Practice in Growth Curve Modeling - SAS Group... · 2016-03-11 · Practice in Growth Curve...

Practice in Growth Curve Modelingfor Women’s Menstural Cycle

- Fundamental Overview -

Chel Hee Lee1 Angela Baerwald2

1Clinical Research Supporting UnitCollege of Medicine

University of Saskatchewan

2Department of Obstetrics, Gynecology and Reproductive SciencesCollege of Medicine

University of Saskatchewan

Saskatoon SAS Users Group MeetingSeptember 16, 2015

Chel Hee Lee, Angela Baerwald (U of S) Practice in Growth Curve Modeling 2015-09-16 1 / 17

1 IntroductionFacing ProblemsLinear Mixed-effect Model

2 Parameteric Approach to Modeling CurvesUseful ModelsUseful Techniques

3 Trial and Error1st trial: Oh No! ...2nd trial: Making a progress ...3rd trial: It’s not done yet ...4th trial: Much better now ...

4 Questions


Introduction

“Essentially, all models are wrong, but some are useful.”

(Box and Draper, 1986, p. 424)

It is a best guess.

It is an approximation.

It would be a translation of observation into a model.


Introduction Facing Problems

Hormones Involved in Women’s Menstrual Cycle

Subject-specific Trajectory (in Logarithm Scale)

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Introduction Linear Mixed-effect Model

Linear Mixed-Effect Models (Singer, 1998)

Level 1 (within person)

Yij =π0j +π1j(TIME)ij + rij, where rij ∼ N(0,σ2)

Level 2 (between-person)

π0j = β00 +u0j,

π1j = β10 +u1j,

where[

u0j

u1j

]∼ N

[(00

),

(τ00 τ01

τ10 τ11

)]

Unconditional Linear Growth Curve

Yij = [β00 +β10TIMEij]︸︷︷︸FIXED

+ [u0j +u1jTIMEij + rij]︸︷︷︸RANDOM

Note that a 3-level model can be made if individuals within groups aretracked over time.



Using PROC MIXED in SAS

Commands for a linear time trend model

PROC MIXED DATA = d a t a f i l e ;CLASS id time ;MODEL y = time / SOLUTION CHISQ ;REPEATED time / TYPE=UN SUBJECT=id ;

RUN ;

Commands for a subject-specific model

PROC MIXED DATA = d a t a f i l e ;CLASS id ;MODEL y = time / SOLUTION CHISQ ;RANDOM intercept time / TYPE=UN SUBJECT=id ;

RUN ;



Concerns Bothering Us

We are working with ...

A small size of participants with many repeated measurements,

An unbalanced data set observed at unequally spaced times, and

An incomplete data due to missing values.



Concerns Bothering Us

We are faced with difficulties of ...

Determining the level of a model of interest,

Incorporating a person-level covariate,

Addressing a difference between groups,

Dealing with missing data and dropout,

Binning data or using data as it is,

Centralizing, normalizing, orthogonalizing data, and

Applying either or both models,

Choosing an appropriate pattern of covariance structure, etc.


Parameteric Approach to Modeling Curves Useful Models

Parameteric Approaches to Modeling Curves

Nonlinear Model

yi = u(xi)+εi, i = 1,2, . . . ,n

p-th Degree Polynomial Regession Model (Johnson et al., 2013)

u(xi) =β0 +β1xi +β2x2i +·· ·+βpxp

i

Exponential Growth Curve Model (Zwietering M H et al., 1990)

u(xi) =β0eβ1xi

Trigonometric Model (Cornelissen Germaine, 2014)

u(xi) =µ+Acos(wxi +φ)


Parameteric Approach to Modeling Curves Useful Techniques

Modeling Techniques

Piecewise Modeling Approach (Naumova et al., 2001)

u(xi) =β0 +β1xi +β2(xi − c)δ,

where δ= 1 if xi > c and δ= 0 if xi <= c.

Cubic-Splines Modeling Approach (Gurrin et al., 2005)

u(x;θ) =β0 +β1x+β2x2 +β3x3 +m∑

i=1θi(x− ci)

3+


Trial and Error 4th trial: Much better now ...

OH No! This is not what I wanted. (Problems?)

3rd degree polynomial regression

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OH No! This is not what I wanted. (Problems?)

3rd degree polynomial regression

Predln_fsh

ioi1

ln_

fsh

id = ues112id = twf049

id = tgs105id = smh100id = slb017id = paf120id = ljw126

id = keb118id = jgc015id = jdm097id = img020id = hjr085

id = dps106id = dmp095id = dks091id = cmd109id = cbd086

id = b_j101id = ava119id = arb092id = adl115id = aam076

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Oh! It’s better! However, ... Disadvantages???

Piecewise-4th degree polynomial regression

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Predln_fsh

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Conditional Residuals for ln_fsh

BIC 305.98AICC 316.22AIC 315.98Objective 305.98

Fit Statistics

Std Dev 0.3154Maximum 1.2595Mean -2E-15Minimum -0.859Observations 269

Residual Statistics

-3 -2 -1 0 1 2 3

Quantile

-1.0

-0.5

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1.0

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idua

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-1 -0.6 -0.2 0.2 0.6 1 1.4

Residual

0

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Per

cent

0.5 1.0 1.5 2.0 2.5 3.0

Predicted

-1.0

-0.5

0.0

0.5

1.0

Res

idua

l



Let’s do more work! Still, ... Disadvantages???

3rd degree polynomial regression with trigonometric functions

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10mra



Let’s do more work! Still, ... Disadvantages???

3rd degree polynomial regression with trigonometric functions

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Much Better Now! Disadvantages???

Natural Cubic Splines

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Much Better Now! Disadvantages???

Natural Cubic Splines

Predln_fshBand

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Questions

Questions

QUESTIONS?


Questions

References I

Box, G. E. P. and Draper, N. R. (1986). Empirical Model-building andResponse Surface. John Wiley & Sons, Inc., New York, NY, USA.

Cornelissen Germaine (2014). Cosinor-based rhythmometry. TheoreticalBiology & Medical Modelling, 11:16–16.

Fitzmaurice, G., Laird, N., and Ware, J. (2004). Applied LongitudinalAnalysis. Wiley Series in Probability and Statistics - Applied Probabilityand Statistics Section Series. Wiley.

Gurrin, L. C., Scurrah, K. J., and Hazelton, M. L. (2005). Tutorial inbiostatistics: spline smoothing with linear mixed models. Statistics inMedicine, 24(21):3361–3381.

Johnson, W., Balakrishna, N., and Griffiths, P. L. (2013). Modeling physicalgrowth using mixed effects models. American Journal of PhysicalAnthropology, 150(1):58–67.


Questions

References II

Miles, J. and Shevlin, M. (2001). Applying Regression and Correlation: AGuide for Students and Researchers. SAGE Publications.

Naumova, E. N., Must, A., and Laird, N. M. (2001). Tutorial in Biostatistics:Evaluating the impact of ‘critical periods’ in longitudinal studies ofgrowth using piecewise mixed effects models. International Journal ofEpidemiology, 30(6):1332–1341.

Singer, J. D. (1998). Using SAS PROC MIXED to Fit Multilevel Models,Hierarchical Models, and Individual Growth Models. Journal ofEducational and Behavioral Statistics, 23(4):323–355.

Zwietering M H, Jongenburger I, Rombouts F M, and van ’t Riet, K. (1990).Modeling of the Bacterial Growth Curve. Applied and EnvironmentalMicrobiology, 56(6):1875–1881.


Practice in Growth Curve Modeling - SAS Group... · 2016-03-11 · Practice in Growth Curve...

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