statistics

Statistics with Python 3 - Parametric Significance Tests (Two way ANOVA)

Posted by Yanchao MURONG on 2021-09-10

For two way ANOVA, we have one dependent variable and two independent variable.

Preconditions

The prerequisite of two way ANOVA is the same as one way ANOVA:

  • inner-group Independence:
    • random samples
    • sample size < 10% total population
  • inter-group Independence:
    • not pairwise
  • Normal Distribution
    • if sample size >= 10, then we could still use ANOVA if condition not met
    • if sample size < 10, we should use non parametric method if condition not met
  • Homogeneity of variance test => variances of each group should be equal
    • if sample size of each group are equal then we could use ANOVA if condition not met
    • if sample size of each group are not equal, but the std ratio between the largest and smallest > 2, then we could still use ANOVA if condition not met

Example

DV: math score IV: teaching method(r=2), scholarship(s=2) Sample Size: nAN = nAY = nBN = nBY = 10; n = 40

A A B B
N Y N Y
77 96 93 74
88 87 94 88
77 94 95 77
85 90 83 93
81 80 94 91
72 99 94 95
80 100 85 85
80 87 91 88
76 96 90 93
84 95 96 79
  1. calculate the average math result of each group (AN,AY,BN,BY)
y‾ij=1nij∑k=1nijyijk\overline{y}_{ij} = \frac {1}{n_{ij}} \sum_{k=1}^{n_{ij}}y_{ijk}

  1. calculate the total average of all math results y‾=1n∑i=1r∑i=1snij∗y‾ij\overline{y} = \frac {1}{n}\sum _ {i=1}^ {r}\sum _ {i=1}^ {s} n_{ij}*\overline{y}_ {ij}

  2. calculate the average of group A and group B y‾i.=1∑j=1snij∑i=1snij∗y‾ijy‾.j=1∑j=1rnij∑i=1rnij∗y‾ij\overline{y}_{i.} = \frac {1}{\sum_{j=1}^{s}n_{ij}}\sum_{i=1}^ {s} n_{ij}*\overline{y}_{ij} \overline{y}_{.j} = \frac {1}{\sum_{j=1}^{r}n_{ij}}\sum_{i=1}^ {r} n_{ij}*\overline{y}_{ij}

  3. calculate the effect of teaching method y‾i.−y‾\overline{y}_{i.} - \overline{y}

  4. calculate the effect of scholarship y‾.j−y‾\overline{y}_{.j} - \overline{y}

  5. hypothesis of teaching method

  • H0: μA = μB => μA - μ = μB - μ = 0 only when μA = μB = μ
  • HA: μA ≠ μB => μA - μ, μB - μ are not all 0
  1. hypothesis of scholarship
  • H0: μN = μY => μN - μ = μY - μ = 0 only when μN = μY = μ
  • HA: μN ≠ μY => μN - μ, μY - μ are not all 0
  1. hypothesis of interaction effect y‾ij−y‾i.−y‾.j+y‾=(y‾ij−y‾)−(y‾i.−y‾)−(y‾.j−y‾)\overline{y}_{ij} - \overline{y}_{i.} - \overline{y}_{.j} + \overline{y} = (\overline{y}_{ij} - \overline{y}) - (\overline{y}_{i.} - \overline{y}) - (\overline{y}_{.j} - \overline{y})
  • H0: μij - μi - μj + μ is equal to 0 to all i,j combinations
  • H1: μij - μi - μj + μ is not equal to 0 to some i,j combinations

  1. calculate the sum of squares total (SST) SStotal=∑i=1r∑j=1s∑k=1nij(yijk−y‾)2SS_{total} = \sum^{r}_{i=1}\sum^{s}_{j=1}\sum^{n_{ij}}_{k=1}(y_{ijk} - \overline{y})^2

  2. calculate the sum of squares for teaching method (SSmethod) SSmethod=∑i=1rni.(y‾i.−y‾)2SS_{method} = \sum^{r}_{i=1}ni.(\overline{y}_{i.} - \overline{y})^2

  3. calculate the sum of squares for scholarship(SSreward) SSreward=∑j=1sn.j(y‾.j−y‾)2SS_{reward} = \sum^{s}_{j=1}n.j(\overline{y}_{.j} - \overline{y})^2

  4. calculate the sum of squares for interaction effect SSM×R=∑i=1r∑j=1snij(y‾ij−y‾i.−y‾.j+y‾)SS_{M \times R} = \sum^{r}_{i=1}\sum^{s}_{j=1}n_{ij}(\overline{y}_{ij}-\overline{y}_{i.}-\overline{y}_{.j} + \overline{y})

  5. calculate the sum of squares error SSerror=∑i=1r∑j=1s∑k=1nij(yijk−y‾ij)2SS_{error} = \sum^{r}_{i=1}\sum^{s}_{j=1}\sum^{n_{ij}}_{k=1}(y_{ijk} - \overline{y}_{ij})^2

  6. calculate the mean of squares method MSmethod=SSmethodr−1MS_{method} = \frac{SS_{method}}{r-1}

  7. calculate the mean of squares reward MSreward=SSrewards−1MS_{reward} = \frac{SS_{reward}}{s-1}

  8. calculate the mean of squares interaction MSM×R=SSM×R(r−1)×(s−1)MS_{M \times R} = \frac{SS_{M \times R}}{(r-1) \times (s-1)}

  9. calculate the mean of squares error MSerror=SSerrorn−rsMS_{error} = \frac{SS_{error}}{n - rs}

  10. calculate the F-value F=MSmethodMSerror∼(dfm,dfe)F=MSrewardMSerror∼(dfr,dfe)F=MSM×RMSerror∼(dfm∗r,dfe)F = \frac{MS_{method}}{MS_{error}} \sim {(dfm,dfe)} F = \frac{MS_{reward}}{MS_{error}} \sim {(dfr,dfe)} F = \frac{MS_{M \times R}}{MS_{error}} \sim {(dfm*r,dfe)}

  11. ANOVA Table

- Df Sum sq Mean sq F value Pr(>F)
method 1 72.9 72.9 2.16 0.15
reward 1 129.6 129.6 3.83 0.06
Method:Reward 1 774.4 774.4 22.91 2.9E-05
Residuals 36 1215.0 33.8 - -

As a result,

  • math score is not influenced by teaching method
  • math score is not influenced by scholarship reward
  • math score is influenced by the interaction of teaching method and scholarship reward
    • the influence of teaching method on math score could be regulated by scholarship reward (A)
    • the influence of scholarship reward on math score could be regulated by teaching method (B)

we need post hoc t-test to know exactly whether A is true or B is true

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