Repeated Measures Designs

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 y 11 y 12 y 13 … y 1t y 21 y 22 y 23 … y 2t y n1 y n2 y 13 … y nt subjects Repeated measures

The Model y ij = the j th repeated measure on the i th subject =  +  i +  j +  ij where  = the mean effect,  i = the effect of subject i,  j = the effect of time j,  ij = random error. ̴ ̴

The Analysis of Variance The Sums of Squares - used to measure the variability of  i (between subject variability) - used to test for the differences in  j (time) - used to measure the variability of  ij (within subject variability)

ANOVA table – Repeated measures (no grouping factor, 1 repeated measures factor (time)) SourceS.S.d.f.M.SF Between Subject Error n - 1 Timet - 1 Between Subject Error (n - 1)(t - 1)

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

Analysis Using SPSS - the data file The repeated measures are in columns

Select General Linear model -> Repeated Measures

Specify the repeated measures factors and the number of levels

Specify the variables that represent the levels of the repeated measures factor There is no Between subject factor in this example

The ANOVA table

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

The general Repeated Measures Design g groups of n subjects t repeated measures

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)

The Model - Repeated Measures Design

ANOVA table for the general repeated measures design Sourced.f. Main effects and interactions of grouping factors g - 1 Between subject Error g(n – 1) interactions of grouping factors with repeated measures factors (t – 1)(g – 1) Main effects and interactions of repeated measures factors t - 1 Within subject Error g(t – 1)(n – 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 surgery for 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 between subject error – Error 1 and the within subject error – Error 2

The Model y ikj = the j th repeated measure on the i th subject in the k th group =  +  k +  kj (1) +  j  ki +  kij (2) where  = the mean effect,  k = the effect of group i,  j = the effect of time j,  ij (1) = between subject error.

(  ) kj = the group-time interaction effect  ij (2) = within subject error.

Tables of means DrugDay 0Day 1Day 2Day 7Overall Control A B Overall

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 number of errors committed at each instance is tabulated below.

The Model y kmji = the i th repeated measure on the j th subject when Anxiety (A) is at the k th level and Tension (T) is at the m th level =  +  k +  m + (  km +  kmj (1) +  i  ki  mi  kmi +  ikmji (2) where  = the mean effect,  k = the effect of Anxiety k,  kmj (1) = between subject error. kmj  m = the effect of Tension m, (  ) km = Anxiety–Tension interaction m,

(  ) ki = the anxiety-time interaction effect  kmji (2) = within subject error.  j = the effect of time j, (  ) mi = the tension-time interaction effect (  ) kmi = the tension-time interaction effect kmji

The Anova Table

The Multivariate Model for a Repeated measures design

The Anova (univariate) Model y ij = the j th repeated measure on the i th subject =  +  j +  j +  ij where  = the mean effect,  j = the effect of subject i,  j = the effect of time j,  ij = random error.

Implications of The Anova (univariate) Model  j = the mean of y ij =   j + 0 =  +  j

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 Here Let denote a sample of n from the p-variate normal distribution with mean vector  and covariance matrix . Allows for arbitrary correlation structure amongst the repeated measures – y i1, y i2, …, y it

Test for equality of repeated measures

Repeated measures equal X repeated measures t 123 …

Let Then

Consider the data This is a sample of n from the (t – 1)-variate normal distribution with mean vector  and covariance matrix. The test for equality of repeated measures:

Hotelling’s T 2 test for equality of variables if H 0 is true than Thus we reject H 0 if F > F   with 1 = p – 1 and 2 = n – t + 1 has an F distribution with 1 = t – 1 and 2 = 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 T 2 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 surgery for 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

Analysis