Statistical Modeling of Inter-Individual Differences

Parsimonious Statistical Modeling of Inter-Individual Response

Differences to Sleep Deprivation

Greg MaislinPrincipal Biostatistician

Biomedical Statistical Consulting &Director, Biostatistics and Data Management Core

Center for Sleep and Respiratory NeurobiologyUniversity of Pennsylvania School of Medicine

Acknowledgements and Thanks Special thanks to Dr. David F. Dinges whose experiments

exploring the consequences of partial and total sleep deprivation with and without counter measures provided fertile ground for deep thinking about some very interesting statistical issues.

To Dr. Hans Van Dongen for challenging me to be clear and compelling in my thinking.

And to Bob Hachadoorian, Senior Statistical Programmer for his dedication in programming SAS production runs capable of mass processing multitudes of assessments across many domains in the PSD and TSD protocols.

Introduction

Inter-individual differences in response to sleep loss are substantial. Sleep deprivation protocols often involve neurobehavioral and physiological testing over multiple days. There is need for conceptually simple, yet quantitatively valid statistical methods that recognize inter-individual variability and accommodate assay-specific non-linearity of time trajectories.

Intraclass Correlation Coefficient (ICC)

Between-subject variance (bs2 )

Within-subject (ws2) variance

Total2 = bs

2 + ws2

ICC is defined as follows to quantify trait-like inter-individual variability:

ICC = bs2

bs2 + ws

2

Intraclass Correlation Coefficient (ICC)

ICC varies by population since bs2 varies by

population. The magnitudes of bs

2 and ws2 should be

interpreted, not just ICC. Mixed effects ANOVA can be used to estimate

ICC-like measures that include multiple sources of variance and that filter out ‘fixed’ effects such as demographic factors and experimental conditions.

Evidence of Trait-Like Variance

Change in Total PVT lapses after two exposures 2-4 wks apart of 36 h sleep deprivation.Van Dongen HP, Kijkman M, Maislin G, Dinges D. Phenotypic aspect of vigilance decrement during sleep deprivation. Physiologist 1999; 42:A-5.


Preliminary data from Heritability of Sleep Homeostasis, Drs. Allan Pack and Samuel Kuna,

Division of Sleep Medicine.

PVT Transformed Lapses Linear SlopesOver 38 Hours (19 trials) of Sleep Deprivation

Monozygotic Twins

Twin Pair

2 82 9 46 37 14 310

1 97 64 121 91 118

109 56 47 122 95 22 67 63 23 51 27 93 136 13 17 129 78 28 31 42 74 79 87 120 62 54 108 73 103 44 53 116 48 49 38

PV

T T

ran

sfo

rmed

Lap

ses

Lin

ear

Slo

pes

-0.1

0.0

0.1

0.2

0.3

0.4

0.5ICC = 56.0% (N=48 pairs)

Var(B) = 3.51*10E-3

Var(W) = 2.76*10E-3


PVT Transformed Lapses Linear SlopesOver 38 Hours (19 trials) of Sleep Deprivation

Dizygotic Twins

Twin Pair

132 98 135 29 117

142

126

106 25 128

134 10 148

100 94 149 41 4 58 113

104 16 124

130

145

127 57 5

138 52

PV

T T

ran

sfo

rmed

Lap

ses

Lin

ear

Slo

pes

-0.1

0.0

0.1

0.2

0.3

0.4

0.5ICC = 24.5% (N=30 pairs)

Var(B) = 0.93*10E-3

Var(W) = 2.87*10E-3

‘Test-bed’ Experiment1

This sleep restriction experiment involved one adaptation day and two baseline days with 8 h sleep opportunities (TIB 23:30–07:30), followed by randomization to 8 h, 6 h or 4 h periods for nocturnal sleep (TIB ending at 07:30) for 14 days. 13 Subjects randomized to 4 hrs TIB for 14 days 13 Subjects randomized to 6 hrs TIB for 14 days 9 Subjects randomized to 8 hrs TIB for 14 days

Assessments every 2 hrs during wakefulness1 Van Dongen HP, Maislin G, Mullington JM, and Dinges DF. The cumulative cost of additional wakefulness: Dose-response effects on neurobehavioral functions and sleep physiology from chronic sleep restriction and total sleep restriction. Sleep 2003, 26(2):117-126.

Neurobehavioral Test Battery (NAB)(1) Psychomotor vigilance task (PVT) (Dinges & Powell 1985)(2) Probed recall memory (PRM) test that controls for reporting bias and evaluates free

recall/retention (Dinges et al 1993)

(3) Digit symbol substitution task (DSST) assesses cognitive throughput (speed/accuracy)

(4) Time estimation task (TET)

(5) Performance evaluation and effort rating scales (PEERS) to track self monitoring, compensatory effort, and motivation (Dinges et al 1992)

(1) Karolinska Sleepiness Scale (KSS) (Akerstedt & Gillberg 1990) (2) Stanford Sleepiness Sale (SSS) (Hoddes et al 1973)

(3) Visual analog scale (VAS) for mental and physical exhaustion

(4) Activation-Deactivation Checklist (AD-ACL)(Thayer 1986)

(5) Profile of Mood States (POMS) (McNair, Lorr, & Druppleman 1971)

Psychomotor vigilance task (PVT)

Simple, high-signal-load reaction time (RT) test designed to evaluate the ability to sustain attention and respond in a timely manner to salient signals.

10 minute duration Yields six primary metrics on the capacity for

sustained attention and vigilance performance.

Psychomotor vigilance task (PVT)

Frequency of lapses (RT>500 msec) Duration of the lapse domain (mean of 10% slowest

reciprocal RTs) Optimum response times (mean of 10% fastest RTs) False response frequency (errors), Frequency of non-responses (caused by spontaneous

sleep episodes) Fatigability function (slope computed from 1 minute

bins of mean 1/RTs).

PVT Lapses Among 9 Subjects withTime in Bed Restricted to 8 Hours

Day

0 2 4 6 8 10 12 14

PV

T L

apse

s

0

5

10

15

20

25


Day

0 2 4 6 8 10 12 14

PV

T L

apse

s

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50


Day

0 2 4 6 8 10 12 14

PV

T L

apse

s

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10

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30

40

Average Daily PVT Lapses/Trial

Karolinska Sleepiness Scale (KSS)Karolinska Sleepiness Scale (KSS)

Place the X next to the ONE statement that best describes your SLEEPINESS during the PREVIOUS 5 MINUTES. You may also use the intermediate steps.

X __ 1. very alert __ 2. __ 3. alert, normal level __ 4. __ 5. neither alert nor sleepy __ 6. __ 7. sleepy, but no effort to keep awake __ 8. __ 9. very sleepy, great effort to keep awake, fighting sleep

Use {UP/ DOWN} cursor keys to move X block, then press {ENTER}

Karolinska Sleepiness Scale (KSS)

Karolinska Sleepiness Score Among 13 Subjectswith Time in Bed Restricted to 4 Hours

Day

0 2 4 6 8 10 12 14

KS

S

0

2

4

6

8

10


Day0 2 4 6 8 10 12 14

KS

S

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10


Day0 2 4 6 8 10 12 14

KS

S

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8

10

Observations For the subjective measure, end-of-study values

depend heavily on baseline values. The objective measure increased linearly throughout

the PSD protocol. The increase in the subjective measure was non-

linear with decelerating increases. Most of the increase was very early.

There was substantial variability among subjects in both objective and subjective responses to PSD.

Substantial Variability in Responses to PSD


Day

0 2 4 6 8 10 12 14

PV

T L

apse

s

0

10

20

30

40

Statistical Approachesfor Growth Curves

Classical repeated measures analysis fails to recognize individual response variance. It is a model for the mean response with no recognition of true biological variability among subjects in the magnitudes of their response.

Mixed effects models for each individual observation include random subject effects and can allow for a variety of covariance structures that can reflect many different assumptions concerning the nature of within subject correlations overtime (e.g. AR(1)). Although theoretically appealing, concern has been raised about the robustness of these models1. They also require sophisticated statistical approaches that may not be immediately accessible to all researchers. 1 Ahnn, Tonidandel, and Overall. Issues in use of Proc.Mixed to test the significance of treatment effects in controlled clinical trials. J of Biopharm Stat 10(2):265-286, 2000

Standard Two Stage (STS) regression

Using simple linear regression, a slope (and intercept) for each subject are determined at the first stage. Second stage group comparisons are made by comparing mean slopes.

STS gives each subject’s first stage slope estimate equal weight, which is not appropriate if the sample size or layout of time values varies widely among subjects.

STS disguises residual error, pooling it with between-subject variance and biasing the latter upward. If residual variance is small or the numerical values of variance components are not themselves of interest, this is not a problem.

Standard Two Stage (STS) regression (cont.)

STS does not account for the covariance between slopes and intercepts.

Parsimony: It is desirable to reduce the response curve to a single number (eliminating the intercept).

Slopes assume constant accumulations of deficit over time. However, accumulation of deficits can be decelerating or accelerating.

Mixed linear model determination of slopes

The simultaneous determination of subject specific slopes using maximum likelihood incorporates the assumption that the slopes are normally distributed with condition-specific mean values and a common variance.

STS does not make this assumption, computing each subject-specific slope independently from all other subjects (robustness?).

Proposed Model: Two-Stage Non-LinearMixed Model Regression

(t)i(j) = Bi(j) · t + (t)i(j)

(t)i(j) = performance deficit for subject i in group j at

time t

is a curvature parameter reflecting the nature of non-

linearity of growth in deficits

Bi(j) are subject-specific “non-linear” slopes

(t)i(j) are residual errors.

Reference

Van Dongen HPA, Maislin G, Dinges DF. Dealing with inter-individual differences in the temporal dynamics of fatigue and performance: Importance and techniques. Aviat Space Environ Med 2004: 75:A147–A154.

Proposed Model: Two-Stage Non-LinearMixed Model Regression

The (non-linear) slopes are combinations of group specific

mean values and random effects reflecting individual

susceptibilities to the deprivation challenge.

Bi(j) = j + bi(j)

j is the mean response in group j

bi(j) ~ Normal(0, 2b).

2b is a subject specific variance contribution.

Bi(j) ~ Normal(j,, 2b ).

Three Methods of Estimating Bi(j)

Two-stage Random Effects Regression1 with grid

search varying .

REML2 (treating as fixed)

MLE3 (estimating )1 Feldman. Families of lines: random effects in linear regression, J Appl. Physiol.

64(4):1721-1732, 1988.2 Diggle, Kiang, Zeger. Analysis of Longitudinal Data, Oxford: Clarendon Press, 1996,

pages 64-68. 3 Vonesh: Nonlinear Models for the Analysis of Longitudinal Data. Stat. in Med, 11,

1929 – 1954, 1992.

Two-stage Random Effects Regression with Grid search varying

Simple linear regression for each subject varying from 0.1 to 2.0.

The that minimizes the average MSE is selected.

Fig. 1 Grid Search Over Average MSEfrom First Stage Linear Regressions

on Delta PVT Lapses Values

Curvature (Theta)0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

Me

an

MS

E

10

12

14

16

18

20

22

24

Theta = 0.78

Two-stage Random Effects Regression with Grid search varying

REML Mixed Linear Model (fixed )

The 2-stage approach disguises residual error, pooling it with between-subject variance and biasing the latter upward (compare SD’s in Table 1 and Table 2).

Conditional on the value of the optimal obtained from the grid search, the model is no longer non-linear.

When is assumed known, a mixed linear model can be used to simultaneously derive subject specific slopes (e.g., SAS Proc Mixed).

ML Mixed Non-Linear Model (simultaneous estimation of )

Has greatest theoretical appeal. Requires specialized software

(e.g., SAS Proc NLMIXED). Model sometimes does not converge. More precise estimate of (Table 3).

Examples of Second Stage AnalysisFig. 2 Delta PVT Lapses

REML (fixed theta) Non-linear Slopesby Study Group (Theta=0.78)

Group

4 hr 6 hr 8 hr

No

n-l

inea

r S

lop

e

-1

0

1

2

3

4

5

6

F2,30=3.67, p=0.037

Examples of Second Stage Analysis

Table 1. Two-stage Non-linear Slopes (=0.7753)

N Mean SD Min Max

4 hr 13 1.9269 1.3493 0.0638 3.8784

6 hr 13 1.2897 1.6999 -0.5383 5.5608

8 hr 9 0.3345 0.6851 -0.3228 2.0517


Table 2. REML Mixed Model Non-linear Slopes (=0.7753)

N Mean SD Min Max

4 hr 13 1.9269 1.3245 0.0981 3.8425

6 hr 13 1.2897 1.6686 -0.5046 5.4822

8 hr 9 0.3345 0.6721 -0.3107 2.0201


Table 3. ML Non Linear Mixed Model Results

StandardLabel Estimate Error DF t Value Pr > |t|theta 0.7753 0.0397 34 19.51 <.0001 4hr Slope 1.9267 0.4028 34 4.78 <.00016hr Slope 1.2896 0.3820 34 3.38 0.00198hr Slope 0.3352 0.4390 34 0.76 0.4504BTW Subj SD 1.3010 0.1985 34 6.55 <.0001Resid SD 3.2918 0.1096 34 30.03 <.0001Total Var 12.5288 0.8848 34 14.16 <.0001Total SD 3.5396 0.1250 34 28.32 <.0001


Parameter Estimates Standard Parameter Estimate Error DF t Value Pr > |t| Alpha Lower beta0 0.3352 0.4390 34 0.76 0.4504 0.05 -0.5569 s2beta 1.6927 0.5166 34 3.28 0.0024 0.05 0.6428 theta 0.7753 0.03974 34 19.51 <.0001 0.05 0.6946 s2e 10.8361 0.7216 34 15.02 <.0001 0.05 9.3697 bcond4 1.5915 0.5876 34 2.71 0.0105 0.05 0.3974 bcond6 0.9544 0.5762 34 1.66 0.1069 0.05 -0.2167

2-Stage Regression Non-linear SlopesPVT Lapses

Delta PVT LapsesTwo-Stage Non-linear Slopes

by Study Group (Theta=0.7753)

Sleep Condition

4 hour 6 hour 8 hour

No

n-l

inea

r S

lop

e

-1

0

1

2

3

4

5

6

REML: PVT LapsesAssuming Known

Delta PVT LapsesREML (fixed theta) Non-linear Slopes


Sleep Condition

4 hour 6 hour 8 hour

No

n-l

inea

r S

lop

e

-1

0

1

2

3

4

5

6

Two-stage vs. REML Mixed ModelDelta PVT Lapses

Attenuation of Extremes Produce by REML

Two-Stage Non-linear Slope

-3 -2 -1 0 1 2 3 4 5 6 7 8 9

RE

ML

No

n-l

inea

r S

lop

e

-3-2-10123456789

Attenuation Slope=0.9999

2-Stage Regression Non-linear SlopesPVT Lapses

Delta PVT LapsesTwo-Stage Non-linear Slopesby Study Group (Theta=1.1)

Group

Active Placebo

No

n-l

inea

r S

lop

e

-4

-2

0

2

4

6

8

10

REML: PVT LapsesAssuming Known

Delta PVT LapsesREML (fixed theta) Non-linear Slopes


Group

Active Placebo

No

n-l

inea

r S

lop

e

-4

-2

0

2

4

6

8

10

Two-stage vs. REML Mixed ModelDelta PVT Lapses

Attenuation of Extremes Produce by REML

Two-Stage Non-linear Slope

-3 -2 -1 0 1 2 3 4 5 6 7 8 9

RE

ML

No

n-l

inea

r S

lop

e

-3-2-10123456789

Attenuation Slope=0.925

Analysis for KSSQ (theta=0.1607)

N COND Obs Variable N Mean Std Dev Minimum Maximum ---------------------------------------------------------------------- 4 13 SLOPE 13 1.3488 0.8470 0.2265 3.1963 SLOPE_MIXED 13 1.3488 0.7402 0.3679 2.9635 6 13 SLOPE 13 1.0061 0.6925 -0.0035 2.3645 SLOPE_MIXED 13 1.0061 0.6052 0.1237 2.1934 8 9 SLOPE 9 0.1297 0.5857 -0.6041 1.1054 SLOPE_MIXED 9 0.1315 0.5112 -0.5114 0.9827 ----------------------------------------------------------------------

Note that the mean 2-stage slope and the mean mixed model slope are not identical for KSSQ in the 8 hour condition because there was a missing value. The mixed model slopes correctly adjust for the reduced precision caused by the single missing value.


Parameter Estimates Standard Parameter Estimate Error DF t Value Pr > |t| Alpha Lower beta0 0.1304 0.2335 34 0.56 0.5803 0.05 -0.3442 s2beta 0.4662 0.1292 34 3.61 0.0010 0.05 0.2037 theta 0.1607 0.03084 34 5.21 <.0001 0.05 0.09800 s2e 0.5868 0.03907 34 15.02 <.0001 0.05 0.5074 bcond4 1.2185 0.3129 34 3.89 0.0004 0.05 0.5825 bcond6 0.8758 0.3082 34 2.84 0.0075 0.05 0.2494

Contrasts Num Den Label DF DF F Value Pr > F Overall Slope Diff 2 34 7.76 0.0017 4 vs 6 Slope 1 34 1.55 0.2217 4 vs 8 Slope 1 34 15.16 0.0004 6 vs 8 Slope 1 34 8.07 0.0075


Additional Estimates Standard Label Estimate Error DF t Value Pr > |t| Alpha Lower 4hr Slope 1.3488 0.2110 34 6.39 <.0001 0.05 0.9201 6hr Slope 1.0062 0.2032 34 4.95 <.0001 0.05 0.5933 8hr Slope 0.1304 0.2335 34 0.56 0.5803 0.05 -0.3442 BTW Subj SD 0.6828 0.09460 34 7.22 <.0001 0.05 0.4905 Resid SD 0.7660 0.02550 34 30.03 <.0001 0.05 0.7142 Total Var 1.0530 0.1346 34 7.83 <.0001 0.05 0.7795 Total SD 1.0261 0.06557 34 15.65 <.0001 0.05 0.8929

Note that estimated slopes using ML are identical to REML because the ML theta was used.

Results from Twin Study

Pooled Mean Change inTransformed PVT Lapses

Hours Post 7:30am Awakening

4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38

Ch

ang

e in

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nsf

orm

ed P

VT

Lap

ses

Per

Tri

al

-1

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7MZDZ


E(it) =

i0 + i1*t + ai*cos(2**t/24) + bi*sin(2**t/24)


Predicted Change in PVT LapsesLinear plus Single Harmonic Model

Hours Post 7:30am Awakening

4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38

Ch

ang

e in

PV

T L

apse

s P

er T

rial

-5

0

5

10

15

20

25 Tw 1/A Tw 1/B Tw 2/A Tw 2/B Tw 3/A Tw 3/B Tw 4/A Tw 4/B Tw 5/A Tw 5/B Tw 9/A Tw 9/B Tw 10/A Tw 10/B Tw 13/A Tw 13/B Tw 14/A Tw 14/B Tw 16/A Tw 16/B


Heritability Indices Classical ApproachPVT Linear Slopes

Monozygotic Twins Dizygotic Twins

2B 2

W 2T ICC 2

B 2W 2

T ICC h2

TransformedLapses 3.51 2.76 6.27 0.560 0.93 2.87 3.80 0.245

0.631

An Application in Another Area Berkowitz RI, Stallings VA, Maislin G, Stunkard AJ.

Growth of children at high risk for obesity during the first six years: Implications for prevention. American Journal of Clinical Nutrition. Am J Clinical Nutrition 2005;81:140–6.

Body size and composition of high and low risk groups were measured repeatedly from 3 mo. to 6 yrs of age at CHOP. Subjects included 33 children at high risk for and 37 children at lower risk for obesity on the basis of mothers’ overweight1

1 high risk mean (SD) BMI = 30.2 (4.2), low risk mothers’ BMI = 19.5 (1.1).

An Application in Another Area

At year 2, there were no differences between high and low risk groups in any measure of body size and composition1

(Energy intake and sucking behaviors at Month 3 were predictive of 2 year weight in both groups.)

1 Stunkard AJ, Berkowitz RI, Schoeller D, Maislin G, Stallings VA. Predictors of body size in the first 2 years of life: a high-risk study of human obesity. International Journal of Obesity 2004 1-11.

Weight Over TimeFrom Month 24 to Month 72

High Risk Group

Month

24 30 36 42 48 54 60 66 72

Wei

gh

t (k

g)

0

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50

High Risk

Weight Over TimeFrom Month 24 to Month 72

Low Risk Group

Month

24 30 36 42 48 54 60 66 72

Wei

gh

t (k

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Low Risk

Average Mean Squared Errorfrom First Stage Linear Regressions

on Delta Weight from Month 24 to Month 72

Curvature (Theta)0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4

Me

an

MS

E

0

2

4

6

8

10

12

Theta = 2.4

Change in Weight from Month 24 to Month 72REML (fixed theta) Non-linear Slopes

by Risk Group (Theta=2.4)

Group

High Risk Low Risk

No

n-l

inea

r S

lop

e

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07Mean (SD) Non-Linear SlopesHigh Risk 0.024 (0.011)Low Risk 0.018 (0.004)Unequal Variance t-test p=0.010

Parsimony of InterpretationsChange in Weight from Month 24 to Month 72

REML (fixed theta) Non-linear Slopesby Group (Theta=2.4)

Group

High GT 85th High LE 85th Low

No

n-l

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0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Generally monotonic but varying in direction:

It is possible that trajectories are generally monotonic but vary in direction (i.e., there are subgroups of individuals with generally increasing values and others with generally decreasing values over time).

In this case, can be set to 1 with slope estimated individually using two-stage random effects regression or simultaneously by Proc Mixed.

Generally non-monotonic changes: If trajectories are non-monotonic (e.g. quadratic), mixed model analyses of

longitudinal changes can be performed that incorporate multiple time variables such as linear plus quadratic terms or appropriately constructed sets of time indicator variables.

Correlations among observations within subject can be accounted for by using appropriately constructed covariance matrices1. A particular covariance structure is the ‘random intercept plus AR(1)’ structure. This structure induces within subject correlations by assuming both systematic variance in overall levels between subjects plus a component that diminishes over time.

1 Littell RC, Pendergast J, Natarajan R. Tutorial in biostatistics: Modeling covariance structure in the analysis of repeated measures data. Statistics in Medicine. 19:1793-1819, 2000.

General Linear Mixed Model:

Verbeke G and Molenberghs G (2000). Linear Mixed Models for Longitudinal Data. Springer Series in Statistics. New-York: Springer.

Yi = Xiβ + Zibi + εi bi ~ N(0,D), εi ~ N(0, ∑i) b1, . . . , bN, εi , . . . , εN independent Terminology

Fixed effects: β Random effects: bi Variance components: Elements in D and ∑i


Hierarchical model can be rewritten as:Yi|bi ~ N(Xiβ + Zibi, ∑i); bi ~ N(0,D)

Marginal model can be rewritten as:Yi ~ N(Xiβ, ZibiZ'i +∑i)

The hierarchical model is most naturally interpreted through a Bayesian perspective

Only the hierarchical model explicitly assumes inter-individual variability


Prior distribution: f(bi) = N(0,D)

Likelihood function: f(yi|bi) = N(Xiβ + Zibi, ∑i)

Posterior distribution:f(bi | Yi = yi) α f(yi|bi) * f(bi)

Posterior mean: ∫ bi f(bi | yi) dbi is the Empirical Bayes estimate of bi

Conclusions The STS, REML, and ML approaches have advantages and

disadvantages. “The great advantage of the STS, aside from conceptual and

computational simplicity, is the availability of valid small-sample statistics. STS can thus be relied upon, whereas WLS and REML cannot, to produce accurate P values in the cases of very few subjects, so long as the assumptions of the small-sample model (e.g., normality) are met1”

The ML approach requires starting values (guesses) for every parameter and sometimes the ML optimization does not converge.1 Feldman HA. Families of lines: random effects in linear regression analysis, J. Appl. Physiol. 64(4):1721-1732, 1988.

Conclusions The “grid search plus STS” method provides good

solutions that in most cases are very similar to the optimal ML solutions, facilitate analysis of inter-individual variability in responses to sleep deprivation, and is easy to implement.

The model: (t)i(j) = Bi(j) · t + (t)i(j) can be generally recommended for sets of responses that are generally monotonic.

Other methods are needed for non-monotonic trajectories. The cost of non-monotonicity is greater analytical complexity.

Conclusions The Linear Mixed Model provides a

comprehensive platform for evaluation and estimation of inter-individual variability including the evaluation of prior and posterior distributions of subject specific parameters. It may be possible to update subject specific parameters reflecting individual performance, and then sum over these individual performance estimates to obtain a summary prediction of unit performance.

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Statistical Modeling of Inter-Individual Differences