Analysis of Variance ANOVA

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Analysis of Variance (ANOVA)

• Engineering Experimental Design
• Valerie L. Young

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Example Problem

• As a chicken nutritionist, you need reliable
  measures of lasalocid sodium levels in chicken
  feed, so you decide to test the analytical labs.
• You send a sample of feed containing 85 mg/kg
  lasalocid sodium to each of three independent
  labs, and get back the results at left.
• Do the analyses agree with each other?
• Do the analyses agree with the true value?

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Graph it First
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Interpret the Graph

• Laboratory B results appear to be higher
• All labs results overlap with each other
• All labs results overlap with the true value (85
  mg/kg)
• Based on the graph alone, I cannot tell whether
  any of the labs differ significantly from each
  other or from the true value.

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ANOVA What does it tell me?

• ANOVA Analysis of Variance
• ANOVA will tell me whether I have sufficient
  evidence to say that measurements from at least
  one lab differ significantly from at least one
  other.
• It will not tell me which ones differ, or how
  many differ.

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ANOVA vs. t-test

• ANOVA is like a t-test among multiple data sets
  simultaneously
• t-tests can only be done between two data sets,
  or between one set and a true value
• ANOVA uses the F distribution instead of the
  t-distribution
• ANOVA assumes that all of the data sets have
  equal variances
• Use caution on close decisions if they dont
• Consult a professional

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ANOVA a Hypothesis Test

• H0 There is no significant difference among the
  results provided by these three laboratories.
• H1 At least one of these laboratories provides
  results significantly different from at least one
  other.

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Excel and ANOVA

• Tools gt Data Analysis gt
• ANOVA Single Factor
• ANOVA Two-Factor with Replication
• ANOVA Two-Factor without Replication
• So how many factors do we have here?
• Factor Independent Variable
• The I.V. here is Laboratory
• We have a SINGLE factor with THREE levels (A,B,C)

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ANOVA Results
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Focus on ANOVA Table
F MSTr / MSE 59.233 / 6.2407
Treatments
MSTr
Error
MSE

• F ratio of variability (between groups) due to
  treatment to variability (within groups) due to
  random error
• P probability of getting an F value at least
  this large if these were 3 sets of 10
  measurements from the same population

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Decision Based on ANOVA
F MSTr / MSE 59.233 / 6.2407
Treatments
MSTr
Error
MSE

• F gt F critical
• Reject H0
• P lt 0.05 (chosen significance level)
• Reject H0
• If H0 were true, the probability of getting 3
  sets of data like this is less than 0.1

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Where Does the Difference Lie?

• ANOVA only shows that a difference exists
• To find the difference, consider
• Graphical representation
• Mean with confidence limits
• Effects plot
• Analysis of Means (ANOM)
• For one factor at multiple levels, ANOM is a
  better technique than ANOVA
• For multiple factors, ANOVA is required. We are
  showing ANOVA for one factor so you can better
  understand it when it is properly applied for
  multiple factors.

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Descriptive (Summary) Statistics
Note I have not used the pooled variance (MSe)
to calculate the confidence limits. Since the
analysis was performed by different labs, I
decided to allow for them to have different
uncertainties.
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Focus on Descriptive Statistics
Mean of 10 replicate measurements by laboratory
A. This sample mean is an estimate of the true
concentration.
Standard deviation of 10 replicate measurements
by laboratory A. This sample std dev is an
estimate of the true std dev. for lab A
95 confidence interval on the mean.
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Graphical Representation
95 CI on mean
Mean
Target Value
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Interpretation of Graph

• Results from Lab B differ significantly from Lab
  C and from the known value (85 mg/kg)
• Results from Lab A and Lab C agree with one
  another and with the known value
• Lab B analysis is unacceptable
• This is the appropriate conclusion even though
  analyses by Labs A and B are apparently not
  significantly different

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Example Lead Contamination

• Lead was banned as a gasoline additive in 1978.
  Soil lead levels were monitored at 80 randomly
  chosen locations in the United States to
  determine whether the lead ban resulted in
  significant reduction of environmental lead
  levels over time. The following results were
  obtained.

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ANOVA Lead Contamination

• You must have all of the original data to
  calculate the values for SSyear and SStotal .
• F-crit can be found on Table B.7, last row (?),
  3rd column (2).

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CRD Completely Randomized Design

• An experiment with multiple independent groups of
  data is a Completely Randomized Design (CRD)
• ANOVA is like a t-test among multiple independent
  groups of data
• If specific data points between groups are linked
  (like in a paired t-test) then it is not
  single-factor ANOVA
• Linking data between groups with some other
  factor is called blocking