Analysis of variance - Wikipedia, the free en...

Analysis of variance

From Wikipedia, the free encyclopedia

Jump to: navigation, search

In statistics, analysis of variance (ANOVA) is a collection of statistical models, and their associated procedures, in which the observed variance is partitioned into components due to different explanatory variables. In its simplest form ANOVA gives a statistical test of whether the means of several groups are all equal, and therefore generalizes Student's two-sample t-test to more than two groups. ANOVAs are helpful because they possess a certain advantage over a two-sample t-test. Doing multiple two-sample t-tests would result in a largely increased chance of committing a type I error. For this reason, ANOVAs are useful in comparing three or more means.

Contents [hide] 1 Overview 2 Models 2.1 Fixed-effects models (Model 1) 2.2 Random-effects models (Model 2) 3 Assumptions of anova 3.1 A model often presented in textbooks 3.2 Randomizaton-based analysis 3.2.1 Statistical models for observational data 4 Logic of ANOVA 4.1 Partitioning of the sum of squares 4.2 The F-test 4.3 ANOVA on ranks 4.4 Effect size measures 4.5 Follow up tests 4.6 Power analysis 5 Examples 6 History 7 See also 8 Notes 9 References 10 External links

[edit] Overview

There are three conceptual classes of such models:

1. Fixed-effects models assume that the data came from normal populations which may differ only in their means. (Model 1)
2. Random effects models assume that the data describe a hierarchy of different populations whose differences are constrained by the hierarchy. (Model 2)
3. Mixed-effect models describe situations where both fixed and random effects are present. (Model 3)

In practice, there are several types of ANOVA depending on the number of treatments and the way they are applied to the subjects in the experiment:

• One-way ANOVA is used to test for differences among two or more independent groups. Typically, however, the one-way ANOVA is used to test for differences among at least three groups, since the two-group case can be covered by a t-test (Gosset, 1908). When there are only two means to compare, the t-test and the F-test are equivalent; the relation between ANOVA and t is given by F = t².
• Two-way ANOVA is used when the subjects are subjected to repeated measures, in which the same subjects are used for each treatment. Note that this method can be subject to carryover effects.
• Factorial ANOVA is used when the experimenter wants to study the effects of two or more treatment variables. The most commonly used type of factorial ANOVA is the 2² (read "two by two") design, where there are two independent variables and each variable has two levels or distinct values. However, such use of ANOVA for analysis of 2^k factorial designs and fractional factorial designs is "confusing and makes little sense"; instead it is suggested to refer the value of the effect divided by its standard error to a t-table.^[1] Factorial ANOVA can also be multi-level such as 3³, etc. or higher order such as 2×2×2, etc. but analyses with higher numbers of factors are rarely done by hand because the calculations are lengthy. However, since the introduction of data analytic software, the utilization of higher order designs and analyses has become quite common.
• Mixed-design ANOVA. When one wishes to test two or more independent groups subjecting the subjects to repeated measures, one may perform a factorial mixed-design ANOVA, in which one factor is a between-subjects variable and the other is within-subjects variable. This is a type of mixed-effect model.
• Multivariate analysis of variance (MANOVA) is used when there is more than one dependent variable.

[edit] Models

[edit] Fixed-effects models (Model 1)

Main article: Fixed effects model

The fixed-effects model of analysis of variance applies to situations in which the experimenter applies several treatments to the subjects of the experiment to see if the response variable values change. This allows the experimenter to estimate the ranges of response variable values that the treatment would generate in the population as a whole.

[edit] Random-effects models (Model 2)

Main article: Random effects model

Random effects models are used when the treatments are not fixed. This occurs when the various treatments (also known as factor levels) are sampled from a larger population. Because the treatments themselves are random variables, some assumptions and the method of contrasting the treatments differ from ANOVA model 1.

Most random-effects or mixed-effects models are not concerned with making inferences concerning the particular sampled factors. For example, consider a large manufacturing plant in which many machines produce the same product. The statistician studying this plant would have very little interest in comparing the three particular machines to each other. Rather, inferences that can be made for all machines are of interest, such as their variability and the mean.

[edit] Assumptions of anova

There are several approaches to the analysis of variance.

[edit] A model often presented in textbooks

Many textbooks present the analysis of variance in terms of a linear model, which makes the following assumptions:

• Independence of cases – this is an assumption of the model that simplifies the statistical analysis.
• Normality – the distributions of the residuals are normal.
• Equality (or "homogeneity") of variances, called homoscedasticity — the variance of data in groups should be the same. Model-based approaches usually assume that the variance is constant. The constant-variance property also appears in the randomization (design-based) analysis of randomized experiments, where it is a necessary consequence of the randomized design and the assumption of unit treatment additivity (Hinkelmann and Kempthorne): If the responses of a randomized balanced experiment fail to have constant variance, then the assumption of unit treatment additivity is necessarily violated.

Levene's test for homogeneity of variances is typically used to examine the plausibility of homoscedasticity. The Kolmogorov–Smirnov or the Shapiro–Wilk test may be used to examine normality.

When used in the analysis of variance to test the hypothesis that all treatments have exactly the same effect, the F-test is robust (Ferguson & Takane, 2005, pp. 261–2).^[2] The Kruskal–Wallis test is a nonparametric alternative which does not rely on an assumption of normality.

The separate assumptions of the textbook model imply that the errors are independently, identically, and normally distributed for fixed effects models, that is, that the errors are independent and

[edit] Randomizaton-based analysis

In a randomized controlled experiment, the treatments are randomly assigned to experimental units, following the experimental protocol. This randomization is objective and declared before the experiment is carried out. The objective random-assignment is used to test the significance of the null hypothesis, following the ideas of C. S. Peirce and Ronald A. Fisher. This design-based analysis was advocated and developed by Oscar Kempthorne at Iowa State University. Kempthorne and his students make an assumption of unit treatment additivity, which is discussed in the books of Kempthorne and David R. Cox. Kempthorne uses the randomization-distribution and the assumption of unit treatment additivity to produce a derived linear model, very similar to the textbook model discussed previously.

The test statistics of this derived linear model are closely approximated by the test statistics of an appropriate normal linear model, according to approximation theorems and simulation studies by Kempthorne and his students (Hinkelmann and Kempthorne). However, there are differences. For example, the randomization-based analysis results in a small but (strictly) negative correlation between the observations (Hinkelmann and Kempthorne, volume one, chapter 7; Bailey chapter 1.14). In the randomization-based analysis, there is no assumption of a normal distribution and certainly no assumption of independence. On the contrary, the observations are dependent!

The randomization-based analysis has the disadvantage that its exposition involves tedious algebra and extensive time. Since the randomization-based analysis is complicated and is closely approximated by the approach using a normal linear model, most teachers emphasize the normal linear model approach. Few statisticians object to model-based analysis of balanced randomized experiments.

[edit] Statistical models for observational data

However, when applied to data from non-randomized experiments or observational studies, model-based analysis lacks the warrant of randomization. For observational data, the derivation of confidence intervals must use subjective models, as emphasized by Ronald A. Fisher and his followers. In practice, the estimates of treatment-effects from observational studies generally are often inconsistent (Freedman). In practice, "statistical models" and observational data are useful for suggesting hypothesis that should be treated very cautiously by the public (Freedman).

[edit] Logic of ANOVA

[edit] Partitioning of the sum of squares

The fundamental technique is a partitioning of the total sum of squares (abbreviated SS) into components related to the effects used in the model. For example, we show the model for a simplified ANOVA with one type of treatment at different levels.

So, the number of degrees of freedom (abbreviated df) can be partitioned in a similar way and specifies the chi-square distribution which describes the associated sums of squares.

See also Lack-of-fit sum of squares.

[edit] The F-test

Main article: F-test

The F-test is used for comparisons of the components of the total deviation. For example, in one-way, or single-factor ANOVA, statistical significance is tested for by comparing the F test statistic

where

I = number of treatments

and

n_T = total number of cases

to the F-distribution with I − 1,n_T − I degrees of freedom. Using the F-distribution is a natural candidate because the test statistic is the quotient of two mean sums of squares which have a chi-square distribution.

[edit] ANOVA on ranks

See also: Kruskal-Wallis one-way analysis of variance

When the data do not meet the assumptions of normality, the suggestion has arisen to replace each original data value by its rank (from 1 for the smallest to N for the largest), then run a standard ANOVA calculation on the rank-transformed data. Conover and Iman (1981) provided a review of the four main types of rank transformations. Commercial statistical software packages (e.g., SAS, 1985, 1987, 2008) followed with recommendations to data analysts to run their data sets through a ranking procedure (e.g., PROC RANK) prior to conducting standard analyses using parametric procedures.

This rank-based procedure has been recommended as being robust to non-normal errors, resistant to outliers, and highly efficient for many distributions. It may result in a known statistic (e.g., Wilcoxon Rank-Sum / Mann-Whitney U), and indeed provide the desired robustness and increased statistical power that is sought. For example, Monte Carlo studies have shown that the rank transformation in the two independent samples t test layout can be successfully extended to the one-way independent samples ANOVA, as well as the two independent samples multivariate Hotelling's T² layouts (Nanna, 2002).

Conducting factorial ANOVA on the ranks of original scores has also been suggested (Conover & Iman, 1976, Iman, 1974, and Iman & Conover, 1976). However, Monte Carlo studies by Sawilowsky (1985a; 1989 et al.; 1990) and Blair, Sawilowsky, and Higgins (1987), and subsequent asymptotic studies (e.g. Thompson & Ammann, 1989; "there exist values for the main effects such that, under the null hypothesis of no interaction, the expected value of the rank transform test statistic goes to infinity as the sample size increases," Thompson, 1991, p. 697), found that the rank transformation is inappropriate for testing interaction effects in a 4x3 and a 2x2x2 factorial design. As the number of effects (i.e., main, interaction) become non-null, and as the magnitude of the non-null effects increase, there is an increase in Type I error, resulting in a complete failure of the statistic with as high as a 100% probability of making a false positive decision. Similarly, Blair and Higgins (1985) found that the rank transformation increasingly fails in the two dependent samples layout as the correlation between pretest and posttest scores increase. Headrick (1997) discovered the Type I error rate problem was exacerbated in the context of Analysis of Covariance, particularly as the correlation between the covariate and the dependent variable increased. For a review of the properties of the rank transformation in designed experiments see Sawilowsky (2000).

A variant of rank-transformation is 'quantile normalization' in which a further transformation is applied to the ranks such that the resulting values have some defined distribution (often a normal distribution with a specified mean and variance). Further analyses of quantile-normalized data may then assume that distribution to compute significance values. However, two specific types of secondary transformations, the random normal scores and expected normal scores transformation, have been shown to greatly inflate Type I errors and severely reduce statistical power (Sawilowsky, 1985a, 1985b).

[edit] Effect size measures

Several standardized measures of effect are used within the context of ANOVA to describe the degree of relationship between a predictor or set of predictors and the dependent variable. Effect size estimates are reported to allow researchers to compare findings in studies and across disciplines. Common effect size estimates reported in bivariate (e.g. ANOVA) and multivariate (MONOVA, CANOVA, Multiple Discriminant Analysis) statistical analysis includes eta-squared, partial eta-squared, omega, and intercorrelation (Strang, 2009).

η² ( eta-squared ): Eta-squared describes the ratio of variance explained in the dependent variable by a predictor while controlling for other predictors. Eta-squared is a biased estimator of the variance explained by the model in the population (it only estimates effect size in the sample). On average it overestimates the variance explained in the population. As the sample size gets larger the amount of bias gets smaller. It is, however, an easily calculated estimator of the proportion of the variance in a population explained by the treatment. Note that earlier versions of statistical software (such as SPSS) incorrectly reports Partial eta squared under the misleading title "Eta squared".

Partial η² (Partial eta-squared): Partial eta-squared describes the "proportion of total variation attributable to the factor, partialling out (excluding) other factors from the total nonerror variation" (Pierce, Block & Aguinis, 2004, p. 918). Partial eta squared is normally higher than eta squared (except in simple one-factor models).

Several variations of benchmarks exist.

The generally-accepted regression benchmark for effect size comes from (Cohen, 1992; 1988): 0.20 is a minimal solution (but significant in social science research); 0.50 is a medium effect; anything equal to or greater than 0.80 is a large effect size (Keppel & Wickens, 2004; Cohen, 1992).

Because this common interpretation of effect size has been repeated from Cohen (1988) over the years with no change or comment to validity for contemporary experimental research, it is questionable outside of psychological/behavioural studies, and more so questionable even then without a full understanding of the limitations ascribed by Cohen. Note: The use of specific partial eta-square values for large medium or small as a "rule of thumb" should be avoided.

Nevertheless, alternative rules of thumb have emerged in certain disciplines: Small = 0.01; medium = 0.06; large = 0.14 (Kittler, Menard & Phillips, 2007).

Omega Squared Omega squared provides a relatively unbiased estimate of the variance explained in the population by a predictor variable. It takes random error into account more so than eta squared, which is incredibly biased to be too large. The calculations for omega squared differ depending on the experimental design. For a fixed experimental design (in which the categories are explicitly set), omega squared is calculated as follows:^[3]

Cohen's ƒ This measure of effect size is frequently encountered when performing power analysis calculations. Conceptually it represents the square root of variance explained over variance not explained.

[edit] Follow up tests

A statistically significant effect in ANOVA is often followed up with one or more different follow-up tests. This can be done in order to assess which groups are different from which other groups or to test various other focused hypotheses. Follow up tests are often distinguished in terms of whether they are planned (a priori) or post hoc. Planned tests are determined before looking at the data and post hoc tests are performed after looking at the data. Post hoc tests such as Tukey's test most commonly compare every group mean with every other group mean and typically incorporate some method of controlling for Type I errors. Comparisons, which are most commonly planned, can be either simple or compound. Simple comparisons compare one group mean with one other group mean. Compound comparisons typically compare two sets of groups means where one set has at two or more groups (e.g., compare average group means of group A, B and C with group D). Comparisons can also look at tests of trend, such as linear and quadratic relationships, when the independent variable involves ordered levels.

[edit] Power analysis

Power analysis is often applied in the context of ANOVA in order to assess the probability of successfully rejecting the null hypothesis if we assume a certain ANOVA design, effect size in the population, sample size and alpha level. Power analysis can assist in study design by determining what sample size would be required in order to have a reasonable chance of rejecting the null hypothesis.

[edit] Examples

In a first experiment, Group A is given vodka, Group B is given gin, and Group C is given a placebo. All groups are then tested with a memory task. A one-way ANOVA can be used to assess the effect of the various treatments (that is, the vodka, gin, and placebo).

In a second experiment, Group A is given vodka and tested on a memory task. The same group is allowed a rest period of five days and then the experiment is repeated with gin. The procedure is repeated using a placebo. A one-way ANOVA with repeated measures can be used to assess the effect of the vodka versus the impact of the placebo.

In a third experiment testing the effects of expectations, subjects are randomly assigned to four groups:

1. expect vodka—receive vodka
2. expect vodka—receive placebo
3. expect placebo—receive vodka
4. expect placebo—receive placebo (the last group is used as the control group)

Each group is then tested on a memory task. The advantage of this design is that multiple variables can be tested at the same time instead of running two different experiments. Also, the experiment can determine whether one variable affects the other variable (known as interaction effects). A factorial ANOVA (2×2) can be used to assess the effect of expecting vodka or the placebo and the actual reception of either.