1 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 If the nuisance variable is known and controllable, we use blocking If the nuisance factor is known and uncontrollable, sometimes we can use the analysis of covariance to remove the effect of the nuisance factor from the analysis If the nuisance factor is unknown and uncontrollable, use randomization to balance out its impact across the experiment

2 The Blocking Principle Blocking is a technique to systematically eliminate the effect of nuisance factors Failure to block is a common flaw in designing an experiment (consequences?) The randomized complete block design or the RCBD – one nuisance variable Extension of the ANOVA to the RCBD Several sources of variability can be combined in a block, to form an aggregate variable

3 The Latin Square Design Text reference, Section 4-2, pg. 136 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 Latin squares are not used as much as the RCBD in industrial experimentation

4 The Rocket Propellant Problem – A Latin Square Design Five different formulations of a rocket propellant Five different materials, and five operators Two nuisance factors This is a 5x5 Latin square design

5 Examples of Latin squares ABDC BCAD CDBA DACB ADBEC DACBE CBEDA BEACD ECDAB 4X4 5X5 Examples of standard Latin squares ABCD BCDA CDAB DABC ABCDE BAECD CDAEB DEBAC ECDBA 456# of standard Latin squares ,280Total # of Latin squares First row and column consist of the letters written in alphabetical order

6 Statistical Analysis of the Latin Square Design The statistical (effects) model is The statistical analysis (ANOVA) is much like the analysis for the RCBD SS T = SS Rows + SS Columns + SS Treatments + SS E Respective degrees of freedom p 2 – 1 = p-1 + p-1 + p-1 + (p-2)(p-1)

7 Mean squares, F o = MS Treatments /MS E F o is compared with F p-1,(p-2)(p-1) for testing the null hypothesis of equal treatments The effects of rows and columns can also be tested using the ratios of MS Rows and MS Columns to MS E, but may be inappropriate

8 Example: The Rocket Propellant Problem Coding (by subtracting 25 from each observation)

9 There is a significant difference in the means of formulations There is also an indication that there are differences between operators There is no strong evidence of a difference between batches of raw materials Model adequacy can be checked by plotting residuals

10 The observations in a Latin square should be taken in random order Select a Latin square from a table Arrange the order of the rows, columns, and letters at random The Latin square design assumes that there is no interaction between the factors Small Latin squares provide a relatively small number of error degrees of freedom (p-2)(p-1), therefore, replication is often needed to increase error DOF. The analysis of variance depends on the method of replication

11 The Latin square is replicated n times – Case 1: same levels of the row and column blocking factors are used in each replicate (repeat the square). Total number of observations: N = np 2. ANOVA: Table 4-13 Case 2: same row (column) but different columns (rows) are used in each replicate (fix the level of the row factor, and change the level of the column factor). N = np 2. ANOVA: Table 4-14 Case 3: different rows and columns are used in each replicate. N = np 2. ANOVA: Table 4-15

12  Identifying Outliers  Calculating standardized residuals.  Values outside 95% of confidence interval are regarded as outliers.  These outliers should be taken away so the model is not affected. DATA PREPARATION LoadRate (A) ButtonDia. (B) HoldTime (C) PeakLoadResidualStd. Residual Note: this example is taken from the peak load observations of Material 01.

13  Filling Missing Points  The missing observation x is estimated by minimizing its contribution to the error sum of squares.  If there are several missing points, the equation is iteratively used to estimate the missing values, until convergence is obtained.

14 If several observations are missing, they may be estimated by writing the error sum of squares as a function of the missing values, differentiating with respect to each missing value, equating the results to zero, and solving the resulting equations. Alternatively, the equation can be iterated to estimate the missing values. Suppose that two values are missing. Arbitrarily choose the first missing value, and then the equation is used with this assumed value along with the real data for estimating the second. The equation is then used to re-estimate the first missing value using the real observations and the estimated second value. Following this, the second is re-estimated. This process is continued until convergence is obtained. In any missing value problem, the error degrees of freedom are reduced by one for each missing observation.  Filling Multiple Missing Points