1 / 36

Randomized block designs

Randomized block designs. Ó Environmental sampling and analysis (Quinn & Keough, 2002). Blocking. Aim : Reduce unexplained variation, without increasing size of experiment. Approach : Group experimental units (“replicates”) into blocks.

alden-wynn
Download Presentation

Randomized block designs

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Randomized block designs Ó Environmental sampling and analysis (Quinn & Keough, 2002)

  2. Blocking • Aim: • Reduce unexplained variation, without increasing size of experiment. • Approach: • Group experimental units (“replicates”) into blocks. • Blocks usually spatial units, one experimental unit from each treatment in each block.

  3. Null hypotheses • No main effect of Factor A • H0: m1 = m2 = … = mi = ... = m • H0: a1 = a2 = … = ai = ... = 0 (ai= mi - m) • no effect of shaving domatia, pooling blocks • Factor A usually fixed

  4. Null hypotheses • No effect of factor B (blocks): • no difference between blocks (leaf pairs), pooling treatments • Blocks usually random factor: • sample of blocks from populations of blocks • H0: 2 = 0

  5. Randomised blocks ANOVA • Factor A with p groups (p = 2 treatments for domatia) • Factor B with q blocks (q = 14 pairs of leaves) • Source general example • Factor A p-1 1 • Factor B (blocks) q-1 13 • Residual (p-1)(q-1) 13 • Total pq-1 27

  6. Randomised block ANOVA • Randomised block ANOVA is 2 factor factorial design • BUT no replicates within each cell (treatment-block combination), i.e. unreplicated 2 factor design • No measure of within-cell variation • No test for treatment by block interaction

  7. Expected mean squares If factor A fixed and factor B (Blocks) random: MSAs2 + sab2 + nå(ai)2/p-1 MSBlockss2 + nsb2 MSResiduals2 + sab2

  8. Residual • Cannot separately estimate s2 and sab2: • no replicates within each block-treatment combination • MSResidual estimates s2 + sab2

  9. Testing null hypotheses • Factor A fixed and blocks random • If H0 no effects of factor A is true: • then F-ratio MSA / MSResidual 1 • If H0 no variance among blocks is true: • no F-ratio for test unless no interaction assumed • if blocks fixed, then F-ratio MSB / MSResidual 1

  10. Assumptions • Normality of response variable • boxplots etc. • No interaction between blocks and factor A, otherwise • MSResidual increase proportionally more than MSA with reduced power of F-ratio test for A (treatments) • interpretation of main effects may be difficult, just like replicated factorial ANOVA

  11. Checks for interaction • No real test because no within-cell variation measured • Tukey’s test for non-additivity: • detect some forms of interaction • Plot treatment values against block (“interaction plot”)

  12. Sphericity assumption • Pattern of variances and covariances within and between “times”: • sphericity of variance-covariance matrix • Equal variances of differences between all pairs of treatments : • variance of (T1 - T2)’s = variance of (T2 - T3)’s = variance of (T1 - T3)’s etc. • If assumption not met: • F-ratio test produces too many Type I errors

  13. Sphericity assumption • Applies to randomised block and repeated measures designs • Epsilon (e) statistic indicates degree to which sphericity is not met • further e is from 1, more variances of treatment differences are different • Two versions of e • Greenhouse-Geisser e • Huyhn-Feldt e

  14. Dealing with non-sphericity If e not close to 1 and sphericity not met, there are 2 approaches: • Adjusted ANOVA F-tests • df for F-ratio tests from ANOVA adjusted downwards (made more conservative) depending on value e • Multivariate ANOVA (MANOVA) • treatments considered as multiple response variables in MANOVA

  15. Sphericity assumption • Assumption of sphericity probably OK for randomised block designs: • treatments randomly applied to experimental units within blocks • Assumption of sphericity probably also OK for repeated measures designs: • if order each “subject” receives each treatment is randomised (eg. rats and drugs)

  16. Sphericity assumption • Assumption of sphericity probably not OK for repeated measures designs involving time: • because response variable for times closer together more correlated than for times further apart • sphericity unlikely to be met • use Greenhouse-Geisser adjusted tests or MANOVA

  17. Partly nested ANOVA Ó Environmental sampling and analysis (Quinn & Keough, 2002)

  18. Partly nested ANOVA • Designs with 3 or more factors • Factor A and C crossed • Factor B nested within A, crossed with C

  19. Partly nested ANOVA Experimental designs where a factor (B) is crossed with one factor (C) but nested within another (A). A 1 2 3 etc. B(A) 1 2 3 4 5 6 7 8 9 C 1 2 3 etc. Reps 1 2 3 n

  20. ANOVA table Source df Fixed or random A (p-1) Either, usually fixed B(A) p(q-1) Random C (r-1) Either, usually fixed A * C (p-1)(r-1) Usually fixed B(A) * C p(q-1)(r-1) Random Residual pqr(n-1)

  21. Linear model yijkl = m + ai + bj(i) + dk + adik + bdj(i)k + eijkl m grand mean (constant) ai effect of factor A bj(i) effect of factor B nested w/i A dk effect of factor C adik interaction b/w A and C bdj(i)k interaction b/w B(A) and C eijkl residual variation

  22. Expected mean squares Factor A (p levels, fixed), factor B(A) (q levels, random), factor C (r levels, fixed) Source df EMS Test A p-1 2 + nr2 + nqr2 MSA/MSB(A) B(A) p(q-1) 2 + nr2 MSB/MSRES C r-1 2 + n2 + npq2 MSC/MSB(A)C AC (p-1)(r-1) 2 + n2 + nq2 MSAC/MSB(A)C B(A) C p(q-1)(r-1) 2 + n2 MSBC/MSRES Residual pqr(n-1) 2

  23. Split-plot designs • Units of replication different for different factors • Factor A: • units of replication termed “plots” • Factor B nested within A • Factor C: • units of replication termed subplots within each plot

  24. Analysis of variance • Between plots variation: • Factor A fixed - one factor ANOVA using plot means • Factor B (plots) random - nested within A (Residual 1) • Within plots variation: • Factor C fixed • Interaction A * C fixed • Interaction B(A) * C (Residual 2)

  25. ANOVA Source of variation df Between plots Site 2 Plots within site (Residual 1) 3 Within plots Trampling 3 Site x trampling (interaction) 6 Plots within site x trampling (Residual 2) 9 Total 23

  26. Repeated measures designs • Each whole plot measured repeatedly under different treatments and/or times • Within plots factor often time, or at least treatments applied through time • Plots termed “subjects” in repeated measures terminology

  27. Repeated measures designs • Factor A: • units of replication termed “subjects” • Factor B (subjects) nested within A • Factor C: • repeated recordings on each subject

  28. Repeated measures design [O2] Breathing Toad 1 2 3 4 5 6 7 8 type Lung 1 x x x x x x x x Lung 2 x x x x x x x x ... ... ... ... ... ... ... ... ... ... Lung 9 x x x x x x x x Buccal 10 x x x x x x x x Buccal 12 x x x x x x x x ... ... ... ... ... ... ... ... ... ... Buccal 21 x x x x x x x x

  29. ANOVA Source of variation df Between subjects (toads) Breathing type 1 Toads within breathing type (Residual 1) 19 Within subjects (toads) [O2] 7 Breathing type x [O2] 7 Toads (Breathing type) x [O2] (Residual 2) 133 Total 167

  30. Assumptions • Normality & homogeneity of variance: • affects between-plots (between-subjects) tests • boxplots, residual plots, variance vs mean plots etc. for average of within-plot (within-subjects) levels

  31. No “carryover” effects: • results on one subplot do not influence results one another subplot. • time gap between successive repeated measurements long enough to allow recovery of “subject”

  32. Sphericity • Sphericity of variance-covariance matrix • variances of paired differences between levels of within-plots (or subjects) factor equal within and between levels of between-plots (or subjects) factor • variance of differences between [O2] 1 and [O2] 2 = variance of differences between [O2] 2 and [O2] 2 = variance of differences between [O2] 1 and [O2] 3 etc.

  33. Sphericity (compound symmetry) • OK for split-plot designs • within plot treatment levels randomly allocated to subplots • OK for repeated measures designs • if order of within subjects factor levels randomised • Not OK for repeated measures designs when within subjects factor is time • order of time cannot be randomised

  34. ANOVA options • Standard univariate partly nested analysis • only valid if sphericity assumption is met • OK for most split-plot designs and some repeated measures designs

  35. ANOVA options • Adjusted univariate F-tests for within-subjects factors and their interactions • conservative tests when sphericity is not met • Greenhouse-Geisser better than Huyhn-Feldt

  36. ANOVA options • Multivariate (MANOVA) tests for within subjects or plots factors • responses from each subject used in MANOVA • doesn’t require sphericity • sometimes more powerful than GG adjusted univariate, sometimes not

More Related