1. ADHD Reaction Times: Densities, Mixed Effects, and PCA

2. ADHD: Attention Deficit (Hyperactive) Disorder ADHD kids have difficulty in focusing on tasks. ADHD kids have difficulty in focusing on tasks. But true ADHD is rarer than is generally believed, and drug treatments are used much too often. But true ADHD is rarer than is generally believed, and drug treatments are used much too often. How can we correctly identify ADHD children? How can we correctly identify ADHD children?

3. Reaction Time Experiment 17 ADHD children 17 ADHD children 16 age-matched controls 16 age-matched controls 1. Warning of cue appears on computer screen. 2. Delay of about 10 seconds. 3. Cue actually appears. 4. Measure the time it takes to react to the cue. 5. Repeat and get about 70 reaction times for each child. Longer reaction times for ADHD than for controls. But what do they look like? How to quantify? Longer reaction times for ADHD than for controls. But what do they look like? How to quantify?

4. How do reaction times differ between ADHD and controls? ADHD child has many reaction times beyond 1 sec. Not so for control. ADHD child has many reaction times beyond 1 sec. Not so for control. How can we represent histogram as a smooth density? How can we represent histogram as a smooth density? What are differences in shape, mean, mode, etc., between groups? What are differences in shape, mean, mode, etc., between groups? How can we account for child-to-child differences when comparing the groups? How can we account for child-to-child differences when comparing the groups?

5. How can we represent the histogram as a smooth curve? Simple answer: Find the probability density function using standard methods. Simple answer: Find the probability density function using standard methods. Problem with that: Standard textbook densities dont capture characteristics like Problem with that: Standard textbook densities dont capture characteristics like Initial lag Initial lag Extreme peak immediately after lag Extreme peak immediately after lag Long right tail with many outliers Long right tail with many outliers New answer: Use flexible modeling of density functions to create a functional data object New answer: Use flexible modeling of density functions to create a functional data object

6. How can we create a functional density object from a histogram? Use tools from before: Use tools from before: Basis expansion: linear combination of splines Basis expansion: linear combination of splines Roughness Penalty: making explicit the competing goals Roughness Penalty: making explicit the competing goals Basis expansion with Basis expansion with 34 B-splines of order 5 34 B-splines of order 5 Equally spaced knots Equally spaced knots Competing goals are Competing goals are Fitting density curve exactly to histogram Fitting density curve exactly to histogram Wanting curve to be close to a normal density Wanting curve to be close to a normal density

7. What about the constraints of 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: Transformation. New tool: Transformation. For any function W(t), can build a density function: For any function W(t), can build a density function: p(t) = C exp{W(t)}, for C = appropriate function of W. Transforms estimation problem from constrained p(t) to unconstrained W(t)! Transforms estimation problem from constrained p(t) to unconstrained W(t)!

8. Hey, the original data arent really functional, are they? Idea again: Transformation. Idea again: Transformation. The functional object is really indirectly related to the data. The functional object is really indirectly related to the data. Data: reaction times t – nonfunctional Data: reaction times t – nonfunctional What we want: reaction time densities p(t) – functional What we want: reaction time densities p(t) – functional Related through Related through

9. What do the group densities look like? Definite shift in mode between groups. Definite shift in mode between groups. Bimodality, or even trimodality? Bimodality, or even trimodality? ADHD has large shoulder and long tail. ADHD has large shoulder and long tail. But what about individual differences in children? But what about individual differences in children?

10. Are there inter-child differences? Examples of four ADHD children: Dashed line is group density for ADHD. Dashed line is group density for ADHD. Solid line is individual density for child. Solid line is individual density for child. Definite child-to- child variability. Shouldnt ignore this.

11. How can we estimate the densities and account for individual differences? Transform first: Transform first: Subtract 120 from each reaction time Subtract 120 from each reaction time Initial dead period not helpful Initial dead period not helpful Logarithm Logarithm Effects are likely multiplicative, not additive Effects are likely multiplicative, not additive Z = log 10 (t-120) Z = log 10 (t-120) Functional Mixed Effects Linear Model

12. What is our functional mixed effects model? Build mixed effects model Build mixed effects model Child i; trial j; group k Child i; trial j; group k Z ijk = transformed reaction time density (functional) Z ijk = transformed reaction time density (functional) μ k = typical performance of all children in group k (functional) μ k = typical performance of all children in group k (functional) α i|k = individual performance of child i within group k (functional) α i|k = individual performance of child i within group k (functional) U ijk = leftover variation in density (functional) U ijk = leftover variation in density (functional) Z ijk = μ k + α i|k + U ijk α i|k = 0 α i|k = 0

13. ADHD have greater variability in residuals. ADHD have greater variability in residuals. ADHD have greater mean residuals (952 vs 645 msec). ADHD have greater mean residuals (952 vs 645 msec). Modality an artifact of instruments. Modality an artifact of instruments.

14. How can we explore variability across subjects within a group? Goal: Goal: explore how densities change from child to child. explore how densities change from child to child. Idea: Idea: Principal components (harmonics) are like empirical basis functions. Want to expand our densities with these harmonics. Principal components (harmonics) are like empirical basis functions. Want to expand our densities with these harmonics. Problem: Problem: Hard to ensure that the densities are positive. Hard to ensure that the densities are positive. Solution: Solution: Transformation! Explore the derivatives instead. Transformation! Explore the derivatives instead. Functional Principal Components Analysis

15. What do the harmonics look like? Used weighted fPCA Used weighted fPCA minimizes importance of variation when density is small. Back-transformed Back-transformed to get harmonics in original density scale. Harmonic Interpretations: Harmonic Interpretations: 1 st : More weight on central peak. 2 nd : More weight on early reaction times. 3 rd : Highlights periodic effect from instrumentation.

16. What have we learned? Transformations Transformations The functional object can be indirectly related to the data, such as the probability density function The functional object can be indirectly related to the data, such as the probability density function Functional Linear Model Functional Linear Model Can add random effects Can add random effects Functional Principal Components Analysis Functional Principal Components Analysis Can be done on a transformation, such as the log density derivative Can be done on a transformation, such as the log density derivative

17. References Applied Functional Data Analysis (2002), Ch. 5. Applied Functional Data Analysis (2002), Ch. 5. Experiment and data described in in Leth- Steenson, et al, in Acta Psychologica (2000). Experiment and data described in in Leth- Steenson, et al, in Acta Psychologica (2000).