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Factorial Experimental Design A simple strategy to design and analyze experiments.

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Observation = Mean + Hypotheses Observation = Mean + Factors Are the Factors Significant? Start Thinking in Equations! Size = mean size + age Growth = mean growth + ?

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Questions on Observations What are available objects of observation? What are available observations on objects? Which type of observations fit into equations?

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Growth and Development Size = mean + time Growth = Size/time = mean /time Growth = Size/time = mean + factors Primary grade observation Middle School experimentation Middle school calculation on observations.

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Graph of separate male and female effect Observe size Factors oMean oMale oFemale

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Graph of the sex_effect contrast Observe size Factors oMean oSex (as a contrast) Male and female effects are combined into a single contrast. How do we do that?

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Growth = mean + sex_effect + error Symbolically: Y = u + s i, ( i = female or male) + err y 1 = 1u + 1s + err 1 y 2 = 1u + 1s + err 2 y 3 = 1u - 1s + err 3 y 4 = 1u - 1s + err 4 where y j = individual j’s growth measurement, and err j = error associated with individual j’s growth.

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Growth = mean + sex_effect + error Y = u + s i, i = female or male Y = X ß + err obs design factors y 1 = 1 1 u er 1 y 2 = 1 1 s +er 2 y 3 = 1 -1 er 3 y 4 = 1 -1er 4

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Ideal_Growth = mean + sex_effect Y = u + s i, i = female or male Ideally the data observations have no error: Y = X ß = (the truth) expected design factors y 1 = 1 1 u y 2 = 1 1 s y 3 = 1 -1 y 4 = 1 -1 The design matrix has a column for estimating each factor, a row for estimating each datum.

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Estimate the true factors: mean & sex_effect Use the data observations and design matrix: Y & X are used to estimate the truth. estimate u estimate s u & s estimate u & s 1 y 1 1 y 1 u = ∑y j x 1j /n 1 y 2 1 y 2 s = ∑ y j x 2j /n 1 y 3 -1 y 3 with a 1 y 4 -1 y 4 certain error y 1+ y 2+ y 3+ y 4 y 1 + y 2 - y 3 - y 4 = sum y i x 1i = sum y i x 2i Design matrix columns are multiplied times the data column and summed.

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Error = (Observed - Expected) Why square the error? ∑(Y - Y) 2 = squared error (obs - exp) (obs - exp) 2 y 1 - y 1 (y 1 - y 1 ) 2 y 2 - y 2 ( y 2 - y 2 ) 2 SD = SS e /n y 2 - y 2 (y 3 - y 3 ) 2 y 4 - y 4 (y 4 - y 4 ) 2 The standard deviation is equal to the square root of the mean squared error. + + 0 errors sum to zero. Sum of Squares of Error (SS e )

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The sex_effect contrast with error Observe size Factors oMean oSex (as a contrast) If the error bar excludes zero you are confident that the sex_effect is significant.

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The Dragonfly, Sympetrum pallidum, has measureable wings. Are the female and male wings the same size and shape? There is some reason to suspect a difference because the female carries the heavier burden of eggs.

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Dragonfly wings: the hind wing length and width was measured.

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Sample Data Set: Dragon fly wings Sexlengthwidth Male 1122.4920.852 Male 1132.3710.831 Male 1142.5080.905 Male 1152.4940.871 Male 1162.2380.804 Female 1062.4100.841 Female 1112.2290.801 Female 1192.3050.816 Female 1232.5480.920 Female 1242.1850.808

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Sample Data Set: Dragon fly wings Ratio Design Matrix U S

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Graph with mean W/L-ratio plotted. The Mean L/W ratio is Meaningless. It exists but detracts from seeing the Female-Male effect. Therefore, remove the mean as a graphed factor as seen in the next slide.

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Graph Focus on the Sex-Difference and the Standard Error. The Mean Hindwing Width/Length is Meaningless. It is not of interest. Therefore, do not plot it as a factor. This graph shows that the female hindwing is significantly broader than the male’s, perhaps to provide the lift to carry its eggs in flight.

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