Which of the following might contribute to within-treatments variance check all that apply

Analysis of Variance

Analysis of variance (ANOVA) is used to test for differences among three or more population means. It allows for multiple comparisons while holding the probability of a type I error (rejection of a true null hypothesis) at a preselected level. ANOVA works by comparing variance estimates: one due to chance factors alone and one due to chance plus treatment effect (if there is a treatment effect). ANOVA can also be used to study two or more treatment variables simultaneously. Although many researchers routinely use ANOVA in combination with post hoc comparisons to make all possible pair-wise comparisons, one should consider using planned comparisons in place of ANOVA if it is known in advance which comparisons are important to the study.

Analysis of Variance

When the mean values of more than two different groups are to be compared, the process is termedanalysis of variance (ANOVA) (Dawson-Saunders & Trapp, 1994). This analysis can be thought of as extending thet-test beyond two independent samples to three or more. The null hypothesis in this situation is that the mean values of all groups are the same. The alternative hypothesis is that not all the means are equal (some could be the same, but others different). The test statistic is theF-ratio of the mean squares among all groups (MSA) to the error mean square (MSE):

(Eq. 10.9)F−ratio=MS AMSE.

It compares the variance of the group means with the mean of all the data (numerator) and the variance of individual data points within each group (denominator). If the group means differ from one another (signal) more than the variation within groups (noise), then theF-ratio will exceed a critical value for significance.

An example of ANOVA is the comparison of serum albumin values from patients at two inpatient sites, an outpatient (OP) clinic and a student health (SH) clinic (Fig. 10.8). One hundred consecutive specimens from each site were recorded. The horizontal line shows the grand mean of all 400 values of 3.17 g/dL. Within each group, the diamonds indicate group means (midline) and 95% confidence intervals on those means (upper and lower vertices). TheF-ratio is 279, for which thep value is less than 0.0001. Thus, the null hypothesis can be rejected with the conclusion that at least some of the means are different. This approach is more conservative and realistic than comparing each group with every other group using a series of differentt-tests (with four groups, six comparisons could be made). The problem with too many comparisons is the possibility of “accidentally finding significance” that is not true. To extend ANOVA, comparisons of group means by such procedures as Tukey’s honestly significant difference (HSD) can be done. In the example inFigure 10.8, Tukey’s HSD indicates that IP1 and IP2 (inpatient wards 1 and 2) are not different from each other but that OP and SH are both different from all other groups. At this stage, the investigator is free to elaborate on potential reasons for these observed differences without putting further statistical significance on the individual differences.

Abstract

Analysis of variance (ANOVA) is a conceptually simple, powerful, and popular way to perform statistical testing on experiments that involve two or more groups. ANOVA is especially suited for experimental designs that involve pairing or blocking, repeated measures on the same subjects, or when looking to see if different factors in the experiment interact with each other. We discuss how the ANOVA works, review the most common types of ANOVAs, and discuss how to interpret the meaning of an ANOVA that achieves statistical significance. A positive ANOVA says that one or more groups are different from the others but does not specify which! Therefore, follow-up analyses must be carried out to identify which groups are significantly different. And, it is necessary to correct the resulting P-values to reflect the fact that multiple comparisons are being made. A variety of correction methods are discussed; each has advantages and disadvantages, and none is suitable for all experiments.

3.1.b N-way ANOVA

The goal of ANOVA is to explain (to model) as much variation in a continuous variable as possible, by using one or more categoric variables to predict the variation. If only one independent variable is tested in a model, it is called an F-test, or a one-way ANOVA. If two or more independent variables are tested, it is called a two-way ANOVA or anN-way ANOVA (theN specifying how many independent variables are used).

In ANOVA, if one variable for an F-test is gender (seeBox 11.1), the total sum of squares (SS) in the dependent variable can be partitioned into a contribution from gender and residual noise. If two independent variables are tested in a model, and those variables are treatment and gender, the total amount of variation is divided into how much variation is caused by each of the following: independent effect of treatment, independent effect of gender, interaction between (i.e., joint effect of) treatment and gender, and error. If more than two independent variables are tested, the analysis becomes increasingly complicated, but the underlying logic remains the same. As long as the research design is “balanced” (i.e., there are equal numbers of observations in each of the study groups),N-way ANOVA can be used to analyze the individual and joint effects of categorical independent variables and to partition the total variation into the various component parts. If the design is not balanced, most computer programs provide an alternative method to do an approximate ANOVA; for example, in SAS, the PROC GLM procedure can be used. The details of such analyses are beyond the scope of this book.

As an example,N-way ANOVA procedures were used in a study to determine whether supplementing gonadotropin-releasing hormone with parathyroid hormone would reduce the osteoporosis-causing effect of gonadotropin-releasing hormone.14 The investigators used ANOVA to examine the effects of treatment and other independent variables on the bone loss induced by estrogen deficiency.

8.1.1.4 One-way ANOVA

One-way ANOVA (analysis of variance) was proposed by Ronald Fisher.679 ANOVA proposes the null hypothesis that all the means of compared groups are equal. ANOVA assumes that the underlying analysis data are normally distributed. However, most of microbiome community composition data, especially multivariate data, are not normally distributed; thus, the application of ANOVA needs to be careful in microbiome studies. In the case that the microbiome data are not normally distributed, either the nonparametric alternative Wilcoxon rank-sum test or other suitable statistical methods are applied. One example of ANOVA was used to identify significant differences in phylogenetic diversity and species richness indexes.680

Analysis of variance (ANOVA) models apply to data that occur in groups. The fundamental ANOVA model is the one-way model that specifies a common mean value for the observations in a group. The analysis of variance associated with the one-way model is presented. When the groups in a one-way ANOVA are identified as combinations of two or more factors, models incorporating factor main effects and factor interactions provide a useful device for exploring the underlying structure of the data. The main effects only two-way ANOVA model is discussed along with its interpretation and how it differs from the interaction model. Three factor models and their interpretations are discussed. Appropriate models can be identified by fitting sequences of successively smaller models and using general testing procedures to identify models within each sequence that fit well. ANOVA models are also useful in generalized linear models. These are discussed with examples involving logistic regression. Again, appropriate models can be identified by fitting sequences of models.

ANOVA F-Test

Analysis of variance (ANOVA) is a tool to compare the means of several populations, based on random, independent samples from each population. It provides a statistical test to determine if population means are equal or not (i.e. came from the same distribution). ANOVA is a parametric test that assumes a normal distribution of values (null hypothesis).

F-test is a class of statistical test that calculates the ratio between variances. F-Test is used with ANOVA to measure the ratio between explained and unexplained variances.

Three assumptions must be satisfied with ANOVA F-Test: samples are independents, from a normally distributed population and standard deviations of the groups are all equal (homoscedasticity). It permits the measure of the linear dependency between two variables. In Johnson and Synovec (2002) a feature selection based on ANOVA and PCA is done to classify jet fuel mixture. In Yakub et al. (2016), a feature selection based on one way ANOVA is used to classify microarray data. The main advantage of ANOVA F-Test is its straightforward computation and interpretation. The limiting factor is that its applicability is only valid with the specific assumptions.

Analysis of variance (ANOVA)

All of the previous tests have concerned two groups of observations. When there are three or more groups, one tests the null hypothesis that there is no difference in the group means by examining the variances.

Consider the data, shown in Table 9.7, from three groups of four patients. There are several sources of variation in this sample. First, there is the total variation of the whole sample. To calculate this value one calculates the sample variance, ignoring the group to which each measurement belongs. This is sometimes called the total mean squares (MStotal) or total variance. Second, there is the variation of the group means about a grand mean of all observations. This is sometimes called the between-groups variance (MSbetween) or sometimes MStreat. Finally, there is the variation between the group measurements and their individual group means. This is sometimes called the within-groups mean squares or residual mean squares (MSresidual) or within-groups variance. The relationship between these sources of variation is:

SStotal=SSbetween+SSresidual(MS=SSdf)

If each of the groups is drawn from the same population with equal population means, the between-groups variance will be comparable to the within-groups variation (MSresidual). Alternatively, if the three groups have different population means, the between-groups variation will be large compared with the within-groups variance. In order to test which one of these situations is more likely, we use the test statistic F.

F=Variation between samplesVariation within samples=MSbetweenMSresidual

In the example above the total variance is calculated from the whole sample (as if they were not in groups). The between groups sum of squares is the sum of squared deviations between the group means and the overall mean multiplied by the number of observations in each group. The residual sum of squares is calculated usually by subtraction. Mean squares are the sum of squares (SS) divided by the appropriate degrees of freedom.

Most statistical packages when performing an ANOVA will produce an output similar to that shown in Table 9.8. The sum of squares is just the sum of the squared difference between each value and its corresponding mean. By dividing by the degrees of freedom, we can calculate the within-groups (MSresidual), between-groups (MSbetween) and total variance.

In the above example, F = 11.9, which is significant, being less than 0.05. We can therefore reject the null hypothesis that there is no difference between the groups.

ANOVA can also be extended to the analysis of data which can be classified in a number of ways. For example, in an observational study measuring memory performance scores, patients may be classified by diagnosis, sex and treatment. If one wished to compare memory score by diagnosis, a one-way ANOVA (as above) could be conducted. However, if gender and treatment also affected memory score, the difference might not be due solely to the effect of diagnosis alone. To avoid this potential pitfall, a factorial ANOVA can include any number of the factors in a single experiment. The resulting analysis could give the effect of each factor independently, but can also provide information about interactions between factors. For example, a factorial ANOVA could detect that memory scores may be impaired in males with schizophrenia but not females, whereas a one-way ANOVA might fail to detect any differences.

ANOVA

Is a parametric statistical test

Tests the null hypothesis that the mean values of three or more independent groups are equal

The test statistic F is the ratio of the between-groups to within-groups variance

The value of F and the two degrees of freedom should always be stated

ANOVA has a non-parametric equivalent called the Kruskal–Wallis test

An analysis of variance can be extended to include paired values from the same samples when it is called a repeated measures ANOVA. Where data can be classified in several ways (e.g. by group and gender) the appropriate statistical test is the factorial ANOVA. An analysis of covariance (ANCOVA) is used when we wish to see if the mean of a variable differs across three or more groups, while taking into account a possible confounder. If, for example, one examined cognition in the three groups of subjects above, their performance may be confounded by their premorbid general intellectual ability or their age. In order to take into account these factors we can either use a regression analysis (see later) or use ANCOVA, where IQ or age or both would be covariates.

Table 9.9 summarises tests for differences, for various data types.

Analysis of Variance, Analysis of Covariance, and Multivariate Analysis of Variance

Analysis of variance (ANOVA) is the statistical procedure of comparing the means of a variable across several groups of individuals. For example, ANOVA may be used to compare the average SAT critical reading scores of several schools. The name of the technique arises from the fact that the first step in an ANOVA is to partition the variance present in the observations into several components. The ANOVA method was the second most frequently used data-analysis procedure in a survey of articles published between 1971 and 1998 in three reputed educational-research journals (Hsu, 2005). Generalizability theory (Cronbach et al., 1963), which is a competitor to the classical theory of reliability of tests, usually applies ANOVA procedures to test scores.

Analysis of covariance (ANCOVA) is used when, like in ANOVA, the interest is in comparing several means, but the investigator also has the values of an additional variable that influences the variable of interest. For example, ANCOVA may be used to compare the average SAT critical reading scores of several schools where the preliminary scholastic aptitude test/national merit scholarship qualifying test (PSAT/NMSQT) critical reading score of each examinee is available in addition to the SAT critical reading score. (The PSAT/NMSQT is supposed to provide firsthand practice for the SAT.)

Multivariate analysis of variance (MANOVA) is used to compare means of several variables simultaneously across several groups of individuals. For example, one could apply MANOVA to simultaneously compare the average scores on several subjects across several schools. Longford (1990) provides such an example.