Introduction to Factorial ANOVA

Transcription

1 Introduction to Factorial ANOVA Read from the bottom up!!!! Two factor factorial ANOVA Two factors ( predictor 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 1

2 Quinn (1988) - fecundity of limpets Factor A - season with levels: spring, summer Factor B - density with 4 levels: 8, 15, 30, 45 per 5cm n = 3 fences in each combination: each combination is termed a cell (8 cells) Response variable: fecundity (no. egg masses per limpet) Fecundity of limpets

3 Stehman & Meredith (1995) - growth of fir tree seedlings Factor A is nitrogen with 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 Response variable: growth of Douglas fir trees seedlings Data layout Factor A 1... i Factor B 1 j 1 j 1 j Reps y 111 y ij1 y 11 y ij y 11k y ijk Cell means y 11 y ij Note levels in factor B are the same for all levels in Factor A (not nested) 3

4 Linear model where i j () ij ijk y ijk = + i + j + () ij + ijk overall mean effect of factor A effect of factor B effect of interaction between A & B unexplained variation (error term) Worked example - Limpets Season Spring Summer Density Reps n = 3 in each of 8 groups (cells) p = seasons, q = 4 densities 4

5 Worked example Density Season Marginal means Spring Summer Density marginal means Grand mean Cell means Main effect: Null hypotheses effect of one factor, pooling over levels of other factor effect of one factor, independent of other factor Factor A marginal means (pooling B): 1,... i Factor B marginal means (pooling A): 1,... j 5

6 Main effect A H 0 : no difference between marginal means of factor A, pooling levels of B H 0 : 1 = = i = H 0 : no main effect of factor A, pooling levels of B ( 1 = = = i = 0) Example: No difference between season marginal means No effect of season, pooling densities Cell means Density Season means Spring spring Summer summer Density means overall

7 Main effect B H 0 : no difference between marginal means of factor B, pooling levels of A H 0 : 1 = = j = H 0 : no main effect of factor B, pooling levels of A ( 1 = = = j = 0) Example: No difference between density marginal means No effect of density, pooling seasons Cell means Density Season means Spring spring Summer summer Density means overall 8 = 15 = 30 = 45 7

8 Interaction An interaction between factors: effect of factor A dependent on level of factor B and viceversa H 0 : no interaction between factor A & factor B: effects of factor A & factor B are independent of each other no joint effects of A & B acting together ( ij = 0) ij - i - j + = 0, which is equal to ij ( i + j ) + = 0 Factors A and B are additive ij ( i + j ) + = 0 µ+α+β+ αβ ( µ+α+µ+β)+ µ = 0 µ+α+β+ αβ µ-α-µ-β+ µ= 0 µ+α+β+ αβ µ-α-µ-β+ µ= 0 αβ = 0 8

9 No interaction Root:shoot ratio of invasive grass: 0.5 C. maculosa 0.4 Two factors: AM fungus (present/absent) Type of phosphorous (organic/inorganic) Factorial design Root:shoot ratio AM fungi - AM fungi Inorganic P Organic P replicate pots Interaction Root:shoot ratio of native grass: 3 F. idahoensis Two factors: AM fungus (present/absent) Invasive competitor C. maculosa Factorial design Root:shoot ratio 1 0 F. idahoensis C. maculosa Competitor + AM -AM replicate pots 9

10 If interaction absent Factors A & B affect Y independently of each other Examine/test main effects and marginal means Interaction examination Organic Inorganic Row means Root:shoot ratio Inorganic P Organic P +AM Fungi -AM Fungi Column means Overall mean = AM fungi - AM fungi ij ( i + j ) + = ( ) =

11 If interaction present Factors A & B interact in their effect on Y Factors A & B do NOT affect Y independently of each other Difficult to examine/test main effect and marginal means Must examine results to determine if main effects are interpretable Interaction examination Root:shoot ratio F. idahoensis C. maculosa Competitor + AM -AM +AM Fungi -AM Fungi Column means F. idahoensis C. maculosa Row means Overall mean = 1.65 ij ( i + j ) + = ( )+1.65 =

12 Residual variation Variation between replicates within each cell Pooled across cells if homogeneity of variance assumption holds ( y y ij) ijk Partitioning total variation SS Total SS A + SS B + SS AB + SS Residual SS A SS B SS AB SS Residual variation between A marginal means variation between B marginal means variation due to interaction between A and B variation between replicates within each cell 1

13 Factorial ANOVA table Source SS df MS Factor A SS A p-1 SS A p-1 Factor B SS B q-1 SS B q-1 Interaction SS AB (p-1)(q-1) SS AB A X B (p-1)(q-1) Residual SS Residual pq(n-1) SS Residual pq(n-1) Expected mean squares Both factors fixed: MS A MS B MS A X B MS Residual nq np n p 1 q 1 i i ( ) ij ( p 1)( q 1) 13

14 If no interaction: H 0 : interaction ( ij ) = 0 true F-ratio: MS AB / MS Residual 1 H 0 : no interaction nq p 1 i np q 1 i n ( ) ij ( p 1)( q 1) H 0 : no main effect If no main effect of factor A: H 0 : 1 = = i = ( i = 0) is true F-ratio: MS A / MS Residual 1 If no main effect of factor B: H 0 : 1 = = j = ( j = 0) is true F-ratio: MS B / MS Residual 1 nq np n p 1 q 1 i i ( ) ij ( p 1)( q 1) 14

15 Worked example Density Season Marginal means Spring Summer Density marginal means Grand mean Cell means Testing of H 0 s Test H 0 of no interaction first: no significant interaction between density and season (P = 0.84) 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 try to limit unplanned comparisons 15

16 Interpreting interactions 1. Hopefully you will have a hypothesis or set of hypotheses for the interaction term. Plot cell means (This is usually the most informative thing to do) 3. Test hypotheses concerning the interaction (using Specify command) 4. If appropriate examine hypotheses concerned with main effects (using contrast or specify) Interaction plot No. egg masses per limpet Spring Summer Density Effect of density same for both seasons Difference between seasons same for all densities Parallel lines in cell means (interaction) plot 16

17 Worked example II Low shore Siphonaria larger limpets Two factors Season (spring and summer) Density (6, 1, 4 limpets per 5cm ) Response variable: no. egg masses per limpet n = 3 enclosures per season/density combination Worked example II Source df MS F P Season < Density Interaction Residual Total 17 17

18 Interaction plot No. egg masses per limpet Spring Summer Density Effect of density different for each season Difference between seasons varies for each density Non-parallel lines in cell means (interaction) plot Is the effect of Season interpretable?? Complex interaction Behavioural response of larval newts in lab Factor A Chemical cues from adult newts Factor B Earthworm prey Factorial design % larval newts in open Newt present Newt absent + worms -worms replicate aquaria 18

19 Multiple comparisons 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 Simple main effects Tests across levels of one factor for each level of second factor separately. Is there effect of density for spring? Is there effect of density for summer? Alternatively Is there effect of season for density = 6? etc. Equivalent to series of one factor ANOVAs Use df Residual and MS Residual from original factorial ANOVA 19

20 Worked example II: Low shore Siphonaria Source df MS F P Season < Density Interaction Residual Total 17 Worked example II: Low shore Siphonaria Source df MS F P Density Season <0.001 Density x Season Simple main effects Density in spring Density in summer <0.001 Residual

21 Worked example II: Low shore Siphonaria No. egg masses per limpet Density Spring Summer Simple main effects df MS F P Density in spring Density in summer <0.001 Mixed Models Traditional ANOVA approach At least one of the Factors is Random At least one of the Factors is Fixed Not too difficult if only one Factor is Random Very complex if more than one Factor is Random (realm or psuedo F stats) 1

22 Palm seedlings in Peru (Losos 1995) Mixed model Survivorship of palm seedlings in Peru Factor A - fixed 4 successional zones early-seral, mid-seral, late-seral, Heliconia Factor B - random randomly located transects 5 replicate plots of seedlings within each zone-transect combination (cell)

23 Mixed model Age at metamorphosis of copepod larvae Factor A - fixed 4 food treatments high food, high to low, etc. Factor B - random 15 randomly chosen sibships 4 replicate dishes of larvae within each food-sibship combinations (cell) Expected mean squares Factor A fixed, B random: MS A MS B MS A X B MS Residual n n np nq p 1 i 3

24 Tests in mixed model H 0 : no effect of random interaction A*B: F-ratio: MS AB / MS Residual H 0 : no effect of random factor B: F-ratio: MS B / MS Residual H 0 : no effect of fixed factor A: F-ratio: MS A / MS AB n n np nq p 1 i Palm seedlings Source df MS F P Denom. Zone Z x T Transect Residual Zone x transect <0.001 Residual Residual

25 Assumptions of factorial ANOVA Assumptions apply to y ijk within each cell Normality boxplots etc. Homogeneity of residual variance residual plots, variance vs mean plots etc. Independence Assumptions not met? Robust if equal n Transformations important No suitable non-parametric (rankbased) test 5

26 More complex ANOVAs Three or more factors Three factor ANOVA: 3 main effects 3 two-way interactions 1 three-way interaction Test three-way interaction first, then two ways, then main effects Canola (Brassica) germination Factor A - seed type 3 levels (different genetic lines & controls) Factor B - light 3 levels (full, dark, shade) Factor C - nutrients levels (low, high) Residual: 5 petri dishes (with seeds) in each of 18 cells Response variable: average time to germinate 6

27 3 factor example Source df F P Seed type (S) 17.3 <0.001 Light (L) Nutrient (N) S x L S x N N x L S x N x L Residual 30 Complex mixed models Source df F-test denominator Season 1 Season x Trans Zone 3 Zone x Trans Transect (random) 1 Residual Season X Zone 3 Season x Zone x Trans Season X Trans 1 Residual Zone x Trans 3 Residual Season X Zone x Trans 3 Residual Residual 55 7

28 General scheme for testing ANOVA models with fixed factors Source df df denomonator F A, B A df A = p-1 df Residual MS A /MS Residual B df B = q-1 df Residual MS B /MS Residual AB df AB = (p-1)(q-1) df Residual MS AB /MS Residual Residual pq(n-1) Source df df denomonator F A df A = p-1 df Residual MS A /MS Residual A, B, C B df B = q-1 df Residual MS B /MS Residual C df C = r-1 df Residual MS C /MS Residual AB df AB = (p-1)(q-1) df Residual MS AB /MS Residual AC df AC = (p-1)(r-1) df Residual MS AC /MS Residual BC df BC = (q-1)(r-1) df Residual MS BC /MS Residual And so on ABC df ABC = (p-1)(q-1)(r-1) df Residual MS ABC /MS Residual Residual pqr(n-1) General scheme for testing Mixed ANOVA models (fixed and random factors) Source df df denomonator F A = fixed B = random A df A = p-1 df AB MS A /MS AB B df B = q-1 df Residual MS B /MS Residual AB df AB = (p-1)(q-1) df Residual MS AB /MS Residual Residual pq(n-1) Source df df denomonator F A df A = p-1 df AC MS A /MS AC A = fixed B = fixed C = random And so on B df B = q-1 df BC MS B /MS BC C df C = r-1 df Residual MS C /MS Residual AB df AB = (p-1)(q-1) df ABC MS AB /MS ABC AC df AC = (p-1)(r-1) df Residual MS AC /MS Residual BC df BC = (q-1)(r-1) df Residual MS BC /MS Residual ABC df ABC = (p-1)(q-1)(r-1) df Residual MS ABC /MS Residual Residual pqr(n-1) 8

29 Effect of Blocking on Power of Test Relationship between supplemental watering and oak seedling germination H o : No difference in seedling number in watered and control plots How to set this up!!! Options Fully randomized Randomized Block - with no replication Account for underlying but unknown spatial variation Randomized block - with replication Account for underlying but unknown spatial variation Tradeoff between number of Blocks and replicates within Block Constraints - we can only logistically handle 4 replicate plots 9

30 Watered Control Completely Random Watered Control Randomized Block - no replication 30

31 Watered Control Randomized Block - Maximize Blocks minimize replication within Blocks Watered Control Randomized Block - Minimize Blocks Maximize replication within Blocks 31

32 Completely randomized design 4 replicated plots (1 per TTT) Randomized Block design Blocks 6 reps of each TTT per Block SEEDINGS SEEDINGS Mean and SEM Control Water TTT Randomized Block design 6 Blocks reps of each TTT per Block Control Water TTT Randomized Block design 1 Blocks (no TTT reps in each Block) SEEDINGS SEEDINGS Control Water TTT 10.0 Control Water TTT Compare Models Design Source df 1 df MS F P Test Notes CR TTT$ MS TTT /MS Residual No Blocks CR Residual 8.7 RB - no rep TTT$ MS TTT /MS Residual 1 Blocks RB - no rep Block No test RB - no rep Residual RB - reps TTT$ MS TTT /MS TTT*Block 6 Blocks RB - reps Block MS Block /MS Residual RB - reps TTT$*Blk MS TTT*Block /MS Residual RB - reps Residual RB - 6 reps TTT$ MS TTT /MS TTT*Block Blocks RB - 6 reps Block MS Block /MS Residual RB - 6 reps TTT$*Blk MS TTT*Block /MS Residual RB - 6 reps Residual

33 Tradeoff between blocking and degrees of freedom RB,, 6 Randomized block, blocks 6 reps RB,, 6 of each treatment per block Randomized block, 6 blocks reps RB, 6, of each treatment per block RB, 1, 1 Randomized block, 1 blocks 1 reps of each treatment per block CR Completely randomized design CR RB, 6, RB, 1, P =0.05 DF Denominator 33