4' Analysis of Variance III Anlisis de Varianza III

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4. Analysis of Variance III Análisis de
Varianza III

• Profesor Simon Wilson
• Departamento de Estadística y Econometría

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Testing that more than 2 means are equal /
Contrastar que las medias son iguales

• In the last section, we tested for 2 means to be
  equal
• In our example, this does not really tell us if
  all 3 means together are equal (only tells us
  about each pair of means)
• The method of Analysis of Variance tells us how
  to do this
• In general, we have I levels of the factor so
• H0 m1 m2 ... mI m (null hypothesis /
  hipótesis nula)
• H1 not all the mi are equal

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Testing the Equality of Means (1)

• The logic of the test is easy (recall 3 plots
  from Section 2)
• If the differences between the group means are
  large relative to the experimental error s, we
  conclude that the means are different
• Si las diferencias entre las medias de grupo son
  grandes con relación a la variabilidad
  experimental s, concluimos que las medias son
  diferentes

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Testing the Equality of Means (2)

• Three groups with means 20, 22 and 24. First,
  the error s is large relative to the difference
  in group means
• o observations
• x group means
• Here we accept H0

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Testing the Equality of Means (3)

• Second, the error s is small relative to the
  difference in group means
• o observations
• x group means
• Here we reject H0

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Testing the Equality of Means (4)

• Remember, we estimate the experimental error s2
  by
• The variability between the groups can be
  measured using the sum of squares / suma de
  cuadrados

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A diversion Decomposition of the Variance (1)

• Suppose H0 is true (all the means equal)
• Then our estimate of this mean would be the mean
  of all observations together
• Our estimate of the variance would use the sum of
  squares
• This is called the total variability /
  variabilidad total (TV)

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Decomposition of the Variance (2)

• Now we can write
• And therefore

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Decomposition of the Variance (3)

• Now! clearly,
• (Why?)
• And so we have

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Testing the Equality of Means (5)

• Recall that is the sum of squares
• used to estimate s2 when we think that the group
  means are different.
• This is called the unexplained variability /
  variabilidad no explicada (UV) of the data or
  unexplained variance (i.e. difference between the
  observations and their estimated expected value
  in the group)

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Testing the Equality of Means (6)

• The other part, measures the
• variability between the means of each groups.
• We call this term the explained variability /
  variabilidad explicada (EV)
• If there are large differences in the means of
  each group then EV is large.
• Note that TV EV UV

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Testing the Equality of Means (7)

• Further, the following is true
• If H0 is true then the explained variability is
  0.
• If H1 is very true -- large differences
  between group means -- then the explained
  variability is large relative to the unexplained
  variability (go back to example)

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Testing the Equality of Means (8)

• So look at the ratio / cociente of explained to
  unexplained variability
• If this ratio is small then means are more or
  less equal (accept H0)
• If this ratio is large then means are different
  (accept H1)

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The ANOVA Table (1)

• All this information about variances is usually
  put into a table called the ANalysis Of VAriance
  (ANOVA) table
• This table is on the next page
• It shows the three types of variability and the
  degrees of freedom of each
• Finally, it shows the estimate of the variance
  (sum of squares divided by the degrees of freedom)

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The ANOVA Table (2)
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Distributions of EV and UV

• When H0 is true
• This helps us to say when F is small or large

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The F test (1)

• We can show that the following ratio of the EV
  and UV (ratio of two c2 distributions)
• Has a distribution called the F-distribution with
  I-1 and n-I degrees of freedom

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The F test (2)

• If the value of F is large according to the
  F-distribution then we reject H0.
• We therefore conduct the test as follows
• Compute F
• Decide on the signifance level a
• Look up (in tables) the value Fa I-1,n-I ,such
  that
• P(F gt Fa I-1,n-I ) a
• If F gt Fa I-1,n-I then reject H0

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The F test / contraste de la F for the whisky
data (1)
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The F test for the whisky data (2)

• First, we calculate everything in the ANOVA
  table.
• Using the table of information that we have
  already
• Then the EV is

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The F test for the whisky data (3)

• For the UV
• We already have calculated
• So

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The F test for the whisky data (4)
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The F test for the whisky data (5)

• So F 57.755 / 9.3556 6.173
• If H0 is true, F has the F distribution with 2
  and 18 degrees of freedom.
• Decide to use a 5 level of significance.
• F0.05 2,18 3.55 (from tables)
• Since F gt 3.55, so we reject H0
• Conclude that the means are different

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Class Example

• Sales of a fast food company have increased in
  the last year. A director of the company wants
  to know if the increase has been the same in the
  4 regions of the country (North, South, East,
  West).
• Five establishments from each region are randomly
  chosen. The percentage increase in sales is
  observed.
• Construct the ANOVA table and test to see if
  there is a difference in mean percentage increase
  in sales between the 4 regions (use 5 level of
  significance).

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Class Example Data
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ANOVA Table
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The F test

• F _________
• If H0 is true, F has the F distribution with ___
  and ___ degrees of freedom.
• Decide to use a 5 level of significance.
• F0.05 __,__ ______ (from tables)
• Conclude that