TWO WAY ANOVA WITHOUT REPLICATION Also known as Randomized Block Design (RBD) In RBD there is one factor or variable that is of primary interest. However, there are also several other nuisance factors. Nuisance factors are those that may affect the measured result, but are not of primary interest. The way to control nuisance factor is by blocking them to reduce or eliminate the contribution to experimental error contributed by nuisance factors Blocking variable – A second treatment variable that when included in ANOVA analysis will have the effect of reducing the SSE term.

The model for a randomized block design with one nuisance variable can be written as: j th observation from i th treatment, Overall mean, i th effect of treatment, j th effect of block random error.

We can use the following layout for this kind of two-way classification: Blocks Total Treatment 1 Treatment 2 Treatment i Treatment k

In the two way analysis of variance where each treatment is represented once in each block, the major objective is to test: The effect of treatment: The effect of block:

We reject if: The effect of treatment: 2. The effect of block:

Example 4.3 SOLUTION

Solution; 1. Construct the table of calculation, we have k = 4 and n = 3: 2. Set up the hypothesis: Engine 1 Engine 2 Engine 3 Totals Detergent A 45 43 51 47 46 52 48 50 55 42 37 49 139 Detergent B 145 153 Detergent C Detergent D 128 Totals 182 176 207 565

3. Construct ANOVA table:

Source of Variation Sum of Squares Degree of Freedom Mean Square F Calculated Treatments 111 3 Blocks 135 2 Error 19 6 Total 265 11 4. At = 0.01, from the statistical table for f distribution, we have (treatments) and (blocks). 5. Since , thus we reject and conclude that there are differences in the effectiveness of the 4 detergents at = 0.01 and also since , thus we reject and conclude that there are differences among the results obtained for the 3 engines are significant

SUMMARY A two factor randomized complete block design is complete balanced two-factor design in which the effects of one factor (the treatment factor) are of interest, while the effects of other factor (the blocking factor) are not of interest. The blocking factor is included to reduce the uncertainty in the main effects estimates of the treatment factor. A two way analysis variance is used to estimate effects and to perform the hypothesis tests on the main effects of the treatment factor. A randomized complete block design provides a great advantage when the blocking factor strongly affects the response and provides a relatively small disadvantage when the blocking factor has little or no effect.

SUMMARY In a two way analysis of variance : If the additive model is not rejected, then the hypothesis tests for the main effects can be used to determine whether the row or column factors that affect the outcome. If the additive model is rejected, then the hypothesis tests for the main effects should not be used. Instead the means must be examined to determine how various combinations of rows and column levels affect the outcome.

Exercise 4.3 A chemical engineer is studying the effects of various reagents and catalysts on the yield of a certain process. There is a combination among three reagents and four catalysts. The results are presented below: Develop a complete analysis of variance table for two factor and it is possible the main effects of catalyst (Treatments) and reagent (blocks) are all differences at ? Catalyst Reagent 1 2 3 A 84.85 89.13 85.28 B 75.35 79.40 84.65 C 70.30 76.65 78.20 D 73.18 81.10 77.23

Exercise 4.4 WARTA, the Warren Area Regional Transit Authority, is expanding bus service from the suburb of Starbrick into the central business district of Warren. There are four routes being considered from Starbrick to downtown Warren. WARTA conducted several tests to determine whether there was a difference in the mean travel times along the four routes. Because there will be many different drivers, the test was set up so each driver drove along each of the four routes. Below is the travel time, in minutes for each driver-route combination.

Continue…. At , is there any possible differences in treatments and also in blocks? Driver U.S. 6 West End Hickory St. Rte. 59 Deans 18 17 21 22 Snaverly 16 23 Ormson 26 Zollaco 29 25 Filbeck 24 28