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