3. Comparison of sources of planting material

3.1 Arrangements of progenies to compare sources of planting material

The following layout in Fig. 1 presents one replication of a trial to compare four sources of planting material, A, B, C and D, each represented by 10 progenies.

Progenies are arranged in plots of 27 palms consisting of two rows between the inspection path and the collection road. Sources of planting material are surrounded by a double guard row with a mixture of progenies from the bordering seed source. The fruit bunches per plot of two half-rows are assembled at the collection road and weighed in bulk. A signboard indicating the number of the progeny (1 to 10) is placed at the side of the collection road.

3.2 Statistical analysis for comparing sources of planting material

There is much scope for improving trials to compare planting material. Sources of planting material are often compared by using a mixture of progenies planted in unreplicated commercial blocks; in the worst case sources are not even planted at the same time, and are also often compared under different environmental conditions. Results are more reliable, if sets of identified crosses representing each source of planting material are compared in a replicated randomized block design; with S sources and with P progenies per source; each replication is in this way a complete block accomodating S*P progenies. However the sources are on main-plots and the progenies are randomized over the sub-plots in a main-plot (see Fig. 1).

ANOVA Table in this chapter presents the Analysis of Variance (ANOVA) table of such source comparison experiment with R replications:

Before showing how to enhance the power of the tests, a basic aspect of statistical analysis is first briefly reviewed. A significant treatment effect (in this case sources of planting material) means that the F-value in the ANOVA-table should be above a critical F-value; this critical value is mainly determined by the Degrees of Freedom (DF) of the denominator Mean Square in the test-statistic. As can be seen in tables of the F-distribution, F-values required for significance are very high if the number of DF of the denominator is below 7, then values decrease sharply and level off when DF becomes 10. In practice therefore the design of experiments should ensure that the number of DF of the denominator is at least 10.

Because we have a design where per replication main-plots have been used to compare sources of planting material, and within a main-plot the progenies of a source are allotted at random to the sub-plots, we have an Analysis of Variance according to a so-called mixed model. There are three variance components, namely Var(RepxSource) for the main-plot error, Var(Prog (Source)) for the variation caused by taking a random sample of all possible progenies from a source, and Var(Error) for the sub-plot error, (see for example Verdooren (1988) or Searle, Casella & McCullogh (1992)).

The expected values of the Mean Squares (EMS) in the ANOVA table are as follows:

EMS(Rep)= Var(Error) + P*Var(RepxSource) + Q(Rep),

where Q(Rep) is a quadratic function of the fixed replication effects;

EMS(So)= Var(Error) + R*Var(Prog(Source)) + P*Var(RepxSource) + Q(Source),

where Q(Source) is a quadratic function of the fixed source-effects;

EMS(RepxSo) = Var(Error) + P* Var(RepxSource) ;EMS(Prog(So)) = Var(Error) + R*Var(Prog(Source)) ; EMS(E)= Var(Error) .

Notice that sometimes one uses an ANOVA table where replicate effects are taken to be random instead of fixed. In that case in the EMS(Rep) the term Q(Rep) is replaced by P*S*Var(Rep). But this change has no effect in the follwing test.

From these EMS it follows that to test the hypothesis H0: " Var(Prog(Source)) = 0" we must use MS(E) as the denominator in the test statistic, MS(Prog(So))/MS(E), because under H0 EMS(Prog(So)) is equal to EMS(E). This test-statistic has under H0 an F distribution with S(P-1) and S(P-1)(R-1) degrees of freedom.

To test the hypothesis H0: "The sources are alike in effect, hence Q(Source)=0 " we must use a combination of Mean Squares as

MS(RepxSo) +MS(Prog(So)) - MS(E)

= MS(Comb) as the denominator in the test statistic, MS(So)/MS(Comb), because under H0 EMS(So) is equal to

EMS(RepxSo) + EMS(Prog(So)) - EMS(E).

This test-statistic has an approximate F-distribution with S-1 and C degrees of freedom, where C is to the nearest integer rounded value of

[MS(Comb)]² /{ [MS(RepxSo)]²

/(R-1)(S-1) + [MS(Pro(So))]²

/S(P-1) + [MS(E)]² /S(P-1)(R-1)}.

The test of comparing progenies-within-source (with test-statistic MS(Pro(So))/MS(E) ) be comes more powerful, in terms of DF, with increasing the number of replications (R).

The main objective is, however, to test whether sources of planting material differ significantly. The power of the test of comparing sources (with test-statisitc MS(So)/MS(Comb)) is with a given number of sources enhanced by increasing the number of progenies within sources (P). Thus, as may be expected, an adequate number of identified progenies is crucial for a meaningful comparison of sources of planting material.