Analyzing Factorial Experiments: Banana Skin Blackening Study
School
RGV College**We aren't endorsed by this school
Course
A&P 101
Subject
Statistics
Date
Dec 10, 2024
Pages
2
Uploaded by hede9362
10.8 Factorial Experiments 319 Table 10.12 Percentage blackened banana skin Experimenter Light Storage Percentage of (Block) C D blackened skin (yp;;) I 1 1 30 30 17 43 1 2 43 35 36 64 2 1 37 38 23 53 2 2 22:)350( 30.1:+38 il 1 1 49 60 41 61 1 2 57 46 31 34 2 1 20 63 64 34 2 2 40 47 62 42 1] 1 1 21 45 38 39 1 2 42711137112 P47" 26 2 1 41 74 24 51 2 2 38151227 31,1155 Table 10.13 Analysis of variance for the banana experiment Source of Degreesof ~ Sumof Mean Ratio p-value Variation Freedkom Squares Square Block (Experimenter) 2 1255.79 627.89 - C (Light) 1 80.08 80.08 0.42 0.5218 D (Storage) 1 154.08 154.08 0.80 0.3754 cD 1 24.08 2408 0.13 0.7250 Error 42 8061.88 191.95 Total 47 9575.92 hypothesis HOC D of no interaction between the treatment factors Light and Storage, using a Type I error probability of & = 0.01, is ms(C D) msE where ms(C D) = ss(CD)/(c — 1)(d — 1) and df is the number of error degrees of freedom calculated below. Since there are equal numbers of observations per cell, we use rule 4 of Chapter 7, page 202; so ss(CD)=bs Y Y 3% —bds Y 3% —bes Yy 37, +beds 7 = 24.0833. i J i J reject HY P if > Fle—1xd-1).4£,001 » Similarly, ssC = bds ) " y*; —beds 7. = 80.08, J ssD = bcs Z?zl — beds y* = 154.08, ss0 =cds ) ¥y —beds ¥ = 125579, h
320 Chapter 10 Complete Block Designs sstot = 9575.92. So, sSE = sstot — ss§ — ssC — ssD — ss(CD) = 8061.88, and df =(bcds —1)—(b—=1)—(c—1)=(d—1)—(c— 1)d — 1) =47-2-1-1-1=42. These values are shown in the analysis of variance table, Table 10.13. We can see that the mean square for blocks is much larger than the error mean square, so it was worthwhile designing this experiment as a block design. We also see that the mean square for the LightxStorage interaction is a lot smaller than the error mean square. As mentioned in the context of the resting metabolic rate experiment (Example 10.4.1, page 302), this is unusual when the model fits well, since the Light and Storage measurements include the error measurement. It suggests that the error mean square may have been inflated by some other source of variability, such as block x treatment interaction, that has been omitted from the model. An interaction plot of the two factors Light and Storage (averaged over blocks and the Lightx Storage interaction) is shown in Figure 10.5. There is no indication that hanging bananas (Storage level 1) might retard the ripening process. In fact, Storage level 1 seems to have given a higher percentage of blackened skin on average than Storage level 2. However, this apparent difference may be due to chance, as the treatment effects are not significantly different from each other. The experimenters commented that it was difficult to select the correct threshold levels for the image analysis and also that the bananas themselves seemed extremely variable. The experimenters felt that rather than draw firm conclusions at this stage, it might be worthwhile working to improve the experimental procedure to reduce variability and then to repeat the experiment. =] 10.9 Using SAS Software Figure 10.5 Interaction plot for the banana experiment The analysis of variance table for a complete block design can be obtained from any computer package that has an analysis of variance routine or a regression routine. It is good practice Vi, ‘ Light, i Vi : Storage, j ol - P 4“1 1 0+ . 0+ - ! 2 RO 2 1 2 5 W . 36 + 36 + 1 2 1 2 Storage, j Light, i