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Analysis of Variance (ANOVA)

Analysis of Variance (ANOVA). Randomized Block Design. First, let’s consider the assumptions (Handouts: Assumptions Handout).

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Analysis of Variance (ANOVA)

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  1. Analysis of Variance (ANOVA) Randomized Block Design

  2. First, let’s consider the assumptions (Handouts: Assumptions Handout) When using one-way analysis of variance, the process of looking up the resulting value of F in an F-distribution table, is reliable under the following assumptions: • The values in each of the groups (as a whole) follow the normal curve, • with possibly different population averages (though the null hypothesis is that all of the group averages are equal) and • equal population variances.

  3. Normal Distribution • While ANOVA is fairly robust with respect to normality, a highly skewed distribution may have an impact on the validity of the inferences derived from ANOVA. • Check this with a histogram, stem-and-leaf display, or normal probability plot for the response, y, corresponding to each treatment.

  4. Equal Population Variances • Book lists a few formal tests of homogeneity of variances available (p. 635, 636) • We can approximately check this by using as bootstrap estimates the sample standard deviations. • In practice, statisticians feel safe in using ANOVA if the largest sample SD is not larger than twice the smallest.

  5. Randomized (Complete) Block Design • Sample Layout: Each horizontal row represents a block. There are 4 blocks (I-IV) and 4 treatments (A-D) in this example. Block I A B C D Block II D A B C Block III B D C A Block IV C A B D

  6. Randomized Block Designs (Formulas: p. 575) Total SS = Treatment SS + Block SS + Error SS SS(Total) = SST + SSB + SSE In Minitab, we will use the Two-Way ANOVA option or the General Linear Model option.

  7. We will have two F values • For testing treatments: • For testing blocks: These values are equivalent to the values in our nested F

  8. Why test blocks? • If we want to determine if blocking was effective in removing extraneous sources of variation • Is there evidence of difference among block means? • If there are no differences between block means, we lose information by blocking • Blocking reduces the number of degrees of freedom associated with the estimated variance of the model • BUT it is okay to use the block design if you believe they may have an effect (p. 574)

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