Presentation on theme: "TWO WAY ANOVA WITHOUT REPLICATION"— Presentation transcript:
1TWO WAY ANOVA WITHOUT REPLICATION Also known as Randomized Block Design (RBD)In RBD there is one factor or variable that isof primary interest. However, there are alsoseveral other nuisance factors.Nuisance factors are those that may affect themeasured result, but are not of primary interest.The way to control nuisance factor is by blockingthem to reduce or eliminate the contribution toexperimental error contributed by nuisance factorsBlocking variable – A second treatment variable that when included in ANOVA analysis will have the effect of reducing the SSE term.
2The 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 blockrandom error.
3We can use the following layout for this kind of two-way classification: BlocksTotalTreatment 1Treatment 2Treatment iTreatment k
4In 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:
8Solution; 1. Construct the table of calculation, we have k = 4 and n = 3: 2. Set up the hypothesis: Engine 1Engine 2Engine 3TotalsDetergent A454351474652485055423749139Detergent B145153Detergent CDetergent D128Totals182176207565
10Source of VariationSum of SquaresDegree of FreedomMean SquareF CalculatedTreatments1113Blocks1352Error196Total265114. 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
11SUMMARYA 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.
12SUMMARY 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.
13Exercise 4.3A 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 ?CatalystReagent123A84.8589.1385.28B75.3579.4084.65C70.3076.6578.20D73.1881.1077.23
14Exercise 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.DriverU.S. 6West EndHickory St.Rte. 59Deans18172122Snaverly1623Ormson26Zollaco2925Filbeck2428
15Continue…. At , is there any possible differences in treatments and also in blocks?