Stat Computing > Seminars > MLM Longitudinal

Statistical Computing Seminars
Analyzing Longitudinal Data using Multilevel Modeling

You can view movies of this seminar via the links below.

• Analyzing Longitudinal Data using Multilevel Modeling: Part 1
• Analyzing Longitudinal Data using Multilevel Modeling: Part 2
• Analyzing Longitudinal Data using Multilevel Modeling: Part 3
• See how we made these movies

The aim of this seminar is to help you learn about the use of Multilevel Modeling for the Analysis of Longitudinal Data.  The seminar will feature examples from Applied Longitudinal Data Analysis: Modeling Change and Event Occurrence by Judith D. Singer and John B. Willett The seminar will address the following issues

• A comparison of strategies for analyzing longitudinal data, including repeated measures ANOVA, mixed models analysis, regression, and multilevel modeling

• Multilevel models for analyzing longitudinal data

• Models for evaluating changes in "elevation" and "slope" over time.

• Using multilevel models to analyze "treatment effects" over time.

The seminar will focus on the construction and interpretation of these models with the aims of appealing to users of all multilevel modeling packages (e.g. HLM, SAS PROC MIXED, MLwiN, SPSS Mixed, etc.). For the sake of realism, many examples will be run using HLM, but examples of using SAS PROC MIXED and MLwiN will also be included.

A comparison of strategies for analyzing longitudinal data

An Example : Kids alcohol use measured at 3 time points, age 14, 15, 16

• Everyone has the same number of waves of data (3 waves of data)
• All waves of data were measured at the same time (all measured on their birthday)
• Measures across time are probably not independent.

Strategies for Analyzing Longitudinal Data

• Traditional Repeated Measures Anova
  • It assumes that all kids have the same number of waves of data.
  • It assumes all kids measured at the same time points (e.g. Sally cannot be measured at age 14,15.5, 16.5).
  • Handles correlations among time points, assuming CS or UN.

    • proc glm data=alc ;
        model alcuse14 alcuse15 alcuse16 = coa / nouni ;
        repeated age 3 polynomial / summary;
      run;

• Mixed Models Anova
  • It is OK if some kids have more waves of data than others.
  • The "repeated" statement assumes kids all measured at the same time points (for computing covariance structures).
  • Handles correlations among time points, using "mixed" can even handle many different kinds of covariance structures.

    • proc mixed data=alcpp ;
        class age;
        model alcuse = age coa age*coa / solution ;
        repeated age / subject=id type=ar(1) ;
      run;

• Regression
  • It is OK if some kids have more waves of data than others
  • Do not need people measured on the same schedule
  • Assumes no correlations among time points for a given person.

    • proc reg data=alcpp ;
        model alcuse = age coa ageBYcoa  ;
      run;

• Multilevel Modeling
  • It is OK of some kids have more waves of data
  • It is OK if kids are measured on different schedules
  • The covariance structure is based on the distance (in time) from each other.

    • proc mixed data=alcpp ;
        class id;
        model alcuse = coa age coa*age / solution ;
        random intercept age / type=un sub=id;
      run;

Multilevel models for analyzing longitudinal data

• Applying the multilevel model to longitudinal data
  • Studying alcohol use over time.
  • Time is level 1, Person is level 2.  Time is nested within persons.
• Just like students within classrooms
  • We can ask about student level (level 1) characteristics the affect performance
  • We can ask about "classroom" characteristics (level 2) that influence student performance.
  • We can ask about "classroom" characteristics that influence the student level intercepts and slopes.
• In this example we can ask
  • What time varying characteristics (level 1) affect alcohol use (e.g. time)
  • What "person" characteristics (level 2) influence alcohol use (e.g. Peer drinking at start of study, being a child of an alcoholic).
  • We can ask about "person" characteristics (level 2) that influence initial alcohol use (intercept) and change over time (slope).

Consider Figure 4.1, Page 1

Consider Table 4.2, Page 2

Consider Figure 4.1, Page 3

• We will examine Models A-E. 
• We could use HLM, MLwiN, SAS Proc Mixed, SPSS Mixed, Splus, R, or Mplus to solve these
• Running the multilevel model is not hard, it is constructing and interpreting the model that is tricky.  Once you have constructed the model, then putting it into the package is generally easy.
• We will show examples using HLM, but also show SAS Proc Mixed and MLwiN examples for final model.

Table 4.1, Model A

This is a simple "intercept only" model, predicting alcohol use from the overall mean.

Level 1/Level 2 model

ALCUSE_ij =  π_0i + ε_ij  
π_0i =  γ₀₀ + ζ_0i

Composite model

ALCUSE_ij = γ₀₀ + (ε_ij + ζ_0i ) 

Running model in HLM

 Final estimation of fixed effects:
 ----------------------------------------------------------------------------
                                       Standard             Approx.
    Fixed Effect         Coefficient   Error      T-ratio   d.f.     P-value
 ----------------------------------------------------------------------------
 For       INTRCPT1, B0
    INTRCPT2, G00           0.921955   0.095707     9.633        81    0.000
 ----------------------------------------------------------------------------

 Final estimation of variance components:
 -----------------------------------------------------------------------------
 Random Effect           Standard      Variance     df    Chi-square  P-value
                         Deviation     Component
 -----------------------------------------------------------------------------
 INTRCPT1,       U0        0.75091       0.56386    81     328.92616    0.000
  level-1,       R         0.74950       0.56175
 -----------------------------------------------------------------------------

Results as Level 1/Level 2 model

Fixed Effects

ALCUSE_ij =  π_0i + ε_ij  
π_0i = .92 + ζ_0i

Variance Components

Level 1
  Within Person, V(ε) = .563
Level 2
  Initial Status, V(ζ₀) = .561

Table 4.1, Model B

This model predicts alcohol use from the intercept and time, both of which randomly vary across children.

Level 1/Level 2 model

ALCUSE_ij =  π_0i + π_1iTIME +  ε_ij  
π_0i =  γ₀₀ + ζ_0i
π_1i =  γ₁₀ + ζ_1i

Composite model

ALCUSE_ij = γ₀₀ + γ₁₀ +  (ε_ij + ζ_0i + ζ_1iTIME ) 

Running Model in HLM

 Final estimation of fixed effects:
 ----------------------------------------------------------------------------
                                       Standard             Approx.
    Fixed Effect         Coefficient   Error      T-ratio   d.f.     P-value
 ----------------------------------------------------------------------------
 For       INTRCPT1, B0
    INTRCPT2, G00           0.651304   0.105080     6.198        81    0.000
 For   AGE_14 slope, B1
    INTRCPT2, G10           0.270651   0.062455     4.334        81    0.000
 ----------------------------------------------------------------------------

 Final estimation of variance components:
 -----------------------------------------------------------------------------
 Random Effect           Standard      Variance     df    Chi-square  P-value
                         Deviation     Component
 -----------------------------------------------------------------------------
 INTRCPT1,       U0        0.79016       0.62436    81     264.14675    0.000
   AGE_14 slope, U1        0.38885       0.15120    81     155.51848    0.000
  level-1,       R         0.58077       0.33729
 -----------------------------------------------------------------------------

Results as Level 1/Level 2 model

Fixed Effects

ALCUSE_ij =  π_0i + π_1iTIME +  ε_ij  
π_0i = .65 + ζ_0i
π_1i = .27 + ζ_1i

Variance Components

Level 1
  Within Person, V(ε) = .34
Level 2
  Initial Status, V(ζ₀) = .62
  Rate of Change, V(ζ₁) =.15
  Cov(ζ_{0 ,} ζ₁) = -.07

Table 4.1, Model C

This model predicts alcohol use from the intercept and time.  It also asks whether the intercept and slope (for time) are affected by being a child of an alcoholic.

Level 1/Level 2 model

ALCUSE_ij =  π_0i + π_1iTIME +  ε_ij  
π_0i =  γ₀₀ + γ₀₁COA + ζ_0i
π_1i =  γ₁₀ + γ₁₁COA + ζ_1i

Composite model

ALCUSE_ij = γ₀₀ + γ₀₁COA + γ₁₀TIME + γ₁₁COA*TIME +  (ε_ij + ζ_0i + ζ_1iTIME ) 

Running Model in HLM

 Final estimation of fixed effects:
 ----------------------------------------------------------------------------
                                       Standard             Approx.
    Fixed Effect         Coefficient   Error      T-ratio   d.f.     P-value
 ----------------------------------------------------------------------------
 For       INTRCPT1, B0
    INTRCPT2, G00           0.315952   0.130695     2.417        80    0.016
         COA, G01           0.743212   0.194566     3.820        80    0.000
 For   AGE_14 slope, B1
    INTRCPT2, G10           0.292955   0.084228     3.478        80    0.001
         COA, G11          -0.049430   0.125389    -0.394        80    0.693
 ----------------------------------------------------------------------------

 Final estimation of variance components:
 -----------------------------------------------------------------------------
 Random Effect           Standard      Variance     df    Chi-square  P-value
                         Deviation     Component
 -----------------------------------------------------------------------------
 INTRCPT1,       U0        0.69827       0.48758    80     224.24420    0.000
   AGE_14 slope, U1        0.38807       0.15060    80     155.22430    0.000
  level-1,       R         0.58077       0.33729
 -----------------------------------------------------------------------------

Results as Level 1/Level 2 model

Fixed Effects

ALCUSE_ij =  π_0i + π_1iTIME +  ε_ij  
π_0i = .32 +  .74COA + ζ_0i
π_1i = .29 + -0.05COA + ζ_1i

Variance Components

Level 1
  Within Person, V(ε) = .34
Level 2
  Initial Status, V(ζ₀) = .48
  Rate of Change, V(ζ₁) =.15
  Cov(ζ_{0 ,} ζ₁) = -.06

We have graphed the results below, adding some annotations to help with interpretation.

Table 4.1, Model D

This model predicts alcohol use from the intercept and time.  It also asks whether the intercept and slope (for time) are affected by being a child of an alcoholic and by amount of peer drinking (at the start of the study).

Level 1/Level 2 model

ALCUSE_ij =  π_0i + π_1iTIME +  ε_ij  
π_0i =  γ₀₀ + γ₀₁COA + γ₀₂PEER + ζ_0i
π_1i =  γ₁₀ + γ₁₁COA + γ₁₂PEER + ζ_1i

Composite model

ALCUSE_ij = γ₀₀ + γ₀₁COA + γ₁₀TIME + γ₁₀COA*TIME + γ₀₂PEER + γ₁₂PEER*TIME +  (ε_ij + ζ_0i + ζ_1iTIME ) 

Running Model in HLM

 Final estimation of fixed effects:
 ----------------------------------------------------------------------------
                                       Standard             Approx.
    Fixed Effect         Coefficient   Error      T-ratio   d.f.     P-value
 ----------------------------------------------------------------------------
 For       INTRCPT1, B0
    INTRCPT2, G00          -0.316514   0.148062    -2.138        79    0.032
         COA, G01           0.579165   0.162486     3.564        79    0.001
        PEER, G02           0.694296   0.111533     6.225        79    0.000
 For   AGE_14 slope, B1
    INTRCPT2, G10           0.429429   0.113689     3.777        79    0.000
         COA, G11          -0.014032   0.124765    -0.112        79    0.911
        PEER, G12          -0.149815   0.085641    -1.749        79    0.080
 ----------------------------------------------------------------------------

 Final estimation of variance components:
 -----------------------------------------------------------------------------
 Random Effect           Standard      Variance     df    Chi-square  P-value
                         Deviation     Component
 -----------------------------------------------------------------------------
 INTRCPT1,       U0        0.49082       0.24091    79     152.28049    0.000
   AGE_14 slope, U1        0.37298       0.13911    79     149.63981    0.000
  level-1,       R         0.58077       0.33729
 -----------------------------------------------------------------------------

 Statistics for current covariance components model
 --------------------------------------------------
 Deviance                       = 588.690655
 Number of estimated parameters = 10

Results as Level 1/Level 2 model

Fixed Effects

ALCUSE_ij =  π_0i + π_1iTIME +  ε_ij  
π_0i = -.31 +  .57COA + .69PEER + ζ_0i
π_1i =  .43 + -.01COA+ -.15PEER + ζ_1i

Variance Components

Level 1
  Within Person, V(ε) = .34
Level 2
  Initial Status, V(ζ₀) = .24
  Rate of Change, V(ζ₁) =.14
  Cov(ζ_{0 ,} ζ₁) = -.006

Table 4.1, Model E

This model is the same as model D, but removes COA as a predictor of the slope for age_14.

Level 1/Level 2 model

ALCUSE_ij =  π_0i + π_1iTIME +  ε_ij  
π_0i =  γ₀₀ + γ₀₁COA + γ₀₂PEER + ζ_0i
π_1i =  γ₁₀ + γ₁₂PEER + ζ_1i

Composite model

ALCUSE_ij = γ₀₀ + γ₀₁COA + γ₁₀TIME + γ₀₂PEER + γ₁₂PEER*TIME +  (ε_ij + ζ_0i + ζ_1iTIME ) 

Running Model using HLM

 Final estimation of fixed effects:
 ----------------------------------------------------------------------------
                                       Standard             Approx.
    Fixed Effect         Coefficient   Error      T-ratio   d.f.     P-value
 ----------------------------------------------------------------------------
 For       INTRCPT1, B0
    INTRCPT2, G00          -0.313821   0.146118    -2.148        79    0.032
         COA, G01           0.571196   0.146228     3.906        79    0.000
        PEER, G02           0.695183   0.111258     6.248        79    0.000
 For   AGE_14 slope, B1
    INTRCPT2, G10           0.424687   0.105590     4.022        80    0.000
        PEER, G11          -0.151377   0.084513    -1.791        80    0.073
 ----------------------------------------------------------------------------

 Final estimation of variance components:
 -----------------------------------------------------------------------------
 Random Effect           Standard      Variance     df    Chi-square  P-value
                         Deviation     Component
 -----------------------------------------------------------------------------
 INTRCPT1,       U0        0.49086       0.24095    79     152.28729    0.000
   AGE_14 slope, U1        0.37305       0.13916    80     149.66517    0.000
  level-1,       R         0.58076       0.33729
 -----------------------------------------------------------------------------

Results as Level 1/Level 2 model

Fixed Effects

ALCUSE_ij =  π_0i + π_1iTIME +  ε_ij  
π_0i = -.31 + .57COA + .70PEER + ζ_0i
π_1i = .42 + -.15PEER + ζ_1i

Variance Components

Level 1
  Within Person, V(ε) = .34
Level 2
  Initial Status, V(ζ₀) = .24
  Rate of Change, V(ζ₁) =.14
  Cov(ζ_{0 ,} ζ₁) = -.006

Graphing

We treat a value for Peer of .65 as being low peer drinking and Peer of 1.38 as high peer drinking in the graph below.  For ease of interpretation, graph was touched up in a graph editing program.

Here is how we made the graph in HLM.

Running Model E using SAS

Level 1/Level 2 model

ALCUSE_ij =  π_0i + π_1iTIME +  ε_ij  
π_0i =  γ₀₀ + γ₀₁COA + γ₀₂PEER + ζ_0i
π_1i =  γ₁₀ + γ₁₂PEER + ζ_1i

Composite model

ALCUSE_ij = γ₀₀ + γ₀₁COA + γ₁₀TIME + γ₀₂PEER + γ₁₂PEER*TIME +  (ε_ij + ζ_0i + ζ_1iTIME ) 

We then convert the composite model into a model statement with the fixed effects and random statement with the random effects.

proc mixed data="c:\alda\alcohol1_pp" method=ml covtest;
  class id;
  model alcuse = coa age_14 peer peer*age_14 / solution ;
  random intercept age_14 / type=un sub=id;
run;

And here is the output

                  Covariance Parameter Estimates
                                    Standard         Z
Cov Parm     Subject    Estimate       Error     Value        Pr Z
UN(1,1)      ID           0.2409     0.09259      2.60      0.0046
UN(2,1)      ID         -0.00614     0.05501     -0.11      0.9111
UN(2,2)      ID           0.1392     0.05481      2.54      0.0056
Residual                  0.3373     0.05268      6.40      <.0001

                    Solution for Fixed Effects
                           Standard
Effect         Estimate       Error      DF    t Value    Pr > |t|
Intercept       -0.3138      0.1461      79      -2.15      0.0348
COA              0.5712      0.1462      82       3.91      0.0002
PEER             0.6952      0.1113      82       6.25      <.0001
AGE_14           0.4247      0.1056      80       4.02      0.0001
PEER*AGE_14     -0.1514     0.08451      82      -1.79      0.0770

Composite model

ALCUSE_ij = γ₀₀ + γ₀₁COA + γ₁₀TIME + γ₀₂PEER + γ₁₂PEER*TIME +  (ε_ij + ζ_0i + ζ_1iTIME ) 

with values of parameter estimates filled in.

ALCUSE_ij = -.31+ .57COA + .42TIME + .69PEER + -.15PEER*TIME +  (ε_ij + ζ_0i + ζ_1iTIME ) 

Results as Level 1/Level 2 model

ALCUSE_ij =  π_0i + π_1iTIME +  ε_ij  
π_0i =  γ₀₀ + γ₀₁COA + γ₀₂PEER + ζ_0i
π_1i =  γ₁₀ + γ₁₂PEER + ζ_1i

With values of parameter estimates filled in.

ALCUSE_ij =  π_0i + π_1iTIME +  ε_ij  
π_0i = -.31 + .57COA + .69PEER + ζ_0i
π_1i = .42 + -.15PEER + ζ_1i

Running Model E using MLwiN

Level 1/Level 2 model

ALCUSE_ij =  π_0i + π_1iTIME +  ε_ij  
π_0i =  γ₀₀ + γ₀₁COA + γ₀₂PEER + ζ_0i
π_1i =  γ₁₀ + γ₁₂PEER + ζ_1i

Composite model

ALCUSE_ij = γ₀₀ + γ₀₁COA + γ₁₀TIME + γ₀₂PEER + γ₁₂PEER*TIME +  (ε_ij + ζ_0i + ζ_1iTIME ) 

We then form the MLwiN model with all of the fixed effects listed above.  A random effects at level 1 is specified for the intercept to specify ε_ij and a random effect for the intercept at level 2 is specified to create ζ_0i .  A random effect for age_14 is specified to create ζ_1iTIME.

and the results are

Using multilevel models to analyze "treatment effects" over time.

• Studying effect of "antidepressant" on mood over 7 days, 3 measurements per day (21 total measures).
• We could number time 0, 1, 2... 20 but then lose the meaning of day.
• If we divide by 3, then a "one unit increase" corresponds to a one day increase in time, so time goes 0, .33, .67, 1 ... 6, 6.33, 6.67
• Patients randomly assigned to "placebo" or "treatment" groups.

Table 5.10, Model A: TIME centered at 0 (start of study)

Level 1/Level 2 model

POS_ij =  π_0i + π_1iTIME +  ε_ij  
π_0i =  γ₀₀ + γ₀₁TREAT + ζ_0i
π_1i =  γ₁₀ + γ₁₁TREAT + ζ_1i

Composite model

POS_ij = γ₀₀ + γ₀₁TREAT + γ₁₀TIME + γ₁₁TREAT*TIME + (ε_ij + ζ_0i + ζ_1iTIME ) 

 Final estimation of fixed effects:
 ----------------------------------------------------------------------------
                                       Standard             Approx.
    Fixed Effect         Coefficient   Error      T-ratio   d.f.     P-value
 ----------------------------------------------------------------------------
 For       INTRCPT1, B0
    INTRCPT2, G00         167.463464   9.326039    17.957        62    0.000
       TREAT, G01          -3.109319  12.332266    -0.252        62    0.801
 For     TIME slope, B1
    INTRCPT2, G10          -2.418121   1.730814    -1.397        62    0.162
       TREAT, G11           5.536805   2.277796     2.431        62    0.015
 ----------------------------------------------------------------------------

 Final estimation of variance components:
 -----------------------------------------------------------------------------
 Random Effect           Standard      Variance     df    Chi-square  P-value
                         Deviation     Component
 -----------------------------------------------------------------------------
 INTRCPT1,       U0       45.94957    2111.36260    62     629.96490    0.000
     TIME slope, U1        7.98338      63.73439    62     315.00708    0.000
  level-1,       R        35.07037    1229.93068
 -----------------------------------------------------------------------------

Treatment effect is difference between groups at start of study, see page 4, Figure 5.5

Model B: TIME-3.33 (middle of study)

See Page 5, Table 5.9

Level 1/Level 2 model

POS_ij =  π_0i + π_1i(TIME-3.33) +  ε_ij  
π_0i =  γ₀₀ + γ₀₁TREAT + ζ_0i
π_1i =  γ₁₀ + γ₁₁TREAT + ζ_1i

Composite model

POS_ij = γ₀₀ + γ₀₁TREAT + γ₁₀(TIME-3.33) + γ₁₁TREAT*(TIME-3.33) + (ε_ij + ζ_0i + ζ_1i(TIME-3.33) )

 Final estimation of fixed effects:
 ----------------------------------------------------------------------------
                                       Standard             Approx.
    Fixed Effect         Coefficient   Error      T-ratio   d.f.     P-value
 ----------------------------------------------------------------------------
 For       INTRCPT1, B0
    INTRCPT2, G00         159.403064   8.764424    18.188        62    0.000
       TREAT, G01          15.346694  11.544537     1.329        62    0.184
 For  TIME333 slope, B1
    INTRCPT2, G10          -2.418118   1.730816    -1.397        62    0.162
       TREAT, G11           5.536811   2.277798     2.431        62    0.015
 ----------------------------------------------------------------------------

 Final estimation of variance components:
 -----------------------------------------------------------------------------
 Random Effect           Standard      Variance     df    Chi-square  P-value
                         Deviation     Component
 -----------------------------------------------------------------------------
 INTRCPT1,       U0       44.81918    2008.75915    62    2052.73713    0.000
  TIME333 slope, U1        7.98338      63.73439    62     315.00698    0.000
  level-1,       R        35.07037    1229.93106
 -----------------------------------------------------------------------------

Treatment effect is difference between groups half-way through study, see page 4, Figure 5.5

Model C: TIME-6.67 (end of study)

See Page 5, Table 5.9

Level 1/Level 2 model

POS_ij =  π_0i + π_1i(TIME-6.67) +  ε_ij  
π_0i =  γ₀₀ + γ₀₁TREAT + ζ_0i
π_1i =  γ₁₀ + γ₁₁TREAT + ζ_1i

Composite model

POS_ij = γ₀₀ + γ₀₁TREAT + γ₁₀(TIME-6.67) + γ₁₁TREAT*(TIME-6.67) + (ε_ij + ζ_0i + ζ_1i(TIME-6.67) )

 Final estimation of fixed effects:
 ----------------------------------------------------------------------------
                                       Standard             Approx.
    Fixed Effect         Coefficient   Error      T-ratio   d.f.     P-value
 ----------------------------------------------------------------------------
 For       INTRCPT1, B0
    INTRCPT2, G00         151.342994  11.542088    13.112        62    0.000
       TREAT, G01          33.802434  15.157594     2.230        62    0.026
 For  TIME667 slope, B1
    INTRCPT2, G10          -2.418048   1.730569    -1.397        62    0.162
       TREAT, G11           5.536745   2.277472     2.431        62    0.015
 ----------------------------------------------------------------------------

 Final estimation of variance components:
 -----------------------------------------------------------------------------
 Random Effect           Standard      Variance     df    Chi-square  P-value
                         Deviation     Component
 -----------------------------------------------------------------------------
 INTRCPT1,       U0       57.64097    3322.48125    62     931.61025    0.000
  TIME667 slope, U1        7.98347      63.73582    62     315.00715    0.000
  level-1,       R        35.07036    1229.92993
 -----------------------------------------------------------------------------

Treatment effect is difference between groups at end of study, see page 4, Figure 5.5

See summary of results on Page 6, 5.10

Models for evaluating changes in "elevation" and "slope" over time.

See Figure 6.1, page 7 for different kinds of change in elevation and slope.

We will consider the following 4 models

See Figure 6.2, page 8 for the 4 models we will test.

• Figure 6.2a: Change in elevation

• Figure 6.2b: Change in slope

• Figure 6.2c: Change in elevation and slope

• Figure 6.2d: Change in elevation and slope (method 2)

Change in elevation, Figure 6.2a, Page 8

Level 1 Model (focusing on "exper" and "ged", ignoring covariates)

LNW_ij =  π_0i + π_1iEXPER + π_2iGED +  ε_ij  

Running Model in HLM

 Final estimation of fixed effects:
 ----------------------------------------------------------------------------
                                       Standard             Approx.
    Fixed Effect         Coefficient   Error      T-ratio   d.f.     P-value
 ----------------------------------------------------------------------------
 For       INTRCPT1, B0
    INTRCPT2, G00           1.734212   0.011800   146.971       887    0.000
 For    EXPER slope, B1
    INTRCPT2, G10           0.043224   0.002621    16.490       886    0.000
       BLACK, G11          -0.018200   0.004471    -4.071       886    0.000
 For      GED slope, B2
    INTRCPT2, G20           0.061321   0.018449     3.324       887    0.001
 For    HGC_9 slope, B3
    INTRCPT2, G30           0.038334   0.006265     6.119      6396    0.000
 For     UE_7 slope, B4
    INTRCPT2, G40          -0.011609   0.001788    -6.494      6396    0.000
 ----------------------------------------------------------------------------

  Final estimation of variance components:
 -----------------------------------------------------------------------------
 Random Effect           Standard      Variance     df    Chi-square  P-value
                         Deviation     Component
 -----------------------------------------------------------------------------
 INTRCPT1,       U0        0.20882       0.04361   103     146.13199    0.004
    EXPER slope, U1        0.04075       0.00166   102     177.35923    0.000
      GED slope, U2        0.16823       0.02830   103     109.96355    0.301
  level-1,       R         0.30686       0.09416
 -----------------------------------------------------------------------------

Model D: Change in slope, Figure 6.2b, Page 8

In addition to exper (time in work force) also have postexp (time in work force after getting GED).

See page 9, Table 6.1 for data example.

Level 1 Model (focusing on "exper" and "postexp")

LNW_ij =  π_0i + π_1iEXPER + π_3iPOSTEXP +  ε_ij  

Running Model in HLM

 Final estimation of fixed effects:
 ----------------------------------------------------------------------------
                                       Standard             Approx.
    Fixed Effect         Coefficient   Error      T-ratio   d.f.     P-value
 ----------------------------------------------------------------------------
 For       INTRCPT1, B0
    INTRCPT2, G00           1.749372   0.011399   153.461       887    0.000
 For    EXPER slope, B1
    INTRCPT2, G10           0.040653   0.002777    14.637       886    0.000
       BLACK, G11          -0.019495   0.004475    -4.357       886    0.000
 For  POSTEXP slope, B2
    INTRCPT2, G20           0.014591   0.004565     3.196       887    0.002
 For    HGC_9 slope, B3
    INTRCPT2, G30           0.039868   0.006354     6.275      6396    0.000
 For     UE_7 slope, B4
    INTRCPT2, G40          -0.011839   0.001791    -6.612      6396    0.000
 ----------------------------------------------------------------------------

 Final estimation of variance components:
 -----------------------------------------------------------------------------
 Random Effect           Standard      Variance     df    Chi-square  P-value
                         Deviation     Component
 -----------------------------------------------------------------------------
 INTRCPT1,       U0        0.22488       0.05057    95      97.32352    0.415
    EXPER slope, U1        0.03805       0.00145    94     128.15709    0.011
  POSTEXP slope, U2        0.02891       0.00084    95     124.13282    0.024
  level-1,       R         0.30764       0.09464
 -----------------------------------------------------------------------------

Model F: Change in elevation and slope, Figure 6.2c, Page 8

Level 1 Model (focusing on "exper" and "postexp")

LNW_ij =  π_0i + π_1iEXPER + π_2iGED + π_3iPOSTEXP +  ε_ij  

Running Model in HLM

 Final estimation of fixed effects:
 ----------------------------------------------------------------------------
                                       Standard             Approx.
    Fixed Effect         Coefficient   Error      T-ratio   d.f.     P-value
 ----------------------------------------------------------------------------
 For       INTRCPT1, B0
    INTRCPT2, G00           1.738550   0.011949   145.498       887    0.000
 For    EXPER slope, B1
    INTRCPT2, G10           0.041475   0.002798    14.824       886    0.000
       BLACK, G11          -0.019620   0.004470    -4.389       886    0.000
 For      GED slope, B2
    INTRCPT2, G20           0.040940   0.022022     1.859       887    0.063
 For  POSTEXP slope, B3
    INTRCPT2, G30           0.009412   0.005546     1.697       887    0.089
 For    HGC_9 slope, B4
    INTRCPT2, G40           0.039032   0.006243     6.252      6395    0.000
 For     UE_7 slope, B5
    INTRCPT2, G50          -0.011723   0.001783    -6.575      6395    0.000
 ----------------------------------------------------------------------------

 Final estimation of variance components:
 -----------------------------------------------------------------------------
 Random Effect           Standard      Variance     df    Chi-square  P-value
                         Deviation     Component
 -----------------------------------------------------------------------------
 INTRCPT1,       U0        0.20336       0.04135    60      79.33686    0.048
    EXPER slope, U1        0.03690       0.00136    59      70.25123    0.150
      GED slope, U2        0.13172       0.01735    60      45.99434    >.500
  POSTEXP slope, U3        0.05801       0.00336    60      66.49441    0.263
  level-1,       R         0.30637       0.09386
 -----------------------------------------------------------------------------

Figure 6.2d: Change in elevation and slope (method 2)

• See page 9, Table 6.1.  If we include ged*exper to reflect change in slope after ged, then what is the meaning of ged effect?  The effect of getting ged on the day you enter the labor force.

Summary

Multilevel modeling offers a unique framework for analyzing longitudinal data because

• It tolerates missing waves of data
• It tolerates differently spaced waves of data from different subjects
• It accounts for correlations of observations across time
• It allows you to study changes over time, such as changes in elevation and slope.
• You are not limited just to linear changes, but can explore a variety of functional forms of change over time.

For more information

• Applied Longitudinal Data Analysis: Modeling Change and Event Occurrence by Judith D. Singer and John B. Willett
• Multilevel Modeling Portal
• Downloadable Papers on Multilevel Models

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