Mouse Livers: Derivatives and Functional Linear Models

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Mouse LiversDerivatives and Functional Linear
Models
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How does cholesterol get metabolized in the liver?
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What questions can we ask of the data?

• What does the real, smooth process look like?
• Do shapes differ among groups?
• Do rates of change differ among groups?

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What do the flow curves look like as functional
objects?

• Took the derivative of the smoothed curves.
• Still retain curve-to-curve variability, but now
  much smoother.

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How can I graphically explore the data?
Phase-Plane Plots

• Have
• flow curves x(t).
• rate of change of flow curves Dx(t).
• Plot Dx(t) vs x(t). No longer an explicit
  function of time!
• Overlay time points on the curve for
  interpretation.
• Gives information about how function is linked
  with its derivative.

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What do we see in these phase-plane plots?

• Difference in curves between receptors and no
  receptors
• Cusps or change-points when there are receptors
• Minute 9 for Receptor A Minute 15 for Receptor B
• Minute 9 for Both Receptors Interactive Effect?

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What is the relationship between the covariates
and response curves?
Functional Linear Models

• Functional response Scalar predictors.
• Regression coefficients are functional.
• Use basis expansion methods.

X(t) ß0(t) ß1(t)A ß2(t)B ß3(t)AB e(t),
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• Receptors affect steady state.
• B stronger than A.
• Effects strongest after minute 9.
• A and B have inhibitory relationship after minute
  9.

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• Receptors affect steady state.
• B stronger than A.
• Effects strongest after minute 9.
• A and B have inhibitory relationship after minute
  9.

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• 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 ß0(t) ß1(t)A ß2(t)B ß3(t)AB e(t),
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• A kicks in earlier than does B.
• A kicks in at minute 9, B at minute 15.
• When together, see push only at minute 9 (from
  A?)

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What have we learned?

• Creating a functional object
• Smoothing with basis expansions to reduce noise
• Examining derivatives graphically
• Phase-plane plots
• Building functional linear models
• Functional regression coefficients
• Derivatives helpful here, too

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References

• Data courtesy of Allen Cooper and Kenny Yu, Palo
  Alto Medical Foundation Stanford University.
• For functional linear models, see Functional Data
  Analysis (1996), Ch. 9 11.
• For phase-plane plots, see Applied Functional
  Data Analysis (2002), Ch. 3.