Michael Fahey

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Identifying dietary patterns in a multivariate
mixture non-normality versus correlation

• Michael Fahey
• Ian White
• July, 2007

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Outline

• motivation
• semi-normal mixture model
• sensitivity to
• marginal non-normality
• correlation among responses

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Motivation

• diet assessment in epidemiology
• foods eaten together interact
• holistic description using multiple dietary
  responses
• latent characteristic (dietary pattern) induces
  correlation among responses

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Notation
is the multivariate probability density function
for a set of food responses y
probability of membership to latent class x
class-specific density, given latent class x
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Mixture distribution
Define dietary patterns as sub-groups having
different food consumption probability
distributions. To find sub-groups decompose the
mixture distribution into sum of K class-specific
distributions
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Food consumption data

• 8000 women in the EPIC-Norfolk cohort who
  completed a dietary questionnaire
• eight food groups reflecting different dietary
  components and characteristics in data
• consumption calculated in (g/d) or (g/d 1) on
  log, square root and Box-Cox scales

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(No Transcript)
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Semi-continuous data
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Semi-normal (SMN) mixture

• two-part density for semi-continuous responses
• binary part (whether food was consumed)
• continuous part (if consumed, how much)
• the two parts considered together if food not
  consumed, continuous part set to missing

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Two-part density
Define d 1 if y gt 0, and d 0 if y 0, and
the PDF for the two-part density is
Here, p is the probability of consumption, g(y)
is scale dependent, and in the mixture f(y x)
is replaced by f(y, d x) for semi-continuous
data.
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Sensitivity to non-normality

• fit SMN mixture on raw, log, square root,
  Box-Cox, rankit scales
• means and variances of responses vary by class,
  covariances constrained to zero
• vary K 1 to 4 and evaluate model BIC,
  pseudo-class residual analysis

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Rankit transformation
Raw scale data are ranked, ri, and assigned their
expected value as a standard normal deviate
For example, the median on the raw scale
has rankit 0 and the 95th percentile has rankit
1.65.
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Change in BIC by scale
?BIC BIC(K-1) - BIC(K)
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Pseudo-residual analysis

• randomly assign women to one latent class using
  p(x y, d) and compute residuals
• pseudo-class residuals have good properties (Wang
  et al, JASA, 2005)
• repeat random assignments M times for each woman
  to create N x M data
• use Q-Q plots to evaluate class-specific
  multivariate normality on log and rankit scale

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Pseudo-residuals log scale
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Pseudo-residuals rankit scale
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Sensitivity to correlation

• vary class-specific covariance matrix, SK
• diagonal SK DK
• linear factor SK ??T DK
• where ? is the coefficient of a class-constant
  linear factor, f, such that E(y) µ ?f
• unconstrained SK CK DK
• where CK is a matrix of covariances among
  continuous responses, and zero elsewhere

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Covariance matrix rankit scale
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Pseudo residuals rankit, SK CK DK
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Concluding remarks 1

• marginal normality leads to
• 1) lower latent dimensionality
• 2) better fitting class-specific models
• SMN model removes non-normality due to clumping
  at zero identification shifts to association
  involving binary responses

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Concluding remarks 2

• two latent classes identified in associations
  involving binary responses
• correlation among continuous responses may be
  uninteresting, e.g. measurement error,
  association not caused by dietary pattern
• shifting emphasis from non-normality to
  correlation localises identification

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Notation
is the multivariate probability density
function for a set of food variables y given
covariates z and parameter vector ? (?,?)
is the probability of membership to latent class
x given covariates z and parameter vector ?
is the class-specific density, given latent class
x, covariates z and parameter vector ?
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Norfolk dietary questionnaire
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Model details (1)

• means and variances of y vary by class
• covariances among the y constrained to zero
• latent class probabilities and the mean of the y
  depend on one covariate, z
• z the log of total energy intake (log kcal)

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Model details (2)

• Model dependence on covariates as follows

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Sensitivity to non-normality (OLD)

• fit SMN mixtures on raw, log, square root,
  Box-Cox, rankit scales vary K 1 to 4
• evaluate model fit BIC, pseudo-class residual
  analysis
• examine classification agreement between any two
  candidate solutions Rand index

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Rand index (RI) 1

• compares two modal classifications by considering
  all pairs of women
• agreement if a pair are either
• in the same class on each classification
• or in different classes on each classification
• RI (count of agreements) / (number of pairs)

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Rand index (RI) 2

• values usually range from 0 to 1
• probability that classifications on a randomly
  chosen pair agree
• adjusted RI takes chance agreement into account
• invariant to permutation of class labels

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Classification agreement (RI) among scales, K 4
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Classification log vs others
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Covariance matrix log scale
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Number of parameters
Semi-normal models with K and SK as given
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MVN patterns deviation from grand mean
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SMN deviation from grand mean
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Posterior probabilities
Average modal posterior probabilities by class

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Hard versus soft classification
hard modal classification soft mixing
proportions estimated from model
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Predicting cancer risk
Cox regression crude effect of SMN latent class
patterns on incidence of 49 cancers in
EPIC-Norfolk