The Quick and handy Anova Guide
One Way Anova
One Discrete Variable (3 or more levels) & One Continuous Variable.
A. Assumptions of Anova
1. Observations are drawn from normally distributed populations and are themselves normally distributed.
-Pearson (1931) & Lindquist (1953).
- F is robust even when the distribution is asymmetrical (skewed), e.g. Type I and Type II error are not inflated.
- Platykurtic (flat) and Leptokurtic (peaked) distributions don’t affect Type I error appreciably, but they do reduce power when n is low.
2. Observations are random samples from the population, or the subjects are randomly assigned to treatment conditions.
3. The numerator and denominator of the F statistic are independent.
- This is true when the population is normally distributed.
4. The numerator and denominator of the F statistic are estimates of the same population variance. (e.g. = Homogeneity of variance for each treatment group).
- Box 1954. Anova for Heterogeneous Variances is robust if :
1) Equal # of subjects in each group.
2) the populations are normally distributed.
3) the ratio of the largest variance/smallest variance does not exceed 3.
B. The statistic
F = MS between grous/ MS within groups = Treatment Variance / Error Variance
Total Variance = Treatment Variance + Error Variance
Thus, the F statistic represents a ratio of the variance attributable to the treatment (or quasi-treatment) with respect to the variance that naturally occurs due to random variation.
MS btw = SS btw/dfbtwdf between = K-1
MS win = SS win/dfwindf within = N-K
df total = (N-K) + (K-1) = N-1
C. Reporting the Statistic:
F(df btw, df within) = ?.??, p < .0?.
- Once you report the statistic then report the tests of the differences between the groups either using Planned Comparisons (Ms contrast/MS within) or using Post Hoc tests (See the Multiple Comparisons Handout). The Means and Standard deviations for the comparisons (usually each group) should be presented either in the text or in a table. If in a table use subscripts to indicate the which means are significantly different.
For Example, Assume that we have 4 groups representing 4 different types of man animal romantic relationship styles (MARRS, labeled Secure, Dismissing, Preoccupied and Fearful). Further assume that we are testing the association between theses classifications and a series of dependent variables. Table 1 displays the results
Table 1
Means, Standard Deviations, and Anova Results for Relationship Factor Scales Separate by
MARRS Classifications.
______
Secure Dismissing Preoccupied Fearful F
______
APA
UAA
SA
Curiosity
3.40a
(.8064)
3.64
(.6509)
4.95b
(.8218)
4.49
(.7933)
3.25ab
(.6277)
3.09
(.8874)
4.03a
(1.4352)
4.99
(0.5909)
3.94bc
(.5841)
3.68
(1.07)
4.55ab
(.9513)
4.15
(.4545)
4.07c
(.8965)
3.83
(0.9127)
4.31a
(1.1686)
4.32
(1.1294)
3.70*
1.14
3.36*
1.43
______
Note: df = 3, 64. (* = p .05). Means within rows with differing subscripts are significantly different at least p
.05 with respect to Fisher’s LSD post hoc analyses. Standard Deviations appear in parentheses below means. APA
= Animal Physical Attractiveness, UAA = Unprotected Animal Amorous, SA = Sexual Ambivalence.
- Once you have provided the overall test statistic and the means and standard deviations, then you need to describe to the readers what the results indicate with respect to people and their behaviors.
- For Example: Participants classified as Secure report that they are less influenced by the physical attractiveness of their animal partners compared to Fearful and Preoccupied participants. Also, Dismissing participants report that they are less influenced by physical attractiveness compared to Fearful participants. Further, Secure participants reported more ambivalence regarding sexuality compared to Dismissing and Fearful participants. Finally, with respect to unprotected animal amorous and curiosity self reports, none of the MARRS classifications differed significantly.