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## PowerPoint Slideshow about 'Repeated Measures Designs' - marciano-boyton

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In a Repeated Measures Design

We have experimental units that

- may be grouped according to one or several factors (the grouping factors)

Then on each experimental unit we have

- not a single measurement but a group of measurements (the repeated measures)
- The repeated measures may be taken at combinations of levels of one or several factors (The repeated measures factors)

Example 1

- No grouping factors
- One repeated measure factor (time)

Example

In the following study the experimenter was interested in how the level of a certain enzyme changed in cardiac patients after open heart surgery.

- The enzyme was measured
- immediately after surgery (Day 0),
- one day (Day 1),
- two days (Day 2) and
- one week (Day 7) after surgery
- for n = 15 cardiac surgical patients.

The data is given in the table below.

Table: The enzyme levels -immediately after surgery (Day 0), one day (Day 1),two days (Day 2) and one week (Day 7) after surgery

The subjects are not grouped (single group).

- There is one repeated measures factor -Time – with levels
- Day 0,
- Day 1,
- Day 2,
- Day 7
- This design is the same as a randomized block design with
- Blocks = subjects

The Anova Model for a simple repeated measures design

Repeated measures

y11 y12 y13 … y1t

y21 y22 y23 … y2t

subjects

yn1 yn2 y13 … ynt

The Model

yij = the jth repeated measure on the ithsubject

= m + ai + tj + eij

where m = the mean effect,

ai = the effect of subject i,

̴

tj = the effect of time j,

̴

eij = random error.

The Analysis of Variance

The Sums of Squares

- used to measure the variability of ai (between subject variability)

- used to test for the differences in tj (time)

- used to measure the variability of eij (within subject variability)

ANOVA table – Repeated measures (no grouping factor, 1 repeated measures factor (time))

The Anova Table for Enzyme Experiment

The Subject Source of variability is modelling the variability between subjects

The ERROR Source of variability is modelling the variability within subjects

Specify the variables that represent the levels of the repeated measures factor

There is no Between subject factor in this example

The Anova Table for Enzyme Experiment

The Subject Source of variability is modelling the variability between subjects

The ERROR Source of variability is modelling the variability within subjects

In a Repeated Measures Design

We have experimental units that

- may be grouped according to one or several factors (the grouping factors – df = g - 1)

Then on each experimental unit we have

- not a single measurement but a group of measurements (the repeated measures)
- The repeated measures may be taken at combinations of levels of one or several factors (The repeated measures factors – df = t - 1)
- There are also the interaction effects between the grouping and repeated measures factors – df = (g -1)(t -1)

Example:(Repeated Measures Design - Grouping Factor)

- In the following study, similar to example 3, the experimenter was interested in how the level of a certain enzyme changed in cardiac patients after open heart surgery.
- In addition the experimenter was interested in how two drug treatments (A and B) would also effect the level of the enzyme.

The 24 patients were randomly divided into three groups of n= 8 patients.

- The first group of patients were left untreated as a control group while
- the second and third group were given drug treatments A and B respectively.
- Again the enzyme was measured immediately after surgery (Day 0), one day (Day 1), two days (Day 2) and one week (Day 7) after surgery for each of the cardiac surgical patients in the study.

Table: The enzyme levels - immediately after surgery (Day 0), one day (Day 1),two days (Day 2) and one week (Day 7) after surgeryfor three treatment groups (control, Drug A, Drug B)

The subjects are grouped by treatment

- control,
- Drug A,
- Drug B
- There is one repeated measures factor -Time – with levels
- Day 0,
- Day 1,
- Day 2,
- Day 7

The Anova Table

There are two sources of Error in a repeated measures design:

The betweensubject error – Error1 and

the withinsubject error – Error2

The Model

yikj = the jth repeated measure on the ithsubject

in the kth group

= m + ak +ekj (1)+ tj+ (at)ki + ekij(2)

where m = the mean effect,

ak = the effect of group i,

eij(1)= between subject error.

tj = the effect of time j,

(at)kj = the group-time interaction effect

eij(2)= within subject error.

Tables of means

Drug Day 0 Day 1 Day 2 Day 7 Overall

Control 118.63 77.88 60.50 55.75 78.19

A 103.25 68.25 52.00 51.50 68.75

B 103.38 69.38 54.13 51.50 69.59

Overall 108.42 71.83 55.54 52.92 72.18

Example: Repeated Measures Design - Two Grouping Factors

- In the following example , the researcher was interested in how the levels of Anxiety (high and low) and Tension (none and high) affected error rates in performing a specified task.
- In addition the researcher was interested in how the error rates also changed over time.
- Four groups of three subjects diagnosed in the four Anxiety-Tension categories were asked to perform the task at four different times patients in the study.

The Model

ykmji = the ith repeated measure on the jthsubject

when Anxiety (A) is at the kth level and Tension (T) is at the mthlevel

= m + ak + bm + (ab)km +ekmj (1)+ ti

+ (at)ki + (bt)mi + (abt)kmi + eikmji(2)

where m = the mean effect,

ak = the effect of Anxiety k,

bm = the effect of Tension m,

(ab)km = Anxiety–Tension interaction m,

ekmj(1)= between subject error.

kmj

(at)ki = the anxiety-time interaction effect

(bt)mi = the tension-time interaction effect

(abt)kmi = the tension-time interaction effect

ekmji(2)= within subject error.

kmji

The Anova (univariate) Model

yij = the jth repeated measure on the ithsubject

= m + aj + tj + eij

where m = the mean effect,

aj = the effect of subject i,

tj = the effect of time j,

eij = random error.

The implication of the ANOVA model for a repeated measures design is that the correlation between repeated measures is constant.

The multivariate model for a repeated measures design

Let denote a sample of n from the p-variate normal distribution with mean vector and covariance matrix S.

Here

Allows for arbitrary correlation structure amongst the repeated measures – yi1, yi2, … , yit

Let

Then

The test for equality of repeated measures:

Consider the data

This is a sample of n from the (t – 1)-variate normal distribution with mean vector and covariance matrix .

Hotelling’s T2 test for equality of variables

if H0 is true than

has an F distribution with n1= t – 1 and n2= n - t + 1

Thus we reject H0 if F > Fawith n1= p – 1 and

n2= n – t + 1

To perform the test, compute differences of successive variables for each case in the group and perform the one-sample Hotelling’s T2 test for a zero mean vector

Example

In the following study the experimenter was interested in how the level of a certain enzyme changed in cardiac patients after open heart surgery.

- The enzyme was measured
- immediately after surgery (Day 0),
- one day (Day 1),
- two days (Day 2) and
- one week (Day 7) after surgery
- for n = 15 cardiac surgical patients.

The data is given in the table below.

Table: The enzyme levels -immediately after surgery (Day 0), one day (Day 1),two days (Day 2) and one week (Day 7) after surgery

Example:(Repeated Measures Design - Grouping Factor)

- In the following study, similar to example 3, the experimenter was interested in how the level of a certain enzyme changed in cardiac patients after open heart surgery.
- In addition the experimenter was interested in how two drug treatments (A and B) would also effect the level of the enzyme.

The 24 patients were randomly divided into three groups of n= 8 patients.

- The first group of patients were left untreated as a control group while
- the second and third group were given drug treatments A and B respectively.
- Again the enzyme was measured immediately after surgery (Day 0), one day (Day 1), two days (Day 2) and one week (Day 7) after surgery for each of the cardiac surgical patients in the study.

Table: The enzyme levels - immediately after surgery (Day 0), one day (Day 1),two days (Day 2) and one week (Day 7) after surgeryfor three treatment groups (control, Drug A, Drug B)

The subjects are grouped by treatment

- control,
- Drug A,
- Drug B
- There is one repeated measures factor -Time – with levels
- Day 0,
- Day 1,
- Day 2,
- Day 7

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