Time series analysisand the individual in

psychological research

Ellen HamakerMethods and Statistics

Faculty of Social Sciences

Utrecht University

Outline

History

Problem

Time series analysis

Examples: - academic performance

- extraversion & neuroticism states

- dyadic interaction

Discussion

The subject of psychological research

Psychological research is often nomothetic, i.e.,

based on studying characteristics of the population.

We may distinguish between:

general psychology

and

differential psychology

General and differential psychology

General psychology concerns meanst-test

ANOVAMANOVA

Differential psychology concerns covariance structure

correlation

regression analysisfactor analysispath analysis

Means and covariance structures combined in SEM

How did it all start?

1878 Charcot began to demonstrate the effects of hypnosis on hysterical patients.Subject: psychologically disturbed mind

1879 Wundt founded the first psychological laboratory in Leipzig. Subject: general mind

1884 Galton established his anthropometric laboratory and measured mental faculties and physical appearances of 9000 visitors.Subject: variation in the population

Variation in the population

Galton believed most mental and physical features were inherited.

He was worried that the protection of the weak (i.e., the poor) would interfere with the mechanisms of natural selection.

Galton is the founder of eugenics.

Other important eugenicists

Pearson follower of Galton, and inventor of the product-moment correlation coefficient

Spearman student of Wundt, and inventor of factor analysis, and the concept of general intelligence

Fisher mathematician, and inventor of: ANOVA, experimental designs, principle of maximum likelihood, inferential statistics, null-hypothesis testing, F-test, Fisher information, non-parameteric statistics, et cetera, et cetera…

Mathematical statistics

The statistical techniques used in the social sciences were developed to study heredity.

Hence, they have two important features:

a. heredity operates at level of population: same holds for these techniques

b. biometrics is concerned with studying trait-like variables, not processes

Outline

History

Problem

Time series analysis

Examples: - academic performance

- extraversion & neuroticism states

- dyadic interaction

Discussion

Psychological processes

Many psychological theories concern processes.

Examples are:

learning; habituation; conditioning

adaptation; coping; affect regulation

interacting; communication

problem solving; information processing

development; decline

Process implies some form of change at the level of the individual.

What is the problem?Our standard techniques focus on characteristics of

the population: means, correlations, and proportions.

Results are not always generalizable to the individual.

For instance:- if we find a beneficial effect of therapy at the group level, this does

not guarantee that every individual improved

- if we find a smooth change at the group level, it is possible that at the individual level there is a sudden change

- if 20% of clients are cured after treatment, this does not imply that an individual has a 20% change of being cured

E.g., correlation

words perminute

mis

take

s

words perminute

mis

take

s

interindividual intraindividual

Who makes this mistake?

sociable

shy

Personality processes, by definition, involve some change in thoughts, feelings and actions of an individual; all these intra-individual changes seem to be mirrored by interindividual differences in characteristic ways of thinking, feeling and acting.

McCrae & John (1992)

The same in formulas…

))(( 2,,

2,,

,,,

2222

iyiyixix

ixyiyix

yx

xyxy

2,

2,

,,

iyix

ixyixy

Questions about processes

Is the relationship at the INTRAindividual level identical to the relationship at the INTERindividual level?

If not, is there an universal relationship?

If not, can the differences between individuals with respect to their dynamics be related to other individual differences?

Outline

History

Problem

Time series analysis

Examples: - academic performance

- extraversion & neuroticism states

- dyadic interaction

Discussion

Dynamic system

A DS is a set of equations that describe how the state of the system changes as a function of its previous state.

Characteristics of a DS:- 1 or more variables- state = values of the variables- stochastic/deterministic- discrete or continuous time- linear or nonlinear

Time series analysis is a technique to study uni- or multivariate, stochastic systems in discrete time, which may be linear or nonlinear.

Autoregressive models

xt-2 xt-1 xt xt+1 xt+2

at-2 at-1 at at+1 at+2

yt-2 yt-1 yt yt+1 yt+2

ut-2 ut-1 ut ut+1 ut+2

Time series

-3-2

-10

12

3

Se

rie

s 1

-3-2

-10

12

0 10 20 30 40 50 60 70

Se

rie

s 2

Time

x

Unrelated series:first series contains autocorrelation second series is white noise

Two related series:first contains positive autocorrelationsecond contains negative autocorreclation

-2-1

01

2

Se

rie

s 1

-3-2

-10

12

0 10 20 30 40 50 60 70

Se

rie

s 2

Time

x

Stationarity

We can distinguish between two kinds of processes:

- stationary processes:variability but no structural changes

- nonstationary processes: sudden or less sudden changes, which may bereversible or not

Time

c

0 20 40 60 80 100

-3-2

-10

12

3

Time

Se

rie

s 1

0 20 40 60 80 100

-10

-8-6

-4-2

02

Time

Se

rie

s 1

0 20 40 60 80 100

-20

24

6

Outline

History

Problem

Time series analysis

Examples: - academic performance

- extraversion & neuroticism states

- dyadic interaction

Discussion

Example 1

Academic performance

Schmitz & Skinner (1993) investigated academic performance of children.

control

effort

performance

evaluation

+

+

+

+

Child 1Lag 0:

eff per eva con

eff -R = per .08 -

eva -.17 .57* -con .03 .11 .22 -

Lag 1:eff per eva con

eff .22 .20 -.49* .27* = per - -.15 .18 .38*

eva - - .23 .32*con - - - .32*

con

eff

per

eva+

+

+

+

-

Child 2Lag 0:

eff per eva con

eff -R = per .42* -

eva .53* .96* -con .30* .24 .33* -

Lag 1:eff per eva con

eff -.18 .09 -.13 .12 = per - -.07 .10 -.25

eva - - .12 -.16con - - - .49*

con

eff

per

eva

+

++

+

+

Example 2

Daily measures of E & N

90 repeated measures in 22 subjecten

0

1

2

3

irritable emotionallystable

calm badtempered

resistant vulnerable

Extraversion items Neuroticism items

total variancestate variancetrait variance

0

1

2

3

dynamic sociable shy silent lively reserved

The model

yt+1 yt+1 yt+1 yt+1 yt+1 yt+1yt yt yt yt yt ytyt-1 yt-1 yt-1 yt-1 yt-1 yt-1

yt+1 yt+1 yt+1 yt+1 yt+1 yt+1yt yt yt yt yt ytyt-1 yt-1 yt-1 yt-1 yt-1 yt-1

Nt Nt+1Nt-1

Et Et+1Et-1

et-1 et-1 et-1 et-1 et-1 et-1 et et et et et et et+1 et+1 et+1 et+1 et+1 et+1

et-1 et-1 et-1 et-1 et-1 et-1 et et et et et et et+1 et+1 et+1 et+1 et+1 et+1

et cetera et cetera

at-1

ut-1

at

ut

at+1

ut+1

Results

1. Does every one have the same 2-factor structure?- 3 persons out of 22 not- only small groups with same factor loadings

2. Are there similarties in dynamics?

NtNt-1

EtEt-1

at-1

ut-1

at

ut

+

+

--+

-

Example 3

Dyadic interaction

Gottman and Murray study spousal interaction- 15 minutes

- code affect (sum 6 seconds): -24 to +24

- bivariate timeseries of 150 points

W1 W2 W3

H1 H2 H3

f(W1) f(W2)

f(H1) f(H2)

Scores of W and H against time

-15

-10

-50

51

0

Se

rie

s 1

-20

-10

01

02

0

0 50 100 150

Se

rie

s 2

Time

dat

State space and null clines

H

W

1

2

34

5

State space and null clines

H

W

Influence function

Ht

Wt+1

positieve threshold

negatieve threshold

Finding the steady states

H

W

A less happy couple

H

W

And next...

H

W

Simulation resultssimulations 2stage simultaneous

parameter true Mean

XSD Mean

SERMSE Mean

XSD Mean

SERMSE

const. Wife .31 .22 1.86 1.53 1.86 .24 .50 .44 .50

autor. Wife .39 .38 .41 .32 .42 .37 .06 .06 .07

neg.th. wife -6.00 -6.12 1.03 - 1.03 -6.00 .33 - .33

pos.th. wife 7.00 6.27 2.00 - 2.13 6.40 1.70 - 1.80

neg.lev. wife -6.20 -6.07 3.87 .66 3.88 -6.20 .81 .81 .81

pos.lev. wife 5.30 5.06 3.96 1.77 3.97 5.11 2.60 1.75 2.60

const. husb. .43 .55 1.77 1.55 1.78 .31 .55 .46 .56

autor. husb. .40 .38 .38 .32 .38 .39 .06 .06 .06

neg.th. husb. -5.00 -5.15 1.26 - 1.27 -5.01 .24 - .24

pos.th. husb. 8.00 6.66 2.51 - 2.84 6.80 2.40 - 2.69

neg.l. husb. -6.50 -6.57 3.30 .64 3.30 -6.39 .86 .80 .87

pos.l. husb. 4.70 4.04 3.92 1.86 3.98 4.38 2.99 1.88 3.01

Outline

History

Problem

Time series analysis

Examples: - academic performance

- extraversion & neuroticism states

- dyadic interaction

Discussion

There is more…

What we saw:

- vector autoregressive models (Examples 1&2)

- threshold autoregressive models (Example 3)

Other possibilities:

- deterministic trends/cycles

- difference scores

- intervention analysis

- latent regime switching

- ordinal data

In sum

Time series analysis is a powerful tool to study processes at the level of the individual (or dyad).

There are different ways of combining the information obtained from multiple subjects:

- can parameters be constrained across individuals?

- individual parameters can be used to compare individuals

In this way time series analysis can also contribute to nomothetic knowledge.

Was it all for nothing?

Populations do not change: individuals do

Thelen & Smith, 1994

So change at the level of the population must imply at least one individual changed.

BUT… beware of the generalization trap!

Thank youemail: e.l.hamaker@fss.uu.nl