One Way ANOVA Flashcards
One way Anova overview
ANOVA = analysis of variance – tests for difference between group means (2 or more groups)
One-way / single-factor ANOVA = investigate means of several groups differing by 1 explanatory variable or factor
- Asks if more variation explained by treatment / groups than unexplained error, is variation between groups greater than expect due to chance
- Can conclude whether indivs in group on average more similar than between groups
- Used to compare means of multiple groups rather than testing multiple pairs of means separately as this causes high Type I error rates (falsely rejecting a true null)
Hypothesis
- Null = no difference between group means (any difference due ot random sampling error) -> MSG=MSE
- Alternative = at least one mean differs -> MSG>MSE
Calculation overview
1) Calculate SST, SSE, SSG
2) Work out DF
3) CAlculate MSE and MSG
4) CAlculate the F ratio
5) Look up critical value in F table
CAlculation: Sums of squares
SST= SSE + SSG
SST:
Deviation between each observation and the grand mean
SST= Sum of (data point- grand mean)^2
SSE:
Deviation between each observation and the mean of its group = error / residual
SSE= Sum of ni x (group mean- grand mean)^2
SSG:
Deviation between an observation’s group mean and the grand mean = group
SSG= Sum of (everydata point - group mean)^2
Calculation: DF
o Groups: K-1
o Error: N-K
o Total: N-1
K= groups
N= Total number of individuals across all groups (Total number of observations)
CAlculate: mean square
SS / DF
Calculation: F ratio
MSG/ MSE
- Under the null hypothesis, variation between groups (MS group) will on average (MSgroup) be the same as the error variation (MSerror).
o Expect F ratio to be close to 1. - Under the alternative hypothesis, variation between groups (MSgroup) will on average exceed the error variation (MSerror).
o Expect F-ratio to exceed 1
COmpare to F disitrbution -> (Group DF, Error DF)
STATISTICAL SIGNIFICANCE
Calculation: R^2
R^2= SSgroup/ SStotal
- When R 2 is close to zero (SSgroup<SStotal -> MS error large), the group means are all very similar and most of the variability is within groups.
- When R2 is close to one (SSgroup=SStotal -> MS error small) the explanatory variable explains most variations
Example: 0.43
- 43% of the variation is explained by differences due to experimental units.
BIOLOGICAL SIGNIFICANCE
ANOVA assumptions
- Measurement in each group represents a random sample from the corresponding pop’n
- Variable is normally distributed in each of the k pop’ns
- Variance is the same in all k pop’ns
Robustness of ANOVA
Pretty robust to deviations from assumption of normality
- > especially if sample sizes are large due to the central limit theorem (within each group the sampling distrib of means is approximately normal when sample size is large, even if the variable itself does not have a normal distribution)
Robust to departures from assumption of equal variance in the k pop’ns
-> Only if the sample sizes are large and about the same size tho (no more than 10 fold difference)
Non-parametric alternative
The Kruskal–Wallis test
- Based on ranks
- Assumptions
o All group samples are random samples from the corresponding populations.
o To use Kruskal–Wallis as a test of differences among populations in means or medians, the distribution of the variable must have the same shape in every population.
little power when sample sizes are very small.
Planned vs unplanned comparison
Planned:
- a comparison between means identified as being of crucial interest during the design of the study, identified prior to obtaining the data.
- Same method as 2 sample T test but SE worked out in a different way using MSerror.
Generally if t > 2SE then NOT significant and CANNOT reject null – better to use df and look at table
Unplanned:
- multiple comparisons, such as between all pairs of means, carried out to help determine where differences between means lie.
- Method must make adjustnent for false positives (Type I error)
- Method: Tukey-Kramer method
