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Factorial Experimental Design

Factorial Experimental Design. A simple strategy to design and analyze experiments. Start Thinking in Equations!. Observation = Mean + Hypotheses. Size = mean size + age. Observation = Mean + Factors. Growth = mean growth + ?. Are the Factors Significant?. Questions on Observations.

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Factorial Experimental Design

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

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

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

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

  5. Graph of separate male and female effect • Observe size • Factors • Mean • Male • Female

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

  7. Growth = mean + sex_effect + error Symbolically:Y = u + si, ( i = female or male) + erry1 = 1u + 1s + err1y2 = 1u + 1s + err2y3 = 1u - 1s + err3y4 = 1u - 1s + err4whereyj = individual j’s growth measurement, and errj = error associated with individual j’s growth.

  8. Growth = mean + sex_effect + error Y = u + si, i = female or male Y = X ß + errobsdesign factors y1= 1 1 u er1 y2 = 1 1 s + er2 y3= 1 -1 er3 y4= 1 -1 er4 ( ( ( ( ( ( ( (

  9. Ideal_Growth = mean + sex_effect Y = u + si, i = female or maleIdeally the data observations have no error:Y = X ß = (the truth)expecteddesign factorsy1= 1 1 uy2 = 1 1 sy3= 1 -1 y4= 1 -1 ( ( ( ( ( ( The design matrix has a column for estimating each factor, a row for estimating each datum.

  10. Estimate the true factors:mean & sex_effect Use the data observations and design matrix:Y & X are used to estimate the truth.estimate uestimate s u & sestimate u & s1 y1 1 y1u = ∑yjx1j/n 1 y2 1 y2s = ∑ yjx2j/n 1 y3 -1 y3with a 1 y4 -1 y4 certain error y1+y2+y3+y4 y1 +y2 -y3 -y4 = sum yi x1i = sum yi x2i ( ( ( ( ( ( Design matrix columns are multiplied times the data column and summed.

  11. Error = (Observed - Expected)Why square the error? ∑(Y - Y)2 = squared error(obs - exp)(obs - exp)2 y1 - y1 (y1 - y1)2 y2 - y2 (y2 - y2)2 SD = SSe/n y2 - y2 (y3 - y3)2 y4 - y4 (y4 - y4)2 ( ( ( ( The standard deviation is equal to the square root of the mean squared error. + + Sum of Squares of Error (SSe) 0 errors sum to zero.

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

  13. 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.

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

  15. Sample Data Set: Dragon fly wings

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

  17. 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.

  18. 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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