Báo cáo khoa học: "Consequences of reducing a full model of variance analysis in tree breeding experiments"

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Original article Consequences of reducing a full model of variance analysis in tree breeding experiments M H Van De Giertych 2 Sype 1 Institute of 62-035 Kornik, Poland; 2 INRA, Station d’Amélioration Dendrology, des Arbres Forestiers, Ardon, 45160 Olivet, France 20 30 June (Received July 1988; accepted 1989) Summary — An analysis of variance was performed on height measurement of 11-year-old trees (7 in the field), using the results of a non-orthogonal progeny within provenance experiment establi- shed for Norway spruce (Picea abies (L.) Karst.) at 2 locations in Poland. The full model including locations, provenances, progenies within provenances, blocks within locations and trees within plots is used assuming all sources of variation to be random. This model is compared with various models reduced by 1 factor or the other within the model. Theoretical modifications of estimated variance components and heritabilities are tested with experimental data. By referring to the original model it is shown how changes came to be and where the losses of information occurred. A method is pro- posed to reduce the factor level number without bias. The general conclusion is that it pays to make the effort and work with the full model. Piceas abies / height / provenance / progeny / variance analysis / method / genetic parameter Résumé — Conséquences de la réduction d’un modèle complet d’analyse de variance pour des expériences d’amélioration forestière. La hauteur totale à 11 ans, après 7 ans de plantation, a été mesurée en Pologne dans deux sites pour 12 provenances d’Epicéa commun originaires de Pologne, avec environ 8 familles par provenance. Les différents termes et indices sont explicités dans le tableau 1. L’analyse de la variance selon un modèle complet (localité, bloc dans localité, provenance, famille dans provenance, et les diverses interactions) a été réalisée en considérant les facteurs comme aléatoires (tableau 2). Elle est comparée à des analyses selon des modèles simplifiés qui ignorent successivement les niveaux provenance, famille ou bloc, ou les valeurs indi- viduelles. Dans le cas du modèle simplifié sans facteur provenance, les nouvelles espérances des carrés moyens (tableau 3) peuvent être strictement comparées à celles obtenues avec le modèle complet. Les modifications théoriques ont été calculées et sont présentées de façon schématique pour l’estimation des composantes de la variance (tableau 4) et des paramètres génétiques (tableau 5). Les résultats théoriques associés aux autres modèles sont également reportés dans ces deux derniers tableaux. En outre, l’implication du nombre de niveaux par facteur sur les biais entraînés a été précisée. En général, les simplifications surestiment fortement les composantes de la variance et augmentent de façon illicite les gains espérés. Les résultats obtenus avec les don- nées expérimentales montrent effectivement des changements au niveau des composantes de la variance ou des tests associés (tableau 7) et de légères modifications pour les paramètres géné- *Correspondence and reprints.
tiques (tableau 8). Les biais que de telles simplifications peuvent entraîner dans un programme d’amélioration forestière sont discutés. En conclusion, une proposition est formulée pour réduire par étapes mais de façon fiable le nombre de niveaux à étudier. Picea abies / hauteur / provenance / descendance / analyse de variance / méthode / para- mètre génétique progeny, the experiment was established in an INTRODUCTION incomplete block design. Not only were the maternal trees selected at random, but the pro- venances were also a random choice of Forest breeding experiments, In complicated tree Districts in the area and cone collections were particularly when one deals with non- carried out from fellings which were being made orthogonal and unbalanced design, and in the Forest District at the time we arrived this is often the case, the temptation there for cone collection. arises to reduce the model to only those Since all our Polish experiments were concentrated in regions near Kornik and Gol- parts that are of particular interest at a dap, the choice of locations could also be consi- given time. Such reductions from the full dered as random. The blocks in our locations model create certain consequences that are just part of the areas, and therefore cover we are not always fully aware of. The aim all variations of the site, and may also be consi- of the present paper is to show on one dered as random. Details of the study were pre- sented in an earlier paper (Giertych and Kroli- experiment how different reductions of the kowski, 1982). The designations used in the experimental model affect the results and study are shown in table I. conclusions derived from them. As the design is far from orthogonal, ana- lyses of data (height in 1983) were performed in France using the Amance ANOVA programs (Bachacou et al., 1981). Furthermore, the num- MATERIAL ber of factor levels was larger than the compu- ter capacity, and accordingly analyses were done in several stages. The experiment discussed here is a Norway spruce (Picea abies (L.) Karst.) progeny within provenance study established at 2 locations in ANALYSES WITH DIFFERENT MODELS Poland, in Kornik and in Goldap, in 1976 using 2+2 seedlings raised in a nursery in Kornik. The experiment includes half-sib progenies from 12 The full model provenances from the North Eastern range of the spruce in Poland. Originally, cones were collected from 10, randomly selected trees from The full model has been used to extract each of the provenances. However, due to an the maximum amount of information from inadequate number of seeds or seedlings per
In an orthogonal system, these heritabi- in the material explained (symbols are lities can be estimated from the Fvalue of table I): the Snedecor’s test by 1 -(1/F). In fact, due to non-orthogonality and unbalanced design, they were calculated from the variance components. For this half-sib experiment, another Since all elements of the experiment approach is to calculate single tree herita- considered to be random, the were bility (narrow sense) based on the bet- degrees of freedom and expected mean ween-families’ additive variance and the squares for the variance analysis are as phenotypic variance (V): Ph shown in table II obtained through the pro- h=4&2/V where: S sigma; 2 FPh cedure described by Hicks (1973). The theoretical degrees of freedom for an orthogonal model and the expected mean squares are shown in table II. On the basis of this full model, it is pos- sible to calculate heritabilities by the for- Heritability (h variance (V), selection ), 2 mula proposed by Nanson (1970) for: intensity (i) and expected genotypic gain (ΔG =i h &jadnr;V) depend on the aim of the provenances: h σ / V where: P P, 2 2P 2 = - selection and the type of material used. For example, it is possible to estimate the genotypic gain which will be expected for reforestation with the same seeds which F 2 h= and families within provenance: gave the material selected in this experi- - &2/V where: ment. The best provenance may be selec- FF sigma; , ted from a total of twelve, so the expected gain will be estimated with heritability and
variance at the provenance phenotypic up of families into provenances. In a fully level, and i 1.840. The selection of the 2 orthogonal model with p provenances and = best families within each provenance will f families (within provenances), the num- and ber of families is pf with a new subscript k’ family parameters use i 1.289. At the end of these 2 steps, 2 instead of k(i). The model now becomes: = families within the best provenance will be selected; this will be compared to a 1 step selection with i 2.417. Another method is = to select the 50 best individuals from the 9122 trees of this experiment, to propaga- The distribution of the degrees of free- te them, and to establish a seed orchard. dom and the expected mean squares are The expected genetic gain of the seed as shown in table III. In order to use the orchard offsprings will be estimated from variance components estimated from the phenotypic variance (V narrow sense full model, we must combine the sum of ), Ph heritability (h and i 2.865. It is assu- s) 2 squares from table II as follows: = med here that these last values are ones that utilize the maximum amount of data Degrees of freedom and are therefore the best that can be and SS from model 2 SS from model 1 obtained. Let us now examine the changes pro- duced with simpler models when a part of the information is not used. Model ignoring the provenance factor The total sum of squares remains unaf- For increasing estimation of genetic para- fected. Working from the bottom of this list meters, it may be tempting to treat the we can identify, on the left hand-side, the families, altogether, disregarding the split-
of squares with the expected new sum squares multiplied by the degrees of mean freedom indicated above (and in table III), and on the right hand-side, the combina- tion of sums of squares with their expec- ted mean squares multiplied by their own degrees of freedom from table II. The pro- cedure is shown for the 2 first factors. 1/ new residual This implies modifications for variance The degrees of freedom are Ibpf(x-1) for components and total variance as shown both sides of the equation. The equation in table IV. The true variance components SS= SS is transformed as Ibpf(x-1)σ E’ 2 E’E of interactions between provenance and , E 2 Ibpf(x-1)σ thus leads to: = locality (σ or block (σ are each ) PL 2 ) PB 2 E’ sigma; 2E σ &2(relation 1). = split in 2 parts. The largest part enters in 2/ family x block interaction new the component of interactions between The degrees of freedom are I(b-1)(pf-1) family and locality (σ or block (σ ), F’B’ 2 ) F’L’ 2 for the new expected mean square and and the smallest one enters in the locality I(b-1)p(f-1) for the full model. SSF’B’ = ) L’ 2 (σ or block (σ components. For the ) B’ 2 SS + SS becomes: FB PB true variance component for provenance (σ the largest part enters in the family ), P, 2 component (σ and the smallest one is ) F’ 2 lost altogether, so the total variance (V ’) T is lowered by σ (f-1)/(pf-1 P 2 ). Compared to the full model, ignoring the provenance level introduces modifica- tions for estimation of genetic parameters (table V). The mean family variance(V is ) F increased by the largest part of all the considering 1 (σ E’ 2 ) E &2 sigma; and simplifying = components of provenance effect and by (pf-1)x gives: interactions (σ + σ + σ P PL PB 222 /I /Ib) (p-1)f/(pf-1).The family heritability (h ) F 2 decreases slightly and the expected gain is higher (ΔG At the individual level, the ). F phenotypic variance (V is lowered by ) Ph The same procedure is followed for the smallest part of variance components other equalities of sums of squares. To for provenance effects (σ + σ + P 2 PL 2 summarize, when we decide to speak of )(f-1)/(pf-1). PB 2 σThe narrow sense heri- families only, instead of provenances and tability (h and the expected genetic gain ) s 2 families (within provenances), we obtain for additive effect (ΔG) are increased by a the following changes in variance compo- part of the non-additive effects, originating nents: from provenance variations. One point of interest is to observe the changes which occur in relation to the number of provenances (p) or families per provenance (f). For the same total number of families (pf),the larger the number of provenances, the lower the loss of total
variance and the larger phenotypic ) T’ (V The total variance and the locality and heritabilities and expected variances, block (within locality) variance compo- gains at family or individual levels. By nents remain unaffected (table IV). Prove- increasing the total number of families (pf), nance and provenance-locality variance the same modifications occur. components are increased respectively by the smallest part of family and family-loca- lity components. The provenance-block Ignoring the family factor interaction is modified by a small part of a combination of all family components while the main part is included in the resi- When comparing provenances, it seems dual. For genetic estimations (table V), the easier to ignore the family variation, and to mean provenance variance (V remains ) P reduce the experiment to a simple prove- unchanged and the provenance heritability nance trial. We then obtain the following ) P 2 (h is higher with the increase of the model, where a new subscript (n’) is used variance component of provenance. instead of (k) family and of (n) tree ones: Consequently, the expected gain for a pro- venance selection is higher. All these modifications depend on the number of families per provenance only (f). The As with the previous model, expected higher the number, the lower the bias. squares can be constructed with the mean new sums of squares and new degrees of freedom and then compared to the original Model ignoring block effect full model 1. Change is observed for the residual level only, which now includes all Sometimes authors have no interest in the family-dependent variations: SS SS + E’ E = variation between blocks and they place SS + SS + SS The degrees of free- FB FL F . all block effects into the residual. A new dom become Ibp(fx-1) Ibpf(x-1) + = subscript n’ must be used instead of m for I(b-1)p(f-1) + (I-1)p(f-1) + p(f-1) with fx block and n for tree, the model will then the new number of trees per plot. Using be: the same procedure as for model 2°, we obtained the following values for the new variance components in terms of those of the original full model: The new sums of squares, compared to the original ones, will change for residual only: SS E’ E SS FB SS PB SS . B SS + + + = bx becomes the new number of trees now per element of the experiment and the new degrees of freedom for the error term become: Following the befo- procedure same as re, we obtain new values of variance com- ponents:
family x block interaction nent for the model becomes: ). FB 2 (σ The The number of trees per plot (x) is the unit of measurement and does not new enter into the degrees of freedom. Since the analysis is performed on the basis of plot means (sums per plot divided by x), original sums of squares must be divided by x The new sums of squares will be . 2 constructed as below: Degrees of freedom When ignoring the block effect, the total variance and the variance components for provenance and family levels remain unaf- fected (table IV). The variance compo- provenance-locality locality nents at or part of the block or small levels include a provenance-block component. The main part of the block and block-interaction variance components enters into the resi- dual. For genetic parameters (table V), variance of means, heritability and expec- ted gain are not changed for provenance the family levels. At the individual level, or the phenotypic variance is increased by main part of the block component (σ B 2 (b-1)/b), so the single tree heritability is lower than that in the full model and the expected genetic gain decreases. The 3 and simplifying: Considering more blocks (b), the smaller the bias for variance components, and the larger for mass selection option. Model with plot averages Computations and results are similar remaining variance components, for all Another method used is to work on plot thus: averages only. This is generally used for traits such as mortality or productivity per unit area. In these cases, SS is not avai- E lable and the only approach is to use SS FB as the new residual. It is therefore impos- sible to estimate the true variance compo-
models. The experimental factor levels are indicated in table I. With the full model (model 1),the result of the analysis of variance (table VI) The decrease of variance components shows that 3 factors are in significant. It is depends on the number of trees per plot very surprising that no locality effect and (x), and reflects the use of plot means as no provenance x locality interaction effect, compared with individual data (table IV). exists. The climatic conditions in Kornik For the total variance, a part of losses is a and Goldap are very different, but grand «logical» reduction due to a lower number means are identical for the 2 locations of data (V Another part is a loss of /x). T (232.8 cm and 234.6 cm respectively). information (σ(x-1)/x2). The higher the E 2 Unfortunately, locality and provenance x number of trees per plot (x), the lower the locality variance components have negati- total variance (V also lower is the relati- ), T’ ve values (they are indicated between ve loss due to lack of information (σ ). E 2 brackets in tables VI and VII), but are For provenance and family levels, in considered as zero for the estimation of comparison with the full model, heritabili- genetic parameters. The non-significance ties are unaffected while means variances of the provenance effect is not unders- are divided by x, and expected gains are tood. For these reasons, the demonstrati- divided by &jadnr;x (table V). Without individual ve aspect of our experimental data will be data, phenotypic variance (V and single ) Ph weaker. Consequences for improvement tree heritability (h cannot be estimated. s) 2 are indicated in table VIII. The choice of the best provenance is uncertain here Experimental data since F is not significant. Reforestation P with the best 2 families, if seed supply is sufficient, would give an expected gain of data (tree height at 7 years Experimental 17 cm. With the seed orchard option, the in the field) analyzed with the 5 were
expected genetic gain would be 18 cm (or so the single tree heritability component, 8%). s) 2 (h and the expected genetic gain With model 2, ignoring the provenance decrease (ΔG). level, the total variance (V calculated, ) T’ For the last model, with plot means ins- including the negative values for some tead of individual data, the family x locality variances, declined from 6842.4 to 6839.0, interaction becomes more significant (from which corresponds exactly to the lost part 1 % to 0.1 %) due to the change of deno- of the variance P (2 σ minator level (new residual instead of provenance (f-1)/(pf-1) 46.5* 0.074 3.4). Changes family x block level). For the total variance = = observed at the family or family interaction (V 339), the reduction of information T’ = levels depend on values of variance com- E] 22 ([σ (x-1)/x 466) represents more = ponents at provenance or provenance than half of the «logical» variance (V /x T = interaction levels (table VII). Thus, the 805). The decrease of genetic parameters F-test can increase (family x block level) is important but corresponds to the reduc- or decrease (family x locality level from tion in the number of measured data. 1 % probability level to non-significant). Compared to the full model, ignoring the provenance level introduces large modifi- DISCUSSION AND CONCLUSION cations for estimation of the genetic para- meters (table VIII). At the family level, the In practice, reduction of a full model to a heritability (h decreases slightly while ) F 2 simpler one is often due to computation the expected genotypic gain (ΔG ) F limitations. This may result in the calcula- becomes higher. At the individual level, tion for unbalanced design with non-ortho- the narrow sense heritability (h and the s) 2 gonal data which necessitates special pro- expected genetic gain for additive effects cedures such as Amance programs used (ΔG) are increased by a part of the non- for this study. Another possible limitation is additive effects originating from provenan- the kind of design treated by a computer ce variations. program, such as nested and cross classi- For the third model, without the family fication. Another difficulty is the maximum level, changes are not large and F-tests number of levels which is compatible with have the same probability level (table VII). computer capacity; in this experiment, the Provenance heritability (h increases ) P 2 number is 1188. All these limitations incite from 0.27 to 0.40 with a part of the family authors to reduce their statistical model to heritability (table VIII). Consequently, the expected gain for provenance selection is a simpler one which they are able to ana- higher (ΔG 6.6 to 8.9 cm) even if the lyse. : P choice for provenance is uncertain (F-test The material used here is just an for provenance is not significant). example. On the hand, it gives one no Ignoring the block level (model 4), intro- significant effects for locality, provenance duces changes for the provenance x loca- and provenance x locality interaction and lity level (F-test from non-significant to 2 variance components are negative. On significant at 0.1 % level) and for the fami- the other hand, our experimental data ly x locality level (table VII). As was expec- gives a very low single tree heritability (h s 2 ted in table V, most genetic parameters 0.080) and a small expected genetic = remain the same (table VIII). However, gain (+ 8 %) for total height at 7 years in changes are observed at the individual the field (11 from the seed). It is possible level. The phenotypic variance (V is that the experimental design, described by ) Ph increased by the main part of the block Giertych and Krolikowski (1982), is not
with zed gain can be very low compared suited to give the best estimation of gene- tic parameters. In spite of the weak the expected one. Working on plot means, all variance demonstrative value or our experimental components are changed in the same data, our results clearly indicate that any ratio but heritabilities for provenance and deviation from the full model introduces family remain the same, while genotypic significant changes. For example, interac- gains are reduced. This method causes a tion estimations are very different, and very important loss of information, but may may lead to opposite conclusions depen- be necessary for traits like mortality or pro- ding on the adopted model. ductivity per unit area. the provenance level leads to Ignoring Since every model reduction leads to of information (total variance loss a modifications, it can prove useful to find decrease) and an unjustified increase of way to change the model in order to obtain single tree heritability and consequently the technical means to treat it. One possi- expected genetic gain. We have shown bility consists of discarding blocks by that these increases result in the addition adjusting individual data to the block of non-additive variance from provenance effect. A first step can be an analysis to an additive one from half-sib families. without the family level, which is the only Thus, this method can only be used if pro- model to give good estimates of locality venances originate from the same ecotype and block variance components. The and if genetic structures are comparable. second step consists of adjusting indivi- This can best be tested by Bartlett’s or dual data to block and/or locality effect. At Hartley’s tests for 1 trait or by comparison the same time, the total degrees of free- of variance-covarience matrix with Kull- dom must be reduced by those for block back’s test (1967) for multitrait analysis. and/or locality. The next steps, comprising Furthermore, we must bear in mind that interaction studies, can be achieved by dif- the bias is lower when the numbers of pro- ferent analyses, with more facility for com- venances and of families per provenance putation. In any case, expected mean are low. squares of simplified models must be By ignoring the family level, provenan- compared with the full model ones. heritability increases because it ce Working on a full model may involve includes a part of the family variance. The extra work, and require some stepwise correlated expected gain is also higher. procedures, due to the limited capacity of The bias is low with a large number of computers, but the effort is worthwhile families per provenance and is unaffected because otherwise unreliable or incomple- by the number of provenances. Compared te results would be obtained. with others, this model introduces less modification. When ignoring the block level, heritability and expected gains at family or provenance levels remain the REFERENCES same. At the single tree level, the modifi- cation appears to have few conse- Bachacou J, Masson JP, Millier C (1981) quences. In fact, with this procedure, the Manuel de la programmathèque statistique. expected genotypic gain is obtained by Amance 1981. Département de Biométrie, selection of the best trees, located in the INRA, 516 pp richest part of the site. Accordingly, the Z (1982) Doswiadcze- Giertych M, Krolikowski selected material does not necessarily nie nad zmiennoscia populacyjna i rodowa swierka pospolitego (Picea abies (L.) Karst.) have the best genetic value, and the reali-
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