Notes › EENG 3421: Probability and Statistics for Engineers Lecture 28
One-Way Analysis of Variance
253 words 2 min Modified
Table of Contents
F Distribution
Definition 28.1 (F Distribution).
The F distribution is continuous and skewed right It has two degrees of freedom
- numerator degrees of freedom
- denominator degrees of freedom It has support on nonnegative values
ANOVA Hypotheses
- Compare means of more than two populations
Theorem 28.2 (One Way ANOVA Hypotheses).
$$
H_0:\ \mu_1 = \mu_2 = \cdots = \mu_k
$$$$
H_1:\ \text{not all } \mu_i \text{ are equal}
$$
Variance Decomposition and Test Statistic
- ANOVA compares two variance estimates
- variance between samples
- variance within samples
- Mean square between samples
- $MSB$
- Mean square within samples
- $MSW$
- One way ANOVA test is right tailed
Theorem 28.3 (One Way ANOVA Test Statistic).
$$
F = \frac{MSB}{MSW}
$$
Sums of Squares
Theorem 28.4 (Total Sum of Squares Decomposition).
$$
SST = SSB + SSW
$$
Definition 28.5 (Between and Within Sums of Squares).
Between samples
$$ SSB = \left(\frac{T_1^2}{n_1} + \frac{T_2^2}{n_2} + \cdots + \frac{T_k^2}{n_k}\right) - \frac{T^2}{n} $$Within samples
$$ SSW = \left(\sum x_{1j}^2 - \frac{T_1^2}{n_1}\right) + \left(\sum x_{2j}^2 - \frac{T_2^2}{n_2}\right) + \cdots + \left(\sum x_{kj}^2 - \frac{T_k^2}{n_k}\right) $$Mean Squares and Degrees of Freedom
Theorem 28.6 (Mean Squares).
$$
MSB = \frac{SSB}{k-1}
$$$$
MSW = \frac{SSW}{n-k}
$$
where
- $k-1$ is numerator degrees of freedom
- $n-k$ is denominator degrees of freedom
ANOVA Table
Definition 28.7 (ANOVA Table).
- Between
- df $k-1$
- sum of squares $SSB$
- mean square $MSB$
- Within
- df $n-k$
- sum of squares $SSW$
- mean square $MSW$
- Total
- df $n-1$
- sum of squares $SST$
Corollary 28.8 (Reported Test Statistic).
$$
F = \frac{MSB}{MSW}
$$
References
- Course lecture slides 23–26
Sources
- Course lecture slides 23–26