##### Single record animal model in ASReml

19 days ago

A single record animal model is the simplest, and probably most important, linear mixed model (LMM) used in animal breeding. This model is called animal model (or individual model) because we estimate a breeding value for each animal defined in the pedigree. The term single record refers to the fact that animals have only one phenotypic observation available (in contrast with repeated measures, where we have multiple records for each individual).

Breeding values are random effects estimated as part of fitting the LMM, and these correspond to the genetic worth of an individual. In the case of progenitors, this is obtained as the mean deviation of the offspring of the given parent against the population mean. Breeding values are used to select individuals to constitute the next generation.

In this blog we will present an analysis from a swine breeding program where the model has several fixed effects and a single additive genetic effect. The base model is:

where

is the phenotypic observation of animal
is the overall mean
is a fixed effect such as contemporary group
is the random additive genetic value of animal
is the random residual terms

The fixed effects  in this case correspond to contemporary groups, but these effects can be any general nuisance factor or continuous variables of interest to control for, such as herds, year, season (also constructed as a single factor often called hys), sex, or some continuous variables to be used as covariates such as initial size or weight.

In the above model, we have distributional assumptions associated with the random effects, specifically, a ~ MVN(0A) and e ~ MVN(0I). Here, the vector of breeding values a has a mean of zero and they have a variance-covariance matrix A that corresponds to the numerator relationship matrix, and  is the variance of additive effects (i.e.,  = VA). In addition, the vector of residuals has a mean of zero and they have an identity variance-covariance matrix (i.e., independent effects) with  being the error or residual variance.

The above model is fitted with ASReml v4.1 using residual maximum likelihood (REML) to estimate variance components: fixed and random effects (i.e., BLUEs and BLUPs) are obtained by solving the set of mixed model equations.

In ASReml we write a text file (with the extension ‘.as’) with all the specifications of the dataset to be used, together with the details of the model to be fitted. The full code is presented below:

There are many elements in the above code, only a few of which we will discuss here, but more information can be found in the manual. The structure of the dataset ‘pig_production.dat’ and a brief description is presented in lines 2 to 11, and for reference we present a few of the lines of this dataset below:

Some of the descriptions of these variables in the ‘.as’ file are critical. For example, !P is used to indicate that the factor animID is associated with a pedigree file that will be the first file to read. The use of !I and !A is used for specification of factors coded as integers or alphanumeric, respectively. When no qualifier is present the data column will be considered as a continuous real variable, such as with BW100d.

In this example, the use of !P requires the reading of a pedigree file, which is critical for fitting an animal model, and in our example this file is the same as the data file. This is possible because the first three columns of the dataset correspond to the ones required for the pedigree: Animal, Sire and Dam. Additional qualifiers are used for reading these files, such as !SKIP and !ALPHA.

In addition, there are two important qualifiers associated with the analysis. These are: !MVINCLUDE, which is required to keep the missing records associated with some of the factors, and the !DDF 1 qualifier that requests the calculation of an ANOVA table associated with our model using approximated degrees of freedom.

Finally, we find the model specification lines:

The first term BW100d is the response variable. Then the following four terms BreedPOBMOB.YOBand SEXare fixed effect factors. Here MOB.YOB corresponds to a combined factor of all combinations of MOB and YOB (recall these are month and year of birth, respectively). Then the use of !r precedes the definition of random effects; here the only term used is ped(animID), which is the animal effect associated with pedigree. The use of ped() is optional here, as the model term animID was previously read with the qualifier !P before, but this is good practice. Also note that we added a value 10 after this random effect; this is to assist the software with an initial guess of the additive variance component.

The second line is required to define complex variance structures for residual terms. However, in this case we have a simple structure based on independent errors with a single variance (i.e., idv) and this is defined for all units that correspond to each observation. Other structures for random effects and the residual term are possible, and further details can be found in the manual.

After fitting the model, a series of output files are produced with the same basename file, but with different filename extensions. The most important outputs for the animal model of above are: ‘.asr’, ‘.sln’, and ‘.pvc’.

The ‘.asr’ file contains a summary of data, the iteration sequence, estimates of the variance parameters, and the analysis of variance table together with estimates of the fixed effects, among many other things, and also messages. In our dataset, the additive genetic and residual variances for BW200d were estimated to be 11.14 and 79.49 kg, respectively. Fixed effect tests for this trait show highly significant differences (p < 0.01) for most factors, as shown below in an excerpt of this file.

Note that there is additional output in the file ‘.asr’ and probably more that you normally will need. Refer to the manual for additional details and definitions.

The file ‘.sln’ has the solutions (BLUEs and BLUPs) from our analysis. A few lines of this output are presented below:

There are columns to identify the factor and its levels followed by the estimated effect and associated standard error. For example, for animal 477, we note that its BLUE effect (in this case breeding value) is 1.022 kg above the mean. The complete list should help to select the best individuals for this swine breeding study.

There was one additional element from the ‘.as’ file that we did not describe, and this corresponds to the command !VPREDICT that is used to request the additional estimation of the narrow-sense heritability; this will be reported in the ‘.pvc’ file. The lines used corresponded to:

Here we are requesting ASReml to generate a ‘.pin’ file with our variance prediction function request. In this case, we will first take the variance associated with ped(animID) and call it AddVar, then we sum the variance for animID and the residual variance (identified as idv(units)). Finally, we take these two elements and divide them. Hence, we just defined the expression: /(). In the file ‘.pvc’ you will notice the following output:

The heritability for BW100d is 0.123 ± 0.034. Note that, as indicated, standard errors are approximated as this calculation uses the Delta method.

There is another relevant output found in the file ‘.aif’ that reports calculations of each individual’s inbreeding coefficient. This is relevant for selection and control of inbreeding in a program. ASReml produced this additional output because we used the !AIF qualifier, but we have not presented the output in this blog.

ASReml has many other options and it can handle large databases and fit many complex linear models. Here we only presented a few of its capabilities, but if you want to learn more about ASReml check the online resources here. You can find more details of this product at https://www.vsni.co.uk/software/asreml.

Dr. John Rogers

3 months ago

The VSNi Team

a month ago

The VSNi Team

2 months ago

The VSNi Team

a month ago
##### What is a p-value?

A way to decide whether to reject the null hypothesis (H0) against our alternative hypothesis (H1) is to determine the probability of obtaining a test statistic at least as extreme as the one observed under the assumption that H0 is true. This probability is referred to as the “p-value”. It plays an important role in statistics and is critical in most biological research. ![alt text](https://web-global-media-storage-production.s3.eu-west-2.amazonaws.com/blog_p_value_7e04a8f8c5.png) #### **What is the true meaning of a p-value and how should it be used?** P-values are a continuum (between 0 and 1) that provide a measure of the **strength of evidence** against H0. For example, a value of 0.066, will indicate that there is a probability that we could observe values as large or larger than our critical value with a probability of 6.6%. Note that this p-value is NOT the probability that our alternative hypothesis is correct, it is only a measure of how likely or unlikely we are to observe these extreme events, under repeated sampling, in reference to our calculated value. Also note that this p-value is obtained based on an assumed distribution (e.g., t-distribution for a t-test); hence, p-value will depend strongly on your (correct or incorrect) assumptions. The smaller the p-value, the stronger the evidence for rejecting H0. However, it is difficult to determine what a small value really is. This leads to the typical guidelines of: p \< 0.001 indicating very strong evidence against H0, p \< 0.01 strong evidence, p \< 0.05 moderate evidence, p \< 0.1 weak evidence or a trend, and p ≥ 0.1 indicating insufficient evidence $1$, and a strong debate on what this threshold should be. But declaring p-values as being either significant or non-significant based on an arbitrary cut-off (e.g. 0.05 or 5%) should be avoided. As [Ronald Fisher](https://mathshistory.st-andrews.ac.uk/Biographies/Fisher/) said: “No scientific worker has a fixed level of significance at which, from year to year, and in all circumstances he rejects hypotheses; he rather gives his mind to each particular case in the light of his evidence and his ideas” $2$. A very important aspect of the p-value is that it **does not** provide any evidence in support of H0 – it only quantifies evidence against H0. That is, a large p-value does not mean we can accept H0. Take care not to fall into the trap of accepting H0! Similarly, a small p-value tells you that rejecting H0 is plausible, and not that H1 is correct! For useful conclusions to be drawn from a statistical analysis, p-values should be considered alongside the **size of the effect**. Confidence intervals are commonly used to describe the size of the effect and the precision of its estimate. Crucially, statistical significance does not necessarily imply practical (or biological) significance. Small p-values can come from a large sample and a small effect, or a small sample and a large effect. It is also important to understand that the size of a p-value depends critically on the sample size (as this affects the shape of our distribution). Here, with a very very large sample size, H0 may be always rejected even with extremely small differences, even if H0 is nearly (i.e., approximately) true. Conversely, with very small sample size, it may be nearly impossible to reject H0 even if we observed extremely large differences. Hence, p-values need to also be interpreted in relation to the size of the study. #### References $1$ Ganesh H. and V. Cave. 2018. _P-values, P-values everywhere!_ New Zealand Veterinary Journal. 66(2): 55-56. $2$ Fisher RA. 1956. _Statistical Methods and Scientific Inferences_. Oliver and Boyd, Edinburgh, UK.

Kanchana Punyawaew and Dr. Vanessa Cave

3 months ago
##### Mixed models for repeated measures and longitudinal data

The term “**repeated measures**” refers to experimental designs or observational studies in which each experimental unit (or subject) is measured repeatedly over time or space. "**Longitudinal data**" is a special case of repeated measures in which variables are measured over time (often for a comparatively long period of time) and duration itself is typically a variable of interest. In terms of data analysis, it doesn’t really matter what type of data you have, as you can analyze both using mixed models. Remember, the key feature of both types of data is that the response variable is measured more than once on each experimental unit, and these repeated measurements are likely to be correlated. ### Mixed Model Approaches To illustrate the use of mixed model approaches for analyzing repeated measures, we’ll examine a data set from Landau and Everitt’s 2004 book, “_A Handbook of Statistical Analyses using SPSS”. Here, a double-blind, placebo-controlled clinical trial was conducted to determine whether an estrogen treatment reduces post-natal depression. Sixty three subjects were randomly assigned to one of two treatment groups: placebo (27 subjects) and estrogen treatment (36 subjects). Depression scores were measured on each subject at baseline, i.e. before randomization (predep_) and at six two-monthly visits after randomization (_postdep_ at visits 1-6). However, not all the women in the trial had their depression score recorded on all scheduled visits. In this example, the data were measured at fixed, equally spaced, time points. (_Visit_ is time as a factor and _nVisit_ is time as a continuous variable.) There is one between-subject factor (_Group_, i.e. the treatment group, either placebo or estrogen treatment), one within-subject factor (_Visit_ or _nVisit_) and a covariate (_predep_). ![alt text](https://web-global-media-storage-production.s3.eu-west-2.amazonaws.com/repeated_measures_data_4f63d505a9_20e39072bf.png) Using the following plots, we can explore the data. In the first plot below, the depression scores for each subject are plotted against time, including the baseline, separately for each treatment group. ![alt text](https://web-global-media-storage-production.s3.eu-west-2.amazonaws.com/repeated_measures_1_4149bce2a1_20e3c0f240.png) In the second plot, the mean depression score for each treatment group is plotted over time. From these plots, we can see variation among subjects within each treatment group that depression scores for subjects generally decrease with time, and on average the depression score at each visit is lower with the estrogen treatment than the placebo. ![alt text](https://web-global-media-storage-production.s3.eu-west-2.amazonaws.com/repeated_measures_2_92810e7fc9_da9b1e85ff.png) ### Random effects model The simplest approach for [analyzing repeated measures data](https://www.theanalysisfactor.com/repeated-measures-approaches/) is to use a random effects model with _**subject**_ fitted as random. It assumes a constant correlation between all observations on the same subject. The analysis objectives can either be to measure the average treatment effect over time or to assess treatment effects at each time point and to test whether treatment interacts with time. In this example, the treatment (_Group_), time (_Visit_), treatment by time interaction (_Group:Visit_) and baseline (_predep_) effects can all be fitted as fixed. The subject effects are fitted as random, allowing for constant correlation between depression scores taken on the same subject over time. The code and output from fitting this model in [ASReml-R 4](https://www.vsni.co.uk/software/asreml-r) follows; ![alt text](https://web-global-media-storage-production.s3.eu-west-2.amazonaws.com/4_020d75dee9.png) ![alt text](https://web-global-media-storage-production.s3.eu-west-2.amazonaws.com/5_ef250deb61.png) ![alt text](https://web-global-media-storage-production.s3.eu-west-2.amazonaws.com/6_15e353865d.png) The output from summary() shows that the estimate of subject and residual variance from the model are 15.10 and 11.53, respectively, giving a total variance of 15.10 + 11.53 = 26.63. The Wald test (from the wald.asreml() table) for _predep_, _Group_ and _Visit_ are significant (probability level (Pr) ≤ 0.01). There appears to be no relationship between treatment group and time (_Group:Visit_) i.e. the probability level is greater than 0.05 (Pr = 0.8636). ### Covariance model In practice, often the correlation between observations on the same subject is not constant. It is common to expect that the covariances of measurements made closer together in time are more similar than those at more distant times. Mixed models can accommodate many different covariance patterns. The ideal usage is to select the pattern that best reflects the true covariance structure of the data. A typical strategy is to start with a simple pattern, such as compound symmetry or first-order autoregressive, and test if a more complex pattern leads to a significant improvement in the likelihood. Note: using a covariance model with a simple correlation structure (i.e. uniform) will provide the same results as fitting a random effects model with random subject. In ASReml-R 4 we use the corv() function on time (i.e. _Visit_) to specify uniform correlation between depression scores taken on the same subject over time. ![alt text](https://web-global-media-storage-production.s3.eu-west-2.amazonaws.com/7_3f3a2b825a.png) Here, the estimate of the correlation among times (_Visit_) is 0.57 and the estimate of the residual variance is 26.63 (identical to the total variance of the random effects model, asr1). Specifying a heterogeneous first-order autoregressive covariance structure is easily done in ASReml-R 4 by changing the variance-covariance function in the residual term from corv() to ar1h(). ![alt text](https://web-global-media-storage-production.s3.eu-west-2.amazonaws.com/8_27fce61956.png) ### Random coefficients model When the relationship of a measurement with time is of interest, a [random coefficients model](https://encyclopediaofmath.org/wiki/Random_coefficient_models) is often appropriate. In a random coefficients model, time is considered a continuous variable, and the subject and subject by time interaction (_Subject:nVisit_) are fitted as random effects. This allows the slopes and intercepts to vary randomly between subjects, resulting in a separate regression line to be fitted for each subject. However, importantly, the slopes and intercepts are correlated. The str() function of asreml() call is used for fitting a random coefficient model; ![alt text](https://web-global-media-storage-production.s3.eu-west-2.amazonaws.com/9_ec27199248.png) The summary table contains the variance parameter for _Subject_ (the set of intercepts, 23.24) and _Subject:nVisit_ (the set of slopes, 0.89), the estimate of correlation between the slopes and intercepts (-0.57) and the estimate of residual variance (8.38). ### References Brady T. West, Kathleen B. Welch and Andrzej T. Galecki (2007). _Linear Mixed Models: A Practical Guide Using Statistical Software_. Chapman & Hall/CRC, Taylor & Francis Group, LLC. Brown, H. and R. Prescott (2015). _Applied Mixed Models in Medicine_. Third Edition. John Wiley & Sons Ltd, England. Sabine Landau and Brian S. Everitt (2004). _A Handbook of Statistical Analyses using SPSS_. Chapman & Hall/CRC Press LLC.

Kanchana Punyawaew

3 months ago
##### Linear mixed models: a balanced lattice square

This blog illustrates how to analyze data from a field experiment with a balanced lattice square design using linear mixed models. We’ll consider two models: the balanced lattice square model and a spatial model. The example data are from a field experiment conducted at Slate Hall Farm, UK, in 1976 (Gilmour _et al_., 1995). The experiment was set up to compare the performance of 25 varieties of barley and was designed as a balanced lattice square with six replicates laid out in a 10 x 15 rectangular grid. Each replicate contained exactly one plot for every variety. The variety grown in each plot, and the coding of the replicates and lattice blocks, is shown in the field layout below: ![alt text](https://web-global-media-storage-production.s3.eu-west-2.amazonaws.com/lattice_layout_7f57633d37_892b6cf234.png) There are seven columns in the data frame: five blocking factors (_Rep, RowRep, ColRep, Row, Column_), one treatment factor, _Variety_, and the response variate, _yield_. ![alt text](https://web-global-media-storage-production.s3.eu-west-2.amazonaws.com/lattice_data_bd9f4ee008_06c8a6e6fc.png) The six replicates are numbered from 1 to 6 (_Rep_). The lattice block numbering is coded within replicates. That is, within each replicates the lattice rows (_RowRep_) and lattice columns (_ColRep_) are both numbered from 1 to 5. The _Row_ and _Column_ factors define the row and column positions within the field (rather than within each replicate). ### Analysis of a balanced lattice square design To analyze the response variable, _yield_, we need to identify the two basic components of the experiment: the treatment structure and the blocking (or design) structure. The treatment structure consists of the set of treatments, or treatment combinations, selected to study or to compare. In our example, there is one treatment factor with 25 levels, _Variety_ (i.e. the 25 different varieties of barley). The blocking structure of replicates (_Rep_), lattice rows within replicates (_Rep:RowRep_), and lattice columns within replicates (_Rep:ColRep_) reflects the balanced lattice square design. In a mixed model analysis, the treatment factors are (usually) fitted as fixed effects and the blocking factors as random. The balanced lattice square model is fitted in [ASReml-R4](https://www.vsni.co.uk/software/asreml-r) using the following code: plaintext &gt; lattice.asr &lt;- asreml(fixed = yield ~ Variety, random = ~ Rep + Rep:RowRep + Rep:ColRep, data=data1)  The REML log-likelihood is -707.786. The model’s BIC is: ![alt text](https://web-global-media-storage-production.s3.eu-west-2.amazonaws.com/lattice_2_ac553eac69_6d6d40e073.jpg) The estimated variance components are: ![alt text](https://web-global-media-storage-production.s3.eu-west-2.amazonaws.com/lattice_3_69e11e2dff_c34641a3a9.jpg) The table above contains the estimated variance components for all terms in the random model. The variance component measures the inherent variability of the term, over and above the variability of the sub-units of which it is composed. The variance components for _Rep_, _Rep:RowRep_ and _Rep:ColRep_ are estimated as 4263, 15596, and 14813, respectively. As is typical, the largest unit (replicate) is more variable than its sub-units (lattice rows and columns within replicates). The _"units!R"_ component is the residual variance. By default, fixed effects in ASReml-R4 are tested using sequential Wald tests: ![alt text](https://web-global-media-storage-production.s3.eu-west-2.amazonaws.com/lattice_4_e237aed045_274881533e.jpg) In this example, there are two terms in the summary table: the overall mean, (_Intercept_), and _Variety_. As the tests are sequential, the effect of the _Variety_ is assessed by calculating the change in sums of squares between the two models (_Intercept_)+_Variety_ and (_Intercept_). The p-value (Pr(Chisq)) of  \< 2.2 x 10-16 indicates that _Variety_ is a highly significant. The predicted means for the _Variety_ can be obtained using the predict() function. The standard error of the difference between any pair of variety means is 62. Note: all variety means have the same standard error as the design is balanced. ![alt text](https://web-global-media-storage-production.s3.eu-west-2.amazonaws.com/lattice_5_575ede3e94_5b9209f7c3.jpg) Note: the same analysis is obtained when the random model is redefined as replicates (_Rep_), rows within replicates (_Rep:Row_) and columns within replicates (_Rep:Column_). ### Spatial analysis of a field experiment As the plots are laid out in a grid, the data can also be analyzed using a spatial model. We’ll illustrate spatial analysis by fitting a model with a separable first order autoregressive process in the field row (_Row_) and field column (_Column_) directions. This is often a useful model to start the spatial modeling process. The separable first order autoregressive spatial model is fitted in ASReml-R4 using the following code: plaintext &gt; spatial.asr &lt;- asreml(fixed = yield ~ Variety, residual = ~ar1(Row):ar1(Column), data = data1)  The BIC for this spatial model is: ![alt text](https://web-global-media-storage-production.s3.eu-west-2.amazonaws.com/lattice_6_3b978358f9_e792bcc2bd.jpg) The estimated variance components and sequential Wald tests are: ![alt text](https://web-global-media-storage-production.s3.eu-west-2.amazonaws.com/lattice_7_82255b3b94_b5bc40e6ab.jpg) ![alt text](https://web-global-media-storage-production.s3.eu-west-2.amazonaws.com/lattice_8_544d852c25_53b792377f.jpg) The residual variance is 38713, the estimated row correlation is 0.458, and the estimated column correlation is 0.684. As for the balanced lattice square model, there is strong evidence of a _Variety_ effect (p-value \< 2.2 x 10-16). A [log-likelihood ratio test](https://www.statisticshowto.com/likelihood-ratio-tests/) cannot be used to compare the balanced lattice square model with the spatial models, as the variance models are not nested. However, the two models can be compared using BIC. As the spatial model has a smaller BIC (1415) than the balanced lattice square model (1435), of the two models explored in this blog, it is chosen as the preferred model. However, selecting the optimal spatial model can be difficult. The current spatial model can be extended by including measurement error (or nugget effect) or revised by selecting a different variance model for the spatial effects. #### References Butler, D.G., Cullis, B.R., Gilmour, A. R., Gogel, B.G. and Thompson, R. (2017). _ASReml-R Reference Manual Version 4._ VSN International Ltd, Hemel Hempstead, HP2 4TP UK. Gilmour, A. R., Anderson, R. D. and Rae, A. L. (1995). _The analysis of binomial data by a generalised linear mixed model_, Biometrika 72: 593-599..

Arthur Bernardeli

5 days ago
##### Accounting for spatial heterogeneity in plant breeding field trials: a path of no return

Some statistical approaches can cope with spatial heterogeneity in different ways, but special attention must be given to the AR1 x AR1 error modeling. This type of spatial analysis can be performed in the ASReml-R package version 4 (Butler et al., 2017), and it is particularly directed at modeling the residual effect of a genetic/statistical model, by estimating the autoregressive correlation of residuals $(\\xi)$ between  columns and rows in a field. This specific random effect can be defined as  $\\mathbf \\xi = \\{\\xi\_m\\}$ ~ $N(0, R)$ and $R = \\sigma\_\\xi^2$ **$**AR1(\\rho\_c)\\otimes AR1 (\\rho\_r)**$**,  and another effect, such as an independent error or local error $(\\eta)$ can be added as another residual term.  A recent study elaborated by Bernardeli et al. (2021) showed the benefits of performing spatial analysis in plant breeding studies. The authors evaluated seed composition traits (protein, oil, and storage protein) in a set of soybean field trials and compared several statistical models within and across trials. The models use were a baseline randomized complete block design (RCB), which is widely used in this type of studies, and four variants considering different spatial-structured residual terms.  Despite the slightly greater computational needs in fitting the analysis, the spatial approaches resulted in greater genetic gains, heritability and accuracy than the baseline model (RCB), and this can be verified in the table below (adapted from Bernardeli et al., 2021).  ![alt text](https://web-global-media-storage-production.s3.eu-west-2.amazonaws.com/blog_spatial_heterogeneity_table_e131ddc01a.jpg) It is important to highlight that the analytical criteria of BIC (Bayesian Information Criteria) was chosen to assist on model selection. In cases where the spatial models were chosen based on BIC, the heritability and accuracy were superior. When the baseline model was the one selected, the above genetic parameters remained unchanged.  In summary, plant breeders should keep in mind that: phenotype-based field trial analyses through the use of AR1 x AR1 spatial models are at least equal, but often better, and never worse than traditional analyses with independent errors. **References** Butler, D. G., Cullis, B.R., A. R. Gilmour, Gogel, B.G. and Thompson, R. 2017. ASReml-R Reference Manual Version 4. VSN International Ltd, Hemel Hempstead, HP1 1ES, UK. Bernardeli A, Rocha JRASdC, Borém A, et al. Modeling spatial trends and enhancing genetic selection: An approach to soybean seed composition breeding. _Crop Science_. 2021;1–13. https://doi.org/10.1002/csc2.20364.

The VSNi Team

10 days ago

The VSNi Team

a month ago