chapter 4: randomized blocks

Chapter 4: Chapter 4: Randomized blocksRandomized blocks

Petter MostadPetter [email protected]

Blocking and randomization

• Last time: Comparing the expectations of several populations. several populations.

• Earlier: The idea of blocking: Controlling a • Earlier: The idea of blocking: Controlling a parameter in such a way that its influence on the estimation of the effect of another on the estimation of the effect of another factor is minimized.

• The use of randomization.• The use of randomization.

Example

• The yield in a process for manufacturing penicillin is investigated. penicillin is investigated.

• Yield is influenced by treatment A, B, C, D • Yield is influenced by treatment A, B, C, D (columns) This effect is of interest.

• Yield is also influenced by blend 1,2,..,5 • Yield is also influenced by blend 1,2,..,5 (rows) This effect is not of interest.

• (Example from Box, Hunter, Hunter: • (Example from Box, Hunter, Hunter: Statistics for Experimenters)

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Sums of squares

• The sums of squares for treatments, for blocks, the total sum of squares and the blocks, the total sum of squares and the error sum of squares are defined similarly to before. before.

• They add up, as before. • They add up, as before.

Degrees of freedom

• The degrees of freedom for each sum of squares is defined similarly to beforesquares is defined similarly to before

• They add up, as before! • They add up, as before!

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Sums of squares:

560 = 264 + 70 + 226

Degrees of freedom:

19 = 4 + 3 + 1219 = 4 + 3 + 12

Mean squares, and test statistic

• As before, the mean squares are the sums of squares divided by their respective degrees squares divided by their respective degrees of freedom

• The test statistic is as before the mean square for the treatments divided by the square for the treatments divided by the mean square for the errors.

ANOVA table

The hypothesis test

• Assumptions: We have independent random samples from normal distributions with the same samples from normal distributions with the same variance, and each expectation is a sum of a group mean and a block mean. mean and a block mean.

• Null hypothesis: The group means are the same. Alternative hypothesis: At least two are different. Alternative hypothesis: At least two are different.

• Test statistic as abovehas; it has, under the null hypothesis, an F distribution with degrees of hypothesis, an F distribution with degrees of freedom found in the ANOVA table

• Reject for large values of the test statistic. • Reject for large values of the test statistic.

Checking the assumptions

• Check that it is reasonable to assume that the data are independent random samplesthe data are independent random samples

• Check the RESIDUALS to see if the group • Check the RESIDUALS to see if the group variances seem to be similar, if the data seem to be normally distributed, and check seem to be normally distributed, and check for independence of experimental runs.

Table 4.3 (p. 125)Randomized Complete Block Design for the Vascular Graft Experiment

Design and Analysis of Experiments, 6/E by Douglas C. Montgomery

Randomized Complete Block Design for the Vascular Graft Experiment

Table 4.4 (p. 125)Analysis of Variance for the Hardness Testing Experiment


Analysis of Variance for the Hardness Testing Experiment

Figure 4.4 (p. 129)Normal probability plot of residuals for Example 4-1.for Example 4-1.


Figure 4.5 (p. 129)Figure 4.5 (p. 129)Plot of residuals versus yij for Example 4-1.


Figure 4.6 (p. 129)Plot of residuals by extrusion pressure (treatment) and by batches of resin (block) for Example 4-1.


Example 4-1.

How to conclude –How to conclude –handling multiple testinghandling multiple testing

• If the p-value is small, it is customary to reject the null hypothesis and declare that reject the null hypothesis and declare that there is a difference between some of the group effects. group effects.

• To find which ones, you may need to do • To find which ones, you may need to do pairwise testing, which can lead to problems with multiple testing. with multiple testing.

• Ways to handle this exist: See textbook.

Additivity

• The null hypothesis means that the effects of the blocks and of the treatments are of the blocks and of the treatments are additive!

• Whether there is an effect from the treatments is checked under this assumptiontreatments is checked under this assumption

• If there is an interaction between the treatment and block effects, use factorial treatment and block effects, use factorial designs!

Missing data

• The computational formulas above reqire that there is exactly one observation for that there is exactly one observation for each combination of block and treatment.

• In practice, there may be missing data.• Possible to interpolate, however, it is • Possible to interpolate, however, it is

unnecessary: Use instead a formulation of a linear model, and computer computations! linear model, and computer computations!

Linear models• In general, we can ”explain” our data as a linear

combination of unknown parameters, plus an ”error term” combination of unknown parameters, plus an ”error term” that is normally distributed, with a fixed unknown variance.

• Such models are examples of linear models. • General mathematical solutions make it possible to test

values and equalities of parameters. values and equalities of parameters. • The methods we discuss above and below are special

cases, where the math is simpler. cases, where the math is simpler. • When there is missing data, it is still possible to use the

general framework of linear models. general framework of linear models.

Controlling for more than one Controlling for more than one factorfactor

• Above, we investigated one factor, and controlled for another. controlled for another.

• One may also control for two or more • One may also control for two or more factors.

• The trick is to construct an experiment • The trick is to construct an experiment where each controlled factor has a neutral influence on the estimation of the effect of influence on the estimation of the effect of interest.

Latin squares

• They are squares that indicate how to set various factors when performing an various factors when performing an experiment.

• They are usually written with latin letters, whence the name. whence the name.

• They can fairly easily be constructed for different sizes. different sizes.

Example

• Car pollution of a certain type depend on several factors: Car type, driver, and several factors: Car type, driver, and additives in the fuel. Assume we have 4 different cars, 4 different drivers, and 4 different cars, 4 different drivers, and 4 different additives.

• How can we test if there is an effect from the additives, while neutralizing the the additives, while neutralizing the (additive) effects from the cars and drivers?

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Sums of squares: 312 = 24 + 216 + 40 + 32312 = 24 + 216 + 40 + 32

Degrees of freedom: 15 = 3 + 3 + 3 + 615 = 3 + 3 + 3 + 6

Table 4.8 (p. 136)Latin Square Design for the Rocket Propellant Problem


Latin Square Design for the Rocket Propellant Problem

Table 4.9 (p. 137)Analysis of Variance for the Latin Square Design


Analysis of Variance for the Latin Square Design

Table 4.11 (p. 139)Analysis of Variance for the Rocket Propellant Experiment


Analysis of Variance for the Rocket Propellant Experiment

chapter 4: randomized blocks

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