ANOVA and Regression

FIGURE 2.19 Antimicrobial death kinetics curve. ( ) Actual collected data points (—) predicted data points (regression analysis) that should be confirmed by the 

For ANOVA employed in regression, three primary sum-of-squares values are needed the total sum of squares, SSj, the sum of squares explained by the regression SSr, and the sum of squares due to the random error, SSe- The total sum of squares is merely the sum of squares of the differences between actual y,- observations and the y mean  

The total sum of squares, to be useful, is partitioned into the sum of squares due to regression (SSr) and the sum of squares due to error (SSe) or unexplained variability. The sum of squares, due to regression (SSr), is the sum-of-squares value of the predicted values (y,) minus the y mean value  

Finally, the sum-of-squares error term (SSe) is the sum of the squares of the actual y, values minus the predicted y, value  

The degrees of freedom for these three parameters, as well as the mean square error, are presented in Table 2.4. The entire ANOVA table is presented in Table 2.5. 

---

Analysis of covariance (ANCOVA) employs both analysis of variance (ANOVA) and regression analyses in its procedures. In the present author s previous book Applied Statistical Designs for the Researcher), ANCOVA was not reported mainly because it presented statistical analysis that did not require the use of a computer. For this book, a computer with statistical software is a requirement hence, ANCOVA is discussed here, particularly because many statisticians refer to it as a special t5q>e of regression. 

Statistical analyses of results (one-way ANOVA and regression) were performed using Statistica 8.0. software (StatSoft, Inc. (2007). Natural litter and remnants data had no normal... 

Analysing for statistical significance using ANOVA and regression... 

An alternative approach, which can be used when there are no or only one or two degrees of freedom, is to use the graphical method of half normal plotting to show which effects are statistically significant. The results for Alloy 800 are analysed by half normal plots instead of analysis of variance or regression as there are only two degrees of freedom for the experimental error variance. Similar to ANOVA and regression, analysis by half normal plots still requires that the data are independent, approximately normal and with constant variance. 

This link applies also to the p-value from the unpaired t-test and the confidence interval for p, the mean difference between the treatments, and in addition extends to adjusted analyses including ANOVA and ANCOVA and similarly for regression. For example, if the test for the slope b of the regression line gives a significant p-value (at the 5 per cent level) then the 95 per cent confidence interval for the slope will not contain zero and vice versa. 

The t-tests and their extensions ANOVA, ANCOVA and regression all make assumptions about the distribution of the data in the background populations. If these assumptions are not appropriate then strictly speaking the p-values coming out of those tests together with the associated confidence intervals are not valid. 

In the paired t-test setting it is the normality of the differences (response on A — response on B) that is required for the validity of the test. The log transformation on the original data can sometimes be effective in this case in recovering normality for these differences. In other settings, such as ANOVA, ANCOVA and regression, log transforming the outcome variable is always worth trying, where this is a strictly positive quantity, as an initial attempt to recover normality. 

Extending non-parametric tests to more complex settings, such as regression, ANOVA and ANCOVA is not straightforward and this is one aspect of these methods that limits their usefulness. 

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. 

Analysis of variance (ANOVA and MANOVA) has been used to investigate the influence of location on forms of metals in roadside soil (Nowak, 1995). Multiple regression analysis has proved valuable in processing sequential extraction data to obtain information on plant availability of trace metals in soils (Qian et al, 1996 ... 

Now that we added the assumption that the errors follow a normal distribution to our hypotheses, we can return to the ANOVA and use the mean square values to test if the regression equation is statistically significant. When Pi = 0, that is, when there is no relation between X and y, it can be demonstrated that the ratio of the MSr and MSr mean squares... 

Data from the experiment were analyzed using ANOVA and multiple regression techniques. 

The OAVs and the basic statistics of the OSs were presented as the mean SD of three samples for each ageing time. ANOVA and linear regression analysis were performed on the triplicated samples by using the Statgraphics 5.0 computer program (STSC Inc., Rockville, MD, USA). 

Each single combination of the factors has been replicated twiee to evaluate the significance of the factor modification on every dependent variable. Furthermore, the factorial analysis of variance (ANOVA) and the response surfaee method (RSM) have been applied, and the first-order regression models linking the dependent variable to the two control factors have been found and analyzed with an ANOVA. 

Where the b s are the regression coefficients (as in Chapter 2). This model allows us to estimate a response inside the experimental domain defined by the levels of the factors and so we can search for a maximum, a minimum or a zone of interest of the response. Complete factorial designs have two main disadvantages. First, when many factors are defined or when each factor has many levels, a large number of experiments is required. Remember the equation Number of experiments = replicates X (levels) (e,g, with 2 replicates, 3 levels for each factor and 3 factors we need 2x3 = 54 experiments). The second disadvantage is the need to use ANOVA and the least-squares method to analyse the responses, two techniques involving no simple calculi. Of course, this is not a problem if proper statistical software is available, but it may be cumbersome otherwise. 

A comparative study of tissues (obtained using LCM) from primary breast cancer with and without axillary lymph node metastasis was carried out with SELDI-TOF-MS and analyzed using ANOVA and multivariate logistic regression. 

Half normal plots give a visual indication of which factors are statistically significant. The technique is useful when there are few or no degrees of freedom available for a residual mean square in ANOVA or regression. Half normal plots are therefore useful when the experimental design is saturated and all effects are of interest. The half normal plot is a type of probability plot where a numerical value for the factor effects is plotted on the vertical axis against the expected normal order statistics on the horizontal axis (see Grove and Davis, 1997). 

Regression and ANOVA An Integrated Approach Using SAS Software by Keith E. Muller and Bethel A. Fetterman... 

For the basic evaluation of a linear calibration line, several parameters can be used, such as the relative process standard deviation value (Vxc), the Mandel-test, the Xp value [28], the plot of response factor against concentration, the residual plot, or the analysis of variance (ANOVA). The lowest concentration that has been used for the calibration curve should not be less than the value of Xp (see Fig. 4). Vxo (in units of %) and Xp values of the linear regression line Y = a + bX can be calculated using the following equations [28] ... 

Other statistical parameters that can be used include examination of residuals and the output from the ANOVA table of regression statistics. This may indicate that a non-linear response function should be checked [9]. 

---