The nature and purpose of analysis of variance 

• Analysis of variance is a generalization of the t-test
  • The t-test is used to test the null hypothesis: µ₁ =µ₂
  • The t-test is appropriate for only two groups 
  • Analysis of variance, and the associated F-test, can test for differences between any number of means for categorized variables.
  •  Formally, the null hypothesis is µ₁ = µ₂ = µ₃ = µ₃ ... = µ_n 
• Example: CLASS by grade on 2/3 Examination in C10 Statistics
  • T-Test results for a previous class (not this year's class)
    • Mean score for undergraduates = 22.3
    • Mean score for graduates = 26.7
  • T-Test showed that this was significant far beyond the .05 level using a one-tailed test.
• Analysis of variance gives a general test for effect of class on grade 

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The underlying structural model of Analysis of variance 

• Each observation in the combined distribution represents a linear combination of components: 

  X_ij = µ_grand + a_j + e_{i j}

  Where:  a_j = effects of group and e_{i j} = effects of random error 

• If the treatment or "group" effect is 0,
  • then observations will vary around the mean depending on e_{i j} (errors around the mean)
  • these errors are assumed to be random, normally distributed, and sum to 0.
•  This model--which accounts for individual observations in the distribution--can be expressed in terms of the total variation of individual observations from the grand mean. 
•  Partitioning the total variation (also known as sum of squares, SS):

• Calculation of sums of squares:

(individual score - grand mean) = total sum of squares
	(individual score - group mean) = within group sum of squares
		(group mean - grand mean) = between group sum of squares

• If the group means were really different, then the BSS (explained SS) would be large relative to the WSS (unexplained SS).
• If the group means were not much different, the WSS would be large relative to the BSS.

 The uses of BSS and WSS

• They are measures of the source of the total variation, and we will eventually use them to measure the strength of the relationship between the independent and dependent variable.
• But they have another purpose: to test the hypothesis that the k groups are really random samples from equivalent populations
  • If they are really drawn from the same population, then

    µ₁ = µ₂ = µ₃ = µ₄ ... = µ_n -- as we hypothesized 

  • But if they really are drawn from equivalent populations, then also

    1 = 2 = 3 = 4 .... n 

• To test if they are from equivalent populations, we actually test for equality of estimates of population variance. 

BSS and WSS as tests of population variance 

• Each of these provide the basis for two independent estimates of the population variance 
  • One estimates the population variance based on the variance within each of the samples
  • The other estimates the population variance based on variance between the sample means 
• The F test is simply a ratio between these two estimates of the population variance: 

• If this ratio is small, then the two estimates agree closely, and we conclude that the groups represent random samples from equivalent populations: i.e., same means and variances 

How do we get these estimates of the population variances? 

• They must be divided by the appropriate degrees of freedom
  • BSS / k-1 = between groups mean square (read "mean square" as the mean of the sum of squares)
  • WSS / N-k = within groups mean square