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An introduction to Models ANOVA, Regression and Models

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fertilyield 16.27 15.36 16.39 14.85 15.99 17.14 15.08 14.07 14.35 14.95 23.07 23.29 24.04 24.19 23.41 23.75 24.87 23.94 26.28 23.15 34.04 33.79 34.56 34.55 34.53 33.53 33.71 37.00 Error variation is caused by forces other than fertiliser Treatment variation is caused by fertiliser

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FertilMFYMYMFFY 114.645.456.271.630.800.82 214.645.455.360.720.80-0.09 314.645.456.391.750.800.94 414.645.454.850.210.80-0.60 514.645.455.991.350.800.54 614.645.457.142.500.801.69 714.645.455.080.440.80-0.37 814.645.454.07-0.570.80-1.38 914.645.454.35-0.290.80-1.10 1014.645.454.950.310.80-0.50 1124.644.003.07-1.57-0.64-0.93 1224.644.003.29-1.35-0.64-0.71 1324.644.004.04-0.60-0.640.04 1424.644.004.19-0.45-0.640.19 1524.644.003.41-1.23-0.64-0.59 1624.644.003.75-0.89-0.64-0.25 1724.644.004.870.23-0.640.87 1824.644.003.94-0.70-0.64-0.06 1924.644.006.281.64-0.642.28 2024.644.003.15-1.49-0.64-0.85 2134.644.494.04-0.60-0.16-0.45 2234.644.493.79-0.85-0.16-0.70 2334.644.494.56-0.08-0.160.07 2434.644.494.55-0.09-0.160.06 2534.644.494.55-0.09-0.160.06 2634.644.494.53-0.11-0.160.04 2734.644.493.53-1.11-0.16-0.96 2834.644.493.71-0.93-0.16-0.78 2934.644.497.002.36-0.162.51 3034.644.494.61-0.03-0.160.12

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FERTILISER 123 YIELD PER PLOT

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Mean yield PLOT NUMBER

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Mean of Y

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PLOT NUMBER Mean of A Mean of B Mean of C

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PLOT NUMBER Mean of Y

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PLOT NUMBER Means of A, B & C

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MS (fertil) estimate (Error variation + Treatment variation) MS (error) estimates (Error variation) So if there is no effect of treatment, both MSs estimate the same thing, and their ratio should be about one. In other words, the F-ratio should be about one. If there is an effect of treatment, then MS (fertil) estimates something bigger than MS(error), so the F- ratio should be bigger than one. Question: is the F-ratio bigger than one, to a greater extent than would occur just by chance?

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0 10 20 30 40 50 60 70 80 62.56567.57072.57577.58082.58587.5 Height (feet) Volume (cubic feet) Volume (cubic feet) Volume of timber plotted against height of a tree

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DIAMHTVOL 8.37010.3 8.66510.3 8.86310.2 10.57216.4 10.78118.8 10.88319.7 11.06615.6 11.07518.2 11.18022.6 11.27519.9 11.37924.2 11.47621.0 11.47621.4 11.76921.3 12.07519.1 12.97422.2 12.98533.8 13.38627.4 13.77125.7 13.86424.9 14.07834.5 14.28031.7 14.57436.3 16.07238.3 16.37742.6 17.38155.4 17.58255.7 17.98058.3 18.08051.5 18.08051.0 20.68777.0

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Y X Positive deviation Negative deviation

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Y X Residual deviation

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0 10 20 30 40 50 60 70 80 62.56567.57072.57577.58082.58587.5 Volume of timber plotted against height of a tree

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Model formulae aid communication

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Linear Model categorical and/or continuous as many x-variables as we like interactions does hypothesis testing (whether) and estimation (what) covers many existing tests with separate names… General

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Minitab Model Syntax +addition operator is (optionally) placed between terms in a list. e.g. A+B+C is factors A, B and C *interaction operator placed between terms. e.g. A*B is the interaction of the factors A and B ( )brackets indicate nesting. When B is nested within A, it is expressed as B(A). When C is nested within both A and B, it is expressed as C(A B). |a model may be abbreviated using a | or ! to indicate factors and their interaction terms. e.g. A|B is equivalent to A+B+A*B. -operator to exclude some of the higher level interactions. e.g. if you want A+B+C+A*B+A*C+B*C, you could use A|B|C-A*B*C.

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Examples of Model Specifications Two factors crossed: A B A*B or A|B Three factors crossed: A B C A*B A*C B*C A*B*C or A|B|C Three factors nested: A B(A) C(A B) Crossed and nested (B nested within A, and both crossed with C): A B(A) C A*C B*C(A)

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Generality is good because one set of principles covers a diversity of tests the relationships between the tests are clear you can learn about assumptions and model criticism once, and it covers all the tests you can construct tests that dont have separate names, now you can use GLMs

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Four assumptions of GLM If an assumption is not met, then all the results of the GLM are in question. Independence Homogeneity of variance Linearity/Additivity Normality of Error

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How can we tell if assumptions are met? Normality of error Homogeneity of variance Linearity/additivity Independence Histogram of residuals Fitted values vs residuals Fitted values vs residuals and continuous x-variable vs residuals No easy answer These techniques are called model criticism

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