Statistics 303

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Statistics 303

• Chapter 12
• ANOVA

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ANOVA Comparing Several Means

• The statistical methodology for comparing several
  means is called analysis of variance, or ANOVA.
• In this case one variable is categorical.
• This variable forms the groups to be compared.
• The response variable is numeric.
• This methodology is the extension of comparing
  two means.

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ANOVA Comparing Several Means

• Example
• An experimenter is interested in the effect of
  sleep deprivation on manual dexterity.
  Thirty-two subjects are selected and randomly
  divided into four groups of size 8.
• After differing amount of sleep deprivation, all
  subjects are given a series of tasks to perform,
  each of which requires a high amount of manual
  dexterity. A score form 0 to 10 is obtained for
  each subject. Test at the a 0.05 level the
  hypothesis that the degree of sleep deprivation
  has no effect on manual dexterity.

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ANOVA Comparing Several Means

• Information Given

Sample size N 32
If H0 is true, there is no difference among the
group means, the two variations will be almost
equal. This is the idea of ANOVA.
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ANOVA Comparing Several Means

• Information Given

Variation Within Groups
Average Within Group Variation (MSE)
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ANOVA Comparing Several Means

• Information Given

Average Between Group Variation (MSG)
Variation Between Groups
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ANOVA Comparing Several Means

• The F-statistic
• where MSG mean squares group, MSE mean
  squares error.
• This compares the variation between groups to
  the variation within groups. This is what gives
  it the name Analysis of Variance.
• The degrees of freedom for the F test are
• df1 I 1 (number of groups minus 1)
• df2 N I (total sample size minus number of
  groups).
• Q Under null hypothesis, what should F
  approximately equal to?
• Under the alternative?

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ANOVA Table
is the proportion of the total variation
explained by the difference in means
We can get this table by SPSS.
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Assumptions for ANOVA

• Suppose we have I populations,
• 1. Each of the I population or group
  distributions is normal.
• -check with a normal quantile (Q-Q) plot of
  each group
• 2. These distributions have identical
  variances
• -check if largest std. dev. is gt 2 times
  smallest std. dev.
• 3. Each of the I samples is a random sample.
• 4. Each of the I samples is selected
  independently of one another.

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ANOVA Comparing Several Means

• Step 1 The null hypothesis for comparing several
  means is

where I is the number of populations to be
compared

• Step 2 The alternative hypothesis (step 2) is

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ANOVA Comparing Several Means

• Step 3 State the significance level
• Step 4 Calculate the F-statistic
• Step 5 Find the P-value
• The P-value for an ANOVA F-test is always
  one-sided.
• The P-value is
• where df1 I 1 and df2 N I.

F-distribution
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ANOVA Comparing Several Means

• Step 6. Reject or fail to reject H0 based on the
  P-value.
• If the P-value is less than or equal to a, reject
  H0.
• It the P-value is greater than a, fail to reject
  H0.
• Step 7. State your conclusion.
• If H0 is rejected, There is significant
  statistical evidence that at least one of the
  population means is different from another.
• If H0 is not rejected, There is not significant
  statistical evidence that at least one of the
  population means is different from another.

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ANOVA Comparing Several Means

• Go back to Example
• Categorical sleep deprivation (4 levels).
• Numeric performance in dexterity
• of groups 4 (I 4)
• total sample size 32 (N 32)
• 8 for each group (ni 8)
• Test at the a 0.05 level the hypothesis that
  the degree of sleep deprivation has no effect on
  manual dexterity.

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Side by Side Boxplots
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Check assumptions

• Normality normal quantile plots

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Check assumptions
2. Equal variances
3. Each of the I samples is a random sample.
4. Each of the I samples is selected
independently of one another.
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ANOVA Comparing Several Means

• Step 1 The null hypothesis is

Step 2 The alternative hypothesis is
Step 3 The significance level is a 0.05
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ANOVA Comparing Several Means

• Step 4 Calculate the F-statistic

MSG and MSE are found in the ANOVA table when the
analysis is run on the computer
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ANOVA Comparing Several Means

• Step 5 Find the P-value
• The P-value is

where df1 I 1 (number of groups minus 1)
4 1 3 and df2 N I (total sample size
minus I) 32 4 28
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ANOVA Comparing Several Means

• Step 6. Reject or fail to reject H0 based on the
  P-value.
• Because the P-value is less than a 0.05, reject
  H0.
• Step 7. State your conclusion.
• There is significant statistical evidence that
  at least one of the population means is different
  from another.

An additional test will tell us which means are
different from the others.
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ANOVA

• Notice that
• (Sum of Squares Between Groups) (Sum of
  Squares Within Groups) (Sum of Squares Total)
• 71.928 18.789 90.717
• Also notice that the Mean Square Column is
  Calculated by dividing the Sum of Squares by the
  associated Degrees of Freedom (df).
• Ex. 71.928 / 3 23.976 for Between Groups
• F (MS Between Groups) / (MS Within Groups)
  MSG/MSE
• 23.976 / .671 35.730