ANOVA Analysis of Variance A Short Introduction by Brad Morantz

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ANOVAAnalysis of VarianceA Short Introduction
byBrad Morantz
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Example

• Have data for miss distances for 3 types of
  weather
• Clear and sunny
• Rain
• Fog
• The question
• Does the weather have effect on miss distances?
• Are the population means for each condition equal
  (within allowable tolerance)?

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The Problem

• There is variability in the system.
• Each time a missile is fired
• Many variables wind, brightness of sun,
  countermeasures, precipitation, much more
• Expect to get different values each time
• How can we tell if certain factors actually are
  causing a difference?
• Each repetition is different
• How do we know when some variance is too much
• How do we know if a certain factor is having an
  affect

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The Solution

• ANOVA ANalysis Of VAriance
• This is for a single dependent variable
• Can also be blocked to control other things,
  called noise reducing
• For example, to group flights by distance or over
  time
• Need more data/observations to do this
• Must be of comparable variance
• Can also be used for two factor test
• e.g velocity and weather

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Overview Purpose

• Null Hypothesis H0 is that all means are equal
  (population means as estimated by sample means)
• µ1 µ2 . . . . µn
• If we reject the null, it signifies that we could
  not prove that all are equal within allotted
  variability in system
• Does NOT mean that all are different
• Use another test (Tukeys HSD) to see which
  one(s) is/are different

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Components

• SSE is sum of square error
• SST is total sum of squares
• SST SSTreatment SSError
• MST SST/(k-1)
• MSE SSE/(N-k)
• F MST/MSE
• The test criteria to reject or fail to reject
  null hypothesis
• k number of treatments
• N number of observations

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Interpretation

• Program will usually give critical value
• Depending on specified allowed tail
• If F value is more than critical value
• Then reject null hypothesis
• If F value is less than critical value
• Then Fail to reject the null hypothesis
• Check to make sure that variances are
  approximately equal/close
• Look at graph of data
• is it approximately bell shaped?

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Blocked ANOVA

• Variance and noise reducing technique
• Use when there are more than one factor
• Ex. Day of the week has affect
• Ex. Type of launch aircraft
• Would still allow to see if weather had affect
• Requires more observations

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Manova

• Multivariate Analysis of Variance
• When there are two or more dependent variables
• Need specialized high power (read expensive)
  software

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Limitations

• Assumes that all of the data is approximately
  normally distributed
• Assumes that all of the data has about the same
  variance
• Is only concerned with the estimates of the
  population means that were calculated from the
  samples

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Example
These value are made up values
Note that the variances are close
F critical given as 3.35 and the calculated value
is 5.32 so we reject the null hypothesis that all
means are equal.
The P value is the probability of obtaining a
result at least as extreme as a given data point,
under the null hypothesis. Note that the P value
is .011 which indicates that if we had chosen an
alpha of .01, the null would not be rejected.