Lecture Notes

Chapter 10 Analysis of Variance: One-Way Classification (Modified: 2003-08-05)

In the world of experimentation, researchers want to be able to analyze more than two groups at a time. The analysis of variance or ANOVA gives them the ability to do so. Your second course in statistics (now there is a scary thought!) is often a course on ANOVA. In this chapter we will look at how the ANOVA works with different levels of one independent variable (i.e, two or more levels). We will also see how FASTAT handles the ANOVA. In your next course, Research Methods I, you will use the ANOVA in more complex designs.

Introduction
- Why not just run multiple t tests on all pairwise comparisons?
  - Too many t tests (N(N - 1)/2), where N is the number of groups
    - For N = 15 that comes to 105 tests!
  - Type I error increases as you conduct those tests
  - You ignore interactions among the set of subsamples tested
- The ANOVA allows you address all of the concerns above
Logic of the ANOVA
- A new sampling distribution--F
  - Compares two estimates of population variance to each other: Between and Within
    - Between--variation of group means from grand mean (mean of the means of all groups)
    - Within--variability of each group's scores (also known as error term)
  - Formula
    - F = B / W
    - If the groups are random, then both B and W are unbiased estimates of the population variance, and thus F = 1
  - Assumptions of F
    - Random sampling of individuals from normally distributed population
    - Homogeneity of variance--the variance of all groups should be equal
    - Independent samples
  - Relative importance of assumptions
    - The first and third are really design issues in experimentation
    - The second is most important and can be tested prior to running an ANOVA
- The hypotheses
  - The null hypothesis
    - the means of each group are the same
  - The alternative hypothesis (upon rejection of null hypothesis)
    - one or more of the means are different
  - In ANOVA, never specify direction of alternative hypothesis (i.e., greater than or less than)
- The F distribution described
  - Another sampling distribution
  - A family of curves
  - Has two degrees of freedom (one for numerator or between; one for denominator or within)
  - F = t ²
  - As the degrees of freedom approach ° , F approaches the normal distribution (like t )
Necessary Terminology
- Sum of squares
  - SS or Sx² or S(X-Xbar)² or sum of the squared deviations
  - SS_tot = SS_treat + SS_error (use for error correction)
- Mean square
  - MS or the SS/df or a variance term for the ANOVA
  - df _tot = df _treat+ df _error
  - df _tot = N_tot- 1
  - df _treat = K - 1
  - df _error = N_tot - K
  - MS_treat = SS_treat / df _treat
  - MS_error = SS_error / df _error
  - MS_totis not used in ANOVA calculations
- Grand mean
  - Mean of all scores
- K
  - Number of treatments or the number of levels of IV (they are the same thing)
- Subscripts
  - _tot after a symbol signifies that the symbol is for all such numbers in the experiment
  - _t after a symbol signifies that the symbol is for all such numbers in the treatment group
Finding F in Table
- F = MS_treat / MS_error
- The null hypothesis assumes that F = 1 (can you say why?)
- Because of the formula above, every F will have 2 dfs one for the numerator or treatment and one for the denominator or error
- The table gives the dfs for the numerator across the top (for both a = .05 and .01) the dfs for the denominator are given down the side of the table
- Find the critical value at the intersection of the appropriate dfs
- If the calculated value is equal or greater than the table value, then reject the null hypothesis
- See below for what to do next (Post ANOVA tests)
Using FASTAT to Compute One-Way ANOVAs
- Here is how you would enter the data from Table 9.3

EXTINCTI	LEVEL
3.000	1.000
5.000	1.000
6.000	1.000
2.000	1.000
5.000	1.000
5.000	2.000
7.000	2.000
9.000	2.000
8.000	2.000
11.000	2.000
10.000	2.000
6.000	2.000
8.000	3.000
12.000	3.000
13.000	3.000
11.000	3.000
10.000	3.000
10.000	4.000
14.000	4.000
15.000	4.000
13.000	4.000
11.000	4.000

First, graph the data using Graph using the Box option. Making a graph like the one below should always be your first step. After a while, you will be able to predict whether or not to reject the null hypothesis on the basis of the graph alone (but still do the ANOVA!). Here is how the data should look:

Then use the ANOVA menu choice under Stats, pick EXTINCTI as the dependent variable and LEVEL as the Factor(s)

The Summary Table

Here is the Summary Table from FASTAT, compare to Table 9.4, page 231

ANOVA
 
Dep var:EXTINCTI      N:   22    Multiple R:  .861    Squared Multiple R:  .741

Analysis of Variance

Source   Sum-of-Squares    DF  Mean-Square     F-Ratio       P
 
    LEVEL           202.473    3       67.491      17.159       0.000
 
    Error            70.800   18        3.933
 
Note that FASTAT computes the actual p value

Now What? Post ANOVA Testing
- You rejected your null hypothesis. So, what can you say?
- All you can say at this point is:
  - "It is not likely that the samples have a common population mean."
- But you would like to know more, specifically, what is the pattern of mean differences which led to the significant F
- Making such a statement is the role of post anova tests, but first let's look graphically at all of the possible patterns of data with three groups

A is significantly different from both B and C

A and B are significally different from C

B is significally different from A and C

B and C are significally different from A

C is significantly different from both A and B

C and A are significantly different from B

What Would the Graph of the Null Hypothesis Look Like?

Draw it in below:

Types of Post Anova Tests

Tukey Honestly Significantly Difference Test (HSD)

Compares all pairwise comparisons

Formulas:

N = the number in each treatment

Example using the means above:

HSD_.05 critical value (4 groups, 18 df ) = 4.00

Group 1 to Group 2 = 4.62*

Group 2 to Group 3 = 3.16

Group 3 to Group 4 = 2.03

Group 2 to Group 4 = 5.19*

* reject null hypothesis

URLs
- Design of Experiments and ANOVA--Text, contents page for 14 topics on ANOVA, from Statistics Glossary
  - http://www.stats.gla.ac.uk/steps/html.glossary/dexanova.html
- Pharmacy 300-ANOVA--Text, simple discussion of logic of ANOVA with example from college course
  - http://137.82.133.98/pharm/pharm300/statsl9.html
- ANOVA-Testing Equality of Means--Text, contents page for ANOVA
  - http://www-prophet.bbn.com/statguide/anova.html
- An Intuitive Approach to ANOVA--Text, from a presentation by Chris Spatz, Hendrix College
  - http://spsp.clarion.edu/RDE3/C7/Lecture72.html
- Analysis of Variance Between Groups--Text, discusses use of ANOVA, link to calculation page below
  - http://www.physics.csbsju.edu/stats/anova.html
- ANOVA: How Many Groups? Size of Largest Group?--Interactive, text, calculates one-way ANOVAs for N groups and N variables
  - http://www.physics.csbsju.edu/stats/anova_NGROUP_NMAX_form.html
- One-way Fixed Effects ANOVA--Java, interactive, text, calculates statistics for one-way ANOVA (not easy to use)
  - http://www.stat.uiowa.edu/~rlenth/Power/OnewayApp.html
- Stats: One-Way ANOVA--Text, discussion of rationale of ANOVA, includes terminology necessary (intermediate)
  - http://www.richland.cc.il.us/james/lecture/m170/ch13-1wy.html
- ANOVA with SYSTAT 5.2.1--Text, instructions on how to use SYSTAT's MGLH module
  - http://www.psych.nwu.edu/help/systat.html
- How ANOVA Works--Text, discusses rationale of ANOVA (intermediate to advanced)
  - http://www.sytsma.com/phad530/anovaworks.html
- The One-Way ANOVA Table--Text, tables, shows how to set up table and why (intermediate)
  - http://www.stats.ox.ac.uk/~dlunn/lectures/b8web_15/node1.html
- ANOVA Data Sets--Text, five sets of data suitable for ANOVA
  - http://www.itl.nist.gov/div898/wrkshops/stats_sci/anova_data.html

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