Analysis of Covariance

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Analysis of Covariance

• Comparison of Head-Lengths for Trout from 2 Lakes
• Source C. McC. Mottley (1941). The Covariance
  Method of Comparing the Head-Lengths of Trout
  From Different Environments, Copeia, Vol.1941,
  3, pp.154-159.

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Analysis of Covariance

• Goal To Compare 2 or More Group/Treatment Means
  for One Variable, after controlling for one or
  more concomitant variables/Covariates
• Procedure Fit a Regression Model Relating
  Response to Group (Using Dummy Variables) and
  Covariate(s)
• Adjusted Means Fitted Values from regression for
  each group evaluated at common value(s) of
  covariate(s), typically at the mean value(s)
• Treatments can be combinations of factors in a
  Factorial Structure
• Model

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Example Fale Trout Heads from 2 Lakes
(Environments)

• Units Female Fish Sampled from Lakes
• Response Variable Head Length (mm)
• Groups Environments (Kootenay Lake, Wilson Lake)
• Covariate Standard-Length (mm)
• Samples n1 50 from Kootenay, n2 10 from
  Wilson
• Transformation Base10 logarithms of Head-Length
  and Fish-Length Taken for Biological Reasons
• Summary Statistics

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T-test Ignoring Covariate
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T-test Ignoring Covariate
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Adjusting for Body Length
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Adjusted Means

• Goal Compare Means Head Lengths after adjusting
  for Body Lengths
• Start with the Unadjusted Mean, and Subtract off
  the product of the slope from the regression and
  the difference between the group mean and the
  overall mean for the covariate. Effect of
  adjustment (assuming positive slope)
• Adjust down for groups with high levels of
  covariate
• Adjust up for groups with low levels of covariate

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Trout Head-Length Data
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Testing for Lake Effect, Controlling for Body
Length

• Null Hypothesis No difference in mean head
  lengths for 2 lakes, controlling for Body Length
  (b1b2)
• Alternative Hypothesis Lake Means Differ (b1?b2)
• Step 1 Fit Full Model Under HA and obtain
  SSE(F), dfE(F)
• Step 2 Place Restriction under H0 and fit model
  forcing b1b2 and obtain SSE(R) and dfE(R)

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Trout Head Size Data
ANOVA for Full Model
ANOVA for Reduced Model