Longitudinal Experiments
Larry V. Hedges
Northwestern University
Prepared for the IES Summer Research
Training Institute July 28, 2010
What Are Longitudinal Experiments?
Longitudinal experiments are experiments with repeated
measurements of an outcome on the same people
Examples
Experiments with immediate and delayed posttests
Experiments that track individuals over many performance
periods (e.g., school years)
Experiments that intend to impact growth rate
Experiments that make repeated measurements of
behavior (e.g., teacher behavior) to increase precision of
measurement
Why Do Longitudinal Experiments?
Three reasons for doing longitudinal experiments
1.
More than one discrete endpoint is of interest (e.g., immediate vs
delayed outcome)
2.
Several measures of the outcome are necessary to increase
precision or reduce variation (e.g., teacher behavior is averaged
over many occasions)
3.
The time course of treatment effect (growth trajectory) is of
interest (e. g., an intervention is intended to increase the rate of
vocabulary acquisition in preschool children)
In all three cases, linear combinations of outcomes may be of interest
Why Do Longitudinal Experiments?
Unless different outcomes are being
compared, there is no need to use
longitudinal methods!
But, if different outcomes are being
compared, outcomes are not independent
Thus, longitudinal methods must be used
Modeling Longitudinal Experiments
We can describe models for longitudinal experiments via
ANOVA or HLM notation
We can analyze longitudinal experiments via either ANOVA
or HLM
There are big advantages to using HLM notation for these
models
There are even bigger advantages to using HLM for
analyses of these models
Hence we will primarily use HLM notation in our discussion
of longitudinal experiments
Discrete Endpoints
The design will typically have at least three levels
Measures are nested (clustered) within individuals,
individuals are nested (clustered) within schools
Level 1 (measures within individuals)
Level 2 (individuals within schools)
Level 3 (schools)
Let Yijk, the observation on the kth measure for the
jth person in the ith school
Discrete Endpoints, Schools Assigned
(No Covariates)
Level 1 (measure level)
Yijk = β0ij + εijk
ε ~ N(0, σW2)
Level 2 (individual level)
β0ij = γ00i + η0ij
η ~ N(0, σI2)
Level 3 (school level)
γ00i = π00 + π01Ti + ξ0i
ξ ~ N(0, σS2)
Where we code the (centered) treatment Tj = ½ or - ½ , so
that π01 is the ANOVA treatment effect
Discrete Endpoints
(Unconditional Model)
Note that the εijk’s are not just measurement errors but also
contain differences between outcomes for each
individual
Similarly the η0ij‘s are between individual differences in
these quantities
Then the ξ0i‘s are between-school differences on these
quantities
That makes the unconditional model difficult to interpret
Discrete Endpoints, Schools Assigned
(Comparing Early and Delayed Outcome)
Level 1 (measure level)
Yijk = β0ij + β1ijDijk + εijk
ε ~ N(0, σW2)
Level 2 (individual level)
β0ij = γ00i + η0ij
β1ij = γ10i + η1ij
η ~ N(0, ΣI)
Level 3 (school level)
γ00i = π00 + π01Ti + ξ0i
γ10i = π10 + π11Ti + ξ1i
ξ ~ N(0, ΣS)
Note that the η0ij’s and η1ij’s can be correlated as can the ξ0i’s and ξ1i’s
Discrete Endpoints, Individuals Assigned
(Comparing Early and Delayed Outcome)
Level 1 (measure level)
Yijk = β0ij + β1ijDijk + εijk
Level 2 (individual level)
β0ij = γ00i + γ01iTi + η0ij
β1ij = γ10i + γ11iTi + η1ij
Level 3 (school level)
γ00i = π000 + ξ00i
γ01i = π010 + ξ01i
γ10i = π100 + ξ10i
γ11i = π110 + ξ11i
ε ~ N(0, σW2)
η ~ N(0, ΣI)
ξ ~ N(0, ΣS)
Note that the η0ij’s and η1ij’s can be correlated as can the ξ’‘s and ξ’s
Discrete Endpoints
(Comparing Early and Delayed Outcome)
Note that, in this model, the εijk’s can be interpreted as
measurement errors
Similarly the η0ij‘s are between individual differences in
these quantities and the intraclass correlation
ρI = σI2/(σs2 + σI2 + σW2) is a true (individual level) reliability
coefficient
Then the ξ0i‘s are between-school differences on these
quantities and the intraclass correlation
ρS = σS2/(σs2 + σI2 + σW2) is a true (school level) reliability
coefficient
Discrete Endpoints, Schools Assigned
(Comparing Early and Delayed Outcome)
Covariates can be added at any level of the design
But remember that covariates must be variables
that cannot have been impacted by treatment
assignment
Thus time varying covariates (at level 1) are
particularly suspect since they may be measured
after treatment assignment
Average of Several Measures
The design will typically have at least three levels
Measures are nested (clustered) within individuals,
individuals are nested (clustered) within schools
Level 1 (measures within individuals)
Level 2 (individuals within schools)
Level 3 (schools)
Let Yijk, the observation on the kth measure for the jth person
in the ith school with p measures per individual
Average of Several Measures
(Treatment Assigned at the School Level)
Level 1 (measure level)
Yijk = β0ij + εijk
ε ~ N(0, σW2)
Level 2 (individual level)
β0ij = γ00i + η0ij
η ~ N(0, σI2)
Level 3 (school level)
γ0i = π00 + π01Ti + ξ0i
ξ ~ N(0, σS2)
Where we code the (centered) treatment Tj = ½ or - ½ , so
that π01 is the treatment effect
Average of Several Measures
(Treatment Assigned at the Individual Level)
Level 1 (measure level)
Yijk = β0ij + εijk
ε ~ N(0, σW2)
Level 2 (individual level)
β0ij = γ00i + γ01iTij + η0ij
η ~ N(0, σI2)
Level 3 (school level)
γ00i = π00 + ξ0i
ξ ~ N(0, ΣS)
γ01i = π01 + ξ1i
Where we code the (centered) treatment Tj = ½ or - ½ , so
that π01 is the treatment effect
Average of Several Measures
Note that, in this model, the εijk’s can be interpreted as like
(item level) measurement errors
Then the β0ij‘s can be interpreted as individual level
measures (for the jth person in the ith school)
Thus the η0ij‘s are between individual differences in these
quantities and the quantity ρI = σI2/(σs2 + σI2 + σW2/p) is a
true (individual level) reliability coefficient
Then the ξ0i‘s are between-school differences on these
quantities and the quantity ρS = σS2/(σs2 + σI2 + σW2/p) is a
true (school level) reliability coefficient
Growth Trajectories
The problem of fitting growth trajectories is more
complicated
It requires choosing a form for the growth trajectories
It also requires choosing a form for the model of individual
differences in these growth trajectories
Many forms are possible, but polynomials are conventional
for two reasons:
• Any smooth function is approximately a polynomial
(Taylor’s Theorem)
• Polynomials are simple
What is a Polynomial Model?
Yijk = β0ij + β1ijtijk + β2ijtijk2 + β3ijtijk3 + εijk
Typically, tijk is a measure of time at the measurement for
the jth person in the ith school at the tth measurement
We typically center the measurements at some point for
convenience (often the middle of the time span)
Centering strategy determines the interpretation of the
coefficients of the growth model
Note that the measurements do not have to be at exactly
the same time for each person
Understanding a Polynomial Model
Yijk = β0ij + β1ijtijk + β2ijtijk2 + β3ijtijk3 + εijk
How do we interpret the coefficients?
β0ij is the intercept at the centering point
β1ij is the linear rate of growth at the centering point
β2ij is the acceleration (rate of change of linear growth) at
the centering point
β3ij is the rate of change of the acceleration (often negative
leading to flattening out of growth curves at the
extremes)
Understanding a Polynomial Model
Consider the quadratic growth model to understand
acceleration with mean centering
Yijk   0ij  1ij  t  t    2ij  t  t

  0ij   1ij   2ij  t  t    t  t
2

Thus you can see that the linear growth rate at time t is
 1ij   2ij  t  t  
In other words, the linear growth rate increases with t and
the only place where the linear growth rate is β1ij is the
middle
Understanding a Polynomial Model
Thus β1ij is the linear rate of growth at the centered value
(here, the middle)
If β2ij > 0, the linear growth rate will be larger above the
centered value and smaller below the centered value
Centering at other values than the middle can make sense
if that is where growth trajectory is of interest and if the
model fits the data
For example, centering at the end gives coefficients with
interpretable rates at the end of the growth period
Understanding a Polynomial Model
Consider the quadratic growth model to understand
acceleration with mean centering
2
3
Yijk   0ij  1ij  t  t    2ij  t  t   3ij  t  t 

  0ij  1ij    2ij  3ij  t  t    t  t
  t  t 
Thus you can see that the acceleration at time t is
  2ij  3ij  t  t    t  t

In other words, the acceleration increases with t and the
only place where the acceleration is β2ij is the middle
Understanding a Polynomial Model
Thus β2ij is the acceleration of growth at the centered value
(here the middle)
If β3ij > 0, the acceleration will be larger above the centered
value and smaller below the centered value
Centering at other values than the middle can make sense
if that is where growth trajectory is of interest and if the
model fits the data
For example, centering at the end gives coefficients with
interpretable rates at the end of the growth period
No Growth (Centered)
β0 = 5, β1 = 0.00, β2 = 0.00, β3 = 0.00
5.005
5.004
5.002
f0( t )
5
4.998
4.996
4.9954.994
0
0
2
4
6
8
t
10
12
14
13
Linear Growth (Centered)
β0 = 5, β1 = 1, β2 = 0.00, β3 = 0.00
13
14
12
10
8
f1( t )
6
4
2
0
0
0
0
2
4
6
8
t
10
12
14
13
Quadratic Growth (Centered)
β0 = 5, β1 = 1, β2 = 0.05, β3 = 0.00
16.2
18
16
14
12
10
f2( t )
8
6
4
2
1.25
0
0
0
2
4
6
8
t
10
12
14
13
Cubic Growth (Centered)
β0 = 5, β1 = 1, β2 = 0.05, β3 = -0.01
11.08
12
10
8
f3( t )
6
4
2.44
2
0
0
2
4
6
8
t
10
12
14
13
Linear, Quadratic, and Cubic Growth (Centered)
β0 = 5, β1 = 1, β2 = 0.05, β3 = -0.01,
16.2
18
16
14
12
f1( t )
10
f2( t )
f3( t )
8
6
4
2
0
0
0
0
2
4
6
8
t
10
12
14
13
Selecting Growth Models
Several considerations are relevant in selecting a growth model
First is how many repeated measures there are: The maximum degree
is one less than the number of measures
(linear needs 2, quadratic needs 3, etc.)
However the estimates of growth parameters are much better if there
are a few additional degrees of freedom
But the most important consideration is whether the model fits the data!
Unfortunately, this is not always completely unambiguous
Selecting Growth Models
Individual growth trajectories are usually poorly estimated
HLM models estimate average growth trajectories (via
average parameters) and variation around that average:
These are much more stable
Estimates of individual growth curves can be greatly
improved by using empirical Bayes methods to borrow
strength from the average
This may make sense if there all the individuals in the
groups are sampled from a common population
It can be problematic if some individuals are dramatically
different
Selecting Analysis Models
One issue is selecting the growth model to characterize
growth
A different, but related, issue is selecting how treatment
should impact growth
Should it impact linear growth term?
Should it impact the acceleration?
Which impact is primary?
How does looking at multiple impacts weaken the design?
What if impacts are in opposite directions?
Longitudinal Experiments Assigning
Treatment to Schools
In the language of experimental design, adding repeated
measures adds another factor to the design: A measures
factor
The measures factor is crossed with individuals,
treatments, and clusters
Schools are nested within the treatment factor and
individuals are nested within school by treatments
Repeated measures analysis of variance can be used to
analyze these designs, but we will not pursue that point
of view
Instead we will use the HLM notation
Longitudinal Experiments Assigning
Treatment To Schools
Level 1 (measures)
Yijk = β0ij + β1ijtijk + β2ijtijk2 + εijk
Level 2 (individuals)
β0ij = γ00j + η0ij
β1ij = γ10j + η1ij
β2ij = γ20j + η2ij
η ~ N(0, ΣI)
Level 3 (schools)
γ00j = π00 + π01Ti + ξ0j
γ01j = π10 + π11Ti + ξ1j
γ20j = π20 + π21Ti + ξ2j
ξ ~ N(0, ΣS)
Longitudinal Experiments Assigning
Treatment To Schools
This model has three trend coefficients in each growth
trajectory
Note that there are 3 random effects at the second and
third level
This means that 6 variances and covariances must be
estimated at each level
This may require more information to do accurately than is
available at the school level
It is often prudent to fix some of these effects because they
cannot all be estimated accurately
Longitudinal Experiments Assigning
Treatment Within Schools
In the language of experimental design, adding repeated
measures adds another factor to the design: A measures
factor
The measures factor is crossed with individuals,
treatments, and clusters
The treatment factor is crossed with schools and
individuals are nested within school by treatments
Repeated measures analysis of variance can be used to
analyze these designs, but we will not pursue that point
of view
Longitudinal Experiments Assigning
Treatment Within Schools
Level 1 (measures level)
Yijk = β0ij + β1ij t + β2ij t2 + εijk
ε ~ N(0, σW2)
Level 2 (individual level)
β0ij = γ00j + γ01jTj + η0ij
β1ij = γ10j + γ11jTj + η1ij
β2ij = γ20j + γ21jTj + η2ij
η ~ N(0, ΣC)
Level 3 (school level)
γ00j = π00 + ξ00j
γ01j = π10 + ξ01j
γ10j = π00 + ξ10j
γ11j = π00 + ξ11j
γ20j = π00 + ξ20j
γ21j = π00 + ξ21j
ξa0 ~ N(0, ΣS)
ξa1 ~ N(0, ΣTxS)
Longitudinal Experiments Assigning
Treatment Within Schools
This model has three trend coefficients in each growth
trajectory
Note that there are 6 random effects at the third level
This means that 15 variances and covariances must be
estimated at the third level
This requires a great deal of information to do accurately
It is often prudent to fix some of these effects because they
cannot all be estimated accurately
However there is some art in this, and sensitivity analysis is
a good precaution
Longitudinal Experiments
Covariates can be added at any level of the design
But remember that covariates must be variables
that cannot have been impacted by treatment
assignment
Thus time varying covariates (at level 1) are
particularly suspect since they may be measured
after treatment assignment
Power Analysis
Power computations for longitudinal experiments are doable, but
depend on parameters that may not be well known
For example reliability of trend coefficients
When parameters such as these are known, the computations are
straightforward, but there is relatively little information about them
that can be used for planning
To make matters worse, the values of some parameters (such as
reliability) depend on the number of measures
Thus it is often necessary to rely on values of variance components
Power Analysis
Still some generalizations are possible
• Power increases with the number of measures
• Power increases with the length of time over which
measures are made (except for β0ij)
• Power increases with the precision of each individual
measure
These factors impact different trend coefficients differently
Clustering increases the complexity of computations
Power Analysis
Pilot data (or data from related studies, perhaps
non-experimental ones) is more important in
planning longitudinal experiments
Longitudinal experiments to look at growth
trajectories are attractive, but this is an area at
the frontier of practical experience
Research is ongoing to produce better methods for
power analysis of longitudinal experiments that
will be practically useful
Good Luck!