Further inference from an ANOVA table: residual variance to standard errors and confidence intervals

Further inference from an ANOVA table: residual variance to standard errors and confidence intervals

Dr. Vanessa Cave

31 May 2022

Below is an example of a 2-way Analysis of Variance (ANOVA) for a randomised complete block design. From the ANOVA table, we’re going to see how to calculate:

• The standard error of a mean (SEM).
• The confidence interval (CI) for a mean.
• The standard error of the difference between two means (SED).
• The least significant difference between two means (LSD).
• The confidence interval for the difference between two means.

The ANOVA table gives us an estimate of the residual mean square () – also known as the mean square error, residual error or residual variation. This is the variation in the data unexplained by the ANOVA model.  In the example above, this is the variation remaining after the block effects (block) and treatment effects (Nitrogen, Sulphur and the Nitrogen by Sulphur interaction, Nitrogen.Sulphur) have been accounted for.

The standard error of a mean (SEM) is calculated using the following formula:

where n is the number of replicates (or sample size).

The SEM describes the uncertainty in our estimate of the population mean for a treatment from the available data. The bigger the sample size (i.e., the larger n), the smaller the SEM and the more precise our estimate is.

In our example, there are 12 unique treatment combinations: the 3 levels of Nitrogen by the 4 levels of Sulphur. Note, we can obtain the number of levels of each treatment, and number of blocks, from the degrees of freedom in the ANOVA table.

Furthermore, the total number of degrees of freedom + 1, gives us the number of experimental units.  Thus, in our example, there are 36 experimental units, with each of the 12 unique treatment combinations occurring exactly once in each of the 3 blocks.

Therefore…

• The 12 Nitrogen by Sulphur means have replication of 3 (1 replicate per block).

For example, the 3 replicates of the treatment corresponding to the first level of Nitrogen and the second level of Sulphur (Nitrogen 1 Sulphur 2) are highlighted yellow in the schematic below of our randomised complete block design:

Thus, the standard error for the Nitrogen by Sulphur means is:

• The 3 Nitrogen means, pooled over the four Sulphur levels, have replication of 12 (3 blocks x 4 levels of Sulphur).

For example, the replicates of the first level of Nitrogen (Nitrogen 1) are highlighted in yellow:

(Note, within each of the 3 blocks, a given level of Nitrogen corresponds to 4 unique treatment combinations: 1 at each level of Sulphur).

Thus, the standard error for the Nitrogen means is:

• The 4 Sulphur means, pooled over the 3 Nitrogen levels, have replication of 9 (3 blocks x 3 levels of Nitrogen).

For example, the replicates of the second level of Sulphur (Sulphur 2) are highlighted yellow:

Thus, the standard error for the Sulphur means is:

The confidence interval (CI) for a mean is

x

where is the critical value of the  distribution with degrees of freedom. For a confidence interval of C%, . For example, for a 95% confidence interval, . The refers to the residual degrees of freedom. This can be read directly from the ANOVA table.

In a nutshell, a C% confidence interval for a mean is a range of values that you can be C% certain contains the true population mean. Although, strictly speaking, the confidence level C% represents a long-run percentage: the C% confidence interval gives an estimated range of values that we would expect the true, but unknown, population parameter to lie within C% of the times, should we repeat our experiment a large number of times.

The tables of means for our example are given below:

For example, the 95% confidence interval for the overall mean for Sulphur level 4 is:

x

x

Similarly, the 99% confidence interval for the Nitrogen level 1, Sulphur level 3 mean is:

x

x

The standard error of the difference between two means (SED) is calculated using the following formula:

Note: The formula is different when the sample sizes of the means being compared are unequal.

The SED describes the uncertainty in our estimate of the difference between two population means.

For our example, the SED between …

a) two Nitrogen by Sulphur means is:

b) two overall Nitrogen means is:

and

c) two overall Sulphur means is:

Two means can be compared using their least significant difference (LSD). The LSD gives the smallest value in which the absolute difference between the two means is deemed to be is statistically significant at the α level of significance. The LSD is given by:

( x 100%) x

Thus, for our example, the LSD(5%) for comparing …

a)  two Nitrogen by Sulphur means is:

% = x

= x

=

b)  two overall Nitrogen means is:

% = x

= x

=

c)  two overall Sulphur means is:

% = x

= x

=

Using the overall means for Nitrogen as an example, at the 5% significance level…

• there is statistical evidence that the Nitrogen 1 and Nitrogen 2 means differ

, the LSD(5%)

• there is statistical evidence that the Nitrogen 1 and Nitrogen 3 means differ

, the LSD(5%)

• there is NO statistical evidence that the Nitrogen 2 and Nitrogen 3 means differ

, the LSD(5%)

The difference between two means can be also compared, and more fully described, using a confidence interval for the difference between two means. This is given by:

x

or, equivalently,

x x 100%

where and are the two means being compared.

Once again, using the overall means for Nitrogen as an example, the 95% confidence interval for the difference between:

• the Nitrogen 1 and Nitrogen 2 means is:

• the Nitrogen 1 and Nitrogen 3 means is:

• the Nitrogen 2 and Nitrogen 3 means is:

Notice that the CIs comparing Nitrogen 1 with Nitrogen 2, and Nitrogen 1 with Nitrogen 3, both exclude zero. Hence, we can conclude, at the 5% significance level, that the mean for Nitrogen 1 is significant differently from both the Nitrogen 2 and Nitrogen 3 means. (In this case the mean for Nitrogen 1 is lower than that of Nitrogen 2 and Nitrogen 3). Conversely, as the CI comparing Nitrogen 2 and Nitrogen 3 includes zero, we conclude that there is no evidence of a difference between these two means (at the 5% significance level).

Luckily for us, we rarely need to calculate these quantities ourselves, as they are generated by most statistical software packages. However, it is useful to understand how they are calculated and how they are related. For example, in order to scrutinize reported results, or to calculate, at a later date, a quantity that you’ve forgotten to generate.

Genstat has a very powerful set of ANOVA tools, that are straightforward and easy to use. In addition to the ANOVA table, you can readily output the treatment means, SEMs, SEDs, LSDs and CIs.

Dr. Vanessa Cave is an applied statistician interested in the application of statistics to the biosciences, in particular agriculture and ecology, and is a developer of the Genstat statistical software package. She has over 15 years of experience collaborating with scientists, using statistics to solve real-world problems.  Vanessa provides expertise on experiment and survey design, data collection and management, statistical analysis, and the interpretation of statistical findings. Her interests include statistical consultancy, mixed models, multivariate methods, statistical ecology, statistical graphics and data visualisation, and the statistical challenges related to digital agriculture.

Vanessa is currently President of the Australasian Region of the International Biometric Society, past-President of the New Zealand Statistical Association, an Associate Editor for the Agronomy Journal, on the Editorial Board of The New Zealand Veterinary Journal and an honorary academic at the University of Auckland. She has a PhD in statistics from the University of St Andrew.

The VSNi Team

27 April 2021

Evolution of statistical computing

It is widely acknowledged that the most fundamental developments in statistics in the past 60+ years are driven by information technology (IT). We should not underestimate the importance of pen and paper as a form of IT but it is since people start using computers to do statistical analysis that we really changed the role statistics plays in our research as well as normal life.

In this blog we will give a brief historical overview, presenting some of the main general statistics software packages developed from 1957 onwards. Statistical software developed for special purposes will be ignored. We also ignore the most widely used ‘software for statistics’ as Brian Ripley (2002) stated in his famous quote: “Let’s not kid ourselves: the most widely used piece of software for statistics is Excel.” Our focus is some of the packages developed by statisticians for statisticians, which are still evolving to incorporate the latest development of statistics.

Ronald Fisher’s Calculating Machines

Pioneer statisticians like Ronald Fisher started out doing their statistics on pieces of paper and later upgraded to using calculating machines. Fisher bought the first Millionaire calculating machine when he was heading Rothamsted Research’s statistics department in the early 1920s. It cost about £200 at that time, which is equivalent in purchasing power to about £9,141 in 2020. This mechanical calculator could only calculate direct product, but it was very helpful for the statisticians at that time as Fisher mentioned: "Most of my statistics has been learned on the machine." The calculator was heavily used by Fisher’s successor Frank Yates (Head of Department 1933-1968) and contributed to much of Yates’ research, such as designs with confounding between treatment interactions and blocks, or split plots, or quasi-factorials.

Frank Yates

Rothamsted Annual Report for 1952: "The analytical work has again involved a very considerable computing effort."

Beginning of the Computer Age

From the early 1950s we entered the computer age. The computer at this time looked little like its modern counterpart, whether it was an Elliott 401 from the UK or an IBM 700/7000 series in the US. Although the first documented statistical package, BMDP, was developed starting in 1957 for IBM mainframes at the UCLA Health Computing Facility, on the other side of the Atlantic Ocean statisticians at Rothamsted Research began their endeavours to program on an Elliot 401 in 1954.

Programming Statistical Software

When we teach statistics in schools or universities, students very often complain about the difficulties of programming. Looking back at programming in the 1950s will give modern students an appreciation of how easy programming today actually is!

An Elliott 401 served one user at a time and requested all input on paper tape (forget your keyboard and intelligent IDE editor). It provided the output to an electric typewriter. All programming had to be in machine code with the instructions and data on a rotating disk with 32-bit word length, 5 "words" of fast-access store, 7 intermediate access tracks of 128 words, 16 further tracks selectable one at a time (= 2949 words – 128 for system).

Computer paper tape

fitting constants to main effects and interactions in multi-way tables (1957), regression and multiple regression (1956), fitting many standard curves as well as multivariate analysis for latent roots and vectors (1955).

Although it sounds very promising with the emerging of statistical programs for research, routine statistical analyses were also performed and these still represented a big challenge, at least computationally. For example, in 1963, which was the last year with the Elliott 401 and Elliott 402 computers, Rothamsted Research statisticians analysed 14,357 data variables, and this took them 4,731 hours to complete the job. It is hard to imagine the energy consumption as well as the amount of paper tape used for programming. Probably the paper tape (all glued together) would be long enough to circle the equator.

Development of Statistical Software: Genstat, SAS, SPSS

The above collection of programs was mainly used for agricultural research at Rothamsted and was not given an umbrella name until John Nelder became Head of the Statistics Department in 1968. The development of Genstat (General Statistics) started from that year and the programming was done in FORTRAN, initially on an IBM machine. In that same year, at North Carolina State University, SAS (Statistical Analysis Software) was almost simultaneously developed by computational statisticians, also for analysing agricultural data to improve crop yields. At around the same time, social scientists at the University of Chicago started to develop SPSS (Statistical Package for the Social Sciences). Although the three packages (Genstat, SAS and SPSS) were developed for different purposes and their functions diverged somewhat later, the basic functions covered similar statistical methodologies.

The first version of SPSS was released in 1968. In 1970, the first version of Genstat was released with the functions of ANOVA, regression, principal components and principal coordinate analysis, single-linkage cluster analysis and general calculations on vectors, matrices and tables. The first version of SAS, SAS 71, was released and named after the year of its release. The early versions of all three software packages were written in FORTRAN and designed for mainframe computers.

Since the 1980s, with the breakthrough of personal computers, a second generation of statistical software began to emerge. There was an MS-DOS version of Genstat (Genstat 4.03) released with an interactive command line interface in 1980.

Genstat 4.03 for MSDOS

Around 1985, SAS and SPSS also released a version for personal computers. In the 1980s more players entered this market: STATA was developed from 1985 and JMP was developed from 1989. JMP was, from the very beginning, for Macintosh computers. As a consequence, JMP had a strong focus on visualization as well as graphics from its inception.

The Rise of the Statistical Language R

The development of the third generation of statistical computing systems had started before the emergence of software like Genstat 4.03e or SAS 6.01. This development was led by John Chambers and his group in Bell Laboratories since the 1970s. The outcome of their work is the S language. It had been developed into a general purpose language with implementations for classical as well as modern statistical inferences. S language was freely available, and its audience was mainly sophisticated academic users. After the acquisition of S language by the Insightful Corporation and rebranding as S-PLUS, this leading third generation statistical software package was widely used in both theoretical and practical statistics in the 1990s, especially before the release of a stable beta version of the free and open-source software R in the year 2000. R was developed by Ross Ihaka and Robert Gentleman at the University of Auckland, New Zealand, and is currently widely used by statisticians in academia and industry, together with statistical software developers, data miners and data analysts.

Software like Genstat, SAS, SPSS and many other packages had to deal with the challenge from R. Each of these long-standing software packages developed an R interface R or even R interpreters to anticipate the change of user behaviour and ever-increasing adoption of the R computing environment. For example, SAS and SPSS have some R plug-ins to talk to each other. VSNi’s ASReml-R software was developed for ASReml users who want to run mixed model analysis within the R environment, and at the present time there are more ASReml-R users than ASReml standalone users. Users who need reliable and robust mixed effects model fitting adopted ASReml-R as an alternative to other mixed model R packages due to its superior performance and simplified syntax. For Genstat users, msanova was also developed as an R package to provide traditional ANOVA users an R interface to run their analysis.

What’s Next?

We have no clear idea about what will represent the fourth generation of statistical software. R, as an open-source software and a platform for prototyping and teaching has the potential to help this change in statistical innovation. An example is the R Shiny package, where web applications can be easily developed to provide statistical computing as online services. But all open-source and commercial software has to face the same challenges of providing fast, reliable and robust statistical analyses that allow for reproducibility of research and, most importantly, use sound and correct statistical inference and theory, something that Ronald Fisher will have expected from his computing machine!

Dr. Vanessa Cave

10 May 2022

The essential role of statistical thinking in animal ethics: dealing with reduction

Having spent over 15 years working as an applied statistician in the biosciences, I’ve come across my fair-share of animal studies. And one of my greatest bugbears is that the full value is rarely extracted from the experimental data collected. This could be because the best statistical approaches haven’t been employed to analyse the data, the findings are selectively or incorrectly reported, other research programmes that could benefit from the data don’t have access to it, or the data aren’t re-analysed following the advent of new statistical methods or tools that have the potential to draw greater insights from it.

An enormous number of scientific research studies involve animals, and with this come many ethical issues and concerns. To help ensure high standards of animal welfare in scientific research, many governments, universities, R&D companies, and individual scientists have adopted the principles of the 3Rs: Replacement, Reduction and Refinement. Indeed, in many countries the tenets of the 3Rs are enshrined in legislation and regulations around the use of animals in scientific research.

Replacement

Use methods or technologies that replace or avoid the use of animals.

Reduction

Limit the number of animals used.

Refinement

Refine methods in order to minimise or eliminate negative animal welfare impacts.

In this blog, I’ll focus on the second principle, Reduction, and argue that statistical expertise is absolutely crucial for achieving reduction.

The aim of reduction is to minimise the number of animals used in scientific research whilst balancing against any additional adverse animal welfare impacts and without compromising the scientific value of the research. This principle demands that before carrying out an experiment (or survey) involving animals, the researchers must consider and implement approaches that both:

1. Minimise their current animal use – the researchers must consider how to minimise the number of animals in their experiment whilst ensuring sufficient data are obtained to answer their research questions, and
2. Minimise future animal use – the researchers need to consider how to maximise the information obtained from their experiment in order to potentially limit, or avoid, the subsequent use of additional animals in future research.

Both these considerations involve statistical thinking. Let’s begin by exploring the important role statistics plays in minimising current animal use.

Statistical aspects to minimise current animal use

Reduction requires that any experiment (or survey) carried out must use as few animals as possible. However, with too few animals the study will lack the statistical power to draw meaningful conclusions, ultimately wasting animals. But how do we determine how many animals are needed for a sufficiently powered experiment? The necessary starting point is to establish clearly defined, specific research questions. These can then be formulated into appropriate statistical hypotheses, for which an experiment (or survey) can be designed.

Statistical expertise in experimental design plays a pivotal role in ensuring enough of the right type of data are collected to answer the research questions as objectively and as efficiently as possible. For example, sophisticated experimental designs involving blocking can be used to reduce random variation, making the experiment more efficient (i.e., increase the statistical power with fewer animals) as well as guarding against bias. Once a suitable experimental design has been decided upon, a power analysis can be used to calculate the required number of animals (i.e., determine the sample size). Indeed, a power analysis is typically needed to obtain animal ethics approval - a formal process in which the benefits of the proposed research is weighed up against the likely harm to the animals.

Researchers also need to investigate whether pre-existing sources of information or data could be integrated into their study, enabling them to reduce the number of animals required. For example, by means of a meta-analysis. At the extreme end, data relevant to the research questions may already be available, eradicating the need for an experiment altogether!

Statistical aspects to minimise future animal use: doing it right the first time

An obvious mechanism for minimising future animal use is to ensure we do it right the first time, avoiding the need for additional experiments. This is easier said than done; there are many statistical and practical considerations at work here. The following paragraphs cover four important steps in experimental research in which statistical expertise plays a major role: data acquisition, data management, data analysis and inference.

Above, I alluded to the validity of the experimental design. If the design is flawed, the data collected will be compromised, if not essentially worthless. Two common mistakes to avoid are pseudo-replication and the lack of (or poor) randomisation. Replication and randomisation are two of the basic principles of good experimental design. Confusing pseudo-replication (either at the design or analysis stage) for genuine replication will lead to invalid statistical inferences. Randomisation is necessary to ensure the statistical inference is valid and for guarding against bias.

Another extremely important consideration when designing an experiment, and setting the sample size, is the risk and impact of missing data due, for example, to animal drop-out or equipment failure. Missing data results in a loss of statistical power, complicates the statistical analysis, and has the potential to cause substantial bias (and potentially invalidate any conclusions). Careful planning and management of an experiment will help minimise the amount of missing data. In addition, safe-guards, controls or contingencies could be built into the experimental design that help mitigate against the impact of missing data. If missing data does result, appropriate statistical methods to account for it must be applied. Failure to do so could invalidate the entire study.

It is also important that the right data are collected to answer the research questions of interest. That is, the right response and explanatory variables measured at the appropriate scale and frequency. There are many statistical related-questions the researchers must answer, including: what population do they want to make inference about? how generalisable do they need their findings to be? what controllable and uncontrollable variables are there? Answers to these questions not only affects enrolment of animals into the study, but also the conditions they are subjected to and the data that should be collected.

It is essential that the data from the experiment (including meta-data) is appropriately managed and stored to protect its integrity and ensure its usability. If the data get messed up (e.g., if different variables measured on the same animal cannot be linked), is undecipherable (e.g., if the attributes of the variables are unknown) or is incomplete (e.g., if the observations aren’t linked to the structural variables associated with the experimental design), the data are likely worthless. Statisticians can offer invaluable expertise in good data management practices, helping to ensure the data are accurately recorded, the downstream results from analysing the data are reproducible and the data itself is reusable at a later date, by possibly a different group of researchers.

Unsurprisingly, it is also vitally important that the data are analysed correctly, using the methods that draw the most value from it. As expected, statistical expertise plays a huge role here! The results and inference are meaningful only if appropriate statistical methods are used. Moreover, often there is a choice of valid statistical approaches; however, some approaches will be more powerful or more precise than others.

Having analysed the data, it is important that the inference (or conclusions) drawn are sound. Again, statistical thinking is crucial here. For example, in my experience, one all too common mistake in animal studies is to accept the null hypothesis and erroneously claim that a non-significant result means there is no difference (say, between treatment means).

Statistical aspects to minimise future animal use: sharing the value from the experiment

The other important mechanism for minimising future animal use is to share the knowledge and information gleaned. The most basic step here is to ensure that all the results are correctly and non-selectively reported. Reporting all aspects of the trial, including the experimental design and statistical analysis, accurately and completely is crucial for the wider interpretation of the findings, reproducibility and repeatability of the research, and for scientific scrutiny. In addition, all results, including null results, are valuable and should be shared.

Sharing the data (or resources, e.g., animal tissues) also contributes to reduction. The data may be able to be re-used for a different purpose, integrated with other sources of data to provide new insights, or re-analysed in the future using a more advanced statistical technique, or for a different hypothesis.

Statistical aspects to minimise future animal use: maximising the information obtained from the experiment

Another avenue that should also be explored is whether additional data or information can be obtained from the experiment, without incurring any further adverse animal welfare impacts, that could benefit other researchers and/or future studies. For example, to help address a different research question now or in the future. At the outset of the study, researchers must consider whether their proposed study could be combined with another one, whether the research animals could be shared with another experiment (e.g., animals euthanized for one experiment may provide suitable tissue for use in another), what additional data could be collected that may (or is!) of future use, etc.

Statistical thinking clearly plays a fundamental role in reducing the number of animals used in scientific research, and in ensuring the most value is drawn from the resulting data. I strongly believe that statistical expertise must be fully utilised through the duration of the project, from design through to analysis and dissemination of results, in all research projects involving animals to achieving reduction. In my experience, most researchers strive for very high standards of animal ethics, and absolutely do not want to cause unnecessary harm to animals. Unfortunately, the role statistical expertise plays here is not always appreciated or taken advantage of. So next time you’re thinking of undertaking research involving animals, ensure you have expert statistical input!

Dr. Vanessa Cave is an applied statistician interested in the application of statistics to the biosciences, in particular agriculture and ecology, and is a developer of the Genstat statistical software package. She has over 15 years of experience collaborating with scientists, using statistics to solve real-world problems.  Vanessa provides expertise on experiment and survey design, data collection and management, statistical analysis, and the interpretation of statistical findings. Her interests include statistical consultancy, mixed models, multivariate methods, statistical ecology, statistical graphics and data visualisation, and the statistical challenges related to digital agriculture.

Vanessa is currently President of the Australasian Region of the International Biometric Society, past-President of the New Zealand Statistical Association, an Associate Editor for the Agronomy Journal, on the Editorial Board of The New Zealand Veterinary Journal and an honorary academic at the University of Auckland. She has a PhD in statistics from the University of St Andrew.

Kanchana Punyawaew and Dr. Vanessa Cave

01 March 2021

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).

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.

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.

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;

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.

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().

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;

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.