Repeated Measures Designs Single-factor

Elements Utilise the same subject (person, store, plant, market, etc) Each treatment is given once to each subject The subject can be treated as a ‘block’ and the experimental units within the block are viewed as: Repeat visits Repeat medication to each subject Progressively giving different doses to cancer patients The same subject is measured repeatedly

elements Example 200 persons with known migraine headache are each given 3 type of drugs The drugs are randomly picked and administered

Disadv & advantages Advantages Disadvantages Good precision Treatment order effect Can see the effect of time-longitudinal Previous treatment residual effect

randomisation Randomly assign the treatment allocation per subject e.g. effect of cereals, pulses, legumes of weight gain -randomly assign the start food type, the 2nd food type and the 3rd food type per subject

examples Food tasting and different judges Food_score Judge Food_type 20 1 1 15 2 1 18 3 1 26 4 1 22 5 1 19 6 1 24 1 2 18 2 2 19 3 2 26 4 2 24 5 2 21 6 2 28 1 3 23 2 3 24 3 3 30 4 3 28 5 3 27 6 3 28 1 4 24 2 4 23 3 4 30 4 4 26 5 4 25 6 4

Single factor model with repeated Single factor are the experimental animals, persons, plants, markets, etc Subjects are a random sample from a pop Treatments are fixed

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Hypothesis Testing .

Nested Designs Study vs Control Site

Nested designs Applicable when you want to assess either the quality of products from different markets or suppliers market/supplier Batches based on (1) Sample some products within each batch 2-stage nested design

Nested Experiments In some two-factor experiments the level of one factor , say B, is not “cross” or “cross classified” with the other factor, say A, but is “NESTED” with it. The levels of B are different for different levels of A. For example: 2 Areas (Study vs Control) 4 sites per area, each with 5 replicates. There is no link from any sites on one area to any sites on another area.

That is, there are 8 sites, not 2. Study Area (A) Control Area (B) S1(A) S2(A) S3(A) S4(A) S5(B) S6(B) S7(B) S8(B) X X X X X X X X X X X X X X X X X X X X X X X X X = replications Number of sites (S)/replications need not be equal with each sites. Analysis is carried out using a nested ANOVA not a two-way ANOVA.

A Nested design is not the same as a two-way ANOVA which is represented by: A1 A2 A3 B1 X X X X X X X X X X X X X X X B2 X X X X X X X X X X X X X X X B3 X X X X X X X X X X X X X X X Nested, or hierarchical designs are very common in environmental effects monitoring studies. There are several “Study” and several “Control” Areas.

Objective The nested design allows us to test two things: (1) difference between “Study” and “Control” areas, and (2) the variability of the sites within areas. If we fail to find a significant variability among the sites within areas, then a significant difference between areas would suggest that there is an environmental impact. In other words, the variability is due to differences between areas and not to variability among the sites.

In this kind of situation, however, it is highly likely that we will find variability among the sites. Even if it should be significant, however, we can still test to see whether the difference between the areas is significantly larger than the variability among the sites with areas.

Statistical Model i indexes “A” (often called the “major factor”) Yijk = m + ri + t(i)j + e(ij)k i indexes “A” (often called the “major factor”) (i)j indexes “B” within “A” (B is often called the “minor factor”) (ij)k indexes replication i = 1, 2, …, M j = 1, 2, …, m k = 1, 2, …, n

Model (continue)

Model (continue) Or, TSS = SSA + SS(A)B+ SSWerror = Degrees of freedom: M.m.n -1 = (M-1) + M(m-1) + Mm(n-1)

Example M=3, m=4, n=3; 3 Areas, 4 sites within each area, 3 replications per site, total of (M.m.n = 36) data points M1 M2 M3 Areas 1 2 3 4 5 6 7 8 9 10 11 12 Sites 10 12 8 13 11 13 9 10 13 14 7 10 14 8 10 12 14 11 10 9 10 13 9 7 Repl. 9 10 12 11 8 9 8 8 16 12 5 4 11 10 10 12 11 11 9 9 13 13 7 7 10.75 10.0 10.0 10.25

Example (continue) SSA = 4 x 3 [(10.75-10.25)2 + (10.0-10.25)2 + (10.0-10.25)2] = 12 (0.25 + 0.0625 + 0.625) = 4.5 SS(A)B = 3 [(11-10.75)2 + (10-10.75)2 + (10-10.75)2 + (12-10.75)2 + (11-10)2 + (11-10)2 + (9-10)2 + (9-10)2 + (13-10)2 + (13-10)2 + (7-10)2 + (7-10)2] = 3 (42.75) = 128.25 TSS = 240.75 SSWerror= 108.0

ANOVA Table for Example Nested ANOVA: Observations versus Area, Sites Source DF SS MS F P Area 2 4.50 2.25 0.158 0.856 Sites (A)B 9 128.25 14.25 3.167 0.012** Error 24 108.00 4.50 Total 35 240.75 What are the “proper” ratios? E(MSA) = s2 + V(A)B + VA E(MS(A)B)= s2 + V(A)B E(MSWerror) = s2 = MSA/MS(A)B = MS(A)B/MSWerror

Summary Nested designs are very common in environmental monitoring It is a refinement of the one-way ANOVA All assumptions of ANOVA hold: normality of residuals, constant variance, etc. Can be easily computed using MINITAB. Need to be careful about the proper ratio of the Mean squares. Always use graphical methods e.g. boxplots and normal plots as visual aids to aid analysis.

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Design & Analysis of Split-Plot Experiments (Univariate Analysis)

Elements of Split-Plot Designs Split-Plot Experiment: Factorial design with at least 2 factors, where experimental units wrt factors differ in “size” or “observational points”. Whole plot: Largest experimental unit Whole Plot Factor: Factor that has levels assigned to whole plots. Can be extended to 2 or more factors Subplot: Experimental units that the whole plot is split into (where observations are made) Subplot Factor: Factor that has levels assigned to subplots Blocks: Aggregates of whole plots that receive all levels of whole plot factor

Examples Agriculture: Varieties of a crop or gas may need to be grown in large areas, while varieties of fertilizer or varying growth periods may be observed in subsets of the area. Engineering: May need long heating periods for a process and may be able to compare several formulations of a by-product within each level of the heating factor. Behavioral Sciences: Many studies involve repeated measurements on the same subjects and are analyzed as a split-plot (See Repeated Measures lecture)

Design Structure Blocks: b groups of experimental units to be exposed to all combinations of whole plot and subplot factors Whole plots: a experimental units to which the whole plot factor levels will be assigned to at random within blocks Subplots: c subunits within whole plots to which the subplot factor levels will be assigned to at random. Fully balanced experiment will have n=abc observations

Data Elements (Fixed Factors, Random Blocks) Yijk: Observation from wpt i, block j, and spt k m : Overall mean level a i : Effect of ith level of whole plot factor (Fixed) bj: Effect of jth block (Random) (ab )ij : Random error corresponding to whole plot elements in block j where wpt i is applied g k: Effect of kth level of subplot factor (Fixed) (ag )ik: Interaction btwn wpt i and spt k (bc )jk: Interaction btwn block j and spt k (often set to 0) e ijk: Random Error= (bc )jk+ (abc )ijk Note that if block/spt interaction is assumed to be 0, e represents the block/spt within wpt interaction

Model and Common Assumptions Yijk = m + a i + b j + (ab )ij + g k + (ag )ik + e ijk

Two-Stage Nested Design Example 14-1 (pg. 528) Three suppliers, four batches (selected randomly) from each supplier, three samples of material taken (at random) from each batch Response: purity of raw materials Objective: determine the source of variability in purity Data is coded Mixed model, assume restricted form

Table 14-3 Table 14-4 14-1

Minitab Analysis –Table 14-6 Page 530 Factor Type Levels Values Supplier fixed 3 1 2 3 Batch(Supplier) random 4 1 2 3 4 Analysis of Variance for purity Source DF SS MS F P Supplier 2 15.056 7.528 0.97 0.416 Batch(Supplier) 9 69.917 7.769 2.94 0.017 Error 24 63.333 2.639 Total 35 148.306 Source Variance Error Expected Mean Square for Each Term component term (using restricted model) 1 Supplier 2 (3) + 3(2) + 12Q[1] 2 Batch(Supplier) 1.710 3 (3) + 3(2) 3 Error 2.639 (3)

Practical Interpretation – Example 14-1 What if we had incorrectly analyzed this experiment as a factorial? Batches differ significantly Batches x suppliers is significant. How to explain? Table 14-5 Incorrect Analysis of the Two-Stage Nested Design in Example 14-1 as a Factorial (Suppliers Fixed, Batches Random)