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MANE 6313

Week 4, Module A

Student Learning Outcome

Analyze simple comparative experiments and experiments with a single factor.

Module Learning Outcome

Explain the model, parameters, strategy and tools used in an One-way Analysis of Variance (ANOVA).

One-way Analysis of Variance (ANOVA)

  • Based upon the (fixed effects) model

    \[ y_{ij}=\mu+\tau_i+\varepsilon_{ij}\;\left\{\begin{array}{l} i=1,2,\ldots,a\\j=1,2,\ldots,n\\ \end{array}\right. \]

  • There are \(a\) different factors. \(\tau_i\) is the parameter associated with the \(i\)-th factor level

  • There are \(n\) observations for each factor level

  • Test the hypothesis

\[ \begin{aligned} H_0&:&\tau_1=\tau_2=\cdots=\tau_a=0\\ H_1&:&\tau_i\neq 0 \mbox{ for at least one }i\\ \end{aligned} \]

with the requirement that \(\sum_{i=1}^a\tau_i=0\).

Important Points

  • The statistical model is the fixed effects model
  • The design used for the experimentation is called the completely randomized design
  • A design is said to be balanced if the number of replicates for treatment level are equal.
  • We will not deal with unbalanced designs in this course (but it is in your textbook for reference)

  • Strategy is to partition variability.

    • Sum of Squares total
\[ SS_T=\sum_{i=1}^a\sum_{j=1}^n\left(y_{ij}-\bar{y}_{..}\right)^2 \]
-   Factor Sum of Squares:
\[ SS_{\mbox{Factor}}=n\sum_{i=1}^a\left(\bar{y}_{i.}-\bar{y}_{..}\right)^2 \]
-   Error Sum of Squares:
\[ SS_E=\sum_{i=1}^a\sum_{j=1}^n\left(y_{ij}-\bar{y}_{i.}\right)^2 \]
  • Note: \(SS_T=SS_{\mbox{Factor}}+SS_E\)

  • There are two method of calculating sum of squares for the one-way ANOVA.

    1. By hand using the convenience formulas:
\[ \begin{aligned} SS_T&=&\sum_{i=1}^a\sum_{j=1}^n y^2_{ij}-\frac{y^2_{..}}{an}\\ SS_{\mbox{Factor}}&=&\sum_{i=1}^a\frac{y^2_{i.}}{n}-\frac{y^2{..}}{an}\end{aligned} \]
2.  Using software such as R, MS Excel, or Minitab (there are others).

ANOVA Table

Source of Variation Sum of Squares Degrees of Freedom Mean Square \(F_0\)
Between Factors \(SS_{Factor}\) \(a-1\) \(MS_{Factor}=\frac{SS_{Factor}}{a-1}\) \(F_0= \frac{MS_{Factor}}{MS_E}\)​
Within \(SS_E\) \(a(n-1)\) \(MS_E=\frac{SS_E}{a(n-1)}\)
Total \(SS_T\) \(an-1\)

Reject \(H_0\) if \(F_0>F_{\alpha,a-1,a(n-1)}\)

Model Parameters

  • Point estimators
\[ \begin{aligned} \hat{\mu}&=&\bar{y}_{..}\\ \hat{\tau}_i&=&\bar{y}_{i.}-\bar{y}_{..}\;\;i=1,2,\ldots,a\\ \hat{\mu}_i&=&\hat{\mu}+\hat{\tau}_i=\bar{y}_{i.} \end{aligned} \]

  • Properties of \(\varepsilon_{ij}\)

    • It is assumed that \(\varepsilon_{ij}\) is a NID\((0,\sigma^2)\) random variable

  • Properties of the ANOVA

    • \(E(MS_{\mbox{E}})=\sigma^2\)

    • \(E(MS_{\mbox{Treatments}})=\sigma^2+\frac{n\sum_{i=1}^a\tau_i^2}{a-1}\)

Fixed vs. Random Factor?

  • For the fixed model, the experimenter specifically chooses the \(a\) treatments.

  • We estimate the parameters \((\mu,\tau_i,\sigma^2)\)

  • If the treatment levels were generated by randomly sampling from a population of treatments, the model is the random effects model

  • In the random effects model, we generalize the sample results to the entire population of treatments.