analysing time-course data kathyruggiero...

Kathy Ruggiero [email protected]

Department of Statistics

Analysing correlated time-course data

Analysing correlated time-course data

9 July 2015

Kathy Ruggiero and Kevin Chang


Time-course experiments• Used to investigate temporal changes in a

measured response

Time (minutes)


Time-course experiments

• Important to know how measurements across time are obtained– All samples from a different experimental unit (EU) Independent observations

– Multiple samples from each EU Correlated observations

– Repeated measures experiments: Each EU contributes samples across all of the same time points

– Longitudinal studies: Each EU contributes samples through time, although not necessarilythe same time points


Main focus today

Repeated measures experiments

• Example: Rat lymph metabolomics experiment– Description of the statistical design– Statistical model and analysis for:

1. One time-point2. Time-course, ignoring data are correlated3. Time-course, accounting for correlated repeated measures


Phase 1 part of experiment

• Three surgical conditionings of lymph– Sham– IRI (Ischaemia Reperfusion Injury)– IRI + IPC (Ischaemic Pre-Conditioning)

• Lymph collected at time– 0 (baseline), 10, 30, 60 and 240



Baseline

10 20 80 320

I R

I R I R

Sham

IRI

IPC

I = Ischaemia; R = Reperfusion

n = 18 rats per surgical conditioning (disease group)



• GC-MS metabolomics analysis• 16 samples analysed per day• Statistical design took this “batch effect”

into account• We will ignore this today, but…

It does matter!


One time-point: 320 minutes

Statistical linear model

response = systematic component + error component

• response = one numerical outcome

• systematic component = describes relationship between experimental conditions (e.g. surgeries) and response– May be described by one, or more, explanatory variables

• error component = variation in response that is not explained by systematic component



Linear model for hexanoic acid (C6_0)

log(hexacid) = systematic component + error component

• response = log-abundance of hexanoic acid

• systematic component = Disease (surgical conditioning)

• error component = biological variation + measurement error



What does the raw data look like?



What about the log-transformed data?






• systematic component = Disease (surgical conditioning)


Df Sum Sq Mean Sq F value Pr(>F) Disease 2 0.63 0.3151 1 0.375 Residuals 51 16.07 0.3150

Analysis of Variance (ANOVA)



ANOVA

• Think of it is a “generalized” two-sample t-test• It assumes:

– Observations are independent of one another– Data are normally distributed– Disease groups have equal variances

• Conclusion: No evidence of a difference in log(hexacid) abundance between surgeries




ANOVA

• Disease Mean Sq (MS)?– Variation between Disease group means

+ (Biological variation + Measurement error)

• Residuals MS?– (Biological variation + Measurement error)

• F value = Ratio of Disease MS to Residuals MS



Time-course, ignoring correlation

Factorial experiment• Two or more sets of experimental conditions• A factor is a set of experimental conditions• The values of a factor are called its levels• Our time-course experiment:

– Disease status (factor): Sham, IRI, IPC (levels)– Time (factor): 0,10, 20, 80, 320 minutes (levels)



Factorial experiment• Study all factor-level (treatment) combinations

DiseaseTime (minutes)

0 10 20 80 320Sham S.0 S.10 S.20 S.80 S.320

IRI IRI.0 IRI.10 IRI.20 IRI.80 IRI.320IPC IPC.0 IPC.10 IPC.20 IPC.80 IPC.320



Factorial experiment• Study all factor-level (treatment) combinations

0 min 320 min

Mea

n lo

g-A

bund

ance

ShamIRI

.

.

.

Estimated from data

Interaction between Disease and Time

80min

Disease group:

.



0 min 320 min

Mea

n lo

g-A

bund

ance

ShamIRI

.

.

.

Estimated from data

Interaction between Disease and Time

80min

Disease group:

.

Interaction means the magnitude of the difference between IRI and Sham depends on the time at which sample was taken



Statistical linear model

response = systematic component + error component

• response = one numerical outcome

• systematic component = describes relationship between experimental conditions (e.g. surgeries and times) and response– May be described by one, or more, explanatory variables

• error component = variation in response that is not explained by systematic component






• systematic component = Disease + Time + Disease:Time



Actual hexanoic acid time-course data

Log-transformed abundances

Each line represents one rat






• systematic component = Disease + Time + Disease:Time




Two-way ANOVA

• Conclusion: Strong time effect (p < 0.0001) but little statistical evidence of an interaction between Disease and Time (p = 0.089).

But observations are not independent!


Time-course, accounting for correlation




• systematic component can be partitioned further, i.e.

syst. comp = explanatory component + structural component– explanatory component = Disease + Time + Disease:Time– structural component = Rat (biological variation)

• error component = variation in response that is not explained by systematic component (other, not biological, variation)



Systematic component• Explanatory component

– Main effects – FIXED effects Disease (average over levels of Time) Time (average over levels of Disease) Disease:Time interaction effect

• Structural component– Two RANDOM effects Rat (between rats variation) in addition to

Error component (within rats variation)

• Called a Linear MIXED Model (LMM)



Linear MIXED model for hexanoic acid

log(hexacid) = Disease + Time + Disease:Time+ Rat+ error component

• Also known as:

– Linear mixed effects

– Multilevel (or hierarchical) mixed models

• Provide a very flexible class of statistical models



Data tableDisease Time Rat AbundanceSham 0 1 9.40Sham 10 1 8.83Sham 20 1 10.33Sham 80 1 10.49Sham 320 1 9.74IRI 0 2 10.98IRI 10 2 7.92IRI 20 2 9.37IRI 80 2 8.69IRI 320 2 11.31IPC 0 3 7.01IPC 10 3 9.29

Not actual data



Multilevel (or multi-stratum) ANOVA(Can think of it as a generalized paired t-test)

• Conclusion: There is statistical evidence of an interaction between Disease and Time (p = 0.009).



Profiles of mean log-Abundance

Each line represents disease group meanError bars are 95% confidence intervals


Let’s compare the two analyses

Bottom half of multi-stratum ANOVA table

Complete two-way ANOVA table



Why the difference?

Complete two-way ANOVA table



Why the difference?

• The linear mixed model enables us to partition the variation between rats and the variation within rats (error component)


Experimental units of different sizes

The Rat (main plot)• Largest experimental unit to which an individual

treatment (Disease) is independently applied– Samples taken across time-points within the same rat are

subjected to homogeneous conditions

The samples across time-points (subplot)• Smaller experimental unit, derived by subdivision of the

main plot, to which a different individual treatment (Time) is independently applied

Sometimes called a split-plot in time


Pairwise comparisons of means

Comparison Difference SED LSD Lower Upper Ratio Lower Upper p‐valueiri:1‐sha:1 ‐0.2462 0.2006 0.3955 ‐0.6417 0.1493 ‐1.2791 ‐1.8996 ‐0.8613 0.2212iri:2‐sha:2 0.0618 0.2006 0.3955 ‐0.3337 0.4573 1.0638 0.7163 1.5798 0.7583iri:3‐sha:3 0.0595 0.2006 0.3955 ‐0.3360 0.4550 1.0613 0.7146 1.5762 0.7670iri:4‐sha:4 0.3408 0.2006 0.3955 ‐0.0547 0.7363 1.4061 0.9468 2.0882 0.0908iri:5‐sha:5 ‐0.1915 0.2006 0.3955 ‐0.5870 0.2040 ‐1.2110 ‐1.7985 ‐0.8154 0.3409iri:1‐ipc:1 ‐0.0852 0.2006 0.3955 ‐0.4807 0.3103 ‐1.0889 ‐1.6172 ‐0.7332 0.6715iri:2‐ipc:2 ‐0.2961 0.2006 0.3955 ‐0.6915 0.0994 ‐1.3445 ‐1.9968 ‐0.9053 0.1415iri:3‐ipc:3 0.2244 0.2006 0.3955 ‐0.1711 0.6199 1.2515 0.8427 1.8587 0.2646iri:4‐ipc:4 ‐0.1397 0.2006 0.3955 ‐0.5352 0.2558 ‐1.1500 ‐1.7078 ‐0.7743 0.4868iri:5‐ipc:5 0.0625 0.2006 0.3955 ‐0.3330 0.4580 1.0644 0.7167 1.5808 0.7558ipc:1‐sha:1 ‐0.1610 0.2006 0.3955 ‐0.5564 0.2345 ‐1.1746 ‐1.7445 ‐0.7909 0.4233ipc:2‐sha:2 0.3579 0.2006 0.3955 ‐0.0376 0.7534 1.4303 0.9631 2.1241 0.0759ipc:3‐sha:3 ‐0.1649 0.2006 0.3955 ‐0.5604 0.2306 ‐1.1792 ‐1.7513 ‐0.7940 0.4121ipc:4‐sha:4 0.4805 0.2006 0.3955 0.0850 0.8760 1.6169 1.0888 2.4014 0.0175ipc:5‐sha:5 ‐0.2539 0.2006 0.3955 ‐0.6494 0.1416 ‐1.2891 ‐1.9144 ‐0.8680 0.2070iri:1‐iri:2 ‐0.5877 0.1636 0.3225 ‐0.9102 ‐0.2651 ‐1.7998 ‐2.4849 ‐1.3036 0.0004iri:2‐iri:3 0.1536 0.1636 0.3225 ‐0.1690 0.4761 1.1660 0.8445 1.6098 0.3489iri:3‐iri:4 ‐0.8436 0.1636 0.3225 ‐1.1662 ‐0.5211 ‐2.3247 ‐3.2096 ‐1.6838 0.0000iri:4‐iri:5 0.1955 0.1636 0.3225 ‐0.1271 0.5180 1.2159 0.8807 1.6787 0.2335ipc:1‐ipc:2 ‐0.7985 0.1636 0.3225 ‐1.1211 ‐0.4760 ‐2.2223 ‐3.0682 ‐1.6096 0.0000ipc:2‐ipc:3 0.6740 0.1636 0.3225 0.3515 0.9966 1.9621 1.4212 2.7090 0.0001ipc:3‐ipc:4 ‐1.2077 0.1636 0.3225 ‐1.5303 ‐0.8852 ‐3.3459 ‐4.6194 ‐2.4234 0.0000ipc:4‐ipc:5 0.3977 0.1636 0.3225 0.0751 0.7202 1.4884 1.0780 2.0549 0.0159sha:1‐sha:2 ‐0.2797 0.1636 0.3225 ‐0.6023 0.0428 ‐1.3228 ‐1.8263 ‐0.9581 0.0888sha:2‐sha:3 0.1513 0.1636 0.3225 ‐0.1713 0.4738 1.1633 0.8426 1.6061 0.3562sha:3‐sha:4 ‐0.5623 0.1636 0.3225 ‐0.8849 ‐0.2398 ‐1.7547 ‐2.4227 ‐1.2710 0.0007sha:4‐sha:5 ‐0.3368 0.1636 0.3225 ‐0.6593 ‐0.0142 ‐1.4004 ‐1.9335 ‐1.0143 0.0408

95% Confidence LimitsLog‐Abundance

95% Confidence LimitsBack‐transformed



• Back-transform to original scale– iri:4 – sha:4: Ratio = exp(0.3408) = 1.401 (Differences ≥ 0 Ratios ≥ 1)– iri:1 – sha:1: Ratio = exp(-0.2462) = 0.7817 (Differences 0 0 < Ratios 1)

Too small to see on graphs, so we calculate: -1/Ratio = -1/exp(-0.2462) = -1.279 = sha:1 / iri:1 (Ratios -1)

Comparison Difference SED LSD Lower Upperiri:1‐sha:1 ‐0.2462 0.2006 0.3955 ‐0.6417 0.1493iri:2‐sha:2 0.0618 0.2006 0.3955 ‐0.3337 0.4573iri:3‐sha:3 0.0595 0.2006 0.3955 ‐0.3360 0.4550iri:4‐sha:4 0.3408 0.2006 0.3955 ‐0.0547 0.7363iri:5‐sha:5 ‐0.1915 0.2006 0.3955 ‐0.5870 0.2040

95% Confidence LimitsLog‐Abundance



Comparisoniri:1‐sha:1iri:2‐sha:2iri:3‐sha:3iri:4‐sha:4iri:5‐sha:5

Ratio Lower Upper p‐value‐1.2791 ‐1.8996 ‐0.8613 0.22121.0638 0.7163 1.5798 0.75831.0613 0.7146 1.5762 0.76701.4061 0.9468 2.0882 0.0908‐1.2110 ‐1.7985 ‐0.8154 0.3409

95% Confidence LimitsBack‐transformed



Profiles of mean log-Abundance

Each line represents disease group meanError bars are 95% confidence intervals


Why is repeated measures ANOVA different…

…to two-way ANOVA ignoring correlated observations?• ANOVA assumes independent errors• Split-plot in time ANOVA assumes

– Observations on different experimental units are uncorrelated (i.e. independent of one another)

– Observations between all pairs of observations on the same experimental unit have the same covariance

• In repeated measures designs– Adjacent observations are likely to be more highly correlated

than more distant observations– Problem involves the covariance structure

i.e. error variance-covariance matrix


What else could we have done?• Adjust for each rat’s baseline levels of hexanoic

acid• Allow for non-constant correlations between

pairs of time-points– Variance models

• Model the shapes of the profiles– Random coefficient regression– Smoothing splines


What else could we have done?• Multivariate linear mixed model

– Simultaneous analysis of data from all metabolites– Each metabolite draws on information from other

metabolites– Each metabolite has its own variance

• Beyond the scope of this talk


The End

Thank you!


Principles of design: Structure in the experimental units


Disease status main effect• Comparisons between levels of Disease status,

averaging over levels of Organ• Which source(s) of non-systematic variation contribute to

protein abundance for each Disease status?Organ main effect• Comparisons between levels of Organ, averaging over

levels of Disease status• Which source(s) of non-systematic variation contribute to

protein abundance for each Organ?




Disease:Organ interaction• There are four (factorial) treatment groups:

– D:I, D:O, H:I and H:O

• Which sources of non-systematic variation contribute to these means?

• And, which sources of non-systematic variation contributed to these two pairs of comparisons between means?

1. D:I-D:O and H:I-H:O2. D:O-H:O and D:I-H:I




Disease:Organ interaction• Sources of variation contributing to these comparisons?

1. D:I-D:O and H:I-H:O2. D:O-H:O and D:I-H:I

1. Between subplots within the same main plot Variation in within Animals + measurement error

2. Between subplots in different main plots Variation between- and within-Animals + measurement error



Linear MIXED model for hexanoic acid

log(hexacid) = Disease + Time + Disease:Time+ Rat+ error component

• Comparisons between Diseases are between rats

• Comparisons between Times are within rats

• Comparisons between Times for the same Disease are within rats

• Comparisons between Diseases for the same time-point involve a combination of between and within rats

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