Cheat Sheet Module 5

Cheat Sheet Module 5#

Notations#

  • Factors - \(F_A\), \(F_B\), and so on

  • CTQs - \(x\) (as always)

  • ‘Levels’ in factors - \(\ell_A\), \(\ell_B\)

Hypothesis Testing on Two-Samples#

\(x_1 \sim \mathcal{N}(\mu_1,\sigma_1)\), \(x_2 \sim \mathcal{N}(\mu_2,\sigma_2)\)

\(H_0: |\mu_1 - \mu_2| = \Delta_0\)

  • Known \(\sigma_1^2\) and \(\sigma_2^2\); Two-sample Z-test
    \(Z_0 = |\frac{\bar{x}_1 - \bar{x}_2| - \Delta_0}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2} }}\)

  • Unknown \(\sigma_1^2\) and \(\sigma_1^2\); Two-sample t-test (d.o.f \(n_1 + n_2 - 2\))

\(t_0 = \frac{|\bar{x}_1 - \bar{x}_2| - \Delta_0}{s_p\sqrt{\frac{1}{n_1} + \frac{1}{n_2} }}\) where \(s_p = \sqrt{\frac{(n_1-1)s_1^2 + (n_2-1)s_2^2 }{n_1 + n_2 -2}}\)

*  $H_a: \mu_1 - \mu_2 \neq \Delta_0 \implies $ Two-sided
*  $H_a: \mu_1 - \mu_2 > \Delta_0 \implies $ One-sided (Upper-Tail)
*  $H_a: \mu_1 - \mu_2 < \Delta_0 \implies $ One-sided (lower-Tail)

Hypothesis Testing on More than Two-Samples (Single factor)#

Notations#

  • \(x_{ij}\) - \(i^{th}\) level, \(j^{th}\) observation

  • Let \(\ell\) denote the number of levels.

  • Recollect: \(n\) observations form one sample.

    • Assume we collect same sample size (\(n\)) at all levels.

  • \(H_0: \mu_1 = \mu_2 = ... = \mu_a \), \(H_a:\)d At least one \(\mu_i\) is different.

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

      • \(F_0 = \frac{SS_{levels}/(\ell-1)}{SS_{E}/(\ell(n-1))}\)

      • \(MS_{levels} = SS_{levels}/(\ell-1)\)

      • \(MS_E = SS_{E}/(\ell(n-1))\)

    • Linear Statistical Model (“view”)

      • \(x_{ij} = \mu + \tau_i + \epsilon_{ij}\) \(\forall i\)

      • \(\mu\) - “base” mean

      • \(\tau_{i}\) - Effect of a level (also known as “treatment”)

      • \(\epsilon_{ij}\) - Random Effect (or Random Error)

      • \(SS_{levels} = n \sum_{i=1}^{\ell} (\bar{x}_{i\cdot} - \bar{x}_{\cdot\cdot})^2\)

      • \(SS_{Errors} = SS_E = \sum_{i=1}^{\ell} \sum_{j=1}^{n} ({x}_{ij} - \bar{x}_{\cdot\cdot})^2\)

      • \(SS_{Total} = SS_T = SS_{levels} + SS_E\)

ANOVA Table (Single Factor Experiment)#

Source of Variation

Sum of Squares

Degree of Freedom

Mean Square

\(F_0\)

Levels

\(SS_{levels}\)

\(\ell-1\)

\(MS_{levels}\)

\(\frac{MS_{levels}}{MS_E}\)

Error

\(SS_{E}\)

\(\ell(n-1)\)

\(MS_{E}\)

Total

\(SS_{T}\)

\(\ell n-1\)

Hypothesis Testing for Varying Factors#

  • \(H_0\): Corresponding Factor has no effect on the CTQ;

  • \(H_a\): Corresponding Factor has an effect on the CTQ

    • Reject \(H_0\) if \(F_0 > F_{\alpha,dof1,dof2}\)

    • dof1 - Numerator d.o.f., dof 2- Denominator d.o.f.

ANOVA Table for Two-factors (A and B)#

Source of Variation

Sum of Squares

Degree of Freedom

Mean Square

\(F_0\)

Decision

A

\(SS_{A}\)

\(\ell_A-1\)

\(MS_{A}\)

\(\frac{MS_{A}}{MS_E}\)

Factor A effect

B

\(SS_{B}\)

\(\ell_B-1\)

\(MS_{B}\)

\(\frac{MS_{B}}{MS_E}\)

Factor B effect

AB (interaction)

\(SS_{AB}\)

\((\ell_A-1)(\ell_B-1)\)

\(MS_{AB}\)

\(\frac{MS_{AB}}{MS_E}\)

AB interaction effect

Error

\(SS_{E}\)

\(\ell_A\ell_B(n-1)\)

\(MS_{E}\)

Total

\(SS_{T}\)

\(\ell_A \ell_B n-1\)

  • \(SS_{T} = \sum_{i=1}^{\ell_A}\sum_{j=1}^{\ell_b} \sum_{k=1}^{n} (x_{ijk} - \bar{x}_{...})^2\)

  • \(SS_A = \ell_Bn\sum_{i=1}^{\ell_A} (\bar{x}_{i..} - \bar{x}_{...})^2\)

  • \(SS_B = \ell_An \sum_{i=1}^{\ell_B} (\bar{x}_{.j.} - \bar{x}_{...})^2\)

  • \(SS_{AB} = n\sum_{i=1}^{\ell_A}\sum_{j=1}^{\ell_B} (\bar{x}_{ij.} - \bar{x}_{i..}- \bar{x}_{.j.} + \bar{x}_{...})^2\)

  • \(SS_{E} = SS_{T}- SS_{A} - SS_{B} - SS_{AB} = \sum_{i=1}^{\ell_A}\sum_{j=1}^{\ell_B}\sum_{k=1}^{n}(x_{ijk} - \bar{x}_{ij.})^2 \)

Linear Regression and Response Surface#

  • \(\hat{x} = \beta_0 + \beta_A f_A + \beta_B f_B +\beta_{AB}f_A f_B\)

    • \(\beta_0 = \bar{x}_{...}\)

    • For \(2^k\) Factorial Experiments: \(\beta_A = \frac{\bar{x}_{A^+} - \bar{x}_{A^-}}{2}\)

    • For \(2^k\) Factorial Experiments: \(\beta_B = \frac{\bar{x}_{B^+} - \bar{x}_{B^-}}{2}\)

    • For \(2^k\) Factorial Experiments: \(\beta_{AB} = \frac{(\frac{[\bar{x}_{A^+B^+}) - (\bar{x}_{A^-B^+})] - [(\bar{x}_{A^+B^-}) - (\bar{x}_{A^-B^-})]}{2})}{2}\)

    • \(\frac{\partial \hat{x}}{\partial f_A} = \beta_A + \beta_{AB} f_B\)

    • \(\frac{\partial \hat{x}}{\partial f_B} = \beta_B + \beta_{AB} f_A\)

  • Steepest Ascent (Maximizing the CTQ):

    • \(\Delta = c (\frac{\partial \hat{x}}{\partial f_A},\frac{\partial \hat{x}}{\partial f_B}) \)

  • Steepest Descent (Minimizing the CTQ):

    • \(\Delta = - c (\frac{\partial \hat{x}}{\partial f_A},\frac{\partial \hat{x}}{\partial f_B}) \)

  • Design Points

    • Center \(+ t \Delta\), where \(t = 1,2,3,...,\)

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