Regression and interpretation low R-squared!, Exercises of Literature

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Regression and interpretation

low R-squared!

Social Research Network 3

nd^

Meeting

Noosa

April 12-13, 2012Kenshi Itaoka

Mizuho Information & Research Institute, Inc.

Contents

z^

Motivation z^

Motivation z^

About r z^

Purpose of regressionPurpose

of regression

z^

Example z^

ConclusionConclusion

Fitness function in regression

z^

R-squared= (1- SSE) / SSTDefined as the ratio of the sum of squares explained by a regressionmodel and the "total" sum of squares around the mean.Interpreted as the ration of variance explained by a regression modelAdjuseted R squared= (

MSE) / MST

z^

Adjuseted

R

-squared= (1- MSE) / MST

MST = SST/(n-1)

MSE = SSE/(n-p-1)

z^

Other indicators such as AIC, BIC etc. also sometimeused for model selection.used for model selection.

Purpose of regression

z^

Analysis

of

partial

correlation

between

factors:

to avoid risks of interpretation of simple correlationsto avoid risks of interpretation of simple correlationsin multi-variable analyses^ z

Examination

of

influential

factors

on

phenomena

to

be

l i

d

explained z^

Causal inference?

Yes / No: need to check of logic

of causalityof causality.

Model estimation in regression

z^

True model

: y = b0 + b1 x1 + b2 x2 + u

z^

Estimated model:

y = a0 + a1 x1 + v

z^

X1 and x2 should be independentN^

l ti

b t

1

d^

th

th

f th

l^

t

z^

No correlation between x1 and v as the theory of the leastsquare methodology In this case,

s case, a

is equal to

b

But variance of error term is influenced by

x

Variance

of

u is

smaller

in

than

in

v^

by

x

Variance

of

u is

smaller

in

than

in

v^

by

x

R squared should be smaller in the estimated model than

that in the true model due to

x2.

z^

To examine the effectiveness of a1, the size of R

squared

does not matter

.

z^

Only significance of at matter z^

Only significance of at matter z^

In practice, examination of correlation between

x

and

x

(if

observed) is important.

Classification of independent variables

Examples are from a CCS related public perception study“Understanding how individuals perceive carbon dioxide: its“Understanding how individuals perceive carbon dioxide: its

relevance to CCS acceptance”

z^

Exogenous

variables

:

Exogenous

variables

:

• Example: age, education….

z^

Indigenous variables:^ z

Not directly related factors to dependent variable to beexplained

• Example: value and beliefs, CO2 knowledge….. z^

Directly

related

factors

to

dependent

variable

to

be

explained• Example: CCS knowledge, CCS perception…..

Example: CCS knowledge, CCS perception…..

Change of R squared

in regression analyses in Understanding how individuals perceive carbon

dioxide: its relevance to CCS acceptance”

p

Independent variable

Country

On shore

Off shore

Opinion 1

DemographicsValue and beliefsCO2 k

l d^

0 137

0 134

0 173

CO2 knowledgeCCS awarenessTrustworthy sources

0.

0.

0.

Opinion change(ANOVA)

Only type providedinformation package

0.

0.

0.

(ANOVA)

information

package

Opinion change(Regression)

Only perception ofpieces of informationincluded in provided

0.

0.

0.

information package

Opinion 2

Full set (not includingCCS impression

0.

0.

0.

z^

In the case we add CCS impression variables (positivity cleanness, usefulness, safety and maturity), R squared increase more than 0.6 but they tendt^

hid

ff^

t^

f^

f^

th^

i bl

p variables )

to hide effects of some of other variables.

Example of literature

z^

In the case exogenous variables are mainly used:

g^

y

literature cited: Geologic Storage of Carbon Dioxide: Risk Analyses and Implications forPublic AcceptancebyGregory R. Singleton B.S., Systems Engineering, University of Virginia, 2002

Example of literature

z^

A researcher’s comment: ……….in many social science settings, an Rsquare of 9% is considered respectable.

That's about as good as it gets in most psychology studies where two distinctvariables are correlated with each other. Example: extraversion explains onlyabout that much of the variation is sales effectiveness.When you measure variables with error, that can lower your Rsquare. What isS&P500 taken to be a measure of? The overall health of the economy, or just thestock market? Or just itself? http://www.marketingprofs.com/ea/qst_question.asp?qstID=

Example of factors to influence of fitness

z^

Accuracy of measurements (size of error) z^

Resolution of measurements z^

Less number of unobserved factors z^

Strength of causality z^

Fundamental randomness z^

…

Conclusion 2

In social science settingIn

social science setting, zList all potentially influential factors zCheck simple correlationzCheck simple correlation zConduct multiple regression zCheck residual (linearity)zCheck residual (linearity) zAgain try to find hidden factors zIf the list of variables for input of regression is

p^

g

defendable and there is not much multi-colinearity, themodel is considered to be fine even with low R-squared.M

b^

b tt

d^

t SEM (

th

l i )

zM

aybe better conduct SEM (path anaylsis)

Th

k

Th

ank you!

Contact: kenshi.itaoka@mizuho-ir.co.jp

jp

Example of literature

z^

Analysis literature cited:The public perspective of carbon capture and storage for CO

emission 2

reductions in ChinaH Duan - Energy Policy, 2010

To what extent

R ecognition on m easures against global w arm ing

'03 V i

sit (1003) '07 V i

sit （

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Energy-savi

ng

P -val

ue in χ square test<03 v.

s 07>

do publicknow about

CCS?

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sit （

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Low f

uel- efficient H ydrogenvehi

cle

CCS?^ （

2007 survey)

（^

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vehicl

e N ucl

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ey)

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B iom assenergyC O 2 sink &fixation by

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S olorenergyC O 2 C aptu re &

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t （334）

W ind energy

Iron

6 0%^ 20%

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S hare of respondents [%]

I^ know to som e extent

I^ have heard of

it^

I^ don'

t know at al

l

dispersali

n *(N um bers i

n parentheses indicate val

id responses.