Factorial Analysis of variance

Factorial Analysis of variance

Orthogonal Is the property that every level of one factor is present in the experiment in combination with every level of the other factor

Effects of season (winter/spring and summer/fall) and adult density (8,15, 30 y 45 animales/225 cm2) on egg production of Siphonaria diemenensis

Spring-30 Summer- 15 Summer- 45 Summer- 30 Spring-8 Spring-15 Summer- 15 Summer- 8 Spring-15 Summer- 8 Spring-45 Summer- 45 Spring-45 Summer- 45 Spring-30 Summer- 30 Spring-8 Spring-45 Spring-15 Summer- 15 Spring-8 Summer- 8 Summer- 30 Spring-30

Two single-factor design animales/225 cm2 Winter/Spring Summer Fall Density treatment 8 15 30 45

Null hypothesis No effects of treatment A No effects of treatment B No effects of the interaction

Data Density 8 15 30 45 Spring Summer 2.875 2.125 2.600 0.867 2.230 1.267 1.400 0.711 2.625 1.50 1.866 0.933 1.466 0.467 1.022 0.356 1.750 1.875 2.066 1.733 1.00 0.700 1.177

Consider the entire analysis as though it were a single factor factorial experiment with ab experimental treatments Source Sum of Squares d f AB Σia Σjb Σjn (Xij-X)2 ab-1 Residual Σia Σjb Σjn (Xijk-Xij)2 ab(n-1) Total Σia Σjb Σkn(Xijk-X)2 abn-1

Now start again and ignore any differences among the data that might be due to factor B. Equivalent to a single-factor analysis of variance of means of the levels of factor A (with a treatments each replicated bn times) Source Sum of Squares d f A Σia Σjb Σjn (Xi-X)2 a-1 Residual Σia Σjb Σjn (Xijk-Xij)2 a(bn-1) Total Σia Σjb Σkn(Xijk-X)2 abn-1 A symmetrical argument can be made to analyze the data as a single factor B

The remaining differences can be identified empirically as Not all the differences among means of the ab combinations of treatments have been accounted for by the two single factor analyses The remaining differences can be identified empirically as SS among all treatments- SS factor A- SS factor B= Σia Σjb(Xij-Xi-Xj+X)2

Expected mean squares for test of null hypothesis for two factorial analysis (A fixed, B fixed) Source df Sum of squares Mean square Expected mean square F ratio Factor A a-1 Factor B b-1 Interaction AxB (a-1)(b-1) Within groups ab (n-1) Total abn-1

Source Type III Sum of Squares df Mean Square F Sig. DENSITY 5.284 3 1.761 9.669 .001 SEASON 3.250 1 17.842 DENSITY * SEASON .165 .055 .301 .824 Error 2.915 16 .182 Total 11.614 23

Expected mean squares for test of null hypothesis for two factorial analysis (A random, B random) Source df Sum of squares Mean square Expected mean square F ratio Factor A a-1 Factor B b-1 Interaction AxB (a-1)(b-1) Within groups ab (n-1) Total abn-1