Randomized Block and Repeated Measures Designs. Block Designs In the Types of Studies presentation we discussed the use of blocking to control for a source

Randomized Block and Randomized Block and Repeated Measures Repeated Measures

DesignsDesigns

Block DesignsBlock Designs

• In the In the Types of StudiesTypes of Studies presentation we presentation we discussed the use of discussed the use of blockingblocking to control to control for a source of variation we think may for a source of variation we think may effect the response (e.g. slope in field, day, effect the response (e.g. slope in field, day, subject, etc.).subject, etc.).

• When the When the blocks blocks are subjects we often are subjects we often times refer to the experiment as a times refer to the experiment as a repeated measures design because we are repeated measures design because we are taking taking repeated measurements on the repeated measurements on the same subjectssame subjects..

Example: Measuring Blood Example: Measuring Blood Serum Serum

The goal of this study was to determine if 4 different The goal of this study was to determine if 4 different methods for determining blood serum levels differ in terms methods for determining blood serum levels differ in terms of the readings they give. Suppose we plan to have 6 of the readings they give. Suppose we plan to have 6 readings from each method which we will then use to make readings from each method which we will then use to make our comparisons. One approach we could take would be to our comparisons. One approach we could take would be to find 24 volunteers and randomly allocate six subjects to each find 24 volunteers and randomly allocate six subjects to each method and compare the readings obtained using the four method and compare the readings obtained using the four methods. methods. (Note: this is a (Note: this is a completely randomized designcompletely randomized design). ).

There is one There is one majormajor problem with this approach, what problem with this approach, what is it? is it?

Any significant differences between the methods could be due to differences between the subjects within the methods and NOT the methods themselves !!!

Example: Measuring Blood Example: Measuring Blood SerumSerum

Instead of using a completely randomized design Instead of using a completely randomized design it would clearly be better to use each method on it would clearly be better to use each method on the same subject, the same subject,

This removes subject to subject variation from This removes subject to subject variation from the results and will allow us to get a clearer the results and will allow us to get a clearer picture of the actual differences in the methods. picture of the actual differences in the methods. Also if we truly only wish to have 6 readings for Also if we truly only wish to have 6 readings for each method, this approach will only require the each method, this approach will only require the use of 6 subjects (i.e. blood samples) versus the use of 6 subjects (i.e. blood samples) versus the 24 subjects the completely randomized approach 24 subjects the completely randomized approach requires, thus reducing the “cost” of the requires, thus reducing the “cost” of the experiment. Also if we still used 24 subjects the experiment. Also if we still used 24 subjects the number of replicates number of replicates 6 6 24 24, which , which increases increases POWERPOWER and PRECISIONand PRECISION (i.e. smaller margins of (i.e. smaller margins of error).error).

i.e. SUBJECTS = BLOCKS.


• The design described on the The design described on the previous slide is called a previous slide is called a randomized complete block randomized complete block (RCB) design(RCB) design with subjects/blood with subjects/blood sample as blocks.sample as blocks.

• The results are shown below:The results are shown below:

13001144134011316

5024715204635

7905837035814

8045917506323

12301002115210352

5023914353601

4321Subject

METHOD

BLOCKS Methods are used in a random order for each blood sample.


• An An incorrect analysisincorrect analysis would treat would treat these results as coming from a these results as coming from a completely randomized design and completely randomized design and we would compare readings across we would compare readings across method only using method only using one-way one-way ANOVAANOVA..

Clearly between group variation is very small relative to within group variation. Not surprisingly we fail to find method differences(p = .7871).


• What is lost in the previous analysis What is lost in the previous analysis is that the same blood samples are is that the same blood samples are being used for each method.being used for each method.

Clearly much of the within group variation is due to subject differences! Our analysis needs to consider this source of variation as well!

After removing the subject to subject variation the differences between the methods look like this. Clearly the methods differ!

Model and AssumptionsModel and Assumptions• The The modelmodel for the observed response is given for the observed response is given

by: by:

• We assume that the errors are normally We assume that the errors are normally distributed with constant variance.distributed with constant variance.

• This implies that the populations being This implies that the populations being sampled are also sampled are also normally distributednormally distributed with with equal variancesequal variances!!

ijjiijx

error random

,...,1 effect, block

1 effect, treatment

block and treatmentfrom valueobserved

ij

j

bjj

,...,k i i

jix

i

ij

HypothesesHypothesesHHoo: treatment means are all equal: treatment means are all equal

HHAA: at least two treatment means differ: at least two treatment means differ

For the blood serum level example we For the blood serum level example we can state things much more precisely…can state things much more precisely…

HHoo: the four methods give the same mean serum : the four methods give the same mean serum

level when measuring the same blood samples.level when measuring the same blood samples.

HHAA: at least two methods differ on average : at least two methods differ on average

when measuring the same blood samples. when measuring the same blood samples.

SS and MS FormulaeSS and MS Formulae

)1)(1(

1 )(

1 )(

1or 1 )(

1

2

1

2

1 1

2

bkdf

SSBlockSSTreatmentSSTotalSSError

bdfxxkSSBlock

kdfxxbSSTreatment

NkbdfxxSSTotal

b

jj

k

ii

k

i

b

jij

The Mean Squares for Treatment, Block and Error are the SS divided by their df, i.e. MSeffect = SSeffect/dfeffect

Test Statistic for Treatment Test Statistic for Treatment EffectEffect

)1)(1( and 1

on distributi-F ~

rdenominatonumerator

bkdfkdf

MS

MSF

Error

Treato

Fo

p-value

Large values of Fo support the alternative and correspond to small p-values.


• Data FormatsData Formats JMPJMP SPSSSPSS

Subject and Method must be nominal!


Blocking factor goes here.

Factor of interest goes here.

In JMP - Analyze > Fit Y by X


• ANOVA TableANOVA Table

SSTre

at

+ SSBlock

+ SSError

= SSTotal

MSTrea

tMSErr

or

= Fo

The methods significantly differ when testing the same blood sample (p < .0001).

Don’t test blocks because they are not randomized.

Subject to subject variation is LARGE! Blocking was effective!!


Use Tukey’s for Pair-wise Comparisons Use Tukey’s for Pair-wise Comparisons of the Methods… of the Methods… Compare Means > All Compare Means > All Pairs, Tukey’s HSDPairs, Tukey’s HSD

Methods 2 and 4 differ significantly from methods 1 and 3. Also methods 2 and 4 do not significantly differ nor do methods 1 and 3.

These intervals can be used to quantify differences between the methods when measuring the same blood sample.


Factors of interest go here.

In JMP - Fit Model Approach (we will use this lots later)

Make sure these boxes are as shown. We will change them later when we consider multiple and logistic regression.Respons

e

For a JMP demo of this and the previous approaches click here.


SPSS: Analyze > General Linear Model > SPSS: Analyze > General Linear Model > Univariate…Univariate…

Dependent Variable – Serum Level

Fixed Factors: Subject (blocks)Method (factor of

interest) Click here to specify the model

Click Custom and add both Subject and Method to be in the model.

Click here to specify Tukey comparisons of the different Methods.

Repeated Measures Repeated Measures AnalysisAnalysis

• The previous example can also be thought The previous example can also be thought of as a repeated measures design because of as a repeated measures design because repeated measurements of the same blood repeated measurements of the same blood sample were taken, each coming using a sample were taken, each coming using a different method. different method.

• Another example would be a situation Another example would be a situation where different treatments are used on the where different treatments are used on the same subjects, obviously this is not always same subjects, obviously this is not always possible. possible.

• Yet another common situation is where the Yet another common situation is where the repeated repeated measurements are taken over measurements are taken over time.time.

Example 2: ESR for Arthritic Example 2: ESR for Arthritic PatientsPatients

• In a study of arthritic patients their In a study of arthritic patients their response to sulfasalazine was response to sulfasalazine was investigated. investigated.

• The erythocyte sedimentation rate The erythocyte sedimentation rate (ESR) was measured at baseline (0 (ESR) was measured at baseline (0 months) and again 3, 6, and 12 months months) and again 3, 6, and 12 months after treatment with sulfasalazine. after treatment with sulfasalazine.

Q: Is there evidence that the mean ESR Q: Is there evidence that the mean ESR changes over time? If so, researchers changes over time? If so, researchers wish to compare mean ESR for the wish to compare mean ESR for the three follow-up periods to baseline. three follow-up periods to baseline.


PatientPatient

(blocks(blocks))

0 0 MonthMonthss

3 3 MonthMonthss

6 6 MonthMonthss

12 12 MonthsMonths

11 8.838.83 3.743.74 3.163.16 2.832.83

22 6.636.63 2.242.24 2.002.00 3.003.00

33 5.575.57 4.244.24 2.242.24 2.452.45

44 5.005.00 3.323.32 1.411.41 2.002.00

55 9.069.06 3.163.16 2.002.00 1.411.41

66 6.936.93 4.244.24 2.002.00 1.731.73

77 7.077.07 5.745.74 5.105.10 3.163.16

88 3.873.87 3.463.46 4.004.00 3.743.74

Example 2: ESR for Example 2: ESR for Arthritic PatientsArthritic Patients

Mean ESR changes over time following treatment

The mean ESR is significantly lower than baseline for all three follow-up periods.

Example 2: ESR for Example 2: ESR for Arthritic PatientsArthritic Patients

There is no evidence that the mean ESR’s past baseline significantly differ.

Use these Tukey intervals to quantify changes in mean ESR from baseline.


• The response ESR appears to be normally distributed for each time period.

• The equality of variance tests do not suggest that the variation in the ESR’s differ over time (p > .10).

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Randomized Block and Repeated Measures Designs. Block Designs In the Types of Studies presentation we discussed the use of blocking to control for a source