Post Hock Tests in Complete Randomized Design lecture - 03

 

Post Hock Tests in Complete Randomized Design

Example

Determinations are made on the yield using three methods of catalyzing a chemical process.

Methods	observations
1	47.2	49.8	48.5	48.7
2	50.1	49.3	51.5	50.9
3	49.1	53.2	51.2	52.8	52.3

Write the appropriate statistical model and test the hypothesis that the three methods differ significantly at 5 % level of significance? Further examine which methods are significant by using Post Hock tests.

Solution:

Let Yij of three catalyzing process is considered the treatments and nothing is stated about the other variation in the statement of the problem, so it is pure CRD Problem and represent by the following linear statistical model.

Yij = μ + τj + ϵij

Statement of hypothesis

i. H0 : μ1 = μ2 = μ3    Vs.   H1 : μ1 ≠ μ2 ≠ μ3

ii. The significance level; α = 0.05

iii. The test statistic: CR Design

F =MST / MSE ~ F (2, 10)

vi. Reject H0, when F ≥ F0.05 (2, 10) = 4.10

v. Computation:

Observation	Method 1	Method 2	Method 3
1	47.2	50.1	49.1
2	49.8	49.3	53.2
3	48.5	51.5	51.2
4	48.7	50.9	52.8
5			52.3
T.j	194.2	201.8	258.6	654.6
T.j ^2	37713.64	40723.24	66873.96

ANOVA Table

Source of Variation	d.f	Sum of Squares	Mean Squares	F - ratio
Treatment	2	22.392	11.196	6.59
Error	10	16.988	1.6988
Total	12	39.38

F = MST / MSE = 6.59

vi. Remarks: 

As F calculated value falls in the rejection region, so we have not sufficient evidence to accept  thus we conclude that the three methods are significant

Post Hock Tests

1.    LSD Test

The method 1 and method 2 have identical effects but the method 3 effect is significantly different from method 1. 

Now if it is desired to construct (1 -  α ) % confidence interval for μi  -  μj

Then it is given by:

 

In the same manner construct confidence interval for other pairs of treatment. 

DMR Test

(Duncan’s Multiple Range Test)

The same result is provide by DMRT. 

• Read More: Estimation of Parameters