3: INDEPENDENT 1 WAY ANOVA Flashcards
one-way ANOVA
- used when we have 1IV with more than 2 levels
- estimates whether the population means under the different levels of the IV are different (estimate is based on the difference between the measured sample means)
- independent one-way ANOVA: between participants
- repeated measures one-way ANOVA: within participants
- just an extension of the T-test
- reduces type I error, by reducing number of tests
familywise error rate
the probability that at least one of a ‘family’ of comparisons, run on the same data will result in a type I error
- it provides a corrected significance level (a’), expressing the probability of making a type I error
- ombnibus tests (e.g. ANOVA) control the familywise error rate
a’ = 1 - (1-a)^c
c = number of comparisons
F-ratio
null: there is no difference between the populatino means under the different levels of the IV
F = variance between IV levels / variance within IV levels
F close to 0 - small variance between IV lvls realtive to within IV lvls
F further from 0 - large variance between IV lvls relative to within IV lvls
independent ANOVA designs: what contributes to between IV level variance
- manipulation of IV (treatment effects)
- individual differences
- experimental error (random/constant error)
independent ANOVA designs: what contributes to variance within IV levels
- individual differences
- experimental error (random error)
t/F ratio
t/F = variance between IV levels/ variance within IV levels
variance between: includes the variance ‘caused’ by our manipulation of the IV and error variance
variance within: includes only error variance
partitioning the variance
- calculates the means for each IV level
- calculates the grand mean (sum of IV levels, divided by the number of IV levels)
- calculates within IV lvls variance (sum of squared differences between individual values and the corresponding IV level mean)
- calculates the between IV levels variance (sum of squared differences between each IV level mean and the grand mean)
assumptions: independent one-way ANOVA
- normality: the DV should be normally distributed, under each level of the IV
- homogeneity of variance: the variance in the DV, under each level of the IV, should be (reasonably) equivalent (SPSS checks with levenes test, welch F statistic can correct for this)
- equivalent sample size: sample size under each level of the IV should be roughly equal
- independence of observations: scores under each level of the IV should be independent
if our data seriously violate these assumptions we should use the non-parametric equivalent - Kruskal Wallis Test
equality of variance
homogenous - the same
heterogenous - different
levenes:
- null - there is no difference between the variance under each level of the IV (homogeneity)
- if P < (or equal) .05 we reject null (heterogeneity)
(report result of Welch’s F test instead of ANOVA F)
unequal variances
where the assumption of homogeneity of variance has been violated we should report the result the result of Welch’s F test instead of ANOVA F
model sum of squares (SSm)
sum of squared differences between IV level means and grand mean (i.e. between IV level variance)
in SPSS SSm can be found in a table in the grid - sum of squares x between groups
SSm + SSr = SSt
residual sum of squares (SSr)
sum of squared differences between individual values and corresponding IV level mean (i.e. within IV level variance)
in SPSS SSr can be found in a table in the grid - sum of squares x within groups
SSr + SSm = SSt
model mean square (MSm)
MSm = SSm/DFm
relates to between IV variance
- in SPSS it is in the mean square x between groups grid
F = MSm / MSr
residual mean square (MSr)
MSr = SSr / DFr
relates to within IV variance
- in SPSS it is in the mean square x within groups grid
F = MSm / MSr
df for independent 1 way ANOVA
need to calculate df for our estimates of :
- between IV level (model) variance
- within IV level (error/residual) variance
N = total sample size, k = IV level no.
between IV:
DFm = k - 1
within IV:
DFr = N - k
