There were small but consistent differences in predicted breeding values between the base and best standard models (Table 6-4). As expected, the correlations were all higher than 0.95 (except in trial 2), and the relative genetic gains were mostly less than 6%. There was almost no gain from the extended spatial model above the best standard model: parental gains were usually zero and the largest offspring gain was around 1%.
Table 6-4 Correlation of breeding values and gains in selection between models.
The models are: Base – base model, Best - best standard model, and Extended - extended spatial model (where fitted).
Model 1 Model 2 Breeding value correlation Selection gain Trial -
Trait (Age)
Parents Trees Parents
1 in 5
Trees 1 in 20
Base Best 0.970 0.968 5.69% 1.67%
Trial 1 -
Defol (2) Best Extended 0.997 0.998 0.00% 0.27%
DBH (14) Base Best 0.998 0.994 0.00% 0.94% Base Best 0.922 0.936 2.77% 5.06% Trial 2 - DBH (5) Best Extended 0.986 0.987 0.00% 0.91% Trial 3 - DBH (5) Base Best 0.965 0.966 3.42% 4.36% Base Best 0.989 0.988 0.04% 1.72% Trial 4-
Form (3) Best Extended 0.994 0.993 0.43% 0.98%
Base Best 0.991 0.988 1.11% 1.57%
Trial 5 -
DBH (4) Best Extended 0.999 0.999 0.14% 0.15%
6.5 Discussion
On the basis of these results, we advocate an initial combined model for spatial analysis of forest genetic trials which adds an autoregressive error term to the design model and retains an independent error term. In most instances, this was a
considerably better model, and although the differences in selected trees were not large, the model does change the trees that are selected. Although not so different
Chapter 6 – Spatial Methodology
from the alternative models that we tried, it is simple to apply and does not inflate the additive variance. The elimination of non-significant effects from the combined model may reduce the combined model to one without design effects terms, but in a few instances they are retained. While there is no a priori way of identifying which ones this might be, our results, and those of Costa e Silva et al. (2001), suggest that where variation is purely continuous in nature, then most design effects will be much reduced if not eliminated. However, the design effects may remain significant in the presence of autoregressive error terms when there is inter-tree competition or some other unusual effects present.
Gilmour et al. (1997a) suggested an initial AR1⊗AR1 model for agricultural variety plot trials. Early generation agricultural variety trials are often unreplicated and may have no design features to retain (e.g., Cullis and Gleeson (1989); Cullis et al. (1998); Moreau et al. (1999)). However, Qiao et al. (2000) found that in most of the 33 designed wheat trials examined design features were retained, and others have also advocated a combined approach (Federer 1998).
Both an independent and an autoregressive error are necessary. These data, and those of Kusnandar and Galwey (2000) and Costa e Silva et al. (2001), clearly show that in forestry trials the independent error is always present, is large, and accounting for it is always necessary. When fitting an individual tree additive genetic model, without an independent error term, the additive genetic variance can be substantially inflated and the actual patterns of spatial variation may be obscured.
In variety trials where the experimental unit is the plot, an independent error is assumed to represent measurement error. If it is modelled, it is often significant but usually small (Basford et al. 1996; Gilmour et al. 1997a; Cullis et al. 1998). In forestry trials, while measurement error may exist, variation from tree to tree will also be due to microsite and non-additive genetic effects, two sources of variation that are not present in the plot means of the inbred lines used in variety trials. As the large independent error persists in clonal trials (Costa e Silva et al. 2001) and across a range of types of measurement data, this suggests that microsite variation is its primary cause.
We advocate no routine further modelling of trends detected in the initial spatial model. While the tools we used greatly aided in the understanding of the nature of variation at a trial site, the extended spatial modelling required a great deal of effort and did not change the breeding values or selections substantially. Our lack of
success compared with such efforts in variety trials (Gilmour et al. 1997a; Cullis et al. 1998; Qiao et al. 2000) may be due to the much larger array sizes we have used, the more heterogeneous nature of forestry trials and the high proportion of independent error present. Also, the AR1⊗AR1 model seems to be extremely robust and able to adequately accommodate a wide variety of the situations, as Figure 6-1 demonstrates. However, one cannot be dogmatic; other effects can be detected, and where they lead to estimation of significant effects and changes in breeding values, they should be considered.
Not fitting detected trends may lead to problems. Global trends and effects aligned with rows or columns will inflate autocorrelations, which may mask local trends. Brownie and Gumpertz (1997) recommended on the basis of simulation studies that, where present, global trends should be fitted. Failure to do so may lead to estimates of precision that are too small. On the other hand, there is a danger of overfitting effects and artificially reducing the estimates of precision. Although we compared a large number of models, including a great number to arrive at the best block size for postblocking, our measure of model fit (AIC) does not account for multiple
comparisons. There is also a difficulty in correctly defining the number of degrees of freedom in a spatial analysis. However, the large AIC differences from adding autoregressive residuals to the design model and the consistency of the results with the data patterns gives confidence to the conclusions we have drawn.
Previous spatial analyses of forestry data have indicated that the autocorrelations were low. Magnussen (1990) found autocorrelations of between 0 and 0.4 for height plot means in Jack Pine and used values up to 0.5 in simulation studies (Magnussen 1993, 1994). Anekonda and Libby (1996) found values up to 0.4 in a variety of clonal redwood traits, and Dutkowski et al. (unpublished data) found values up to 0.25 in plot basal area for Eucalyptus globulus. However, our much higher autocorrelations are consistent with these studies because they are calculated on a different basis. Our model separates independent error from the much smaller autocorrelated
Chapter 6 – Spatial Methodology
component, whereas previous studies have used a single autocorrelated error. Fitting a family model with no independent error term (not shown) led to much lower autocorrelations.
Our results indicated that the spatial model had no consistent effect on the additive genetic variance. This is broadly consistent with the results from analysis of plot data by Magnussen (1990) but not with his simulation results (Magnussen 1993, 1994), which indicated that patchiness inflated the additive genetic variance. This may again be due to the very high independent error that we have found.
Post-blocking was better than the design model in many cases, confirming the results of Fu et al. (1999), who found that row or column factors explained more variation in Douglas Fir progeny trials than the replicates did. Even though some of our selected cases displayed patterns aligned with rows or columns which would suit post-
blocking, it was also better than the design model where continuous trend dominated. It, however, was not as good as the combined model. Where computing resources are limited and large across-site breeding value predictions are being undertaken (e.g., Jarvis et al. (1995)), post-blocking may be an appropriate approach. However, such computing restrictions are rapidly decreasing, and if they do apply, then for each site the fitted spatial surface may be simply subtracted from the data to greatly simplify across-site predictions. However, both of these approaches may be inefficient where replication at any site is insufficient to properly estimate the environmental surface. The problems inherent in multiple comparisons of spatial models also apply to the post-blocking procedure that we have used.
Finally, the confirmation of efficiency and validity that has been demonstrated through simulation in the agricultural variety trial situation (Lill et al. 1988) (Baird and Mead 1991; Cullis et al. 1992; Brownie and Gumpertz 1997; Azais et al. 1998) needs to be verified for forest genetic trials. We are currently looking at these issues through simulations with assumptions that better match the situations we have found.
6.6 Conclusion
We believe that spatial analysis has a major role to play in the analysis of forest genetic trials. While the gains presented here have been modest, other analyses (Costa e Silva et al. 2001) have indicated that much more substantial gains can be achieved. We advocate analysis at the individual tree level, retaining the design effects, including an independent error term, and not pursuing extended spatial models. This approach differs from that advocated for agricultural variety field trials because of their differences from forest genetic trials. Although the advantage of using spatial analysis is diminished with good design, an autoregressive error structure usually accommodates the spatial variation more naturally than designed blocks. There are many trials designed with relatively large blocks for which it would be vital. Even for well-designed trials adding spatial terms may improve the model. The power of spatial analysis to reveal hitherto undetected trends and improve understanding of variation is an added bonus. Understanding the nature of variation can only help in our design, measurement and analysis of trials.
6.7 Acknowledgements
We thank the Estação Florestal Nacional, Lisbon, Portugal; WA Plantation Resources, Manjimup, Western Australia; Hancock Victorian Plantations, Melbourne, Victoria, Australia; and Instituto Nacional de Tecnología Agropecuaria (INTA), Buenos Aires, Argentina, in association with Sociedade Portuguesa de Papel (SOPORCEL),
Portugal, for making the data available. For his useful comments on the manuscript, we thank Luis Apiolaza. We also thank the Fundação para a Ciência e Tecnologia, Lisbon, Portugal, for their financial support of the second author.
Chapter 7 – Spatial Application