# Presentation: 154729

## 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**
**Two Level Hierarchical Linear Models**
**Latent Growth Curve Models**

## 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.**

## 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*

## 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

## 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

## 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

## 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

## 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

## 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

## 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

## Analytic methods to address nesting

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

## 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)**

##  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

## Longitudinal Data

## Traditional Growth Curve Analysis

**"Univariate" Analysis  (“Mixed Model”)**
**	****  ****General model for one grouping variable and linear change related to age.  **
***Y******ijk******  = ************0k****** + ************1k****** Age******ijk****** +************ik****** Person******ik****** + ************ijk***
**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*****;**

## “Univariate” Growth Curves

## 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*

## 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**
***Y******ijk******  = ************0ik****** + ************1ik****** Age******ijk****** + ************ijk***
***(individual growth curve)***
***E(Y******jk******)  = ************0k****** + ************1k****** Age******ijk ***
***(population growth curve******)***
**for ****	****i=1,...,n individuals, **
**	****	****j=1,...,p occasions,****	**
**	****	****k=1,...,r groups;**

## Yijk  = 0ik + 1ik Ageijk +ijk
E(Yjk)  = 0k + 1k Ageijk 
where Yijk represents the j-th assessment of the i-th individual in the k-th group,
	0ik is the intercept for the i-th subject in the
		 k-th group
	0k is the intercept for the k-th group -  the
		 unweighted mean of the 0ik within
		 the k-th group
	1ik is the slope for the regression of Y on Age for
		 the i-th individual in the k-th group
	1k is the slope for the regression of Y on Age for
		 the k-th group -  the	 unweighted mean of
		 the 1ik within the k-th group

***E(Y******jk******)  = ************0k****** + ************1k****** Age******ijk****** ***
**where *****Y******ijk****** *****represents the j-th assessment of the i-th individual in the k-th group,**
**	*********0ik***** is the intercept for the i-th subject in the**
**	****	**** ****k-th group**
**	***********0k****** *****is the intercept for the k-th group -  the**
**	****	**** **_***unweighted***_** mean of the *********0ik****** *****within**
**	****	**** ****the k-th group**
**	***********1ik***** ****is the slope for the regression of *****Y***** on *****Age *****for**
**	****	**** ****the i-th individual in the k-th group**
**	***********1k***** is the slope for the regression of *****Y***** on *****Age***** for**
**	****	**** ****the k-th group -  the****	**** **_***unweighted ***_**mean of**
**	****	**** ****the ***********1ik***** ****within the k-th group**

## Profile Analysis

## 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*

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

**General model for one between-subjects categorical factor and linear change related to age.  **
***Y******ijk ****** = (************0k****** + ************0ik******) + (************1k****** + ************1ik******) Age******ijk****** + ************ijk***
*** ******or ***
***Y******ijk******  = ************0ik****** + ************1ik****** Age******ijk****** + ************ijk***
***(Level 1 or individual growth curve)***
***E(Y******jk******)  = ************0k****** + ************1k****** Age******ijk****** ***
***(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*****;**

## Yijk  = (0k + 0ik) + (1k + 1ik) Ageijk + ijk
where Yijk represents the j-th assessment of the i-th individual in the k-th group,
0k is the intercept for the k-th group- estimated  as weighted mean of 0ik, 
0ik is the increment to the intercept for the i-th individual in the k-th group 
1k is the slope for the regression of Y on Age for the k-th group- estimated as weighted mean of 1ik, 
1ik is the increment to the slope for the i-th individual in the k-th group
ijk represents the random error of the j-th assessment of the i-th individual in the k-th group

**where *****Y******ijk***** represents the j-th assessment of the i-th individual in the k-th group,**
*********0k***** is the intercept for the k-th group- estimated  as **_***weighted***_*** *****mean of ***********0ik,***** **
*********0ik***** is the increment to the intercept for the i-th individual in the k-th group **
*********1k***** ****is the slope for the regression of Y on Age for the k-th group- estimated as **_***weighted***_** mean of ***********1ik,***** **
*********1ik *****is the increment to the slope for the i-th individual in the k-th group**
*********ijk***** ****represents the random error of the j-th assessment of the i-th individual in the k-th group**

## Hierarchical Linear Model

## Hierarchical Linear ModelAdvantages

**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**

## Hierarchical Linear ModelsDisadvantages

**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 ***

## 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.**

## 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%

## Model 

***Y ******ij = ***********0i**** + **********1i**** Age****ij****+ **********2ik**** Age****ij****2**** +**
********1****Dep****ij**** + **********2**** Dep****ij ****x ****Age****ij**** + **********3**** Dep****ij ****x ****Age****2****j**** **
********ijk **
**(individual component of growth curve)**
********0**** + **********4**** AGE****ij**** +  **********5**** AGE****ij****2 + **
********6****Med****i**** + **********7**** Med****i**** x Age****ij**** + **********7**** Med****i**** x AGE****ij****2 + **
********8****Male****i**** + **********9****Male****i**** x Age****ijk**** + **********10**** Male****i**** x AGE****ij****2 +**
**(group component of growth curve).**

## 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

## Continuous Predictors

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

## Categorical Predictors

- Mother's Sensitivity for Male versus Female Children

## Categorical Predictors

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

## 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

## Latent Growth Curves

**HLM Level 1 corresponds to LISREL measurement model for Y****	**
**	****HLM: Y****ip **** = ********op ****+ ********1p**** time ****I ****+ ********ip**
**	****LGC:   **_**Y**_**p**** = [**_**1**_** **_**t**_**p**** ]  **_****_** **** ****+ **_**e**_**p**
**	****	**** ****= **_**0**_** +  [**_**1**_** **_**t**_**p**** ] **_****_** ****+ **_****_**p ****  ( endogenous variable **_**Y) **_** **
- 	**  ****	****= **_****_**Y ****+**_****_**Y ******_****__** **_**+ **_****_** p**
**	****where **_**Y**_**p**** is vector of observed values for person p**
_****_****_****_** ****the vector of latent growth curve parameters for person p**
_****_** ****p**** is individual-specific vector of unknown measurement error**
**and unlike the usual practice of LISREL analysis, **_****_** Y ****& **_****_**Y**** parameter matrices are constrained to contain only known values**
_****_**Y ****= **_**0**_** **
_****_**Y **** =  [**_**1**_** **_**t**_**p**** ] - this passes the Level 1 growth curve parameters into the LISREL endogenous constraintsLatent Growth Curves **

## Latent Growth Curves

**HLM Level 2 corresponds to LISREL structural model**
**HLM: **_****_** = **_**X**__****_** + **_**r**_
**LGC****:******_****_****_****_****_****_****_****_****_****_****_****_****_****_****
**which has the form of a reduced LISREL structural model**
****
****_****_****_****_****
******the group growth curve parameters **
****_****_****_****_

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

## 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**

## 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**

## Latent Growth CurvesExample – 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**

## SECCYD – Maternal Sensitivity

**Bold **indicates sign. at p<0.05

## 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).

## 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**