Statistical procedures for agricultural research pdf

Statistical Procedures for Agricultural Research, Second Edition will prove equally useful to students and professional researchers in all agricultural and biological disciplines. A wealth of examples of actual experiments help readers to choose the statistical method best suited for their needs, and enable even the most complicated procedures to be easily understood and directly applied.

Gomez, published by Unknown which was released on Gomez,Arturo A. Download or read online Statistical procedures for agricultural research written by A. Gomez Kwanchai, published by Unknown which was released on Get Statistical procedures for agricultural research Books now! This classic book will meet the needs of food and agricultural industries in both their research and business needs. Learn the fundamentals of applying statistics to the business and research needs in the food and agricultural industries.

Statistical Methods for Food and Agriculture is a practical, hands-on resource that explores. Better experimental design and statistical analysis make for more robust science. A thorough understanding of modern statistical methods can mean the difference between discovering and missing crucial results and conclusions in your research, and can shape the course of your entire research career. IV 1 3 2 3 1 2 2 1 1 3 3 1 3 2 2 As shown, forreplication I,block I of the basic plan becomes block 3of the new plan, block 2retains the same pnsition, and block 3of the basic plan becomes block I of the new plan.

For replication I1,block 1of the basic plan becomes block 2 of the new plan, block 2 of the basic plan becomes block 1ofthe new plan, and so on. IV 1 2 3 44 Single. Randomize the treatment arrangement within each incomplete block. For our example, randomly reassign the three treatments in each of the 12 incomplete blocks, following the samerandomization scheme used in steps 4 and 5. After 12 independent randomization processes, the reassigned treatment sequences may be shown as: Reassigned Treatment Sequence in New Plan Treatment Sequence Rep.

Iv Sequence Rep. III in Basic Plan Block1 Block2 Block3 Block I Block2 Block 3 1 3 3 1 1 3 2 2 2 1 2 3 1 3 3 1 2 3 2 2 1 In thiscase, for incomplete block 1of replication I,treatment sequence 1 of the basic plan treatment 3 becomes treatment sequence 2 of the new plan, treatment sequence 2 of the basic plan treatment 6 becomes treat- ment sequence 3ofthe new plan, and treatment sequence3of the basicplan treatment 9 becomes treatment sequence 1of the new plan, and so on.

Apply the final outcome ofthe randomization process ofstep 6 to the field layout ofFigure 2. Note that an important feature in the layout of a balanced lattice design is that every pair of treatments occurs together only once in the same block. TO ofreplication I; with treatments 2 and 3in block 2 of replication It; with treatments 6 and 8in block 2 of replication III;and with treatments5and 9 inblock 1ofreplication IV. As a consequence ofthis feature, thedegree of precision for comparing each pair oftrea'ments in a balanced lattice design is the samefor all pairs.

There are four sources of variation thatcan be accounted for in a balanced lattice design: replication, treatment, incom- plete block, and experimental error. Relative to the RCJ; design, the incom- plete block is an additional source of variation and re,1ects the differences imong incomplete block, cf the samereplication.

The computational procedure for the analysis of variance of a balanced ,atticedesign is illustrated using data on tiller count from a field experiment nvolving 16 rice fertilizer treatments.

The experiment followed a 4 X4 bal- inced lattice design with five replications. The data are shown in Table 2. Such a rearrangement is not iecessary for the computation of the analysis ofvariance but we doit here to 'acilitate the understanding of the analytical procedure to be presented.

Calculate the block totals B and replication totals R , as shown in Table 2. Calculate the treatment totals T and the grand total G , as shown incolumn 2 ofTable2. For each treatment, calculate the B, value as the sum of block totals over all blocks in which the particular treatment appears.

For exam- ple, treatment 5in our example was tested in blocks 2,5, 10, 15, and 20 Table 2. TheB,values for all 16 treatments are shown in column 3ofTable 2. Notethat thesum ofB,valuesover all treatments mustequal k G , where k is the block size. Note that the sum of W values over all treatments must be zero. FactorExperiments 13 sTnp 6. Compute the block adjusted SS i. SS - Block adj. Compute the block adj.

SS Block adj. MS] Note that ifthe intrablock error MS isgreater than the block adj. MS, 11 is taken to be zero and no adjustme't for treatment nor any further adjustment is necessary.

The F test for significance of treatment effect is thenmade in the usual manner as the ratio ofthe treatment unadj. MS and intrablock error MS, and steps 10 to 14 and step 17 can be ignored. Forour example, the intrablock error MS is smaller than the block adj.

Hence, the adjustment factor pis computed as: - - 0. The results of T' values for all 16 treatments are shown in column 5of Table 2. Theresults ofM' values forall 16 treatments are presented in, the last column ofTable 2.

Compute the adjusted treatment mean square as: Treatment adj. Compute theFvaluefor testing thetreatment difference as: F Treatment adj. Enter all values computed in steps 6 to 9 and 12 to 15 in the analysis ofvariance outlineofstep 5. Thefinal resultis shown inTable2. Estimate the gain in precisionofa balanced latticedesign relative to the RCB design as: R. Block adj. Intrablock error Treatment adj. While the partially balanced lattice design requires that the number of treatments must be a perfect square and that the block size is equal to the square root of this treatment number, thenumber of replications is not prescribed as a function of the number of treatments.

In fact, any number ofreplications can be used in a partiallybalanced lattice design. With tworeplications, the partially balanced lattice design is referred toas a simple lattice; with three replications, a triple lattice; with four replications, a quadruple lattice; and so on. However, such flexibility in the choice of the number of replications results in a loss of symmetry in the arrangement of treatments over blocks i.

Consequently, the treatment pairs that are tested in the same incomplete block arecompared with a level ofprecision thatis higher than for those that are not tested in the same incomplete block. Because there is more than one level of precision for comparing treatment means, data analysis becomes more complicated.

The procedures for randomization and layout of a partially balanced lattice design are similar to those for a balanced lattice design described in Section 2. For example, with a 3x 3simple lattice i. With a triple lattice i.

For example, for a 5X5quadruple lattice design i. In general, the procedure of using the basic plan without repetition is slightly preferred because it comes closer to the symmetry achieved in a balanced lattice design. For a partially balanced lattice design with p repeti- tions, the process of randomization will be done p times, separately and independently.

Two sample field layouts of a 5x 5 quadruple lattice design, one with repetition and another without repetition, aie shown in Figures 2. The procedure for the analysis of variance of a partially balanced lattice design is discussed separately for a case with repetition and one without repetition.

A 9x 9 triple lattice design is used to illustrate the case without repetition; a 5x 5 quadruple lattice is used to illustrate the case with repetition. T 5 Rephcaton Mn Replication X A sample layout of a 5 x 5 quadruple lattice design without repetuion,involving 25 Figure 2.

To illustrate the analysis ofvariance of a partially balanced lattice design without repetition, we use a 9X9triple lattice design that evaluates the performance of 81 rice varieties. The yield data, rearranged according to the basic plan ofAppendix L, aregiven in Table 2.

Thesteps in theanalysis ofvariance procedure are: o STEP 1. Calculate the block totals B and the replication totals R as shown in Table 2. Calculate the treatment totals T asshown inTable2.

E3 STEP 3. Compute thetotal SS,replication SS,and treatment unadj. SS in the standard manner: C. Forexample, blo. LatticeDesign 59 Table 2. Foreach replication, calculate thesum ofC b values over allblocks i. Calculate the block adj. SS as: Block adj. Calculate the intrablock error mean square and block adj. Calculate the adjustment factorI. The Ftest forsignificance of treatmenteffect is made in the usual manner as the ratio of treatment unadj. MS and intrablock error MS, and steps 10 to14 and step 17 can be ignored.

For eac'. They are computed simply by dividing these individual adjusted treatment totals by the number of replications. Compute the adjusted treatment SS: Treatment adj. SS - Treatment unadj. Compute the treatment adj. SS k- 1 MS IntrablockerrorMS 2. Enter all values computed in steps 4 to 9 and 12 to 15 in the analysisofvariance outline ofstep 3.

The finalresult isshown inTable2. LatticeDesign 63 Table 2. Estimate the gain in precision of a partially balanced lattice design relative to the RCBdesign, as follows: A. Compute the effective error mean square. For a partially balanced lattice design, there are two error terms involved: one for comparisons between treatments appearing in the sameblock [i. And, for simplicity, the average error MS may be computed and used for comparing any pair of means i. For a triple lattice, the formula is: 9 Av.

Factor Experiments For our examp computed as: le, the value of the two error mean squares are 6 ErrorMS i 9 0. It is computed as: 9 Av. For the analysis of variance of a par- tiallybalanced lattice design with repetition, we use a 5x 5quadruple lattice LatticeDesign 65 whose basic plan is obtained by repeating a simple lattice design i. Data on grain yield for the 25 rice varieties used as treatments rearranged according to the basic plan are shown in Table 2.

The steps involved in the analysis of variance are: 0 STEP 1. Calculate the block totals B and replication totals R as shown in Table 2. Ol STEP 3. As before, k is the block size. Compute the total SS, replication SS,and treatment unadj. SS, in thestandard manner as: C. For our example, there are two repetitions, each consisting of two replications-replication Iand replication III in repetition 1and replication II and replication IV in repetition 2.

Because the five treatments in block 1 of repetition I are treatments 1,2, 3,4, and 5 Table 2. Compute the total Cvalues overall blocksinarepetition i. Forour example, the two R, values are 9, forrepetition 1and - 9, forrepetition 2. The sumofall R, values must bezero. Let B denote the block total; D, the sum of S values for each repetition; and A, thesum ofblock totals foreach replication. Compute the Table 2.

SS as: A. SS -Block adj. SS np k- 1. For example, using thedata ofTables 2. Compute theadjusted treatment sumofsquares as: Treatment adj. Compute the adjusted treatmentmean square as: Treatment adj. O STEP Enter all values computed in steps 4 to 9 and 12 to 14 in the analysis ofvariance outline ofstep 3. Thefinal result is shown inTable 2. Component a Component b Treatment unadj. Total "cv - 5.

Compute thevalues ofthe two effective error mean square as: A. Whereas the lattice design achieves homogeneity within blocks by grouping experimentalplotsbased on some known patterns ofheterogeneity in the experimental area, the group balanced block design achieves the same objective by grouping treatmentsbased on some known characteristics of the treatments. In a group balanced block design, treatments belonging to the same group arealways testedin the sameblock, but those belonging todifferent groups are never tested together in the same block.

Hence, the precision with which the different treatments are compared is not the same for all comparisons. Treat- ments belonging to the same group are compared with a higher degree of precision than those belonging to different groups. The group balanced block design is commonly used in variety trials where varieties with similar morphological characters are put together in the same group.

Another type oftrials usingthe group balanced block design is that involving chemical insect control in which treatments may be subdivided into similar sprayoperations to facilitate the field application of chemicals. We outline procedures for randomization, layout, and analysis of v. Based on growth duration, varieties aredivided into group A forvarieties with less than daysin growth duration, groupB for to days, and group C for longer than days. Each group consists of 15 varieties.

For ourexample, the varieties are grouped into three groups, A,B, and C each consisting of 15 varieties, according to their expected growth duration. Divide the experimental area into rreplications, each consisting of t experimental plots. Usingone of the randomization schemes described in Section 2. Then, independently repeat the process for the remaining replications. The ,esult is shown in Figure 2.

For ourexample, startingwith thefirst block ofreplication I, randomly assign the 15 varieties of group A to the 15 plots in theblock. Repeat this process for the remaining eight blocks, independently ofeach other.

The final result isshown in Figure2. Compute the treatment totals T , replication totals R , and the grand total G , as shown in Table 2. Construct the replication x group two-way table of totals RS and compute the group totals S , as shown in Table 2. III S A Compute themean square foreach source ofvariation by dividing the SS by its d. Enterall values obtained in steps 3to9 in the analysis ofvariance outline of step 1, as shown in Table 2. Results indicate a significant difference among the means of the three groups of varieties and significant differences among thevarieties in each of the threegroups.

Because an organism'sresponse toany single factor may vary with the level of the other factors, single-factor experiments are often criticized fortheir narrowness. Indeed, the result ofa single-factor experiment is, strictly speaking, applicable only to the particular level in which the other factors were maintained in the trial.

Thus, when response to the factor of interest is expected to differ under different levels of the other factors, avoid single-factor experiments and con- sider instead the use of a factorial experiment designed to handle simulta- neously two ormore variable factors.

We shall define and describe the measurement of the interaction effect based on an experiment with two factors A and B, each with two levels a o and a, for factor A and b o and b,for factor B.

The four treatment combinations are denoted by a 0 b o ,alb o ,a 0 bl, and albl. Inaddition, wedefine anddescribe themeasurement ofthesimpleeffect and themaineffect ofeach of the two factors A and B because these effects are closely related to, and are in fact an immediate step toward the computation of, theinteraction effect.

Toillustrate the computation ofthesethree types ofeffects, consider the two setsofdata presented in Table 3. O3 cup1. Compute the simple effect offactorA as thedifference between its two levels ata given levelof factor B. Nointeraction X 1. Cases with lower and highe, irteraction effects than 1. Figure 3. From theforegoing numerical computation and graphical representations of the interaction effects, three points should be noted: 1.

An interaction effect between two factors can be measured only if the two factors are tested together in the same experiment i. When interaction is absent as in Figure 3. For our example, the simple effects of variety at N o and N, are both 1. That is, when interaction is absent, the results from separate single-factor experiments i. In our example, the varietal effect would have been estimated at 1.

FactorialExperiment 89 3. When interaction is present as in Figures 3. Consequently, the main effect isdifferent from the simple effects. Forexample, in Figure 3. In other words, although there was a large response to nitrogen application in variety Y,there was none in variety X. Or, in Figure 3. If the mean yields of the two varieties were calculated over the two nitrogen rates, the two variety means would be the same i.

Thus, if we look at the difference between these two variety means i. It istherefore clear that when an interaction effect between two factors is present: " The simple effects and not the main effects should be examined. Its treatments consist of the following four possible combinations of the two levelsin each of the twofactors. In contrast, the term incomplete factorial experiment isused when onlyafractionrof all the combinationsistested.

Throughout this book, however, we refer to complete factorial experiments as factonal experiments and use the term incomplete factorial,otherwise. For example, if the foregoing 23 factorial experiment is in a randomized complete block design, then thecorrect description of the experiment would be 2' factorialexperiment in a randomized completeblock dcsign. The number of treatments increases rapidly with an increase in the number of factors or an increase in the level in each factor.

Thus, avoid indiscriminate use of factorial experiments because of their large size, complexity, and cost. Furthermore, it is not wise to commit oneself to a large experiment at the beginning of the investigation when several small preliminary experiments may offer promising results.

For example, a plant breeder has collected 30 new rice varieties from a neighboring c intry and wants to assess their reaction to the local environment.

Because the environ- ment is expected to vary in terms of soil fertility, moisture levels, and so on, the ideal experiment would be one that tests the 30 varieties in a factorial experiment involving such other variable factors as fertilizer, moisture level, and population density. Such an experiment, however, becomes extremely large as variable factors other than varieties are added.

Even if only one factor, say nitrogen fertilizer with three levels, were included the number of treatments would increase from 30 to Such a large experiment would mean difficulties in financing, in obtaining an adequate experimental area, in controlling soil heterogeneity, and so on.

Complete Block Design 91 Thus, the more prac,. For example, the initial single-factor experiment may show that only five varieties are outstanding enough to warrant further testing. These five varieties could then be put into a factorial experiment with three levels of nitrogen, resulting in an experiment with 15 treatments rather than the 90 treatments needed with a factorial experiment with 30 varieties.

Thus, although a factorial experiment provides valuable information on inter- action, and is without question more informative than a single-factor experi- ment, practical consideration may limit its use. For most factorial experiments, the number of treatments is usually too large foran efficient use of a complete block design.

Furthermore, incomplete block designs such as the lattice designs Chapter 2, Section 2. There are, however, special types of design, developed specifically for factorial experiments, that are comparable to theincomplete block designs forsingle-factor experiments.

Suchdesigns, which are suitable for two-factor experiments and are commonly used in agricultural research, are discussed here. The procedures for ran- domization and layout of the individual designs are directly applicable by simply ignoring the factorcomposition ofthe factorial treatments and consid- ering all the treatments as if they were unrelated For the analysis ofvariance, the computations discussed for individual designs are also directly applicable.

However, additional computational steps are required to partition the treat- ment sum of squares into factorial components corresponding to the main effects of individual factors and to their interactions. The procedure for such partitioning is the same for all complete block designs and is, therefore, illustrated for only cne case, namely, that of a randomized complete block RCB design.

We illustrate the step-by-step procedures for the analysis ofvariance of a two-factor experiment in a RCB design with an experiment involving five rates of nitrogen fertilizer, three rice varieties, and four replications. The list ofthe 15 factorial treatment combinations is shown in Table 3. Denote the numberof replications by r,the level offactor A i.

Construct the Table 3. Compute treatment totals T , replication totals R , and the grand total G , as shown in Table3. Construct the factor A x factor B two-way table of totals, with factor A totals and factor B totals computed.

For our example, the variety x nitrogen table of totals AB with variety totals A and nitrogen totals B computed is shown inTable 3. CompleteBlock Design 95 Table 1. Computethe mean square foreach source ofvariation by dividing the SS by itsd. Enterall values obtained in steps 4 to8in thepreliminary analysis ofvariance of step 2,as shown in Table 3. The results show a nonsignifi- cant interaction between variety and nitrogen, indicating that the varietal difference was not significantly affected by the nitrogen level applied and that the nitrogen effect did not differ significantly with the varieties tested.

Main effectsbothofvariety and ofnitrogen were significant. In a 98 Two-FactorExperiments split-plot design, oneof the factors is assigned to the main plot.

The assigned factor is called the main-plotfactor. The main plot is divided into subplots to which the second factor, called the subplotfactor,is assigned. Thus, each main plot becomes a block for the subplot treatments i. With a split-plot design, the precision for the measurement of the effects of the main-plot factor is sacrificed to improve that of the subplot factor.

Measurement of the main effect of the subplot factor and its interaction with the main-plot factor is more precise than that obtainable with a randomized complete block design. On the other hand, the measurement of the effects of the main-plot treatments i. Because, with the split-plot design, plot size and precision of measurement ofthe effects are not the same for both factors, the assignment ofa particular factor to either the main plot or the subplot is extremely important.

To make such achoice, the following guidelines are suggested: 1. For agreater degree oi precision forfactor B than for factorA, assign factor B to the subplot and factorA to the main plot.

For example, a plant breeder who plans to evaluate 10 promising rice varieties with three levels of fertilizatiui in a 10 x 3 factorial experiment would probably wish to have greater precision for varietal comparison than for fertilizer response. Thus, he would designate variety as the subplot factor and fertilizer as the main-plot factor. On the other hand, an agronomist who wishes to study fertilizer responses of the 10 promising varieties developed by the plant breeder would probably - want greater precision for fertiliz; response than for varietal effect and would assign variety to main plot and fertilizer to subplot.

Relative Size of the Main Effects. If the main effect of one factor factor B is expected to be much larger and easier to detect than that of the other factor factor A , factor B can be assigned to the main plot and factor A to the subplot. This increases the chance of detecting the difference among levels of factor A which has a smaller effect.

For example, in a fertilizer X variety experiment, the researcher may assign variety to the subplot and fertilizer to the main plot because he expects the fertilizer effect to be much larger than. The cultural practices required by a factor may dictate the use of large plots. For practical expediency, such a factor may be assigned to the main plot.

For example, in an experiment to evaluate water management and variety, it may be desirable to assign water management to the main plot to minimize water movement betwcen adjacent plots, facilitate the simulation ofthe water level required, and reduce border effects.

Or, in an experiment to evaluate the performance of several rice varieties with different fertilizer rates, the researcher may assign themain plot to fertilizer to minimize the need to separate plots receiving different fertilizer levels. Replicotion I Figure3.

In a split-plot design, both the procedure for randomization and that for analysis of variance are accomplished in two stages-one on the main-plot level and another on the subplot level.

In each replication, main-plot treatments are first randomly assigned to the main plots followed by a random assignment of the subplot treatments within each main plot. Each is done by any of the randomization schemes ofChapter 2, Section 2. The steps in the randomization and layout of a split-plot design are shown, using a as the number of main-plot treatments, bas the number of subplot treatments, and r as the number of replications.

For illustration, a two-factor experiment involvingsix levels ofnitrogen main-plot treatments and fourrice varieties subplot treatments in three replications is used. The assignment of the main-plot factor can, in fact, follow any of the complete block designs, namely, completely randomized design, randomized complete block, and latin square; but we consider only the randomized complete block because it is the most appropriate and the most commonly used for agricultural experiments.

The result may beas shown in Figure3. L-- I ,. The result maybeas shown inFigure 3. Note that field layout of a split-plot design as illustrated by Figure 3. The size of the main plot isbtimes the size of the subplot. Each main-plot treatment is tested r times whereas each subplot treatment istested a r times. Thus,the number of timesa subplot treatment istested will always be larger than that for the main plot and isthe primary reason for more precision for the subplot treat- ments relative to the main-plo, treatments.

In our example, each of the 6levels of nitrogen was teLed 3times but each ofthe4 varieties was tested 18 times. We show the computations involved in the analysis with data from the two-factor experiment six levels of nitrogen and four rice varieties shown in Figure 3.

Grain yield data are shown in Table 3. Let A denote the main-plot factor and B, the subplot factor. Compute analysis ofvariance: 0 sup 1. Construct two tablesof totals: A. The replication XfactorAtwo-way table oftotals,with thereplication totals, factor Atotals, and grand totalcomputed.

For ourexample, the Table 3. III A No 15, 17, 15, 48, N 1 21, 21, 22, 65, N 2 22, 24, 23, 70, N 3 22, 24, 23, 70, N 4 23, 23, 23, 69, N 5 22, 24, 22, 69, Rep.

The factor A x factor B two-way table of totals, with factor B totals computed. For our example, the nitrogen x variety table of totals A B ,with thevariety totals B computed, is shown in Table 3. Compute the correction factor and sums of squares for the main- plot analysis as: G 2 G2 C.

Foreach source ofvariation, compute themean squareby dividing theSS by its corresponding d. The value of cv b indicates the precision of the subplot factor and its interaction with the main-plot factor. The value of cv b is expected to be smaller than that of cv a because, as indicated earlier, the factor assigned to the main plot is expected to be measured with less precision than that assigned to thesubplot. This trend doesnot always hold, however, asshown by thisexample in which thevalue ofcv b is larger than that of cv a.

The cause for such an unexpected outcome is beyond the scope ofthis book. If such results occur frequently,a competent statistician should be consulted. Enter all values obtained from steps 3 to 8 in the analysis of variance outline ofstep 1,as shown in Table 3. For our example, all the three effects the two main effects and the interaction effect are highly significant.

With a significant interaction, caution mustbe exercised when interpreting the results see Section 3. For proper comparisons between treatment means when the interaction effect is present, see Chapter 5,Section 5. Table 3. This is accomplished with the use of three plot sizes: the verticalfactor 1. Vertical-strip plotfor the first factor- the horizontal factor 2. Horizontal-strip plotfor the second factor- 3. Intersection plotforthe interaction between the two factors The vertical-strip plot andthe horizontal-strip plot are always perpendicular to each other.

However, there is no relationship between their sizes, unlike the case of main plot and subplot ofthe split-plot design. Theintersection plot is, of course, the smallest. Thus, in a strip-plot design, the degrees of precision with the main effects of both factors are sacrificed in order to associated improve theprecision of the interaction effect.

The order in which these performed is immaterial. Let A represent the horizontal factor and B the vertical factor, and aand b their levels. As in all previous cases, r represents the number of represent a tw3-factor experiment replications.

We illustrate the steps involved with rice varieties horizontal treatments and three nitrogen rates involving six vertical treatments tested ina strip-plot design with three replications. The result is shown in Figure 3. The final layout is shown in Figure 3. We show the computational procedure with data from a two-factor experiment involving six rice varieties horizontal factor and three nitrogen levels vertical factor tested in three replications.

The field layout is shown in Figure 3. Construct three tables oftotals: 1. Thereplication x horizontal-factor tableoftotalswith replication totals, horizontal-factor totals,and grandtotal computed. Forour example, the replication Xvariety table of totals RA with replication totals R , variety totals A , and thegrand total G computed is shown in Table 3. I 0 N 1 2, 60 N 2 4, N 3 7, 0 4, 60 5, 7, 0 2, 60 4, 7, 0 2, 60 4, 6, 0 4, 60 5, 6, 0 2, 60 3, 1, Rep. III 4, 4, 8, 5, 7, 6, 5, 7, 8, 3, 4, 6, 4, 6, 6, 3, 4, 3, Strip-PlotDesign 2.

The replication Xvertical-factor table of totals with the vertical-factor totals computed. For our example, the replication Xnitrogen table of totals RB withnitrogen totals B computed isshown inTable 3.

The horizontal-factor Xvertical-factor table of totals. For our example, the variety Xnitrogen tableoftotals AB isshown inTable3. Compute the correction factor and the totalsum of squares as: G 2 C.

III A V, 13, 17, 17, 48, V 2 16, 21, 18, 56, V 3 14, 18, 21, 54, V 4 14, 20, 15, 50, V 5 16, 13, 16, 47, V 6 8, 9, 10, 28, Rep. Compute the mean square for each sourceofvariation by dividing theSS by its d.

The value of cv c is expected to be the smallest and the precision for measuring the interaction effect is, thus, the highest. Forcv a and cv b , however, there is no basis to expect oneto be greater orsmaller then the other. Forourexample, because thed.

Enter all values computed in steps 3 to 10 in the analysis of variance outline ofstep 1,as shown in Table 3. Compare each computed Fvalue withits corresponding tabular Fvalues and designate thesignificant results with the appropriate asterisk notation see Chapter 2,Section 2. For our example, both Fvalues, one corresponding to the main effect of variety and another to the interaction between variety and nitrogen, are significant. With a significant interaction, caution must be exercised when mean interpreting the results.

See Chapter 5,Section 5. This is done by applying the rules for grouping of treatments described in Section 2. Thus, the group balanced block design can be superimposed on the split-plot design resulting in what is generally called thegroup balanced block in split-plot design; orit can be superimposed onthe strip-plot design resulting in a groupbalanced block in strip-plot design. We limit our discussicn to a group balanced block in split-plot design and illustrate it using an.

Hunter and J. Stuart Hunter This fresh approach to statistics focuses on applications in the physical, engineering, biological, and social sciences. Written for the non-statistician, the book emphasizes the need for the investigator to make his research as effective as possible through the proper choice and conduct of experiments, and the appropriate analysis of data.

Included are numerous worked examples, exercises with answers , and end-of-chapter questions and problems. From the Publisher Designed for the nonexpert in statistics, this text presents a clear discussion of statistical methods for agricultural research and how to use them.

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