Research Methods IV
One-Way Independent ANOVA (between-subjects)
Why use ANOVA?
Using a T Test would create 5% chance of error for comparing multiple tests
How does ANOVA work?
Same logic as a T Test (between group variance/within group variance)
Calculating ANOVA
Sums of Squares
SS = variance in dataset
Degrees of Freedom
Df = number of observations that can vary
SS(total) = total variability
SS(between or M) = Variability between groups
SS(within or R) = variability within groups
df(total) = N - 1
df(between) = k - 1 (k = conditions)
df(within or error) = df(total) - df(between)
Why calculate df? = so SS can be adjusted according to design
Mean Squares (MS) = SS / df = average amount of variance
MS(within or R) = average amount of within groups variance
MS(between or M) = average amount of between groups variance
MS(between) = SS(between) / df(between)
MS(within) = SS(within) / df(within)
F-ratio = systematic variance:unsystematic variance
MS(between)/MS(within)
ANOVA in SPSS
Analyze
Compare Means / generalized linear model
One-Way Anova / univariate
Options...
Tick descriptive, Homogeneity of Variance, Means plot
In ANOVA output: top MS = systematic variation and bottom MS is unsystematic. sig = probability if the null hyp was true
Effect size = eta squared (η2) = ss(between)/SS(total)
APA format: F(df between, df error) = XX.XX, p =.XXX, η2 =.XX
Repeated-Meaures ANOVA (within subjects)
Why Repeated Measures?
More powerful
Eliminate individual differences
However may have carry over effects
How does repeated measures work?
Same logic as independent ANOVA: ratio of unexplained to explained variance
In repeated measures, experimental manipulation affects the within participant variation. Variation consists of:
Manipulation effect
Individual differences in performance
Independent ANOVA: F = Variance between group means / random variance
Repeated-measures ANOVA: F = Variance between condition means (no indiv dif bc the groups are same people) / random variance (indiv dif are removed to make more powerful)
Calculating ANOVA
Sums of Squares
Total SS (SSt) = total variability in the data - total deviation of all scores from the overall mean
Within-participant SS (SSw) = variability within participants - total deviation of each participant's score from each participant's mean
Model SS (SSm) = variability due to the independent variable - toal deviation of the condition means from the grand mean
Error/residual SS (SSr) = variability within the participants that cannot be explained by the independent variable = SSw - SSm
Degrees of Freedom
df(total) = N - 1
Within-participants df/ df(w) = k - 1 for all participants
eg. 100 participants & 3 conditions, df(w) = (3-1) x 100 = 200
df(m) = k - 1
df(r) = df(w) - df(m)
Mean Squares = average amount of variance
MS(M) = average amount of variance explained by experimental manipulation
MS(M) = SS(M) / DF(M)
MS(R) = average amount of unexplained/residual variance
MS(R) = SS(R) / DF(R)
F-ratio
MS(M) / MS(R)
Assumptions of repeated measures ANOVA
normally distributed
Assumption of sphericity = the variances of the differences between all combinations of conditions are equal (basically homogeneity of variances)
Can test for sphericity by using Mauchly's Test in SPSS
ANOVA in SPSS
1 Analyze
2 General Linear Model
3 Repeated Measures
4 Give the number of levels (conditions)
5 for example time could have 3 levels, before, after, 6 months. martial arts could also have 3 levels, no training, little training and fully experienced
6 you then allocate these levels in the next box it's pretty self explanitory
7 options
8 put your factor in the display means box
9 tick compare main effects
10 confidence dropdown Bonferrroni
11 tick descriptive stats, estimates of effect size and homogeneity
12 Mauchly's Test box, you want significance to be ABOVE .05 because the null hypothesis tests that the variances of the differences are equal
13 APA format: F(df within (the first row df thing), df error) = XX.XX, p =.XXX, η2 =.XX
14 If the assumption of sphericity is broken, use greenhouse-geisser instead (even for df) and report to 2dp(?)
15 Pairwise comparisons -check the significance for all the different pairs of conditions, if they're below .05 they're good, if they're above it's not significant/you can't say that value is different from the other
ANOVA assumptions
Normally distributed
Analyze
Desc Stat
Explore
Plots...
Normality plots with tests
Sig should be ABOVE .05
Homogenity of variance
(Variances of groups should be equal)
Tested with Levene's Test
One way ANOVA
Options...
Homogeneity of variance
Sig should be HIGHER than .05
This means variances are equal
If it's broken (less than .05)
F Statistic can be corrected,
Welch
Brown-Forsythe
F and α
How to increase the power of F?
Reduce the number of treatments
Reduce the number of unexplained variance
Increase differences across treatments
Increase number of paritcipants
ANOVA is a hypothesis testing strategy
Null is always that the population means are equal
Alternative is that at least one is different
We know what differences are significant by using a t-test
But we only use t-tests is possible contrasts are low
α increases when you increase the
number of comparisons
Therefore increased chance of false positives
An approximation for α growth comes from Bonferroni
Want to reduce the potential false positives without increasing false negatives
Follow-Up Tests for Multiple Hypotheses
Planned contrasts (priori)
Post-Hoc (posteriori)
Predicted by the hypothesis/before the experiement
Not predicted. Tests are exploratory. Hypotheses arise after to explain inspected data.
Allow comparison of one or more groups with another group(s)
Group are combined using coefficients (-1, 1, 0)
coefficients can be used to compare groups of data (-1, -1, 2)
Planned contrasts in spss
Analyze
Compare means
One-way ANOVA
Factor is the group variable
if now you click ok, you will see in the ANOVA table the sig. if it's less than .05, there are differences between the groups
If you click contrasts...
then enter your coefficients, for example comparing groups 1 & 2 (1, -1, 0)
you are looking for ANOVA to be less than .05 and descriptives mean to be as low as possible
They compare every level against the rest. Different techniques to avoid α growth
Normal t-test
Bonferroni
Studentized t-test
Tucky's Honestly Significant Difference (THSD)
Least Significant Difference (LSD)
Student-Newman-Keuls
Strict
Strict
Powerful
Powerful
compares all possible levels, doesn't prevent α growth, but very powerful(produces many significant results)
compares all possible levels, prevents α growth, very strict
organises variables in sets of equal means in ascending order, then compares the smallest mean to the rest of the sub-groups. Prevents growth of α by reducing the number of comparisons needed
Identical to Student-Newman, but more strict.
Post-Hoc in SPSS
One-way ANOVA
Post Hoc...
Tick Bonferroni
Multiple Comparisons output table
sig should be less than .05
Polynomial Contrasts (Repeated-Measures Post-Hoc)
Can use these contrasts to test the 'shape' of the line that connects these variables
IV needs to be: quantitative, continuous and the levels need to be equally spaced (1, -1, 0)
If you use the coefficients -2, -1, 1, 2, you get a straight line
If you use the coefficients -1, +1, +1, -1, you get a quadratic function (a pyramid with the top cut off)
If you use the coefficients -1, 3, -3, 1, you get a cubic function (a zig-zag like ~ )
linear: -2, -1, 1, 2
Quadratic: -1, +1, +1, -1
Cubic: -1, 3, -3, 1
These contrasts should be independent (no information from one contrast is in the other)
To see if two contrasts are independent, multiply their coefficients for each level and add them:
-3, -1, 1, 3 AND 1, -1, -1, 1 (so linear and quadratic)
(-3x1)+(-1x-1)+(1x-1)+(3x1)=0
Don't worry SPSS does this for you
There could be an overall shape (like linear) but might break down into sub-shapes
They're powerful and can replace repeated measures ANOVA
Sig should be above .05, if not use Greenhouse-Geisser
should be above .05
Within subjects variance comes from
Individual differences
Experimental Error
Between groups variances comes from:
Treatment effects
Individual differences
Experimental error