Factorial Treatment Structure: Part I. Lukas Meier, Seminar für Statistik

Transcription

1 Factorial Treatment Structure: Part I Lukas Meier, Seminar für Statistik

2 Factorial Treatment Structure So far (in CRD), the treatments had no structure. So called factorial treatment structure exists if the g treatments are the combination of the levels of two or more factors. In the case that we see all the possible combinations of the levels of the two factors, we call the factors crossed. Examples Biomass of crop: different fertilizers and different crop varieties. Battery life: Different temperature levels and different plate material. (Montgomery, 1991, Example 7-3.1).

3 Example (Linder, A. und W. Berchtold, 1982) Response: Needleweight of 20 three-week old pine seedlings [in 1/100 g]. 2 factors: A = origin, B = exposure to light We denote by y ijk kth response of the treatment formed by the ith level of factor A and the jth level of factor B. y ijk Factor A on level i (here:origin) Factor B on level j (here: exposure) Replicate number (here: k = 1, 2) 2

4 Two Factor Design: Generic Data Table B 1 B 2 B 3 A 1 y 111 y 112 y 113 y 114 A 2 y 211 y 212 y 213 y 214 y 121 y 122 y 123 y 124 y 221 y 222 y 223 y 224 y 131 y 132 y 133 y 134 y 231 y 232 y 233 y 234 3

5 Data Table of Our Example Taglieda Pfyn Rheinau Short Long Permanent

6 Visualization As for one-way ANOVA situation for all treatment combinations factor-wise summaries ( marginal summaries ) More useful: Interaction plot (see R-code) 5

7 Factorial Treatment Structure The structure of the treatment influences the analysis of the data. Setup: Factor A with a levels Factor B with b levels n replicates for every combination Total of N = a b n observations a so called balanced design We could analyze this with the usual cell means model (ignoring the special treatment structure) Typically, we have research questions about both factors and their possible interaction (interplay). 6

8 Factorial Treatment Structure Examples: Is effect of light exposure location specific? ( interaction between light exposure and location) What is the effect of light exposure averaged over all locations? ( main effect of light exposure) What is the effect of location averaged over all exposure levels? ( main effect of location) We could use the cell means (one-way ANOVA) model and try to answer these questions with appropriate contrasts ( complicated). Easier: Use a model that incorporates the factorial structure of the treatments. 7

9 Factorial Model because two factors involved The two-way ANOVA model with interaction is where Y ijk = + α i + β j + αβ ij + ε ijk α i is the main effect of factor A at level i. β j is the main effect of factor B at level j. αβ ij is the interaction effect between A and B for level combination i, j (not the product α i β j!) ε ijk are i.i.d. N(0, σ 2 ) errors. Typically, sum-to-zero constraints are being used, i.e. a i=1 a i=1 b α i = 0, j=1 β j = 0. b (αβ) ij = 0, j=1 (αβ) ij = 0. a 1 and b 1 degrees of freedom a 1 (b 1) degrees of freedom 8

10 Interpretation of Main Effects Main-effects are nothing else than the average effect when moving from row to row (column to column). Interaction effect is the difference to the main-effects model, i.e. it measures how far the treatment means differ from the main-effects model. If there is no interaction, the effects are additive. In our example it would mean: No matter what location we are considering, the effect of light exposure is always the same. 9

11 Visualization of Model Factor A, Factor B with two levels each E Y ijk = + α i + β j + αβ ij + α 2 + β 2 + αβ 22 E[Y] + α 2 + β 1 + αβ 21 + α 1 + β 2 + αβ 12 Effect of B 1 B 2 at A 1 Effect of (A 1 A 2 ) at B 1 + α 1 + β 1 + αβ 11 A 1 A 2 10

12 Visualization of Model If no interaction is present, lines will be parallel. E Y ijk = + α i + β j Interaction plot is nothing else than empirical version of this plot. E[Y] + α 2 + β 2 + α 1 + β 2 + α 2 + β 1 Effect of B 1 B 2 at A 1 Effect of (A 1 A 2 ) at B 1 + α 1 + β 1 A 1 A 2 11

13 Two-Way ANOVA Step-By-Step Short Long Permanent Short Long Permanent Taglieda Taglieda β 1 β 2 β 3 β 1 β 2 β 3 Pfyn Pfyn β 1 β 2 β 3 β 1 β 2 β 3 Rheinau = Short Long Permanent Rheinau β 1 β 2 β 3 β 1 β 2 β 3 + Short Long Permanent Taglieda Taglieda αβ 11 αβ 12 αβ 13 αβ 11 αβ 12 αβ 13 Pfyn Pfyn αβ 21 αβ 22 αβ 23 αβ 21 αβ 22 αβ 23 Rheinau + Short Long Permanent Rheinau αβ 31 αβ 32 αβ 33 αβ 31 αβ 32 αβ 33 + Short Long Permanent Taglieda α 1 α 1 α 1 α 1 α 1 α 1 Taglieda ε 111 ε 121 ε 131 ε 112 ε 122 ε 132 Pfyn α 2 α 2 α 2 α 2 α 2 α 2 Pfyn ε 211 ε 221 ε 231 ε 212 ε 222 ε 232 Rheinau α 3 α 3 α 3 α 3 α 3 α 3 + Rheinau ε 311 ε 321 ε 331 ε 312 ε 322 ε

14 Parameter Estimates Estimates for the balanced case (and the sum-to-zero constraints) are Parameter Estimator = y α i α i = y i y β j β j = y j y αβ ij (αβ) ij = y ij α i β j This means: we estimate the main effects as if the other factors wouldn t be there (see next slides). 13

15 Main Effect of Location Original data-set Short Long Permanent Taglieda Pfyn Rheinau Ignore factor exposure, estimate effect as in one-way ANOVA model Taglieda Pfyn Rheinau α i 14

16 Main Effect of Light Exposure Original data-set Short Long Permanent Taglieda Pfyn Rheinau Ignore factor location, estimate effect as in one-way ANOVA model Short Long Permanent β j 15

17 Sum of Squares Again, total sum of squares can be partitioned into different sources, i.e. SS T = SS A + SS B + SS AB + SS E. Source df Sum of squares (SS) A a 1 B b 1 AB a 1 (b 1) Error n 1 ab Total abn 1 a i=1 a i=1 a i=1 b a j=1 b i=1 b j=1 j=1 b n α i 2 b a n β j 2 j=1 n k=1 n k=1 2 n (αβ) ij y ijk y ij 2 y ijk y 2 squared effect # observations with that effect product of df s of involved factors #observations 1 sum(df above) degrees of freedom of total 16

18 ANOVA Table As before, we can construct an ANOVA table Source df SS MS F A a 1 SS A SS A a 1 B b 1 SS B SS B b 1 AB a 1 (b 1) SS AB SS AB a 1 b 1 MS A MS E MS B MS E MS AB MS E Error ab (n 1) SS E SS E n 1 ab Under the corresponding (global) null-hypothesis, the Fratio is again F-distributed with degrees of freedom defined through the numerator / denominator. 17

19 F-Tests: Overview Interaction AB H 0 : αβ ij = 0 for all i, j H A : At least one αβ ij 0 Under H 0 : MS AB MS E F a 1 b 1, ab n 1 Main effect A H 0 : α i = 0 for all i H A : At least one α i 0 Under H 0 : MS A MS E F a 1, ab n 1 Main effect B H 0 : β j = 0 for all j H A : At least one β j 0 Under H 0 : MS B MS E F b 1, ab n 1 18

20 Analyzing the ANOVA Table Typically, the F-tests are analyzed from bottom to top (in the ANOVA table). Here, this means we start with the F-test of the interaction. If we reject: Conclude that we need an interaction in our model, i.e. the effect of A depends on the level of B. Don t continue testing the main-effects (principle of hierarchy) Perform individual analysis for every level of factor A (or B). Have a look at interaction plot to get a better understanding of what is going on. Interaction might be based on a single cell. Transformation might help in getting rid of the interaction. If we can t reject, continue testing the main effects. 19

21 Two-Way ANOVA model in R Define model with interaction in formula interface 20

22 More than Two Factors Model can be easily extended to more than two factors, say 3 (A, B, C). Model: Y ijkl = + α i + β j + γ k + main effects αβ ij + αγ ik + βγ jk + αβγ ijk + ε ijkl two-way interactions three-way interaction with ε ijkl i.i.d. N(0, σ 2 ) and the usual sum-to-zero constraints. Two-way interaction describes how a main-effect depends on the level of the other factor. Three-way interaction describes how a two-way interaction depends on level of third factor 21

23 Interpretation The higher-order interactions are quite difficult to interpret. Parameter estimation as before. Degrees of freedom of an interaction is the product of the df s of the involved factors. E.g., for the three-way interaction: a 1 b 1 (c 1) 22