The VSNi Team

2 months agoLinear mixed effects models provide a powerful tool for modelling temporally correlated data. The goal of correlation modelling is to describe how the dependence between measurements changes as the separation in time between them increases. For example, if we measure a patient’s blood pressure each month, we expect measurements on consecutive months to be more alike than measurements several months apart.

Four commonly used correlation models for repeated measures data are:

**Simple**correlation structure, also known as**uniform**correlation. This model assumes the correlation () between measurements is constant regardless of how far apart in time they are.

**Autoregressive**model of order 1. This model allows the correlations between measurements to decrease as the length of time between them increases. This is a more realistic correlation structure for most repeated measures data sets than assuming constant correlation between all pairs of measurements. However, the autoregressive model should only be used when the repeated measures data is at equally spaced time intervals.

**Power**model of order 1. This is an alternative to the autoregressive model of order 1 which accommodates unequally spaced repeated measures. As with the autoregressive model, this allows the correlations between measurements to decrease as the length of time between them increases.

Note: is the Euclidean distance between the and measurements (i.e., time points).

**General**correlation structure, also known as**unstructured**correlation. This is the most flexible correlation structure. It allows a separate correlation for every pair of measurements.

Note: is the Euclidean distance between the and measurements (i.e., time points).

In addition, another important correlation structure is the **identity** structure. This assumes that the observations are uncorrelated, or independent of one another.

Let’s look at an example.

Our example is from an orthodontic growth rate study of children (Potthoff and Roy, 1964). In this study, researchers at the University of North Carolina Dental School tracked the orthodontic growth of 27 children by measuring the distance between the pituitary and the pterygomaxillary fissure every 2 years from the ages of 8 to 14 years. The graph below plots the orthodontic growth profiles of the individual children, showing the distance data in terms of sex and age.

The data set contains 2 variates:

`distance`

, the response variable`age`

, age of the child in years

and 3 factors :

`Subject`

, the individual children from whom repeated measures have been taken`Sex`

, sex of the child`agef`

, also age of the child in years but stored as a factor

Our goal is to model the `distance`

data allowing male and female children to have different growth patterns as they age. However, as this is *repeated measures* data, we must take into account the correlated nature of the distance measurements taken on the same child (i.e., `Subject`

) as they age. For this study, distance measurements taken on the **same** child at consecutive ages should be * correlated* but measurements on

This correlation is imposed by specifying the appropriate correlation structures on the residual.

The orthodontic growth data set contains a total of 108 observations arising from 27 children (factor `Subject`

) each measured 4 times, at ages 8, 10, 12 and 14 (factor `agef`

). Thus, for this data set the residual corresponds to the combination of the factors `Subject`

and `agef`

.

Our residual model has an I ⊗ C covariance structure, where the identity matrix I corresponds to the independent children, and the covariance matrix C corresponds to the correlated measurements over age within a child. |

Since the distance measurements taken on different children are assumed to be *independent*, the correlation structure that should be associated with the `Subject`

term in the residual is the **identity** correlation structure.

However, as the distance measurements taken on the same child over consecutive ages should be *correlated*, we need to select a different correlation structure for the `agef`

term in the residual.

Recall that:

- the simple correlation structure assumes that the correlation between distance measurements taken on the same child is constant regardless of how far apart in age the measurements were taken
- the autoregressive correlation structure of order 1 and the power model of order 1 allow the correlation between distance measurements taken on the same child to decrease as the age difference between the measurements increase.
- the general correlation structure allows a separate correlation for every pair of ages

So, for example, if we were to impose a general correlation structure on the `agef`

term in the residual we would be allowing the correlation to be different between every pair ages, i.e.,

In summary, if we model the distance data using a repeated measures model with an identity correlation structure on the `Subject`

term and a general correlation structure on the `agef`

term in the residual, we are saying that the residuals between different children (`Subject`

) are independent, but the residuals originating from measurements taken on the same child but at different levels of `agef`

are correlated according to the general correlation structure.

Related Reads

The VSNi Team

7 months agoOutliers are sample observations that are either much larger or much smaller than the other observations in a dataset. Outliers can skew your dataset, so how should you deal with them?

Imagine Jane, the general manager of a chain of computer stores, has asked a statistician, Vanessa, to assist her with the analysis of data on the daily sales at the stores she manages. Vanessa takes a look at the data, and produces a boxplot for each of the stores as shown below.

Vanessa pointed out to Jane the presence of outliers in the data from Store 2 on days 10 and 22. Vanessa recommended that Jane checks the accuracy of the data. *Are the outliers due to recording or measurement error?* If the outliers can’t be attributed to errors in the data, Jane should investigate what might have caused the increased sales on these two particular days. Always investigate outliers - this will help you better understand the data, how it was generated and how to analyse it.

Vanessa explained to Jane that we should never drop a data value just because it is an outlier. The nature of the outlier should be investigated before deciding what to do.

Whenever there are outliers in the data, we should look for possible causes of error in the data. If you find an error but cannot recover the correct data value, then you should replace the incorrect data value with a missing value.

However, outliers can also be real observations, and sometimes these are the most interesting ones! If your outlier can’t be attributed to an error, you shouldn’t remove it from the dataset. Removing data values unnecessarily, just because they are outliers, introduces bias and may lead you to draw the wrong conclusions from your study.

- Transform the data: if the dataset is not normally distributed, we can try transforming the data to normalize it. For example, if the data set has some high-value outliers (i.e. is right skewed), the log transformation will “pull” the high values in. This often works well for count data.
- Try a different model/analysis: different analyses may make different distributional assumptions, and you should pick one that is appropriate for your data. For example, count data are generally assumed to follow a Poisson distribution. Alternatively, the outliers may be able to be modelled using an appropriate explanatory variable. For example, computer sales may increase as we approach the start of a new school year.

In our example, Vanessa suggested that since the mean for Store 2 is highly influenced by the outliers, the median, another measure of central tendency, seems more appropriate for summarizing the daily sales at each store. Using the statistical software Genstat, Vanessa can easily calculate both the mean and median number of sales per store for Jane.

Vanessa also analyses the data assuming the daily sales have Poisson distributions, by fitting a log-linear model.

Notice that Vanessa has included “Day” as a blocking factor in the model to allow for variability due to temporal effects.

From this analysis, Vanessa and Jane conclude that the means (of the Poisson distributions) differ between the stores (p-value < 0.001). Store 3, on average, has the most computer sales per day, whereas Stores 1 and 4, on average, have the least.

There are other statistical approaches Vanessa might have used to analyse Jane’s sales data, including a one-way ANOVA blocked by Day on the log-transformed sales data and Friedman’s non-parametric ANOVA. Both approaches are available in Genstat’s comprehensive menu system.

There are many ways to deal with outliers, but no single method will work in every situation. As we have learnt, we can remove an observation if we have evidence it is an error. But, if that is not the case, we can always use alternative summary statistics, or even different statistical approaches, that accommodate them.

Dr. John Rogers

9 months agoEarlier 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.

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.

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.

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!

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)

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.

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

Kanchana Punyawaew

9 months agoThis 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:

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*.

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).

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:

The estimated variance components are:

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:

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.

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*).

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:

The estimated variance components and sequential Wald tests are:

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.

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..