Chapter 𝟒 Randomized Blocks, Latin Squares, and Related Designs Pages 𝟏𝟑𝟗−𝟏𝟖𝟐

Topics Randomized complete block design (RCBD) Latin square designs Graeco Latin square designs

Blocking Principle Blocking is a technique for dealing with nuisance factors. A nuisance factor is a factor that probably has some effect on the response, but it’s of no interest to the experimenter…however, the variability it transmits to the response needs to be minimized. Examples include batches of raw material, operators, pieces of test equipment, time (shifts, days, etc.), different experimental units.

If the nuisance variable is known and controllable, we use blocking If the nuisance variable is known and controllable, we use blocking. Many industrial experiments involve blocking (or should). Failure to block is a common flaw in designing an experiment (consequences?). If the nuisance factor is known and uncontrollable, sometimes we can use the analysis of covariance (Chapter 15) to remove the effect of the nuisance factor from the analysis. If the nuisance factor is unknown and uncontrollable (a “lurking” variable), we hope that randomization balances out its impact across the experiment.

Example Consider a hardness testing machine that presses a rod with a pointed tip into a metal test coupon with a known force. By measuring the depth of the depression caused by the tip, the hardness of the coupon is determined. We wish to determine whether or not four different tips produce different readings on a hardness testing machine.

Structure of a completely randomized single factor experiment : 4 ⅹ 4 = 16 runs, requiring 16 coupons. Suppose the metal coupons are taken from ingots produced in different heats so that coupons differ slightly in their hardness. The coupons will contribute to the variability observed in the hardness data.

The experimental error reflects both random error and variability between coupons. The test coupons are a source of nuisance variability. The experimenter may want to test the tips across coupons of various hardness levels ⇒ the need for blocking.

RCBD To conduct this experiment as a RCBD, assign all 4 tips to each coupon. Each coupon is called a “block”; that is, it’s a more homogenous experimental unit on which to test the tips. Variability between blocks can be large, but variability within a block should be relatively small. In general, a block is a specific level of the nuisance factor.

A complete replicate of the basic experiment is conducted in each block. A block represents a restriction on randomization. All runs within a block are randomized. RCBD improves the comparison accuracy among tips by eliminating the variability among coupons.

We use 𝑏 = 4 blocks:

Notice the two-way structure of the experiment. Once again, we are interested in testing the equality of treatment means, but now we have to remove the variability associated with the nuisance factor (the blocks).

Fixed Effects Model for RCBD There are 𝑎 treatments and 𝑏 blocks. A fixed effects model for the RCBD is 𝑦 𝑖𝑗 = 𝜇+ 𝜏 𝑖 + 𝛽 𝑗 + 𝜀 𝑖𝑗 𝑖=1,2,…,𝑎 𝑗=1,2,…,𝑏  : overall mean 𝜏 𝑖 : 𝑖 𝑡ℎ treatment effect 𝛽 𝑗 : 𝑗 𝑡ℎ block effect 𝜀 𝑖𝑗 : random error iid 𝑁(0, 𝜎 2 ) 𝑖=1 𝑎 𝜏 𝑖 =0 and 𝑗=1 𝑏 𝛽 𝑗 =0

The relevant hypotheses are 𝐻 0 : 𝜇 1 = 𝜇 2 =…= 𝜇 𝑎 where 𝜇 𝑖 = 1 𝑏 𝑗=1 𝑏 𝜇+ 𝜏 𝑖 + 𝛽 𝑗 = 𝜇+ 𝜏 𝑖 Alternatively, 𝐻 0 : 𝜏 1 = 𝜏 2 =…= 𝜏 𝑎 =0

Extension of ANOVA to RCBD 𝑖=1 𝑎 𝑗=1 𝑏 𝑦 𝑖𝑗 − 𝑦 .. 2 = 𝑖=1 𝑎 𝑗=1 𝑛 𝑦 𝑖. − 𝑦 .. + 𝑦 .𝑗 − 𝑦 .. + 𝑦 𝑖𝑗 − 𝑦 𝑖. − 𝑦 .𝑗 + 𝑦 .. 2 =𝑏 𝑖=1 𝑎 𝑦 𝑖. − 𝑦 .. 2 +𝑎 𝑖=1 𝑏 𝑦 .𝑗 − 𝑦 .. 2 + 𝑖=1 𝑎 𝑗=1 𝑏 𝑦 𝑖𝑗 − 𝑦 𝑖. − 𝑦 .𝑗 + 𝑦 .. 2 𝑆𝑆 𝑇 = 𝑆𝑆 𝑇𝑟𝑒𝑎𝑡𝑚𝑒𝑛𝑡 + 𝑆𝑆 𝐵𝑙𝑜𝑐𝑘𝑠 + 𝑆𝑆 𝐸

The degrees of freedom for the sums of squares 𝑆𝑆 𝑇 = 𝑆𝑆 𝑇𝑟𝑒𝑎𝑡𝑚𝑒𝑛𝑡 + 𝑆𝑆 𝐵𝑙𝑜𝑐𝑘𝑠 + 𝑆𝑆 𝐸 are as follows: 𝑎𝑏−1=𝑎−1+𝑏−1+(𝑎−1)(𝑏−1) Ratios of sums of squares to their degrees of freedom result in mean squares.

𝐸 𝑀𝑆 𝑇𝑟𝑒𝑎𝑡𝑚𝑒𝑛𝑡𝑠 = 𝜎 2 + 𝑏 𝑖=1 𝑎 𝜏 𝑖 2 𝑎−1 𝐸 𝑀𝑆 𝑏𝑙𝑜𝑐𝑘𝑠 = 𝜎 2 + 𝑎 𝑗=1 𝑏 𝛽 𝑗 2 𝑏−1 𝐸 𝑀𝑆 𝐸 = 𝜎 2

The ratio of the mean square for treatments to the error mean square is an 𝐹 statistic that can be used to test the hypothesis of equal treatment means.

Example Consider a medical device manufacturer producing vascular grafts. Grafts are produced by extruding billets of PTFE resin combined with a lubricant into tubes. Some of the tubes in a production run contain small protrusions on the external surface, known as flicks. The product developer suspects that the extrusion pressure affects the occurrence of flicks.

The resin is manufactured by an external supplier and is delivered in batches. The engineer also suspects there may be significant batch-to-batch variation. The engineer decides to investigate the effect of four different levels of extrusion pressure on flicks using a RCBD, considering batches of resin as blocks.

To conduct this experiment as a RCBD, assign all 4 pressures to each of the 6 batches of resin. Each batch of resin is called a “block”; that is, it’s a more homogenous experimental unit on which to test the extrusion pressures. The response variable is the yield, i.e, the percentage of tubes in the production run that did not contain any flicks.

Parameter Estimation Point estimate 𝜇 = 𝑦 .. 𝜏 𝑖 = 𝑦 𝑖. − 𝑦 .. 𝛽 𝑗 = 𝑦 .𝑗 − 𝑦 .. Fitted values of 𝑦 𝑖𝑗 𝑦 𝑖𝑗 = 𝑦 𝑖. + 𝑦 .𝑗 − 𝑦 .. Residuals 𝑒 𝑖𝑗 = 𝑦 𝑖𝑗 − 𝑦 𝑖𝑗 = 𝑦 𝑖𝑗 −( 𝑦 𝑖. + 𝑦 .𝑗 − 𝑦 .. )

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Basic residual plots indicate that normality and constant variance assumptions are satisfied. No obvious problems with randomization No patterns in the residuals vs block Can also plot residuals vs the pressure (residuals by factor) These plots provide more information about the constant variance assumption and possible outliers.

Multiple Comparisons

Other Aspects of the RCBD The RCBD utilizes an additive model – no interaction between treatments and blocks. May consider treatments and/or blocks as random effects What are the consequences of not blocking if we should have?

Latin Square Design These designs are used to simultaneously control (or eliminate) two sources of nuisance variability. A significant assumption is that the three factors (treatments, nuisance factors) do not interact. If this assumption is violated, the latin square design will not produce valid results.

𝑦 𝑖𝑗𝑘 = 𝜇+ 𝛼 𝑖 + 𝜏 𝑗 + 𝛽 𝑘 + 𝜀 𝑖𝑗𝑘 The statistical (effects) model is 𝑦 𝑖𝑗𝑘 = 𝜇+ 𝛼 𝑖 + 𝜏 𝑗 + 𝛽 𝑘 + 𝜀 𝑖𝑗𝑘 𝑖=1,2,…,𝑝 𝑗=1,2,…,𝑝 𝑘=1,2…,𝑝 𝑦 𝑖𝑗𝑘 : observation in row 𝑖, column 𝑘, treatment 𝑗 𝜇 : overall mean 𝛼 𝑖 : row i (block 1) effect 𝜏 𝑗 : treatment j effect 𝛽 𝑘 : column k (block 2) effect 𝜀 𝑖𝑗𝑘 : random error iid 𝑁(0, 𝜎 2 )

Example An experimenter is studying the effects of five different formulations of a rocket propellant used in aircrew escape systems on the observed burning rate. Each formulation is mixed from a batch of raw material that is only large enough for five formulations to be tested. Furthermore, the formulations are prepared by several operators and there may be substantial differences in the skills and experience of the operators.

𝑆𝑆 𝑇 = 𝑆𝑆 rows + 𝑆𝑆 columns + 𝑆𝑆 Treatment + 𝑆𝑆 𝐸 𝑝 2 −1= 𝑝−1 + 𝑝−1 + 𝑝−1 +(𝑝−2)(𝑝−1) 𝐹 0 = 𝑀𝑆 𝑇𝑟𝑒𝑎𝑡𝑚𝑒𝑛𝑡𝑠 𝑀𝑆 𝐸

Example

Residuals 𝑒 𝑖𝑗𝑘 = 𝑦 𝑖𝑗𝑘 − 𝑦 𝑖𝑗𝑘 = 𝑦 𝑖𝑗𝑘 − 𝑦 𝑖.. − 𝑦 .𝑗. − 𝑦 ..𝑘 +2 𝑦 …

Graeco-Latin Square Design Consider a p  p Latin square and superimpose on it a second p  p Latin square in which the treatments are denoted by Greek letters. If each Greek letter appears once and only once with each Latin letter, the design is called a Graeco-Latin square. This is a basic course blah, blah, blah…

The statistical model is 𝑦 𝑖𝑗𝑘𝑙 =𝜇+ 𝜃 𝑖 + 𝜏 𝑗 + 𝜔 𝑘 + Ψ 𝑙 + 𝜀 𝑖𝑗𝑘 𝑖=1,2,…,𝑝 𝑗=1,2,…,𝑝 𝑘=1,2…,𝑝 𝑙=1,2…,𝑝 𝑦 𝑖𝑗𝑘𝑙 : observation in row 𝑖 and column 𝑙 for Latin letter 𝑗 and Greek letter 𝑘 𝜃 𝑖 : row 𝑖 effect 𝜏 𝑗 : Latin letter treatment j effect 𝜔 𝑘 : Greek letter k effect Ψ 𝑙 : column l effect 𝜀 𝑖𝑗𝑘 : random error ~ iid 𝑁(0, 𝜎 2 ) This is a basic course blah, blah, blah…

Example In the rocket propellant experiment, additional factor, test assemblies, could be of importance. Let there be five test assemblies denoted by the Greek letters 𝛼, 𝛽, 𝛾, 𝛿, 𝜀.

HW (09/30/2017)