logo
Software
Sector
Learning
Resources
Blogs
About us
Licensing
Contact
Single record animal model in ASReml

The VSNi Team

4 months 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:

alt text

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:

alt text

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:

alt text

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.

alt text

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:

alt text

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:

alt text

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:

alt text

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.

Related Reads

READ MORE

Kanchana Punyawaew

7 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

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

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 using the following code:

> lattice.asr <- 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

The estimated variance components are:

alt text

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

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

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:

> spatial.asr <- asreml(fixed = yield ~ Variety,
                        residual = ~ar1(Row):ar1(Column),
                        data = data1)

The BIC for this spatial model is:

alt text

The estimated variance components and sequential Wald tests are:

alt text

alt text

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 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..

READ MORE

Dr. John Rogers

6 months ago
50 years of bioscience statistics

Earlier this year I had an enquiry from Carey Langley of VSNi as to why I had not renewed my Genstat licence. The truth was simple – I have decided to fully retire after 50 years as an agricultural entomologist / applied biologist / consultant. This prompted some reflections about the evolution of bioscience data analysis that I have experienced over that half century, a period during which most of my focus was the interaction between insects and their plant hosts; both how insect feeding impacts on plant growth and crop yield, and how plants impact on the development of the insects that feed on them and on their natural enemies.

Where it began – paper and post

My journey into bioscience data analysis started with undergraduate courses in biometry – yes, it was an agriculture faculty, so it was biometry not statistics. We started doing statistical analyses using full keyboard Monroe calculators (for those of you who don’t know what I am talking about, you can find them here).  It was a simpler time and as undergraduates we thought it was hugely funny to divide 1 by 0 until the blue smoke came out…

After leaving university in the early 1970s, I started working for the Agriculture Department of an Australian state government, at a small country research station. Statistical analysis was rudimentary to say the least. If you were motivated, there was always the option of running analyses yourself by hand, given the appearance of the first scientific calculators in the early 1970s. If you wanted a formal statistical analysis of your data, you would mail off a paper copy of the raw data to Biometry Branch… and wait.  Some months later, you would get back your ANOVA, regression, or whatever the biometrician thought appropriate to do, on paper with some indication of what treatments were different from what other treatments.  Dose-mortality data was dealt with by manually plotting data onto probit paper. 

Enter the mainframe

In-house ANOVA programs running on central mainframes were a step forward some years later as it at least enabled us to run our own analyses, as long as you wanted to do an ANOVA…. However, it also required a 2 hours’ drive to the nearest card reader, with the actual computer a further 1000 kilometres away.… The first desktop computer I used for statistical analysis was in the early 1980s and was a CP/M machine with two 8-inch floppy discs with, I think, 256k of memory, and booting it required turning a key and pressing the blue button - yes, really! And about the same time, the local agricultural economist drove us crazy extolling the virtues of a program called Lotus 1-2-3!

Having been brought up on a solid diet of the classic texts such as Steele and Torrie, Cochran and Cox and Sokal and Rohlf, the primary frustration during this period was not having ready access to the statistical analyses you knew were appropriate for your data. Typical modes of operating for agricultural scientists in that era were randomised blocks of various degrees of complexity, thus the emphasis on ANOVA in the software that was available in-house. Those of us who also had less-structured ecological data were less well catered for.

My first access to a comprehensive statistics package was during the early to mid-1980s at one of the American Land Grant universities. It was a revelation to be able to run virtually whatever statistical test deemed necessary. Access to non-linear regression was a definite plus, given the non-linear nature of many biological responses. As well, being able to run a series of models to test specific hypotheses opened up new options for more elegant and insightful analyses. Looking back from 2021, such things look very trivial, but compared to where we came from in the 1970s, they were significant steps forward.

Enter Genstat

My first exposure to Genstat, VSNi’s stalwart statistical software package, was Genstat for Windows, Third Edition (1997). Simple things like the availability of residual plots made a difference for us entomologists, given that much of our data had non-normal errors; it took the guesswork out of whether and what transformations to use. The availability of regressions with grouped data also opened some previously closed doors. 

After a deviation away from hands-on research, I came back to biological-data analysis in the mid-2000s and found myself working with repeated-measures and survival / mortality data, so ventured into repeated-measures restricted maximum likelihood analyses and generalised linear mixed models for the first time (with assistance from a couple of Roger Payne’s training courses in Hobart and Queenstown). Looking back, it is interesting how quickly I became blasé about such computationally intensive analyses that would run in seconds on my laptop or desktop, forgetting that I was doing ANOVAs by hand 40 years earlier when John Nelder was developing generalised linear models. How the world has changed!

Partnership and support

Of importance to my Genstat experience was the level of support that was available to me as a Genstat licensee. Over the last 15 years or so, as I attempted some of these more complex analyses, my aspirations were somewhat ahead of my abilities, and it was always reassuring to know that Genstat Support was only ever an email away. A couple of examples will flesh this out. 

Back in 2008, I was working on the relationship between insect-pest density and crop yield using R2LINES, but had extra linear X’s related to plant vigour in addition to the measure of pest infestation. A support-enquiry email produced an overnight response from Roger Payne that basically said, “Try this”. While I slept, Roger had written an extension to R2LINES to incorporate extra linear X’s. This was later incorporated into the regular releases of Genstat. This work led to the clearer specification of the pest densities that warranted chemical control in soybeans and dry beans (https://doi.org/10.1016/j.cropro.2009.08.016 and https://doi.org/10.1016/j.cropro.2009.08.015).

More recently, I was attempting to disentangle the effects on caterpillar mortality of the two Cry insecticidal proteins in transgenic cotton and, while I got close, I would not have got the analysis to run properly without Roger’s support. The data was scant in the bottom half of the overall dose-response curves for both Cry proteins, but it was possible to fit asymptotic exponentials that modelled the upper half of each curve. The final double-exponential response surface I fitted with Roger’s assistance showed clearly that the dose-mortality response was stronger for one of the Cry proteins than the other, and that there was no synergistic action between the two proteins (https://doi.org/10.1016/j.cropro.2015.10.013

The value of a comprehensive statistics package

One thing that I especially appreciate about having access to a comprehensive statistics package such as Genstat is having the capacity to tease apart biological data to get at the underlying relationships. About 10 years ago, I was asked to look at some data on the impact of cold stress on the expression of the Cry2Ab insecticidal protein in transgenic cotton. The data set was seemingly simple - two years of pot-trial data where groups of pots were either left out overnight or protected from low overnight temperatures by being moved into a glasshouse, plus temperature data and Cry2Ab protein levels. A REML analysis, and some correlations and regressions enabled me to show that cold overnight temperatures did reduce Cry2Ab protein levels, that the effects occurred for up to 6 days after the cold period and that the threshold for these effects was approximately 14 Cº (https://doi.org/10.1603/EC09369). What I took from this piece of work is how powerful a comprehensive statistics package can be in teasing apart important biological insights from what was seemingly very simple data. Note that I did not use any statistics that were cutting edge, just a combination of REML, correlation and regression analyses, but used these techniques to guide the dissection of the relationships in the data to end up with an elegant and insightful outcome.

Final reflections

Looking back over 50 years of work, one thing stands out for me: the huge advances that have occurred in the statistical analysis of biological data has allowed much more insightful statistical analyses that has, in turn, allowed biological scientists to more elegantly pull apart the interactions between insects and their plant hosts. 

For me, Genstat has played a pivotal role in that process. I shall miss it.

Dr John Rogers

Research Connections and Consulting

St Lucia, Queensland 4067, Australia

Phone/Fax: +61 (0)7 3720 9065

Mobile: 0409 200 701

Email: john.rogers@rcac.net.au

Alternate email: D.John.Rogers@gmail.com

READ MORE

Kanchana Punyawaew and Dr. Vanessa Cave

7 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

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

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

Random effects model

The simplest approach for analyzing repeated measures data 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 follows;

alt text

alt text

alt text

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

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

Random coefficients model

When the relationship of a measurement with time is of interest, a random coefficients model 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

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.

plant
plant
plant
A world leader in the advancement and application of algorithmic and analytical content for the smart/precision biotech sector

Follow us

youtube     twitter     linkedin
Copyright © 2000-2021 VSN International Ltd. | Privacy Policy | EULA | Terms & Conditions | Sitemap
VSN International Limited is registered in England & Wales, company number: 4027977 VAT number: GB750 0348 63