Continuous data ANOVA

ANOVA is used for comparison of three or more groups of continuous data when the variances are homogeneous and the data are independent and normally distributed. 

The practical consequence from this is that in the study type under consideration, always the dam/litter rather than the individual fetus is the basic statistical unit (see Chapters 23, 33, 34 and 35). Six malformed fetuses from six different litters in a treated group of dams is much more likely to constitute a teratogenic effect of the test substance than ten malformed fetuses all from the same litter. It is, therefore, important to report all fetal observations in this context and to select appropriate statistical tests (e.g., Fisher s exact test with Bonferroni correction) based on litter frequency. For continuous data, a procedure to calculate the mean value over the litter means (e.g., ANOVA followed by Dunnet s test) is preferred. An increase in variance (e.g., standard deviation), even without a change in the mean, may indicate that some animals were more susceptible than others, and may indicate the onset of a critical effect. 

We saw in the previous chapter how to account for centre in treatment comparisons using two-way ANOVA for continuous data and the CMH test for binary, categorical and ordinal data. These are examples of so-called adjusted analyses, we have adjusted for centre differences in the analysis. 

It probably comes as no surprise to learn that there are mathematical connections between ANOVA and ANCOVA for continuous data. 

In Chapter 10 we saw that there are various methods for the analysis of categorical (and mostly binary) efficacy data. The same is true here. There are different methods that are appropriate for continuous data in certain circumstances, and not every method that we discuss is appropriate for every situation. A careful assessment of the data type, the shape of the distribution (which can be examined through a relative frequency histogram or a stem-and-leaf plot), and the sample size can help justify the most appropriate analysis approach. For example, if the shape of the distribution of the random variable is symmetric or the sample size is large (> 30) the sample mean would be considered a "reasonable" estimate of the population mean. Parametric analysis approaches such as the two-sample t test or an analysis of variance (ANOVA) would then be appropriate. However, when the distribution is severely asymmetric, or skewed, the sample mean is a poor estimate of the population mean. In such cases a nonparametric approach would be more appropriate. 

If the data are recorded at corresponding time values, an alternative is to treat them in a way similar to paired differences as in a paired f-test or in an ANOVA, where time is not considered as continuous independent variable but only as a class effect. The result is a model-independent index, which... 

In Chapter 6 we covered methods for adjusted analyses and analysis of covariance in relation to continuous (ANOVA and ANCOVA) and binary and ordinal data (CMH tests and logistic regression). Similar methods exist for survival data. As with these earlier methods, particularly in relation to binary and ordinal data, there are numerous advantages in accounting for such factors in the analysis. If the randomisation has been stratified, then such factors should be incorporated into the analysis in order to preserve the properties of the resultant p-values. 

FIGURE 10.2 (CONTINUED) and unattached larvae. N = six (6) replicates (dishes) were done for all treatments. The results of the assay are expressed as percentage settlement of the seawater (untreated) control. Data are mean + S.E. Treatments lacking error bars indicate 100% settlement in all replicates. Statistical analysis of the data (separate one-factor analysis of variance ANOVA for each of Figure 10.2A and 10.2B, followed by Tukey s post-hoc comparison among means) showed that only extracts from D. pulchra significantly deterred settlement (at both natural and twice natural concentrations). 

In this first section, we will consider the statistical methods to process data, originating in the observation of a single continuous random variable. We will distinguish three possible situations, with one, two or more than two data sets of the observed variable. In the last case, we will present the Analysis of Variance (ANOVA) for one or more factors. 

Normal Distribution is a continuous probability distribution that is useful in characterizing a large variety of types of data. It is a symmetric, bell-shaped distribution, completely defined by its mean and standard deviation and is commonly used to calculate probabilities of events that tend to occur around a mean value and trail off with decreasing likelihood. Different statistical tests are used and compared the y 2 test, the W Shapiro-Wilks test and the Z-score for asymmetry. If one of the p-values is smaller than 5%, the hypothesis (Ho) (normal distribution of the population of the sample) is rejected. If the p-value is greater than 5% then we prefer to accept the normality of the distribution. The normality of distribution allows us to analyse data through statistical procedures like ANOVA. In the absence of normality it is necessary to use nonparametric tests that compare medians rather than means. 

Sometimes two of the modes of a three-way array are formed by a two-way ANOVA layout in qualitative variables and the three-way structure comes from measuring a continuous variable (e.g. a spectrum) in each cell of the ANOVA layout. In such cases each cell of the ANOVA has a multitude of responses that not even MANOVA can handle. When quantitative factors are used, one or two modes of the three-way array may result from an experimental design where the responses are noisy spectra that behave nonlinearly. Such data are treated in this section. 

While more information about the functional relationship between continuous variables can be obtained from regression techniques than from ANOVA, there are some pitfalls associated with this analytic approach. One must always determine if an obtained equation is representative of the raw data... 

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