advanced models and methods in behavioral research

Advanced Methods and Models in Behavioral Research –

Advanced Models and Methods in Behavioral Research

• Chris Snijders• [email protected]

• 3 ects

• http://www.chrissnijders.com/ammbr (=studyguide)

• literature: Field book + separate course material

• laptop exam (+ assignments)

ToDo (if not done yet):

Enroll in 0a611


The methods package

• MMBR (6 ects)– Blumberg: questions, reliability, validity, research design– Field: SPSS: factor analysis, multiple regression, ANcOVA,

sample size etc

• AMMBR (3 ects) - Field (1 chapter): logististic regression - literature through website:

conjoint analysis multi-level regression


Models and methods: topics• t-test, Cronbach's alpha, etc• multiple regression, analysis of (co)variance and

factor analysis

• logistic regression• conjoint analysis / repeated measures

– Stata next to SPSS– “Finding new questions”– Some data collection

In the background: “now you should be able to deal with data on your own”


Methods in brief (1)

• Logistic regression: target Y, predictors Xi. Y is a binary variable (0/1).

- Why not just multiple regression?- Interpretation is more difficult- goodness of fit is non-standard- ...

(and it is a chapter in Field)


Methods in brief (2)

• Conjoint analysis

Underlying assumption: for each user, the "utility" of an offer can be written as

U(x1,x2, ... , xn) = c0 + c1 x1 + ... + cn xn

- 10 Euro p/m- 2 years fixed- free phone- ...

How attractive is thisoffer to you?

Conjoint analysis as an “in between method”

BetweenWhich phone do you like and why?What would your favorite phone be?

And:Let’s keep track of what people buy.

We have:



Local Master Thesis example:

Fiber to the home

Speed: really fastPrice: sort of highInstallation: free!Your neighbors: are in!

How attractive is this to you?

(Roel Schuring)

Coming up with new ideas (3)


“More research is necessary”

But on what?

YOU: come up with sensible new ideas, given previous research

Stata next to SPSS


• It’s just better (faster, better written, more possibilities, better programmable …)

• Multi-level regression is much easier than in SPSS

• It’s good to be exposed to more than just a single statistics package (your knowledge should not be based on “where to click” arguments)

• More stable

• BTW Supports OSX as well… (anybody?)

Every advantage has a disadvantage• Output less “polished”

• It takes some extra work to get you started

• The Logistic Regression chapter in the Field book uses SPSS (but still readable for the larger part)

• (and it’s not campus software, but subfaculty software)

• Installation …



If on Windows, try downloading

• www.chrissnijders.com/ammbr/TUeStata12-zip.exe

Logistic Regression Analysis

Credit where credit is due:slides adapted from Gerrit Rooks

That is: your Y variable is 0/1: Now what?

The main points

1. Why do we have to know and sometimes use logistic regression?

2. What is the underlying model? What is maximum likelihood estimation?

3. Logistics of logistic regression analysis1. Estimate coefficients2. Assess model fit3. Interpret coefficients4. Check residuals

4. An SPSS example

Advanced Methods and Models in Behavioral Research

Suppose we have 100 observations with information about an individuals age and wether or not this indivual had some kind of a heart disease (CHD)

ID age CHD1 20 02 23 03 24 04 25 1…98 64 099 65 1

100 69 1

A graphic representation of the data

CHD

Age

Let’s just try regression analysis

pr(CHD|age) = -.54 +.022*Age

... linear regression is not a suitable model for probabilities

pr(CHD|age) = -.54 +.0218107*Age

In this graph for 8 age groups, I plotted the probability of having a heart disease (proportion)

A nonlinear model is probably better here

Something like this

This is the logistic regression model

)( 111011)|Pr(

XbbeXY

Predicted probabilities are always between 0 and 1

)( 111011)|Pr(

XbbeXY

similar to classic regressionanalysis

Side note: this is similar to MMBR …


Suppose Y is a percentage (so between 0 and 1).

Then consider

…which will ensure that the estimated Y will vary between 0 and 1and after some rearranging this is the same as

… (continued)


And one “solution” might be:

- Change all Y values that are 0 to 0.001- Change all Y values that are 1 to 0.999

Now run regression on log(Y/(1-Y)) …

… but that really is sort of higgledy-piggledy …

Logistics of logistic regression

1. How do we estimate the coefficients? 2. How do we assess model fit?3. How do we interpret coefficients? 4. How do we check regression assumptions?

Kinds of estimation in regression

• Ordinary Least Squares (we fit a line through a cloud of dots)

• Maximum likelihood (we find the parameters that are the most likely, given our data)

We never bothered to consider maximum likelihood in standard multiple regression, because you can show that they lead to exactly the same estimator (in MR, that is, normally they differ).

Actually, maximum likelihood has superior statistical properties (efficiency, consistency, invariance, …)


Maximum likelihood estimation• Method of maximum likelihood yields values

for the unknown parameters that maximize the probability of obtaining the observed set of data

)( 111011)|Pr(

XbbeXY

Unknown parameters

Maximum likelihood estimation• First we have to construct the “likelihood

function” (probability of obtaining the observed set of data).

Likelihood = pr(obs1)*pr(obs2)*pr(obs3)…*pr(obsn)

Assuming that observations are independent

Log-likelihood

• For technical reasons the likelihood is transformed in the log-likelihood (then you just maximize the sum of the logged probabilities)

LL= ln[pr(obs1)]+ln[pr(obs2)]+ln[pr(obs3)]…+ln[pr(obsn)]


Some subtleties

• In OLS, we did not need stochastic assumptions to be able to calculate a best-fitting line (only for the estimates of the confidence intervals we need that). With maximum likelihood estimation we need this from the start

(and let us not be bothered at this point by how the confidence intervals are calculated in

maximum likelihood)

Note: optimizing log-likelihoods is difficult• It’s iterative (“searching the landscape”)

it might not converge it might converge to the wrong answer



Nasty implication: extreme cases should be left out

(some handwaving here)


SPSS output

Estimation of coefficients: SPSS Results

Variables in the Equation

B S.E. Wald df Sig. Exp(B)

Step 1a age ,111 ,024 21,254 1 ,000 1,117

Constant -5,309 1,134 21,935 1 ,000 ,005

a. Variable(s) entered on step 1: age.

)11.3.5( 111)|Pr( Xe

XY

)11.3.5( 111)|Pr( Xe

XY

)11.3.5( 111)|Pr( Xe

XY

This function fits best: other values of b0 and b1 give worse results (that is, other values have a smaller likelihood value)

Illustration 1: suppose we chose .05X instead of .11X

)05.3.5( 111)|Pr( Xe

XY

)40.3.5( 111)|Pr( Xe

XY

Illustration 2: suppose we chose .40X instead of .11X


• Estimate the coefficients (and their conf.int.)• Assess model fit

– Between model comparisons– Pseudo R2 (similar to multiple regression)– Predictive accuracy

• Interpret coefficients • Check regression assumptions

41

Model fit: comparisons between models

)]baseline()New([22 LLLL

The log-likelihood ratio test statistic can be used to test the fit of a model

The test statistic has achi-square distribution

reduced modelfull model

NOTE This is sort of similar to the variance decomposition tables you see in MR!

Between model comparisons: the likelihood ratio test

)( 11011)(P Xbbe

Y


reduced modelfull model

)( 011)(P be

Y

The model including only an interceptIs often called the empty model. SPSS uses this model as a default.

)]baseline(2)New(22 LLLL

Omnibus Tests of Model Coefficients

Chi-square df Sig.

Step 1 Step 29,310 1 ,000

Block 29,310 1 ,000

Model 29,310 1 ,000

Model Summary

Step -2 Log likelihood

Cox & Snell R

Square

Nagelkerke R

Square

1 107,353a ,254 ,341

a. Estimation terminated at iteration number 5 because

parameter estimates changed by less than ,001.

This is the test statistic,and it’s associated significance

Between model comparison: SPSS output

45

Overall model fitpseudo R2

Just like in multiple regression, pseudo R2 ranges 0.0 to 1.0

– Cox and Snell• cannot theoretically

reach 1

– Nagelkerke• adjusted so that it

can reach 1

)(2)(2

LOGIT2

EmptyLLModelLLR

log-likelihood of modelbefore any predictors wereentered

log-likelihood of the modelthat you want to test

NOTE: R2 in logistic regression tends to be (even) smaller than in multiple regression

46

Overall model fit: Classification table

We predict 74% correctly

Classification Tablea

Observed

Predicted

chd

0 1

Percentage

Correct

Step 1 chd 0 45 12 78,9

1 14 29 67,4

Overall Percentage 74,0

a. The cut value is ,500

47


14 cases had a CHD while according to our modelthis shouldnt have happened


Observed

Predicted

chd

0 1

Percentage

Correct

Step 1 chd 0 45 12 78,9

1 14 29 67,4



48


12 cases didn’t have a CHD while according to our modelthis should have happened


Observed

Predicted

chd

0 1

Percentage

Correct

Step 1 chd 0 45 12 78,9

1 14 29 67,4




• Estimate the coefficients • Assess model fit• Interpret coefficients

– Direction– Significance– Magnitude

• Check regression assumptions

50

The Odds Ratio

)...(

)...(

)...( 1110

1110

1110 111)(

nn

nn

nn XbXbb

XbXbb

XbXbb ee

eYp

We had:

And after some rearranging we can get

Magnitude of association: Percentage change in odds

event

event

prob1probOddsi

Probability Odds25% 0.3350% 175% 3

52

Interpreting coefficients: direction

• original b reflects changes in logit: b>0 implies positive relationship

• exponentiated b reflects the “changes in odds”: exp(b) > 1 implies a positive relationship

nnxbxbxbbypyp

...)(1

)(lnlogit 22110

53

3. Interpreting coefficients: magnitude

• The slope coefficient (b) is interpreted as the rate of change in the "log odds" as X changes … not very useful.

• exp(b) is the effect of the independent variable on the odds, more useful for calculating the size of an effect

nnxbxbxbbypyp

...)(1

)(lnlogit 22110

nnxbxbxbb eeeeypyp

...)(1

)(Odds 22110



Step 1a age ,111 ,024 21,254 1 ,000 1,117

Constant -5,309 1,134 21,935 1 ,000 ,005

a. Variable(s) entered on step 1: age.

• For the age variable:

– Percentage change in odds = (exponentiated coefficient – 1) * 100 = 12%, or “the odds times 1,117”

– A one unit increase in age will result in 12% increase in the odds that the person will have a CHD

– So if a soccer player is one year older, the odds that (s)he will have CHD is 12% higher

Magnitude of association

Ref=0 Ref=1

Another way to get an idea of the size of effects: Calculating predicted probabilities

)11.3.5( 111)|Pr( Xe

XY

For somebody of 20 years old, the predicted probability is .04

For somebody of 70 years old, the predicted probability is .91

But this gets more complicatedwhen you have more than a single X-variable

(see blackboard)

Conclusion: if you consider the effect of a variable on the predicted probability, the size of the effect of X1 depends on the value of X2! (yuck!)


Testing significance of coefficients

• In linear regression analysis this statistic is used to test significance

• In logistic regression something similar exists

• however, when b is large, standard error tends to become inflated, hence underestimation (Type II errors are more likely)

b

bSE

Wald

t-distribution standard error of estimate

estimate

Note: This is not the Wald Statistic SPSS presents!!!

Interpreting coefficients: significance

• SPSS presents

• While Andy Field thinks SPSS presents this (at least in the 2nd version of the book):

bSEb

2

2

Wald

b

bSE

Wald


• Estimate the coefficients • Assess model fit• Interpret coefficients • Check regression assumptions

Checking assumptions

• Influential data points & Residuals– Follow Samanthas tips

• Hosmer & Lemeshow– Divides sample in subgroups– Checks whether there are differences between observed and

predicted between subgroups– Test should not be significant, if so: indication of lack of fit

Hosmer & Lemeshow

Test divides sample in subgroups, checks whether difference between observed and predicted is about equal in these groups

Test should not be significant (indicating no difference)

Examining residuals in logistic regression

1. Isolate points for which the model fits poorly2. Isolate influential data points

Residual statistics: Field’s rules of thumb


Time for a summary …

Logistic regression

• Y = 0/1• Multiple regression (or ANcOVA) is not right• You consider either the odds or the log(odds)

• It is estimated through “maximum likelihood”• Interpretation is a bit more complicated than normal• Assumption testing is a bit more concrete than in

multiple regression


Advanced Methods and Models in Behavioral Research – 2008/2009 68

Make sure to

• enroll in studyweb (0a611)• Read the Field chapter on logistic

regression• Go through the slides as well• Bring your laptop next time: we’ll go

through a logistic regression in Stata



69

Illustration with SPSS (without the outlier part)

• Penalty kicks data, variables:

– Scored: outcome variable,• 0 = penalty missed, and 1 = penalty scored

– Pswq: degree to which a player worries– Previous: percentage of penalties scored by a particular

player in their career

70

Case Processing Summary

75 100,00 ,0

75 100,00 ,0

75 100,0

Unweighted Casesa

Included in AnalysisMissing CasesTotal

Selected Cases

Unselected CasesTotal

N Percent

If weight is in effect, see classification table for the totalnumber of cases.

a.

Dependent Variable Encoding

01

Original ValueMissed PenaltyScored Penalty

Internal Value

SPSS OUTPUT Logistic Regression

Tells you somethingabout the number of observations and missings

71

Classification Tablea,b

0 35 ,00 40 100,0

53,3

ObservedMissed PenaltyScored Penalty

Result of PenaltyKick

Overall Percentage

Step 0

MissedPenalty

ScoredPenalty

Result of Penalty KickPercentage

Correct

Predicted

Constant is included in the model.a.

The cut value is ,500b.


,134 ,231 ,333 1 ,564 1,143ConstantStep 0B S.E. Wald df Sig. Exp(B)

Variables not in the Equation

34,109 1 ,00034,193 1 ,00041,558 2 ,000

previouspswq

Variables

Overall Statistics

Step0

Score df Sig.

Block 0: Beginning Blockthis table is based on the empty model, i.e. onlythe constant in the model

)( 011)(P be

Y

these variableswill be enteredin the modellater on

72

Block 1: Method = Enter

Omnibus Tests of Model Coefficients

54,977 2 ,00054,977 2 ,00054,977 2 ,000

StepBlockModel

Step 1Chi-square df Sig.

Model Summary

48,662a ,520 ,694Step1

-2 Loglikelihood

Cox & SnellR Square

NagelkerkeR Square

Estimation terminated at iteration number 6 becauseparameter estimates changed by less than ,001.

a.


Block is useful to check significance of individual coefficients, see Field

New model

this is the test statistic

after dividing by -2

Note: Nagelkerkeis larger than Cox

73


,065 ,022 8,609 1 ,003 1,067-,230 ,080 8,309 1 ,004 ,7941,280 1,670 ,588 1 ,443 3,598

previouspswqConstant

Step1

a


Variable(s) entered on step 1: previous, pswq.a.


30 5 85,77 33 82,5

84,0



Overall Percentage

Step 1

MissedPenalty

ScoredPenalty


Correct

Predicted

The cut value is ,500a.

Block 1: Method = Enter (Continued)

Predictive accuracy has improved (was 53%)

estimatesstandard errorestimates

significance based on Wald statistic

change in odds

74


,065 ,022 8,609 1 ,003 1,067-,230 ,080 8,309 1 ,004 ,7941,280 1,670 ,588 1 ,443 3,598

previouspswqConstant

Step1

a


Variable(s) entered on step 1: previous, pswq.a.


30 5 85,77 33 82,5

84,0



Overall Percentage

Step 1

MissedPenalty

ScoredPenalty


Correct

Predicted


How is the classification table constructed?

)*230,0*065,028,1(11)(P Pred. pswqpreviouse

Y

# cases not predictedcorrrectly

# cases not predictedcorrrectly

75


)*230,0*065,028,1(11)(P Pred. pswqpreviouse

Y

pswq previous scored Predict. prob.

18 56 1 .68

17 35 1 .41

20 45 0 .40

10 42 0 .85

76


pswq previous

scored Predict. prob.

predicted

18 56 1 .68 117 35 1 .41 020 45 0 .40 010 42 0 .85 1


30 5 85,77 33 82,5

84,0



Overall Percentage

Step 1

MissedPenalty

ScoredPenalty


Correct

Predicted


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