Step 1. Denote the number of replication byr, the number of levels of factor A (i.e., spacing) bya,and that of factor B (i.e., age) byb.Construct the outline of the analysis of variance asfollows:Table 4.16. Schematic representation of ANOVA of a factorial experiment with two levels offactor A, three levels of factor B and with three replications in RCBD.SourceofvariationDegreesoffreedom(df)Sum ofsquares(SS)Mean squareComputedReplicationr-1SSRTreatmentab- 1SSTfMSRMST

MSAMSBMSE

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Step 2.Compute treatment totals(Tij), replication totals (Rk), and the grand total (G), as shown inTable 4.15 andcomputetheSSTO,SSR,SSTandSSEfollowingtheproceduredescribedSection4.3.3.Letyijkrefer to the observation corresponding to theith level of factor A andlevel factor B in thekth replication.injth

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22

23SST(4.25)= 14251.87SSE = SSTO - SSR - SST (4.26)= 17479.10 - 2040.37 - 14251.87= 1186.86The preliminary analysis of variance is shown in Table 4.17.Table 4.17. Preliminary analysis of variance for data in Table 4.15.SourceofvariationDegreeoffreedomSumofsquaresMeansquareComputedFTabularF5%Replication22040.371020.1878.59567*4.10Treatment514251.872850.37324.01609*3.33Error101186.86118.686Total1717479.10*Significant at 5% level.1.8 Degrees of FreedomTo determine the degrees of freedom (the number of variables whose values may beindependently specified) in our model we could simply count the number of independentvariables (the number of

24variables which remain on the right- hand side) in our modified equations. This suggests apossible definition:degrees of freedom = # variables - # equationsDefinition:The degrees of freedom for a given problem are the number of independent problem variableswhich must be specified to uniquely determine a solution. In our distillation example, there are:16 equations 16 variables (recall that F and XF are fixed by upstream processes). This seems toindicate that there are no degrees of freedom.Consider the three equations relating QC, QR, and qvapour:QR - QC = 0QR - DHvap qvapour= 0 QC - DHvapqvapour = 0Notice that if we subtract the last from the second equation:QR - DHvap qvapour = 0- QC - DHvap qvapour = 0QR - QC = 0 the result is the first equation.It seems that we have three different equations, which contain no more information than two ofthe equations. In fact any of the equations is a linear combination of the other two equations. Werequire a clearer, more precise definition for degrees of freedom.Measures that ignore time value of moneynet profitpayouttimereturn on investment, ROIMeasures that recognize the time value of moneyInternal rate ofreturn, IRR Netpresent value,NPV Discounted return on investment,DROI A measure of the totalprofitabilityâ€¢Alternative, Benefits-to-cost ratioâ€¢Strengths

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