A19 Research Internship Results Charlie Cloutier-Langevin & - - PowerPoint PPT Presentation
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion A19 Research Internship Results Charlie Cloutier-Langevin & Julien Corriveau-Trudel Universit e de Sherbrooke Tuesday, December
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Charlie Cloutier-Langevin & Julien Corriveau-Trudel
Universit´ e de Sherbrooke
Tuesday, December 10th 2019
Supervisors: F´ elix Camirand Lemyre, Alan A. Cohen, Nancy Presse Collaborators: V´ eronique Legault, Val´ erie Turcot, Alistair Senior
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Context New approach to study aging through Physiological Dysregulation (Phys. Dys.) with the Mahalanobis distance [4][5][9] Advent of the NuAge Dataset Task Study the potential relationship between nutrients intake of an individual and the deviance of his or her biological profile from a reference population.
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
The NuAge dataset in numbers: 1 754 elderly women and men, from age 68 to 81; 6 586 visits, between 1 and 4 visits per person; 23 186 24h recalls, 1 to 3 recalls per timepoints, for 5 timepoints; 188 medical variables and 43 nutritional variables; 364 421 missing values out of 1 238’168 entries (29.4%); Each year, a set of of biological, nutritional, functional, medical, and social traits is measured for each participant.
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Different physiological systems considered:
1 Oxygen Transport 2 Liver/Kidney functions 3 Hematopoiesis 4 Micronutrients 5 Lipids
System information comes from a previous study on how to regroup these biomarkers and the effects of using different subsets
Global Phys. Dys. score has been computed, which is the sum
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
1
Transformation to normality
2
Longitudinal imputation Intrapolation Extrapolation Results
3
Clustering
4
Measurement error and regression Additive Error Model CoCoLasso Deconvolution Nonparametric Regression
5
Conclusion
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Statistical methods + normality = Better performances Classic transformations As best transform provided by V´ eronique Legault Examples: sqrt(), log(), exp() Parametric transformation methods Provide an accurate and simplified process for transformation. Parametric transformation methods BoxCox[1] Yeo-Johnson[12] Manly[8]
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Best transformation? ... = ⇒ BoxCox transformation! BoxCox transformation (Box & Cox, 1964) Parametric power transformation Strictly positive observation values λ ǫ [-5, 5] Defined as: y (λ) :=
λ , if λ = 0
ln(y), if λ = 0
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Since the BoxCox transformation requires strictly positive data, Box & Cox proposed a shifted modification. Shifted BoxCox transformation Parametric power transformation λ1 ǫ [-5, 5] New shifting parameters λ2 Defined as: y(λ)
i
:= (yi+λ2)λ1−1
λ1
, if λ1 = 0 ln(yi + λ2), if λ1 = 0 Remark Shift the data by λ2 = 1 does not impact the result because it will not impact the variable distribution
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Let’s compare best transform and BoxCox for 2 examples EXAMPLE 1 Biomarker name: Creatinine Name in data set: CREAT Best transform applied: log(x) BoxCox λ applied: λ = -0.5 (equivalent to
1 √x )
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Creatinine (CREAT) No transformation histogram: Right skewed
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Creatinine (CREAT) Best transform transformation histogram: Approaching normality
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Creatinine (CREAT) BoxCox transformation histogram: Almost normal
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Creatinine (CREAT) No transformation Q-Q plot: No line shape
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Creatinine (CREAT) Best transform transformation Q-Q plot: Ends does not fit the line
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Creatinine (CREAT) BoxCox transformation Q-Q plot: Approaching a line shape
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Creatinine (CREAT) Shapiro-Wilk[11] normality test comparison: H0: Data are from a normally distributed population No transformation: p-value < 2.2e-16 = ⇒ Reject H0 Best transform: p-value = 3.844e-15 = ⇒ Reject H0 BoxCox: p-value = 0.004968 = ⇒ Reject H0
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
EXAMPLE 2 Biomarker name: Weight Name in data set: weight Best transform applied: x (no transformation) BoxCox λ applied: λ = 0.1
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Weight No transformation (& Best transform) histogram: Right skewed
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Weight BoxCox transformation histogram: Almost normal
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Weight No transformation (& Best transform Q-Q plot: No line shape
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Weight BoxCox transformation Q-Q plot: Almost a line shape
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Weight Shapiro-Wilk[11] normality test comparison: H0: Data are from a normally distributed population No transformation: p-value < 2.2e-16 = ⇒ Reject H0 BoxCox: p-value = 0.005793 = ⇒ Reject H0
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
General results 119 continuous biomarker variables transformed Average difference of λ1 between Best transform and BoxCox = 0.7831933 The only biomarker that was not transformed by BoxCox is lipids tot, compare to 75 for Best transform Limitations In most case, we still have to reject normality. BoxCox search for the best power transformation in NuAge data set, it could vary on other data sets. Not necessarily the best results for every variable, but the best overall
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
What is imputation? Imputation is the act of substituting missing values. Replacing NAs by a plausible value. Why imputation? Having more data means stronger statistical model, since dataimputed > datainitial, and a consistent statistical model is one that is ”stronger” when n → ∞.
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Risk: Introducing bias to subsequent statistical estimations. Mean imputation Imputing variable X with the mean of non missing values of X attenuates correlation in the data. Linear regression imputation On the opposite, imputing with linear regression based on non missing values of X will strengthen correlation.
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
In NuAge biomarkers: 1 238 168 entries, with 364 421 of them missing. In the light of our objectives: Conservative approach ⇒ only impute the necessary, without negatively impacting the computation of Mahalanobis Distance(MHBD).
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Definition Let X be the set of n observations in a p dimensional space, with mean µ and covariance matrix Σ. Let xi be an observation from this set. The Mahalanobis Distance is defined as follow: DM(x) =
A generalization of measuring ”how many standard deviation the
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Using the population = ⇒ introducing correlation between individuals (something we don’t want.) To dodge this effect, we relied on longitudinal data instead. Time reference: The first visit.
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Using all variables to impute data? We rejected this approach since correlation between variables was too close to 0, and the small number of timepoints (up to 4) introduced too much variability. Using only the few data points in one variable = ⇒ think about intrapolation vs extrapolation.
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Intrapolation is predicting data points that are within the bounds
In our data, intrapolating Phys. Dys. values will never generate negative values.
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Extrapolation is predicting data points that are OUTSIDE the bounds of the data. Extrapolating with too few points becomes a slippery slope. The next slides includes some illustrations.
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Extrapolating with only 2 points, we have seen, can result in
Our solution was to extrapolate with the mean, which can be justified by the fact that the slope computed on all subjects of each variables were not significantly non-zero.
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Out of 1 238’168 entries, 364’421 were missing (29.4%). Our imputation method successfully substituted 128’367 NAs, which represents 35.2% of the missing values. The remaining missing values are either of non-continuous and non-categorical type, or are still there because the subject had no values to rely on for imputation.
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Clustering - general goals Have a better understanding of data Discover / describe known or hidden patterns in the data Clustering - Our goals Searching for nutritional tendency and describing them. Looking if these patterns can provide knowledge on our models and on physiological dysregulation
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Unsupervised learning technique ⇒ observation data only. Partitioning observation into different groups (clusters) Regroup according to the similarity or dissimilarity among the data Remarks
(a)
More than a new method publish on arXiv.org / day
(b)
There is not a better technique or algorithm
(c)
Choice of methods = ⇒ Considerable impact on results
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
In the case of our research, we decided to perform clustering on nutritional data. However, typical clustering questions occur: Does it have ”real” patterns into our data? Is it possible to subdivide the observations into groups? How many groups can we subdivide our
Which clustering algorithm (with what parameters) we should use? An approach that can brings us possible answers to those question is the CONSENSUS CLUSTERING methods.
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Consensus clustering (Monti et al., 2003) [10] Clustering methods that search to present a consensus between several run of a given clustering algorithm. Observation Consensus clustering Best number of cluster & Quantification of cluster’s stability
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Consensus clustering procedure (Monti et al., 2003)[10] Given a set of observation D, a clustering algorithm C, a number
resampling to do.
1 For each number of cluster k in K
For each iteration of resampling h in H
Resample D Cluster the resample using C Compute a connectivity matrix M(h) (to be define)
Compute a consensus matrix M (to be define)
2 Find the best number of cluster k* in K based on consensus
distribution of M (one for each k)
3 Partition D into k* clusters based on M of k*
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Connectivity matrix Given a subsample h and two observations i & j Connectivity matrix is define as M(h)(i, j) = 1 if items i and j belong to the same cluster,
Consensus Matrix Given H subsample and a connectivity matrix for each Consensus matrix is define as M(i, j) =
M(i, j) = 0.5 = ⇒ No consensus
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Example Let do an example on the board: For three observations, a number of cluster k=2 and two subsamples. Suppose that we obtain these connextivity matrix after clustering. M(1) = 1 1 1 1 1 and M(2) 1 1 1 1 1 After applying the consensus matrix equation, we obtain M = 1 1
1 2
1 1
1 2
1
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
DiceR(2007)[3] Compute consensus clustering for several algorithms Compare the results and give the most relevant and stable algorithm and k. To choose the better algorithm, DiceR test the validity of the algorithms on the three criteria:
tiny
Compactness
tiiny Connectivity tiiiny Separation
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Some details on clustering experiments: Taking the average intake at T1 for each individual Selection of variable to cluster on:
1
Selection 1: CHOL, TRANS, SATS, MONO, POLY = ⇒ DONE
2
Selection 2: Selection 1 / ENERGY = ⇒ ON ANALISYS
3
Selection 3: All nutriments = ⇒ ON GOING
Using DiceR with these algorithm:
i
Agglomerating hierarchical clustering (H CLUST)
ii
Divisive hierarchical clustering (DIANA)
iii
K-means (KM)
iv
Partitioning around medoids (PAM)
Remark Since these algorithms are distance based, then we have also to decide on which distance we will run the algorithms and where the distance between the different cluster will be measure (linkage)
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Using the ”proportion of ambiguous clustering” (PAC) [10] measure the algorithm give k = 2 as the number of cluster.
Table 1: Selection 1 - PAC measure
K H clust PAM DIANA KM 2 0.1609881 0.2841789 0.001832891 0.03700218 3 0.1896322 0.2702275 0.002043847 0.29046358 4 0.2580985 0.2692805 0.007719303 0.30591599 5 0.1172091 0.3833785 0.008524355 0.29816908 6 0.1856891 0.2404978 0.014866197 0.22395310 ... ... ... ... ... PAC measure: the lower, the better!
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
According the clustering validation criteria of the consensus clustering, the better algorithms are:
1 Agglomerative hierarchical (H clust) 2 Divisive hierarchical (DIANA) 3 K-means (KM) 4 Partitioning around medoids (PAM)
HOWEVER
Table 2: Selection 1 - Observations per cluster
Clusters H clust PAM DIANA KM Cluster 1 1744 1128 1461 1135 Cluster 2 1 617 284
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Partitioning around medoids (PAM) Must know the number of cluster a prior. Choose k medoids (random or semi-random) = ⇒ Results depend of initialization Calculate every distances medoid-observation Place the observation in the cluster of the closest medoids Recalculate medoids and repeat until it converged (stable). Divisive hierarchical clustering (DIANA) Select all the observation as a cluster Split it in two Repeat with the bigger cluster (high variance whithin the cluster) Can continue until all observations are a cluster.
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Describing clusters: Nutriment medians & means = ⇒ higher in cluster 2 T-test and C.I on means difference = ⇒ Significant differences DIANA & PAM has found 2 significant different cluster!
Example: SATS mean box plot for each cluster
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Impact on Phys. Dys.:
⇒ low variation between clusters T-test and C.I on means difference = ⇒ No significant differences
Example: PAM Lipid Physiological Dysregulation box plot for each cluster
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
F´ elix made a good point in justifying the measurement error model in 24H recalls (24HR) data in a presentation at the CRCHUS [2]. Thus, we dug this avenue.
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Additive Measurement Error Model with repeated values Let there be n observations with m repeated values, written Wij, such that they follow the model: Wij = Ti + Uij, where Ti is the variable of interest and Uij is an error variable, with: variance: Var [Uij] = σ2
U,
mean: E [Uij] = 0. = ⇒ Wij is the repeated contaminated version of Ti, and Ti is empirically unavailable.
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Nutrient measurement error ↔ deviation from the long-term mean intake (LTMI). Wij = Ti + Uij Wij: Observed values in the 24HR, Ti: Long-term mean intake (LTMI), Uij: deviation from the LTMI (variability between days).
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
(Image is a courtesy of F´ elix CL, from his talk at CRCHUS [2].)
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Generalization: The data is one transformation apart from an additive model: h (Wij) = Ti + Uij, with h(·) is a real valued bijective and monotone function from a family of function H. Example If data under Multiplicative error model (Wij = mij · Ti) = ⇒ log(Wij) = Ti + log(mij). (Additive!)
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
One transformation we already know: Box-Cox power transformation. With a criteria to validate if data is subject to additive error, we search for the best λ parameter of: Box-Cox h(y|λ) = y(λ) = yλ−1
λ
, if λ = 0 ln(y), if λ = 0 , where y > 0
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Ratio of measurement error variance on total variance of some nutritional values:
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Convex Conditioned Lasso Regression (CoCoLasso): penalized linear regression, conditioned for measurement error Deconvolution non-parametric regression: generalization of the Nadaraya-Watson kernel non-parametric regression.
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
CoCoLasso in a sentence: Fit a multidimensional linear function with weight decay on the parameters, while correcting the model for additive error model. [6] Weight decay is a way to encode our preference for simpler models, which, in the case of Lasso, means selecting few variables that affect the linear function.
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
For every physiological system, the resulting optimal linear function was the mean. Since the effect seems nonlinear, = ⇒ nonparametric model might capture the effects.
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Deconvolution nonparametric (NP) regresssion: generalization
The initial model relies on the distribution of the prediction
distribution information for additive error using Deconvolution Kernel Density Estimation (DKDE). [7] Downsides: I did not find a multidimensional equivalent Intuitively, I can tell that it will suffer from multidimensionality.
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
10 20 30 100 200
PROTEIN global
Non parametric regression between mean daily PROTEIN intake and global dysregulation score
Data points removed for confidentiality
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
10 20 30 100 200 300 400 500
CARBO global
Non parametric regression between mean daily CARBO intake and global dysregulation score
Data points removed for confidentiality
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
10 20 30 50 100 150
LIPID global
Non parametric regression between mean daily LIPID intake and global dysregulation score
Data points removed for confidentiality
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Our work includes: Transforming biomarker data to approach normality, Imputing biomarker data longitudinally, Exploratory Clustering of nutritional data, Transformation to approach additive measurement error model, Exploration of a multivariate penalized linear regression model, designed for additive error, Exploration of a univariate nonparametric regresssion model, also designed for additive error.
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
To explore: More clustering! (On other data subset), Discretized version of the difference of Phys. Dys. compared to time T1, Correction of Phys. Dys. or nutritional values for confounders, such as BMI or Energy(KJ) intake, Other nonparametric regression models, designed for additive error model.
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
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Cross-population validation of statistical distance as a measure of physiological dysregulation during aging. Experimental gerontology 57 (Sep 2014), 203–210.
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
[5] Cohen, A. A., Milot, E., Yong, J., Seplaki, C. L., F¨ ul¨
A novel statistical approach shows evidence for multi-system physiological dysregulation during aging. Mechanisms of Ageing and Development 134, 3–4 (Mar 2013), 110–117. [6] Datta, A., and Zou, H. Cocolasso for high-dimensional error-in-variables regression. The Annals of Statistics 45, 6 (Dec 2017), 2400–2426. Zbl: 06838137. [7] Fan, J., and Truong, Y. K. Nonparametric regression with errors in variables. The Annals of Statistics 21, 4 (Dec 1993), 1900–1925. Zbl: 0791.62042. [8] Manly, B. F. J. Exponential data transformations.
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
Journal of the Royal Statistical Society. Series D (The Statistician) 25, 1 (1976), 37–42. [9] Milot, E., Morissette-Thomas, V., Li, Q., Fried,
Trajectories of physiological dysregulation predicts mortality and health outcomes in a consistent manner across three populations. Mechanisms of ageing and development 0 (2014), 56–63. [10] Monti, S., Tamayo, P., Mesirov, J., and Golub, T. Consensus clustering: A resampling-based method for class discovery and visualization of gene expression microarray data. Machine Learning 52, 1 (Jul 2003), 91–118. [11] Shapiro, S. S., and Wilk, M. B. An analysis of variance test for normality (complete samples). Biometrika 52, 3/4 (1965), 591–611.
Transformation to normality Longitudinal imputation Clustering Measurement error and regression Conclusion
[12] Yeo, I.-K., and Johnson, R. A. A new family of power transformations to improve normality
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