Appunti Statistica per la sperimentazione e le previsioni in ambito tecnologico (parte 1)

Statistics for experiments and forecasts

in the field of technology

Anno: 2026-2027

Sommario

1. General concepts .................................................................................................... 3

2. Linear regression model ........................................................................................... 5

2.1 Model with k regressors ................................................................................................... 5

2.2 Parameter estimation model fitting .............................................................................. 6

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2.3 Estimating ................................................................................................................. 9

2.4 Estimator properties ..................................................................................................... 10

3. Test of significance ................................................................................................ 11

3.1 Test of the model ........................................................................................................... 11

3.2 regression SS: SSR ........................................................................................................ 13

3.3 The Determination Index .......................................................................................... 14

3.4 Testing any single coeIicient ......................................................................................... 14

3.5 Testing subset of coeIicients ........................................................................................ 15

4. Confidence interval, prediction interval and diagnostics .......................................... 16

4.1 Confidence Interval ...................................................................................................... 16

4.2 Prediction of new responses .......................................................................................... 17

4.3 Diagnostic on residuals ................................................................................................. 18

4.4 Test Lack-of-fit .............................................................................................................. 20

5. DoE ....................................................................................................................... 21

5.1 First concepts about DoE (and ANOVA) ........................................................................... 22

5.2 Plantulae example ........................................................................................................ 22

5.3 Anova one-way .............................................................................................................. 25

5.4 ANOVA one-way steps ................................................................................................... 25

5.5 Example (Logothetis e Wynn, 1989) ................................................................................ 27

5.6 Example (Logothetis e Wynn, 1989) ................................................................................ 28

5.7 ANOVA model (Two-way) for the RCB design ................................................................... 29

5.8 Full factorial design – example by Montgomery (1991) ..................................................... 30

5.9 Interaction concept ....................................................................................................... 31

5.10 ANOVA – cornered point parametrization ...................................................................... 32

6. Fractional Factorial Design ..................................................................................... 33

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6.1 Example – full factorial design ................................................................................... 33

6.2 Electric welding example – experimental planning .......................................................... 37

7. Resolution criterium .............................................................................................. 39

7.1 Complete defining relation ............................................................................................ 41

8. RSM – Response Surface Methodology .................................................................... 44

8.1 RSM – Theoretical modelling concepts ........................................................................... 45

8.2 RSM – Ist order steepest ascent/discent optimization method ........................................ 47

8.3 RSM – 2 order estimated surface .................................................................................. 50

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8.4 Rotability ...................................................................................................................... 51

9. RSM – 1 order model ............................................................................................. 52

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9.1 The method of steepest ascent ...................................................................................... 53

9.2 Passage from the 1 to the 2 order ............................................................................... 61

st nd

10. Steepest ascent/discent – theory integrated by an example ................................... 63

11. Central Composite Design CCD ............................................................................ 71

12. Fractional Factorial at three levels ........................................................................ 74

13. 2 order RSM models ........................................................................................... 76

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14. Split Plot Design ................................................................................................... 77

14.2 Examples .................................................................................................................... 79

14.3 Summary .................................................................................................................... 82

15. Mixed Linear Model (MLM) .................................................................................... 83

16. Mixed RSM ........................................................................................................... 84

16.1 Split Plot and mixed RSM model ................................................................................... 85

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1. General concepts

DoE – Design of Experiments: DoE is a set of statistical methodology to plan, execute and

analyse experiments in a systematic way, with the aim of study the eCect of one or more

factor on the response variable (RV), by reducing the number of tests and maximizing the

obtained information.

Statistical data: there are two diCerent kind of data:

- Observation data: no dataset planned in advance, and the source is unknown. The

research do not manipulate any variables, they just observe what’s happening.

- Experimental data: the researchers have planned an experimental design to sample in

a structured framework. They manipulate variables to control the conditions and test

specific hypotheses, to identify cause-eCect relationships.

These are totally diCerent kind of data.

DiCerence between variable and factor:

- Variables (2° step of DoE, statistical modelling): they can be quantitative or qualitative

and can assume all values contained in X.

- Factors (1° step of DoE, planning the experiment): they represent SoVs, they can be

categorical or quantitative, and they can assume a specific set of values called

“levels”, which define the X.

RV – Response Value.

SoVs: Sources of Variability (or Variation), these are variables that can influence the results of

an experiment. A source of variability is a fraction of RV variability captured by a factor.

If a factor is categorical:

- RV is estimated to each level of F

- Replace it with a quantitative factor, where each set of value identifies a level 1

estimation of RV.

Interaction: measures how the eCect of a factor on the RV depends/ is aCected by changing

the level of another one. If it’s null or negligible, it means that the eCect of a factor on the RV is

the same across diCerent levels of another factor.

2 FS: we may define AB as the average diCerence between the eCect of B at the high level of A

and the eCect of B a the low level of A.

ECect: eCect is diCerent from variable. Direct (or indirect) impact of a factor on the RV, is

measured through a coeCicient. It can be:

- Fixed: variables whose values are known and fixed, because we control them.

- Random: variables whose values are not known in advance, they’re not fixed. They

cannot be controlled a priori (eg. Xz method). 3

Statistical model: it describes the relationship (expressed by the model parameters) between

variables (independent) and the response variable (dependent) with the goal of making

inference, you don’t fit all the data point, but you try to minimize residuals.

Dof - degrees of freedom represents the amount of independent information available to

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estimate variability after accounting for model constraints.

Classical replication: performing the same experimental combination under the same

experimental conditions. 4

2. Linear regression model

LRM is a model which models the relationship between a set of independent variables

(regressors) and a dependent variable (RV) by fitting a linear equation to the data

(observational or experimental).

understand and quantify the linear relationship and male predictions.

àGOAL:

Assumptions:

- Linearity

~

- independently and normally distributed, whit homoscedasticity and mean 0.

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Example:

Let’s start by fitting linear regression models. To illustrate, we can consider an example

relating the yield of a chemical reaction to the temperature and the catalyst feed rate.

The basic model is:

Where y represents the yield, x1 represents the temperature, and x2 represents the catalyst

feed rate. X1 and x2 are the two independent variables (also named predictor variables or

regressors).

the term linear is used because the equation is a linear function of the unknown

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parameters , and .

" # $

Partial regression coeCicients and the unit of measurement is important.

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# $

This model describes a plane in the two-dimensional x1 and x2 space. The parameter "

de