Linear statistical models as a first statistics course for math majors
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Linear Statistical Models as a First Statistics Course for Math Majors. George W. Cobb Mount Holyoke College [email protected] CAUSE Webinar October 12, 2010. Overview. A. Goals for a first stat course for math majors B. Example of a modeling challenge

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Linear Statistical Models as a First Statistics Course for Math Majors

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Linear statistical models as a first statistics course for math majors

Linear Statistical Modelsas a First Statistics Coursefor Math Majors

George W. Cobb

Mount Holyoke College

[email protected]

CAUSE Webinar

October 12, 2010


Overview

Overview

A. Goals for a first stat course

for math majors

B. Example of a modeling challenge

C. Examples of methodological challenges

D. Some important tensions

E. Two geometries

F. Gauss-Markov Theorem

G. Conclusion


A goals for a first statistics course for math majors

A. Goals for a First Statistics Course for Math Majors :

1. Minimize prerequisites

2. Teach what we want students to learn

- Data analysis and modeling

- Methodological challenges

- Current practice

3. Appeal to the mathematical mind

- Mathematical substance

- Abstraction as process

- Math for its own sake


B a modeling challenge pattern only how many groups

B. A Modeling Challenge:Pattern only – How many groups?


B a modeling challenge pattern plus context two lines

B. A Modeling Challenge:Pattern plus context: two lines?


B a modeling challenge lurking variable 1 the five solid dots

B. A Modeling Challenge:Lurking variable 1: The five solid dots


B a modeling challenge lurking variable 1 the five solid dots1

B. A Modeling Challenge:Lurking variable 1: The five solid dots


B a modeling challenge lurking variable 2 confounding

B. A Modeling Challenge:Lurking variable 2 – confounding


B a modeling challenge lurking variable 2 confounding1

B. A Modeling Challenge:Lurking variable 2 – confounding


B a modeling challenge lurking variable 2 confounding2

B. A Modeling Challenge:Lurking variable 2 -- confounding


B the modeling challenge summary

B. The Modeling Challenge: summary

  • Do the data provide evidence of discrimination?

  • Alternative explanations

    based on classical economics

  • Additional variables:

    percent unemployed in the subject

    percent non-academic jobs in the subject

    median non-academic salary in the subject

  • Which model(s) are most useful ?


C methodological challenges

C. Methodological Challenges

  • How to “solve” an inconsistent linear system?

    Stigler, 1990: The History of Statistics

  • How to measure goodness of fit?

    (Invariance issues)

  • How to identify influential points?

    Exploratory plots

  • How to measure multicolinearity?

    Note that none of these require any assumptions about probability distributions


D some important tensions

D. Some Important Tensions

  • Data analysis v. methodological challenges

  • Abstraction: Top down v. bottom up

  • Math as tool v. math as aesthetic object

  • Structure by dimension v. structure by assumptions

    Distribution Number of covariates

    assumptions One Two Many

    A. None

    B. Moments

    C. Normality

    5. Two geometries


D4 structure by assumptions

D4. Structure by assumptions

  • No distribution assumptions about errors

    1. Inconsistent linear systems; OLS Theorem

    2. Measuring fit and correlation

    3. Measuring influence and the Hat matrix

    4. Measuring multicolinearity

    B. Moment assumptions: E{e}=0, Var{e}=s2I

    1. Moment Theorem: EV and Var for OLS estimators

    2. Variance Estimation Theorem: E{MSE} = s2

    3. Gauss-Markov Theorm: OLS = BLUE

    C. Normality assumption: e ~ N


D4 structure by assumptions1

D4. Structure by assumptions

  • Normality assumption: e ~ N

    1. Herschel-Maxwell Theorem

    2. Distribution of OLS estimators

    3. t-distribution and confidence intervals for bj

    4. Chi-square distribution and confidence interval for s2

    5. F-distribution and nested F-test


E two geometries the crystal problem tom moore primus 1992

E. Two Geometries:the Crystal Problem (Tom Moore, Primus, 1992)

y1 = b + e1y2 = 2b + e2


E two geometries crystal problem individual space variable space

E. Two Geometries: Crystal ProblemIndividual Space Variable Space

Point = Case Vector = Variable

Axis = Variable Axis = Case


F gauss markov theorem

F. Gauss-Markov Theorem:

  • Crystal Example: y1 = b + e1, y2 = 2b + e2

  • LINEAR: estimator = a1y1 + a2y2

  • UNBIASED: a1b+ a22b = b, i.e. a1+ 2a2= 1.

    • Ex: y1 = 1y1+ 0y2a=(1,0)T

    • Ex: (1/2)y2= 1y1 + 0y2a=(0, 1/2)T

    • Ex: (1/5)y1+ (2/5)y2a=(1/5, 2/5)T

  • BEST: SD2(aTy) = s2|a|2 best = shortest a

  • THEOREM:OLS = BLUE


F gauss markov theorem coefficient space for the crystal problem

F. Gauss-Markov Theorem:Coefficient Space for the Crystal Problem


F gauss markov theorem estimator y 1 1 0 y 1 y 2 t

F. Gauss-Markov Theorem:Estimator y1 = (1,0)(y1,y2)T

B.


F gauss markov theorem estimator y 2 2 0 1 2 y 1 y 2 t

F. Gauss-Markov Theorem: Estimator y2/2 = (0,1/2)(y1,y2)T

B.


F gauss markov theorem ols estimator 1 5 2 5 y 1 y 2 t

F. Gauss-Markov Theorem: OLS estimator = (1/5,2/5)(y1,y2)T

B.


F gauss markov theorem the set of linear unbiased estimators

F. Gauss-Markov Theorem:The Set of Linear Unbiased Estimators

B.


F gauss markov theorem lues form a translate of error space

F. Gauss-Markov Theorem:LUEs form a translate of error space

B.


F gauss markov theorem ols estimator lies in model space

F. Gauss-Markov Theorem:OLS estimator lies in model space

B.


F gauss markov theorem four steps plus pythagoras

F. Gauss-Markov Theorem:Four Steps plus Pythagoras

1. OLS estimator is an LUE

2. LUEs of βj are a flat set in n-space

3. LUEs of 0 = error space

4. OLS estimator lies in model space


G conclusion a least squares course can be

G. Conclusion:A Least Squares Course can be

1. Accessible

- Requires only Calc. I and matrix algebra

2. A good vehicle for teaching data modeling

3. A sequence of methodological challenges

4. Mathematically attractive

- Mathematical substance

- Abstraction as process

- Math as tool and for its own sake

5. A direct route to current practice

- Generalized linear models

- Correlated data, time-to-event data

- Hierarchical Bayes


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