Appunti Statistica per la sperimentazione e le previsioni in ambito tecnologico (parte 2)

5.7 ANOVA model (Two-way) for the RCB design

When the ANOVA one-way result shows that DW is much higher than DB surely, I have missed

some SoVs. Moreover, the factor appears not to have a significant eCect.

CBR

- We introduce a bock Factor, that has the only role to absorb the not explained

variability. It directly influences the experimental factor (that we want to optimize) and

indirectly the RV.

- We must randomize the exp F within each level of the block F. No randomization of the

BFs

- Furthermore it must be verified that no interaction is present among the Fs or that is

negligible.

Two factorsà two additive components

two classification criteria because we have two factors.

à

ANOVA statistical model corresponding to the RCB design is defined as follows:

(where is completely diCerent from the seen before)

!% !%

Where denotes the generic eCect of the i-th level of the experimental factor (at fixed levels),

!

which is also named sub-exponential factor.

IMPORTANT: the model additivity implies that a RCB design can be planned if and only if we

à

can assume a null or a negligible interaction between the experimental factor and the block

factor; in practice there must not be an interaction between these two factors. 29

5.8 Full factorial design – example by Montgomery (1991)

Best charact all factors are important at the same level

à

Interaction a way to measure associations between factors

à

* if I have no replicates, I can’t estimate the interaction because if I have replicate in each cell

I am able to calculate mean and variability in each cell. So, each cell explains the variability of

the corresponding trial. The sum of every variability of cell gives the tot.

- A*B: first order interaction two factors involved

à

- Two order interaction three factors involved

à

- Three order interaction four factors involved

à

- And so on…

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Full factorial design all possible combination of factors’ levels.

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A full factorial design is an experimental plan in which all possible combinations of factor

levels are tested, allowing complete estimation of the main eCects and interactions.

Full factorial design in absence of 1 order interaction between A and B:

st

I always start by the model wich include only the main eCect, so, the basic model without

interactions. If I desire to check the presence of 1 order interaction, the statistical model

st

becomes: 30

() is the interaction. I can apply interaction if and only if at least one factor (of the two) is

!%

significant (both better). If no one factor is significant surely the interaction is significant. If I

apply model with interaction all the main eCects must be included in the model.

Example:

5.9 Interaction concept

Interaction: amount of variability I am able to estimate from residual error after I have

estimated the variability of main eCects (A and B)

The interaction is estimated after estimating the two main eCects; I cannot estimate the

interaction alone, but I must necessarily include the main eCects in the model (this is a

hierarchical estimation). The interaction is a component of systematic variability that is

subtracted from the error once the main eCects have been estimated. 31

The plots above are used to understand the presence of interaction and the total absence of

interaction. Panel (1) shows the total absence of interaction because there is no variation

across levels of B as the levels of A change (completely parallel segments Panel

àp-value=1).

(2) shows the presence of interaction, that is, an association between A and B.

P=3 2 DOF

à

q=3 2 DOF

à

5.10 ANOVA – cornered point parametrization

Let’s suppose to have data on the clotting times of the blood (measured in seconds) of 24

animals randomly subjected to four types of diet (A, B, C, D). The ANOVA statistical model

(one-way) is the following: 32

Linear model estimates; cornered point results

Diet B and diet C show significant eCect on the t.m. of coagulation compared with diet A (level

of reference); no eCect by diet D. Diet B has an average eCect (increment) on t.m. equal to 5

sec; diet C has an average eCect equal to 7 sec.

Nota: we must consider the distinction between the ANOVA model and the corresponding

linear statistical model.

6. Fract