2uData ⮞Single Factor Experiments ⮞Post-hoc analysis

Post-hoc analysis

After realizing analysis of variance, assume `F_o>F`* and we conclude "factor affects response". Indeed, the result only informs us that there are at least two means unequal. It does not mean that all the pairs of means are different.

Hence the next step is to compare all the pairs of means to investigate their difference. This step is known as "post-hoc analysis". There are several ways to realize it, In the following section, we discuss the method of “Least Significant Difference” (LSD) proposed by Fisher.

To conclude about the difference of responses corresponding to two certain levels `u` and `v` of factor, or if there is really difference between `bar y_u` and `bar y_v`, Fisher proposed that we compare the absolute difference `|bar y_u-bar y_v|` with the value of `LSD` defined by:

`LSD=t_(alpha//2,a(n-1))sqrt(MS_E(1/n_u+1/n_v))`

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in which :

• `t_(alpha//2,a(n-1))` is the percentage point of Student's distribution corresponding significance level `alpha` and degree of freedom `a(n-1)`,
• `n_u`, `n_v` are the number of runs in level `u` an `v` respectively.

If the number of runs in both levels are equal (equal `n`), then:

`LSD=t_(alpha//2,a(n-1))sqrt((2MS_E)/n)`

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If :

• `|bar y_u-bar y_v|>LSD` : the difference between level `u` and `v` is significant.
• `|bar y_u-bar y_v|< LSD` : the difference between level `u` and `v` is insignificant.

Example 6

We continue Example 5 in the last page, which result of ANOVA shows that at least two means are unequal.

We know already that `a=3` ; `n_A=n_B=n_C=n=10` ; `MS_E=241,567`.

Besides that : `t_(alpha//2,a(n-1))=t_(0,025,27)=2,0518` (percentage point table of Student's distribution).

Hence :   `LSD=2,0518xxsqrt((2xx241,567)/10)=14,262`

We calculate all the differences between means and show them in Table 1.

Table 1 Absolute differences of means

	`bar y_A=97,5`	`bar y_B=86,1`	`bar y_C=77,1`
`bar y_A=97,5`		11,4	20,4
`bar y_B=86,1`			9,0
`bar y_C=77,1`

Comparing values in Table 1 with `LSD`, we recognize that only the `IQ` difference between colour A and colour C is significant, other differences are insignificant.