ANOVA2484 ANALYSIS OF VARIANCE (ANOVA) considering random factors Variance component analysis

ANOVA2485 Analysis of variance in case of random factors Fixed factors their levels may be set for the experiments Question: is there a difference in the result at different levels of a factor, which level is the best Random factor its levels are randomly chosen from a population imagined Question: does the factor affect fluctuation of the response, what is the expected fluctuation due to the fluctuation in that factor

ANOVA2486 Example 53 A chemical analysis was performed on 3 days. Does the day contribute to the variance of the results? random1.xls ANOVA for one random factor

ANOVA2487 Model  i effect of the i-th level (day i)  is a common value for a fixed factor for a random factor

ANOVA2488 ANOVA table r-1 r(p-1)

ANOVA2489 Open Data Table: random1.xls Analyze>Fit Model Y: Y Add: day Random Effect

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ANOVA2491 Statistics>Advanced Linear/Nonlinear Models> >General Linear Models>One-way ANOVA

ANOVA2492 ANOVA table for one random factor The null hypothesis is accepted.

ANOVA2493 If is rejected, estimation of is required Summary tab: Random effects>Var. comp.

ANOVA2494 If is rejected, estimation of is required

ANOVA2495 Cross-classification for two factors Example 54 A chemical analysis was performed on 3 days by 4 persons. Check and quantify the variance components. random2.xls

ANOVA2496 In the example r=3, q=4, p=2 Model i=1,…,r; j=1,…,q; k =1,…,p (repetition) dayperson interactionrepetition independent

ANOVA2497 Null hypotheses Do they contribute? How much? (day, person, interaction, error)

ANOVA2498 ANOVA table

ANOVA2499 Statistics>Advanced Linear/Nonlinear Models> >General Linear Models>Factorial ANOVA Options tab: Random day,person

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ANOVA2503 Open Data Table: random2.xls Analyze>Fit Model Y: Y Add: day, person (Macros: Full Factorial) Random Effect

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ANOVA2508 Gage R&R as ANOVA part, operator : random factors, crossed partoperatorinteraction repetition error

ANOVA2509 Total variance of measurement data : Variation attributable to the measurement (precision): Reproducibility:

ANOVA2510 (Example 16) Graph> Variability/Gage Chart Example 55 gage2.xls Analyze>Fit Model

ANOVA2511 Random block design One fixed, one random (blocking) factor Block: homogeneous circumstances cannot be assured for the whole set of experiments time (weather) personnel equipment raw material

ANOVA2512 Example 56 Box-Hunter-Hunter: Statistics for Experimenters, J. Wiley, 1978, p. 209 Penicillin manufacturing, 4 technologies are to be compared, batches of corn liquor (blends) are differing no replicates

ANOVA2513 Do the technologies differ in their effect on yield? Does the fluctuation between blends contribute to the fluctuation of the yield? Do they interact? Model technologyblend

ANOVA2514 ANOVA table In the example p=1, df error =0

ANOVA2515 Open Data Table: PENICILLrun.xls Analyze>Fit Model Y: yield Add: blend, treatm (no interaction) Attribute: blend is random but no replicates! (no interaction)

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ANOVA2519 Checking residuals Click on the red triangle next to Response YIELD Choose Row Diagnostics, Plot Residual by Predicted by run order? Plot Residual by Row

ANOVA2520 Sort first: Tables>Sort Repeat the whole analysis, then click on the red triangle next to Response YIELD choose Row Diagnostics, Plot Residual by Row What would be seen if there is a time effect?

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ANOVA2522 Two blocking factors: Latin square Example 57 Box-Hunter-Hunter: Statistics for Experimenters, J. Wiley, 1978, p. 245 Studying possible differences between 4 gasoline additives on the reduction of NO x emission. Cars and drivers are the blocking factors. Latin.sta Driver: 1,…,4Car: I,…,IVAdditives: A, B, C, D

ANOVA2523 Model no replicates The complete model would be:4 3 experiments!

ANOVA2524 Statistics>Industrial Statistics & Six Sigma>Experimental Design> >Latin squares... Statistics>Advanced Linear/Nonlinear Models> >General Linear Models>Main effects ANOVA Options tab: Random factors: Driver, Car>All effects Additive is fixed Car, Driver are random

ANOVA2525 Summary tab: Coefficients same results with fixed factors

ANOVA2526 Open Data Table: Latin.xls Analyze>Fit Model Y: REDUCTIN Add: DRIVER, CAR, ADDITIVE (no interaction)

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ANOVA2528 Open Data Table: Latin.xls Analyze>Fit Model Y: REDUCTIN Add: DRIVER, CAR, ADDITIVE Attributes: DRIVER, CAR Random Effect fixed:

ANOVA2529 Nested designs batches 1 2 … 15 samples … analyses … (1) (2) (1) (2) … (1) (2) (1) (2) (1) (2) (1) (2) (1) (2) … (1) (2) (1) (2) Example 58 Box-Hunter-Hunter: Statistics for Experimenters, J. Wiley, 1978, p. 571 Moisture content of manufactured pigment paste 15 batches are sampled twice each, two repeated analysis Moisture.sta

ANOVA2530 Detail of the data table

ANOVA2531 Model  i effect of batch (i=1,..., r)  j(i) effect of sample j taken from batch i (j=1,..., q, within i)  k(ij) experimental error committed at the k-th repeated analysis (k=1,..., p) all independent

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ANOVA2533 Statistics>Advanced Linear/Nonlinear Models> >General Linear Models>Nested design ANOVA Options tab: Random batch, sample Between effects

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ANOVA2535 Open Data Table: Moisture.xls Analyze>Fit Model Y: Moisture Add: Batch, Sample Attributes: Batch, Sample(Batch) Random Effect

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Minitab>Stat>ANOVA>Fully Nested ANOVA sample is nested into batch, examine df!

ANOVA2538 Example 59 In a pharmaceutical factory each tablet batch is sampled 3 times: front, middle and end, the content of the important component is analysed (assay) with 3 repetition

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ANOVA2541 The day/person example in nested setting (the days are different for the persons)