Agronomy Trials Usually interested in the factors of production: –When to plant? –What seeding rate? –Fertilizer? What kind? –Irrigation? When? How much? –When should we harvest?

Interactions of Treatment Factors Could consider one factor at a time –Hold all other factors constant –This is ok if the factors act independently But often factors are not independent of one another Examples: –Plant growth habit and plant density –Crop maturity group and response to fertilizer or planting date –Breed of animal and levels of a nutritional supplement –Others?

Interactions Yield V1 V2 V3 No interaction V1 V2 V3 Crossover Interactions Consider 3 varieties at four rates of nitrogen V1 V2 V3 Noncrossover Interactions Relative yield of varieties is the same at all fertilizer levels Magnitude of differences among varieties depends on fertilizer level Ranks of varieties depend on fertilizer level

Interactions – numerical example No interaction Positive interaction Negative interaction Effect of two levels of phosphorous and potassium on crop yield Main effects are determined from the marginal means Simple effects refer to differences among treatment means at a single level of another factor

Factorial Experiments If there are interactions, we should be able to measure and test them. –We cannot do this if we vary only one factor at a time We can combine two or more factors at two or more levels of each factor –Each level of every factor occurs together with each level of every other factor –Total number of treatments = the product of the levels of each factor This has to do with the selection of treatments –Can be used in any design - CRD, RBD, Latin Square - etc. –“Designs” generally refer to the layout of replications or blocks in an experiment –A “factorial” refers to the treatment combinations

Advantages and Disadvantages Advantages - IF the factors are independent –Results can be described in terms of the main effects –Hidden replication - the other factors become replications of the main effects Disadvantages –As the number of factors increase, the experiment becomes very large –Can be difficult to interpret when there are interactions

Uses for Factorial Experiments When you are charting new ground and you want to discover which factors are important and which are not When you want to study the relationship among a number of factors When you want to be able to make recommendations over a wide range of conditions

How to set up a Factorial Experiment The Field Plan –Choose an appropriate experimental design –Make sure treatments include combinations of all factors at all levels –Set up randomization appropriate to the chosen design Data Analysis –Construct tables of means and deviations –Complete an ANOVA table –Perform significance tests –Compute appropriate means and standard errors –Interpret the analysis and report the results

Two-Factor Experiments Four spacings at two nitrogen levels (2x4=8 treatments) in three blocks IIIIII Block

Tables of Means Spacing NitrogenMean T 11 T 12 T 13 T 14 A 1. T 21 T 22 T 23 T 24 A 2. MeanB.1 B.2 B.3 B.4 X.. Block I II IIIMean R 1 R 2 R 3 X..

ANOVA for a Two-Factor Experiment (fixed model) SourcedfSSMSF Totalrab-1SSTot  Blockr-1SSRMSR=F R = SSR/(r-1)MSR/MSE Aa-1SSAMSA=F A = SSA/(a-1)MSA/MSE Bb-1SSBMSB=F B = SSB/(b-1)MSB/MSE AB(a-1)(b-1)SSABMSAB=F AB = SSAB/(a-1)(b-1) MSAB/MSE Error(r-1)(ab-1)SSE=MSE= SSTot-SSR-SSASSE/(r-1)(ab-1) -SSB-SSAB Note: F tests may be different if any of the factors are random effects

Definition formulae SS treatment = SSA + SSB + SSAB

Means and Standard Errors A FactorB FactorTreatment (AB) Standard Error MSE/rb MSE/ra MSE/r Std Err Difference 2MSE/rb 2MSE/ra 2MSE/r t statistic

Interpretation If the AB interaction is significant: –the main effects may have no meaning whether or not they test significant –summarize in a two-way table of means for the various AB combinations If the AB interaction is not significant: –test the independent factors for significance –summarize in a one-way table of means for the significant main effects

Interactions V1 V2 Interaction V1 V2 No interaction Avg for V1 Avg for V2 Main effects for varieties Tests for main effects are meaningful because differences are constant across all levels of factor B Avg for V1 Avg for V2 Tests for main effects may be misleading. In this case the test would show no differences between varieties, when in fact their response to factor B is very different Factor B

Factorial Example To study the effect of row spacing and phosphate on the yield of bush beans –3 spacings: 45 cm, 90 cm, 135 cm –2 phosphate levels: 0 and 25 kg/ha

Tables of Means Spacing PhosphateS1S2S3Mean P P Mean Treatment Means BlockIIIIIIMean Mean Block Means

ANOVA SourcedfSSMSF Total Block ** Spacing Phosphate ** S X P ** Error ** Significant at the 1% level. CV = 4.7% StdErr Spacing Mean = StdErr Phosphate Mean = StdErr Treatment (SxP) Mean = 1.498

Report of Statistical Analysis Yield response depends on whether or not phosphate was supplied If no phosphate - yield decreases as spacing increases If phosphate is added - yield increases as spacing increases Blocking was effective Spacing Phosphate45 cm90 cm135 cm None kg/ha