One-way ANOVA

one-way Analysis of Variance (ANOVA) extends the methods of the pooled-variance two-sample t tests to more than two groups
- if we wish to prove that $H_{0} : μ_{1} = μ_{2} = μ_{3} = μ_{k}$
assumptions
- the same as those of the pooled-variance two-sample t
1. The samples are simple random samples from the populations.
2. The populations are normally distributed.
3. The samples are independent.
4. The population variances are equal.
based on the partitioning of the total sum of squares
- split the total sum of squares into SS treatment and SS error
  - sum of squares
    - total variability in the response variable
    - the numerator of the sample variance formula if we were to pool all observations together
  - SS treatment
    - between groups
  - SS error
    - error within/inside each group
calculations
- variables
  - $k$ represents the number of groups.
  - $X_{i j}$ represents the $j$ th observation in the $i$ th group.
  - ${\bar{X}}_{i}$ represents the mean of the $i$ th group.
  - $\bar{X}$ represents the overall mean (or grand mean). $\bar{X} = \frac{Total of all observations}{Total number of observations}$
  - $s_{i}$ represents the standard deviation of the $i$ th group.- $n_{i}$ represent the number of observations in the $i$ th group.
  - $n = n_{1} + n_{2} + \dots + n_{k}$ represents the total number of observations.
  - - mean square = $\frac{sum of squares}{d f}$
    - $F = \frac{M S T}{M S E}$
      - ratio of $\frac{d f}{d f}$
  - $S S (Total) = \sum_{All obs} (X_{i j} - \bar{X})^{2}$
  - $S S T = \sum_{Groups} n_{i} (\bar{X_{i}} - \bar{X})^{2}$
  - $S S E = \sum_{Groups} (n_{i} - 1) s_{i}^{2}$
    - $M S E = \frac{(n_{1} - 1) s_{1}^{2} + (n_{2} - 1) s_{2}^{2} + \dots + (n_{k} - 1) s_{k}^{2}}{n_{1} + n_{2} + \dots + n_{k} - k}$
- $S S (T o t a l) = S S T + S S E$ ("between" + "within")
  - Total $d f$ = Treatment $d f$ + Error $d f$
  - $n - 1 = k - 1 + n - k$
- if $H_{0}$ is true
  - $F$ has an F distribution with $(k - 1, n - k)$
    - $F \sim F (k - 1, n - k)$
  - $M S T$ and $M S E$ are an unbiased estimator of $\sigma {#2}$
    - $F \to 1$
- if $H_{0}$ is false
  - $M S T$ will tend to be greater than $M S E$ and the $F$ statistic will tend to be greater than it would be under $H_{0}$
    - variation between the groups > variation inside the groups
- large values of the $F$ test statistic give evidence against $H_{0}$
  - observed value of the $F$ test in the $F$ distribution gives us the area we use to calculate P-values
    - then there is very strong evidence that the true means of the scores
if we find significant evidence against the $H_{0}$
- we may investigate further, possibly comparing pairs of means
  - $(\binom{k}{_{2}}) =$ number of pairs we can make to compare them
- different approaches
  - pooled-variance t (simplest one)
    - compare each pair of means withe either a confidence interval or hypothesis test
    - with multiple comparisons using confidence intervals, we start to get inaccuracy on the parameter
  - complicated ones