PROBLEM TWO (ALKALOID CONCENTRATIONS IN TEA) 1. Statistical Design

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1 PROBLEM TWO (ALKALOID CONCENTRATIONS IN TEA) 1. Statistical Design The purpose of this experiment was to determine differences in alkaloid concentration of tea leaves, based on herb variety (Factor A) and fertilizer level (Factor B). The 18 temporary fields (F) are not of interest; the fields will be used for control and Figure 1. Hasse Diagram M 1 1 A 3 2 B 4 3 will be helpful in averaging out nuisance factors. Each temporary field has infinitely many "levels" and there is randomness due to both area planted and the process of (F ) AB 12 6 (E) spreading topsoil. Therefore the temporary fields will be thought of as random, drawn from a Normal population of infinitely many fields that might have been created. Although several designs may be appropriate, it is reasonable to consider this experiment as a Split-Plot Design. Figure 1 shows the Hasse Diagram for this model. Split-Plot design is appropriate here, since the experiment includes the 3 main characteristics of the design (Kowalski, 2003). (1) A different randomization scheme is used for herb variety and fertilizer level. Herb varieties are randomly assigned to a field, and fertilizer levels are randomly assigned to quarters of each field. (2) Each experimental unit for herb variety produces 4 tea leaf samples. Each experimental unit for fertilizer level produces only 1 sample. (3) Herb variety can be considered a hard to change factor. The randomization scheme restricts 1 variety of herb to each field in order to "protect from confusion at harvest". While a Split-Plot Design doesn t formally include a blocking variable, the design "may be viewed as a type of incomplete block design, where the whole plots are considered to be the blocks" (Kutner, 2005). In this experimental setup, the 18 fields will play the role of the whole plots. Thus we can interpret the temporary fields as a sort of blocking variable. Variables such as quality and thickness of topsoil, location of field or even construction of the greenhouse may have an effect on the final alkaloid concentration. These so called nuisance variables are troublesome, because we are not interested in their effect on the response variable. The use of a blocking variable helps control for this added source of variance, and can lead to more reliable estimates of the desired effects. The treatments of interest are the 12 combinations of herb variety and fertilizer level. The response is the alkaloid concentration in the processed tea leaf samples. In the following sections, we will determine the combinations of herb variety and fertilizer level which may lead to differences in the response variable. 1

2 PROBLEM TWO (ALKALOID CONCENTRATIONS IN TEA) Coffee 2 2. Analysis of Variance Model We begin by presenting the model (and it s assumptions) described by the Hasse Diagram in Figure 1. (1) Y ijk = µ + ρ i(j) + α j + β k + (αβ) jk + ɛ ijk ρ i(j) iid N(0, σ 2 ρ) independent of ɛ ijk iid N(0, σ 2 ) and 3 α j = j=1 4 β k = k=1 3 j=1 k=1 4 (αβ) jk = 0 Notice that the model inherently assumes there is no interaction between field and the factors of interest; any interaction here will be attributed to the ɛ term. To check this informally, we inspect the interaction plot given in Figure B1. Fortunately, we see the traces generally follow a parallel trend. Table 1 gives the ANOVA table for this model and Table A1 provides details on how these values were computed (Ane, 2009). Table 1. Analysis of Variance for Model (1) Source of Var Sum Sq Df Mean Sq F p-val A Whole-Plot Error B e-12 AB Split-Plot Error Total Note: For Split-Plot Design, the ANOVA table is separated by Whole-Plot and Split-Plot. This serves as a reminder that the denominator for the F-test depends on what we are testing (Zhang, 2009). Figure s 2 and 3 provide interaction plots for Factors A and B. From these plots, we can see that interaction between these two factors is negligible as the traces are very nearly parallel. Formal tests will be considered in Figure 2. Fertilizer as Trace Figure 3. Herb Variety as Trace the next section, but for now we simply comment that the additive effects of Factor B (Fertilizer level) seem to quite significant. There appears to be a strong relationship with increased fertilizer level and alkaloid concentration in the processed sample. On the other hand, the additive effects of Factor A (Herb variety) are present, but the magnitude looks to be negligible. 2

3 PROBLEM TWO (ALKALOID CONCENTRATIONS IN TEA) Coffee 3 Model (1) appears to be a reasonable fit for the data, the studentized residuals are calculated (using r i = e i ) MSE(1 hi) and plotted in Figure 4. They appear to be linear and normally distributed with constant variance; all of the residuals also fall below the Bonferroni critical value (dotted line). We also note that the Shapiro-Wilk test for Normality and Figure 4. Studentized Residuals for Model (1) the Modified-Levene test for constant variance across treatments provide p-values of and respectively. Figures 5 presents a QQ-plot and the distribution of the studentized residuals, both are encouraging. As usual, Figure 5. Normality Assumption checking independence is the most difficult part of the model validation process. We can simply say that it is reasonable to assume the differences due to field are independent of each other. While interaction between fields and the main effects would be tacked onto the primary error term, Figure A1 leads us to believe that these interactions are not a major concern. We conclude by noting that the model has quite good predictive power. This manifests itself in the statistic Ω 2 o = , where Ω 2 o is a generalization of R 2 to linear mixed models (Xu, 2003). 3. Inference We now argue that the interaction effects of the AB term and the main effects of Factor A appear negligible, while the main effects of Factor B appear significant). We supplement this conviction with Box-Plots given in Figures B2 and B3 and with formal F-tests. The F-test for herb variety, the whole plot factor must be constructed with MSWPE as the denominator, while the F-tests for fertilizer level and the interaction are constructed with MSSPE in 3

4 PROBLEM TWO (ALKALOID CONCENTRATIONS IN TEA) Coffee 4 the denominator (Dean, 1999). The distribution used to obtain the p-value must also reflect this (see Table A1 for details). The p-values in Table 1 accurately reflect these ideas, and support our beliefs. We see that neither factor A main effects or the interaction term are particularly significant, but the main effects of factor B are very significant. That is, increasing fertilizer level (g/m 2 ) will have a meaningful effect on the alkaloid concentration in the processed sample. We quickly comment that a drawback of Split-Plot design is that Factor A effects cannot be estimated with as much precision as the split-plot treatments. Perhaps with another design, we may have been able to show substantial herb variety effects. Given Figures 3, 2B, 3B and 6 however, we are confident in assuming no herb variety effect. Finally, we consider multiple comparison of the treatments using Bonferroni procedures (For Tukey, see Fig B4). Since the presence of an interaction effect is negligible, it will be sufficient to consider contrasts of the form L = µ j µ j and K = µ k µ k. This leads to a total of 9 contrasts, for which we desire simultaneous inference. As before, we must be careful with our degree s of freedom and choice of "MSE" depending on inference for L or K. All of the intervals for the L h overlap zero, indicating no difference between means. On the other hand, all of the Figure 6. 90% Bonferroni Intervals for L Contrasts Figure 7. 90% Bonferroni Intervals for K Contrasts intervals for the difference in fertilizer effects avoid zero except for K 1 = µ 4 µ 3. Since we used a Bonferroni correction value of 9, these results hold simultaneously. Hence, all treatment means will be different except those of the form µ j4 µ j 3 for any j and j. Descriptions of each L h and K h contrast and their corresponding p-value is given in Table A2 (Federer, 1991). 4. Summary of Results The Split-Plot Design led to some limitations, but provided precise estimates of the fertilizer level effects. From the analysis, we see that regardless of herb variety, increasing the g/m 2 of fertilizer used tends to increase the final alkaloid concentration. Future researchers may consider choosing a design that would allow the effects due to herb variety to be measured more precisely. Of course, we may also be content knowing that fertilizer is a far more influential factor when it comes to alkaloid concentration in tea leaves. 4

5 PROBLEM TWO (ALKALOID CONCENTRATIONS IN TEA) Coffee 5 Appendix A - Tables Table A1. Calculations for Table 1 Source of Var Sum Sq Df Mean Sq F p-val A a 1 MSA/MSW P E F (2, 15) Whole-Plot Error a(s 1) - B b 1 MSB/MSSP E F (3, 45) AB (a 1)(b 1) MSAB/MSSP E F (6, 45) Split-Plot Error a(s 1)(b 1) - Total abs 1 - Calculations and Computations *These values were calculated using the relationship MS = SS/df ** These values can be easily calculated, but the computer output gives the correct SS here ***Equivalent to MSE, thus it is simply ˆσ 2, provided by the summary of the model. ****SST ot = i j k (Y ijk Ȳ ) 2 *****SSWPE = SSWTot - SSA with SSWTot = b i j Y ij 2 sab Y 2 Table A2. Hypothesis Test P-values for L and K Contrasts (α =.10) L h ˆ Lh SE( ˆ L h ) df p-val K h ˆ Kh SE( ˆ K h ) df p-val L 1 = µ 2 µ K 1 = µ 4 µ L 2 = µ 2 µ K 2 = µ 4 µ e-5* L 3 = µ 1 µ K 3 = µ 4 µ e-10* K 4 = µ 3 µ * K 5 = µ 3 µ e-8* K 6 = µ 2 µ e-4* * implies H o is rejected. H o : L h = 0 Reject if p <.10/9 =.0111 H o : K h = 0 Reject if p <.10/9 =

6 PROBLEM TWO (ALKALOID CONCENTRATIONS IN TEA) Coffee 6 Appendix B - Figures Figure B1. Field vs. Fertillizer Level Figure B2. Box-Plots by Herb Variety Figure B3. Box-Plots by Fertilizer Level Figure B4. 90 %Tukey Intervals for L Contrasts Figure B5. 90% Tukey Intervals for K Contrasts 6

7 PROBLEM TWO (ALKALOID CONCENTRATIONS IN TEA) Coffee 7 References Ane, C Factorial and split plot designs. Dean, A. and D. Voss Design and Analysis of Experiments. Chapter 19. Federer, W. and C.E. McCulloch Multiple Comparisons in Split Block and Split-Split Plot Designs. Kowalski, S. and K. Potcner How to Recognize a Split-Plot Experiment. Kutner, M., Nachtsheim, C., Neter, J. and William Li Applied Linear Statistical Models. Chapter 27. Xu, R Measuring Explained Variance in Linear Mixed Effects Models. Zhang, H Chapter 19 Split-Plot Designs. 7