Analysis of Variance

1
Analysis of Variance

• (read together with document VARIANCE.doc)
• This level of statistical analysis of data
  follows the basic error propagation and
  uncertainty analysis (discussed in
  ERR_PROP.ppt)

2
Secondary Treatment

• analysis of variance (ANOVA) extract information
  about variability
• Why?
• how much of the variation is due to
• assignable causes
• random effects
• need to quantify degree of confidence in cause of
  variation

3
Analysis of Variance

• How?
• Must do several measurements
• t-statistic quantifies distance from mean for
  certain degree of confidence that data will be in
  interval

4
Normal Distribution

• distribution of events for large samples

5
t-statistic

• smaller numbers of samples

6
t-statistic (cont.)

• uncertainty in mean decreases as number of
  measurements (e.g. series of temperature
  measurements)

7
Analysis of Variance (ANOVA)

• from several repeats of an observation under same
  experimental conditions
• but
• variance may change with operating conditions
• may not have time to do repeat experiments under
  all conditions
• equipment may break down, so repeat observations
  may only be available for some conditions.

8
One-Way Analysis of Variance

• one assignable cause varies at a time (e.g.
  flowrate)
• no repeats give information on variance between
  operating conditions, not within an operating
  condition due to uncertainties / errors
• no repeats mean no error/variance estimate

9
One-Way Analysis of Variance (cont.)

• worked examples given (in VARIANCE.doc) for
• measurement uncertainty
• no measurement uncertainty
• mean square (error) variance within conditions
  (estimate of measurement uncertainty)

10
One-Way Analysis of Variance (cont.)

• measurement uncertainty is not the only component
  of error variance (also systematic errors,
  unassigned causes)

11
Two-Way Analysis of Variance

• two assignable causes vary (e.g. steam flowrate,
  reflux ratio)
• again, repeat experiments required to get
  estimate of variance

12
Two-Way Analysis of Variance (cont.)

• worked examples given (in VARIANCE.doc) for
• no repeats (no estimate of error variance)
• repeats (estimate of error variance)
• repeats with no uncertainty/error
• one-way ANOVA of two-way situation still gives
  estimate of error variance
• unbalanced ANOVA

13
Unbalanced ANOVA

• if equipment breaks down, if time runs out, may
  only have some repeat information
• e.g.

14
Unbalanced ANOVA (cont.)

• estimate of error variance not as good as from
  complete, balanced, design

15
Relative and Absolute Errors

• Absolute error e.g. 0.3 - 0.1oC
• appropriate if error / uncertainty independent of
  operating condition
• Relative error e.g 0.3 - 10 kg/s
• error / uncertainty dependent on operating
  condition
• can be analysed in same way as absolute errors by
  using logarithms

16
Concluding Remarks 1

• always possible to estimate errors and
  uncertainties associated with measurements (e.g.
  0.5oC in thermometer with 1oC divisions)
• can always assess how these errors combine

17
Concluding Remarks 2

• analysis of variance (ANOVA) gives more detailed
  information
• need some repeat experiments
• can deal with partial information (e.g. if
  equipment breaks down)
• absolute and relative errors (relative error
  analysis using logarithms)