Repeated measures: Approaches to Analysis Peter T. Donnan Professor of Epidemiology and Biostatistics.

Repeated measures:

Approaches to Analysis Peter T. Donnan

Professor of Epidemiology and Biostatistics

Objectives of sessionObjectives of session

• Understand what is meant by Understand what is meant by repeated measuresrepeated measures

• Be able to set out data in Be able to set out data in required formatrequired format

• Carry out mixed model analyses Carry out mixed model analyses with continuous outcome in with continuous outcome in SPSSSPSS

• Interpret the outputInterpret the output

• Understand what is meant by Understand what is meant by repeated measuresrepeated measures

• Be able to set out data in Be able to set out data in required formatrequired format

• Carry out mixed model analyses Carry out mixed model analyses with continuous outcome in with continuous outcome in SPSSSPSS

• Interpret the outputInterpret the output

Repeated MeasuresRepeated Measures

Repeated Measures arise when: Repeated Measures arise when: • In trials where baseline and several In trials where baseline and several

measurement of primary outcomemeasurement of primary outcome• Example - Trial of Chronic Example - Trial of Chronic

RhinosinusitisRhinosinusitis• Treatment usual care vs 2 weeks Treatment usual care vs 2 weeks

oral steroidsoral steroids• Measurements at 0, 2 , 10, 28 Measurements at 0, 2 , 10, 28

weeksweeks

Repeated Measures arise when: Repeated Measures arise when: • In trials where baseline and several In trials where baseline and several

measurement of primary outcomemeasurement of primary outcome• Example - Trial of Chronic Example - Trial of Chronic

RhinosinusitisRhinosinusitis• Treatment usual care vs 2 weeks Treatment usual care vs 2 weeks

oral steroidsoral steroids• Measurements at 0, 2 , 10, 28 Measurements at 0, 2 , 10, 28

weeksweeks

General PrinciplesGeneral Principles

Battery of methods to analyse Battery of methods to analyse Repeated Measures: Repeated Measures:

• Repeated use of significance Repeated use of significance testing at multiple time pointstesting at multiple time points

• ANOVA - ANOVA - ‘‘a dangerously wrong method’ - a dangerously wrong method’ - David David FinneyFinney

• MANOVAMANOVA• Multi-level models / mixed modelsMulti-level models / mixed models

Battery of methods to analyse Battery of methods to analyse Repeated Measures: Repeated Measures:

• Repeated use of significance Repeated use of significance testing at multiple time pointstesting at multiple time points

• ANOVA - ANOVA - ‘‘a dangerously wrong method’ - a dangerously wrong method’ - David David FinneyFinney

• MANOVAMANOVA• Multi-level models / mixed modelsMulti-level models / mixed models

Significance testing Significance testing at all time pointsat all time points

• Probably most common – multiple t-testsProbably most common – multiple t-tests• Least valid!Least valid!• Sometimes account for multiple testing Sometimes account for multiple testing

by adjusting p-value i.e. 0.05/k with k by adjusting p-value i.e. 0.05/k with k teststests

• Assumes that aim of study is to show Assumes that aim of study is to show significant difference at significant difference at everyevery time point time point

• Most studies aim to show OVERALL Most studies aim to show OVERALL difference between treatments and /or difference between treatments and /or reaching therapeutic target quickerreaching therapeutic target quicker

• PRIMARY HYPOTHESIS IS GLOBALPRIMARY HYPOTHESIS IS GLOBAL

• Probably most common – multiple t-testsProbably most common – multiple t-tests• Least valid!Least valid!• Sometimes account for multiple testing Sometimes account for multiple testing

by adjusting p-value i.e. 0.05/k with k by adjusting p-value i.e. 0.05/k with k teststests

• Assumes that aim of study is to show Assumes that aim of study is to show significant difference at significant difference at everyevery time point time point

• Most studies aim to show OVERALL Most studies aim to show OVERALL difference between treatments and /or difference between treatments and /or reaching therapeutic target quickerreaching therapeutic target quicker

• PRIMARY HYPOTHESIS IS GLOBALPRIMARY HYPOTHESIS IS GLOBAL

Repeated Measures: Repeated Measures: Summary MeasuresSummary Measures

• Post treatment meansPost treatment means• Mean change (post – baseline)Mean change (post – baseline)• ANCOVA or Multiple regression ANCOVA or Multiple regression

account for baseline as covariateaccount for baseline as covariate• Slope of changeSlope of change• Maximum value – with multiple Maximum value – with multiple

endpoints select highest value and endpoints select highest value and compare across treatmentscompare across treatments

• Area under the curve – difference Area under the curve – difference • Time to reach a target or peakTime to reach a target or peak

• Post treatment meansPost treatment means• Mean change (post – baseline)Mean change (post – baseline)• ANCOVA or Multiple regression ANCOVA or Multiple regression

account for baseline as covariateaccount for baseline as covariate• Slope of changeSlope of change• Maximum value – with multiple Maximum value – with multiple

endpoints select highest value and endpoints select highest value and compare across treatmentscompare across treatments

• Area under the curve – difference Area under the curve – difference • Time to reach a target or peakTime to reach a target or peak

Type of Analyses – Type of Analyses – Compare SlopesCompare Slopes

Compare slopes which summarise Compare slopes which summarise changechange

Acti

vit

yA

cti

vit

y

Baseline 3-months

Difference in slopes as summary measuree.g. β1-β2

Advice onlyPedometerControls β1

β2 β3

Type of Analyses – Type of Analyses – Area under the curveArea under the curve

Acti

vit

yA

cti

vit

y

Baseline 3-months

Difference in Area between treatment slopes as summary measure

Advice onlyPedometerControls

6-months

Simple approachSimple approach

• Basically just an extension of Basically just an extension of analysis of variance (ANOVA)analysis of variance (ANOVA)

• Pairing or matching of Pairing or matching of measurements on same unit measurements on same unit needs to be taken into accountneeds to be taken into account

• Method is General Linear Model Method is General Linear Model for continuous measures and for continuous measures and adjusts tests for correlationadjusts tests for correlation

• Basically just an extension of Basically just an extension of analysis of variance (ANOVA)analysis of variance (ANOVA)

• Pairing or matching of Pairing or matching of measurements on same unit measurements on same unit needs to be taken into accountneeds to be taken into account

• Method is General Linear Model Method is General Linear Model for continuous measures and for continuous measures and adjusts tests for correlationadjusts tests for correlation

Simple approachSimple approach

• But simple approach can only But simple approach can only use COMPLETE CASE analysis use COMPLETE CASE analysis where say wk 0 50, wk 2 47, where say wk 0 50, wk 2 47, wk10 36, wk 28 30wk10 36, wk 28 30

• Then analysis is on 30Then analysis is on 30• Assumes data is MCARAssumes data is MCAR• Better approach is MIXED Better approach is MIXED

MODEL which only assumes MODEL which only assumes MAR and uses all dataMAR and uses all data

• But simple approach can only But simple approach can only use COMPLETE CASE analysis use COMPLETE CASE analysis where say wk 0 50, wk 2 47, where say wk 0 50, wk 2 47, wk10 36, wk 28 30wk10 36, wk 28 30

• Then analysis is on 30Then analysis is on 30• Assumes data is MCARAssumes data is MCAR• Better approach is MIXED Better approach is MIXED

MODEL which only assumes MODEL which only assumes MAR and uses all dataMAR and uses all data

Organisation of data Organisation of data (Simple Approach)(Simple Approach)

Generally each unit in one row and repeated measures in separate Generally each unit in one row and repeated measures in separate columns columns

Unit Score 1 Score2 Score3 1 2.8 3.1 4.1

2 5.6 5.7 5.1

3 4.3 4.1 5.4

….

Generally each unit in one row and repeated measures in separate Generally each unit in one row and repeated measures in separate columns columns

Unit Score 1 Score2 Score3 1 2.8 3.1 4.1

2 5.6 5.7 5.1

3 4.3 4.1 5.4

….

Repeated Measures in SPSS: Set factor and number of levels

Within Within subject factorsubject factor

Within Within subject subject factor levelsfactor levels

Within subject Within subject factor namefactor name

Repeated Measures in SPSS: Enter columns of repeated measures

Use arrow to Use arrow to enter each enter each repeated repeated measure measure columncolumn

Between Between subject factor subject factor columncolumn

Repeated Measures in SPSS:

Select options

Use arrow to Use arrow to select display select display of means and of means and Bonferroni Bonferroni corrected corrected comparisonscomparisons

Select other Select other optionsoptions

Select a Select a plot of plot of means means of each of each within within subject subject treatmetreatmentnt

Repeated Measures in SPSS: Select options

Repeated Measures in SPSS: Output - Mean glucose uptake

Means for Means for four four treatmenttreatments and s and 95% CI95% CI

1 = Basal; 2 = Insulin; 1 = Basal; 2 = Insulin; 3 = Palmitate; 4 = 3 = Palmitate; 4 = Insulin+Palmitate Insulin+Palmitate

Estimates

Measure: treat

9.617 .911 7.732 11.502

13.026 1.155 10.636 15.415

7.538 .525 6.453 8.623

8.420 .685 7.004 9.837

f actor11

2

3

4

Mean Std. Error Lower Bound Upper Bound

95% Conf idence Interval

Basal Insulin Palmitate Basal Insulin Palmitate Insulin+Palmitate Insulin+Palmitate

Repeated Measures in SPSS: Output – Plot of Mean glucose uptake

Repeated Measures in SPSS: Output – Comparisons of

Mean glucose uptake

ComparisComparison of on of means means with with BonferronBonferroni i correctiocorrectionn

1 = Basal; 2 = Insulin; 1 = Basal; 2 = Insulin; 3 = Palmitate; 4 = 3 = Palmitate; 4 = Insulin+Palmitate Insulin+Palmitate

Pairwise Comparisons

Measure: treat

-3.409* .637 .000 -5.249 -1.569

2.079 .723 .051 -.007 4.164

1.196 .873 1.000 -1.325 3.717

3.409* .637 .000 1.569 5.249

5.488* 1.013 .000 2.563 8.413

4.605* 1.015 .001 1.677 7.534

-2.079 .723 .051 -4.164 .007

-5.488* 1.013 .000 -8.413 -2.563

-.882 .741 1.000 -3.022 1.257

-1.196 .873 1.000 -3.717 1.325

-4.605* 1.015 .001 -7.534 -1.677

.882 .741 1.000 -1.257 3.022

(J) f actor12

3

4

1

3

4

1

2

4

1

2

3

(I) factor11

2

3

4

MeanDif f erence

(I-J) Std. Error Sig.a

Lower Bound Upper Bound

95% Conf idence Interv al forDif f erence

a

Based on estimated marginal means

The mean dif f erence is signif icant at the .05 lev el.*.

Adjustment f or multiple comparisons: Bonf erroni.a.

Repeated Measures:Repeated Measures:ConclusionConclusion

• Energy intake significantly Energy intake significantly higher with insulin compared higher with insulin compared to all other treatmentsto all other treatments

• Addition of palmitate Addition of palmitate removes this effectremoves this effect

• Energy intake significantly Energy intake significantly higher with insulin compared higher with insulin compared to all other treatmentsto all other treatments

• Addition of palmitate Addition of palmitate removes this effectremoves this effect

Organisation of data Organisation of data (Mixed Model)(Mixed Model)

Note most other programs and Mixed Note most other programs and Mixed Model analyses require ONE row per Model analyses require ONE row per measurement measurement

Note most other programs and Mixed Note most other programs and Mixed Model analyses require ONE row per Model analyses require ONE row per measurement measurement

Unit Score

1 2.8

1 3.1

1 4.1

2 5.6

2 5.7

2 5.1

3 4.3

Etc…….

Repeated Measures in Repeated Measures in SPSSSPSS

• Mixed Model in SPSS is: Mixed Model in SPSS is: • Mixed Model Mixed Model

LinearLinear

• Hence can ONLY be used for Hence can ONLY be used for continuous outcomes.continuous outcomes.

• For binary need other For binary need other Software e.g. SASSoftware e.g. SAS

• Mixed Model in SPSS is: Mixed Model in SPSS is: • Mixed Model Mixed Model

LinearLinear

• Hence can ONLY be used for Hence can ONLY be used for continuous outcomes.continuous outcomes.

• For binary need other For binary need other Software e.g. SASSoftware e.g. SAS

Repeated Measures in SPSS: Mixed: Set within subject factor

Repeated Repeated Within Within subject subject factor factor

Within subject Within subject factor namefactor name

Repeated Measures in SPSS: Enter columns of repeated measures

Use arrow to Use arrow to enter enter subjects and subjects and repeated repeated measure measure columncolumn

Choose Choose covariance covariance type = AR (1)type = AR (1)

Repeated Measures in SPSS:

Select options

Add Add dependentdependentTreatment Treatment factorfactorAnd And covariatescovariates

Select other Select other optionsoptions

Add Add effects effects as fixedas fixedAnd And Main Main EffectsEffects

Repeated Measures in SPSS: Select options

Repeated Measures in SPSS: Output -

Overall Overall test for test for treatmetreatment p = nt p = 0.0240.024

Type III Tests of Fixed Effectsa

1 62.930 27.360 .000

1 60.960 5.398 .024

1 62.995 .138 .712

1 61.041 .020 .888

SourceIntercept

Treatment

age

sexnum

Numerator dfDenominator

df F Sig.

Dependent Variable: polypgradetotv1.a.

Repeated Measures in SPSS: Output –

Estimates of Fixed Effectsb

2.920968 .631679 62.229 4.624 .000 1.658353 4.183584

.740800 .318851 60.960 2.323 .024 .103210 1.378390

0a 0 . . . . .

.004508 .012153 62.995 .371 .712 -.019778 .028794

.046545 .329233 61.041 .141 .888 -.611788 .704879

ParameterIntercept

[Treatment=0]

[Treatment=1]

age

sexnum

Estimate Std. Error df t Sig. Lower Bound Upper Bound

95% Confidence Interval

This parameter is set to zero because it is redundant.a.

Dependent Variable: polypgradetotv1.b.

Mixed Model Repeated Mixed Model Repeated Measures:ConclusionMeasures:Conclusion

• Use of Mixed Models ensures Use of Mixed Models ensures all data used assuming data all data used assuming data is MAR and so more efficient is MAR and so more efficient in presence of missing data in presence of missing data (if MAR) than the simple (if MAR) than the simple repeated measuresrepeated measures

• Other software e.g. SAS can Other software e.g. SAS can also handle binary outcome also handle binary outcome datadata

• Use of Mixed Models ensures Use of Mixed Models ensures all data used assuming data all data used assuming data is MAR and so more efficient is MAR and so more efficient in presence of missing data in presence of missing data (if MAR) than the simple (if MAR) than the simple repeated measuresrepeated measures

• Other software e.g. SAS can Other software e.g. SAS can also handle binary outcome also handle binary outcome datadata

Sample size for repeated Sample size for repeated MeasuresMeasures

Number in each arm = Number in each arm =

Where r = number of post treatment Where r = number of post treatment measuresmeasures

p = number of pre-treatment measures p = number of pre-treatment measures often 1often 1

Frison&Pocock Stats in Med1992; 11: 1685-1704Frison&Pocock Stats in Med1992; 11: 1685-1704

Sample size for repeated Sample size for repeated MeasuresMeasures

Number in each arm = Number in each arm =

Where Where σσ = between treatment = between treatment variancevariance

δδ = difference in treatment means = difference in treatment means

ρρ = pairwise correlation (often 0.5 – = pairwise correlation (often 0.5 – 0.7)0.7)

Sample size for repeated Sample size for repeated MeasuresMeasures

Efficiency increase with number of Efficiency increase with number of measurements (r)measurements (r)

(z(zαα +z +zββ))22 = 7.84 for 5% sig and 80% = 7.84 for 5% sig and 80% powerpower

Methods assumes compound Methods assumes compound symmetry – often wrong but symmetry – often wrong but reasonable for sample sizereasonable for sample size

Example: Sample size for Example: Sample size for repeated Measuresrepeated Measures

For r = 3 post-measures, For r = 3 post-measures, correlation=0.7, p=1,correlation=0.7, p=1,(z(zαα +z +zββ))22 = 7.84 for 5% sig. and 80% = 7.84 for 5% sig. and 80% powerpowerSay Say δδ=0.5=0.5σσ then….. then…..

Example: Sample size for Example: Sample size for repeated Measuresrepeated Measures

Which gives n = 19 in each arm Which gives n = 19 in each arm with 80% power and 5% with 80% power and 5% significance levelsignificance level

ReferencesReferences

Repeated Measures in Clinical Trials: Analysis using Repeated Measures in Clinical Trials: Analysis using mean summary statistics and its implications for mean summary statistics and its implications for design. Statist Med 1992; 11: 1685-1704.design. Statist Med 1992; 11: 1685-1704.

Field A. A bluffers guide to …Sphericity. Field A. A bluffers guide to …Sphericity. J Educational Statistics 13(3): 215-226.J Educational Statistics 13(3): 215-226.

Pallant J. SPSS Survival Manual 3Pallant J. SPSS Survival Manual 3rdrd ed, Open University ed, Open University Press, 2007.Press, 2007.  Field A. Field A. Discovering Statistics using SPSS for WindowsDiscovering Statistics using SPSS for Windows. . Sage publications, London, 2000.Sage publications, London, 2000.  Puri BK. SPSS in practice. An illustrated guide. Arnold, Puri BK. SPSS in practice. An illustrated guide. Arnold, London, 2002.London, 2002.

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