An introduction to analysis of variance (ANOVA) with special reference to data from clinical experiments in optometry

This article is aimed primarily at eye care practitioners who are undertaking advanced clinical research, and who wish to apply analysis of variance (ANOVA) to their data. ANOVA is a data analysis method of great utility and flexibility. This article describes why and how ANOVA was developed, the basic logic which underlies the method and the assumptions that the method makes for it to be validly applied to data from clinical experiments in optometry. The application of the method to the analysis of a simple data set is then described. In addition, the methods available for making planned comparisons between treatment means and for making post hoc tests are evaluated. The problem of determining the number of replicates or patients required in a given experimental situation is also discussed. Copyright (C) 2000 The College of Optometrists.

The application of analysis of variance (ANOVA) to different experimental designs in optometry

Analysis of variance (ANOVA) is the most efficient method available for the analysis of experimental data. Analysis of variance is a method of considerable complexity and subtlety, with many different variations, each of which applies in a particular experimental context. Hence, it is possible to apply the wrong type of ANOVA to data and, therefore, to draw an erroneous conclusion from an experiment. This article reviews the types of ANOVA most likely to arise in clinical experiments in optometry including the one-way ANOVA ('fixed' and 'random effect' models), two-way ANOVA in randomised blocks, three-way ANOVA, and factorial experimental designs (including the varieties known as 'split-plot' and 'repeated measures'). For each ANOVA, the appropriate experimental design is described, a statistical model is formulated, and the advantages and limitations of each type of design discussed. In addition, the problems of non-conformity to the statistical model and determination of the number of replications are considered. © 2002 The College of Optometrists.

Statnote 23: Non-parametric analysis of variance (ANOVA)

To carry out an analysis of variance, several assumptions are made about the nature of the experimental data which have to be at least approximately true for the tests to be valid. One of the most important of these assumptions is that a measured quantity must be a parametric variable, i.e., a member of a normally distributed population. If the data are not normally distributed, then one method of approach is to transform the data to a different scale so that the new variable is more likely to be normally distributed. An alternative method, however, is to use a non-parametric analysis of variance. There are a limited number of such tests available but two useful tests are described in this Statnote, viz., the Kruskal-Wallis test and Friedmann’s analysis of variance.

Statnote 10: the two-way analysis of variance

The two-way design has been variously described as a matched-sample F-test, a simple within-subjects ANOVA, a one-way within-groups ANOVA, a simple correlated-groups ANOVA, and a one-factor repeated measures design! This confusion of terminology is likely to lead to problems in correctly identifying this analysis within commercially available software. The essential feature of the design is that each treatment is allocated by randomization to one experimental unit within each group or block. The block may be a plot of land, a single occasion in which the experiment was performed, or a human subject. The ‘blocking’ is designed to remove an aspect of the error variation and increase the ‘power’ of the experiment. If there is no significant source of variation associated with the ‘blocking’ then there is a disadvantage to the two-way design because there is a reduction in the DF of the error term compared with a fully randomised design thus reducing the ‘power’ of the analysis.

Statnote 9: The one-way analysis of variance (random effects model)

There is an alternative model of the 1-way ANOVA called the 'random effects' model or ‘nested’ design in which the objective is not to test specific effects but to estimate the degree of variation of a particular measurement and to compare different sources of variation that influence the measurement in space and/or time. The most important statistics from a random effects model are the components of variance which estimate the variance associated with each of the sources of variation influencing a measurement. The nested design is particularly useful in preliminary experiments designed to estimate different sources of variation and in the planning of appropriate sampling strategies.

Statnote 11: the two-factor analysis of variance

Experiments combining different groups or factors are a powerful method of investigation in applied microbiology. ANOVA enables not only the effect of individual factors to be estimated but also their interactions; information which cannot be obtained readily when factors are investigated separately. In addition, combining different treatments or factors in a single experiment is more efficient and often reduces the number of replications required to estimate treatment effects adequately. Because of the treatment combinations used in a factorial experiment, the degrees of freedom (DF) of the error term in the ANOVA is a more important indicator of the ‘power’ of the experiment than simply the number of replicates. A good method is to ensure, where possible, that sufficient replication is present to achieve 15 DF for each error term of the ANOVA. Finally, in a factorial experiment, it is important to define the design of the experiment in detail because this determines the appropriate type of ANOVA. We will discuss some of the common variations of factorial ANOVA in future statnotes. If there is doubt about which ANOVA to use, the researcher should seek advice from a statistician with experience of research in applied microbiology.

Statnote 12: the split-plot analysis of variance

In some experimental situations, the factors may not be equivalent to each other and replicates cannot be assigned at random to all treatment combinations. A common case, called a ‘split-plot design’, arises when one factor can be considered to be a major factor and the other a minor factor. Investigators need to be able to distinguish a split-plot design from a fully randomized design as it is a common mistake for researchers to analyse a split-plot design as if it were a fully randomised factorial experiment.

Statnote 13: a more complex analysis of variance incorporating a 'repeated measures' factor

Experiments combining different groups or factors and which use ANOVA are a powerful method of investigation in applied microbiology. ANOVA enables not only the effect of individual factors to be estimated but also their interactions; information which cannot be obtained readily when factors are investigated separately. In addition, combining different treatments or factors in a single experiment is more efficient and often reduces the sample size required to estimate treatment effects adequately. Because of the treatment combinations used in a factorial experiment, the degrees of freedom (DF) of the error term in the ANOVA is a more important indicator of the ‘power’ of the experiment than the number of replicates. A good method is to ensure, where possible, that sufficient replication is present to achieve 15 DF for the error term of the ANOVA testing effects of particular interest. Finally, it is important to always consider the design of the experiment because this determines the appropriate ANOVA to use. Hence, it is necessary to be able to identify the different forms of ANOVA appropriate to different experimental designs and to recognise when a design is a split-plot or incorporates a repeated measure. If there is any doubt about which ANOVA to use in a specific circumstance, the researcher should seek advice from a statistician with experience of research in applied microbiology.

The use of analysis of variance (ANOVA) in applied microbiology

Experiments combining different groups or factors and which use ANOVA are a powerful method of investigation in applied microbiology. ANOVA enables not only the effect of individual factors to be estimated but also their interactions; information which cannot be obtained readily when factors are investigated separately. In addition, combining different treatments or factors in a single experiment is more efficient and often reduces the number of replications required to estimate treatment effects adequately. Because of the treatment combinations used in a factorial experiment, the DF of the error term in the ANOVA is a more important indicator of the ‘power’ of the experiment than the number of replicates. A good method is to ensure, where possible, that sufficient replication is present to achieve 15 DF for each error term of the ANOVA. Finally, it is important to consider the design of the experiment because this determines the appropriate ANOVA to use. Some of the most common experimental designs used in the biosciences and their relevant ANOVAs are discussed by. If there is doubt about which ANOVA to use, the researcher should seek advice from a statistician with experience of research in applied microbiology.

Data analysis methods in optometry Part 4: an introduction to analysis of variance (ANOVA)

In any investigation in optometry involving more that two treatment or patient groups, an investigator should be using ANOVA to analyse the results assuming that the data conform reasonably well to the assumptions of the analysis. Ideally, specific null hypotheses should be built into the experiment from the start so that the treatments variation can be partitioned to test these effects directly. If 'post-hoc' tests are used, then an experimenter should examine the degree of protection offered by the test against the possibilities of making either a type 1 or a type 2 error. All experimenters should be aware of the complexity of ANOVA. The present article describes only one common form of the analysis, viz., that which applies to a single classification of the treatments in a randomised design. There are many different forms of the analysis each of which is appropriate to the analysis of a specific experimental design. The uses of some of the most common forms of ANOVA in optometry have been described in a further article. If in any doubt, an investigator should consult a statistician with experience of the analysis of experiments in optometry since once embarked upon an experiment with an unsuitable design, there may be little that a statistician can do to help.

Data methods in optometry Part 9: experimental design and analysis of variance

The key to the correct application of ANOVA is careful experimental design and matching the correct analysis to that design. The following points should therefore, be considered before designing any experiment: 1. In a single factor design, ensure that the factor is identified as a 'fixed' or 'random effect' factor. 2. In more complex designs, with more than one factor, there may be a mixture of fixed and random effect factors present, so ensure that each factor is clearly identified. 3. Where replicates can be grouped or blocked, the advantages of a randomised blocks design should be considered. There should be evidence, however, that blocking can sufficiently reduce the error variation to counter the loss of DF compared with a randomised design. 4. Where different treatments are applied sequentially to a patient, the advantages of a three-way design in which the different orders of the treatments are included as an 'effect' should be considered. 5. Combining different factors to make a more efficient experiment and to measure possible factor interactions should always be considered. 6. The effect of 'internal replication' should be taken into account in a factorial design in deciding the number of replications to be used. Where possible, each error term of the ANOVA should have at least 15 DF. 7. Consider carefully whether a particular factorial design can be considered to be a split-plot or a repeated measures design. If such a design is appropriate, consider how to continue the analysis bearing in mind the problem of using post hoc tests in this situation.

Statnote 30 : The one-way analysis of variance (ANOVA):fixed effects model

In Statnote 9, we described a one-way analysis of variance (ANOVA) ‘random effects’ model in which the objective was to estimate the degree of variation of a particular measurement and to compare different sources of variation in space and time. The illustrative scenario involved the role of computer keyboards in a University communal computer laboratory as a possible source of microbial contamination of the hands. The study estimated the aerobic colony count of ten selected keyboards with samples taken from two keys per keyboard determined at 9am and 5pm. This type of design is often referred to as a ‘nested’ or ‘hierarchical’ design and the ANOVA estimated the degree of variation: (1) between keyboards, (2) between keys within a keyboard, and (3) between sample times within a key. An alternative to this design is a 'fixed effects' model in which the objective is not to measure sources of variation per se but to estimate differences between specific groups or treatments, which are regarded as 'fixed' or discrete effects. This statnote describes two scenarios utilizing this type of analysis: (1) measuring the degree of bacterial contamination on 2p coins collected from three types of business property, viz., a butcher’s shop, a sandwich shop, and a newsagent and (2) the effectiveness of drugs in the treatment of a fungal eye infection.

In vivo analysis of ciliary muscle morphologic changes with accommodation and axial ametropia

Purpose. To use anterior segment optical coherence tomography (AS-OCT) to analyze ciliary muscle morphology and changes with accommodation and axial ametropia. Methods. Fifty prepresbyopic volunteers, aged 19 to 34 years were recruited. High-resolution images were acquired of nasal and temporal ciliary muscles in the relaxed state and at stimulus vergence levels of -4 and -8 D. Objective accommodative responses and axial lengths were also recorded. Two-way, mixed-factor analyses of variance (ANOVAs) were used to assess the changes in ciliary muscle parameters with accommodation and determine whether these changes are dependent on the nasal–temporal aspect or axial length, whereas linear regression analysis was used to analyze the relationship between axial length and ciliary muscle length. Results. The ciliary muscle was longer (r = 0.34, P = 0.02), but not significantly thicker (F = 2.84, P = 0.06), in eyes with greater axial length. With accommodation, the ciliary muscle showed a contractile shortening (F = 42.9. P < 0.001), particularly anteriorly (F = 177.2, P < 0.001), and a thickening of the anterior portion (F= 46.2, P < 0.001). The ciliary muscle was thicker (F = 17.8, P < 0.001) and showed a greater contractile response on the temporal side. Conclusions. The accommodative changes observed support an anterior, as well as centripetal, contractile shift of ciliary muscle mass.

Conditional volatility and firm size:an empirical analysis of UK equity portfolios

The techniques and insights from two distinct areas of financial economic modelling are combined to provide evidence of the influence of firm size on the volatility of stock portfolio returns. Portfolio returns are characterized by positive serial correlation induced by the varying levels of non-synchronous trading among the component stocks. This serial correlation is greatest for portfolios of small firms. The conditional volatility of stock returns has been shown to be well represented by the GARCH family of statistical processes. Using a GARCH model of the variance of capitalization-based portfolio returns, conditioned on the autocorrelation structure in the conditional mean, striking differences related to firm size are uncovered.

Principal components analysis of Alzheimer’s disease based on neuropathological data: A study of 79 patients

A Principal Components Analysis of neuropathological data from 79 Alzheimer’s disease (AD) cases was performed to determine whether there was evidence for subtypes of the disease. Two principal components were extracted from the data which accounted for 72% and 12% of the total variance respectively. The results suggested that 1) AD was heterogeneous but subtypes could not be clearly defined; 2) the heterogeneity, in part, reflected disease onset; 3) familial cases did not constitute a distinct subtype of AD and 4) there were two forms of late onset AD, one of which was associated with less senile plaque and neurofibrillary tangle development but with a greater degree of brain atherosclerosis.