Dr. Anja Neundorf (University of Nottingham, School of ... · Surveysandmeasurementmodelsingeneral...

Using latent class analysis in survey research

Dr. Anja Neundorf(University of Nottingham, School of Politics & IR)

Nottingham-Warwick-Birmingham Advanced Quantitative Methods

Nottingham, 8th March 2013

Anja Neundorf (Nottingham) Using latent class analysis in survey research 1 / 58

Outline

1 Surveys and measurement models in general

2 What is latent class analysis?

3 Example I: Role conflict

4 Example II: Religiosity

5 Summary: What is this all good for?

6 Practical issues


Surveys and measurement models in general


Using surveys

The purpose of survey research

I Collect information about respondents / consumers

I Some information is straight-forward and not questionable (age,gender, race...)

I Most information – esp. attitudes – is not straight-forward at all

I Examples: Racial prejudice, religious beliefs, partisan affiliations,consumer loyalty....

→ Measurement models needed!


From concepts to measurements

An example:

I Concept: Religious commitment

I Measurement: Church attendance, denomination, praying, believe inGod, importance of religion in one’s life...

→ These are highly correlated


Measurement Model: Religious commitment

Religious

Pray

Belong

Church

Importance

God

є1 є2 є3 є4 є5

Assumption: Observed co-variation between observed variables is due tounobserved, true variable


Measurement Model: General

η

y1 y2 y3 y4 y5

є1 є2 є3 є4 є5


Contrasting different measurement models

Latent Manifest measure: yiConcept: η Categorical Continuous

Categorical Latent Class Latent Profile

Continues Latent Trait/ Factor AnalysisIRT



Latent Manifest measure: yiConcept: η Categorical Continuous

Categorical Latent Class Latent Profile

Continues Latent Trait/ Factor AnalysisIRT



Factor Analysis (FA)

I Latent Variable: continues → normal distribution

I People differ quantitatively along one or more continua

Latent Class Analysis (LCA)

I Latent Variable: categorical → multinominal distribution

I Qualitative differences exist between groups or people


What is latent class analysis?


What is latent class analysis?

1 Measurement instrument

I Confirmatory or exploratoryI Non-causal

2 Using latent variable in causal model

I Can be used as dependent or independent variable


Latent Class Analysis (LCA): A measurement instrument

Overview: Aim of LCA

I Identify clusters of similar "types" of individuals from multivariatecategorical data → Typology-formation!

I Estimating the characteristics of these latent groups

I Estimating the probability that each observation belongs to eachgroup

I Investigating sources of confounding and non-independence amonga set of categorical variables


Latent Class Analysis (LCA): An illustration

Observed relationship

Eval. of economybad good

Approval negative 95 55 150Cameron positive 70 80 150

165 135

→ Significant relationship (chi2 = 8.42, p<.01)


Latent Class Analysis (LCA): An illustration

The confounder: Vote

Vote in last electionLabour Conservatives

Eval. of economy Eval. of economybad good bad good

Approval negative 80 20 15 35Cameron positive 40 10 30 70

→ Non-Significant relationship between Econ. and Cameron, conditionalon vote (chi2 = 0.00)

→ Items are locally independent


Measurement Model: General

η

y1 y2 y3 y4 y5

є1 є2 є3 є4 є5


Latent Class Analysis (LCA)

Aim of LCA

I Identify the unobserved / latent variable that explains systematicrelationship between observed variables

I What is the "vote" variables in your model?


Example I

Stouffer and Toby (1951) “Role conflict and personality”.American Journal of Sociology. 56: 395-406.


Stouffer and Toby (1951)

I Four set of situations “involving conflict between obligations to afriend and more general social obligations” (p. 396).

I Example item: “You are riding in a car that your close friend isdriving, and he hits a pedestrian. You know that he was going atleast 35 miles an hour in a 20-mile-an-hour zone. There are no otherwitnesses. His lawyer says if you testify under oath the speed wasonly 20 miles an hour, it may save him from serious consequences.What right has your friend to expect you to protect him?”

I Universalistic response: “He has no right as a friend to expect meto testify to the lower figure.”

I Particularistic response: “He has a right as a friend to expect meto testify to the lower figure.”


Cross-classification of 216 respondents

Observed Variables

I A: Auto passenger

I B: Insurance doctor

I C: Drama critic

I D: Board of directors

→ Responses

+ Universalistic value

– Particularistic value

Response pattern:2x2x2x2 = 16 possibilities

Response

A B C D Obs. N

+ + + + 42+ + + – 23+ + – + 6+ + – – 25+ – + + 6+ – + – 24+ – – + 7+ – – – 38– + + + 1– + + – 4– + – + 1– + – – 6– – + + 2– – + – 9– – – + 2– – – – 20


Reduction of information

I LCA helps us to reduce this number to a few (optimal) types ofrespondents to these 16 possible responses.

I Allowing some of the response patters as measurement error –treating them as ‘mis-classifications’ – rather than as true responsetypes.

→ Aim: Identify a set of mutually exclusive latent classes that accountfor the distribution of cases that occur within a cross-tab of observeddiscrete variables.


The latent class model (LCM)

Basic unrestricted LCM

πABCDXijklt = πX

t πA|Xit π

B|Xjt π

C |Xkt π

D|Xlt

I Latent variable Xt : (t=1,...,T)

I Item Ai : (i=1,...,I) ... Item Dl : (l=1,...,L)

I πABCDXijklt : Probability of a specific response pattern, depending on

class membership in t

I πXt : Latent class probability

I πA|Xit ...π

D|Xlt : Conditional probabilities



πXt : Latent class probability

I Describes the distribution of classes of the latent variable withinwhich the observed measures are locally independent of one another

I Two important aspects: 1) Number of classes; 2) relative size ofthese classes

Identification restriction:

T∑t=1

πXt = 1



πA|Xit ...π

D|Xlt : Conditional probabilities

I Comparable to factor loadings in factor analysis

I Probability of a respondent to give a specific response to surveyitems A-D

I Gives information about the character of each class t

I Number of distinct probabilities for each indicator depends onnumber of response-options

Identification restriction:

I∑i=1

πA|Xit = 1 ...

L∑l=1

πD|Xlt = 1


Estimation procedure

Maximum likelihood estimation (introduced by Goodman 1972)

I EM (Expectation Maximization)

I Newton-Raphson


Cross-classification of 216 respondents

Observed Variables

I A: Auto passenger

I B: Insurance doctor

I C: Drama critic

I D: Board of directors

→ Responses

+ Universalistic value

– Particularistic value

Response pattern:2x2x2x2 = 16 possibilities

Response

A B C D Obs. N

+ + + + 42+ + + – 23+ + – + 6+ + – – 25+ – + + 6+ – + – 24+ – – + 7+ – – – 38– + + + 1– + + – 4– + – + 1– + – – 6– – + + 2– – + – 9– – – + 2– – – – 20


Results Example I

Probability of ‘Universalistic Response’ (+) and relative frequencyfor two-class model

Observed Response TypeVariables Universalistic Particularistic

Auto passenger 0.99 0.71Drama critic 0.93 0.35Insurance doctor 0.94 0.33Board of directors 0.77 0.13

Relative Class 0.28 0.72Frequency


Results Example I

Probability of Universalistic Response (+) and relative frequencyfor two-class model

Observed Response TypeVariables Universalistic Particularistic

Auto passenger 0.99 0.71Drama critic 0.93 0.35Insurance doctor 0.94 0.33Board of directors 0.77 0.13

Relative Class 0.28 0.72Frequency


Judging the goodness of this model?

Model evaluation

I Pearson’s Chi2

I Likelihood Ratio Chi2

I Akaike Information Criteria (AIC)

I Bayesian Information Criteria (BIC)



Model evaluation

Chi2

No. of Latent Degrees of Goodness LikelihoodModel Classes Freedom of Fit Ratio

M0: Independence 1 11 104.11 81.08M1: Latent class 2 6 2.72 2.72M2: Latent class 3 2 0.42 0.39


Using restrictions

Hypotheses are tested by imposing restrictions and determininghow these restrictions affect the fit of the model to the data.Examples:

I Equality constraint, e.g. parallel indicators or equal error rate

I Deterministic, e.g. setting conditional probability to specific value(usually 1 or 0)



Model evaluation

Chi2

No. of Latent Degrees of Goodness LikelihoodModel Classes Freedom of Fit Ratio

M0: Independence 1 11 104.11 81.08M1: Latent class 2 6 2.72 2.72M2: Latent class 3 2 0.42 0.39M3: Restricted LC 3 9 2.28 2.28


Results Example I

Probability of Universalistic Response (+) and relative frequencyfor three-class model

Observed Response TypeVariables Strict Mixed Strict

Universalistic Particularistic

Auto passenger 1 0.80 0Drama critic 1 0.42 0Insurance doctor 1 0.44 0Board of directors 1 0.18 0

Relative Class 0.17 0.78 0.05Frequency


Example II

Religiosity in post-Socialist Europe

(with Tim Mueller; Social Forces 2012, 91(2): 559-582)


Forced Secularization in Eastern Europe?

Religion in Socialism:

I Based on Marxism-Leninism ideology of ‘scientific materialism’ →suppression of religion.

I Examples: Churches did not play a role in public education, religiousorganizations were monitored or prohibited (Froese 2004), pastorswere imprisoned and individual restrictions of university access(Burgess 1997).

I Success of suppression differs in Protestant, Catholic, and Orthodoxcountries.


Problem of measurement

I People in socialist societies grew up with a suppression of religion

I Overt religiosity lead to direct disadvantages

I Respondents will be reluctant to openly confess religious belief

→ Measurement model needed to measure latent belief


Data and analysis

I Data: International Social Survey Programme (1991, 1998, and2008)

I Dependent variable: Believe in God

I Method: Latent class analysis (using three indicators)

I Number of latent classes: 3 (atheist, agnostic, religious)


Question wording: Item 1

I “Please tick one box below to show which statement comes closestto expressing what you believe about God.“ [Express]

(1) I don’t believe in God.(2) I don’t know whether there is a God and I don’t believethere is any way to find out.(3) I don’t believe in a personal God, but I do believe in aHigher Power of some kind.(4) I find myself believing in God some of the time, but not atothers.(5) While I have doubts, I feel that I do believe in God.(6) I know God really exists and I have no doubts about it.(8) Can’t choose, don’t know; (9) NA.

(1) is coded as ‘atheist’; (2)-(3) and (8) ‘agnostic’; (4)-(6)‘religious’ response; (9) are set to missing.



I “Which best describes your beliefs about God?” [Describe]

(1) I don’t believe in God now and I never have.(2) I don’t believe in God now, but I used to.(3) I believe in God now, but I didn’t used to.(4) I believe in God now and I always have.(8) Can’t choose, don’t know; (9) NA, refused.

(1) is coded as ‘atheist’; (2) and (8) ‘agnostic’; (3)-(4) ‘religious’response; (9) are set to missing.



I “How much do you agree or disagree with each one of thefollowings? There is a God who concerns Himself with every humanbeing personally.” [Concern]

(1) Strongly agree.(2) Agree.(3) Neither agree or disagree.(4) Disagree.(5) Strongly disagree.(8) Can’t choose, don’t know; (9) NA, refused.

(4)-(5) is coded as ‘atheist’; (3) and (8) ‘agnostic’; (1)-(2)‘religious’ response; (9) are set to missing.


Conditional Probabilities on Indicator Variables

Latent Class C1: ??? C2: ??? C3: ???

(Size) (0.26) (0.27) (0.47)

Item Response

Express atheist 0.53 0.01 0.01agnostic 0.45 0.58 0.04religious 0.02 0.41 0.96

Describe atheist 0.83 0.05 0.00agnostic 0.17 0.35 0.00religious 0.00 0.60 1.00

Concern atheist 0.84 0.47 0.10agnostic 0.14 0.45 0.23religious 0.01 0.08 0.68


Conditional Probabilities on Indicator Variables (all countries; 1991)

Latent Class Atheist Agnostic Religious

(Size) (0.26) (0.27) (0.47)

Item Response

Express atheist 0.53 0.01 0.01agnostic 0.45 0.58 0.03religious 0.02 0.41 0.96

Describe atheist 0.83 0.05 0.00agnostic 0.17 0.35 0.00religious 0.00 0.60 1.00

Concern atheist 0.84 0.47 0.10agnostic 0.15 0.45 0.22religious 0.01 0.08 0.68


Comparing the fit of the latent class model


Model fit by year

1991 1998 2008

N (cases) 18,128 15,042 16,296N (parameter) 72 96 96

Classif. errors 0.11 0.07 0.08Entropy R2 0.75 0.81 0.79


Comparing latent classes among different groups


Comparing Religiosity in East and West Germany

West Germany East GermanyReligious Atheist Religious Atheist

Pre-CW Cohort(born before 1932)

1991 64.2 5.7 32.0 36.81998 62.4 11.9 35.2 44.62008 73.9 5.9 32.3 50.2

Cold War Cohort(born 1932-1976)

1991 48.9 13.8 14.2 65.91998 43.0 16.2 15.7 68.82008 50.7 10.2 14.7 68.4

Post-CW Cohort(born after 1976)

1998 27.9 25.9 15.1 75.92008 39.6 18.7 3.3 76.7


Using latent classes probabilities as dependent variable


Explaining Trends in Religiosity

Revival or decline of religiosity?

Positive Economic Thighter Church-Development State Relation

Modernization Theory Decline Increase


Model fit

Table: Fixed Effects Regression Model on Religiosity for Cohorts

Western Europe Eastern Europe

GDP (in 1000s) -0.21∗ 0.08[-0.41; -0.02] [-0.39; 0.54]

Legislations 6.32 6.76∗∗[-7.07; 19.71] [1.47; 12.06]

Intercept 24.48 14.99

σu 23.63 29.79σε 7.05 7.35

R2 (within) 0.207 0.359R2 (between) 0.015 0.186

N (obs) 56 40N (cohorts) 21 15

Significance levels: ∗p<.05, ∗∗p<.01. Data: ISSP (1991, 1998 and 2008).


Summary: What is this all good for?


Summary

The opportunities of LCA

I Classifying people’s opinions and behaviour in descriptive types(latent classes)

I Use these classifications as independent or dependent variables

I Use either latent class probability of sample or post-estimationclassification of respondents

I Compare different groups

I Investigating sources of confounding and non-independence amonga set of categorical variables


Problems of LCA

Sparseness

I Too many items with too many response-options can lead tosparseness

I Leads to difficulties in model evaluation (determining the degrees offreedom)

I Ideally all cells in a cross-table are filled

→ Large-N is needed


Problems of LCA

Number of classes

I Aim is reduction of information from full response-pattern (= KN ;K=number of response-options; N=number of items) to smallernumber of distinct types (=latent classes)

I But what is the exact number of optimal classes?

I Two options to determine this:

1 Theory2 Model fit


Practical issues: Where to go from here?


Reading

I Collins, L.M. and Lanza, S.T. (2010). Latent class and latent transitionanalysis for the social, behavioral, and health sciences. New York: Wiley.

I McCutcheon, A.L. (1987) Latent class analysis. Quantitative Applicationsin the Social Sciences Series No. 64.Thousand Oaks: Sage.

I Hagenaars, J. A. and McCutcheon, A.L. (2002) Applied latent classanalysis . Cambridge: CUP.

I Skrondal, A. and Rabe-Hesketh, S. (2004). Generalized latent variablemodeling : multilevel, longitudinal, and structural equation models.London : Chapman & Hall/CRC.

I Lazarsfeld P.F. and Henry, N.W. (1968) Latent structure analysis.Boston: Houghton Mifflin

I Goodman, L.A. (1974) "Exploratory latent structure analysis using bothidentifiable and unidentifiable models". Biometrika 61 (2): 215–231.


Software

I R (package poLC) (free)I SAS (add-on: PROC LCA & PROC LTA) (free)I Lem (free)I MPlusI Latent Gold


Workshops

I Essex Summer School in Data Analysis and CollectionI Course Convenor: Allan McCutcheon


Thank you for your attention!


Modelling heterogeneity in attitude dynamics

Issues to deal with:

I State dependence

I Measurement error

I Initial condition

I Heterogeneity

→ Mixed Latent Markov Model

P(y i |x i ) =M∑ξ=1

T∑θ0=1

· · ·T∑

θT=1

P(ξ|x i )P(θ0|ξ, x i0)×

T∑t=1

P(θt |θt−1, ξ)T∑

t=1

[P(yit |θt , ξ)]Iit


Simple Markov Model

yt=0 yt=1 yt=2 ...


Latent Markov Model

I Measurement error equation:

log[

P(yit = l |θ = s)P(yit = s|θ = s)

]= δls

yt=0 yt=1 yt=2 ...

...θt=0 θt=1 θt=2


Mover Stayer Model

yt=0 yt=1 yt=2 ...

...θ1,t=0 θ1,t=1

...

1 1θ1,t=2

θ2,t=2θ2,t=1θ2,t=0


Mover Stayer Model

yt=0 yt=1 yt=2 ...

Covariates:

Soc. Class

Union

Housing

Age

Gender

Education

Region

...θ1,t=0 θ1,t=1

...

1 1θ1,t=2

θ2,t=2θ2,t=1θ2,t=0


Modelling heterogeneity in attitude dynamics II

I Transition dynamics equation:

log[P(θt = r |θt−1 = s, ξ)P(θt = s|θt−1 = s, ξ)

]= β0rs + β1rsttimeit + β2rsDξ=2,

I Initial state equation:

log[P(θ0 = 1|x i0, ξi )

P(θ0 = 2|x i0, ξi )

]= α0 +

P∑p=1

αpxi0p + ηsξ

I Mixture allocation equation:

log[P(ξi = 2)P(ξi = 1)

]= γ0 +

Q∑q=1

γqxi


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