Graphs in Statistical Analysis

F. J. AnscombeDept. of Statistics, Yale UnivThe American Statistician, 1973

Jan 2, 2014Hee-gook Jun

2 / 19

Outline

Introduction Regression Analysis Model Graphs in Statistical Analysis Conclusion

3 / 19

Both Calculations and Graphs

Should be made and studied Each will contribute to understanding

4 / 19

Stereotype of Graph

Calculations are exact, but graphs are rough Intricate calculations are virtuous, whereas looking at the data is cheating Some data fits specific statistical calculations

5 / 19

Purpose of Graph

Perceive broad features of the data Look behind those broad features Check if the assumptions of statistical calculation are correct

6 / 19

Good Statistical Analysis is

Not a simple routine way– More than one pass through the computer

Sensitive to specific features in the data Sensitive to general background information about data

7 / 19

Outline

Introduction Regression Analysis Model Graphs in Statistical Analysis Conclusion

8 / 19

Regression Analysis

Explain data Estimate new data

x

y

x

y

𝒚=𝒂+𝒃𝒙

𝑎

9 / 19

Regression Analysis Model

Model

data

12345..

1.52.02.53.02.9 ..

( x , y )(1.0, 1.5)(2.0, 2.0)(3.0, 2.5)(4.0, 3.0) …

𝒇 (𝒙 )=𝟏+𝟏 .𝟓𝑿𝒇 :𝑿→𝒀

𝒚=𝒂+𝒃𝒙

new instance (x=10, y=?) → f(10) = 1 + 1.5 * 10 = 16

10 / 19

Residual Value [1/3]

x

y

𝒑𝒆𝒓𝒇𝒆𝒄𝒕𝒍𝒚 𝒇𝒊𝒕𝒕𝒆𝒅𝒎𝒐𝒅𝒆𝒍

x

y

𝒇 (𝒙)=𝒂+𝒃𝒙

𝒖𝒏𝒆𝒙𝒑𝒍𝒂𝒊𝒏𝒆𝒅 𝒗𝒂𝒓𝒊𝒂𝒕𝒊𝒐𝒏

11 / 19

Residual Value [2/3]

x

y

�̂�=𝒇 (𝒙)=𝒂+𝒃𝒙

(x, y)

(x, )

error

12 / 19

Residual Value [3/3]

�̂� 𝒊=𝜷𝟎+𝜷𝟏𝒙 𝒊

𝒚 𝒊=�̂� 𝒊+𝜺𝒊

x

y (x, y)

(x, )

error

𝒚 𝒊=𝜷𝟎+𝜷𝟏𝒙 𝒊+𝜺𝒊

𝒔𝒖𝒎𝒐𝒇 𝒆𝒓𝒓𝒐𝒓𝒔 ?

13 / 19

Outline

Introduction Regression Analysis Model Graphs in Statistical Analysis Conclusion

14 / 19

Numerical Calculations

10.08.0

13.09.0

11.014.06.04.0

12.07.05.0

8.046.957.588.818.339.967.244.26

10.844.825.68

9.148.148.748.779.268.106.133.109.137.264.74

7.466.77

12.747.117.818.846.085.398.156.425.73

8.08.08.08.08.08.08.0

19.08.08.08.0

6.585.767.718.848.477.045.25

12.505.567.916.89

10.08.0

13.09.0

11.014.06.04.0

12.07.05.0

10.08.0

13.09.0

11.014.06.04.0

12.07.05.0

Data set 1x y

Data set 2x y

Data set 3x y

Data set 4x y

N 11 11 11 11

Mean(x) 9.0 9.0 9.0 9.0

Mean(y) 7.5 7.5 7.5 7.5

Regression line y = 3 + 0.5x y = 3 + 0.5x y = 3 + 0.5x y = 3 + 0.5x

15 / 19

Data set 1

The kind of thing most people would see in their mind’s eye

16 / 19

Data set 2

Does not conform with the theoretical description

17 / 19

Data set 3

One of the observation is far from this line

18 / 19

Data set 4

There was something unsatisfactory about the data set

19 / 19

Conclusion

Both Calculations and Graphs contribute to understanding Thought and ingenuity devoted to devising good graphs are likely to pay off