ADHD Reaction Times:ADHD Reaction Times:Densities, Mixed Effects, and Densities, Mixed Effects, and

PCAPCA

ADHD: Attention Deficit ADHD: Attention Deficit (Hyperactive) Disorder(Hyperactive) Disorder

ADHD kids have difficulty in focusing ADHD kids have difficulty in focusing on tasks.on tasks.

But true ADHD is rarer than is But true ADHD is rarer than is generally believed, and drug generally believed, and drug treatments are used much too often.treatments are used much too often.

How can we correctly identify ADHD How can we correctly identify ADHD children?children?

Reaction Time ExperimentReaction Time Experiment 17 ADHD children17 ADHD children 16 age-matched controls16 age-matched controls

1.1. ““Warning of cue” appears on computer screen.Warning of cue” appears on computer screen.2.2. Delay of about 10 seconds.Delay of about 10 seconds.3.3. Cue actually appears.Cue actually appears.4.4. Measure the time it takes to react to the cue.Measure the time it takes to react to the cue.5.5. Repeat and get about 70 reaction times for each Repeat and get about 70 reaction times for each

child.child.

Longer reaction times for ADHD than for controls. Longer reaction times for ADHD than for controls. But what do they look like? How to quantify? But what do they look like? How to quantify?

How do reaction times differ How do reaction times differ between ADHD and controls?between ADHD and controls?

ADHD child has many ADHD child has many reaction times beyond reaction times beyond 1 sec. Not so for 1 sec. Not so for control.control.

How can we represent How can we represent histogram as a smooth histogram as a smooth density?density?

What are differences in What are differences in shape, mean, mode, shape, mean, mode, etc., between groups?etc., between groups?

How can we account How can we account for child-to-child for child-to-child differences when differences when comparing the groups?comparing the groups?

How can we represent the How can we represent the histogram as a smooth curve?histogram as a smooth curve?

Simple answer: Find the probability Simple answer: Find the probability density function using standard methods.density function using standard methods.

Problem with that: Standard textbook Problem with that: Standard textbook densities don’t capture characteristics likedensities don’t capture characteristics like Initial lagInitial lag Extreme peak immediately after lagExtreme peak immediately after lag Long right tail with many outliersLong right tail with many outliers

New answer: Use flexible modeling of New answer: Use flexible modeling of density functions to create a functional density functions to create a functional data objectdata object

How can we create a functional How can we create a functional density object from a density object from a

histogram?histogram? Use tools from before:Use tools from before:

Basis expansionBasis expansion: linear combination of splines: linear combination of splines Roughness PenaltyRoughness Penalty: making explicit the : making explicit the

competing goalscompeting goals Basis expansion withBasis expansion with

34 B-splines of order 534 B-splines of order 5 Equally spaced knotsEqually spaced knots

Competing goals areCompeting goals are Fitting density curve exactly to histogramFitting density curve exactly to histogram Wanting curve to be close to a normal densityWanting curve to be close to a normal density

What about the constraints of What about the constraints of a probability density function?a probability density function?

Constraints:Constraints: P(t) > 0 over interval of interest.P(t) > 0 over interval of interest. Area under the curve is 1.Area under the curve is 1.

New tool: New tool: TransformationTransformation.. For any function W(t), can build a density For any function W(t), can build a density

function:function:p(t) = C exp{W(t)}, for C = appropriate p(t) = C exp{W(t)}, for C = appropriate

function of W.function of W. Transforms estimation problem from Transforms estimation problem from

constrained p(t) to unconstrained W(t)!constrained p(t) to unconstrained W(t)!

Hey, the original data aren’t Hey, the original data aren’t really functional, are they?really functional, are they?

Idea again: Idea again: TransformationTransformation.. The functional object is really The functional object is really

indirectlyindirectly related to the data. related to the data. Data: reaction times t – nonfunctionalData: reaction times t – nonfunctional What we want: reaction time densities What we want: reaction time densities

p(t) – functionalp(t) – functional Related throughRelated through

0 0{ } { } { }j jP t t t P t P t

What do the group densities look What do the group densities look like?like?

Definite shift in Definite shift in mode between mode between groups.groups.

Bimodality, or Bimodality, or even even trimodality?trimodality?

ADHD has large ADHD has large shoulder and shoulder and long tail.long tail.

But what about But what about individual individual differences in differences in children?children?

Are there inter-child Are there inter-child differences?differences?

Examples of Examples of four ADHD four ADHD children:children:

Dashed line Dashed line is group is group density for density for ADHD.ADHD.

Solid line is Solid line is individual individual density for density for child.child.

Definite child-Definite child-to-child to-child variability. variability. Shouldn’t Shouldn’t ignore this.ignore this.

How can we estimate the How can we estimate the densities and account for densities and account for

individual differences?individual differences?

TransformTransform first: first: Subtract 120 from each reaction timeSubtract 120 from each reaction time

Initial dead period not helpfulInitial dead period not helpful LogarithmLogarithm

Effects are likely multiplicative, not additiveEffects are likely multiplicative, not additive

Z = logZ = log1010(t-120)(t-120)

Functional Mixed Effects Linear Model

What is our functional mixed What is our functional mixed effects model?effects model?

Build mixed effects modelBuild mixed effects model Child i; trial j; group kChild i; trial j; group k ZZijkijk = transformed reaction time density (functional) = transformed reaction time density (functional) μμkk = typical performance of all children in group k = typical performance of all children in group k

(functional)(functional) ααi|ki|k = individual performance of child i within group = individual performance of child i within group

k (functional)k (functional) UUijkijk = leftover variation in density (functional) = leftover variation in density (functional)

ZZijkijk = = μμkk + + αα i|ki|k + U + Uijkijk

∑ ∑ ααi|ki|k = 0 = 0

ADHD have ADHD have greater greater variability in variability in residuals.residuals.

ADHD have ADHD have greater greater mean mean residuals residuals (952 vs 645 (952 vs 645 msec).msec).

Modality an Modality an artifact of artifact of instrumentsinstruments..

How can we explore variability How can we explore variability across subjects within a group?across subjects within a group?

Goal:Goal: explore how densities explore how densities

change from child to change from child to child.child.

Idea:Idea: Principal components Principal components

((harmonicsharmonics)) are like are like empirical basis functions. empirical basis functions. Want to expand our Want to expand our densities with these densities with these harmonics.harmonics.

Problem: Problem: Hard to ensure that the Hard to ensure that the

densities are positive.densities are positive. Solution: Solution:

Transformation! Transformation! Explore Explore the derivatives instead. the derivatives instead.

Functional Principal Components Analysis

What do the harmonics look What do the harmonics look like?like?

Used weighted fPCAUsed weighted fPCA minimizes importance of variation when density is small.minimizes importance of variation when density is small.

Back-transformedBack-transformed to get harmonics in original density scale.to get harmonics in original density scale.

Harmonic Interpretations:Harmonic Interpretations: 11stst : More weight on central peak. : More weight on central peak. 22ndnd : More weight on early reaction times. : More weight on early reaction times. 33rdrd : Highlights periodic effect from instrumentation. : Highlights periodic effect from instrumentation.

What have we learned?What have we learned?

TransformationsTransformations The functional object can be indirectly The functional object can be indirectly

related to the data, such as the probability related to the data, such as the probability density functiondensity function

Functional Linear ModelFunctional Linear Model Can add random effectsCan add random effects

Functional Principal Components Functional Principal Components AnalysisAnalysis Can be done on a transformation, such as Can be done on a transformation, such as

the log density derivativethe log density derivative