Where we are Where we are going

Statistical analyses by number of sample groups

1 - Single sample Z or t tests 2 – T tests (dependent, independent)

3 or more ?Analysis of variance

Between subjects (independent)Equal N (Easy)Unequal N (Harder – structural)

Within subjects (dependent – structural)Two-way (independent - structural)

Analysis of VarianceEqual N (Easy)

Used with 3 or more groupsExtends logic of independent groups t-testSome additional things to think about

ASSUMPTIONS of ANOVA:

equal variances (required for pooling)normality (required for test distribution)

The null hypothesis in ANOVA is always:

This implies that any combination of means are also equal

1 2 3 j

1 2 3( ) / 2

The alternative hypothesis in ANOVA is always- The population means are different (at least one

mean is different from another)

**** Null hypothesis is tested by comparing two estimates of the population variance (σ2 ):

(MSB) between-group estimate of (σ2 ) AFFECTED by whether the null is true

(MSW) within-group estimate of (σ2 )

UNAFFECTED by whether the null is true

ANOVA: Null Hypothesis Is TRUE

Score Distributions

Mean Distributions, N=9

Sample 3 ScoresSample 2 ScoresSample 1 Scores

Sample 3 MeanSample 2 MeanSample 1 Mean

ANOVA: Null Hypothesis Is False

Score Distributions

Mean Distributions (N=9)

-- when the null hypothesis is true (MSB)

F ratio = ------- = About 1 (MSw)

-- when the null hypothesis is not true (MSB)

F ratio = ------- = Much greater than 1 (MSw)

Let: K = # of groups (treatments)J = a particular group 1…KNJ = # of people in group J_Xj = Sample mean for group JNG = Total (grand) number of people_XG = The grand mean

I = Individual = Variance of the means

/ij Gx n

S2_X

Is an estimate of Variance of the meansS2_X

σ2_X

= σ2_X

σ2

--- Because the variance of the means is the n variance of the variable divided by sample size

S2_X (n) estimates the population variance (σ2 )

S2_X (n) = MSB IF groups are same size (n1=n2=n3…)

Between Groups variance

S2_X (n) estimates σ2 just as well as any random samples would

S2_X (n) = MSB will be higher than the populations variance because the means are farther away from each other than would be expected by chance

If Null is true, then these are just random samples

If Null is false, then these are not just random samples

Between Groups variance

S1

2 + S22 + S3

2 +… Sk2

= ----------------------- k

If Null is true, then these are just random samplesSo we can poll the variances just as we did with the independent samples t-test

Within Groups variance

S2pooled

S2pooledMSW =

MSB

F = ---------

MSW

• F Ratio

• Critical values of f depend on dfW and dfB

• Look up in table

Post-hoc tests

ANOVA tells us there is some difference, but it does not tell us which groups are different from each other

ANOVA is like a shotgun – firing many pellets at many different hypothesis like

u1=u2

u2=u3

(u1+u2)/2=u3

Post-hoc tests - Tukey

Tests all the pairwise comparisons – does not test complex hypotheses (such as (u1+u2)/2=u3)

u1=u2

u1=u3

u1=u4

u2=u3

u2=u4

u3=u4

Apriori tests

Also referred to as

“planned comparisons”

“planned contrast”

A rifle instead of a shotgun. Used to test a specific hypothesis that is a subset of all hypotheses. For example, with 3 groups – if you wanted to test if group 3 was different from the other two groups, then you would test the following:

(u1+u2)/2=u3

But WHAT IF the sample sizes are not the same?

Structural Model of ANOVA An alternative way of understanding ANOVA - used whenever Nj is not equal across groups All the basic logic stays the same, computationally however

GX = mean of all the scores For each score, deviation from grand mean is divided into two parts a) deviation of score from mean of its group b) deviation of group mean from grand mean

.

2( )ij GX X =

2( )j GX X + 2( )ij jX X

SST = SSB + SSW

DFB = 1K DFW = gN K

2 ( )

1j G

B B

X XS MS

K

2 ( )ij jW W

G

X XS MS

N K

F = MSB/MSW DFB = 1K DFW = gN K

22

STANDARD WAY OF SETTING UP ANOVA EFFECT SS DF MS F

BETWEEN 2( )j GX X 1K /B BSS DF /B WMS MS

WITHIN 2( )ij jX X gN K /W WSS DF

TOTAL 2( )ij GX X 1gN

2( )ij GX X =

2( )j GX X + 2( )ij jX X

• F Ratio

• Critical values of f depend on dfW and dfB

• Look up in table

Anova Effect size

dM = --------------

How far apart the means are divided by the standard deviation -

Similar to the effect size for independent t-test (mean difference/stdev)

Small Medium Large

.2 .5 .8

_ _Xmax - Xmin

Spooled =