Correlated-Samples ANOVA

The Univariate Approach

An ANOVA Factor Can Be

• Independent Samples– Between Subjects

• Correlated Samples– Within Subjects, Repeated Measures– Randomized Blocks, Split Plot

• Matched Pairs if k = 2

The Design

• DV = cumulative duration of headaches• Factor 1 = Weeks• Factor 2 = Subjects (crossed with weeks)• The first two weeks represent a baseline

period.• The remaining three weeks are the

treatment weeks.• The treatment was designed to reduce

headaches.

The DataSubject Wk1 Wk2 Wk3 Wk4 Wk5

1 21 22 8 6 62 20 19 10 4 43 17 15 5 4 54 25 30 13 12 175 30 27 13 8 66 19 27 8 7 47 26 16 5 2 58 17 18 8 1 59 26 24 14 8 9

Crossed and Nested Factors

• Subjects is crossed with Weeks here – we have score for each subject at each level of Week.

• That is, we have a Weeks x Subjects ANOVA.

• In independent samples ANOVA subjects is nested within the other factor– If I knew the subject ID, I would know which

treatment e got.

Order Effects

• Suppose the within-subjects effect was dose of drug given (0, 5, 10 mg)

• DV = score on reaction time task.• All subject tested first at 0 mg, second at 5

mg, and thirdly at 10 mg• Are observed differences due to dose of

drug or the effect of order• Practice effects and fatigue effects

Complete Counterbalancing

• There are k! possible orderings of the treatments.

• Run equal numbers of subjects in each of the possible orderings.

• Were k = 5, that would be 120 different orderings.

Asymmetrical Transfer

• We assume that the effect of A preceding B is the same as the effect of B preceding A.

• Accordingly, complete counterbalancing will cancel out any order effects

• If there is asymmetrical transfer, it will not.

Incomplete Counterbalancing

• Each treatment occurs once in each ordinal position.

• Latin Square

A B C D E E A B C D D E A B C C D E A B B C D E A

Power

• If the correlations between conditions are positive and substantial, power will be greater than with the independent samples designs

• Even though error df will be reduced• Because we are able to remove subject

effects from the error term• Decreasing the denominator of the F ratio.

Reducing Extraneous Variance

• Matched pairs, randomized blocks, split-plot.

• Repeated measures or within-subjects.• Variance due to the blocking variable is

removed from error variance.

Error

Treatment

Blocks

Error

Treatment

Blocks

Partitioning the SS

• The sum of all 5 x 9 = 45 squared scores is 11,060.

• The correction for the mean, CM, is (596)2 / 45 = = 7893.69.

• The total SS is then 11,060 ‑ 7893.69 = 3166.31.

N

YYSSTOT

22 )(

SSweeks

• From the marginal totals for week we compute the SS for the main effect of Week as: (2012+ 1982+ 842+ 522+ 612) / 9 ‑ 7893.69 = 2449.20.

• Wj is the sum of scores for the jth week.

CMn

WSS j

weeks

2

SSSubjects

• From the subject totals, the SS for subjects is: (632+ 572+ ...... + 812) / 5 ‑ 7893.69 = 486.71.

• S is the sum of score for one subject

CMn

SSS i

subjects

2

SSerror

• We have only one score in each of the 5 weeks x 9 subjects = 45 cells.

• So the traditional within-cells error variance does not exist.

• The appropriate error term is the Subjects x Weeks Interaction.

• SSSubjects x Weeks = SStotal – SSsubjects – SS weeks

• = 3166.31 ‑ 486.71 ‑ 2449.2 = 230.4.

df, MS, F, p

• The df are computed as usual in a factorial ANOVA ‑‑ (s‑1) = (9‑1) = 8 for Subjects, (w‑1) = (5‑1) = 4 for Week, and 8 x 4 = 32 for the interaction.

• The F(4, 32) for the effect of Week is then (2449.2/4) / (230.4/32) = 612.3/7.2 = 85.04, p < .01.

Assumptions

• Normality• Homogeneity of Variance• Sphericity

– For each (ij) pair of levels of the Factor– Compute (Yi Yj) for each subject

– The standard deviation of these difference scores is constant – that is, you get the same SD regardless of which pair of levels you select.

Sphericity

• Test it with Mauchley’s criterion• Correct for violation of sphericity by using

a procedure that adjust downwards the df• Or by using a procedure that does not

assume sphericity.

Mixed Designs

• You may have one or more correlated ANOVA factors and one or more independent ANOVA factors

Multiple Comparisons

• You can employ any of the procedures that we earlier applied with independent samples ANOVA.

• Example: I want to compare the two baseline weeks with the three treatment weeks.

• The means are (201 + 198)/18 = 22.17 for baseline, (84 + 52 + 61)/27 = 7.30 for treatment.

t

• The 7.20 is the MSE from the overall analysis.

• df = 32, from the overall analysis• p < .01

21.18

271

181

20.7

30.717.22

11

jierror

ji

nnMS

MMt

Controlling FW

• Compute– And use it for Tukey or related procedure

• Or apply a Bonferroni or Sidak procedure• For example, Week 2 versus Week 3• t = (22‑9.33)/SQRT(7.2(1/9 + 1/9)) =

10.02, q = 10.02 * SQRT(2) = 14.16.• For Tukey, with r = 5 levels, and 32 df,

critical q.01 = 5.05

2tq

Heterogenity of Variance

• If suspected, use individual error terms for a posteriori comparisons– Error based only on the two levels being

compared.– For Week 2 versus Week 3, t(8) = 10.75, q(8)

= 15.2– Notice the drop in df

SAS

• WS-ANOVA.sas • Proc Anova;• Class subject week;• Model duration = subject week;• SAS will use SSerror =

SStotal – SSsubjects – SSweeks

Source DF Anova SS

Mean Square

F Value Pr > F

subject 8 486.7111 60.83888 8.45 <.0001week 4 2449.200 612.3000 85.04 <.0001

Source DF Sum of Squares

Mean Square

F Value Pr > F

Model 12 2935.91111 244.65925 33.98 <.0001

Error 32 230.40000 7.200000    

Corrected Total

44 3166.31111      

Data in Multivariate Setup

data ache; input subject week1-week5;

d23 = week2-week3; cards;

1 21 22 8 6 6

2 20 19 10 4 4

3 17 15 5 4 5

4 25 30 13 12 17

5 30 27 13 8 6

6 19 27 8 7 4

And data for three more subjects

Week 2 versus Week 3

•proc anova; model week2 week3 = / nouni; repeated week 2 / nom;

•The value of F here is just the square of the value of t, 10.75, reported on Slide 23, with an individual error term.

Source DF Anova SS

Mean Square

F Value Pr > F

week 1 722.0000 722.0000 115.52 <.0001Error(week) 8 50.00000 6.250000    

proc means mean t prt;var d23 week1-week5;

Proc Anova;

Model week1-week5 = / nouni;

Repeated week 5 profile / summary printe;Sphericity Tests

Variables DF Mauchly's Criterion

Chi-Square Pr > ChiSq

Orthogonal Components

9 0.2823546 8.1144619 0.5227

We retain the null that there is sphericity.

Univariate Tests of Hypotheses for Within Subject Effects

Source DF Anova SS

Mean Square

F Value

Pr > F Adj Pr > F

G - G H - F

week 4 2449.2 612.30 85.04 <.0001 <.0001 <.0001

Error(week) 32 230.40 7.2000      

Greenhouse-Geisser Epsilon 0.6845Huynh-Feldt Epsilon 1.0756

Epsilon

• Used to correct for lack of sphericity• Multiply both numerator and denominator

df by epsilon.• For example: Degrees of freedom were

adjusted according to Greenhouse and Geisser to correct for violation of the assumption of sphericity. Duration of headches changed significantly across the weeks, F(2.7, 21.9) = 85.04, MSE = 7.2, p < .001.

Which Epsilon to Use?

• The G-G correction is more conservative (less power) than the H-F correction.

• If both the G-G and the H-F are near or above .75, it is probably best to use theH-F.

Profile Analysis

• Compares each level with the next level, using individual error.

• Look at the output.– Week 1 versus Week 2, p = .85– Week 2 versus Week 3, p < .001– Week 3 versus Week 4, p = .002– Week 4 versus Week 5, p = .29

Multivariate AnalysisMANOVA Test Criteria and Exact F Statistics for the Hypothesis of no week Effect

Statistic Value F Value

Num DF

Den DF

Pr > F

Wilks' Lambda 0.01426 86.39 4 5 <.0001

Pillai's Trace 0.98573 86.39 4 5 <.0001Hotelling-Lawley Trace

69.1126 86.39 4 5 <.0001

Roy's Greatest Root

69.1126 86.39 4 5 <.0001

Strength of Effect

• 2 = SSweeks / SStotal = 2449.2/3166.3 = .774

• Alternatively, if we remove from the denominator variance due to subject,

914.4.2302.2449

2.24492

ErrorConditions

Conditionspartial SSSS

SS

Higher-Order Mixed or Repeated Univariate Models

• If the effect contains only between-subjects factors, the error term is Subjects(nested within one or more factors).

• For any effect that includes one or more within-subjects factors the error term is the interaction between Subjects and those one or more within-subjects factors.

AxBxS Two-Way Repeated Measures

CLASS A B S; MODEL Y=A|B|S;

TEST H=A E=AS;

TEST H=B E=BS;

TEST H=AB E=ABS;

MEANS A|B;

Ax(BxS) Mixed (B Repeated)

CLASS A B S; MODEL Y=A|B|S(A);

TEST H=A E=S(A);

TEST H=B AB E=BS(A);

MEANS A|B;

AxBx(CxS) Three-Way Mixed (C Repeated)

CLASS A B C S;

MODEL Y=A|B|C|S(A B);

TEST H=A B AB E=S(A B);

TEST H=C AC BC ABC E=CS(A B);

MEANS A|B|C;

Ax(BxCxS) Mixed(B and C Repeated)

CLASS A B C S;

MODEL Y=A|B|C|S(A);

TEST H=A E=S(A);

TEST H=B AB E=BS(A);

TEST H=C AC E=CS(A);

TEST H=BC ABC E=BCS(A);

MEANS A|B|C;

AxBxCxS All WithinCLASS A B C S; MODEL Y=A|B|C|S;

TEST H=A E=AS;

TEST H=B E=BS;

TEST H=C E=CS;

TEST H=AB E=ABS;

TEST H=AC E=ACS;

TEST H=BC E=BCS;

TEST H=ABC E=ABCS;

MEANS A|B|C;