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Experimental Design & Analysis

Experimental Design & Analysis. Factorial ANOVA. Two factor factorial ANOVA. Two factors (2 independent variables) Factor A (with p groups or levels) Factor B (with q groups or levels) Crossed design: every level of one factor crossed with every level of second factor

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Experimental Design & Analysis

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  1. Experimental Design & Analysis Factorial ANOVA

  2. Two factor factorial ANOVA • Two factors (2 independent variables) • Factor A (with p groups or levels) • Factor B (with q groups or levels) • Crossed design: • every level of one factor crossed with every level of second factor • all combinations (i.e. cells) of factor A and factor B

  3. Quinn (1988) - fecundity of limpets • Factor A is season with 2 levels: • spring, summer • Factor B is density with 4 levels: • 8, 15, 30, 45 per 225cm2 • n = 3 fences in each combination: • each combination is termed a cell (8 cells) • Dependent variable: • fecundity (no. egg masses per limpet)

  4. Stehman & Meredith (1995) - growth of fir tree seedlings • Factor A is nitrogen with 2 levels • present, absent • Factor B is phosphorous with 4 levels • 0, 100, 300, 500 kg.ha-1 • 8 cells, n replicate seedlings in each cell • Dependent variable: • growth of Douglas fir trees seedlings

  5. Data layout Factor A 1 2 ........ i Factor B 1 2 j 1 2 j 1 2 j Reps y111yij1 y112yij2 y11kyijk Cell means y11yij

  6. Linear model • yijk = m + ai + bj + (ab)ij + eijk where • m is overall mean • ai is effect of factor A • bi is effect of factor B • (ab)ii is effect of interaction between A and B • eijkis unexplained variation

  7. Worked example Season Spring Summer Density 8 15 30 45 8 15 30 45 Reps n = 3 in each of 8 groups (cells) p = 2 seasons, q = 4 densities

  8. Worked example Cell means: Season Density 8 15 30 45marginal means Spring 2.42 2.18 1.57 1.20 1.84 Summer 1.83 1.18 0.81 0.59 1.10 Density marginal 2.13 1.68 1.19 0.891.47 means Grand mean

  9. Null hypotheses • Main effect: • effect of one factor, pooling over levels of other factor • effect of one factor, independent of other factor • Factor A marginal means (pooling B): • m1, m2...mi • Factor B marginal means (pooling A): • m1, m2...mj

  10. HO: no difference between marginal means of factor A, pooling levels of B • HO: m1 = m2 = mi = m • HO: no main effect of factor A, pooling levels of B (a1 = a2 = … = ai = 0) • Example: • No difference between season marginal means • No effect of season, pooling densities

  11. Cell means Density 8 15 30 45 Season means Spring 2.42 2.18 1.57 1.20 1.84 m spring Summer 1.83 1.18 0.81 0.59 1.10 m summer Density means 2.13 1.68 1.19 0.891.47 m overall m 8 m15 m30 m45

  12. HO: no difference between marginal means of factor B, pooling levels of A • HO: m1 = m2 = mj= m • HO: no main effect of factor B, pooling levels of A (b1 = b2 = … = bi = 0) • Example: • No difference between density marginal means • No effect of density, pooling seasons

  13. Cell means Density 8 15 30 45 Season means Spring 2.42 2.18 1.57 1.20 1.84 m spring Summer 1.83 1.18 0.81 0.59 1.10 m summer Density means 2.13 1.68 1.19 0.891.47 m overall m 8 = m 15 = m 30 = m 45

  14. Interaction • An interaction between 2 factors: • effect of factor A is dependent on level of factor B and vice-versa • HO: no interaction between factor A and factor B: • effects of factor A and factor B are independent of each other • no joint effects of A & B acting together (abij = 0) • mij - mi - mj + m = 0

  15. Interaction example from Underwood (1981) • Season: Summer Autumn Winter Spring • Area: 1 2 1 2 1 2 1 2 • n = 3 plankton tows for each season/area combination • DV is no. plankton in each tow • cell means (each season/area combination) • main effect means: • season means pooling areas • area means pooling seasons

  16. 800 600 A1 400 200 A2 800 600 A2 400 200 A1 800 600 400 A2 200 A1 S A W S S A W S 1 2 SEASON SEASON AREA Interaction plot Main effects Cell means Season means Area means MEAN NUMBER PER HAUL

  17. Residual variation • Variation between replicates within each cell • Pooled across cells if homogeneity of variance assumption holds

  18. Partitioning total variation SSTotal SSA + SSB + SSAB + SSResidual SSA variation between A marginal means SSB variation between B marginal means SSAB variation due to interaction between A and B SSResidual variation between replicates within each cell

  19. Factorial ANOVA table Source SS df MS Factor A SSAp-1 SS Ap-1 Factor B SSBq-1 SS B q-1 Interaction SSAB (p-1)(q-1) SS ABA X B (p-1)(q-1) Residual SSResidualpq(n-1) SS Residualpq(n-1)

  20. Worked example Source SS df MS Season 3.25 1 3.25 Density 5.28 3 1.76 Interaction 0.17 3 0.06 Residual 2.92 16 0.18 Total 11.62 23

  21. Expected mean squares Both factors fixed: • MSAs2 + nqåai2/p-1 • MSBs2 + npåbi2/q-1 • MSA X Bs2 + nåå(ab)ij2/(p-1)(q-1) • MSResiduals2

  22. HO: no interaction • If no interaction: • HO: interaction (abij) = 0 true • F-ratio: • MSAB / MSResidual 1 • Compare F-ratio with F-distribution with (a-1)(b-1) and ab(n-1) df • Determine P-value

  23. HO: no main effects • If no main effect of factor A: • HO: m1 = m2 = mi = m(ai= 0)is true • F-ratio: • MSA / MSResidual 1 • If no main effect of factor B: • HO: m1 = m2 = mj= m(bj= 0)is true • F-ratio: • MSB / MSResidual 1

  24. Worked example Source df MS FP Season 1 3.25 17.84 0.001 Density 3 1.76 9.67 0.001 Interaction 3 0.06 0.30 0.824 Residual 16 0.18 Total 23

  25. Testing of HO’s • Test HO of no interaction first: • no significant interaction between density and season (P = 0.824) • If not significant, test main effects: • significant effects of season (P = 0.001) and density (P = 0.001) • Planned and unplanned comparisons: • applied to interaction and to main effects

  26. Interpreting interactions • Plotting cell means • Multiple comparison across interaction term • Simple main effects • Treatment-contrast tests • Contrast-contrast tests

  27. Interaction plot • Effect of density same for both seasons • Difference between seasons same for all densities • Parallel lines in cell means (interaction) plot

  28. Worked example II • Low shore Siphonaria • larger limpets • Two factors • Season (spring and summer) • Density (6, 12, 24 limpets per 225cm2) • DV = no. egg masses per limpet • n = 3 enclosures per season/density combination

  29. Worked example II Source df MS FP Season 1 17.15 119.85< 0.001 Density 2 2.00 13.980.001 Interaction 2 0.85 5.91 0.016 Residual 12 0.14 Total 17

  30. Interaction plot • Effect of density different for each season • Difference between seasons varies for each density • Non-parallel lines in cell means (interaction) plot

  31. Multiple comparison • Use Tukey’s test, Bonferroni t-tests etc.: • compare all cell means in interaction • Usually lots of means: • lots of non-independent comparisons • Often ambiguous results • Not very informative, not very powerful

  32. Simple main effects • Tests across levels of one factor for each level of second factor separately. • Is there an effect of density for winter? • Is there an effect of density for summer? Alternatively • Is there an effect of season for density = 6? • etc. • Equivalent to series of one factor ANOVAs • Use dfResidual and MSResidual from original 2 factor ANOVA

  33. Treatment-contrast interaction • Do contrasts or trends in one factor interact with levels of other factor? • Does the density contrast [6 vs the average of 12 & 24] interact with season?

  34. Worked example II Source df MS FP Density 2 2.00 13.98 0.001 Simple main effects Density in winter 2 0.17 1.21 0.331 Density in summer 2 2.67 18.69 <0.001 Season 1 17.15 119.85 <0.001 Density x Season 2 0.85 5.91 0.016 Treatment-contrast inter. 6 vs (avg 12 & 24) x season 1 1.53 10.66 0.007 Residual 12 0.14

  35. Mixed model Factor A fixed, B random: • MSAs2 + nsab2 + nqåai2/p-1 • MSBs2 + npsb2 • MSA X Bs2 + nsab2 • MSResiduals2

  36. Tests in mixed model • HO: no effect of random interaction A*B: • F-ratio: MSAB / MSResidual • HO: no effect of random factor B: • F-ratio: MSB / MSResidual • HO: no effect of fixed factor A: • F-ratio: MSA / MSAB

  37. Assumptions of factorial ANOVA Assumptions apply to DV within each cell • Normality • boxplots etc. • Homogeneity of residual variance • residual plots, variance vs mean plots etc. • Independence

  38. Factorial ANOVAs in the literature

  39. Barbeau et al. (1994) • J. Exp. Mar. Biol. Ecol.180:103-136 • Experiment on consumption rate of crabs feeding on scallops • Two factors: • Crab size (3 levels - small, medium, large) • Scallop size (3 levels - small, medium, large)

  40. Dependent variable: • consumption rate (number of scallops per predator per day) of crabs (Cancer irroratus) • Sample size: • n = 2 - 4 replicate aquaria in each of 9 cells

  41. Source df MS FP Crab size 2 123.0 2.89 0.082 Scallop size 2 220.3 5.18 0.017 Interaction 4 68.0 1.60 0.218 Residual 18 42.6 Unplanned multiple comparison on main effect of scallop size (pooling crab sizes): S = M < L

  42. McIntosh et al. (1994) • Ecology75 : 2078-2090 • Effects of predatory fish on drifting behaviour of insects (mayflies) in streams • Factors: • Predator cues (3 levels - no fish, galaxids, trout) • Time (2 levels - day, night)

  43. Dependent variable: • number of mayflies drifitng • Sample size: • n=3 stream channels in each cell

  44. Source df FP Predator cues 2 1.45 0.261 Time 1 131.88 <0.001 Interaction 2 1.05 0.369 Residual 18

  45. Assumptions not met? • Robust if equal n • Transformations important • No suitable non-parametric (rank-based) test

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