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Statistical Analysis Overview I Session 1 . Peg Burchinal Frank Porter Graham Child Development Institute, University of North Carolina-Chapel Hill. Overview: Statistical analysis overview I. Linear models Nesting Longitudinal models Mixed Model ANOVA Multivariate Repeated Measures

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Statistical analysis overview i session 1 l.jpg
Statistical Analysis Overview ISession 1

Peg Burchinal

Frank Porter Graham

Child Development Institute,

University of North Carolina-Chapel Hill


Overview statistical analysis overview i l.jpg
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 Analyses

  • 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

        aspecial methods needed if assumption not met



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Traditional Growth Curve Analysis Analyses

"Univariate" Analysis (“Mixed Model”)

General model for one grouping variable and linear change related to age.

Yijk = b0k + b1k Ageijk + aik Personik + eijk

for i=1,...,n individuals,

j=1,...,p occasions,

k=1,...,r groups;

with 2 fixed effect variables - Group and Age

& 3 random variables - Y, Person, E;



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Mixed-Model ANOVA Analyses

  • 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 AnalysesMultivariate 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 i=1,...,n individuals,

j=1,...,p occasions,

k=1,...,r groups;


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Y Analysesijk = p0ik + p1ik Ageijk +eijk

E(Yjk) = b0k + b1k Ageijk

where Yijkrepresents 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

b0kis the intercept for the k-th group - the

unweighted mean of the p0ikwithin

the k-th group

p1ikis 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 p1ikwithin the k-th group



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Profile Analysis Analyses

  • 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 Analyses("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 i=1,...,n individuals,

j=1,...,p occasions,

k=1,...,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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Y Analysesijk = (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 weightedmean of p0ik,

p0ik is the increment to the intercept for the i-th individual in the k-th group

b1kis 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

eijkrepresents 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 AnalysesAdvantages

  • 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 Models AnalysesDisadvantages

  • 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 Analyses

  • 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 Analyses

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 Analyses

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 Analyses

  • 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 Analyses

Mother's Sensitivity for Mothers with High School Degree versus Bachelor’s Degree


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Categorical Predictors Analyses

Mother's Sensitivity for Male versus Female Children


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Categorical Predictors Analyses

Mother's Sensitivity for Mothers with and without Clinical Levels of Depressive Symptoms


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Analytic issues-repeated measures Analyses

  • 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 Analyses

HLM Level 1 corresponds to LISREL measurement model for Y

HLM: Yip = pop + p1p time I + eip

LGC: Yp = [1tp ] p+ ep

= 0 + [1tp ] 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 = [1tp ] - this passes the Level 1 growth curve parameters into the LISREL endogenous constraintsLatent Growth Curves


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Latent Growth Curves Analyses

HLM Level 2 corresponds to LISREL structural model

HLM: p = Xb + r

LGC:p = m + ( 00 ) 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 = (00)


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Latent Growth Curve Model Analyses(same as HLM individual curve)


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Latent Growth Curves: AnalysesAdvantages

  • 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 Analyses

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 Curves AnalysesExample – 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 Analyses

Bold indicates sign. at p<0.05


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SECCYD-LGC Analysis of Maternal Sensitivity Analyses

  • 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 Analyses

  • 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


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