1. Analysis of Variance -ANOVA
At its lowest level it is essentially an extension of the logic of t-tests to those situations where we wish to
compare the means of three or more samples concurrently.

2. ANOVA One-way ANOVA Two-way ANOVA / Factorial
One IV and one DV
Two-way ANOVA / Factorial
Two IVs and one DV
MANOVA (Multivariate Analysis of Variance)
Tests differences between / among group means for 2 DV, while controlling for correlations among DV (use only if DVs correlated)
ANCOVA
Tests differences between / among group means while controlling for a continuous covariate
Repeated Measures
Tests differences, while controlling for error associated with repeated measurements

3. ANOVA Tests the difference among two or more groups.
Ho : There is no differences among the group means.
HA: There is a difference among group means.
IV: Nominal level data (Grouping variable)
DV: Continuous variable
Generates a f ratio : variation between groups is contrasted with variation within groups

4. Comparing means when there are >2 groups  ANOVA
Analysis of Variance – decomposes variation in DV to Between groups and Within groups using “Sums of Squares”
Can a significant amount of variability in DV be attributed to differences between groups?
Variance between groups (BMS)
F = Variance within groups (WMS)
When differences between groups are large relative to variation within groups, the probability is high that the IV is related to or caused the group differences
** cannot just do multiple t-tests because it increases the risk of Type I error **

5. ANOVA: Assumptions ANOVA ANOVA is a fairly robust test
DV is continuous in nature
IV is nominal level data in the form of 2 or more categories
Data should be near normally distributed
Groups are mutually exclusive
Groups have equal variance
ANOVA is a fairly robust test

6. Skewed Distributions
The distributions are asymmetrical (skewed to one side)
ANOVA

7. Variance Not Homogeneous
Dispersion in the red category is greater than in the green
ANOVA

8. ANOVA Terms — Sums of Squares
Sum of squares- proportion of variance due to group differences
Source S.S. d.f. M.S. F P___
Between groups
Within groups
Combined
S.S.—The sum of squared deviations of each data point from some mean value
Between groups—The difference between S.S. combined and S.S. within groups
[variability due to IV]
Within groups—The total squared deviation of each point from the group mean
[variability due to individual differences, measurement error…]
Combined—The total squared deviation of each data point from the grand mean

9. ANOVA Terms — Degrees of Freedom
Source S.S. d.f. M.S. F P___
Between groups
Within groups
Combined
Source S.S. d.f. M.S. F P
Between groups
Within groups
Combined
d.f.—The number of cases minus some "loss" because of earlier calculations.
Between groups d.f.—Equal to the total number of groups minus one.
Within groups d.f.—The total number of cases minus the number of groups.
Combined d.f.—Equal to the total number of cases minus one.

10. Degrees of freedom (df)
u = number of treatments/groups
v = number of replicates.
Source of variance
Sum of squares (S of S)
Degrees of freedom (df)
Mean square = S of S / df
Between treatments
***
u - 1 [2]
Residual
u(v-1) [6]
Total
(uv)-1 [8]
The total df is always one fewer than the total number of data entries]
Assume that we have recorded the biomass of 3 bacteria in flasks of glucose broth, and we used 3 replicate flasks for each bacterium
For u treatments (3 in our case) and v replicates (3 in our case); the total df is one fewer than the total number of data values in the table (9 values in our case)

11. ANOVA Terms — Mean Squares & F-Ratio
Source S.S. d.f. M.S. F P___
Between groups
Within groups
Combined
Source S.S. d.f. M.S. F P
Between groups
Within groups
Combined
M.S. (the variance)—the sums of squares divided by the degrees of freedom
F—the ratio of mean squares between groups to the mean squares within groups.

12. ANOVA Terms — Mean Squares & F-Ratio
Source S.S. d.f. M.S. F P___
Between groups
Within groups
Combined
P value compares the calculated F to a critical value of F
Critical Value of F3,16 = 3.24
Our F-value of exceeds 3.24 so we reject the null hypothesis and indicate that the data supports there are differences in our DV by the IV groups- but we can’t tell which group better

13. Fill in the blank on the ANOVA Table
Source S.S. d.f. M.S. F
Between groups
Within groups
Combined
How many groups are being compared in the above example?
How many subjects are in the sample?
What is the F-ratio?

14. Examples: One-Way ANOVA

15. Mean (SD) Confidence Score
Depression Level N
Mean (SD) Confidence Score
F-Ratio and p-value
Rarely
356
68.89(10.05)
F = (p=.001)
Sometimes
266
58.22(10.83)
Often 47
46.23(12.50)
Research Question: Is there a difference in confidence scores among those who are rarely, sometimes or often in a depressed state of mind?
Null hypothesis: Confidence would be similar regardless of depression level
Interpretation: We see the mean confidence scores vary from 46 among those with odepression often to 69 for those who rarely have depression. There is a statistically significant difference in overall confidence score between depression groups F(3, 676) = , p <.001.

16. Mean (SD) Job Satisfaction Score
Nursing Staff N
Mean (SD) Job Satisfaction Score
F-Ratio and p-value RN 20
45.0(3.95)
F = (p=.001)
LPN
41.7(5.79)
UAP
36.2(6.37)
Research question: Are different nursing staff less satisfied with their job?
Null hypothesis: Job satisfaction would be comparable for all nurse staff.
Interpretation:
We see that the mean job satisfaction score for the RN group is 45.0 (SD 3.95), the LPN group is 41.7 (SD 5.79) and the UAP group is 36.2 (SD 6.37).
The next value is the F-ratio which is and the p-value corresponding to the F-ratio is Since the p-value is <.05 (p=.001) we can reject the null hypothesis.