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One-way random effects ANOVA (Model II)Principles Model Formulae Estimating variance components Assumptions
In one-way ANOVA we have a single 'treatment' factor with several levels (= groups), and replicated observations at each level. In random effects one-way ANOVA, the levels or groups being compared are chosen at random. This is in contrast to fixed effects ANOVA, where the treatment levels are fixed by the researcher. Random effects ANOVA is appropriate in three situations:
The mathematical model for one-way random effects ANOVA is similar to (but not identical to) the model for one-way fixed effects ANOVA. It again describes the effects that the determine the value of any given observation, but this time the 'treatment' factor is random rather than fixed:
Expected mean squares
The methodology for working out sums of squares is identical to that used for fixed-effects ANOVA. Again we are not assuming equal sample sizes in each group.
These values are then inserted into the ANOVA table (see below), along with the degrees of freedom, and mean squares obtained by dividing the sums of squares by their respective degrees of freedom.
The F-ratio for the 'groups effect' is obtained by dividing MSBetween by MSWithin. The P-value of this F-ratio is then obtained for k − 1 and N − k degrees of freedom.
Estimating variance components
Since we are now assuming random 'treatment' effects, there is no point estimating the magnitude of those effects (that is the means), nor the differences between means. For example, if we are making (n =) 2 measurements of weight on each of (k =) 20 subjects, we are not interested in which subject happens to be the heaviest. What is of interest is the amount of variability between subjects compared to the variability between the paired measurements on each subject. In other words, we need to estimate the variance components.
The variance within groups is estimated by MSW. The variance between groups is known as the added variance component and is estimated as shown below:
The added variance component (sA2) can be quoted as an absolute measure of the variability between groups, or it can be quoted relative to the total variability (s2 + sA2). When it is quoted as a proportion of the total variability, it is known as the intraclass correlation coefficient.
The intraclass correlation coefficient
The intraclass correlation coefficient is the proportion the between groups variance comprises of (between groups + residual) variance. When the coefficient is high, it means that most of the variation is between groups. Hence it is a measure of similarity among replicates within a group relative to the difference between groups. When subjects are the
'groups', and the replicates are repeated observations being made on each subject, the intraclass correlation coefficient provides another measure of
The intraclass correlation coefficient is calculated from the variance components derived from a random effects analysis of variance. For now we will only consider its estimation when we are doing a one way analysis of variance.
Note that the intraclass correlation coefficient is sensitive to the nature of the sample used to estimate it. For example, if the sample is homogeneous (that is the between subject variance is very small), then the within subject variance will be proportionally larger and the ICC will be low. In other words it's all relative. So whenever you interpret a correlation, remember to take into consideration the sample that was used to calculate it. The often-reproduced table which shows ranges of acceptable and unacceptable ICC values should not be used as it is meaningless.
One might think the Pearson correlation coefficient could be used to provide a measure of repeatability, at least when group size (n) = 2. Unfortunately that coefficient overestimates the true correlation for small sample sizes (less than ~15). In fact, the intraclass correlation is equivalent to the appropriate average of the Pearson correlations between all pairs of tests.
There are other intraclass correlation coefficients that can be used in special situations. Unfortunately these have resulted in a certain amount of confusion over the correct formulation for the most frequently used version of the ICC given above. For example there is an average measure intraclass correlation coefficient. This is appropriate if one wishes to assess the reliability of a mean measure based on multiple measurements on each subject. Some sources give this as [MSB-MSW]/MSW, or use what is known as the Spearman-Brown Prophecy formula (2*ICC)/1+ICC). One can also use different ANOVA models, for example a two way analysis of variance. Details are given in the references on the ICC given below.
In random effects ANOVA the groups (usually subjects) should be a random sample from a larger population. Otherwise, the same assumptions must hold as for a fixed effects ANOVA if one is to make valid statistical tests such as the F-ratio test, namely:
Note, however, that estimates of ICC for descriptive purposes only are not dependent on either normality or homogeneity of variances. They can for example be done on dichotomous data coded to 0s and 1s to perform the ANOVA. In this case of course the normal approximation confidence interval for the ICC (given by some statistical packages) would not be valid.