Technology & Engineering
Two Way ANOVA
Two-way ANOVA is a statistical method used to analyze the influence of two categorical independent variables on a continuous dependent variable. It allows for the examination of interactions between the two independent variables, providing insights into how they jointly affect the dependent variable. This technique is commonly employed in experimental research to assess the impact of multiple factors on an outcome.
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10 Key excerpts on "Two Way ANOVA"
eBook - ePubFundamental Statistical Principles for the Neurobiologist
A Survival Guide
- Stephen W. Scheff(Author)
- 2016(Publication Date)
- Academic Press(Publisher)
Chapter 7Two-Way Analysis of Variance
Stephen W. Scheff University of Kentucky Sanders–Brown Center on Aging, Lexington, KY, USAAbstract
The two-way analysis of variance (ANOVA) is an extremely powerful and important statistical technique used to look at the possible interaction of two different variables. It differs in several important ways from a one-way ANOVA. The concept of interaction, which is explained in detail in this chapter, totally changes the interpretation of the data. This chapter describes how to interpret these data and how to report the findings for scientific publication. The multifactorial design becomes more complicated if there are repeated measures for one or both of the variables. Specific examples are given for each of these different types of designs.Keywords
Interaction; Interpreting two-way ANOVA; Multifactorial design; Repeated measure two-way ANOVA; Two-way ANOVAOutlineConcept of Interaction 137Difference between One-Way and Two-Way Analysis of Variance 139Interpreting a Two-Way Analysis of Variance 141Two-Way Repeated Measure Analysis of Variance 146Both Factors Repeated 146Single Factor Repeated 154Summary 155One-way ANOVAs are always jealous of two-way ANOVAs because they can have interactions with their variables. AnonymousUp to this point the discussion has really centered on the simplest form of an analysis of variance (ANOVA), the one-way ANOVA, where you have multiple levels of a single factor (e.g., Antioxidant). There are always experimental designs that want to look at outcome after manipulating two or more different factors. If there are only two independent variables (with multiple levels) being manipulated, then this type of ANOVA is called a two-way ANOVA . If there are three or more factors (with different levels) being evaluated, then it is called a multifactorial ANOVA . Often times a biostatistician will call it a two-way factorial design , but it is still just a two-way ANOVA. When only a single dependent variable is quantified, the analysis is termed a univariate analysis . Naturally, if two dependent variables are analyzed simultaneously (e.g., cell firing, cell size), then it is called a bivariate analysis
eBook - PDFApplied Data Analysis for Process Improvement
A Practical Guide to Six Sigma Black Belt Statistics
- James L. Lamprecht(Author)
- 2004(Publication Date)
- ASQ Quality Press(Publisher)
The computational principles outlined for the one-way ANOVA (one variable) can be extended to 2, 3, 4, or N factors. The use of statistical soft- ware has greatly facilitated the sum of squares computations. This next example illustrates how a two-way ANOVA is analyzed. 5 N 112 Part One: Applied Data Analysis Table 4.5 95% confidence intervals for the means. Individual 95% Confidence Intervals for Mean Based on Pooled StDev Level N Mean StDev ------+---------+---------+---------+ A 5 77.80 14.31 (----------*----------) B 5 83.40 13.81 (----------*---------) C 5 66.80 11.43 (----------*---------) ------ --------- --------- --------- Pooled StDev 13.24 60 72 84 96 4.2 TWO-WAY ANOVA The mathematical model for a two-way ANOVA is: i 1, 2, 3, . . . , k (levels of ) X ij X – – i i e ij for j 1, 2, 3, . . . , n (levels of ) where alpha ( ) and beta ( ) represent two factors. The generalized analysis of variance table for two factors is presented in Table 4.6. The equations needed to compute the appropriate sum of squares and the detail of these computations are not shown. The generic two factors are labeled factor A and factor B. The source of variation is labeled “between levels of factor A” and “between levels of factor B.” By levels we mean that the factors can repre- sent, as explained in Example 4.1, either qualitative levels such as inspectors, departments, types of material, or quantitative levels such as different tem- peratures, different speed, or different humidity levels, for example. Example 4.2: In Example 1.6 of Chapter 1, we analyzed various graphs to determine whether or not Workmen or Machine or both had an impact (an effect) on the number of defective parts produced. We can now restate the problem and analyze it as a two-way ANOVA.
eBook - PDF- Deborah J. Rumsey(Author)
- 2021(Publication Date)
- For Dummies(Publisher)
In this chapter, first I give you an example of when you’d need to use a two-way ANOVA. Then I show you how to set up the model, make your way through the ANOVA table, take the F -tests, and draw the appropriate conclusions. Setting Up the Two-Way ANOVA Model The two-way ANOVA model extends the ideas of the one-way ANOVA model and adds an interaction term to examine how various combinations of the two factors affect the response. In this section, you see the building blocks of a two-way ANOVA: the treatments, the main effects, the interaction term, and the sums of squares equation that puts everything together. Determining the treatments The two-way ANOVA model contains two factors, A and B, and each factor has a certain number of levels — say, i levels of Factor A and j levels of Factor B. In the drug study example from the chapter introduction, you have A drug dosage with i 1 2 3 , , or , and B number of times taken per day with j 1 2 or . Each per-son involved in the study is subject to one of the three different drug dosages and will take the drug in one of the two methods given. That means you have 3 2 6 different combinations of Factors A and B that you can apply to the subjects, and you can study these combinations and their effects on blood pressure changes in the two-way ANOVA model. Each different combination of levels of Factors A and B is called a treatment in the model. Table 12-1 shows the six treatments in the drug study. For example, Treat -ment 4 is the combination of 20mg of the drug taken in two doses of 10mg each per day. If Factor A has i levels and Factor B has j levels, you have i j different combina -tions of treatments in your two-way ANOVA model. CHAPTER 12 Finding Your Way through Two-Way ANOVA 207 Stepping through the sums of squares The two-way ANOVA model contains the following three terms. » The main effect A: Term for the effect of Factor A on the response. » The main effect B: Term for the effect of Factor B on the response.
eBook - PDFStatistics Using R
An Integrative Approach
- Sharon Lawner Weinberg, Daphna Harel, Sarah Knapp Abramowitz(Authors)
- 2020(Publication Date)
- Cambridge University Press(Publisher)
SUMMARY OF STEPS TO BE TAKEN IN A TWO-WAY ANOVA PROCEDURE In summary, we may express the sequence of tests in a two-way analysis of variance as follows. 1. Conduct the two-way ANOVA on the dependent variable (dv) using the aov function with the interaction between the two categorical variables x1 and x2 represented as factor variables (i.e., aov(dv~x1*x2)) at the given α level. SUMMARY OF STEPS TAKEN IN A TWO-WAY ANOVA PROCEDURE 465 2. Graph the result using the interaction.plot function as demonstrated in the work- ing example. 3. If the interaction term and both main effects are not statistically significant, do not proceed with post-hoc testing. 4. If the interaction term is not statistically significant, but one or both main effects are statistically significant, examine the marginal effects for each main effect that is statistically significant, adjusting the alpha level or p-value using a multiple com- parisons adjustment (e.g., Bonferroni or Tukey) as described above. 5. If the interaction term is statistically significant do the following steps as in our working example: (a) Test for simple effects of the row factor within the column factor, and if desired, test for simple effects of the column factor within the row factor as well. To control the family-wise Type I error for testing simple effects asso- ciated with the Row factor, use α = α FW /C and for testing simple effects associated with the Column factor use α = α FW /R. (b) Follow all tests of simple effects that are statistically significant with tests of pairwise comparisons on the individual group means within each simple effect adjusting for multiple comparisons using for example, Bonferroni or Tukey adjustments. (c) Finally, if meaningful, conduct all pairwise comparisons on the marginal means for each main effect that is statistically significant. •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• EXAMPLE 14.3 For this example, we turn to the Wages dataset.
- 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 - 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.
No longer available |Learn more- K. Paul Nesselroade, Jr., Laurence G. Grimm, K. Paul Nesselroade, Jr(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
13 Two-Way Analysis of Variance 13.1 The Research Context Factorial Designs In previous chapters, research designs were presented with only one factor, or in experimental terms, one independent variable. 1 However, we do not live in a “ one-variable world. ” Our behavior is constantly affected by the combined influence of two or more sets of conditions. For this reason, many research designs in the social and behavioral sciences allow the researcher to evaluate the interaction between two independent variables and measure their combined influence on behavior. Factorial (or complex) designs are a blend of two or more single-variable designs and serve at least two purposes. For one thing, an investigator can ascertain, in one study, information about the effect of more than one independent variable, thereby saving the time, expense, and effort required to conduct separate, single-variable experiments. Most importantly, by combining independent variables, information can be gained about the combined effect of the independent variables on the dependent variable. When factorial designs are used, there is typically a prediction made regarding the combined effect. The researcher believes that the effect of one independent variable will be altered, depending on the value of the second independent variable. An example serves to illustrate this point. If we saw a person in trouble, would we come to their aid? As we think about this question, we may answer, “ it depends. ” It may depend on how many people are present, how dangerous the situation is, or even what the person in trouble looks like (Latane & Darley, 1970). “ It depends ” qualifies our answer, which, in effect, says our action is determined by the joint presence of certain conditions. 1 As in previous chapters, concepts introduced here will be presented from an experimental perspective, even though two-way ANOVAs can be used to analyze data gathered in nonexperimental designs as well.
eBook - PDF- Ken Black, Ignacio Castillo, Amy Goldlist, Timothy Edmunds(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
In addition, Excel has been used to analyze the data with a two-factor ANOVA. What are the hypotheses for this problem? Study the output in terms of differences. Discuss the results obtained. What conclusions might we reach from this analysis? 452 CHAPTER 13 Analysis of Variance (ANOVA) 13.20 Finish the computations in the ANOVA table shown below determine the critical table F values. Interpret the analysis. Assume that α = 0.05. Two-Factor ANOVA: Dependent Variable versus Row, Column Source df SS MS Rows 2 0.296 0.148 Columns 2 1.852 0.926 Interaction 4 4.370 1.093 Within 18 14.000 0.778 Total 26 20.519 Valve Openings (cm) Shift Shift 1 Shift 2 Shift 3 Machine 1 6.56 6.38 6.29 6.40 6.19 6.23 2 6.54 6.26 6.19 6.34 6.23 6.33 3 6.58 6.22 6.26 6.44 6.27 6.31 4 6.36 6.29 6.21 6.50 6.19 6.58 ANOVA: Two-Factor with Replication A B C D E F G 1 ANOVA 2 3 Source of Variation SS df MS F P value F crit 4 Sample 0.00538 3 0.00179 0.14 0.9368 3.49 5 Columns 0.19731 2 0.09865 7.47 0.0078 3.89 6 Interaction 0.03036 6 0.00506 0.38 0.8760 3.00 7 8 Within 0.15845 12 0.01320 9 Total 0.39150 23 End-of-Chapter Review Decision Dilemma Solved Job and Career Satisfaction of Foreign Self-Initiated Expatriates Is there a difference in the job satisfaction ratings of self- initiated expatriates (SEs) by industry? The data presented in the Decision Dilemma to study this question represent responses on a seven-point Likert-type scale by 24 SEs from five different in- dustries. In Chapter 1 we discussed how Likert-type scales can be analyzed as quantitative or categorical data, and here we have chosen to analyze the Likert scale score as the quantitative vari- able. There is only one categorical variable, industry, with five groups: IT, finance, education, health care, and consulting. If a series of t tests for the difference of two means from independent populations were used to analyze these data, there would be 5 C 2 or 10 different t tests on this one problem.
- 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 );
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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