 '%%%%...'까지 다 긁어서 붙인후 한 번만 엔터쳐도 됨)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 잔차를 종속변수로 두었을 때 R2와 uncentered R2를 비교하기
%
% step 1: resid를 구하는 regression을 수행
% step 2: resid를 종속변수로 두어 다시 regression
% step 3: R2와 uncentered R2의 값을 비교하기
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
J=ones(length(Y),1);
% 절편항을 위한 설명변수인 J 벡터를 생성합니다.
[b, bint, resid]=regress(Y, [J X]);
% Y를 J와 X에 대해 회귀한 후 그것의 잔차를 resid에 저장합니다.
[beta, bint, residhat]= regress(resid, [J X Z]);
% resid를 J와 X 그리고 누락된 변수라 의심하는 Z에 대해서 회귀하여 회귀계수와 '잔차'를 구합니다.
% 여기에서의 '잔차'는 첫번째 단계에서 구한 resid와 혼동되지 않도록 주의
uncenR2= (residhat'*residhat)/(resid'*resid);
% uncentered R2는 yhat'*yhat / y'y 꼴입니다.
residbar=mean(resid)
% centered R2를 위한 resid의 평균을 구합니다.
% 실제 이 값을 확인해보면 굉장히 0에 가까운 작은 값이어서, 이것을 빼는 것이 무의미하다고, 즉 R2나 uncentered R2나 똑같을 거라고 짐작할 수 있습니다.
R2=((residhat-residbar)'*(residhat-residbar))/((resid-residbar)'*(resid-residbar));
% R2는 (yhat-ybar)'*(yhat-ybar) / y'y 꼴입니다.
result=R2-uncenR2
% 두 개의 R2값을 비교하기 위해 빼줍니다.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
5. result의 값이 practically 0인 것을 확인 (실제로는 0.0000000000000002220)
6. 대부분의 컴퓨터 패키지는 uncentered R2를 보여주지 않고 R2부터 보여주기 때문에, 만약 두 값이 다르다면 일일이 위와 같은 계산을 거쳐서 uncentered R2를 구해야 할 것 입니다. 따라서 위의 결과는 우리가 이용하기 굉장히 좋은 결과입니다.
아래는 데이터입니다.
변수명(X,Y,Z)을 포함해서 그대로 긁으신 후 txt파일로 저장하시면 됩니다.
X , Y, Z
3.58,4.93,2.84
3.6,4.94,2.71
3.62,4.94,2.85
3.64,4.94,2.96
3.65,4.95,2.85
3.65,4.95,3.25
3.63,4.95,3.24
3.59,4.96,3.36
3.59,4.95,4
3.59,4.95,4.12
3.59,4.94,4.21
3.65,4.94,4.57
3.68,4.94,4.44
3.67,4.94,3.95
3.66,4.94,3.44
3.65,4.94,3.24
3.65,4.94,3.39
3.64,4.94,2.64
3.63,4.94,2.4
3.63,4.95,2.29
3.62,4.95,2.49
3.62,4.95,2.43
3.61,4.95,2.38
3.59,4.95,2.27
3.59,4.95,2.3
3.59,4.95,2.41
3.59,4.96,2.42
3.62,4.96,2.33
3.63,4.96,2.29
3.65,4.96,2.36
3.66,4.96,2.27
3.67,4.97,2.4
3.66,4.97,2.3
3.68,4.97,2.35
3.7,4.98,2.46
3.71,4.98,2.62
3.7,4.98,2.75
3.72,4.98,2.75
3.72,4.98,2.72
3.72,4.99,2.73
3.72,4.99,2.69
3.72,4.99,2.72
3.73,4.99,2.94
3.73,4.99,2.84
3.74,4.99,2.79
3.74,4.99,2.75
3.74,4.99,2.8
3.74,5,2.86
3.75,5,2.91
3.76,5,2.92
3.77,5.01,2.9
3.78,5.01,2.91
3.79,5.01,2.92
3.79,5.01,2.99
3.79,5.02,3.14
3.79,5.02,3.32
3.8,5.02,3.38
3.8,5.03,3.45
3.81,5.03,3.52
3.81,5.03,3.52
3.82,5.04,3.53
3.82,5.04,3.53
3.82,5.04,3.55
3.84,5.04,3.48
3.85,5.05,3.48
3.85,5.05,3.48
3.85,5.05,3.48
3.86,5.06,3.51
3.86,5.07,3.53
3.85,5.07,3.58
3.88,5.08,3.62
3.89,5.08,3.86
3.9,5.08,3.83
3.91,5.08,3.93
3.92,5.08,3.94
3.93,5.09,3.93
3.94,5.09,3.89
3.94,5.09,3.81
3.95,5.09,3.83
3.96,5.1,3.84
3.96,5.11,3.91
3.97,5.11,4.03
3.97,5.12,4.08
3.99,5.12,4.36
4,5.13,4.6
4,5.13,4.67
4.02,5.14,4.63
4.02,5.15,4.61
4.03,5.14,4.64
4.03,5.15,4.54
4.04,5.14,4.86
4.04,5.14,4.93
4.05,5.15,5.36
4.05,5.14,5.39
4.05,5.14,5.34
4.05,5.15,5.01
4.05,5.15,4.76
4.04,5.15,4.55
4.04,5.16,4.29
4.05,5.16,3.85
4.04,5.17,3.64
4.04,5.18,3.48
4.03,5.18,4.31
4.05,5.19,4.28
4.05,5.2,4.45
4.06,5.2,4.59
4.07,5.21,4.76
4.09,5.21,5.01
4.08,5.21,5.08
4.09,5.22,4.97
4.09,5.22,5.14
4.09,5.23,5.36
4.1,5.24,5.62
4.11,5.24,5.54
4.11,5.25,5.38
4.11,5.26,5.09
4.11,5.26,5.2
4.11,5.27,5.33
4.13,5.28,5.49
4.13,5.29,5.92
4.14,5.29,6.18
4.14,5.29,6.16
4.15,5.3,6.08
4.15,5.3,6.15
4.14,5.3,6.08
4.15,5.3,6.49
4.16,5.31,7
4.16,5.31,7.01
4.16,5.31,7.13
4.16,5.31,7.04
4.15,5.32,7.19
4.15,5.32,7.72
4.13,5.33,7.91
4.13,5.32,7.16
4.13,5.33,6.71
4.13,5.33,6.48
4.12,5.33,7.03
4.12,5.34,6.74
4.12,5.34,6.47
4.12,5.35,6.41
4.11,5.36,6.24
4.09,5.36,5.93
4.09,5.36,5.29
4.11,5.37,4.86
4.12,5.37,4.49
4.12,5.38,3.77
4.12,5.39,3.32
4.12,5.39,3.78
4.13,5.4,4.14
4.13,5.41,4.7
4.13,5.42,5.41
4.12,5.42,5.08
4.14,5.42,4.67
4.14,5.43,4.49
4.15,5.43,4.19
4.16,5.43,4.02
4.18,5.44,3.4
4.19,5.45,3.18
4.2,5.46,3.72
4.21,5.46,3.72
4.21,5.46,3.65
4.22,5.47,3.87
4.21,5.48,4.06
4.23,5.48,4.01
4.24,5.49,4.65
4.25,5.5,4.72
4.26,5.51,4.77
4.27,5.52,5.06
4.27,5.53,5.31
4.29,5.53,5.56
4.29,5.53,6.05
4.29,5.53,6.29
4.3,5.54,6.35
4.3,5.55,7.19
4.31,5.55,8.02
4.31,5.55,8.67
4.32,5.55,8.48
4.32,5.56,7.16
4.32,5.56,7.87
4.3,5.57,7.36
4.29,5.58,7.76
4.29,5.58,7.06
4.29,5.59,7.99
4.29,5.59,8.23
4.3,5.59,8.43
4.3,5.59,8.15
4.3,5.6,7.75
4.3,5.6,8.74
4.3,5.6,8.36
4.29,5.61,7.24
4.26,5.61,7.59
4.22,5.61,7.18
4.19,5.61,6.49
4.18,5.62,5.58
4.16,5.62,5.54
4.17,5.62,5.69
4.17,5.63,5.32
4.18,5.64,5.19
4.19,5.65,6.16
4.2,5.65,6.46
4.21,5.65,6.38
4.22,5.65,6.08
4.23,5.66,5.47
4.24,5.66,5.5
4.25,5.66,4.96
4.26,5.67,4.85
4.26,5.68,5.05
4.27,5.68,4.88
4.28,5.69,5.18
4.28,5.69,5.44
4.28,5.69,5.28
4.29,5.7,5.15
4.29,5.7,5.07
4.3,5.71,4.93
4.31,5.72,4.81
4.32,5.72,4.36
4.32,5.73,4.6
4.33,5.74,4.66
4.34,5.75,4.61
4.35,5.76,4.54
4.36,5.76,4.94
4.37,5.76,5
4.37,5.77,5.15
4.37,5.77,5.5
4.37,5.78,5.77
4.37,5.79,6.19
4.38,5.8,6.16
4.37,5.8,6.06
4.37,5.81,6.45
4.37,5.82,6.46
4.38,5.82,6.32
4.41,5.83,6.31
4.41,5.84,6.43
4.42,5.85,6.71
4.42,5.85,7.07
4.43,5.86,7.04
4.43,5.87,7.84
4.44,5.87,8.13
4.45,5.87,8.79
4.45,5.88,9.12
4.44,5.88,9.35
4.45,5.89,9.27
4.46,5.89,9.46
4.45,5.91,9.49
4.46,5.91,9.58
4.46,5.92,9.05
4.45,5.93,9.26
4.45,5.94,9.45
4.45,5.94,10.1
4.45,5.94,11.4
4.45,5.94,11.8
4.45,5.95,12
4.45,5.96,12
4.46,5.97,12.8
4.46,5.96,15.5
4.44,5.95,14
4.41,5.95,9.15
4.4,5.96,6.99
4.4,5.97,8.13
4.41,5.99,9.26
4.42,6,10.3
4.43,6.02,11.5
4.45,6.02,13.8
4.45,6.01,15.6
4.45,6.02,14.7
4.45,6.03,14.9
4.45,6.04,13.4
4.44,6.06,13.6
4.45,6.05,16.3
4.46,6.05,14.5
4.47,6.05,14.7
4.46,6.06,15.6
4.46,6.06,14.9
4.45,6.06,13.8
4.44,6.07,11.2
4.43,6.08,10.9
4.41,6.09,12.4
4.43,6.09,13.7
4.43,6.09,12.4
4.42,6.1,12.8
4.42,6.1,12.1
4.41,6.1,12.1
4.41,6.1,11.9
4.4,6.11,9.01
4.39,6.13,8.2
4.39,6.14,7.75
4.38,6.16,8.04
4.37,6.16,8.01
4.39,6.17,7.81
4.39,6.18,8.13
4.4,6.19,8.3
4.41,6.2,8.25
4.42,6.21,8.19
4.43,6.22,8.82
4.45,6.23,9.12
4.46,6.24,9.39
4.48,6.24,9.05
4.48,6.25,8.71
4.49,6.25,8.71
4.49,6.26,8.96
R2와 uncentered R2가 같은지 확인해볼 수 있는 데이터와 구하는 방법을 올립니다.
(절편항이 들어있는 회귀분석의 경우 uncentered R2와 R2가 다른 값인 것이 일반적입니다.)
컴퓨터 패키지를 단 한번이라도 사용해보신 분들에게는 극히 쉬운 일입니다만,
그저 노파심에 적어봅니다.
<Matlab-최근버전-의 경우>만 적겠습니다.
1. File/Import data 클릭
2. 저장한 txt 파일을 열기(*아래의 데이터를 쓰는 경우)
3. Import Wizard가 뜨는데, 이때 한 번은 그냥 next 클릭/
'create vectors from each column using column names.'선택 후 Finish 클릭
4. 다음을 입력 (맨 윗줄 '%%%%..'에서부터 맨 아랫줄 '%%%%...'까지 다 긁어서 붙인후 한 번만 엔터쳐도 됨)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 잔차를 종속변수로 두었을 때 R2와 uncentered R2를 비교하기
%
% step 1: resid를 구하는 regression을 수행
% step 2: resid를 종속변수로 두어 다시 regression
% step 3: R2와 uncentered R2의 값을 비교하기
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
J=ones(length(Y),1);
% 절편항을 위한 설명변수인 J 벡터를 생성합니다.
[b, bint, resid]=regress(Y, [J X]);
% Y를 J와 X에 대해 회귀한 후 그것의 잔차를 resid에 저장합니다.
[beta, bint, residhat]= regress(resid, [J X Z]);
% resid를 J와 X 그리고 누락된 변수라 의심하는 Z에 대해서 회귀하여 회귀계수와 '잔차'를 구합니다.
% 여기에서의 '잔차'는 첫번째 단계에서 구한 resid와 혼동되지 않도록 주의
uncenR2= (residhat'*residhat)/(resid'*resid);
% uncentered R2는 yhat'*yhat / y'y 꼴입니다.
residbar=mean(resid)
% centered R2를 위한 resid의 평균을 구합니다.
% 실제 이 값을 확인해보면 굉장히 0에 가까운 작은 값이어서, 이것을 빼는 것이 무의미하다고, 즉 R2나 uncentered R2나 똑같을 거라고 짐작할 수 있습니다.
R2=((residhat-residbar)'*(residhat-residbar))/((resid-residbar)'*(resid-residbar));
% R2는 (yhat-ybar)'*(yhat-ybar) / y'y 꼴입니다.
result=R2-uncenR2
% 두 개의 R2값을 비교하기 위해 빼줍니다.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
5. result의 값이 practically 0인 것을 확인 (실제로는 0.0000000000000002220)
6. 대부분의 컴퓨터 패키지는 uncentered R2를 보여주지 않고 R2부터 보여주기 때문에, 만약 두 값이 다르다면 일일이 위와 같은 계산을 거쳐서 uncentered R2를 구해야 할 것 입니다. 따라서 위의 결과는 우리가 이용하기 굉장히 좋은 결과입니다.
아래는 데이터입니다.
변수명(X,Y,Z)을 포함해서 그대로 긁으신 후 txt파일로 저장하시면 됩니다.
X , Y, Z
3.58,4.93,2.84
3.6,4.94,2.71
3.62,4.94,2.85
3.64,4.94,2.96
3.65,4.95,2.85
3.65,4.95,3.25
3.63,4.95,3.24
3.59,4.96,3.36
3.59,4.95,4
3.59,4.95,4.12
3.59,4.94,4.21
3.65,4.94,4.57
3.68,4.94,4.44
3.67,4.94,3.95
3.66,4.94,3.44
3.65,4.94,3.24
3.65,4.94,3.39
3.64,4.94,2.64
3.63,4.94,2.4
3.63,4.95,2.29
3.62,4.95,2.49
3.62,4.95,2.43
3.61,4.95,2.38
3.59,4.95,2.27
3.59,4.95,2.3
3.59,4.95,2.41
3.59,4.96,2.42
3.62,4.96,2.33
3.63,4.96,2.29
3.65,4.96,2.36
3.66,4.96,2.27
3.67,4.97,2.4
3.66,4.97,2.3
3.68,4.97,2.35
3.7,4.98,2.46
3.71,4.98,2.62
3.7,4.98,2.75
3.72,4.98,2.75
3.72,4.98,2.72
3.72,4.99,2.73
3.72,4.99,2.69
3.72,4.99,2.72
3.73,4.99,2.94
3.73,4.99,2.84
3.74,4.99,2.79
3.74,4.99,2.75
3.74,4.99,2.8
3.74,5,2.86
3.75,5,2.91
3.76,5,2.92
3.77,5.01,2.9
3.78,5.01,2.91
3.79,5.01,2.92
3.79,5.01,2.99
3.79,5.02,3.14
3.79,5.02,3.32
3.8,5.02,3.38
3.8,5.03,3.45
3.81,5.03,3.52
3.81,5.03,3.52
3.82,5.04,3.53
3.82,5.04,3.53
3.82,5.04,3.55
3.84,5.04,3.48
3.85,5.05,3.48
3.85,5.05,3.48
3.85,5.05,3.48
3.86,5.06,3.51
3.86,5.07,3.53
3.85,5.07,3.58
3.88,5.08,3.62
3.89,5.08,3.86
3.9,5.08,3.83
3.91,5.08,3.93
3.92,5.08,3.94
3.93,5.09,3.93
3.94,5.09,3.89
3.94,5.09,3.81
3.95,5.09,3.83
3.96,5.1,3.84
3.96,5.11,3.91
3.97,5.11,4.03
3.97,5.12,4.08
3.99,5.12,4.36
4,5.13,4.6
4,5.13,4.67
4.02,5.14,4.63
4.02,5.15,4.61
4.03,5.14,4.64
4.03,5.15,4.54
4.04,5.14,4.86
4.04,5.14,4.93
4.05,5.15,5.36
4.05,5.14,5.39
4.05,5.14,5.34
4.05,5.15,5.01
4.05,5.15,4.76
4.04,5.15,4.55
4.04,5.16,4.29
4.05,5.16,3.85
4.04,5.17,3.64
4.04,5.18,3.48
4.03,5.18,4.31
4.05,5.19,4.28
4.05,5.2,4.45
4.06,5.2,4.59
4.07,5.21,4.76
4.09,5.21,5.01
4.08,5.21,5.08
4.09,5.22,4.97
4.09,5.22,5.14
4.09,5.23,5.36
4.1,5.24,5.62
4.11,5.24,5.54
4.11,5.25,5.38
4.11,5.26,5.09
4.11,5.26,5.2
4.11,5.27,5.33
4.13,5.28,5.49
4.13,5.29,5.92
4.14,5.29,6.18
4.14,5.29,6.16
4.15,5.3,6.08
4.15,5.3,6.15
4.14,5.3,6.08
4.15,5.3,6.49
4.16,5.31,7
4.16,5.31,7.01
4.16,5.31,7.13
4.16,5.31,7.04
4.15,5.32,7.19
4.15,5.32,7.72
4.13,5.33,7.91
4.13,5.32,7.16
4.13,5.33,6.71
4.13,5.33,6.48
4.12,5.33,7.03
4.12,5.34,6.74
4.12,5.34,6.47
4.12,5.35,6.41
4.11,5.36,6.24
4.09,5.36,5.93
4.09,5.36,5.29
4.11,5.37,4.86
4.12,5.37,4.49
4.12,5.38,3.77
4.12,5.39,3.32
4.12,5.39,3.78
4.13,5.4,4.14
4.13,5.41,4.7
4.13,5.42,5.41
4.12,5.42,5.08
4.14,5.42,4.67
4.14,5.43,4.49
4.15,5.43,4.19
4.16,5.43,4.02
4.18,5.44,3.4
4.19,5.45,3.18
4.2,5.46,3.72
4.21,5.46,3.72
4.21,5.46,3.65
4.22,5.47,3.87
4.21,5.48,4.06
4.23,5.48,4.01
4.24,5.49,4.65
4.25,5.5,4.72
4.26,5.51,4.77
4.27,5.52,5.06
4.27,5.53,5.31
4.29,5.53,5.56
4.29,5.53,6.05
4.29,5.53,6.29
4.3,5.54,6.35
4.3,5.55,7.19
4.31,5.55,8.02
4.31,5.55,8.67
4.32,5.55,8.48
4.32,5.56,7.16
4.32,5.56,7.87
4.3,5.57,7.36
4.29,5.58,7.76
4.29,5.58,7.06
4.29,5.59,7.99
4.29,5.59,8.23
4.3,5.59,8.43
4.3,5.59,8.15
4.3,5.6,7.75
4.3,5.6,8.74
4.3,5.6,8.36
4.29,5.61,7.24
4.26,5.61,7.59
4.22,5.61,7.18
4.19,5.61,6.49
4.18,5.62,5.58
4.16,5.62,5.54
4.17,5.62,5.69
4.17,5.63,5.32
4.18,5.64,5.19
4.19,5.65,6.16
4.2,5.65,6.46
4.21,5.65,6.38
4.22,5.65,6.08
4.23,5.66,5.47
4.24,5.66,5.5
4.25,5.66,4.96
4.26,5.67,4.85
4.26,5.68,5.05
4.27,5.68,4.88
4.28,5.69,5.18
4.28,5.69,5.44
4.28,5.69,5.28
4.29,5.7,5.15
4.29,5.7,5.07
4.3,5.71,4.93
4.31,5.72,4.81
4.32,5.72,4.36
4.32,5.73,4.6
4.33,5.74,4.66
4.34,5.75,4.61
4.35,5.76,4.54
4.36,5.76,4.94
4.37,5.76,5
4.37,5.77,5.15
4.37,5.77,5.5
4.37,5.78,5.77
4.37,5.79,6.19
4.38,5.8,6.16
4.37,5.8,6.06
4.37,5.81,6.45
4.37,5.82,6.46
4.38,5.82,6.32
4.41,5.83,6.31
4.41,5.84,6.43
4.42,5.85,6.71
4.42,5.85,7.07
4.43,5.86,7.04
4.43,5.87,7.84
4.44,5.87,8.13
4.45,5.87,8.79
4.45,5.88,9.12
4.44,5.88,9.35
4.45,5.89,9.27
4.46,5.89,9.46
4.45,5.91,9.49
4.46,5.91,9.58
4.46,5.92,9.05
4.45,5.93,9.26
4.45,5.94,9.45
4.45,5.94,10.1
4.45,5.94,11.4
4.45,5.94,11.8
4.45,5.95,12
4.45,5.96,12
4.46,5.97,12.8
4.46,5.96,15.5
4.44,5.95,14
4.41,5.95,9.15
4.4,5.96,6.99
4.4,5.97,8.13
4.41,5.99,9.26
4.42,6,10.3
4.43,6.02,11.5
4.45,6.02,13.8
4.45,6.01,15.6
4.45,6.02,14.7
4.45,6.03,14.9
4.45,6.04,13.4
4.44,6.06,13.6
4.45,6.05,16.3
4.46,6.05,14.5
4.47,6.05,14.7
4.46,6.06,15.6
4.46,6.06,14.9
4.45,6.06,13.8
4.44,6.07,11.2
4.43,6.08,10.9
4.41,6.09,12.4
4.43,6.09,13.7
4.43,6.09,12.4
4.42,6.1,12.8
4.42,6.1,12.1
4.41,6.1,12.1
4.41,6.1,11.9
4.4,6.11,9.01
4.39,6.13,8.2
4.39,6.14,7.75
4.38,6.16,8.04
4.37,6.16,8.01
4.39,6.17,7.81
4.39,6.18,8.13
4.4,6.19,8.3
4.41,6.2,8.25
4.42,6.21,8.19
4.43,6.22,8.82
4.45,6.23,9.12
4.46,6.24,9.39
4.48,6.24,9.05
4.48,6.25,8.71
4.49,6.25,8.71
4.49,6.26,8.96