• The Big Idea behind ANalysis Of VAriance (ANOVA)
  

• Calculating an ANOVA
• Some Examples
  

• Same logic as before:
• Assume that  is true
• State an expected result based on this assumption
• compute a sample statistic (T-Test, F-Test, etc)
• Treat the statistic as a score in some known sampling distribution
• If the test statistic falls within the rejection region, reject  , otherwise, retain 
  

• What’s new with ANOVA?
• The Basic logic does not change between mean difference tests and ANOVA
• The difference is that they use different statistical tests
• Rather than using means to evaluate the , the F-Test is based on the ratio of the variances
  

• Essentially, ANOVA entails calculating two estimates of the population variance:

1. Variance among treatment means
• For any given treatment, the variance of the scores that compose that treatment could be estimated and used as a measure of the underlying population variance
  
2. Variance among multiple groups from the same underlying population
• Assume homogeneity of variance  
• Thus, multiple groups represent multiple estimates of the common population variance
  

• If the two estimates agree, we have no reason to reject the 
• If they disagree, conclude that the underlying treatment changed the second estimate

• To calculate the F-Test, you must distinguish between two types of Variance
1. Within-group Variance
• Unexplained variance
• Variance caused by individual differences, errors in measurement, etc.
• Sometimes called error variance
2. Between-group Variance
• Same sources as within-group variance (individual differences, measurement error, etc.)
• But also includes any variance due to group differences (i.e. the experimental manipulation)
  

• F = The ratio of 2 independent variance estimates:

• Basically, you divide the first variance estimate by the second variance estimate:

• If  is true:

• If  is false:

• N  = number of total data values.
• ni = number of data values in ith group
• t = number of groups (or “factor levels”)
• yij = jth data value in ith group
• Yi. = sum of data values in ith group
• Y.. = sum of all data values
  

• If Fobtained > Fcritical, reject 
• If Fobtained < Fcritical, fail to reject   
  

• N-1 = (t-1) + (N-t)
• SS(Total) = SS(Between) + SS(Error)
• MSB = SS(Between)/(t-1) 
• MSE = SS(Error)/(N-t)
• The P-value comes from F-distribution with t-1 numerator and N-t denominator d.f.

• ANOVA = Division of the overall variability in data values in order to compare means.
• Overall (or “total”) variability is divided into two components:
• The variability “between” groups, and
• The variability “within” groups
• Summarized in an “ANOVA” table
  

• Data:

• Comparison of Five Tire Brands Stopping Distance at 60 mph:

• Sample Descriptive Statistics

• Hypotheses:
• The null hypothesis is that the group population means are all the same. That is:
• 
• The alternative hypothesis is that at least one group population mean differs from the others.  That is:
• 
  

• Results: Analysis of Variance for comparing all 5 brands
  

• The P-value (significance) is small (0.000, to three decimal places) so reject the null hypothesis.  There is sufficient evidence to conclude that at least one brand is different from the others.

• Research Question: Does learning method affect student’s exam scores?
• Consider 3 methods:
• Standard
• Osmosis
• Shock therapy
  

• Procedures: 
• Convince 15 students to take part.  Assign 5 students randomly to each method
• Wait eight weeks, then test students to get exam scores
  

• Results: One-way Analysis of Variance

• The P-value is pretty large so cannot reject the null hypothesis.  There is insufficient evidence to conclude that the average exam scores differ for the three learning methods.