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Warning! The missing data problem

The VSNi Team

4 months ago

Missing data are a common problem, even in well-designed and managed research studies. It results in a loss of precision and statistical power, has the potential to cause substantial bias and often complicates the statistical analysis. As data analysts, it is crucial that we understand the reasons for the missing data and apply appropriate methods to account for it. Failure to do so leads to trouble! Let’s consider an example.

The following figure illustrates data from a study on the intelligence quotient (IQ) of students living in a city, town, or village. A random selection of students was recruited into the study and their IQs measured. Unfortunately, IQ measurements were not available for all students in the study, resulting in missing data (denoted by NA). To make valid inferences, it is very important that we understand why the data are missing, and what we can do about it.

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There are three main types of missing data:

Missing completely at random (MCAR)

Here missingness has nothing to do with the subject being studied: instead the missing values are an entirely random subset of the data. In our example, the missingness could be considered MCAR if the missing IQ test results were accidentally deleted by the researcher.

Missing at random (MAR)

Here missingness is not related to the value of the missing data but is related to other observed data. For example, we might find that younger students were less likely to complete the IQ test due to some factor unrelated to their IQ, such as a shorter attention span than older students. This missing data can be considered MAR, as failure to complete the test had nothing to do with their IQ (after accounting for age).

Missing not at random (MNAR)

When the missing data are neither of the above, they are MNAR. In other words, missingness is related to the value of the missing data. For example, if failure to complete the IQ test was related to the student’s IQ the missingness would be MNAR.

When the data are MCAR, missingness doesn’t induce bias. However, this isn’t the case for MAR and MNAR, and great care needs to be taken to ensure that this does not lead to biased and misleading inferences.

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Let’s go back to our example.  Assuming the MCAR mechanism, we could analyse the data in Genstat using an unbalanced ANOVA. Assuming that  MAR is conditional on age, an unbalanced ANOVA with age as a covariate would be an appropriate analysis.  But what to do if the missingness is MNAR? This is much more problematical. Indeed, the only way to reduce potential bias is if we can explicitly model the process that generated the missing data. Challenging indeed!

As our example illustrates, when faced with missing data we must apply an appropriate statistical method to accommodate it. The choice of method will depend on the nature of the missingness, in addition to the type of data we have, the aims of our analysis, etc.

But what about imputation?

Imputation replaces missing values with estimated values. That is, the dataset is completed by filling in the missing values. Many different methods of imputation exist, including mean substitution, regression imputation, EM algorithm, and last observation carried forward. Be warned however, imputation may not overcome bias - indeed it may also introduce it! In addition, it does not account for the uncertainty about the imputed missing values, resulting in standard errors that are too low. However, if the proportion of missing values in the dataset is small, imputation can be useful. Many statistical methods can handle datasets with missing values e.g., maximum likelihood, expected maximization, Bayesian models. Others, such as principal component analysis and some spatial or temporal mixed models, require complete datasets. Imputation allows us to apply statistical methods requiring complete datasets.

As a final thought - the best possible solution for missing data is to prevent the problem from occurring. Carefully planning and managing your study will help minimize the amount of missing data (or at least ensure it is MCAR or MAR!).  When you have missing data, use an appropriate statistical technique to accommodate it. Statistical or computational techniques for imputing missing data should be the last resort.

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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:

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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.

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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:

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The estimated variance components are:

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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:

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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.

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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:

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The estimated variance components and sequential Wald tests are:

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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..

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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

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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).

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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.

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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.

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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;

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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.

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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().

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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;

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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.

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