ANOVA

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Lesson 2
2
ANOVA

• Analysis of Variance

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One Way ANOVA

• One variable is measured for many different
  treatments (population)_
• Null Hypothesis all population means are equal
• Alternative Hypothesis not all population means
  are equal (i.e. at least one is different)
• If variance is small (the sample means are close)
  and the null hypothesis is true
• If variance is large (the sample means are far
  apart), the alternative hypothesis is true

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Example 1

• Does the weight class of a car make a difference
  in the number of head injuries sustained by crash
  test dummies?
• A random sample of 5 compact, midsize, and
  full-size cars was obtained and the head injury
  count for these vehicles was recorded below
  Calculate the mean and variance for each sample.
  Calculate the overall mean.

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CAR HEAD INJURY COUNT
Compact
Chevy Cavalier 643
Dodge Neon 655
Mazda 626 442
Pontiac Sunfire 514
Subaru Legacy 525
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• Midsize
• Chevy Camaro 469
• Dodge Intrepid 727
• Ford Mustang 525
• Honda Accord 454
• Volvo S70 259

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• Full-Size
• Audi A8 384
• Cadillac Deville 656
• Ford Crown Vic 602
• Olds Aurora 687
• Pontiac Bonneville 360

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• Mean 1 head injuries for compact cars
• Mean 2 head injuries for midsize cars
• Mean 3 head injuries for full size cars
• Null hypothesis means are all equal
• Alternative hypothesis at least one mean is
  different

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MSTR

• Mean square due to treatment
• Estimate of the variance BETWEEN the treatments
  (populations)
• A good estimate of the variance ONLY when the
  null hypothesis is TRUE
• If the null hypothesis is FALSE, MSTR
  overestimates the variance

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MSTR Formula

• Find the difference between each sample mean and
  the overall mean square this number
• Multiply result of 1st step by n
• Sum these numbers
• Divide by the degrees of freedom
• Steps 1 through 3 are SSTR (sum of the squares
  due to treatment) and the numerator
• Step 4 is k-1 and is the denominator

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• Compact
• Midsize
• Fullsize

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MSE

• Mean Square Due to Error
• Within treatment estimate of the variance
• Average of the individual population variances
• Unaffected by whether the null hypothesis is true
  or not
• Provides an UNBIASED estimate of the population
  variance

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• MSE Average of the sample variances

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F-test

• F is the test statistic for the equality of k
  population means
• F MSTR / MSE has k-1 df in numerator and n-k df
  in denominator
• 6405 / 20096.023 0.319
• n.b. If null hypothesis is T, the value of MSTR
  / MSE should appear to have been selected from
  this F distribution
• If null hypothesis is F, MSTR / MSE will be
  inflated because MSTR overestimates variance

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• Calculate p-value P(F gt F observed) and write
  conclusion
• Summarize in an ANOVA table

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• SOURCE DF SS MS F P
• Factor
• Error
• Total
• SST SSTR SSE