Three Way ANOVA

10 Key excerpts on "Three Way ANOVA"

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  eBook - PDF

  Applied 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 simplest model, known as the one-way ANOVA, allows an experi- menter to analyze the effect of one factor (set at various levels) on a set of observations or measurements. The one-way ANOVA allows us to compare the average of a variable for several categories (or populations). For exam- ple, suppose you are the quality manager in a plant where three inspectors (A, B, and C) are responsible for product inspection. You suspect that each inspector interprets the inspection procedure for product 1021AB differently. To test your theory you collect information on the number of product defects found by each inspector over a period of a week. Your design plan is simple: collect the corrective action reports for each inspec- tor for the previous week and write down the number of nonconformances associated with product 1021AB for each inspector. A generalized version of the data is presented in Table 4.1. Where X ij is the number of product defects found by the ith inspector by the jth day and X i – (X-bar i) is the average for inspector i. For this simple example, the variable (or factor) Inspector is represented by three qualitative levels: Inspector A, Inspector B, and Inspector C. The mathematical model associated with the one-way analysis of vari- ance, or ANOVA, is as follows: i 1, 2, 3, . . . , k groups (or treatments or levels) X ij X i – e ij for (4.1) j 1, 2, 3, . . . , n observations 102 Part One: Applied Data Analysis where e ij is the experimental error or residual representing all uncon- trolled and/or unknown sources of variation affecting a particular meas- urement. From the example of Table 4.1 we would have: i 1, 2, 3 (3 inspectors) X ij X i – e ij for j 1, 2, 3, 4, 5 (5 days) This equation tells us that the value of each observation is equal to the factor average, which can also be viewed as a column average, plus an error term (e).

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  Available until 25 Jan |Learn more

  IBM SPSS Statistics 27 Step by Step

  A Simple Guide and Reference

  • Darren George, Paul Mallery(Authors)
  • 2021(Publication Date)
  • Routledge

    (Publisher)

  General Linear Model: Three-Way ANOVA

  THE PREVIOUS two chapters—Chapter 13 (one-way ANOVA) and Chapter 14 (two-way ANOVA)—have overviewed the goals of analysis of variance. A three-way analysis of variance asks similar questions, and we are not going to reiterate here what we have already presented. We will address in this chapter those things that are unique to three-way analysis of variance (as contrasted with one-way or two-way ANOVAs), and also describe what a covariate is and how it can be used with the General Linear Model command.

  This chapter will be organized in the same manner as all the analysis chapters (Chapters 6 –28 ), but the emphasis will be somewhat different. This introductory portion will deal with the experimental questions that are usually asked in a three-way ANOVA and then explain what a covariate is and how it is typically used. The Step by Step section will include two different types of analyses. A three-way ANOVA will be described based on the grades.sav file, in which total points earned (total) will serve as the dependent variable, and gender, section, and lowup (lower- or upper-division student) will be designated as independent variables. The second analysis will be identical except that it will include one covariate, gpa, to “partial out” variance based on each student’s previous GPA. The output from an analysis of variance is divided into two major sections: (1) The descriptive statistics portion, which lists the mean and frequency for each category created by the ANOVA, and (2) the ANOVA table, which indicates sums of squares, F-values, significance values, and other statistical information. The first portion (the descriptive statistics section) of the analysis will be identical whether or not a covariate is used, but the second portion (the ANOVA table) will often be quite different.

  The major difference between this chapter and previous chapters is that most of the explanation will occur in the Output section. A three-way or higher-order ANOVA is complex by any standards, and a very thorough and careful explanation of the analysis done here will hopefully assist you in untangling any three-way or higher- order ANOVAs that you might conduct. An almost universally practiced procedure for helping to clarify ANOVA results is to graph the cell means for interactions. We will create and examine such graphs in this chapter (instructions for producing these graphs in SPSS are included at the end of the chapter). In fact, even nonsignificant interactions will be graphed and related to the appropriate lines of output so that you can understand visually why there is a significant effect (when one occurs) and why there isn’t (when a significant effect does not

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  eBook - ePub

  Intermediate Statistics

  A Modern Approach, Third Edition

  • James P. Stevens, Keenan A. Pituch, Tiffany A. Whittaker(Authors)
  • 2013(Publication Date)
  • Routledge

    (Publisher)

  9. In a Three Way ANOVA there are 7 hypotheses that are tested. The 3 main effects test whether the population level means are equal. The nature of the two way interactions is ascertained by examining the means for each pair of factors lumped over the third factor. The three way interaction indicates whether the patterns of means (profiles) for any two factors are different for the levels of the third factor.

  10. Power estimation for two and Three Way ANOVA was discussed using Cohen's approach.

  11. An improved Bonferroni type procedure, which makes use of p values, is discussed and illustrated.

  12. A comprehensive computer example, using real data, is used to illustrate and integrate several important concepts from the chapter, as well as indicating some aspects of practical data analysis.

  13. For 3 and 4 way ANOVA there are many hypotheses being tested (7 for three way and 15 for four way). It is important to note that if the .05 level is used for each effect, then the overall a level becomes quite high. Thus, 1 or 2 significant results from such a design, if not hypothesized a priori, could well be spurious.

  14. The distinction between fixed and random factors is illustrated with some examples.

  EXERCISES

  1.  Can you think of a fourth advantage of a factorial design? 2.  Consider the following hypothetical data for an Age by Treatments factorial design:

  TREATMENTS
  1	2	3
  10 yrs	21, 27, 23	24, 32, 30	19, 30, 27
  28, 20	35, 32	20, 21
  AGE
  12 yrs	18, 25, 27	24, 16, 18	34, 28, 21
  20, 23	19, 20	30, 29

  (a) Test each of the effects for significance at the .05 level using the definitional formulas given in the text. Use your calculator to obtain the mean and variance for each cell and then go from there.

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  eBook - ePub

  Practical Statistics for Field Biology

  • 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.

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  User-friendliness of verb syntax in pedagogical dictionaries of English

  • Anna Dziemianko(Author)
  • 2012(Publication Date)
  • De Gruyter

    (Publisher)

  As has been mentioned in the conclusion of section 3.2, results of the ANOVA are of help in this regard. It is those results that are examined in what follows. 3.4. An attempt at generalization - the four-way ANOVA 3.4.1. Preliminaries The range of potentially useful sources of syntactic information in API and CPI as well as the exhaustive discussion of the effects of the incorporation of pattern illustrations into entries on the localization of verb syntax by the subjects, offered in sections 3.2.5.1 and 3.2.5.2, imply the need to exclude the two dictionaries from the ANOVA. The independent variable that comes into play there does not conform to the discussion and any attempt to submit it to scrutiny would enormously and needlessly complicate the analysis. Consequently, it is only the data related to the remaining eight dictionaries that are examined in what follows. All dependent, independent and moderator variables discussed in sections 3.2.2-3.2.4 are taken into account in the ANOVA. The place of codes, the form of codes, the type of definition and the level of proficiency are thus seen as factors and are designated by the first four letters of the alphabet, i.e. A, B, C, and D, respectively. The number of factors considered concurrently in the four-way ANOVA ranges from one to four. Naturally, the effects each of them produced when looked at individually, or the main effects, constitute but part of those the ANOVA brings to light. Since the capacity of this statistical method permits also the investigation of interaction effects when at least two factors are taken into account, there are as many as eleven possible interactions that must be additionally 169 examined. These encompass six interaction effects produced by two factors, i.e. AxB, AxC, AxD, BxC, BxD and CxD, four resulting from three factors, i.e. AxBxC, AxBxD, AxCxD and BxCxD, and one brought about by all four, that is, AxBxCxD.

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  Statistics II For Dummies

  • 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.

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  Statistics

  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.

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  Basic Statistics and Pharmaceutical Statistical Applications

  • 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.

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  Sensory Evaluation of Food

  Statistical Methods and Procedures

  • Michael O'Mahony(Author)
  • 2017(Publication Date)
  • Routledge

    (Publisher)

  12

  Analysis of Variance: Three- and Four-Factor Designs

  12.1 How to Compute a Three-Factor Anova

  With two factors (ABC and αβγ ) there are mean-square terms

  MSA

  (between-treatments ABC ),

  MSα

  (between-treatments αβγ ), and an interaction MS

  α × A

  (interaction between ABC and αβγ ). It is possible to have three factors (ABC , αβγ , and abc ). For instance, in a completely randomized design judges could be scoring wines A , B , and C for quality under different lighting conditions α , β , γ (white, red, and green light) on consecutive days a , b , c (replications effect). Alternatively, one of the factors could be “subjects.” Besides the three between-treatments effects, we can now have three two-way interactions α × A , α × a , a × A , and even a three-way interaction a × A × a . The two-way interactions are exactly what you would expect from the two-factor analysis of variance, while three-way interactions are a little more complex and will be dealt with in more detail later (Section 12.2 ). First, we will run through the calculation and then interpret the results in detail. The calculation is an extension of the calculation for the two-factor ANOVA. There we had a table of the original data (whole matrix) and also a table of the totals of the scores from each combination of treatments (totals matrix or two-way matrix). The totals matrix is used as a means of obtaining the two-way interaction. For a three-factor ANOVA, we have a table of the original data (whole matrix), but we find that we can break down the totals into more than one two-way matrix; we even find that we have a three-way matrix of totals. We will now see how this is done.

  Let us assume that we are obtaining scores (X values) under three combinations of factors: one with treatments A , B , C , a second with treatments α , β , γ , and a third with treatments a , b , c . One of these factors may or may not be subjects (e.g., three subjects named a , b , and c ). There must be more than one score in each combination of treatments for us to be able to get the highest-order (three-way) interaction. In fact, we can generalize and say that we need more than one score in each combination of factors to get the highest possible order interaction (rc-way interaction in an n

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  Statistics 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.

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