ANALYSIS OF VARIANCE (ANOVA)
Guards against familywise error:
- Analyses all the variance in data at once
- Using multiple t-tests inflates type 1 error rate
Is an omnibus technique:
- Tests whether DV varies among the levels of the IV
- Tells us whether there is a significant difference between group means somewhere
- Does not tell us which means are significantly different
ONE WAY ANOVA
A one-way ANOVA is focused on the effect of one IV on the DV, but the IV can have more than two levels
F STATISTIC
The F-statistic aims to compare the variance among the treatments, to the variance within the samples themselves.
If ANOVE doesn’t yield significant response…
We state there is no effect of IV on DV, no follow up tests.
If ANOVA yields significant response…
We do follow up tests to find what means are significantly different
IV - in ANOVA
Called the factor of treatment (if manipulated)
LEVELS
Different conditions that make up a factor
NULL HYPOTHESIS
H0 = u1 = u2 = uk
(if three means then.. u1 = u2 = u3)
ALTERNATIVE HYPOTHESIS
H1 = uk ≠ uk’
(at least two means are different)
LOGIC OF ANOVA
Observed differences relative to expected differences
Separate total variance into two components:
- Between-groups variance
- Within-groups variance
Between-groups variability due to:
Treatment effect/levels of factor
Individual differences
Experimental error
Within-groups variability due to:
Individual differences
Experimental error
How to know if presence of treatment effect?
Therefore everything will cancel out apart from treatment effect
So if between groups variability > within-groups variability = presence of treatment effect
MEAN SQUARES
Use sample variance to estimate population variance
We compare two independent estimates of population variance:
Within-group variability:
- MS error also referred to as MS residual
Between-groups variability:
- MS treatment also referred to as MS model
COMPARING MEAN SQUARES
If H0 is false, there will be more variation among the means that can be accounted for by chance and MS treatment will be larger than MS error
Represented as a ratio F = MS treatment/ MS error
F-DISTRIBUTION
Fobt is compared to the distribution of Fs that would be expected if H0 is true.
Fobt is considered significant if Fobt > Fcrit
F distribution dependent on df
Fobt can only be positive
F distribution is positively skewed
- Most Fs cluster around 1 (H0)
F-test is a one tailed test
- We are only testing for the presence/absence of treatment effect
INDEPENDENT GROUPS ANOVA STEPS
- State H0 and H1 in words and symbols
- Calculate sum of squares
- Calculate degrees of freedom
- Calculate mean squares
- Calculate f-ratio
- Construct summary/source table
- Use F-table to find Fcrit
- Make decision
- Interpret results
STEP 2. Calculating sum of squares
SS total = sum (X - Xbar)^2
SS treatment = sum nk(Xbar k - Xbar)^2
SS error = sum (Xik - Xbar k)^2
X = individual raw score
Xbar = grand mean
nk = no. people in group k
Xbar k = mean of group k
Xik = individual raw score in group k
STEP 2. Calculating sum of squares SHORTCUTS
SS total = SS treatment + SS error
SS treatment = SS total - SS error
SS error = SS total - SS treatment
STEP 3. Calculating degrees of freedom
df treatment = k - 1
df error = df k1 + df k2 + df k3
STEP 4. Calculate mean squares
MS treatment = SS treatment / df treatment
MS error = SS error / df error
STEP 5. Calculate F ratio
F = MS treatment / MS error
STEP 6. Find Fcrit
df numerator = df treatment
df denominator = df error
STEP 9. Interpret results
A one-way independent groups ANOVA revealed that (DV) varied significantly as a function of (levels of IV), F (df treatment, df error) = X, p < .05.
