Analysis of variance ANOVA

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Analysis of variance (ANOVA)

• By Mona Tarek
• Assistant Researcher
• Department of Pharmaceutics
• Statistics
• Supervised by Dr. Amal Fatany
• King Saud University

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• It compares variance (or means) of test groups in
  relation to the associated error.
• Useful in
• Designs for tests comparing more than 2 groups.
• Separation of variation due the treatment from
  variation due to experimental error. (within
  group variation)

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One way ANOVA (completely randomized design)

• This is the ANOVA form used for
• Comparing means from two or more groups.
• Parallel groups design
• It is the multiple analogue of the two
  independent t test (unpaired data)

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• But is more accurate than t-test as

T-test
ANOVA
It gives the difference due to treatment only it
removes any errors ( as variability among
experimental units other experiment error
sources among a single group.) E.g. manufacture,
personal error, and time factor.  
It doesnt differentiate between difference in
means due to treatment and due to error.
More complex calculations that lead to F-value
that is then compared to a tabulated F one.
A t-value is calculated and compared to a
tabulated one.
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• The design of ANOVA ( randomized block design)
• The experimental units are divided into t no.
  of groups that equals no, of applied
  treatments.
• Total no. of observations (experiment units N) is
  conveniently chosen to be divisible by t i.e.
  N/t integral no. that is the no. of units in
  each group.( ie the number of units in each group
  is the same)

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• F-ratio relates (variance due treatment /variance
  due error)
• To calculate the F-ratio, we calculate
• BSS the between sum of squares ----? represent
  the actual difference among the tested treatment
  ---?the lager value numerator
• WSS the within sum of squares ------? represent
  the difference within a single treatment group.
  i.e represent error due to variability among
  experiment units denominator
•  
• As the value of WSS ? F
• meaning more significant difference is declared
  with more confidence due to the treatments not
  error.

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• Example Groups of three subjects were
  given 1 of 10 food regimen showed the weight
  gain in kgs in the following table. These are
  unpaired data its a completely randomized
  experiment. There are only two sources of
  variation the variation between the regimens
  the variation within regimens. Are all the food
  regimens the same?

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Total Number of observations (N) 30 N.B. n N/t
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• First construct the null hypothesis (at p0.05)
• There is no difference between the two
• regimens
• Then construct another table

(?X)²/N is called correction term
160.54/9 17.81
17.81/2.178.22
43.33
43.33/202.17
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• Calculated F-value 8.22
• Then we look at a tabulated F-value at
• DF1 9 (between regimens)
• DF2 20 (within regimens)
• Tabulated F-value 2.39 at p-value 0.05
• ( There are different f-tables depending on the
  p-value we are working at)
• Calculated value is larger than the tabulated
  value at p0.05
• . Null hypothesis is rejected
• i.e. there is a significant difference between
  the regimens.

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