Statistical Analysis Overview I Session 1

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Statistical Analysis Overview ISession 1

• Peg Burchinal
• Frank Porter Graham
• Child Development Institute,
• University of North Carolina-Chapel Hill

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Overview Statistical analysis overview I

• Linear models
• Nesting
• Longitudinal models
• Mixed Model ANOVA
• Multivariate Repeated Measures
• Two Level Hierarchical Linear Models
• Latent Growth Curve Models

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Overview Linear Models

• Most commonly used statistical models
• 1. t-test, Analysis of Variance/Covariance--
  comparing means across groups
• 2. Correlations, Multiple Regression
• estimating associations among continuous
  variables.

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Linear Models

• General Model
• Yij B0 B1 X1ij B2 X2ij eij
• Assumptions
• One source of random variability (eij)
• Normally distributed error terms
• Homogeneity of variance
• Independence of observations

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Linear Models

• Equivalence of models
• T-test and ANOVA (1-way with 2 groups)
• Regression and ANOVA
• t-test Yij B0 B1 X1ij eij
• X1ij 1 if in first group, 0 if in second group
• One-way ANOVA (2 groups)
• Yij B0 B1 X1ij eij
• X1ij 1 if in first group, 0 if in second group

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Linear Models

• One-way ANOVA (p groups)
• Yij B0 B1 X1ij B2 X2ij Bp-1 Xp-1ij
  eij
• X1ij 1 if in first group, 0 otherwise,
• X2ij 1 if in second group, 0 otherwise,
• etc for p-1 groups (last group is reference cell)
• Regression (p predictors)
• Yij B0 B1 X1ij B2 X2ij Bp Xpij
  eij
• X1ij first continuous predictor
• X2ij second continuous predictor
• etc for the p predictors in the model

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Linear Models

• One-way ANCOVA (2 level factor and 1 covariate)
• Yij B0 B1 X1ij B2 X2ij eij
• X1ij 1 if in first group, 0 otherwise,
• X2ij continuous predictor
• Separate slopes ANCOVA (2 level factor and 1
  covariate)
• Yij B0 B1 X1ij B2 X2ij B3 X3ij eij
• X1ij 1 if in first group, 0 otherwise,
• X2ij first continuous predictor
• X3ij X1ij X2ij 0 if not in first group
• value of first continuous predictor if
  in first group

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Linear Models

• Two-way ANOVA (2 2-level factors and
  interaction)
• Yij B0 B1 X1ij B2 X2ij B3 X3ij eij
• X1ij 1 if in first group on first factor, 0
  otherwise,
• X2ij 1 if in first group on second factor, 0
  otherwise,
• X3ij X1ij X2ij
• 1 if in first level of the fist and
  second factor, 0 otherwise

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Linear Models General Issues

• Design parameterization
• Showed Reference Cell Coding
• Effect Coding often preferable (use -.5 and .5
  instead of 0 and 1)
• Centering variables
• Whenever an interaction is included, you should
  center your data so main effects are
  interpretable
• Easiest subtract sample mean from all values
• Nested data- correlated observations

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Correlations among Observations

• Many sources of nesting
• Repeated measures over time
• Clustering of students in a classroom, therapy
  group, etc
• Clustering of individuals in a family
• Consequence of nesting
• Standard errors are under-estimated when
  observations within cluster are positively
  correlated
• P-values are too small when standard errors are
  under-estimated

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Nesting

• Longitudinal models provide the easiest nested
  model to understand
• Obvious that repeated assessments of individuals
  are not independent
• Present various approaches to modeling
  longitudinal data

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Analytic methods to address nesting

• Mixed-model repeated measures
• Multivariate repeated measures
• Hierarchical linear models
• Latent growth curves

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Overview Additional Assumption for Repeated
Measures Analyses

• General assumptions
• An adequate model to describe
• Individual patterns of change (within cluster
  patterns of change)
• Individual differences in developmental patterns
  (between cluster patterns of change)
• Both models must include
• Important covariates relevant interactions
• Represent correlations in nested factors
• (Type I error rate control)

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General statistical assumptions

• Same outcome measured in the same metric over
  time
• Interval or ratio measurement a
• Normally distributed variables a
• Homogeneity of variance a
• Monotonic assessment
• Must be able index amount of change
• Unit change must be uniform across scale and age
• Standard score not great, but can be used
• If same outcome over time
• Identical items not required
• a special methods needed if assumption not met

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Longitudinal Data
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Traditional Growth Curve Analysis

• "Univariate" Analysis (Mixed Model)
• General model for one grouping variable and
  linear change related to age.
• Yijk b0k b1k Ageijk aik Personik eijk
• for i1,...,n individuals,
• j1,...,p occasions,
• k1,...,r groups
• with 2 fixed effect variables - Group and Age
• 3 random variables - Y, Person, E

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Univariate Growth Curves
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Mixed-Model ANOVA

• Advantages
• Estimates individual intercepts
• Corrections are available to avoid inflating test
  statistics
• Disadvantages
• Assumes all slopes are identical
• Deletions of individuals with missing data if
  apply corrections
• Cannot easily accommodate repeated measures of
  predictors or multiple levels of nesting

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Profile Analysis or Multivariate Repeated
Measures Analysis

• Transforms model into separate analyses of
  between- and within-factors
• General model for one grouping variable and
  linear change related to age
• Yijk p0ik p1ik Ageijk eijk
• (individual growth curve)
• E(Yjk) b0k b1k Ageijk
• (population growth curve)
• for i1,...,n individuals,
• j1,...,p occasions,
• k1,...,r groups

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• Yijk p0ik p1ik Ageijk eijk
• E(Yjk) b0k b1k Ageijk
• where Yijk represents the j-th assessment of the
  i-th individual in the k-th group,
• p0ik is the intercept for the i-th subject in
  the
• k-th group
• b0k is the intercept for the k-th group - the
• unweighted mean of the p0ik within
• the k-th group
• p1ik is the slope for the regression of Y on Age
  for
• the i-th individual in the k-th group
• b1k is the slope for the regression of Y on Age
  for
• the k-th group - the unweighted mean of
• the p1ik within the k-th group

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Profile Analysis
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Profile Analysis

• Advantages
• Estimates individual intercepts and slopes
• Standard errors are not inflated with moderate to
  large sample sizes
• Disadvantages
• Case wise deletion of individuals with missing
  data
• Forced to use categorized nesting variable
• Cannot easily accommodate repeated measures of
  predictors or multiple levels of nesting

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Hierarchical Linear Model ("Mixed-Effects Linear
Model")

• General model for one between-subjects
  categorical factor and linear change related to
  age.
• Yijk (b0k p0ik) (b1k p1ik) Ageijk eijk
• or
• Yijk p0ik p1ik Ageijk eijk
• (Level 1 or individual growth curve)
• E(Yjk) b0k b1k Ageijk
• (Level 2 or population growth curve)
• for i1,...,n individuals,
• j1,...,p occasions,
• k1,...,r groups
• with 1 fixed effect variables - Group
• 4 random variables - Y, Individual's mean
  level, Individual's change over Age, E

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• Yijk (b0k p0ik) (b1k p1ik) Ageijk eijk
• where Yijk represents the j-th assessment of the
  i-th individual in the k-th group,
• b0k is the intercept for the k-th group-
  estimated as weighted mean of p0ik,
• p0ik is the increment to the intercept for the
  i-th individual in the k-th group
• b1k is the slope for the regression of Y on Age
  for the k-th group- estimated as weighted mean of
  p1ik,
• p1ik is the increment to the slope for the i-th
  individual in the k-th group
• eijk represents the random error of the j-th
  assessment of the i-th individual in the k-th
  group

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Hierarchical Linear Model
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Hierarchical Linear ModelAdvantages

• Accommodate multiple levels of nesting
• Slopes and intercepts of individual growth curves
  can vary
• Increased precision
• Permits missing or mistimed data
• ignorably missing data
• purposefully missing data designs
• inconsistently timed data
• 5. Allows repeated measures of predictors
• 6. Flexible specification of growth patterns
• 7. Fixed-effect parameter estimates fairly
  robust

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Hierarchical Linear ModelsDisadvantages

• Assumes that an infinite number of individuals
  were observed, but a "large" number is
  sufficient.
• Unclear what is large enough
• 2. Models can get very complicated
• 3. No direct tests of mediation

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SECCYD Example Maternal Sensitivity

• Goal determine whether maternal sensitivity
  between 6m and first grade varies as a function
  of
• maternal education,
• maternal depression
• child gender.

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Analysis Data

• 6 15 24 36 54 G1
• Time-varying
• Maternal sensitivity
• N 1272 1240 1172 1161 1040 1004
• M 3.07 3.13 3.12 3.27 3.23 3.22
• sd .59 .55 .59 .53 .56 .58
• Maternal Depression
• 18 17 18 16 18 14
• Time-Invariant
• Maternal Education
• M (sd) 14.3 (2.49)
• Child Gender
• male 51

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Model

• Y ij p0i p1i Ageij p2ik Ageij2
• b1Depij b2 Depij x Ageij b3 Depij x Age2j
• eijk
• (individual component of growth curve)
• b0 b4 AGEij b5 AGEij2
• b6Medi b7 Medi x Ageij b7 Medi x AGEij2
• b8Malei b9Malei x Ageijk b10 Malei x AGEij2
• (group component of growth curve).

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Results

• Maternal Education Mothers with more education
  show more sensitivity, and show less reduction in
  sensitivity after children enter schools
• Gender mothers more sensitive with girls during
  early childhood, but show increasing levels of
  sensitivity with boys over time
• Maternal depression Depressed mothers show
  less sensitivity during early childhood, but show
  modest gains when children enter school

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Continuous Predictors
Mother's Sensitivity for Mothers with High School
Degree versus Bachelors Degree
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Categorical Predictors
Mother's Sensitivity for Male versus Female
Children
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Categorical Predictors
Mother's Sensitivity for Mothers with and without
Clinical Levels of Depressive Symptoms
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Analytic issues-repeated measures

• Time-varying (within-subjects) and time-invariant
  (between-subjects) data
• Analysis data one record per subject or one
  record per subject per assessment (software
  issue)
• Plotting results
• Interpreting interactions

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Latent Growth Curves

• HLM Level 1 corresponds to LISREL measurement
  model for Y
• HLM Yip pop p1p time I eip
• LGC Yp 1 tp p ep
• 0 1 tp p ep ( endogenous variable
  Y)
• tY lY h e p
• where Yp is vector of observed values for person
  p
• h p the vector of latent growth curve
  parameters for person p
• e p is individual-specific vector of unknown
  measurement error
• and unlike the usual practice of LISREL analysis,
  t Y lY parameter matrices are constrained to
  contain only known values
• tY 0
• lY 1 tp - this passes the Level 1 growth
  curve parameters into the LISREL endogenous
  constraintsLatent Growth Curves

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Latent Growth Curves

• HLM Level 2 corresponds to LISREL structural
  model
• HLM p Xb r
• LGC p m ( 0 0 ) p p - m
• which has the form of a reduced LISREL structural
  model
• h a b h z
• z p - m
• a m the group growth curve parameters
• b (0 0)

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Latent Growth Curve Model(same as HLM individual
curve)
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Latent Growth Curves Advantages

• Allows individual intercepts and slopes to vary.
• Allows for error in predictors
• Easily handles error heterogeneity and
  correlated errors
• Permits latent variables with multiple indicators
• Can examine patterns of change on more than one
  dimension.
• Easily estimates direct and indirect
  (intervening) effects

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Latent Growth Curves

• Disadvantages
• Does not easily accommodate more than one level
  of nesting
• Easy-to-use software requires time-structured
  data (M-Plus)
• Number of estimated parameters gets large quickly
• Less power for testing interactions or moderating
  effects
• Equivalence HLM and LGC can be shown to be
  interchangeable when data are time structured

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Latent Growth CurvesExample SECCYD Maternal
Sensitivity

• Goal - describe developmental patterns in
  maternal sensitivity with target child from six
  months to first grade
• Analysis- Structural Equation Model
• Quadratic individual growth curve
• Maternal education and gender as predictors
• AMOS with FIML - due to missing data

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SECCYD Maternal Sensitivity
Bold indicates sign. at plt0.05
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SECCYD-LGC Analysis of Maternal Sensitivity

• Maternal education related to higher levels of
  sensitivity over time (intercept).
• Mothers are more sensitivity with girls in
  general (intercept), but show nonlinear increases
  in sensitivity toward boys (quadratic slope).

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Conclusions

• Growth curve analyses can provide an appropriate
  and powerful analytic tools for examining
  longitudinal or other types of nested data
• Careful selection of analytic methods and models
  is needed