Exploring social mobility with latent trajectory group

analysis

Patrick Sturgis, University of Southampton and

National Centre for Research Methods

From work co-authored with Louise Sullivan

Motivation

Conventional focus on correspondence between ‘origin’ and ‘destination’ points

Does this overlook potentially interesting information about what goes on in-between?

Our approach aims to uncover latent mobility trajectories

And to model the antecedents of membership of different trajectory groups

Latent curves

ittiiity

ii

ii

Conceptual example

we have one child, size of vocabulary measured each year from age 1 to 5

Plot vocabulary size against time

Vocabulary size child 1, t=5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

1 2 3 4 5

time

sco

re

Add line of best fit

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

1 2 3 4 5

time

sco

re

y = 0.79x + 1.39

Can be expressed as regression equation:

Vocabulary size child 2, t=5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

1 2 3 4 5

time

sco

re y = 0.24x + 1.94

Less rapid growth

Case-by-Case approach

So each individual’s growth trajectory can be expressed as a linear equation:

If we have lots of individual growth equations… We can find the average of the intercepts… …and the average of the slopes And the variances of intercepts and slopes The averages tell us about initial status and rate of

growth for sample as a whole Variances tell us about individual variability around

these averages

ttty

Latent curves

ittiiity

ii

ii

Extend model to examine variability between individuals in initial position and rate of change

Latent Class Growth Analysis (LCGA)

Latent curve approach yields parameters for whole sample/population

But what if there are qualitatively different growth trajectories?

Use latent class analysis to find distinct groupings which possess similar trajectory parameters

Multinomial logistic regression of group membership on fixed covariates

Data

1970 British Cohort Study Every child born in week in 1970 n = Direct Maximum Likelihood

Registrar General’s Social Class

I Professional etc occupations

II Managerial and technical occupations

IIIN Skilled non-manual occupations

IIIM Skilled manual occupations

IV Partly-skilled occupations

V Unskilled occupations

BCS70 latent curve model

Growth Factors Men

Estimate s.e.

Growth factor means

0 -

0.384 0.030

Growth factor variances

2 7.308 0.465

2 2.191 0.172

Growth factor covariances

-1.787 0.183

Sample size 6355

Source: BCS70 1980, 1996, 2000.

How many latent trajectory groups?

BICs for conditional LCGA Models

31600

31800

32000

32200

32400

32600

32800

33000

33200

33400

33600

33800

1 2 3 4 5 6 7 8 Number of latent of classes

BIC

Posterior probability plot for 5 group LCGA

Mean posterior probabilities

Most likely

trajectory

group Group 1 Group 2 Group 3 Group 4 Group 5

Group 1 0.727 0.073 0.114 0.073 0.014

Group 2 0.141 0.623 0.049 0.001 0.186

Group 3 0.116 0.019 0.733 0.003 0.129

Group 4 0.202 0.003 0.014 0.780 0.000

Group 5 0.01 0.053 0.129 0.000 0.807

Source: BCS70, 1980, 1996, 2000 waves

Estimated parameters for the 5 latent groups

Growth parameters Trajectory

Group s.e. s.e. Estimated posterior %

1. lower middle stable 4.846 0.398 -0.253 0.152 21%

2. middle declining 4.108 0.248 -2.103 0.183 9%

3. working rising 0.743 0.209 1.855 0.162 27%

4. upper middle stable 6.290 0.274 0.460 0.750 4%

5. working stable 0 - -0.024 0.042 40%

Source: BCS70, 1980, 1996, 2000 waves; n=6355; coefficients in logit scale.

Lower middle class stable (21%)

0

0.10.2

0.3

0.4

0.50.6

0.7

10 26 30

age

p

0

0.1

0.2

0.3

0.40.5

0.6

0.7

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

p

10 26 30

age

1. middle stable 22%

Professional

Managerial & Technical

Skilled non-manual

Skilled manual

Partly-skilled

Unskilled

Working class rising

0

0.10.2

0.3

0.4

0.50.6

0.7

10 26 30

age

p

0

0.1

0.2

0.3

0.40.5

0.6

0.7

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

p

10 26 30

age

1. middle stable 22%

Professional

Managerial & Technical

Skilled non-manual

Skilled manual

Partly-skilled

Unskilled

Covariate Trajectory Group contrast 1v3 1v4 1v2 3v5 2v4 5v1 3v2 Merit variables General ability

0.209 (0.116)

-0.805* (0.235)

0.537* (0.146)

0.537* (0.062)

-1.341* (0.182)

-0.746* (0.113)

0.328* (0.094)

Academic motivation

0.023 (0.071)

-0.067 (0.168)

0.133 (0.121)

0.286* (0.056)

-0.20 (0.169)

-0.309* (0.064)

0.110 (0.103)

Social advantage variable Private secondary school

2.481* (0.757)

-1.129 (0.633)

0.836 (0.662)

-0.577 (0.867)

-1.965* (0.589)

-1.903* (0.454)

-1.644* (0.802)

Cultural capital variables Father had post-compulsory education

1.474* (0.249)

-0.878 (0.562)

0.273 (0.309)

-0.136 (0.210)

-1.151* (0.405)

-1.338* (0.242)

-1.201* (0.232)

Mother had post-compulsory education

0.608* (0.261)

-0.946* (0.460)

-0.033 (0.313)

0.516* (0.204)

-0.913* (0.340)

-1.124* (0.239)

-0.641* (0.220)

Post-compulsory education anticipated for child

1.242* (0.197)

-2.181 (2.650)

0.929* (0.276)

0.494* (0.131)

-3.11 (2.60)

-1.736* (0.218)

-0.313 (0.193)

Father very interested in child’s education

0.389* (0.158)

-0.450 (0.354)

0.258 (0.222)

0.095 (0.136)

-0.708 (0.361)

-0.485* (0.146)

-0.131 (0.200)

Mother very interested in child’s education

0.211 (0.159)

0.006 (0.377)

0.117 (0.229)

0.250* (0.127)

-0.111 (0.367)

-0.461* (0.137)

-0.094 (0.199)

Trajectory group key: 1 – lower middle stable; 2 – middle declining; 3 – working rising; 4 – upper middle stable; 5 – working stable; *=significantly different from zero with p0.05; Source: BCS70, 1980, 1996, 2000 waves; n=6355.

Covariate coefficient contrasts for trajectory group membership

Predicted probability of trajectory group membership

general ability

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

-4.393 -3.393 -2.393 -1.393 -0.393 0.607 1.607 2.607 3.607

z score

p

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

-4.393 -3.393 -2.393 -1.393 -0.393 0.607 1.607 2.607 3.607

lower middle stable

middle declining

working rising

jpper middle stable

working stable

Predicted probability of trajectory group membership

academic motivatoin

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

-3.199 -2.199 -1.199 -0.199 0.801 1.801 2.801

z score

p

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

-4.393 -3.393 -2.393 -1.393 -0.393 0.607 1.607 2.607 3.607

lower middle stable

middle declining

working rising

jpper middle stable

working stable

Mother interested in child’s education

mother interested in child's education?

0

0.1

0.2

0.3

0.4

0.5

0.6

no yes

0

0.1

0.2

0.3

0.4

0.5

0.6

no yes

lower middle stable

middle declining

working rising

upper middle stable

working stable

Father post-compulsory education

0

0.1

0.2

0.3

0.4

0.5

0.6

no yes

lower middle stable

middle declining

working rising

upper middle stable

working stable

0

0.1

0.2

0.3

0.4

0.5

0.6

no yes

Conclusions

Potentially useful approach But this exercise hasn’t told us much new in

substantive terms Problem = endogeneity of predictors Extension = modelling different cohorts

simultaneously