Model ANOVA

Figure 3. Relationship between leaf area (A), epidermal cell density (B), stomatal density (C) and stomatal index (D) versus altitude for Nothofagus solandri leaves growing on the slope of Mt. Ruapehu, New Zealand (collected in 1999). Black diamonds indicate the mean of ten counting fields on each leaf, white squares are the averages of five to eight leaves per elevation, with error bars of 1 S.E.M. Nested mixed-model ANOVA with a general linear model indicates significant differences for all factors (p = 0.000). Averages per elevation were used for regression analysis A. y = -0.0212 + 73.1 R2 = 0.276 p = 0.147. B. y = 1.70 + 3122 R2 = 0.505 p = 0.048. C. y = 0.164 + 360 R2 = 0.709 p = 0.009. D. linear (dashed) y = 0.004 + 9.33 R2 = 0.540 p = 0.038 non-linear (solid) y = 0.00001 2 - 0.0206 + 21.132 R2 = 0.770.

Table 1. Stomatal density (SD), epidermal cell density (ED) and stomatal index (SI) of sun and shade leaves of Nothofagus solandri var. cliffortioides. Sun and shade leaves were collected at three localities (Fig. 2) Horrible Bog (HOR), Kawatiri Junction (KJ) and St. Arnaud (SA). Values are means of five leaves per light level (seven counts per leaf). The complete data set (total) was analyzed with a nested mixed-model ANOVA based on a general linear model, for comparisons within the individual localities a fully nested ANOVA was used.

Table 2. Stomatal density (SD), Epidermal cell density (ED) and stomatal index (SI) of modern Quercus kelloggii leaves, assigned to light regime during growth by degree of undulation, and p-values from a pairwise comparison using a nested mixed-model ANOVA based on a general linear model.

Figure 6. Stomatal density (SD A), epidermal cell density (ED B) and stomatal index (SI C) of 61 modem Quercus kelloggii leaves collected in California in 2003 (Fig. 1). Leaves were assigned to sun, shade or neutral type by degree of undulation of the epidermal cell walls (e.g., Kiirschner 1997). Nested mixed-model ANOVA showed significant differences for SD (p = 0.001) and SI (p = 0.000), but not ED (p = 0.217). Paired comparison of the means of stomatal density and index in a nested mixed-model ANOVA showed significant differences for SD and SI between sun and shade leaves (p < 0.01 Table 2). Error bars represent 1 S.E.M.

Results were analyzed by nested mixed-model ANOVA s using general linear procedures, in the MINITAB 15 statistical program. Nested mixed-model ANOVA was used when multiple leaves per tree and multiple trees per treatment were available. Additional analyses were linear and quadratic regressions (performed in MINITAB 15 and Excel), and when significant differences occurred, means were compared using Student s t-test or nested mixed-model ANOVA. 

Another commonly used statistical model is the Linear Model, which represents a family of models of a similar structure. The most commonly employed linear model is the analysis of variance model (ANOVA). We shall illustrate this model using the simplest case, the one-way ANOVA model. 

ANOVA was important in determining the adequacy and significance of the quadratic model. ANOVA summary is shown in Table 16.4. The fitness of the model was expressed by the value, which is 0.9938, indicating that 99.38% of the variability in the response can be explained by the model. The adjusted value of 0.9893 suggested that the model was statistically significant. 

Multiplicative multilinear models can be used for modeling ANOVA data. Such multilinear models can be interesting, for example, in situations where traditional ANOVA interactions are not possible to estimate. In these situations GEMANOVA can be a feasible alternative especially if a comparably simple model can be developed. As opposed to traditional ANOVA models, GEMANOVA models suffer from less developed hypothesis testing and often the modeling is based on a more exploratory approach than in ANOVA. 

Analysis of variance (ANOVA) tests whether one group of subjects (e.g., batch, method, laboratory, etc.) differs from the population of subjects investigated (several batches of one product different methods for the same parameter several laboratories participating in a round-robin test to validate a method, for examples see Refs. 5, 9, 21, 30. Multiple measurements are necessary to establish a benchmark variability ( within-group ) typical for the type of subject. Whenever a difference significantly exceeds this benchmark, at least two populations of subjects are involved. A graphical analogue is the Youden plot (see Fig. 2.1). An additive model is assumed for ANOVA. 

ANOVA) if the standard deviations are indistinguishable, an ANOVA test can be carried out (simple ANOVA, one parameter additivity model) to detect the presence of significant differences in data set means. The interpretation of the F-test is given (the critical F-value for p = 0.05, one-sided test, is calculated using the algorithm from Section 5.1.3). 

The OUTSTAT= output data set pvalue contains the p-value in the PROB variable. If you have multiple predictor variables, you need to use the PROC GLM ODS data set OverallANOVA to get the overall model p-value from the ProbF variable. These output data sets contain other variables, such as the degrees of freedom, sum of squares, mean square, and F statistic, if you need them for an ANOVA table presentation. 

ANOVA was developed by Fisher [1925,1935] as a statistical procedure that investigates influences (effects) of factors on a target quantity y according to a linear model which holds in the simplest case... 

By means of Two-way ANOVA two factors can be studied simultaneously. The model... 

Table 5.3. Evaluation scheme of two-way ANOVA (ab model single measurements in each point the index point marks that levels over which is actually added up or averaged, respectively)...

If possible, two-way ANOVA should be applied doing repetitions at each level. In case of double measurements the 2ab model represented in Tables 5.5 and 5.6 is taken as the basis of evaluation and variance decomposition. 

In the case that interactions prove to be insignificant, it should be gone over to the ab model the estimations of which for the various variance components is more reliable than that of the 2ab model. A similar scheme can be used for three-way ANOVA when the factor c is varied at two levels. In the general, three-way analysis bases on block-designed experiments as shown in Fig. 5.1. 

Table 5.7. Variance components in three-way ANOVA (abc model with repetitions)... 

Model-independent techniques compare data pairs observed at corresponding time values, where time is only a class effect, as in a paired -test or in an ANOVA. A data-poor set of only two or three observations, originating from routine quality control of an immediate-release dosage form, cannot be treated other than model independent. 

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... 

The ANOVA test, which is also recommended by the Analytical Methods Committee of The Royal Society of Chemistry (UK), can be generalized to other regression models, and it can be extended to handle heteroscedasticity. For a more detailed prescription and the extension of the test see further reading. 

In this chapter we examine these and other sums of squares and resulting variances in greater detail. This general area of investigation is called the analysis of variance (ANOVA) applied to linear models [Scheff6 (1953), Dunn and Clark (1987), Alius, Brereton, and Nickless (1989), and Neter, Wasserman, and Kutner (1990)]. 

Figure 9.4 emphasizes the relationship among three sums of squares in the ANOVA tree - the sum of squares due to the factors as they appear in the model, SSf (sometimes called the sum of squares due to regression, SS ) the sum of squares of residuals, SS, and the sum of squares corrected for the mean, (or the total sum of squares, SSj, if there is no Pq term in the model). 

---