Technology & Engineering
One Way ANOVA
One Way ANOVA, or analysis of variance, is a statistical method used to compare the means of three or more groups to determine if there are statistically significant differences between them. It assesses whether the variability within groups is similar to the variability between groups. This technique is commonly used in experimental research to analyze the impact of different factors on a dependent variable.
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11 Key excerpts on "One Way ANOVA"
eBook - PDFStatistics
Principles and Methods
- Richard A. Johnson, Gouri K. Bhattacharyya(Authors)
- 2019(Publication Date)
- Wiley(Publisher)
14 Analysis of Variance (ANOVA) 1 Comparison of Several Treatments—One-Way Analysis of Variance 2 Population Model and Inferences for a One-Way Analysis of Variance 3 Simultaneous Confidence Intervals 4 Graphical Diagnostics and Displays to Supplement ANOVA 5 Randomized Block Experiments for Comparing k Treatments © Richard Levine/Alamy Limited Which Brand HDTV Has the Clearest Picture? A good experimental design is to collect samples of several HDTVs of each brand and measure their picture clarity. The statistical technique called “analysis of variance” enables us to verify differences among the brands. In Chapter 10, we introduced methods for comparing two population means. When several means must be compared, more general methods are required. We now become acquainted with the powerful technique called analysis of variance (ANOVA) that allows us to analyze and interpret observations from several populations. This versatile statistical tool partitions the total variation in a data set according to the sources of variation that are present. In the context of comparing k pop- ulation means, the two sources of variation are (1) differences between means or treatments and (2) within population variation (error). We restrict our discussion to this case, although ANOVA techniques apply to much more complex situations. In this chapter, you will learn how to test for differences among several means and to make confidence statements about pairs of means. 1. COMPARISON OF SEVERAL TREATMENTS—ONE-WAY ANALYSIS OF VARIANCE It is usually more expedient in terms of both time and expense to simultaneously compare sev- eral treatments than it is to conduct several comparative trials of two treatments at a time. The term one-way analysis of variance is synonymous with independent random sampling from several populations when each population is identified as the population of responses under a particu- lar treatment.
eBook - ePub- Jim Fowler, Lou Cohen, Phil Jarvis, Philip Jarvis(Authors)
- 2013(Publication Date)
- Wiley(Publisher)
Analysis of variance (ANOVA) overcomes these difficulties by allowing comparisons to be made between any number of sample means, all in a single test. When it is used in this way to compare the means of several samples statisticians speak of a one-way ANOVA.ANOVA is such a flexible technique that it may also be used to compare more than one set of means. Referring to our island races again, it is possible to compare at the same time the mean lengths of samples of males and females of each race obtained from the islands. When the influence of two variables upon a sample mean is being analysed, such as island of origin and sex in our hypothetical example, the technique involved is described as a two-way ANOVA. Three-way, four-way (and so on) treatments are also possible but they get progressively more complicated. We restrict our examples in this text to one-way and two-way treatments.17.2 How ANOVA works
How analysis of variance is used to investigate differences between means is illustrated in the following example.Example 17.1
Compare the individual variances of the three samples below with the overall variance when all 15 observations n = 15 are aggregated.The means of Samples 1 and 2 are similar; the mean of Sample 3 is much lower; the mean of the aggregated observations is intermediate in value. The variances of the three samples are identical (10.00) and therefore the ‘average variance’ is 10.00. The variance of the aggregated observations however is larger (16.0) than the average sample variance. The increase is due to the difference between the means of the samples, in particular, the difference between the mean of Sample 3 and the other two means. The samples thus give rise to two sources of variability:(i) the variability around each mean within a sample (random scatter);(ii) the variability between the samples due to differences between the means of the populations from which the samples are drawn.In other words:ANOVA involves the dividing up or partitioning the total variability of a number of samples into its components. If the samples are drawn from normally distributed populations with equal means and variances, the within variance is the same as the between variance. If a statistical test shows that this is not the case, then the samples have been drawn from populations with different means and/or variances. If it is assumed that the variances are equal (and this is an underlying assumption in ANOVA) then it is concluded that the discrepancy is due to differences between means.
eBook - PDF- Lorena Madrigal(Author)
- 2012(Publication Date)
- Cambridge University Press(Publisher)
6 The analysis of variance (ANOVA) In this chapter we deal with a fundamentally important technique to analyze experimental data. The analysis of variance (ANOVA from now on) is due to Fisher (in whose honor the F distribution – used to test the ANOVA’s H0) is so named (Fisher 1993; Zabell 2001). In experimental, lab-based fields, in which researchers have much control over experimental and control groups, the applications of ANOVA are almost endless. It is for this reason that textbooks written for biology and psychology students include exhaustive chapters on many ANOVA designs (Pagano and Gauvreau 1993; Rosner 2006; Sokal and Rohlf 1995; Zar 1999). In anthropology we are more limited by the fact that we are unable to control with much precision different levels of the independent variable and have (in general) less control over our experiment. My treatment of ANOVA is thus rather limited when compared with textbooks in biology, psychology, or public health. 6.1 Model I and model II ANOVA It is easy for students to be overwhelmed when they read that there are model I, model II, and mixed-model, and one-way and two-way ANOVAs. Let us clarify what models I and II are. Mixed-model ANOVAS will be discussed when we cover two-way ANOVAs, in section 6.5. The experimental design of a model I ANOVA is similar to that of an un-paired t test. Indeed, I mentioned that the t test should be understood as a special form of the analysis of variance. In a one-way ANOVA we have more than two groups which have been subjected to different levels of the treatment. As we did for the t test, let us think of analysis of variance in terms of dependent (Y) and independent (X) variables. The dependent variable is that whose behavior we want to study such as the number of calories consumed by children. The dependent variable (Y) is usually a continuous numeric variable or a discontinuous numeric variable with a broad range of possible outcomes.
- Robert K. Leik(Author)
- 1997(Publication Date)
- SAGE Publications, Inc(Publisher)
··············· 3 One-Way Analysis of Variance The basic question in ANOVA is whether, on the average, two or more populations are different on some dependent variable. That question is answered by appropriate calculations using sample data. This chapter is intended to help you understand (a) how the calculations for the simplest version of ANOVA are done and inter-preted, (b) whether you can reach a conclusion that the populations are or are not significantly different, and (c) how confident you can be that such a conclusion is valid. It is, of course, possible to conduct ANOVA without knowing why the various numbers involved are calculated or what justifies the conclusion about group differences. There are several problems with not understanding what is behind the calculations. The two most critical ones are as follows: e are many variations of ANOVA, each suited to a particular set of research operations. You need to understand the different models on which those variations are based and what assump-tions they make to tailor ANOVA to your research needs. 2. If your research situation does not match exactly the assumptions of one of the standard ANOVA variations, then you need to know how to decide whether or not to trust the results. In a nutshell, if you do not understand why a statistical procedure works (and when it does not work), then you might reach totally inappropriate research conclusions without knowing it. Not only would that be a waste of your time, but it could get pretty embarrass-ing when someone else discovers your goof. S o m e F u n d a m e n t a l F o r m a l i t i e s Before getting tangled in equations, let's review the terminology and notation needed for ANOVA. This can be prosaic, but it is the only 56 · EXPERIMENTAL D E S I G N A N D THE A N O V A way in which you are able to work back and forth between the substantive theory you want to test and the data you get from your research.
- James E. De Muth(Author)
- 2014(Publication Date)
- Chapman and Hall/CRC(Publisher)
205 10 One-Way Analysis of Variance (ANOVA) Where the t-test was appropriate for the one- or two-sample cases (one or two levels of the discrete independent variable), the F-test or one-way analysis of variance provides an extension to k levels of the independent variable. The calculation involves an analysis of variance of the individual sample means around a central grand mean. Like the t-test, the dependent variable represents data from a continuous distribution. The analysis of variance is also referred to as the F-test, after R.A. Fisher, a British statistician who developed this test during the 1920s (Salsburg, 2002). This chapter will focus on the one-way analysis of variance (abbreviated with the acronym ANOVA), which involves only one independent discrete variable and one dependent continuous variable. Hypothesis Testing with the One-Way ANOVA There are numerous synonyms for the one-way ANOVA including: univariate ANOVA , simple ANOVA , single-classification ANOVA , or one-factor ANOVA . The hypotheses associated with the one-way analysis of variance can be expanded to any number ( k ) levels of the discrete independent variable. H 0 : μ 1 = μ 2 = μ 3 ... = μ k H 1 : H 0 is false The null hypothesis states that there are no differences among the population means, and that any fluctuations in the sample means are due to chance variability only. The ANOVA represents a variety of techniques used to identify and measure sources of variation within a collection of observations, hence the analysis of variance name. The ANOVA has the same assumption as those seen with the t-test: 1) sample sizes are relatively equal; 2) the variances are similar (homogeneity of variance); and 3) the dependent variable is sampled from a normally distributed population. In fact the t-test could be considered a special case of the one-way ANOVA where k = 2.
eBook - PDF- Sherri Jackson(Author)
- 2016(Publication Date)
- Cengage Learning EMEA(Publisher)
Please note that although the study used to illustrate the ANOVA procedure in this section has an equal number of subjects in each condition, this is not necessary to the procedure. One-Way Between-Subjects ANOVA: What It Is and What It Does The analysis of variance (ANOVA) is an inferential statistical test for com-paring the means of three or more groups. In addition to helping maintain an acceptable Type I error rate, the ANOVA has the added advantage over using multiple t tests of being more powerful and thus less susceptible to a Type II error. In this section, we will discuss the simplest use of ANOVA—a design with one independent variable with three levels. Let’s continue to use the experiment and data presented in Table 13.1. Remember that we are interested in the effects of rehearsal type on memory. The null hypothesis ( H 0 ) for an ANOVA is that the sample means represent the same population ( H 0 : m 1 5 m 2 5 m 3 ). The alternative hypothesis ( H a ) is that they represent different populations ( H a : at least one m 2 another m ). When a researcher rejects H 0 using an ANOVA, it means that the indepen-dent variable affected the dependent variable to the extent that at least one group mean differs from the others by more than would be expected based on chance. Failing to reject H 0 indicates that the means do not differ from each other more than would be expected based on chance. In other words, there is not enough evidence to suggest that the sample means represent at least two different populations. In our example, the mean number of words recalled in the rote rehearsal condition is 4, for the imagery condition it is 5.5, and in the story condition it is 8. If you look at the data from each condition, you will notice that most subjects in each condition did not score exactly at the mean for that condi-tion. In other words, there is variability within each condition. The overall mean, or grand mean , across all participants in all conditions is 5.83.
eBook - PDF- Jingmei Jiang(Author)
- 2022(Publication Date)
- Wiley(Publisher)
Applied Medical Statistics , First Edition. Jingmei Jiang. © 2022 John Wiley & Sons, Inc. Published 2022 by John Wiley & Sons, Inc. Companion website: www.wiley.comgojiangappliedmedicalstatistics 193 In Chapter 7, we introduced the concept and methods of hypothesis testing and one-sample testing. Chapter 8 extended the ability to make statistical inferences from two sample situations. In this chapter, we introduce the use of analysis of variance ( ANOVA ) for hypothesis testing of more than two sample means with normal distributions. ANOVA was first introduced by the statistician R.A. Fisher in 1921 and is now widely applied in the biomedical and other research fields. ANOVA is a family of hypothesis testing methods for dealing with data obtained from different experimental designs. The basic principles of ANOVA methods are similar despite using different calculation ways with different designs. We focus on one-way ANOVA in this chapter and continue discussing ANOVA in different experimental designs in the next chapter. Further discussion of experimental designs can be found in Chapter 20. 9.1 Overview One-way ANOVA refers to the analysis of variance for a completely randomized design under which all subjects are mutually independent. The experimental sub-jects are randomly assigned to k groups, and each group is assigned to a specific 9 One-way Analysis of Variance CONTENTS 9.1 Overview 193 9.1.1 Concept of ANOVA 194 9.1.2 Data Layout and Modeling Assumption 195 9.2 Procedures of ANOVA 196 9.3 Multiple Comparisons of Means 204 9.3.1 Tukey’s Test 204 9.3.2 Dunnett’s Test 206 9.3.3 Least Significant Difference (LSD) Test 209 9.4 Checking ANOVA Assumptions 211 9.4.1 Check for Normality 211 9.4.2 Test for Homogeneity of Variances 213 9.4.2.1 Bartlett’s Test 213 9.4.2.2 Levene’s Test 215 9.5 Data Transformations 217 9.6 Summary 218 9.7 Exercises 218
- K D Bird(Author)
- 2004(Publication Date)
- SAGE Publications Ltd(Publisher)
2 One-way Analysis of Variance The ANOVA model In this chapter we consider the simplest version of the ANOVA model and examine various procedures for making inferences about the parameters of this model. We can think of an ANOVA model as a theory of the data produced by an experiment. The single-factor fixed-effects ANOVA model, appropriate for simple randomized experiments with J groups ( J ≥ 2), can be written as ij j ij Y = µ + α + ε (2.1a) or ( ) , ij j E Y = µ + α (2.1b) where µ is a constant, j α is an effect parameter referring to treatment j and ij ε is the value of subject i in group j on an error variable ε with a mean of zero within treatment population j . The right hand side of (2.1a) contains three terms (two parameters and the value of a random variable) that account for, or (in a loose sense) explain, the score on the dependent variable of subject i in group j. None of the terms on the right hand side is at any stage known to the experimenter, who must therefore make inferences about them from an appropriate analysis. The error component ij ε is the only term with an i subscript, so it is the only component of the model that can account for individual differences in dependent variable scores between different subjects in the same group. According to (2.1b), the systematic component of the scores of subjects in group j is the sum of two parameters: the constant µ and the effect parameter . j α If the effect of the first treatment on dependent variable scores is different from the effect of the second treatment, then that difference must be reflected in the difference between 1 α and 2 . α Inferences about effect parameters depend on the following assumptions about error distributions: 1 • for any pair of Y values in the experiment, the associated ε values are statistically independent; • the variance of j ε is the same in all J treatment populations 2 2 ( j ε ε σ = σ for all j );
- David Kleinbaum, Lawrence Kupper, Azhar Nizam, Eli Rosenberg(Authors)
- 2013(Publication Date)
- Cengage Learning EMEA(Publisher)
Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 526 Chapter 17 One-way Analysis of Variance a. Identify the appropriate ANOVA table for these data. b. Test the null hypothesis of no differences among the laboratories. c. With large-scale interlaboratory studies, analysts usually make inferences about a large population of laboratories of which only a random sample (e.g., laboratories I, II, and III) can be investigated. In such a case, describe the appropriate random-effects model for the data. d. Two quantities of particular interest in a large-scale interlaboratory study are repeatability (i.e., a measure of the variability among replicate measurements within a single laboratory) and reproducibility (i.e., a measure of the variability between results from different laboratories). Using the random-effects model defined in part (c), define what you think are reasonable measures of repeatability and reproducibility, and obtain estimates of the quantities you have defined using the data in the accompanying computer output. 4. Ten randomly selected mental institutions were examined to determine the effects of three different antipsychotic drugs on patients with the same types of symptoms. Each institution used one and only one of the three drugs exclusively for a one-year period. The proportion of treated patients in each institution who were discharged after one year of treatment is as follows for each drug used: Drug 1: 0.10, 0.12, 0.08, 0.14 1 Y 1 . 0.11, S 1 0.0192 2 Drug 2: 0.12, 0.14, 0.19 1 Y 2 . 0.15, S 2 0.0361 2 Drug 3: 0.20, 0.25, 0.15 1 Y 3 . 0.20, S 3 0.0500 2 a. Determine the appropriate ANOVA table for this data set using the accompany-ing computer output.
eBook - ePub- Bruce J. Chalmer(Author)
- 2020(Publication Date)
- CRC Press(Publisher)
Chapter 14 .The null hypothesis tested in a one-way ANOVA is that the means of several populations are equal. The alternative hypothesis is that at least one mean differs from the others. Note that the one-way ANOVA is inherently a two-tailed test, in the sense that we are looking for any type of difference, in any direction, among the population means.If we have only two populations of interest, the one-way ANOVA is equivalent to the two-sample t test. In fact, when we have only two samples the F statistic used in the one-way ANOVA (discussed below) is simply the square of the t score from the two-sample t test. This suggests that the assumptions for the one-way ANOVA had better be the same as those for the two-sample t test—and indeed they are.Assumptions required for one-way ANOVA
Just as for the two-sample t test, three assumptions are required for one-way ANOVA. First, we require that all individual scores be independent of one another, both within and among samples. Second, we require that all populations have normal distributions. Third, we require that all populations have the same standard deviation (homogeneity).Assumptions should be checked by looking at pictures of the sample histograms, by inspecting the sample standard deviations, and when in doubt, by applying one of the fancier techniques described briefly in Appendix C. Violations of the normality or homogeneity assumptions can sometimes be made less severe by transforming the data, using one of the methods described in Appendix C.Sums of squares
Before we get into it, note that this discussion is directed toward understanding rather than computational detail. Nowadays, we almost always use a packaged computer program to carry out the calculations, right down to the test statistic.
eBook - PDF- Mike Allen, Scott Titsworth, Stephen K. Hunt(Authors)
- 2008(Publication Date)
- SAGE Publications, Inc(Publisher)
There are other potential estimates of effect sizes used in ANOVA (partial η 2 and ω 2 ); however, η 2 is appropriate in most circumstances (see Levine & Hullett, 2002). ONEWAY Analysis of Variance 59 ASSUMPTIONS OF THE ONEWAY ANOVA The ONEWAY ANOVA procedure is commonly used to test for significant differences between groups of randomly selected participants. Results of the ANOVA can provide meaningful evidence to make causal claims provided that certain assumptions are met. All ANOVA procedures (the ONEWAY ANOVA discussed here, the more advanced factorial design discussed in the next chap- ter, and multivariate designs discussed later) meet the “big three” assumptions to allow credible conclusions. First, groups associated with one or more categorical independent vari- ables must have independence. Independence is simply the condition where participants’ scores are influenced only by the treatment involved in the manipulation of variables and any individual differences that participants had before the experiment began. Violations of this assumption often occur because participants were not randomly assigned to groups. For instance, suppose you wanted to conduct an experiment exploring differences between the use of peer narratives and expert narratives in messages trying to get col- lege students to exercise. Rather than randomly assigning people to groups, suppose that you assigned participants for the two narrative groups from the university swim team. When answering questions about their propensity to exercise, they would be influenced by a shared but concealed variable: They are all athletes who would be predisposed to exercise. Given this violation, it would be impossible to discern whether participants’ responses were due to your experimental conditions or to the hidden variable of participation in com- petitive athletics.
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