P.Mean >> Category >> Analysis of variance (created 2007-06-20).

Analysis of variance (ANOVA) is an approach that allows you to compare a continuous outcome variable across a factor representing three or more groups and to examine interactions among factors. Also see Analysis of means, and Linear regression.

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12. P.Mean: Can I focus on a subset of conditions in my experiment? (created 2013-09-10). I got an email from a colleague who was running a 2 by 2 by 6 factorial design. Without giving up too much detail, let's say that the first factor was protein (Y or N), the second was oxygen level (normal or high) and the third was media type (A, B, C, D, E, and F). This person wanted to analyze a subset of factors. He wanted to look at the combinations protein=Y/oxygen=normal, protein=Y/oxygen=high, and protein=N/oxygen=high, but did not seem interested in looking at protein=N/oxygen=normal combination. Also he wanted to compare only four of the media types (B, C, D, and E). Was his approach valid?


11. The Monthly Mean: What's the difference between ANOVA and ANCOVA? (July/August 2011)


10. P.Mean: Use of Likert data with ANOVA (created 2009-10-13). I never quite feel I can offer my students a thoughtful explanation about the use of Likert data with ANOVA. It is recommended that ANOVA be used with interval or ratio data, but, in practice, ANOVA is sometimes used when the data is ordinal (as you'd find when using Likert scales). This confuses some students. Are there any good references out there I can share with my students that might explain the pros and cons of using ordinal data with ANOVA?


10. The Monthly Mean: What's the difference between regression and ANOVA? (November 2008). Someone asked me to explain the difference between regression and ANOVA. That's challenging because regression and ANOVA are like the flip sides of the same coin. They are different, but they have more in common that you might think at first glance.

9. P.Mean: Using ANOVA for a sum of Likert scaled variables (created 2008-10-09). I want to analyse data derived from a questionnaire. The range of possible values that my variable can take goes from 20 to 100. No evidence for rejecting the hypothesis of normality was found. I would therefore apply an ANOVA, but I still have some doubts whether this methods of analysis is valid, since the range of my dependent variable is not [- infinity;+ infinity]. Is the ANOVA a valid method of analysis or are there other approaches I can apply?

Outside resources: (also available at http://www.zotero.org/groups/pmeanreferences/items/collection/2959828)

Martin Kermit, Valerie Lengard. Assessing the Performance of a Sensory Panel - Panelist monitoring and tracking. Abstract: "Sensory science uses the human senses as instruments of measures. This study presents univariate and multivariate data analysis methods to assess individual and group performances in a sensory panel. Green peas were evaluated by a trained panel of 10 assessors for six attributes over two replicates. A consonance analysis with Principal Component Analysis (PCA) is run to get an overview of the panel agreement and detect major individual errors. The origin of the panelist errors is identified by a series of tests based on ANOVA: sensitivity, reproducibility, crossover and panel agreement, complemented with an eggshell-correlation test. One assessor is identified with further need for training in attributes pea flavour, sweetness, fruity and off-flavour, showing errors in sensitivity, reproducibility and crossover. Another assessor shows poor performance for attribute mealiness and to some extent also fruity flavour. Only one panelist performs well to very well in all attributes. The specificity and complementarity of the series of univariate tests are explored and verified with the use of a PCA model. Keywords: Sensory panel performance; ANOVA; Agreement error; Sensitivity; Reproducibility; Crossover; Eggshell plot." [Accessed December 1, 2009]. Available at: http://www.camo.com/resources/casestudies/PMT.pdf.

UCLA Academic Technology Services. Coding systems for categorical variables in regression analysis. Excerpt: "Categorical variables require special attention in regression analysis because, unlike dichotomous or continuous variables, they cannot by entered into the regression equation just as they are. For example, if you have a variable called race that is coded 1 = Hispanic, 2 = Asian 3 = Black 4 = White, then entering race in your regression will look at the linear effect of race, which is probably not what you intended. Instead, categorical variables like this need to be recoded into a series of variables which can then be entered into the regression model. There are a variety of coding systems that can be used when coding categorical variables. Ideally, you would choose a coding system that reflects the comparisons that you want to make. In Chapter 3 of the Regression with SAS Web Book we covered the use of categorical variables in regression analysis focusing on the use of dummy variables, but that is not the only coding scheme that you can use. For example, you may want to compare each level to the next higher level, in which case you would want to use "forward difference" coding, or you might want to compare each level to the mean of the subsequent levels of the variable, in which case you would want to use "Helmert" coding. By deliberately choosing a coding system, you can obtain comparisons that are most meaningful for testing your hypotheses." [Accessed December 1, 2009]. Available at: http://www.ats.ucla.edu/stat/sas/webbooks/reg/chapter5/sasreg5.htm.

David C. Howell. Multiple Comparisons with Repeated Measures. Excerpt: "One of the commonly asked questions on listservs dealing with statistical issue is 'How do I use SPSS (or whatever software is at hand) to run multiple comparisons among a set of repeated measures?' This page is a (longwinded) attempt to address that question. I will restrict myself to the case of one repeated measure (with or without a between subjects variable), but the generalization to more complex cases should be apparent." [Accessed December 1, 2009]. Available at: http://www.uvm.edu/~dhowell/StatPages/More_Stuff/RepMeasMultComp/RepMeasMultComp.html.

Keith A. McGuinness. Of rowing boats, ocean liners and tests of the ANOVA homogeneity of variance assumption. Austral Ecology. 2008;27(6):681-688. Abstract: "One of the assumptions of analysis of variance (ANOVA) is that the variances of the groups being compared are approximately equal. This assumption is routinely checked before doing an analysis, although some workers consider ANOVA robust and do not bother and others avoid parametric procedures entirely. Two of the more commonly used heterogeneity tests are Bartlett's and Cochran's, although, as for most of these tests, they may well be more sensitive to violations of the ANOVA assumptions than is ANOVA itself. Simulations were used to examine how well these two tests protected ANOVA against the problems created by variance heterogeneity. Although Cochran's test performed a little better than Bartlett's, both tests performed poorly, frequently disallowing perfectly valid analyses. Recommendations are made about how to proceed, given these results." [Accessed August 19, 2010]. Available at: http://onlinelibrary.wiley.com/doi/10.1111/j.1442-9993.2002.tb00217.x/abstract.

Data sets:

Data Analysis and Story Library. Nambeware Polishing Times. Excerpt: "Nambe Mills manufactures a line of tableware made from sand casting a special alloy of several metals. After casting, the pieces go through a series of shaping, grinding, buffing, and polishing steps. In 1989 the company began a program to rationalize its production schedule of some 100 items in its tableware line. The total grinding and polishing times listed here were a major output of this program. Number of cases: 59. Variable Names: 1. BOWL: Bowl (1) or not (0); 2. CASS: Casserole (1) or not (0); 3. DISH: Dish (1) or not (0); 4. TRAY: Tray (1) or not (0); 5. DIAM: Diameter of item, or equivalent (inches); 6. TIME: Grinding and polishing time (minutes); 7. PRICE: Retail price ($). Note: Items not classed as bowl, casserole, dish, or tray are plates." [Accessed December 1, 2009]. Available at: http://lib.stat.cmu.edu/DASL/Datafiles/nambedat.html.

Creative Commons License All of the material above this paragraph is licensed under a Creative Commons Attribution 3.0 United States License. This page was written by Steve Simon. Anything below this paragraph represents material from my old website, StATS. Until recently (June 2012), this material was available through Children's Mercy Hospital, but is no longer available there. Although I do not hold clear copyright for this material, I am reproducing it here as a service. See my old website page for more details.


8. Stats: When does heterogeneity become a concern? (June 5, 2008). Dear Professor Mean, I have an ANOVA model and I am worried about heterogeneity--unequal standard deviations in each group. How should I check for this?

7. Stats: What statistic should I use when? (January 4, 2008). Someone was asking about a multiple choice question on a test that reads something like this: A group of researchers investigating in patients with diabetes on the basis of demographic characteristics and the level of diabetic control. Select the most appropriate statistical method to use in analyzing the data: a t-test, ANOVA, multiple linear regression, or a chi-square test. This is one of the more vexing things that people face--what statistic should I use when.


6. Stats: Analyzing data from a simple crossover design (August 22, 2007). A doctor brought me some data from a crossover design and asked me to help analyze it. The analysis was a bit trickier than I had expected, so I reviewed some of the material in Stephen Senn's book.


5. Stats: Post hoc comparisons (March 15, 2006). Dear Professor Mean, I need to run multiple comparisons among all possible pairs of means following an analysis of variance test. What is the best approach? Tukey? Scheffe? Bonferroni?


4. Stats: When the F test is significant, but Tukey is not (September 9, 2005). Someone asked me how to interpret a one factor analysis of variance where the overall F test was significant, but the Tukey folloup test comparing all four group means was not significant for any pair of means.


3. Stats: Multiple degree of freedom tests (September 22, 2004). Someone sent me an email describing a situation where an interaction effect in SPSS had a large p-value, but one of the individual components of that interaction had a small and statistically significant p-value. This can occur in many statistical models where you are testing a factor or interaction that involves multiple degrees of freedom.


2. Stats: Guidelines for ANOVA models (June 20, 2003). Dear Professor Mean, I wanted to compare two groups in my research, those who completed every test battery, and those who completed only some of them. I ran ANOVAs on age, iq, adhd score, and so forth. My professor says that I should have used a t-test instead. Why can't I use ANOVA. Isn't ANOVA better than a t-test?  --Angry Anastasia


1. Stats: Unequal group sizes (November 2, 2001). Dear Professor Mean: I am comparing several groups of subjects, but the number of subjects in each group differ quite a bit. How does this affect the assumptions in analysis of variance?

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Creative Commons License This work is licensed under a Creative Commons Attribution 3.0 United States License. This page was written by Steve Simon and was last modified on 2010-09-14.