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Analysis of Variance (ANOVA). Agenda. Lab Stuff Questions about Chi-Square? Intro to Analysis of Variance (ANOVA). This Thursday: Lab 4. Final lab will be distributed on Thursday Very similar to lab 3, but with different data

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  • Lab Stuff
  • Questions about Chi-Square?
  • Intro to Analysis of Variance (ANOVA)
this thursday lab 4
This Thursday: Lab 4
  • Final lab will be distributed on Thursday
    • Very similar to lab 3, but with different data
    • You will be expected to find appropriate variables for three major tests (correlation, t-test, chi-square test of independence)
    • You will be expected to interpret the findings from each test (one short paragraph per test).
  • We will use the first 15 minutes of class to return lab 3 and discuss common issues and questions
analysis of variance
Analysis of Variance
  • In its simplest form, it is used to compare means for three or more categories.
    • Example:
      • Income (metric) and Marital Status (many categories)
  • Relies on the F-distribution
    • Just like the t-distribution and chi-square distribution, there are several sampling distributions for each possible value of df.
what is anova
What is ANOVA?
  • If we have a categorical variable with 3+ categories and a metric/scale variable, we could just run 3 t-tests.
    • One problem is that the 3 tests would not be independent of each other (i.e., all of the information is known).
    • As number of comparisons grow, likelihood of some differences are expected– but do not necessarily indicate an overall difference.
  • A better approach: compare the variability between groups (treatment variance + error) to the variability within the groups (error)
the f ratio
The F-ratio
  • MS = mean square
  • bg = between groups
  • wg = within groups
  • The numerator and denominator have their own degrees of freedom
  • df = # of categories – 1 (k-1)
interpreting the f ratio
Interpreting the F-ratio
  • Generally, an f-ratio is a measure of how different the means are relative to the variability within each sample
  • Larger values  greater likelihood that the difference between means are not just due to chance alone
null hypothesis in anova
Null Hypothesis in ANOVA
  • If there is no difference between the means, then the between-group sum of squares should = the within-group sum of squares.
f distribution
  • A right-skewed distribution
  • It is a ratio of two chi-square distributions
f distribution12
  • F-test for ANOVA is a one-tailed test.
visual anova and f ratio
Visual ANOVA and f-ratio

anova and t test
ANOVA and t-test
  • How do we know where the differences exist once we know that we have an overall difference between groups?
    • t-tests become important after an ANOVA so that we can find out which pairs are significantly different (post-hoc tests).
    • Certain ‘corrections’ can be applied to such post-hoc t-tests so that we account for multiple comparisons (e.g., Bonferroni correction, which divides p-value by the number of comparisons being made)
    • There are many means comparisons test available (Tukey, Sidak, Bonferroni, etc). All are basically modified means comparisons.
logic of the anova
Logic of the ANOVA
  • Conceptual Intro to ANOVA
  • Class Example:
    • GSS96_small.dta