Multiple measurements

1
Multiple measurements

• So far, weve considered a single variable
  measured on each unit in an experiment. (e.g.
  previous examples)
• Typically, several measurements per unit to
  record response of the units to the various
  treatments in the expt.
• E.g. to compare various varieties of carrots, we
  could measure plant height at various dates,
  total harvested weights, numbers of roots,
  incidence of disease, colour and taste.
• Information stored as a matrix one row per unit,
  each column measures a particular variable for
  all units. (e.g. id ht(t1) ht(t2) wt(t1) wt(t2)
  )
• Other variables can be calculated based on these
  measurements, e.g. increase in height
• incr ht(t2) - ht(t1)

2
Multiple and Repeated measures

• Multiple measurements per unit can be of two
  types
• 1. Different kinds of measurements, e.g. height,
  weight, colour, taste
• 2. The same variable measured at different
  times e.g. height(week1), height(week2),
  height(week3).
• This kind is known as Repeated measures data.

3
Cow medication data

• Cows given medication and observed for 4 weeks
• Does medication affect males and females
  differently ?

4
Standard Analysis of Medication Data

• Combined data from all weeks (12 per sex)
• Confirmatory t-test for difference between sexes
  gives p-value of 0.002 (highly significant)
• With gt2 groups we would use a one-way ANOVA

5
Need for separate analysis

• Two-sample t-test is invalid. It assumes
  independence of all 12 observations in each
  group.
• Data within subject, i.e. 4 repeated measurements
  on each cow are not independent. Because .
• However, data between subjects , i.e.
  measurements on different cows are independent.
• Analysis needs to recognise these two different
  sources of variation.
• Repeated measures ANOVA with 2 tables
• 1. ANOVA between subjects
• 2. ANOVA within subjects

6
Repeated measures ANOVA

• Difference in genders Same conclusion as before
• However p-value differs by factor of 5
• Choice of analysis can lead to different
  conclusions
• Assumption of circularity is implicit.
• Circularity is not valid when in many situations

7
Case Study Growth in rats

• Experiment to see effect of chemicals added to
  drinking water
• Three treatments one control group,
  one thiouracil group and one thyroxin group.
  
• 10, 7 and 10 rats respectively in each group.
• Weights of rats recorded at start of experiment
  (week 0) and after 1, 2, 3 and 4 weeks of
  treatment.
• Object To find if the chemicals retard growth.
• Data shown on next slide

8
Growth in rats data
9
Single timepoint comparisons
Incr
10
Visualising growth curves (over time)
11
Growth - curve analysis

• This approach reflects the time ordering. Model
  chosen to match biological situation and pattern
  of data.
• Analysis is done in two stages
• 1. Examine and summarise the pattern of change
  (growth) over time for each subject. Use
  biological models E.g.
• growth rate (wt(week4) - wt(week0))/4
• 2.Analyse the derived parameter (growth rate)
  across subjects.

12
Quadratic model for growth curves

• Growth model Weight a b. week c. week2
• a intercept, b slope, c curvature
• These quantities convey independent pieces of
  information about the data, i.e. analysis of the
  slope does not affect results about the
  curvature.
• Strategy Break up our analysis of growth curves
  to analysis of the slope and curvature values.
  Individually, these give rise to univariate
  analyses.
• i.e., we will try to distinguish the growth
  curves of rats by examining how their slopes and
  curvatures differ between the groups.

13
Computing individual slopes and curvatures
Data for Rat 1 Week 0 1 2 3
4 Weight 57 86 114 139 172
Want to fit model Weight a b. week c. week2
Method Use regression Do it in SAS, (Weight
week week2 ) or Trick Use algebraic formulae
for regression b (2.wk4 wk3 - wk1 - 2wk0)/10
28.3 c (2.wk4 wk3 - 2.wk2 - wk1 2wk0)/14
0.357
14
Fit for Rat 1 , group 1
Weight 57.7 28.3 week 0.357 week2
15
Fit for Rat 6, group 3
Weight 52.5 21.5week -1.357week2
16
Comparison of slopes and curvatures
17
Confirmatory ANOVAs

• Same conclusion as boxplot comparison
• Substantial difference across groups in both
  slope and curvature parameters

18
Growth models for groups

• Use group means for slopes and curvatures
• ControlWeight 104.7 26.5(week-2)
  .
  0.6(week -2)2
• ThyroxinWeight 104.2 27.6(week-2)
  .
  1.4(week -2)2
• ThiouracilWeight 94.4 17.1(week-2)
  .
  -1.3(week -2)2
• Clearly shows the difference between Thiouracil
  and the other two treatments.
• Form of the model should reflect biological
  situation and pattern of data. E.g. Straight
  lines, sine curves, logistic curves etc. may be
  appropriate in other situations.