1 STA 536 – Experiments with More Than One Factor Experiments with More Than One Factor (Chapter...

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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. 3.10 Analysis of covariance (ANCOVA).

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

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)

Paired t-tests

Unpaired t-tests

Analysis Results : t-tests

Unpaired t-test fails to declare significant difference because its denominator 2.0863 is too large.

Why ? Because the denominator contains the sample-to-sample

variation component.

The p-values areThe p-values are Wrong analysis

Wrong analysis

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.

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.

Conclusion: Both the treatment variable (method) and the blocking variable (sample) are significant.

(Wrong) Conclusion: the treatments (method) are not significant different.

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.

2.2 Randomized Block Design Example

Objective : To compare four methods for predicting the shear strength for steel plate girders (k = 4,b = 9).

Model and Estimation

The zero-sum constraints

Testing and Multiple Comparisons

Simultaneous Confidence Intervals

How about the Bonferroni method?

The F statistic in (7) has the value (1.514/3)/(0.166/24)= 73.03.

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

Analysis of Girder Experiment : Multiple Comparisons

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.

Another example: penicillin yield experiment - DATA

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;Blend1 89 88 97 94Blend2 84 77 92 79Blend3 81 87 87 85Blend4 87 92 89 84Blend5 79 81 80 88;

proc GLM; class blend treatment; model y=blend treatment;run;

SAS Output

The blocking variable (blend) is significant at 5% level; the treatments are not different after block effects are adjusted.

Wrong analysis ignoring blockingproc GLM; class blend treatment;

model y=treatment;

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.

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.

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

Constrains

The interpretations of the parameters and estimates depend on the constraints.

Estimates Under zero-sum constrains

Estimates under baseline constrains

What are the fitted values?

The ANOVA decomposition:

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 df’s.

Another method is considered in the following pages using regression

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.

Two-way layout

For an unreplicated experiment, n = 1 and there is not enough degrees of freedom for estimating both the interaction effects

The SSAB should be used as the residual sum of squares. Alternatively, we can use the following Tukey’s “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.

Practical modeling and estimating the effects

Qualitative factor: baseline constraints are easy to interpret.

Quantitative factor: orthogonal polynomial contrasts are easy to interpret.

Two Qualitative Factors

Regression Model (continued)

for one replicate

Regression Model (continued)Interpretation of parameters

Regression Analysis Results

Adjusted p Values

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.

An interaction plot shows the mean responses of all treatments. It is a useful tool to see the relationship between two factors.

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:

EstimatesLet

are the estimates of the parameters under the zero-sum constraints.

Multi-Way Layout

Estimation, F test, residual analysis are similar to those for two-way layout.

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