1 STA 536 – Experiments with More Than One Factor Experiments with More Than One Factor (Chapter...
-
Upload
colt-estabrook -
Category
Documents
-
view
216 -
download
0
Transcript of 1 STA 536 – Experiments with More Than One Factor Experiments with More Than One Factor (Chapter...
CommandButton1CommandButton1CommandButton1CommandButton1 1STA 536 – Experiments with More Than One Factor
CommandButton1CommandButton1CommandButton1CommandButton1
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).
CommandButton1CommandButton1CommandButton1CommandButton1 2STA 536 – Experiments with More Than One Factor
CommandButton1CommandButton1CommandButton1CommandButton1
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).
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
CommandButton1CommandButton1CommandButton1CommandButton1 3STA 536 – Experiments with More Than One Factor
CommandButton1CommandButton1CommandButton1CommandButton1
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)
CommandButton1CommandButton1CommandButton1CommandButton1 4STA 536 – Experiments with More Than One Factor
CommandButton1CommandButton1CommandButton1CommandButton1
Paired t-tests
CommandButton1CommandButton1CommandButton1CommandButton1 5STA 536 – Experiments with More Than One Factor
CommandButton1CommandButton1CommandButton1CommandButton1
Unpaired t-tests
CommandButton1CommandButton1CommandButton1CommandButton1 6STA 536 – Experiments with More Than One Factor
CommandButton1CommandButton1CommandButton1CommandButton1
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
CommandButton1CommandButton1CommandButton1CommandButton1 7STA 536 – Experiments with More Than One Factor
CommandButton1CommandButton1CommandButton1CommandButton1
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.
CommandButton1CommandButton1CommandButton1CommandButton1 8STA 536 – Experiments with More Than One Factor
CommandButton1CommandButton1CommandButton1CommandButton1
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.
CommandButton1CommandButton1CommandButton1CommandButton1 9STA 536 – Experiments with More Than One Factor
CommandButton1CommandButton1CommandButton1CommandButton1
Conclusion: Both the treatment variable (method) and the blocking variable (sample) are significant.
Conclusion: Both the treatment variable (method) and the blocking variable (sample) are significant.
(Wrong) Conclusion: the treatments (method) are not significant different.
(Wrong) Conclusion: the treatments (method) are not significant different.
CommandButton1CommandButton1CommandButton1CommandButton1 10STA 536 – Experiments with More Than One Factor
CommandButton1CommandButton1CommandButton1CommandButton1
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.
CommandButton1CommandButton1CommandButton1CommandButton1 11STA 536 – Experiments with More Than One Factor
CommandButton1CommandButton1CommandButton1CommandButton1
2.2 Randomized Block Design Example
Objective : To compare four methods for predicting the shear strength for steel plate girders (k = 4,b = 9).
CommandButton1CommandButton1CommandButton1CommandButton1 12STA 536 – Experiments with More Than One Factor
CommandButton1CommandButton1CommandButton1CommandButton1
Model and Estimation
The zero-sum constraints
CommandButton1CommandButton1CommandButton1CommandButton1 13STA 536 – Experiments with More Than One Factor
CommandButton1CommandButton1CommandButton1CommandButton1
ANOVA
CommandButton1CommandButton1CommandButton1CommandButton1 14STA 536 – Experiments with More Than One Factor
CommandButton1CommandButton1CommandButton1CommandButton1
Testing and Multiple Comparisons
CommandButton1CommandButton1CommandButton1CommandButton1 15STA 536 – Experiments with More Than One Factor
CommandButton1CommandButton1CommandButton1CommandButton1
Simultaneous Confidence Intervals
How about the Bonferroni method?
CommandButton1CommandButton1CommandButton1CommandButton1 16STA 536 – Experiments with More Than One Factor
CommandButton1CommandButton1CommandButton1CommandButton1
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
CommandButton1CommandButton1CommandButton1CommandButton1 17STA 536 – Experiments with More Than One Factor
CommandButton1CommandButton1CommandButton1CommandButton1
Analysis of Girder Experiment : Multiple Comparisons
CommandButton1CommandButton1CommandButton1CommandButton1 18STA 536 – Experiments with More Than One Factor
CommandButton1CommandButton1CommandButton1CommandButton1
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.
CommandButton1CommandButton1CommandButton1CommandButton1 19STA 536 – Experiments with More Than One Factor
CommandButton1CommandButton1CommandButton1CommandButton1
Another example: penicillin yield experiment - DATA
CommandButton1CommandButton1CommandButton1CommandButton1 20STA 536 – Experiments with More Than One Factor
CommandButton1CommandButton1CommandButton1CommandButton1
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;
CommandButton1CommandButton1CommandButton1CommandButton1 21STA 536 – Experiments with More Than One Factor
CommandButton1CommandButton1CommandButton1CommandButton1
SAS Output
The blocking variable (blend) is significant at 5% level; the treatments are not different after block effects are adjusted.
CommandButton1CommandButton1CommandButton1CommandButton1 22STA 536 – Experiments with More Than One Factor
CommandButton1CommandButton1CommandButton1CommandButton1
Wrong analysis ignoring blockingproc GLM; class blend treatment;
model y=treatment;
run;
CommandButton1CommandButton1CommandButton1CommandButton1 23STA 536 – Experiments with More Than One Factor
CommandButton1CommandButton1CommandButton1CommandButton1
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.
CommandButton1CommandButton1CommandButton1CommandButton1 24STA 536 – Experiments with More Than One Factor
CommandButton1CommandButton1CommandButton1CommandButton1
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.
CommandButton1CommandButton1CommandButton1CommandButton1 25STA 536 – Experiments with More Than One Factor
CommandButton1CommandButton1CommandButton1CommandButton1
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
CommandButton1CommandButton1CommandButton1CommandButton1 26STA 536 – Experiments with More Than One Factor
CommandButton1CommandButton1CommandButton1CommandButton1
Constrains
The interpretations of the parameters and estimates depend on the constraints.
CommandButton1CommandButton1CommandButton1CommandButton1 27STA 536 – Experiments with More Than One Factor
CommandButton1CommandButton1CommandButton1CommandButton1
Estimates Under zero-sum constrains
Estimates under baseline constrains
What are the fitted values?
CommandButton1CommandButton1CommandButton1CommandButton1 28STA 536 – Experiments with More Than One Factor
CommandButton1CommandButton1CommandButton1CommandButton1
The ANOVA decomposition:
CommandButton1CommandButton1CommandButton1CommandButton1 29STA 536 – Experiments with More Than One Factor
CommandButton1CommandButton1CommandButton1CommandButton1
ANOVA
CommandButton1CommandButton1CommandButton1CommandButton1 30STA 536 – Experiments with More Than One Factor
CommandButton1CommandButton1CommandButton1CommandButton1
Tests
CommandButton1CommandButton1CommandButton1CommandButton1 31STA 536 – Experiments with More Than One Factor
CommandButton1CommandButton1CommandButton1CommandButton1
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
CommandButton1CommandButton1CommandButton1CommandButton1 32STA 536 – Experiments with More Than One Factor
CommandButton1CommandButton1CommandButton1CommandButton1
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.
CommandButton1CommandButton1CommandButton1CommandButton1 33STA 536 – Experiments with More Than One Factor
CommandButton1CommandButton1CommandButton1CommandButton1
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.
CommandButton1CommandButton1CommandButton1CommandButton1 34STA 536 – Experiments with More Than One Factor
CommandButton1CommandButton1CommandButton1CommandButton1
Practical modeling and estimating the effects
Qualitative factor: baseline constraints are easy to interpret.
Quantitative factor: orthogonal polynomial contrasts are easy to interpret.
CommandButton1CommandButton1CommandButton1CommandButton1 35STA 536 – Experiments with More Than One Factor
CommandButton1CommandButton1CommandButton1CommandButton1
Two Qualitative Factors
CommandButton1CommandButton1CommandButton1CommandButton1 36STA 536 – Experiments with More Than One Factor
CommandButton1CommandButton1CommandButton1CommandButton1
Regression Model (continued)
for one replicate
CommandButton1CommandButton1CommandButton1CommandButton1 37STA 536 – Experiments with More Than One Factor
CommandButton1CommandButton1CommandButton1CommandButton1
Regression Model (continued)Interpretation of parameters
CommandButton1CommandButton1CommandButton1CommandButton1 38STA 536 – Experiments with More Than One Factor
CommandButton1CommandButton1CommandButton1CommandButton1
Regression Analysis Results
CommandButton1CommandButton1CommandButton1CommandButton1 39STA 536 – Experiments with More Than One Factor
CommandButton1CommandButton1CommandButton1CommandButton1
Adjusted p Values
CommandButton1CommandButton1CommandButton1CommandButton1 40STA 536 – Experiments with More Than One Factor
CommandButton1CommandButton1CommandButton1CommandButton1
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.
CommandButton1CommandButton1CommandButton1CommandButton1 41STA 536 – Experiments with More Than One Factor
CommandButton1CommandButton1CommandButton1CommandButton1
An interaction plot shows the mean responses of all treatments. It is a useful tool to see the relationship between two factors.
CommandButton1CommandButton1CommandButton1CommandButton1 42STA 536 – Experiments with More Than One Factor
CommandButton1CommandButton1CommandButton1CommandButton1
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:
CommandButton1CommandButton1CommandButton1CommandButton1 43STA 536 – Experiments with More Than One Factor
CommandButton1CommandButton1CommandButton1CommandButton1
EstimatesLet
where
are the estimates of the parameters under the zero-sum constraints.
CommandButton1CommandButton1CommandButton1CommandButton1 44STA 536 – Experiments with More Than One Factor
CommandButton1CommandButton1CommandButton1CommandButton1
ANOVA
CommandButton1CommandButton1CommandButton1CommandButton1 45STA 536 – Experiments with More Than One Factor
CommandButton1CommandButton1CommandButton1CommandButton1
Multi-Way Layout
Estimation, F test, residual analysis are similar to those for two-way layout.