april 25, 2006lecture 14 slide #1 logits and factor analysis homework review logit analysis –logit...

April 25, 2006 Lecture 14 Slide #1

Logits and Factor Analysis

• Homework Review

• Logit Analysis– Logit Interpretation– Logit diagnostics

• Factor analysis-in-brief– Scaling in factor analysis– Using scales


Homework Review

• Predict the choice between Quad and LD-HR– required recodes

• Plot the effect of the risk index and ideology on probability of a shift


Interpreting Logits (again)

• Logits can be used to directly calculate odds:

• Logits can be reversed to obtain the predicted probabilities:

ˆ P =1

1+e−ˆ L

antilog=eˆ L


Interpreting Logits, ContinuedHow would you calculate the effect of a particular combination of independent variables on the probability of Y=1?

• Set all Xj’s at the appropriate values, then calculate

(e=2.71828..)

• The result is the average probability for that “type” of respondents

ˆ P =1

1+e−ˆ L


Example: Effect of ideology, gender on probability of choosing the LD-HR

model for standard setting

• Model: choice (DR_standard) as a function of:– Ideology, gender and certainty

• Types– A=conservative male; B=liberal female

– Set certainty at the average

– A: conservative, male, average level of certainty• Ideology=7, gender = 1, certainty=5.865

– B: liberal, female, average level of certainty• Ideology=1, gender = 0, certainty=5.865

0=chose threshold, 1=choose LD-HR


Logit Model Results

. logit DR_standard DR_cert ideology sex if DR_correct==0

Iteration 0: log likelihood = -549.8376Iteration 1: log likelihood = -506.99023Iteration 2: log likelihood = -506.45582Iteration 3: log likelihood = -506.45549

Logit estimates Number of obs = 891 LR chi2(3) = 86.76 Prob > chi2 = 0.0000Log likelihood = -506.45549 Pseudo R2 = 0.0789

------------------------------------------------------------------------------ DR_standard | Coef. Std. Err. z P>|z| [95% Conf. Interval]-------------+---------------------------------------------------------------- DR_cert | -.2312371 .0316 -7.32 0.000 -.2931719 -.1693022 ideology | -.1996686 .0583874 -3.42 0.001 -.3141059 -.0852313 sex | -.6374052 .207332 -3.07 0.002 -1.043768 -.2310419 _cons | 1.824489 .3209997 5.68 0.000 1.195341 2.453637------------------------------------------------------------------------------


Analyzing TypesL =1.824489 + (-.1996686*(ideology)) + (-.6374052*(sex)) +

(-.2312371*(certainty))

L ProbabilityConservative Males: -1.567 0.173(indep. vars.: 7; 1; 5.865)

Liberal Females: 0.269 0.567(indep. vars.: 1; 0; 5.865)

Hint: Use a spreadsheet to calculate L and P.

In Excel, the formula for probability would be:

P = 1/(1+EXP(-L))

Example from Scientist data


Estimates of Coefficient Strength• In Excel, calculate the difference in probability for

each X at its min and max, holding all other variables constant:

Ideology Sex DR_cert L Prob DifferenceMean 3.613 0.847 5.865 -0.793 0.3115Minimum 1 0.847 5.865 -0.271 0.4326 0.246Maximum 7 0.847 5.865 -1.469 0.1871

Ideology Sex DR_cert L Prob DifferenceMean 3.613 0.847 5.865 -0.793 0.3115Minimum 3.613 0 5.865 -0.253 0.4371 0.146Maximum 3.613 1 5.865 -0.891 0.291

Ideology Sex DR_cert L Prob DifferenceMean 3.613 0.847 5.865 -0.793 0.3115Minimum 3.613 0.847 0 0.5632 0.6372 -0.49Maximum 3.613 0.847 10 -1.749 0.1482


Estimated Logit Probabilities

0

1

2

3

4

Density

0 .2 .4 .6 .8predicted probability


Logit DiagnosticsThe most useful diagnostics are to match “influence” (case-wise dfbetas) with predicted probabilities:

0

.2

.4

.6

Influence

0 .2 .4 .6 .8predicted probability


Logit Outliers and Influencegsort –dBlist case_no dB phat DR_standard ideology sex DR_cert in 1/10

+---------------------------------------------------------------------+ | case_no dB phat DR_sta~d ideology sex DR_cert | |---------------------------------------------------------------------| 1. | 1895 .5331572 .3101657 1 3 1 6 | 2. | 5118 .5331572 .3101657 1 3 1 6 | 3. | 6086 .5331572 .3101657 0 3 1 6 | 4. | 2187 .5331572 .3101657 1 3 1 6 | 5. | 5616 .5331572 .3101657 0 3 1 6 | |---------------------------------------------------------------------| 6. | 7092 .5331572 .3101657 0 3 1 6 | 7. | 8726 .5331572 .3101657 1 3 1 6 | 8. | 7915 .5331572 .3101657 1 3 1 6 | 9. | 2680 .5331572 .3101657 1 3 1 6 | 10. | 3076 .5331572 .3101657 1 3 1 6 | +---------------------------------------------------------------------+

In this instance, the high influence cases are those in the mid-range on key variables (certainty, ideology). This simply makes them hard to predict.


Factor Analysis-in-Brief

• A means for estimating the underlying structure in a set

of “indicator” variables

– A reversal of regression analysis

– What unobserved dependent variable would best explain the

variance in the observed indicators?

• Multiple related approaches

– Exploratory versus Confirmatory

– We focus on confirmatory

• principal factor analysis


Indicators of Societal Risk Management Views

• U/E4_34: When the risk is very small, it is acceptable for the government to impose that risk on individuals without their consent.

• U/E4_35: Even if the potential benefits to society are very large, it is wrong for the government to impose risks on individuals without their consent.

• U/E4_36: It is acceptable for the government to impose risks without consent if the individuals harmed by the policy are compensated for their losses.

• U/E4_37: For society as a whole to survive and prosper, it is necessary that risks and sacrifices be accepted by citizens.


Underlying Structure?

SocietalRisk Mgmt.Perspective

Small risksOK to impose

Wrong to imposeeven if benefits

are large

Acceptable ifcompensated

Risks necessaryfor society to

prosper


Factor procedure


“Rotating” Factors


Saving Factor Scores


Using Factor Scores. logit DR_standard DR_cert ideology sex risk_factor if DR_correct==0

Iteration 0: log likelihood = -546.81911Iteration 1: log likelihood = -483.95752Iteration 2: log likelihood = -482.27287Iteration 3: log likelihood = -482.26589

Logit estimates Number of obs = 885 LR chi2(4) = 129.11 Prob > chi2 = 0.0000Log likelihood = -482.26589 Pseudo R2 = 0.1181

------------------------------------------------------------------------------ DR_standard | Coef. Std. Err. z P>|z| [95% Conf. Interval]-------------+---------------------------------------------------------------- DR_cert | -.2420916 .0328576 -7.37 0.000 -.3064914 -.1776919 ideology | -.1533426 .0599723 -2.56 0.011 -.2708861 -.0357992 sex | -.5656747 .2131953 -2.65 0.008 -.9835298 -.1478196risk_factors | -.675367 .1072255 -6.30 0.000 -.8855251 -.4652088 _cons | 1.618216 .3279038 4.94 0.000 .9755367 2.260896------------------------------------------------------------------------------


Summing Up• Factor analysis is a means of

“data reduction”• Useful when you have

indicators• Best when used in

“confirmatory” mode• Has an “exploratory” side


Final (non-cumulative) Exam

• Will include a simple exercise in– Using factor analysis to construct a factor score– Running a logit model using that score (along with

several other variables)– Plotting the influence of the independent variables

• Will be an extension of the homework completed for today– No new data

• Posted on Friday (April 28) at 5PM• Due on Wednesday (May 3) at 5PM

april 25, 2006lecture 14 slide #1 logits and factor analysis homework review logit analysis –logit...

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