Time series data a nuisance

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Time series data - a nuisance
or the whole story?

• personal experience
• Nic Friggens
• Danish Institute of Agricultural Sciences, Foulum

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The problem with the nice theoretical
presentations is that they usually ignore the
kind of problems that the PhD student will have
to confront
Jørgensen 2002 (pers. com.)

• Dont use too much time on the statistical
  issues, it is the biological focus and your
  practical experience that is most important

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The nuisance

• When time just happens to be the medium in which
  you accumulate replicates
• Repeated measures
• Seriously affect test statistics
• Much less important for estimation of effects

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First example (the good old days)

• Effect of feed type on milk production
• 2 feeds high and low
• 2 periods each of 13 weeks
• Dry matter intake (DMI) measured weekly
• Cross-over design
• i.e. There may be carry-over effects

(Friggens et al. 1998 J.Dairy Sci. 81 2228)
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SED 0.5
carry-over effect?
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Interpreting the interaction

• Use week data and not the period averages.
• Make use of a repeated measures structure
• Include week as a class variable
• Which covariance structure?
• Within feed x period?
• Not easy (statisticians dont come cheap)
• May well be necessary but not always.
• Unsanitized examples DIAS Biometry Research Unit
  Internal Report 2001-06

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Ændringer i mælk ved mastitis
Second example

• K.H.M.N. Sloth, N.C. Friggens, I.R. Korsgaard, J.
  Jensen, P. Løvendahl, P.H. Andersen, K.L.
  Ingvartsen
• Malkekoens Energioptagelse, Mobilisering og
  Sundhed (MEMO), Danmarks JordbrugsForskning
  Intern Rapport 148 87-112. (2001)

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Experimental setup

• 8 milk parameters
• Measured at each milking throughout lactation
• Effect of stage of lactation (also breed, parity,
  feed, etc) on milk composition

Cannot ignore time
BUT what is the purpose of the analysis?
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Purpose To assess the potential of a
multivariate analysis for improving detection and
description of udder health status
Time (stage of lactation) is NOT in itself of
interest, it is just a factor to be accounted for.
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Simple stepwise approach to the analysis

• Treat stage of lactation as a class variable
• Use 5 windows equally spaced through lactation
• Adjust values for fixed effects including stage
  of lactation (mixed REML model)
• Generate components accounting for the variation
  in the adjusted values (PCA)
• Compare grouping of combined milk measures
  component with independently assessed udder
  health status (Cluster analysis)

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Variance reduction in step 1 Adjustment for
fixed effects
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Step 2 Principal Component Analysis on adjusted
values
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Step 3 Incidence of clinical mastitis in the
different clusters
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Forekomst af subklinisk mastitis
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Statistics are used by many just as a drunk uses
a lamp post - more for support than illumination

• Think carefully about the biological
  interpretation when choosing your statistical
  approach.

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Time series - the Whole Story
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Time series - The Whole Story

• Explicitly consider time effects
• Include time as a continuous variable
• In a biologically sensible way
• But be prepared to fall back on more pragmatic
  solutions

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3rd Example Changes in milk composition
immediately after calving

• To achieve a better understanding of the
  physiological processes underlying the transition
  from colostrum to milk production
• To identify the timepoint at which colostrum
  production ceases
• Birgitte Madsen, Morten Dam Rasmussen (speciale)

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Milk protein
Milkings from calving
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Options for describing time

• Linear models
• Simple regression
• Can use Random Regression osv
• Biologically limited
• Non-linear models
• Difficult statistical behaviour
• May be biologically more relevant

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Random Regression

• Random regression
• Ignores time as a repeated measure
• Yi (a Ai) (b Bi)t ..
• Some very nice statistical properties
• BUT, only with linear models
• And..

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Beware Regressions

• End effects
• Adding correlated terms
• eg. curve minimum decreased from 16 to 12 weeks
  on increasing from 4th to 6 order (Legendre)
  polynomial
• Always check against raw data, if possible

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• Y a b.t d.t2 linear but meaningless
  
• Y a.exp(-b.t) linear (in log form) but
  tends to 0

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Milk fat
Milkings from calving
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Last example Predicting the effect of parity on
lactation curves

• To develop a means to predict the potential
  lactation curve of heifers and 2nd parity cows
  from information on mature cows.
• Note the use of the term potential
• 40 cows each with 3 lactations

(Friggens et al., 1999. Livest. Prod. Sci. 62
1-13)
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Woods Function
dY/dt a.(tb).exp(-c.t)

• Linear in log form..random regression
• Robust and generally gives a good fit.
• Does not relate well to the underlying biological
  processes.

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dY/dt a.tb.exp(-ct)
u
b
t
3.00
2.50
2.00
1.50
1.00
-ct
e
0.50
0.00
0
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40
60
80
100
120
140
160
180
200
220
240
Days post calving
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Lactation Curves
Woods function dY/dt a.tb.exp(-ct)
Alternative function (Emmans and Fisher,
1986) dY/dt a.u.exp(-ct) where u
exp(-exp(G0 - Bt))
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Advantages of alternative function

• u tends to 1 as t increases. Realistic values
  for G0 and B mean that u is very close to 1 after
  peak yield. The rate of decline post-peak is
  described by only one parameter, c.

• Can have a positive yield at t 0
• BUT, nonlinear - 2 step analysis
• parameter estimation for each individual curve
• ANOVA on the curve coefficients

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Alternative functions to describe the lactation
curve.
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The effect of Parity on curve coefficients
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Choosing a (more) biologically meaningful
function resulted in

• A clearer understanding of which phases of the
  lactation curve are affected by parity
• A means to simplify the description of parity
  effects on potential milk production
• The cost of using a more complex model was
  justified for this purpose.

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Use your head
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Summary

• Careful consideration of the biological purpose
  is important in deciding how to deal with time in
  data analysis
• Careful consideration of the biological
  interpretation of time series functions is also
  important.

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Having the right tools is not enough!