1 STA 536 – Experiments with More Than One Factor Experiments with More Than One Factor (Chapter 3) 3.1 Paired comparison design. 3.2 Randomized block design. 3.3 Two-way layout. 3.5 multi-way layout. 3.6 Latin square design. 3.7 Graeco-Latin square design. 3.8 Balanced incomplete block design Analysis of covariance (ANCOVA).

2 STA 536 – Experiments with More Than One Factor 3.1 Paired Comparison Design Example: Sewage Experiment Objective : To compare two methods MSI and SIB for determining chlorine content in sewage effluents; y = residual chlorine reading. Experimental Design: Eight samples were collected at different doses and contact times. Two methods were applied to each of the eight samples. It is a paired comparison design because the pair of treatments are applied to the same samples (or units). Mean of d =0.4138

3 STA 536 – Experiments with More Than One Factor Paired Comparison Design vs. Unpaired Design Paired Comparison Design : Two treatments are randomly assigned to each block of two units. Can eliminate block-to-block variation and is effective if such variation is large. Examples : pairs of twins, eyes, kidneys, left and right feet. (Subject-to-subject variation much larger than within-subject variation). Unpaired Design : Each treatment is applied to a separate set of units, or called the two-sample problem. Useful if pairing is unnecessary; also it has more degrees of freedom for error estimation. An unpaired design is a completely randomized design (i.e., one-way layout)

4 STA 536 – Experiments with More Than One Factor Paired t-tests

5 STA 536 – Experiments with More Than One Factor Unpaired t-tests

6 STA 536 – Experiments with More Than One Factor Analysis Results : t-tests Unpaired t-test fails to declare significant difference because its denominator is too large. Why ? Because the denominator contains the sample-to-sample variation component. The p-values are Wrong analysis

7 STA 536 – Experiments with More Than One Factor Paired or Unpaired? Which design is more powerful? The answer depends If there is a large sample to sample variation, a paired design is more effective. Otherwise, an unpaired design is more effective. Recall: For blocking to be effective, the units should be arranged so that the within-block variation is much smaller than the between-block variation.

8 STA 536 – Experiments with More Than One Factor Alternative analysis using ANOVA and F test Wrong to analyze by ignoring pairing. A better explanation is given by ANOVA. F-statistic in ANOVA for paired design equals ; similarly, F-statistic in ANOVA for unpaired design equals. Data can be analyzed in two equivalent ways. In the correct analysis (Table 2), the total variation is decomposed into three components; the largest one is the sample-to-sample variation (its MS = 34.77). In the unpaired analysis (Table 3), this component is mistakenly included in the residual SS, thus making the F-test powerless.

9 STA 536 – Experiments with More Than One Factor Conclusion: Both the treatment variable (method) and the blocking variable (sample) are significant. (Wrong) Conclusion: the treatments (method) are not significant different.

10 STA 536 – Experiments with More Than One Factor 3.2 Randomized Block Design Recall the principles of blocking and randomization in Chapter 1. In a randomized block design (RBD), k treatments are randomly assigned to each block (of k units); there are in total b blocks. Total sample size N = bk. Paired comparison design is a special case with k = 2. (Why ?) Recall that in order for blocking to be effective, the units within a block should be more homogeneous than units between blocks.

11 STA 536 – Experiments with More Than One Factor 2.2 Randomized Block Design Example Objective : To compare four methods for predicting the shear strength for steel plate girders (k = 4,b = 9).

12 STA 536 – Experiments with More Than One Factor Model and Estimation The zero-sum constraints

13 STA 536 – Experiments with More Than One Factor ANOVA

14 STA 536 – Experiments with More Than One Factor Testing and Multiple Comparisons

15 STA 536 – Experiments with More Than One Factor Simultaneous Confidence Intervals How about the Bonferroni method?

16 STA 536 – Experiments with More Than One Factor The F statistic in (7) has the value (1.514/3)/(0.166/24)= Therefore, the p-value for testing the difference between methods is Prob(F(3,24) > 73.03)=0.00. The small p-value suggests that the methods are different after block (girder) effects are adjusted. Analysis of Girder Experiment : F test

17 STA 536 – Experiments with More Than One Factor Analysis of Girder Experiment : Multiple Comparisons

18 STA 536 – Experiments with More Than One Factor Another example: penicillin yield experiment Data from a randomized block experiment in which a process of the manufacture of penicillin was investigated. Want to try 4 variants of the process, called treatments A, B, C, D. Yield is the response. The properties of an important raw material varied considerably, and this might cause considerable differences in yields. A blend of the material could be made sufficient to make 4 runs. Each blend forms a block. The 4 treatments were run randomly within each block; see the superscripts. Note that the randomization is restricted within the blocks.

19 STA 536 – Experiments with More Than One Factor Another example: penicillin yield experiment - DATA

20 STA 536 – Experiments with More Than One Factor Data penicillin; input blend $ x1-x4; y=x1; treatment="A"; output; y=x2; treatment="B"; output; y=x3; treatment="C"; output; y=x4; treatment="D"; output; cards; Blend Blend Blend Blend Blend ; proc GLM; class blend treatment; model y=blend treatment; run;

21 STA 536 – Experiments with More Than One Factor SAS Output The blocking variable (blend) is significant at 5% level; the treatments are not different after block effects are adjusted.

22 STA 536 – Experiments with More Than One Factor Wrong analysis ignoring blocking proc GLM; class blend treatment; model y=treatment; run;

23 STA 536 – Experiments with More Than One Factor 3.3 Two-way layout This is similar to RBD. The only difference is that here we have two treatment factors instead of one treatment factor and one block factor. Interested in assessing interaction effect between the two treatments. In blocking, block×treatment interaction is assumed negligible.

24 STA 536 – Experiments with More Than One Factor Bolt experiment Objective: To determine what factors affect the torque values. Torque is the work (i.e., force×distance) required to tighten a locknut. Two experimental factors: type of plating: heat treated (HT), cadmium and wax plated (C&W), phosphate and oil plated (P&O) test medium: mandrel, bolt Data are the torque values for 10 locknuts for each of the six treatments.

25 STA 536 – Experiments with More Than One Factor two-way layout, which involves two treatment factors. In general, a two-way layout for two factors (A has I levels and B has J levels) has IJ factor level combinations. Each combination is a treatment. Suppose that the number of replicates is n, and for each replicate, IJ units should be randomly assigned to the IJ treatments. Total IJn observations The linear model for the two-way layout is

26 STA 536 – Experiments with More Than One Factor Constrains The interpretations of the parameters and estimates depend on the constraints.

27 STA 536 – Experiments with More Than One Factor Estimates Under zero-sum constrains Estimates under baseline constrains What are the fitted values?

28 STA 536 – Experiments with More Than One Factor The ANOVA decomposition:

29 STA 536 – Experiments with More Than One Factor ANOVA

30 STA 536 – Experiments with More Than One Factor Tests

31 STA 536 – Experiments with More Than One Factor Analysis of Bolt Experiment Conclusions: Both factors and their interactions are significant. Multiple comparisons of C&W, HT and P&O can be performed by using Tukey method with k = 3 and 54 error dfs. Another method is considered in the following pages using regression

32 STA 536 – Experiments with More Than One Factor Comparison with randomized block design RBD has one blocking factor and one experimental factor A two-way layout studies two experimental factors In a 2-way layout, randomization is applied to all IJ units. In a RBD, randomization is applied within each block. The blocks represent a restriction on randomization.

33 STA 536 – Experiments with More Than One Factor Two-way layout For an unreplicated experiment, n = 1 and there is not enough degrees of freedom for estimating both the interaction effects The SS AB should be used as the residual sum of squares. Alternatively, we can use the following Tukeys one- degree of freedom for non-additivity to test interaction effects: where is an unknown constant. If there is no interaction effects, should be zero. This method can also be used to test if there is an interaction effect between treatments and blocks in a RBD.

34 STA 536 – Experiments with More Than One Factor Practical modeling and estimating the effects Qualitative factor: baseline constraints are easy to interpret. Quantitative factor: orthogonal polynomial contrasts are easy to interpret.

35 STA 536 – Experiments with More Than One Factor Two Qualitative Factors

36 STA 536 – Experiments with More Than One Factor Regression Model (continued) for one replicate

37 STA 536 – Experiments with More Than One Factor Regression Model (continued) Interpretation of parameters

38 STA 536 – Experiments with More Than One Factor Regression Analysis Results

39 STA 536 – Experiments with More Than One Factor Adjusted p Values

40 STA 536 – Experiments with More Than One Factor Box-Whisker Plot : Bolt Experiment The plot suggests that the constant variance assumption in (5) does not hold and that the variance of y for bolt is larger than that for mandrel. These are aspects that cannot be discovered by regression analysis alone.

41 STA 536 – Experiments with More Than One Factor An interaction plot shows the mean responses of all treatments. It is a useful tool to see the relationship between two factors.

42 STA 536 – Experiments with More Than One Factor 3.5 Multi-Way Layout Consider a three-way layout (or 3-factor factorial design) A has I levels, B has J levels and C has K levels. There are total IJK treatments, each replicated n times. For each replicate, IJK units are randomly assigned to the IJK treatments. The linear model for the three-way layout is:

43 STA 536 – Experiments with More Than One Factor Estimates Let where are the estimates of the parameters under the zero-sum constraints.

44 STA 536 – Experiments with More Than One Factor ANOVA

45 STA 536 – Experiments with More Than One Factor Multi-Way Layout Estimation, F test, residual analysis are similar to those for two-way layout.