Dr. Salvador A. Gezan18 days ago
Often, we see reported the mean, or average, of a group of data. For example, we hear that the mean cost of a litre of regular gasoline is US $2.98. But what does this really mean? We know that the mean is a summary statistic that tells us about the data of interest, but this statistic can often be quite misleading or ‘mean’ in its interpretation!
Let’s consider another case: the monthly rainfall of approximately 50 mm (600 mm annually) for several cities in the world. Here, the cities of London (United Kingdom) and Seville (Spain) both have this precipitation, but as you might guess their patterns are very different. London has the driest month with 40 mm and wettest of 73 mm, and Seville ranges from 2 to 96 mm. That is why in London it is easier to keep the grass green!
There are many cases like the ones above, where the mean is not sufficient to describe the process of interest properly. To illustrate some of these concepts I am going to use three different data distributions.
Before we go into more detail, let’s go back to the summary statistics. First, there are two main statistics that are commonly presented as a measure of ‘central tendency’, and a measure of ‘dispersion’. The most popular central tendencies are the mean, mode and median.
The mean is often represented by the arithmetic average of all available observations. The mode is the most frequent value found on the distribution of the variable, and the median is the value that is exactly at the middle of all data when they are sorted from smallest to largest. All these central tendency measures are good, but they have their specific properties.
Let's consider plot A, which is a perfectly symmetrical distribution, and this, as an example, is the weight in grams of individual apples. Here we have a symmetric distribution, that resembles what is known as the ‘Normal distribution’. In this instance the mean, median and mode are almost identical with a value of 188 grams, so any could be reported.
The distribution in plot B is a typical case of a skewed distribution representing, for example, the average income in a food factory. Here, there is a small group of managers with large incomes, but the majority will earn much less. For this distribution, the mean will always be larger than the median and the mode, making it a misleading statistic. In our example, the mean is US $223,000 but the median is only US $80,000 and the mode is even lower at US $38,000. Here the median is a much better representation of the centre of the data, indicating that 50% of the people earn less than $80,000 and another 50% earn more than $80,000. The arithmetic mean will be a larger, inflated value that is affected, in this case by, the salary of the CEO corresponding to two million US dollars.
Finally, in the distribution for plot C we have the total height of a random group of students from a given college. The mean is 165.6 cm, and this value is found on the centre of the distribution. But we do have two peaks! The mode of each of these peaks is 162.3 and 174.8 cm, corresponding to the two groups we are expecting to find in this sample: females and males, respectively. In this case, we have more females in the sample than males. Here, the mean is correct, but it fails to describe the fact that we have different groups.
Hence, we see that the mean can be, and often is, a misleading ‘central tendency’ statistic. We can do better, depending on the distribution of the data, by reporting median or mode. However, this is not the full story; we also have the other group of statistics, the measures of ‘dispersion’, ‘spread’ or ‘variability’ of the data. These are also very important and provide a different angle to our data. Some of the more important measures of variability are the standard deviation, range, and percentiles.
The standard deviation is a complicated formula that calculates how the data is dispersed around its mean, where large values indicate more variability. Using our fictional food factory example of managers’ pay, it will be common to report that we have a mean salary of US $223,000 with a standard deviation of US $96,000. If we have a perfect Normal distribution, we often extrapolate that twice the standard deviation covers approximately 95% of the data (US $31,000 to US $415,000).
Often a range is more useful to describe the spread or dispersion of the data. In this same case we have a range of salaries from US $16,000 to US $2,000,000, but as you can see, this is not very informative to describe anything but the extremes; hence, we often use a range associated with percentiles. For example, we can indicate that 90% of the salaries are found between US $27,000 and US $303,000, a better description of the actual dispersion of earnings.
As you can see, the mean is often a misleading statistic which could lead us to some incorrect interpretations, or it can provide us with incomplete pictures of the data of interest. For this reason, we always suggest to first pick the correct measure of centrality, and second, follow it with one measure of dispersion.
Nevertheless, do not forget that these are summary statistics that aim at providing you with a quick and dirty simplification of the data, and in any summarization, there will be always loss of information! So, next time you get some of these statistics think about how mean or misleading they can be!
Salvador Gezan is a statistician/quantitative geneticist with more than 20 years’ experience in breeding, statistical analysis and genetic improvement consulting. He currently works as a Statistical Consultant at VSN International, UK. Dr. Gezan started his career at Rothamsted Research as a biometrician, where he worked with Genstat and ASReml statistical software. Over the last 15 years he has taught ASReml workshops for companies and university researchers around the world.
Dr. Gezan has worked on agronomy, aquaculture, forestry, entomology, medical, biological modelling, and with many commercial breeding programs, applying traditional and molecular statistical tools. His research has led to more than 100 peer reviewed publications, and he is one of the co-authors of the textbook Statistical Methods in Biology: Design and Analysis of Experiments and Regression.
The VSNi Team6 months ago
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.
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.
Rothamsted Annual Report for 1952: "The analytical work has again involved a very considerable computing effort."
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.
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.
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 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.
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!
The VSNi Team5 months ago
A way to decide whether to reject the null hypothesis (H0) against our alternative hypothesis (H1) is to determine the probability of obtaining a test statistic at least as extreme as the one observed under the assumption that H0 is true. This probability is referred to as the “p-value”. It plays an important role in statistics and is critical in most biological research.
P-values are a continuum (between 0 and 1) that provide a measure of the strength of evidence against H0. For example, a value of 0.066, will indicate that there is a probability that we could observe values as large or larger than our critical value with a probability of 6.6%. Note that this p-value is NOT the probability that our alternative hypothesis is correct, it is only a measure of how likely or unlikely we are to observe these extreme events, under repeated sampling, in reference to our calculated value. Also note that this p-value is obtained based on an assumed distribution (e.g., t-distribution for a t-test); hence, p-value will depend strongly on your (correct or incorrect) assumptions.
The smaller the p-value, the stronger the evidence for rejecting H0. However, it is difficult to determine what a small value really is. This leads to the typical guidelines of: p < 0.001 indicating very strong evidence against H0, p < 0.01 strong evidence, p < 0.05 moderate evidence, p < 0.1 weak evidence or a trend, and p ≥ 0.1 indicating insufficient evidence , and a strong debate on what this threshold should be. But declaring p-values as being either significant or non-significant based on an arbitrary cut-off (e.g. 0.05 or 5%) should be avoided. As Ronald Fisher said:
“No scientific worker has a fixed level of significance at which, from year to year, and in all circumstances he rejects hypotheses; he rather gives his mind to each particular case in the light of his evidence and his ideas” .
A very important aspect of the p-value is that it does not provide any evidence in support of H0 – it only quantifies evidence against H0. That is, a large p-value does not mean we can accept H0. Take care not to fall into the trap of accepting H0! Similarly, a small p-value tells you that rejecting H0 is plausible, and not that H1 is correct!
For useful conclusions to be drawn from a statistical analysis, p-values should be considered alongside the size of the effect. Confidence intervals are commonly used to describe the size of the effect and the precision of its estimate. Crucially, statistical significance does not necessarily imply practical (or biological) significance. Small p-values can come from a large sample and a small effect, or a small sample and a large effect.
It is also important to understand that the size of a p-value depends critically on the sample size (as this affects the shape of our distribution). Here, with a very very large sample size, H0 may be always rejected even with extremely small differences, even if H0 is nearly (i.e., approximately) true. Conversely, with very small sample size, it may be nearly impossible to reject H0 even if we observed extremely large differences. Hence, p-values need to also be interpreted in relation to the size of the study.
 Ganesh H. and V. Cave. 2018. P-values, P-values everywhere! New Zealand Veterinary Journal. 66(2): 55-56.
 Fisher RA. 1956. Statistical Methods and Scientific Inferences. Oliver and Boyd, Edinburgh, UK.
The VSNi Team5 months ago
Outliers 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.
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.