Statnote 22: Measuring 'agreement'

If a quantity has been measured by two different methods, the degree of agreement between them should not be tested using Pearson’s correlation coefficient ‘r’. Instead the differences between the two methods should be compared with their mean difference using a Bland and Altman plot. Such a plot illustrates the level of agreement between two methods and enables the degree of bias of one method over the other to be calculated and applied if necessary as a correction factor.

Statnote 35: are the data log-normal?

In many of the Statnotes described in this series, the statistical tests assume the data are a random sample from a normal distribution These Statnotes include most of the familiar statistical tests such as the ‘t’ test, analysis of variance (ANOVA), and Pearson’s correlation coefficient (‘r’). Nevertheless, many variables exhibit a more or less ‘skewed’ distribution. A skewed distribution is asymmetrical and the mean is displaced either to the left (positive skew) or to the right (negative skew). If the mean of the distribution is low, the degree of variation large, and when values can only be positive, a positively skewed distribution is usually the result. Many distributions have potentially a low mean and high variance including that of the abundance of bacterial species on plants, the latent period of an infectious disease, and the sensitivity of certain fungi to fungicides. These positively skewed distributions are often fitted successfully by a variant of the normal distribution called the log-normal distribution. This statnote describes fitting the log-normal distribution with reference to two scenarios: (1) the frequency distribution of bacterial numbers isolated from cloths in a domestic environment and (2), the sizes of lichenised ‘areolae’ growing on the hypothalus of Rhizocarpon geographicum (L.) DC.

Statnote 36: Do the data fit the Poisson distribution

In previous Statnotes, many of the statistical tests described rely on the assumption that the data are a random sample from a normal or Gaussian distribution. These include most of the tests in common usage such as the ‘t’ test ), the various types of analysis of variance (ANOVA), and Pearson’s correlation coefficient (‘r’) . In microbiology research, however, not all variables can be assumed to follow a normal distribution. Yeast populations, for example, are a notable feature of freshwater habitats, representatives of over 100 genera having been recorded . Most common are the ‘red yeasts’ such as Rhodotorula, Rhodosporidium, and Sporobolomyces and ‘black yeasts’ such as Aurobasidium pelculans, together with species of Candida. Despite the abundance of genera and species, the overall density of an individual species in freshwater is likely to be low and hence, samples taken from such a population will contain very low numbers of cells. A rare organism living in an aquatic environment may be distributed more or less at random in a volume of water and therefore, samples taken from such an environment may result in counts which are more likely to be distributed according to the Poisson than the normal distribution. The Poisson distribution was named after the French mathematician Siméon Poisson (1781-1840) and has many applications in biology, especially in describing rare or randomly distributed events, e.g., the number of mutations in a given sequence of DNA after exposure to a fixed amount of radiation or the number of cells infected by a virus given a fixed level of exposure. This Statnote describes how to fit the Poisson distribution to counts of yeast cells in samples taken from a freshwater lake.

Can aberrometry provide rapid and reliable measures of subjective depth of focus following multifocal intraocular lens implantation?

Purpose: To determine whether the ‘through-focus’ aberrations of a multifocal and accommodative intraocular lens (IOL) implanted patient can be used to provide rapid and reliable measures of their subjective range of clear vision. Methods: Eyes that had been implanted with a concentric (n = 8), segmented (n = 10) or accommodating (n = 6) intraocular lenses (mean age 62.9 ± 8.9 years; range 46-79 years) for over a year underwent simultaneous monocular subjective (electronic logMAR test chart at 4m with letters randomised between presentations) and objective (Aston open-field aberrometer) defocus curve testing for levels of defocus between +1.50 to -5.00DS in -0.50DS steps, in a randomised order. Pupil size and ocular aberration (a combination of the patient’s and the defocus inducing lens aberrations) at each level of blur was measured by the aberrometer. Visual acuity was measured subjectively at each level of defocus to determine the traditional defocus curve. Objective acuity was predicted using image quality metrics. Results: The range of clear focus differed between the three IOL types (F=15.506, P=0.001) as well as between subjective and objective defocus curves (F=6.685, p=0.049). There was no statistically significant difference between subjective and objective defocus curves in the segmented or concentric ring MIOL group (P>0.05). However a difference was found between the two measures and the accommodating IOL group (P<0.001). Mean Delta logMAR (predicted minus measured logMAR) across all target vergences was -0.06 ± 0.19 logMAR. Predicted logMAR defocus curves for the multifocal IOLs did not show a near vision addition peak, unlike the subjective measurement of visual acuity. However, there was a strong positive correlation between measured and predicted logMAR for all three IOLs (Pearson’s correlation: P<0.001). Conclusions: Current subjective procedures are lengthy and do not enable important additional measures such as defocus curves under differently luminance or contrast levels to be assessed, which may limit our understanding of MIOL performance in real-world conditions. In general objective aberrometry measures correlated well with the subjective assessment indicating the relative robustness of this technique in evaluating post-operative success with segmented and concentric ring MIOL.

Statnote 15: the non-parametric correlation coefficients

If in a correlation test, one or both variables are small whole numbers, scores based on a limited scale, or percentages, a non-parametric correlation coefficient should be considered as an alternative to Pearson’s ‘r’. Kendall’s t and Spearman’s rs are similar tests but the former should be considered if the analysis is to be extended to include partial correlations. If the data contain many tied values, then gamma should be considered as a suitable test.

Statnote 32 : the partial correlation coefficient

In previous statnotes, the application of correlation and regression methods to the analysis of two variables (X,Y) was described. The most important statistic used to measure the degree of correlation between two variables is Pearson’s ‘product moment correlation coefficient’ (‘r’). The correlation between two variables may be due to their common relation to other variables. Hence, investigators using correlation studies need to be alert to the possibilities of spurious correlation and the methods of ‘partial correlation’ are one method of taking this into account. This statnote applies the methods of partial correlation to three scenarios. First, to a fairly obvious example of a spurious correlation resulting from the ‘size effect’ involving the relationship between the number of general practitioners (GP) and the number of deaths of patients in a town. Second, to the relationship between the abundance of the nitrogen-fixing bacterium Azotobacter in soil and three soil variables, and finally, to a more complex scenario, first introduced in Statnote 24involving the relationship between the growth of lichens in the field and climate.

Statnote 33:the intra-class correlation coefficient

The intra-class correlation coefficient (ICC or ri) is a method of measuring correlation when the data are paired and therefore, should be used when experimental units are organised into groups. A useful analogy is with the unpaired or paired ‘t’ test to compare the differences between the means of two groups. In studies of reproducibility, there may actually be little difference between the ICC and Pearson’s ‘r’ for ‘true’ repeated measurements. If, however, there is a systematic change in the measurements made on the first compared with the second occasion, then the ICC will be significantly less than ‘r’, and less confidence would be placed in the reproducibility of the results.