One-Factor Analysis of Variance

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One-Factor Analysis of Variance

• Between-Subjects Case

2
Illustration

• From an infinite, normally distributed population
  of scores with ? 10 and ? 2, two random
  samples (b1 and b2) of 10 scores each is drawn.
  (Think of the population as millions of students
  who have taken a math test that is scored out of
  20.) Use the sample information to estimate the
  population variance in two different ways.
  (Hint One of the ways utilizes the Central Limit
  Theorem.)

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Two Independent Ways of Estimating ?2

• Method 1
• for a given sample, s2 ? ?2
• to improve estimation, average the variances of
  all available samples
• Method 2
• from the Central Limit Theorem
• Since,
• Then use
• Method 1 and Method 2 should give roughly the
  same estimate of the population variance if all
  samples were drawn randomly from the same
  population.

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Results
As the scores are independent estimates the same
thing, ?2, they are roughly the same. Given this
fact, what should the ratio of the two values be
equal to? Answer Approximately 1.
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The Logic of Analysis of Variance

• Samples are drawn from two or more populations
  with unknown means but similar variances.
• Two variance estimates are taken.
• One is mean square error, or MSE
• The other is mean square treatment, or MST
• MST/MSE F (named in honor of Sir Ronald
  Fisher).
• If population means are the same, MST and MSE
  should be about the same because both are
  measuring sampling error. Thus F is about 1.
• If, however, population means differ, MST
  reflects sampling error and treatment, whereas
  MSE reflects sampling error alone. The ratio of
  MST to MSE, in the case of different means, would
  be larger than 1.
• Thus, think of F as equal to

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Exercise 4.1

• A researcher wants to know if coffee affects math
  performance. Twenty students of comparable math
  skill are randomly assigned to two groups.
  Students in one group are given coffee with
  caffeine and students in the other group are
  given coffee with no caffeine. One hour later,
  all take a math test (scored out of 20). Do the
  data below indicate that amount of caffeine
  affects math performance? (alpha .05)

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Preliminary Computations
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The ANOVA Summary Table
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Hypothesis Test

• Test assumptions
• Populations have normal distributions.
• Populations have equal variances.
• Samples are randomly selected.

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Relationship Between F and t

• In the two sample, between subjects case, either
  an independent t-test or an ANOVA can be
  performed
• The two-tailed t-test and the ANOVA lead to an
  identical decision regarding rejecting or not
  rejecting H0.
• F and t are related, as follows

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Exercise 4.2

• We wish to examine the effectiveness of four
  different types of filter for reducing pollution
  in water. Three randomly selected examples of
  each type are run through the test. Higher
  numbers represent more pollution. Do the filters
  differ in their ability to remove pollution?
  (alpha .05)

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SPSS Output for ANOVA Example