Mouse Livers:Mouse Livers:Derivatives and Functional Derivatives and Functional

Linear ModelsLinear Models

How does cholesterol get How does cholesterol get metabolized in the liver?metabolized in the liver?

What questions can we ask of What questions can we ask of the data?the data?

What does the What does the real, “smooth” real, “smooth” process look process look like?like?

Do shapes Do shapes differ among differ among groups?groups?

Do rates of Do rates of change differ change differ among groups?among groups?

What do the flow curves look What do the flow curves look like as functional objects?like as functional objects?

Took the Took the derivative of the derivative of the smoothed curves.smoothed curves.

Still retain curve-Still retain curve-to-curve to-curve variability, but variability, but now much now much smoother.smoother.

How can I graphically explore the How can I graphically explore the data?data?

Have:Have: flow curves x(t).flow curves x(t). rate of change of rate of change of

flow curves Dx(t).flow curves Dx(t). Plot Dx(t) vs x(t). Plot Dx(t) vs x(t).

No longer an No longer an explicit function of explicit function of time!time!

Overlay time points Overlay time points on the curve for on the curve for interpretation.interpretation.

Gives information Gives information about how function about how function is linked with its is linked with its derivative. derivative.

Phase-Plane Plots

What do we see in these What do we see in these phase-plane plots?phase-plane plots?

Difference in Difference in curves between curves between receptors and no receptors and no receptorsreceptors

Cusps or ‘change-Cusps or ‘change-points’ when points’ when there are there are receptorsreceptors

Minute 9 for Minute 9 for Receptor A; Receptor A; Minute 15 for Minute 15 for Receptor BReceptor B

Minute 9 for Both Minute 9 for Both Receptors: Receptors: Interactive Effect?Interactive Effect?

What is the relationship What is the relationship between the covariates and between the covariates and

response curves?response curves?

Functional response; Scalar predictors.Functional response; Scalar predictors. Regression coefficients are functional.Regression coefficients are functional.

Use basis expansion methods.Use basis expansion methods.

Functional Linear Models

X(t) = X(t) = ββ00(t) + (t) + ββ11(t)(t)AA + + ββ22(t)(t)B B + + ββ33(t)(t)A*BA*B + + εε(t),(t),

Receptors Receptors affect steady affect steady state.state.

B stronger B stronger than A.than A.

Effects Effects strongest strongest after minute after minute 9.9.

A and B have A and B have inhibitory inhibitory relationship relationship after minute after minute 9.9.

Receptors Receptors affect steady affect steady state.state.

B stronger B stronger than A.than A.

Effects Effects strongest strongest after minute after minute 9.9.

A and B have A and B have inhibitory inhibitory relationship relationship after minute after minute 9.9.

Can also do a functional linear model for derivative (rate of change):

FDA allows us to work with derivatives – which are closer to the mechanisms of the process

dX/dt = dX/dt = ββ00(t) + (t) + ββ11(t)(t)AA + + ββ22(t)(t)B B + + ββ33(t)(t)A*BA*B + + εε(t),(t),

A “kicks in” A “kicks in” earlier than earlier than does B.does B.

A kicks in at A kicks in at minute 9, B minute 9, B at minute at minute 15.15.

When When together, together, see push see push only at only at minute 9 minute 9 (from A?)(from A?)

What have we learned?What have we learned? Creating a functional objectCreating a functional object

Smoothing with basis expansions to Smoothing with basis expansions to reduce noisereduce noise

Examining derivatives graphicallyExamining derivatives graphically Phase-plane plotsPhase-plane plots

Building functional linear modelsBuilding functional linear models Functional regression coefficientsFunctional regression coefficients Derivatives helpful here, tooDerivatives helpful here, too