up-to-date longitudinal analysis with individual growth curves warren lambert

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Up-to-date Longitudinal Analysis with Individual Growth Curves

Warren LambertPeabody College & Vanderbilt Kennedy Center

March 2008

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JPSP 1972

1. Intro and Readings 2. S&W Examples 3. Sibs’ Head Size Pilot 4. Jittery Mouse Movies

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)

Longitudinal analysis in Ancient Times . . .

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JPSP 19721. Make table with BMD 8V Fortran program

2. Make “camera ready” graph with Leroy lettering guide + India ink

3. Photographer for “camera-ready glossies”

Drafting: Leroy lettering device


Making charts in Ancient Times . . .

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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.

Computing in in Ancient Times . . . 1972

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Graphics Then

(JCCP 1998, 8 numbers)

Now

Science March 2008

Graphics NowFollows Tufte & Cleveland Complex, hundreds or thousands of numbersSmall multiples of tiny chartsTakes time to assimilateNo distracting chart junkNo wasted space or ink


Business graphics Scientific graphics

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If you use the mouse to run SPSS, Bickel offers concrete instructions. In this way HLM becomes “just another GLM.”

Getting started now

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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.

Getting started now

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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.

1. Intro and Readings 2. S&W Examples 3. Sibs’ Head Size Pilot 4. Jittery Mouse MoviesGetting started now

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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.

Getting started now

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Begin with individual growth curves. Code available in many languages, including SAS, SPSS, R, and STATA.

STATAR

SAS SPSS

1. Intro and Readings 2. S&W Examples 3. Sibs’ Head Size Pilot 4. Jittery Mouse MoviesS&W’s Methods

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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?

Look at individual growth curves fit with nonparametric lines

S&W’s Methods

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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.

See if a linear slope-as-outcome model makes sense.

S&W’s Methods

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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


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6 months, 7 waves, right?

•Families had 7 waves over six months

•We called it “monthly”

•Pattern blurred by Wave 3

•Use a mixed model that likes unequal intervals

6 months 7 waves, right?


Viewing the whole sample of individual subjects is often possible.

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Look at individual curves if you don’t know the model for time


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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

Make spaghetti charts to compare the mean curve (linear or nonlinear) with individual curves.

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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.

Systematically build up the model . . . add group by time interactions.

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Can the reader understand this?

Longitudinal coefficients may need concrete explanations.

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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?

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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.


Outcome experiment’s two questions

1. Do groups A and B start out equal?

2. Do group A and B have the same slopes?

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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.


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1155 babies 397 have normal sib 758 have autistic sib

4160 head circumferencesMany measured at 6, 12, 18, 36 months

Mean min ≈ 37 cmMean max < 52 cm

Growth looks curvilinear – any suggestions?

Pilot Data: Head size and autism

(Study ongoing, data still coming in)

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Slope as outcomeCommon 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?

Running an HLM (hierarchical linear model) is very popular.

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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?

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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)

Try a real nonlinear model

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The bad news: Nonlinear model is complicated!Circumference (age) = B0 + B1 * (1 - EXP((B2 * AGE)));

FIXED EFFECTSB01 = 36.2761 /* BIRTH 36 CM */B11 = 13.6990 /* AVERAGE GROWTH LIMIT 14 CM */B21 = -0.1287 /* RATE OF GROWTH PER MONTH */

RANDOM EFFECTSB0 = 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 ;

Months0 10 20 30 40

HC

(cm

)

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35

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Fabricated infantExponential model

HC = 35 + 17 * (1 - e0.08*Age)

The good newsConceivable that B0, B1, and B2 have biological basis

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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)?


Compare different models with individual animated GIFs

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Let’s start with a two wave pre-post model

Look at each individual mouse.

Compare different models with animated GIFs. Start simple.

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

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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.

Compare different models with animated GIFs. Gradually build up.

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.

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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.

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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.


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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?


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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


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Which model tells a clear story?


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Name 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 ***


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.

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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 two-wave 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

up-to-date longitudinal analysis with individual growth curves warren lambert

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