Introduction

Based on the photoperiod and temperature gradients, the Swedish breeding strategy for Norway spruce [Picea abies (L.) Karst.] delineates the country into 22 breeding zones (populations), with each including 50 founders (Danell 1993). This multiple population strategy was considered to be effective in managing the genetic diversity and adaptation to the future climate change while maintaining genetic gain. The drawback of managing so many breeding populations is the increased costs (Rosvall et al. 2011). Also, the division of populations was mainly based on geo-climatic data. Similarly, the deployment of Norway spruce was also divided into 9 seed orchard zones to maximize adaptation and genetic gain.

To optimize the regionalization in Swedish breeding program, it was suggested that all available Swedish Norway spruce trials with an acceptable genetic connectivity should be used to find the biological base of the current division of breeding and seed zones. Berlin et al. (2014) analyzed 65 trials and estimated genetic correlations across trial within each of 17 test series to explore genotype-by-environment (G × E) interaction patterns. However, no convincing G × E interaction patterns justifying the current division of breeding and seed zones were observed.

Within the Norway spruce breeding program in Sweden, progeny trials at 4 sites as a test series are used for each of the 22 populations. These trials are usually well connected for genetic entries. However, genetic connectivity among test series is usually poor (Johnson 2004). Within each breeding zone, a low or moderate G × E interaction, demonstrated with a high genetic correlation, was usually found using the trials from the same test series for tree height. This was attributed to the relatively small geographic area within each test series (Berlin et al. 2014). It is highly desirable to combine different test series within adjacent breeding zones for a joint analysis to explore the validity of current delineation of breeding zones using biological data.

There are many traditional ways to analyze G × E interactions and detect the patterns including (1) analysis of variance (ANOVA), (2) principal components analysis (PCA), and (3) linear regression (Freeman 1973). These methods are not always adequate to dissect a complex interaction structure with missing connections (Zobel et al. 1988). ANOVA can only test the significance of G × E and the relative size of G × E variance to genetic variance, but cannot provide any insight about its patterns; PCA only considers the multiplicative effects of G × E. The linear regression method combines additive and multiplicative components. Various stability parameters can also be estimated from such regression to examine stability of genotypes/families and test trials to infer the causes of the interactions (Finlay and Wilkinson 1963; McKeand et al. 2003; Wu and Ying 1998). For multiple site progeny trials in forestry, among-trial type B (Burdon 1977) genetic correlations were usually estimated using a mixed linear model (Baltunis et al. 2010). Since the 1980s, singular value decomposition (SVD) was employed to describing the G × E patterns (Gauch 1992), initially in agronomic crops using additive main effects and multiplicative interaction model (AMMI), and later in forestry. Recently, factorial regression using a mixed model approach (factor-analytic method—FA) was introduced to explore the G × E patterns for multiple environmental trials in crops (Burgueño et al. 2008; Fox et al. 2007; Mathews et al. 2007; Piepho 1998; Smith et al. 2001; Smith et al. 2015) to relate underlining factors to the causes of G × E interactions. Besides the linear and nonlinear fixed and mixed models using the parametric approaches to decompose the G × E interactions, there are also nonparametric methods to analyze G × E such as multivariate regression tree (MRT) (Hamann et al. 2011). For index selection with linear combination of many traits, G × E pattern for index is a function of the relative importance of each traits (weights) and traits themselves (Namkoong 1985). In forestry, multiple regression and response surfaces were also commonly used to detect a relationship between population variation and environmental gradients for inferring adaptive variation to describe nonlinear G × E (Campbell and Sorensen 1978; Rehfeldt 1983; Wang et al. 2010; Wu and Ying 2004).

Cullis et al. (2014) summarized three main advantages of FA model in multi-environment trials (MET) analysis: (1) the ability to estimate the unstructured variance-covariance matrix without the use of an excessive number of variance parameters when the number of MET increases; (2) less than full rank variance structure for the G × E effect could be fitted using an appropriate estimation algorithm; and (3) the FA model is a mixture of multiple regression and principle component analysis. It could be easily used to explain the nature and the extent of G × E interaction using graphical tools such as biplots (Kempton 1984), regression plots (Thompson et al. 2003), and heatmap for estimated genetic correlation matrix with rows and columns ordered using cluster analysis (Cullis et al. 2010).

The FA model is also becoming popular in forestry MET analysis (Costa e Silva and Graudal 2008; Costa e Silva et al. 2006; Cullis et al. 2014; Hardner et al. 2010; Ivković et al. 2015). Costa e Silva et al. (2006) successfully used FA models to analyze stem diameter at breast height (DBH) for 15 Eucalyptus globulus progeny trials.

Spatial analysis models have recently been widely used in estimating genetic parameters for tree species (Costa e Silva et al. 2001; Dutkowski et al. 2006; Fu et al. 1999). Ye and Jayawickrama (2008) reported that spatial analysis could generally increase heritability and improve the accuracy of estimates of genetic parameters. Smith et al. (2001) considered that using an FA model and adjustments of spatial local and global field trend will improve the results of MET analysis. The suggested approach has been used in MET analysis of radiata pine (Pinus radiata) and eucalypt hybrids (Eucalyptus camaldulensis Dehnh. × E. globulus Labill. and E. camaldulensis × E. grandis) in Australia (Hardner et al. 2010; Ivković et al. 2015). A clear G × E pattern and some causes of G × E patterns have been explored in the 20 full-sib radiata pine progeny trials (Ivković et al. 2015).

In this paper, we selected 20 relatively well-connected progeny trials from a total of 146 Norway spruce trials to explore G × E patterns in southern and central Sweden. The aim of this study are to (1) estimate additive genetic variance and heritability for height in 20 large trials within six test series in three seed orchard zones, (2) estimate genetic correlations between trials, (3) dissect the G × E patterns in southern and central Sweden, and (4) analyze stabilities of selected parents in these trials.

Materials and methods

Field trials and measurements

In six test series, 20 open-pollinated progeny trials were planted by the Swedish tree breeding organization Skogfork in southern and central Sweden from 1986 to 1996, within three seed orchard zones (Fig. 1 and Table 1). The detailed characteristics of the trials are listed in Table 1. Randomized complete block design (RCB) with single-tree plots was used in test series 1 and 2, and completely randomized design with single-tree plots was used in the other test series. These 20 large trials were selected because of their relatively good parental connectedness. The number of female parents varied from 304 to 1389 among trials. The height of trees was measured at the ages of 7–13 years.

Fig. 1
figure 1

Location of 20 Norway spruce progeny trials in six test series. The numbers correspond to the test series described in Table 1

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Table 1 Summary of trial information
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Climate data

Climate information for these 20 trials is shown in Table 2. Daily mean temperature and precipitation from the year of planting to the year of height measurement were downloaded from climate database (http://luftweb.smhi.se/). Mean annual precipitation (MAP), mean precipitation growing season (April to October) (MPGS), mean annual temperature (MAT), and mean annual heat sum (MAHT) above 5 °C were calculated. In order to estimate the influence of low temperature (related to frost damage) on bud burst and the consequent growth of Norway spruce trees, two climate indices were computed, including the mean temperature of days when daily mean temperature was below 3.2 °C in May and July (MTMJ) and the mean temperature of days when daily mean temperature was below 1.3 °C in September and October (MTSO). The MTMJ and MTSO represent spring and autumn cold indices, respectively.

Table 2 Clusters, trial identification, test series, frost damage, age (age of frost damage measured), rotated trial loadings for first three factors, and calculated climate indices for each trial
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Statistical analysis

Spatial analysis

Spatial analysis based on a two-dimensional separable autoregressive (AR1) model was used to fit the row and column directions for tree height data at each trial using ASReml 3.0 (Gilmour et al. 2009). Block, trial, and extraneous effects were estimated simultaneously and all significant block and spatial effects were removed from the raw data. The spatially adjusted data were used for the MET analysis in this paper.

Statistical model, variance components, and genetic parameters

The following reduced parental linear mixed model was used for multi-environment analysis:

$$ \mathbf{y}=\mathbf{X}\tau +\frac{\mathbf{1}}{\mathbf{2}}{\mathbf{F}}_{\mathbf{p}}{\mathbf{u}}_{{\mathbf{a}}_{\mathbf{p}}}+\mathbf{e} $$
(1)

where y is the vector of observations; τ is the vector of fixed effect, including the grand mean, trial, and age; \( {\mathbf{u}}_{{\mathbf{a}}_{\mathbf{p}}} \)is the vector of random additive genetic effects for female parents (mother); and e is the vector of random residual term. X and F p are known incidence matrices for fixed effects and female parents, respectively. Age effect was confounded with the trial effect; therefore, it was dropped out in the final model because of singularity. The random effects in the model are assumed to follow a multivariate normal distribution with means and variances defined by:

$$ {\mathbf{u}}_{\mathbf{a}\mathbf{p}}\sim \mathbf{N}\left(\mathbf{0},{\sigma}_{\mathbf{a}}^{\mathbf{2}}{\mathbf{A}}_{\mathbf{pp}}\right)\mathbf{a}\mathbf{n}\mathbf{d}\ \mathbf{e}\sim \mathbf{N}\left(\mathbf{0},{\sigma}_{\mathbf{e}}^{\mathbf{2}}\mathbf{I}\right) $$

where 0 is the null vector; A pp is the numerator relationship matrix of female parents; I is the identity matrix, with order equal to the number of trees; and \( {\sigma_a^2\mathrm{and}\ \sigma}_e^2 \) are the additive and residual variances, respectively.

Narrow-sense heritability was calculated as follows:

$$ {\widehat{h}}_i^2=\frac{{\widehat{\sigma}}_a^2}{{\widehat{\sigma}}_p^2}=\frac{{\widehat{\sigma}}_a^2}{{\widehat{\sigma}}_a^2+{\widehat{\sigma}}_e^2} $$
(2)

where \( {\sigma}_a^2 \) is the estimate of additive genetic variance, \( {\sigma}_p^2 \) is the phenotypic variance, and \( {\sigma}_e^2 \) is the residual variance. All variance components were estimated from FA model.

Factor analysis

Smith et al. (2001) have found that a FA variance model provides a good parsimonious approximation to the unstructured genotype-by-environment covariance matrix using a few informative factors. The FA model can be viewed as arising from a multiplicative model for the genetic effect in each trial. The additive genetic effect i at trial j can be expressed as

$$ {u}_{a_{p_{ij}}}={\lambda}_{a_{1 j}}{f}_{a_{1 i}}+{\lambda}_{a_{2 j}}{f}_{a_{2 i}}+\dots +{\lambda}_{a_{kj}}{f}_{a_{ki}}+{\delta}_{a_{ij}} $$
(3)

which includes a sum of k multiplicative terms. Each term is the product of a genetic effect (\( {f}_{a_{ri}}) \), which is known as a factor score, and an environment effect (\( {\lambda}_{a_{rj}}) \), which is known as a loading. The k of the FA models is the number of factors (multiplicative terms), and we denote a FA model with k factors as a FAk model. The last term of \( {\delta}_{a_{ij}} \) represents a lack of fit of the regression model, and so will be termed as a genetic residual.

The model 3 can be written in vector notation as

$$ {u}_{a_p}=\left({I}_m\otimes {\varLambda}_a\right){f}_a+{\delta}_a $$
(4)

where Λ a is the t (site) × k matrix of environment loadings, f a is the mk × 1vector of additive genetic scores, and δ a is mt × 1 vector of genetic regression residuals with t and m representing number of trials and additive effects, respectively. f a and δ a are assumed to be independent and be distributed as multivariate Gaussian distribution with zero means and variance matrices given by

$$ Var\left({f}_a\right)={A}_{pp}\otimes {I}_k\; and\;\mathrm{var}\left({\delta}_a\right)={A}_{pp}\otimes {\psi}_a $$

where ψ a is a t × t diagonal matrix with a variance for each environment, which is known as a specific variance. These assumptions lead to

$$ \mathrm{Var}\left({\mathrm{u}}_{\mathrm{a}}\right)={\mathrm{A}}_{\mathrm{pp}}\otimes \left({\Lambda}_{\mathrm{a}}{\Lambda_{\mathrm{a}}}^{\mathrm{T}}+{\uppsi}_{\mathrm{a}}\right) $$
(5)

So that genetic covariance matrix among trials: G a  = (Λ a Λ a T + ψ a ).

To determine the appropriate value of k, Cullis et al. (2014) suggested a measure similar to an R 2 goodness of fit value for a multiple regression. Thus, it can define, for each trial, a percentage of additive genetic variance accounted for by the k multiplicative terms as follows:

$$ {v}_{a_j}=100\sum_{r=1}^k{\lambda}_{a_{r j}}^2/\left(\sum_{r=1}^k{\lambda}_{a_{r j}}^2+{\psi}_{a_j}\right) $$
(6)

In addition, an overall percentage variance accounted for can be calculated as

$$ {v}_a=100\ trace\left({\varLambda}_a{\varLambda}_a^T\right)/ trace\left({G}_a\right) $$
(7)

The FA models with different levels of r factors were tested using residual maximum likelihood ratio test (REMLRT).

Parental stability

Empirical best linear unbiased predictions (EBLUPs) of breeding value for parent i at trial j could be expressed as

$$ {\widehat{u}}_{a_{p_{ij}}}={\widehat{\lambda}}_{a_{1 j}}{\widehat{f}}_{a_{1 i}}+{\widehat{\lambda}}_{a_{2 j}}{\widehat{f}}_{a_{2 i}}+\dots +{\widehat{\lambda}}_{a_{kj}}{\widehat{f}}_{a_{ki}}+{\widehat{\delta}}_{a_{ij}}={\widehat{\beta}}_{ij}+{\widehat{\delta}}_{a_{ij}} $$
(8)

where \( {\widehat{\beta}}_{ij} \) is the predicted regression component for parent i at trial j. Equation 8 could be used to investigate parental stability (Cullis et al. 2010; Smith et al. 2015). When k in the FA model is greater than 1, using rotated trial loading (\( {\widehat{\lambda}}_{a_{rj}}^{\ast } \)) and scores (\( {\widehat{f}}_{a_{ri}}^{\ast } \)) to model \( {\widehat{\beta}}_{ij} \) as \( {\sum}_{r=1}^k{\widehat{\lambda}}_{a_{r j}}^{\ast }{\widehat{f}}_{a_{r i}}^{\ast } \), it may explain more of biological meaning of the interaction. We could build k plots for each parent using the predicted parental regression component (\( {\widehat{\beta}}_{ij} \)) on each trial and their corresponding trial loading (\( {\widehat{\lambda}}_{a_{rj}}^{\ast } \)) for k factors. Therefore, for parent i, the plot for the first factor has the y- and x-axes corresponding to the values of \( {\widehat{\beta}}_{ij} \) and \( {\widehat{\lambda}}_{a_{1 j}}^{\ast } \), respectively, with a slope representing rotated factor score of \( {\widehat{f}}_{a_{1 i}} \) as shown in the plot. The plot for factor 2 was similar, but with adjusted values from factor 1. Generally, the y-axis for plot b (b = 2,…,k) corresponds to the adjusted value of \( {\widehat{\beta}}_{ij} \)as \( \Big({\widehat{\beta}}_{ij}-{\sum}_{r=1}^{b-1}{\widehat{\lambda}}_{a_{r j}}^{\ast }{\widehat{f}}_{a_{r i}}^{\ast } \)), and the x-axis values are \( {\widehat{\lambda}}_{a_{bj}}^{\ast } \). Similarly, the line drawn on plot b for parent i has a slope given by \( {\widehat{f}}_{a_{bi}}^{\ast } \).

Heatmap and hierarchical clustering

There are several tools for exploring G × E interaction based on the FA model (Cullis et al. 2010). We used heatmap to represent genetic correlation matrix. Cluster analysis was employed using the dissimilarity of genetic correlation between trials, and the dendrogram of cluster analysis is presented in the same heatmap.

All the models were fitted using ASReml-R package (Butler et al. 2009) within the R statistical environment (R Core Team 2014). Heatmap.2 is used to build the heatmap for genetic correlation matrix in gplots package. The dissimilarity used for cluster analysis is calculated by hierarchical clustering algorithm in hclust function within the R package.

Approximate standard errors of genetic correlations were estimated in ASReml 4.1 stand alone (Gilmour et al. 2015). Since Swedish Norway spruce tree breeding program used breeding value (BV) for volume at age 55 years as the standard age for selection, all BV estimates need to be projected to the BV at age 55. Breeding values of female parents at age 55 for volume shown in the 20 trials were predicted using TREEPLAN (McRae et al. 2004).

Results

The parental concurrence between trials is presented in Fig. 2 along with test series. The diagonal elements of the matrix are the number of female parents used in the trials, and the off-diagonal elements are the number of the female parents in common among trials. Test series 4 (F1148-F1150) and 5 (F1184, F1215, and F1216) had the best and the second best connection with all other trials, respectively. Test series 3 had good connection with series 2, 4, and 5. However, test series 1 (F1021-F1024) had no connections with test series 2 and 3 (F1059-F1147), and test series 6 (F1267-F1271) also had weak connection with all other test series (i.e., less than 13 common parents with all other test series).

Fig. 2
figure 2

The common parents between all pairs of trials within six test series in three seed zones, two cluster distributions of either 3 or 6 clusters each, respectively. The value on diagonal of the matrix show the number of parents used in each of 20 trials

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Based on REMLRT, k = 4 was the best model to fit G a (Table 3). However, the models beyond r = 3 did not show greater improvement for the Bayesian and Akaike information criteria (BIC and AIC, respectively). Therefore, the FA3 model was selected based on the improvement of AIC and the overall percentage variance accounted for. The distribution of individual trials based on their percentage variances accounted for the first three factors is shown in Fig. 3. FA1 accounted for about 80% variance for six trials while FA2 accounted for about 80−100% for 7 trials. We also observed that all 20 trials have an individual trial variance explained greater than 70% in FA3 model. The FA1, FA2, and FA3 models explain overall 78.3, 86.1, and 93.0% of the variances, respectively, for all 20 trials combined.

Table 3 Residual log-likelihoods for different models (DIAG, heterogeneous additive variances for each of trials, FAk—factor analytic with k factors) of additive variance-covariance matrix, AIC and BIC, and percentages of variance accounted for
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Fig. 3
figure 3

Distribution of percentage variance accounted for by individual trials in FA models (FA1, FA2, and FA3). Overall percentage for each FA model is given in parenthesis

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Regression interpretation of the FA model is often expressed using loadings that have been rotated to a principal component solution (Cullis et al. 2014). In FA3 model, the three loadings totally explained 93.0% of the additive genetic variance. The rotated loadings for the first three factors account for 56.2, 23.7, and 13.1% of the additive genetic variance, respectively, with each pair of loadings orthogonal.

The distribution of narrow-sense heritabilities for 20 trials is shown in Fig. 4. The mean (median) of narrow-sense heritability for the 20 trials was 0.33 (0.31) with a range from 0.11 to 0.57, based on spatial adjusted height data.

Fig. 4
figure 4

Distribution of estimated narrow-sense heritabilities for 20 trials

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The trial-trial additive genetic correlation matrix for tree height was also obtained using the FA3 model and is represented by a heatmap with dendrograms added to the left and to the top of the heatmap (Fig. 5). Additive trial-trial correlations varied from −0.76 to 0.98 with the mean (median) of additive trial-trial correlations of 0.48 (0.58). Test series 6 (F1267, F1270, and F1271) with two most northern trials had particularly low additive trial-trial genetic correlations with other trials (Fig. 5). All the three trials had very low connection with all other trials. Excluding the three trials, mean (median) of additive trial-trial correlations increases to 0.54 (0.65).

Fig. 5
figure 5

Heat map and dendrogram of cross-trial additive genetic correlations for height

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The relationships between parents in common and estimates of type B genetic correlations (as well as their standard errors) for all pairs of trials are shown in Fig. 6. The estimates of type B genetic correlations were generally higher and less varied when the common parents were more than 75 parents. The standard errors were smaller when common parents were more than 75 parents.

Fig. 6
figure 6

Estimated genetic correlations for tree height on the left and their approximated standard errors on the right

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Average genetic correlation coefficients (standard errors) for tree height among and within 6 test series, derived from factor-analytic model, are shown in Table 4. Average additive genetic correlations within test series (average of 0.76 ± 0.05) were higher than those among test series (average of 0.44 ± 0.18), except that genetic correlation (0.69 ± 0.08) between test series 1 and 4 was slightly higher than that within the test series 4 (0.67 ± 0.05). Test series 6 had average correlation of 0.60 ± 0.11 within the test series, but had the lowest (a negative) correlation of −0.19 ± 0.34 with the test series 2 and had a correlation close to 0 with the test series 1 and 3.

Table 4 Average genetic correlation coefficients (standard errors) for height among the six test series, derived from factor-analytic model
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Cluster analysis could divide the 20 trials into six clusters as shown in Figs. 2 and 5 and Table 2 if a low cutoff value (1.4) is used. There was only one trial (F1271) in cluster I. Cluster II had two trials (F1270 and F1267). The three trials for clusters I and II were all from the same test series 6 with two of the most northern trials. Cluster III had four trials that were from the test series 3 and one trial F1067 from test series 2. Cluster IV had three trials from test series 2. Cluster V included all trials in test series 5, plus one trial from test series 1 (F1024) and two trials from test series 4 (F1148 and 1149). Cluster VI had three trials from test series 1 plus one trial F1150, which is from test series 4. It seemed that the six clusters represented more of same series plus some trials which had good connection with the test series. For example, cluster V included all trials in test series 5 plus two trials of test series 4 which had 352/353 common families with series 5.

Three clusters could be derived if a higher value was used as cutoff value (2.3) (Fig. 2): cluster I including all trials in test series 6, cluster II including all trials of test series 2 and 3, and cluster III including all trials of test series 1, 4, and 5. Again, three clusters more or less reflected trial geography and trial connectedness. Cluster I had two trials (F1270 and F1271) in the northern seed zone (zone 6) and one trial (F1267) in the middle seed zone 7. The reason that these three trials being in cluster I could be explained by geography (F1270 and F1271 in the seed zone 6) and good connection with the third trials F1267 (e.g., same series of seed zone 6). For cluster III, test series 1 had good connection (74 common families) with the test series 4 and the best connection (352 common families) between test series 4 and 5, and plus the geography (6 trials in the seed zone 7).

The correlations between climate variables and the trial rotated loadings for each of three factors may reveal the contribution level of the climatic variable to the observed different performance between trials (Table 2). Under the FA3 model, the trial loadings of the three factors were moderately to highly correlated to mean annual temperature (MAT) (0.41, 0.42, and −0.28), mean annual heat sum (MAHT) (0.30, 0.33, and −0.20), conditional mean daily temperature of May and June (MTMJ) (0.65, 0.44, and −0.46), and conditional mean daily temperature of September and October (MTSO) (0.55, 0.27, and −0.50). Both mean annual precipitation (MAP) and mean precipitation growing season – (April to October) (MPGS) showed low correlations with trial loadings for each of 3 factors (−0.27, 0.02, and 0.17 and −0.30, −0.10, and 0.11, respectively). For geographical variables, only latitude showed significant correlation with trial loadings for each of 3 factors (−0.45, −0.36, and 0.21).

Ten trials were recorded for frost damage (Table 2). The trial F1271 in the northeast had the heaviest frost incidence (39.8%). Pearson correlation between dissimilarity (computed as 1-genetic correlation across trial) and absolute difference of frost damage across trial is 0.33. Absolute differences of MTMJ, MTSO, AMT, and North latitude (NL) showed significant Pearson correlations with the relatively additive genetic correlations across trial (0.49, 0.35, 0.25, and 0.23, respectively). Stepwise regression was used to select the variables to predict the variation of additive genetic correlation matrix. We only found that MTMJ and MTSO had significant effect on G × E and they explained 27.8% of variation.

Parental stability may be best viewed using latent regression plots which show genetic responses to each of trial loadings. Twelve parents with the highest breeding values of volume at age 55 estimated by TREEPLAN (McRae et al. 2004) were selected to show their latent regression plots responding to the first three factors in Figs. 7 and 8. If the progeny of the associated parents were planted in the corresponding trials, the score points on the plot were colored as blue; otherwise, they were colored as red dots. For example, progeny of the parents P1, P3, and P4 was tested in four, seven, and six trials, respectively, as shown in Fig. 7. The line on each plot is a regression slope estimated by the predicted factor scores (rotated) for the parent. Table 5 shows the 3 factor scores for the 12 parents and the percentage representation of each parent planted in the 20 trials.

Fig. 7
figure 7

Latent additive genetic regression plots using the first factor (on the left) and the second factor (on the right) for 12 parents with the largest breeding value for volume in age 55 estimated in TREEPLAN

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Fig. 8
figure 8

Latent additive genetic regression plot for the third factor for 12 parents with the largest breeding value for volume in age 55 estimated in TREEPLAN

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Table 5 Predicted (rotated) three factor scores for 12 parents with the highest breeding value of volume in age 55
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The estimated trial loadings for the first factor were all positive, except for trial F1270 (Table 2). In addition, the first factor explained a large proportion of additive G × E variation (56.2%), indicating that the latent regression on the first factor had the greatest impact on predicted breeding values. The large positive rotated factor scores (slope) for the first factor indicated positive correlation between tree height and trial loading score (Fig. 7, on the left). For example, P5 had the largest slope for factor 1, the predicted breeding values were all positive in all trials (Table 2, see loadings for factor 1), and it had larger breeding values in those trials with higher trial loadings. All 12 parents had positive slopes in Fig. 7, which means that these parents had positive response to trial loadings in the first factor. However, P1, P2, P7, P11, and P12 had more spread around the lines (slope) than other parents.

Latent regression plots for factor 2 are shown in Fig. 7 (on the right). P1, P2, and P12 had strong positive responses to trial loadings while P6, P9, and P10 showed small positive responses to trial loadings. P3, P4, P5, and P11 showed negative responses to trial loadings and P8 showed quite stable responses to trial loadings.

Latent regression plots for factor 3 are shown in Fig. 8. P1, P2, P4, P5, and P12 showed moderate to strong positive response to trial loadings for factor 3. P3, P9, P10, and P11 had moderate to strong negative responses to trial loadings. P6, P7, and P8 were almost stable across trials. The differential parental response to FA2 and FA3 factors in Figs. 7 and 8 may indicate potential genotype by FA factor (trial loading) interaction.

Discussion

Connectedness

Prior to estimating the genetic parameters and exploring G × E interaction in MET analysis, genetic connectedness between trials should be examined and analyzed (Apiolaza 2012; Ivković et al. 2015; Smith et al. 2015). Poor connectedness between trials may bias the estimates of genetic correlations (Apiolaza 2012; Dutkowski et al. 2016). Cullis et al. (2014) reported that in an unpublished simulation study, when an FA model was used for estimation of G × E effect, there was little bias in the estimated genetic correlation for a pair of trials with poor concurrence (even zero concurrence), if there was sufficient linkage through other trials. In our study, genetic correlation within test series was generally higher than that among test series. Genetic correlations were generally low among two series if there were a few direct common parents, and in addition, there were only a few common parents in the third series that connected with the both series. For example, testing series 6 had poor correlations and large standard errors with other series because it has no good connections directly and indirectly with other series. On the other hand, test series 1 had no direct connections with test series 2 and 3, but had more than 74 common parents with test series 4, while the test series 4 had about 20 and 965 common parents with the test series 2 and series 3, respectively. From this, we observed a medium correlation of 0.45 between test series 1 and 2 and a medium standard error of 0.15, and a high correlation of 0.71 between test series 1 and 3 and a low standard error of 0.09. Therefore, we could estimate genetic correlation between series that have no or low direct connection, but have good connection with a common third series.

Estimate of G × E interaction and cluster analysis

To understand and estimate the amount of G × E interaction, genetic correlations between trials are commonly used to explore G × E interaction (Berlin et al. 2014; Burdon 1977; Hannrup et al. 2008; Smith et al. 2015; Wu and Matheson 2005). In Sweden, there are a few published papers that discuss patterns of G × E interaction and their significance for Norway spruce breeding. Bentzer et al. (1988) reported high genotypic correlations of height at age 5 for two studies with six and three clonal trials, respectively. Average genotypic correlations of 0.66 and 0.91 were observed, respectively. Karlsson and Högberg (1998) observed a medium type B genotypic genetic correlation of 0.54 for tree height in several clonal trials at age 11. In another study, Karlsson et al. (2001) showed a similar result using 11 clonal trials in southern Sweden and Demark. However, a univariate model was used to estimate the genotypic genetic correlations in their studies. Kroon et al. (2011) reported that a high average of 0.80 for additive genetic correlation in a study using a bivariate model, indicating a low G × E interaction in Norway spruce. Berlin et al. (2014) used a widely accepted multivariate model to estimate genetic correlations and found a low to moderate G × E interaction with an average of 0.72 trial-by-trial genetic correlation in 65 Norway spruce trials in southern Sweden. Estimation of type B genetic correlation between two trials using conventional approach may be inefficient, and often result in estimates of genetic correlation greater than 1 or less than −1 (Cullis et al. 2014).

To improve MET data analysis, Smith et al. (2001) recommended a new approach combining a FA model and spatial analysis; thus, in this study, we used the more advanced approach to estimate trial-by-trial genetic correlation. The substantial G × E interaction was found with an average (median) of 0.48 (0.58) for genetic correlations between the 20 trials in southern and middle Sweden. Our study indicated stronger G × E than the most studies of Norway spruce except for the study by Karlsson and Högberg (1998) in which they observed a median genotypic correlation of 0.54. Such large G × E interaction indicates that exploring patterns and the drivers of G × E is important in current Norway spruce breeding.

In previous studies for G × E interaction of Swedish spruce, estimates of genetic correlations were all within the same test series; so, exploring G × E pattern is relatively difficult because the area of a breeding test series is not large enough to cover a large area. In this study, we used 20 trials in 6 test series which cover several major southern and central Sweden’s seed orchard zones. Thus, a cluster analysis is recommended to detect if the clusters follow the current seed zones. The clusters partially reflected the three seed orchard zones but with many exceptions that were similarly observed by Karlsson et al. (2001) using 11 clonal trials in Denmark and southern Sweden. The discrepancy between the existing seed zone delineation and our clustering results using biological data may indicate that current delineation of seed zone may not be optimal. The fact that uneven trial distribution of same series among the seed zones, low connectedness between series made the clustering of trials based on type B genetic correlation more complicated, and it may have only partially reflected the real G × E patterns for the region tested. We recommend that more connections among testing series should be used in future progeny testing. For example, St Clair and Kleinschmit (1986) found a simple geographic pattern for Norway spruce in northern Germany with trials having more connections. Their study area was divided into two zones based on the altitude of trials, which indicated effect of spring or autumn frost.

Environment drivers

In a previous study, Berlin et al. (2014) examined 15 climate variables to investigate the possibility of predicting genetic correlations across trials using a linear regression model. However, none of climate variables could explain more than 5% G × E interaction variation. However, they observed that frost damage had a significant influence on across-trial genetic correlation. In almost all the published papers involving the G × E discussion for Norway spruce, frost damage was considered as the main cause in southern and central Sweden (Berlin et al. 2014; Karlsson and Högberg 1998; Karlsson et al. 2001). Thus, in this study, we firstly tried to use the frost damage to explain across-trial genetic correlations and found that it could explain 11% of additive genetic correlations. However, the frost damage was only recorded in ten trials out of the total 20 trials and was measured at different ages. To relate potential frost severity at different trials with G × E interactions, we constructed two climate indices (MTMJ and MTSO) related to the temperature at spring and autumn frost seasons. We observed that MTMJ and MTSO could explain 27.8% variation of additive genetic correlation matrix and these two variables showed significant correlation with the first and second (rotated) trial loadings (r = 0.65, and 0.44 for MTMJ for the first and second loading, respectively, and r = 0.55 and 0.37 for MTSO for the first and second loading, respectively). These may indicate that the climate indices that reflect spring or autumn frost seasons could more reliably predict the G × E interaction. We recommend use of the minimum and maximum daily temperatures in spring and autumn to further examine possible accuracy in predicting the G × E interaction for Norway spruce. Since temperature loggers are routinely placed in all field trials now, the local temperature records should be more reliable.

Stability of parents

In general, tree breeding programs select individual trees with the highest breeding values, which are stable across trials. In this study, 12 parents selected with the highest breeding values from a total of 3160 parents. P3, P4, P5, and P8 had both high stability and high breeding values. They are high valued for breeding and deployment in these 20 trials (three seed zones). In contrast, most of other parents that had the higher breeding value in many trials showed very high incidence of frost damage on frost prone trials, and therefore showed poor stability.

In conclusion, we find a relatively large amount of G × E interactions for tree height of Norway spruce in southern and central Sweden. Revision of the delineation of breeding and seed orchard zones was not recommended at moment due to the limited number of test trials and less satisfactory genetic connection among several series in the study. It is recommended that more trials with better connection among test series should be used to improve delineation of breeding and seed orchard in the future. The climate indices of spring and autumn temperature have been shown to account for a large amount of G × E interaction and frost damage could be an important cause of G × E interaction in Norway spruce of southern and central Sweden.