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

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

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

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

6. 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?

7. 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.

8. 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

9. 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.

10. 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.

11. 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 )

12. 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.

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 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

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

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.