Up-to-date Longitudinal Analysis with
Individual Growth Curves
Warren Lambert
Peabody College & Vanderbilt Kennedy Center
March 2008
1
1.
Intro and Readings
2. S&W Examples
3. Sibs’ Head Size Pilot
Longitudinal analysis in Ancient Times . . .
Mistakes
Median splits on
continuous
variables
ANOVA instead of
ANCOVA
RM ANOVA not
mixed model
(PROC MIXED,
HLM: 20 years).
Individual growth
curves ignored
Obsession with sig.
tests
No confidence
intervals on plot
No effect sizes
(Cohen 1988, due
in 16 years)
2
JPSP 1972
4. Jittery Mouse Movies
1.
Intro and Readings
2. S&W Examples
3. Sibs’ Head Size Pilot
4. Jittery Mouse Movies
Making charts in Ancient Times . . .
1. Make table
with BMD 8V
Fortran
program
2. Make “camera
ready” graph
with Leroy
lettering guide
+ India ink
3. Photographer
for “cameraready
glossies”
3
JPSP 1972
Drafting: Leroy lettering device
1.
Intro and Readings
2. S&W Examples
Computing in in Ancient Times . . . 1972
3. Sibs’ Head Size Pilot
4. Jittery Mouse Movies
1. Keypunch the data & program on
IBM cards at computer center.
2. Sort data cards with “proc sort”
machine. If cards sorted wrong,
answers are wrong.
3. Run with BMDP 8V (balanced
with no missing values).
4. Later, pick up printout from bin.
4
1.
Intro and Readings
2. S&W Examples
Graphics Then
(JCCP 1998, 8 numbers)
3. Sibs’ Head Size Pilot
4. Jittery Mouse Movies
Graphics Now
Follows Tufte & Cleveland
Complex, hundreds or thousands of numbers
Small multiples of tiny charts
Takes time to assimilate
No distracting chart junk
No wasted space or ink
Now
Science March 2008
5
Business graphics
Scientific graphics
1.
Intro and Readings
2. S&W Examples
Getting started now
If you use the mouse to run SPSS,
Bickel offers concrete instructions. In
this way HLM becomes “just another
GLM.”
6
3. Sibs’ Head Size Pilot
4. Jittery Mouse Movies
1.
Intro and Readings
2. S&W Examples
Getting started now
Teach yourself graphic-rich longitudinal
analysis:
Work through S&W chapters 1-5 using your
preferred software and S&W’s:
http://www.ats.ucla.edu/stat/examples/alda/
(Or Google: UCLA ALDA)
ALDA: Applied Longitudinal Data Analysis
PS. W&S flipped coin for first author.
7
3. Sibs’ Head Size Pilot
4. Jittery Mouse Movies
1.
Intro and Readings
2. S&W Examples
3. Sibs’ Head Size Pilot
4. Jittery Mouse Movies
Getting started now
http://www.ats.ucla.edu/stat/seminars/alda/default.htm
Suggestion
Go to this site and start the
slideshow. Then start the movie to
hear & watch the lecture. Advance
the slides to keep up.
8
1.
Intro and Readings
2. S&W Examples
Getting started now
Work S&W’s examples in
your preferred software,
MPLUS, MLwiN, HLM,
SAS, Stata, R, or SPSS.
Use the examples &
W&S book to learn
longitudinal analysis.
Then you’re ready for
original multilevel
research!
KC stats consultants and
biostats clinics will help
P.R.N.
9
3. Sibs’ Head Size Pilot
4. Jittery Mouse Movies
1.
Intro and Readings
2. S&W Examples
3. Sibs’ Head Size Pilot
4. Jittery Mouse Movies
S&W’s Methods
Begin with
individual
growth curves.
Code available in
many languages,
including SAS,
SPSS, R, and
STATA.
10
SAS
SPSS
R
STATA
1.
Intro and Readings
2. S&W Examples
3. Sibs’ Head Size Pilot
S&W’s Methods
Look at individual growth curves fit with nonparametric lines
After looking at the
disconnected
points, fit models
to the points using
few assumptions
(e.g. Loess,
smooth, spline).
Are there patterns
that suggest a
reasonable math
model?
11
4. Jittery Mouse Movies
1.
Intro and Readings
2. S&W Examples
3. Sibs’ Head Size Pilot
S&W’s Methods
See if a linear slope-as-outcome model makes sense.
Always consider a
convenient linear
model (“HLM”), but
only use it if it is
realistic, as it is in
this case.
Next a few outside
examples.
12
4. Jittery Mouse Movies
1.
Intro and Readings
2. S&W Examples
3. Sibs’ Head Size Pilot
4. Jittery Mouse Movies
Viewing the whole sample of individual subjects is often possible.
•Abused women in urban
emergency room
•N = 493
•Each line is a woman and
each dot is reported abuse
incident.
•Chart shows the great range
of abuse in the sample
•Means could be deceiving
13
1.
Intro and Readings
2. S&W Examples
3. Sibs’ Head Size Pilot
4. Jittery Mouse Movies
Viewing the whole sample of individual subjects is often possible.
•Families had 7
waves over six
months
•We called it
“monthly”
•Pattern blurred by
Wave 3
6 months, 7 waves, right?
•Use a mixed
model that likes
unequal intervals
6 months 7 waves, right?
14
1.
Intro and Readings
2. S&W Examples
3. Sibs’ Head Size Pilot
4. Jittery Mouse Movies
Look at individual curves if you don’t know the model for time
15
1.
Intro and Readings
2. S&W Examples
3. Sibs’ Head Size Pilot
4. Jittery Mouse Movies
Make spaghetti charts to compare the mean curve (linear or nonlinear)
with individual curves.
It is quite possible
that no single
individual has the
mean timeline.
These charts can
be made with
code from S&W
Charts help you
understand the
mean curve
without
worshipping it
16
1.
Intro and Readings
2. S&W Examples
3. Sibs’ Head Size Pilot
4. Jittery Mouse Movies
Systematically build up the model . . . add group by time interactions.
Use likelihood ratio
tests (LRTs) to see if
extra terms improve
model fit.
Check pseudo-R2
based on r(observed,
predicted)
Use plots to interpret
slope coefficients
(time, group by time).
Explain the Group by
Time interaction
visually.
17
1.
Intro and Readings
2. S&W Examples
3. Sibs’ Head Size Pilot
4. Jittery Mouse Movies
Longitudinal coefficients may need concrete explanations.
Can the reader
understand this?
18
1.
Intro and Readings
2. S&W Examples
3. Sibs’ Head Size Pilot
Longitudinal coefficients may need concrete explanations.
Outcome experiment’s two questions
1. Do groups A and B start out equal?
2. Do group A and B have the same slopes?
19
4. Jittery Mouse Movies
1.
Intro and Readings
2. S&W Examples
3. Sibs’ Head Size Pilot
4. Jittery Mouse Movies
Q: What does a 2-wave pre-post longitudinal analysis show?
A: Not much
Outcome experiment’s two questions
1. Do groups A and B start out equal?
2. Do group A and B have the same slopes?
Lambert, E. W., A. Doucette & Bickman, L. (2001). "Measuring Mental Health Outcomes with Pre-post Designs." Journal of Behavioral Health Services
Research 28(3): 273-286.
20
1.
Intro and Readings
2. S&W Examples
3. Sibs’ Head Size Pilot
4. Jittery Mouse Movies
Q: What does a 2-wave pre-post longitudinal analysis show?
A: Not much
Lambert, E. W., A. Doucette & Bickman, L. (2001). "Measuring Mental Health Outcomes with Pre-post Designs." Journal of Behavioral Health Services
Research 28(3): 273-286.
21
1.
Intro and Readings
2. S&W Examples
Pilot Data: Head size and autism
(Study ongoing, data still coming in)
1155 babies
397 have normal sib
758 have autistic sib
4160 head
circumferences
Many measured at 6,
12, 18, 36 months
Mean min ≈ 37 cm
Mean max < 52 cm
Growth looks
curvilinear – any
suggestions?
22
3. Sibs’ Head Size Pilot
4. Jittery Mouse Movies
1.
Intro and Readings
2. S&W Examples
3. Sibs’ Head Size Pilot
Running an HLM (hierarchical linear model) is very popular.
Slope as outcome
Common hierarchical
linear model (HLM).
Linear is easy to
understand
Constant growth rate of
red group a little higher
in cm/month.
Anyone see a problem?
Suggest a solution?
23
4. Jittery Mouse Movies
1.
Intro and Readings
2. S&W Examples
3. Sibs’ Head Size Pilot
Use age and age squared as X’s to produce
a smooth quadratic curve.
Linear model didn’t
work because head
size doesn’t keep
growing at a constant
rate forever
Quad-time adds
constant deceleration
to the model.
Any problem?
P.S. What’s a squared month?
24
4. Jittery Mouse Movies
1.
Intro and Readings
2. S&W Examples
Try a real nonlinear model
The good news
Exponential growth fits
better and raises the
hope that the
parameters will mean
something.
What’s a “squared
month?”
Model ~ decelerating
growth that reaches a
limit (asymptote)
25
3. Sibs’ Head Size Pilot
4. Jittery Mouse Movies
1.
Intro and Readings
2. S&W Examples
3. Sibs’ Head Size Pilot
4. Jittery Mouse Movies
The bad news: Nonlinear model is complicated!
Circumference (age) = B0 + B1 * (1 - EXP((B2 * AGE)));
FIXED EFFECTS
B01 =
36.2761
B11 =
13.6990
B21 =
-0.1287
/* BIRTH 36 CM
/* AVERAGE GROWTH LIMIT 14 CM
/* RATE OF GROWTH PER MONTH
*/
*/
*/
RANDOM EFFECTS
B0 = B01 + U0 ; * *** 36 CM BIRTH SIZE (WITH SUBJECT OFFSETS) ;
B1 = B11 + U1 ; * *** FIXED STARTING DELTA OF 14 CM
;
B2 = B21 + U2 ; * *** GROWTH RATE PARM -0.13 PER MONTH
;
55
HC = 35 + 17 * (1 - e0.08*Age)
HC (cm)
50
The good news
Conceivable that B0, B1, and B2 have biological basis
26
45
40
35
Fabricated infant
Exponential model
30
0
10
20
30
Months
40
1.
Intro and Readings
2. S&W Examples
3. Sibs’ Head Size Pilot
Compare different models with individual animated GIFs
1.
Longitudinal experiment with
34 repeated measures per
mouse
2.
12 mice: 6 knockouts + 6
“wild” mice
3.
X is minutes and Y is activity
4.
Evaluate statistical models
using each individual’s
model scores & observed
scores.
5.
Do the knockouts (PV/PV)
respond more to Ritalin than
the normals (WT, wild type)?
27
4. Jittery Mouse Movies
1.
Intro and Readings
2. S&W Examples
3. Sibs’ Head Size Pilot
Compare different models with animated GIFs. Start simple.
Let’s start with a two
wave pre-post model
Look at each individual
mouse.
SAS Code for animated GIF
Proc gplot etc . . . .
by Subject_ID ;
filename MOVEgif '.\hmix1.gif';
goptions device
= gifanim
gsfname
= MOVEgif
gsfmode
= replace
iteration = 0
delay
= 200
xpixels
= 1200
ypixels
= 1200
GEPILOG = '3B'x
display ;
Animated GIF = stack of pictures
28
4. Jittery Mouse Movies
1.
Intro and Readings
2. S&W Examples
3. Sibs’ Head Size Pilot
4. Jittery Mouse Movies
Compare different models with animated GIFs. Gradually build up.
Now a 34 wave RM
ANOVA.
Time is categorical, not
ordinal.
Under compound
symmetry we assume
that minute 20 and
minute 160 are equally
close to minute 180.
Assumptions not found
in most situations.
Nich, C. and K. Carroll (1997). "Now you see it,
now you don't: A comparison of traditional versus
random-effects regression models in the analysis
of longitudinal follow-up data from a clinical trial."
Journal of Consulting and Clinical Psychology
65(2): 252-261.
29
1.
Intro and Readings
2. S&W Examples
3. Sibs’ Head Size Pilot
4. Jittery Mouse Movies
Compare different models with animated GIFs. Gradually build up.
Two slopes fit all?
Are the residuals
(vertical distance from
star to line) random?
Are residuals balanced
(over or under?)
Look at the residuals in
the startup interval 0-30
minutes.
30
1.
Intro and Readings
2. S&W Examples
3. Sibs’ Head Size Pilot
4. Jittery Mouse Movies
Compare different models with animated GIFs. Gradually build up.
Assume the mouse was
randomly sampled from
a population (that’s my
story and I’m sticking to
it).
Mouse is a random
effect
A “random intercept”
gives each mouse its
own personal offset up
or down.
These offsets, like
residuals, sum to zero
for the sample.
Accounts for some mice
being more energetic
than others.
31
1.
Intro and Readings
2. S&W Examples
3. Sibs’ Head Size Pilot
4. Jittery Mouse Movies
Compare different models with animated GIFs. Gradually build up.
We add a slope as a
random effect for each
individual mouse
Model moves up and
down and different
slopes for individuals
Represent
characteristics each
individual arrived with
Are the residuals
looking better?
32
1.
Intro and Readings
2. S&W Examples
3. Sibs’ Head Size Pilot
4. Jittery Mouse Movies
Compare different models with animated GIFs. Gradually build up.
In a “hockeystick model”
different time intervals have
separate slopes
Early slope for habituation
and another slope for Ritalin
A time-varying covariate
marking the times when
injections were given.
00001000001000001000001
33
1.
Intro and Readings
2. S&W Examples
Which model tells a clear story?
34
3. Sibs’ Head Size Pilot
4. Jittery Mouse Movies
1.
Intro and Readings
2. S&W Examples
3. Sibs’ Head Size Pilot
4. Jittery Mouse Movies
What did we get?
We got a theoretically interpretable model, not a perfect fit. RM ANOVA
fits pretty well, but categorical time just doesn’t make sense.
Name
35
Model
Pseudo R2
F
P
Prepost
Two points only
77%
8.5
***
RM
RM ANOVA
71%
4.2
***
Linmix0
Time Linear no random effects
56%
58.7
***
Linmix1
Time linear random intercept
67%
77.5
***
Linmix2
Time linear random int + slope
69%
27.9
***
Hmix3
Hockeystick random int + slope + TVC
76%
28.2
***
Conclusions
1. Individual growth curves make it possible to model time accurately.
2. Good longitudinal statistical models and good software are now
available. Scientists who do hands-on data analysis can now do
up-to-date graphic-enriched longitudinal analysis.
3. Good reference books are available and self-teaching is feasible
with the ALDA site.
4. After initial practice, you can treat a mixed model as “just another
regression or ANOVA.”
5. Multiple repeated measures are valuable, and the pre-post twowave design is generally a bad idea (3 waves ~ truly longitudinal).
6. Time spent understanding the role of time is well spent.
7. It is now possible to show results graphically in ways that explain
the role of time while demonstrating the validity of your
conclusions
36