These Analyses Require the Vertical File, Ch5vertest.Sav

L8 1 L8 - Longitudinal Using MIXED I, p. 162 ff

These analyses require the vertical file, ch5vertest.sav .

We’ve analyzed these data using the traditional repeated measures techniques. For those analyses, we used the file ch5hortest.sav .

Here’s a screen shot of the first few lines of the Data Editor . . .

The major differences between the “hor” and “ver” versions of the data are that

1) test1, test2, and test3 have been replaced by a single column, test, with the values of test1, test2, and test3 placed on successive lines of the data file.

2) a variable called index1 (on the right of the screen shot) has been added to identify each line of the 3 lines per subject.

3) a variable called time, with values 0, 1, and 2 for each of the three successive lines of data has been added.

4) a variable called quadtime with values 0, 1, and 4 (the squares of time) has been added.

The π, β, γ notation will be used for Level 1, Level 2, and Level 3 variables.

This is because meaningful comparisons involving repeated measures require at least 3 levels.

Level 1 = measurements at different time periods. Level 2 = People. Level 3 = Groups of people.

For this lecture, there won’t be any Level 3 variables, so we’ll use only 2 subscripts per symbol. L8 2 The Level 1 model (p. 162) assesses the nature of the change in test performance over time,

2 Yti = p0i + p1i*timeti + p2i*time ti + eij

This is a Level 1 equation, relating test performance to time and to the square of time.

The text used the symbol a and a2 to represent “time-varying variables of interest”. Since the variable of interest is time itself, I’ve simply put time in the equation.

The coefficient of time, p1i, assesses amount of linear change in test performance over time periods for person i. 2 The coefficient of time , p2i, assesses amount of quadratic change over time periods for person i.

(On page 163, the text next presents information on the Level1 variance/covariance matrix. That will be dealt with here later in the lecture.

The Level 2 models are presented beginning in the last paragraph of p. 164 of the text.

Level 2 Model 1.

The first Level 2 model is one that simply assumes that all of the Level 1 parameters, p0i, p1i, and p2i, just vary randomly from person to person.

The intercept model: p0i = B00 + g0i Eq. 5.8, p. 165

The linear slope model: p1i = B10 + g1i Eq. 5.9

The quadratic slope model: p2i = B20 + g2i Eq. 5.10 (The text presents a random component for p2i in Eq. 5.10, then says in the following paragraph that it’s not significant, so they don’t actually include the random component in the tested model.)

This leads to the following combined model of Yti

2 Yti = B00 + g0i + (B10 + g1i)*timeti + (B20 + g2i)*time ti + eti

Multiplying through . . .

2 2 Yti = B00 + g0i + B10*timeti + g1i*timeti + B20*time ti +g2i*time ti + eti

This is the model that is applied in the following. L8 3 Level 2 Model 1 (Text p. 164-165)

Note that there is a field labeled “Repeated” that is empty.

Use of this field allows you to specify the nature of the variance/covariance matrix of the repeated measures.

Later . . . L8 4 L8 5

I asked for random effects associated with both time and quadtime – NOT what was done on p. 168.

Note that I’ve asked for an Unstructured Covariance type. More on this later. L8 6 2 2 Yti = B00 + g0i + B10*timeti + g1i*timeti + B20*time ti +g2i*time ti+ eti (The term in red was not in the text’s model, for reasons shown below.) MIXED test WITH time quadtime /CRITERIA=CIN(95) MXITER(100) MXSTEP(10) SCORING(1) SINGULAR(0.000000000001) HCONVERGE(0, ABSOLUTE) LCONVERGE(0, ABSOLUTE) PCONVERGE(0.000001, ABSOLUTE) /FIXED=time quadtime | SSTYPE(3) /METHOD=REML /PRINT=G SOLUTION TESTCOV /RANDOM=INTERCEPT time quadtime | SUBJECT(id) COVTYPE(UN).

Mixed Model Analysis

[DataSet1] G:\MdbO\html\myweb\PSY5950C\ch5vertest.sav

Warnings Iteration was terminated but convergence has not been achieved. The MIXED procedure continues despite this warning. Subsequent results produced are based on the last iteration. Validity of the model fit is uncertain. I tried running the model again with Iterations = 10,000, to no avail.

Model Dimensionb

Covariance Number of Number of Levels Structure Parameters Subject Variables Fixed Effects Intercept 1 1 time 1 1 quadtime 1 1 Random Effects Intercept + time + quadtimea 3 Unstructured 6 id Residual 1 Total 6 10

a. As of version 11.5, the syntax rules for the RANDOM subcommand have changed. Your command syntax may yield results that differ from those produced by prior versions. If you are using version 11 syntax, please consult the current syntax reference guide for more information. b. Dependent Variable: test. Information Criteriaa -2 Restricted Log Likelihood 189125.000 Akaike's Information Criterion 189139.000 (AIC) Hurvich and Tsai's Criterion 189139.005 (AICC) Bozdogan's Criterion (CAIC) 189203.163 Schwarz's Bayesian Criterion 189196.163 (BIC)

The information criteria are displayed in smaller- is-better forms. a. Dependent Variable: test. L8 7 2 2 Yti = B00 + g0i + B10*timeti + g1i*timeti + B20 *time ti +g2i*time ti+ eti

Fixed Effects Type III Tests of Fixed Effectsa

Source Numerator df Denominator df F Sig. Intercept 1 9344.654 225847.750 .000 time 1 9419.171 510.464 .000 quadtime 1 9388.302 6.432 .011

a. Dependent Variable: test. Estimates of Fixed Effectsa

95% Confidence Interval

Parameter Estimate Std. Error df t Sig. Lower Bound Upper Bound

Intercept 48.632275 .102333 9344.654 475.234 .000 48.431679 48.832870 B00 time 4.719043 .208868 9419.171 22.593 .000 4.309617 5.128469 B10 quadtime -.243999 .096205 9388.302 -2.536 .011 -.432582 -.055416 B20

a. Dependent Variable: test. Covariance Parameters – Estimates of Covariance Parametersb

Parameter Estimate Std. Error Wald Z Sig. Lower Bound Upper Bound

Residual 49.742772 .780814 63.706 .000 48.235706 51.296923 V(eti)

Intercept + time + UN (1,1) 41.050255 1.455393 28.206 .000 38.294589 44.004218 Var(g0i) quadtime [subject = id] UN (2,1) -16.073659 1.941758 -8.278 .000 -19.879435 -12.267883 Cov(g0i,g1i)

UN (2,2) 54.907088 2.239022 24.523 .000 50.689475 59.475628 Var(g1i)

UN (3,1) 5.390135 .827164 6.516 .000 3.768924 7.011345 Cov(g2i,g0i)

UN (3,2) -17.572350 .534529 -32.874 .000 -18.620008 -16.524691 Cov(g2i,g1i)

UN (3,3) 5.630711a .000000 . . . . Var(g2i)

a. This covariance parameter is redundant. The test statistic and confidence interval cannot be computed.

The red line represents a problem in estimation of all the parameters. In this case, it was the variance of the random component associated with quadtime.

There are at least two possible reasons. 1) the variance is negligible. 2) the variance of g2i is so highly correlated with the variances of the other parameters (g0i, g1i) that it was not estimable.

In either case, the solution is to re-apply the model without trying to estimate the offending parameter. This is what the authors of the text discovered and which led to the statement on p. 165 “It turns out that the quadratic component does not vary across individuals, so we can treat it as a fixed parameter by removing the random term (g2i).”

What’s below is what they ended up doing. L8 8 Level 2 Model 1 without random variation in quadtime coefficient.

A model with only randomly varying linear time slope. This is the model the text ultimately used on p. 166 ff.

2 2 Yti = B00 + g0i + B10*timeti + g1i*timeti + B20 *time ti +g2i*time ti+ eti

The model was the same as that above with the exception that only time was included in the Model: field. L8 9 Fixed Effects

Type III Tests of Fixed Effectsa

Source Numerator df Denominator df F Sig. Intercept 1 10597.930 223584.460 .000 time 1 10329.959 526.601 .000 quadtime 1 8669.000 6.176 .013 a. Dependent Variable: test.

Estimates of Fixed Effectsa

Intercept 48.632275 .102850 10597.930 472.847 .000 48.430670 48.833880 B00 time 4.719043 .205643 10329.959 22.948 .000 4.315944 5.122142 B10 quadtime -.243999 .098184 8669.000 -2.485 .013 -.436463 -.051535 B20 a. Dependent Variable: test.

Covariance Parameters Estimates of Covariance Parametersa

Residual 55.719643 .846328 65.837 .000 54.085318 57.403354 Var(eti)

Intercept + time [subject UN (1,1) 35.992459 1.436950 25.048 .000 33.283458 38.921950 Var(g0i) = id] UN (2,1) -1.246911 .764304 -1.631 .103 -2.744918 .251097 Cov(g0i,g1i)

UN (2,2) 4.466856 .648198 6.891 .000 3.361103 5.936386 Var(g1i a. Dependent Variable: test. Note that a parameter “emerged” from the analysis of this model – the covariance of the intercept (g0i) and the slope (g1i). It is often found that persons who start low increase at a rapid rate while persons who start high increase at a lower rate. This would imply a negative correlation between intercept and slope. The actual correlation that was found here was negative (covariance = -1.246911), but it was not significantly different from 0.

Conclusions

Test scores increase with time. The rate of increase is initially high, but levels off (the negative quadtime coefficient, B20). There is significant variability in individual test scores about the predicted value at each time point. There is significant variability between people in initial test scores There is significant variability between people in linear rate of increase across time. L8 10 A model with only randomly varying quadtime slope

2 2 Yti = B00 + g0i + B10*timeti + g1i*timeti + B20 *time ti +g2i*time ti+ eti L8 11 Fixed Effects

Type III Tests of Fixed Effectsa

a. Dependent Variable: test.

Intercept 48.632275 .102844 13510.270 472.874 .000 48.430686 48.833863 B00 time 4.719043 .211335 17488.541 22.330 .000 4.304805 5.133281 B10 B quadtime -.243999 .101546 17504.750 -2.403 .016 -.443039 -.044959 20

Covariance Parameters Estimates of Covariance Parametersb

Residual 59.572977 .637071 93.511 .000 58.337336 60.834790 Var(eti)

Intercept + quadtime UN (1,1) 32.128886 1.077234 29.825 .000 30.085423 34.311144 Var(g0i) [subject = id] UN (2,1) 1.152874 .210592 5.474 .000 .740121 1.565628 Covg0i,g2i)

a UN (2,2) .041400 .000000 . . . . Var(g2i

a. This covariance parameter is redundant. The test statistic and confidence interval cannot be computed. b. Dependent Variable: test. Random Effect Covariance Structure (G)a

Intercept | id quadtime | id Intercept | id 32.128886 1.152874 quadtime | id 1.152874 .041400

Unstructured a. Dependent Variable: test.

I’m not sure why we’re getting the message associated with footnote a. That message is probably why the text dropped attempts to estimate randomly variability in the quadtime coefficient.

Based on these considerations, we’ll continue with a model that includes fixed slopes for time and quadtime but random variation in the slopes only for time. L8 12 Modeling variance/covariance matrices

Text p. 163-164, 171-177

General Discussion

Any time you measure a random variable, you should measure how variable it is. The variance is an acceptable measure for inferential purposes, although the standard deviation is better for description.

If you measure pairs or triplets of random variables on the same entities, you should also consider the extent to which they covary. The measure of how much two variables covary that is analogous to the variance is called the covariance. It, like the variance, is useful for inference. For description of covariation, we often use the correlation.

The are two types of covariance that may need to be considered when doing longitudinal models.

Consider Model 1 for the above data (Ch5vertest.sav) the we ended up with above . . .

2 Yti = B00 + g0i + B10*timeti + g1i*timeti + B20*time ti + eti

The random components of this model are g0i, g1i, and eti.

Note that g0i and g1i are Level 2 characteristics - characteristics of the people. One the other hand, eti is a Level 1 characteristic – of the time of measurement of a person.

It is assumed in virtually all applications of these models that eti is independent of g0i, and g1i, so the covariance of eti with both of them is assumed to be 0. We won’t consider that covariance any more.

But g0i, and g1i are random characteristics of the people in the data. That is, each person has a value of g0i and g1i, associated with him/herself. These random values may covary. So we may want to consider the variances and covariances of the Level 2 characteristics. These are the quantities that we specify in the Random Effects dialog box and that we inspect in the Estimates of Covariance Parameters table.

Buried in the above model, not visible in the equations are a whole collection of other variances and covariances. These variance and covariances are only considered when the Level 1 observations are repeated measures. (They could be considered in the first type of multilevel modeling we just finished considering, but because of the fact that in “cross sectional” multilevel modeling there are usually large numbers of Level 1 observations per group – much larger than the 3 per “group” (person) here – they are not considered.)

The variances and covariances are of the eti values. Since t takes on only 3 values – 0, 1, and 2 – we can consider the variance of e0i, the variance of e1i, and the variance of e2i. We can also consider the covariance of e0i with e1i, the covariance of e0i with e2i, and the covariance of e2i with e3i. These Level 1 variances and covariances may also be formally considered in the models. They can be specified in the very first dialog box in MIXED – by specifying a variable that identifies different Level 1 observations (e.g., Index or Time in the present example). L8 13 Below is a presentation on variances and covariances in general. – p. 162-163, p. 171 in text

Be sure to remember that matrices below might be matrices of variances and covariances of the intercept and slope deviations, g0i and g1i.

Or they might be matrices of variances and covariances of the individual time deviations, eo1, e1i, and e2i.

Types of Variance / Covariance Matrices

A Completely Unstructured variance/covariance matrix for the three measures can be expressed as

Time 1 Time 2 Time 3 (This is the “Unstructured” option for MIXED.) 2 Time 1 σ1 σ12 σ13 2 Time 2 σ2 σ23 (Every value is potentially different from every other value) 2 Time 3 σ3

In standard repeated measures analyses, the variance/covariance matrix of the etis is computed from the sample variances and covariances. In MIXED analyses, however, the variances and covariances are less closely tied to the data and often must be estimated as part of the function maximization process that estimates the other parameters of the model. Thus the Level 1 variances and covariances become estimable parameters of the model. Although it would be desirable to estimate the 3 different variances and the 3 different covariances shown above, it may happen that not enough data are available to obtain reliable estimates of all the values. This is what is reported at the bottom of p. 171 in the text.

Be sure to remember that the variances and covariances referred to above could be of two types . . .

1) variances and covariances of the etis – the random deviations of individual scores at each time period from the Level 1 model.

2) variances and covariances of g0i, g1i, g2i, etc. – the random deviations of Level 1 intercepts and slopes. L8 14 In cases where all variances and covariances are not separately estimable, it may be possible to estimate a restricted model of the Level 1 covariances. The possible restricted models are discussed in the text on page 163-164. For review, they are shown below. The presentation focuses on variances and covariances of the etis. But all of the types of matrices could just as well apply to variances and covariances of g0i, g1i, g2i, etc. the Level 2 random components. Scaled identity Time 1 Time 2 Time 3 Time 1 Time 2 Time 3 Time 1 σ2 0 0 [1 0 ] Time 2 σ2 0 = σ2 | 1 0 | Time 3 σ2 [ 1 ]

Compound Symmetry Time 1 Time 2 Time 3 2 2 2 2 Time 1 σ +σ1 σ σ 2 2 2 Time 2 σ +σ1 σ 2 2 Time 3 σ +σ1

Diagonal (This is the default, “Variance Components” option for MIXED.) Time 1 Time 2 Time 3 2 Time 1 σ1 0 0 2 Time 2 σ2 0 2 Time 3 σ3

Autoregressive Time 1 Time 2 Time 3 Time 1 Time 2 Time 3 2 2 2 2 2 Time 1 σε σε ρ σε ρ [1 ρ ρ ] 2 2 2 Time 2 σε σε ρ = σε | 1 ρ | 2 Time 3 σε [ 1 ] L8 15 SPSS commands to specify the model of the Level 1 variance/covariance matrix between the etis for Repeated measures

Note the repeated covariance structure menu L8 16

Note that only random variation in the linear slopes is specified. L8 17

MIXED test WITH time quadtime /CRITERIA=CIN(95) MXITER(100) MXSTEP(10) SCORING(1) SINGULAR(0.000000000001) HCONVERGE(0, ABSOLUTE) LCONVERGE(0, ABSOLUTE) PCONVERGE(0.000001, ABSOLUTE) /FIXED=time quadtime | SSTYPE(3) /METHOD=REML /PRINT=G SOLUTION TESTCOV /RANDOM=INTERCEPT time | SUBJECT(id) COVTYPE(UN) /REPEATED=time | SUBJECT(id) COVTYPE(DIAG). Mixed Model Analysis

[DataSet1] G:\MdbO\html\myweb\PSY5950C\ch5vertest.sav

Model Dimensionb Number of Covariance Number of Subject Number of Levels Structure Parameters Variables Subjects Fixed Effects Intercept 1 1 time 1 1 quadtime 1 1 Random Effects Intercept + timea 2 Unstructured 3 id Repeated Effects time 3 Diagonal 3 id 8670 Total 8 9 a. As of version 11.5, the syntax rules for the RANDOM subcommand have changed. Your command syntax may yield results that differ from those produced by prior versions. If you are using version 11 syntax, please consult the current syntax reference guide for more information. b. Dependent Variable: test. Information Criteriaa -2 Restricted Log Likelihood 189111.502 Akaike's Information Criterion 189123.502 (AIC) Hurvich and Tsai's Criterion 189123.505 (AICC) Bozdogan's Criterion (CAIC) 189178.499 Schwarz's Bayesian Criterion 189172.499 (BIC) The information criteria are displayed in smaller- is-better forms. a. Dependent Variable: test. L8 18 2 Yti = B00 + g0i + B10*timeti + g1i*timeti + B20 *time ti + eti (Rep=Diag; Ran=UN) Argh!! I had to add a descriptor to the equation to remind us which variance/covariance matrix was being used for the two types of random effects – the Level 1 etis, and the Level 2 g0i, g1i. Fixed Effects Type III Tests of Fixed Effectsa

Parameter Estimate Std. Error df t Sig. Lower Bound Upper Bound Intercept 48.632275 .104309 8669.000 466.231 .000 48.427804 48.836746 B00 time 4.719043 .213341 8669.000 22.120 .000 4.300845 5.137242 B10 quadtime -.243999 .098184 8669.000 -2.485 .013 -.436463 -.051535 B20

Covariance Parameters (Diagonal V/C matrix of Level 1 observations)

Estimates of Covariance Parametersa

Parameter Estimate Std. Error Wald Z Sig. Lower Bound Upper Bound Repeated Var: [time=0] 63.583481 2.025908 31.385 .000 59.734217 67.680792 Var(e1i)

Measures Var: [time=1] 58.778811 1.158060 50.756 .000 56.552321 61.092959 Var(e2i)

Var: [time=2] 35.619135 1.980283 17.987 .000 31.941840 39.719778 Var(e3i)

Intercept + time UN (1,1) 30.749900 1.841123 16.702 .000 27.345054 34.578698 Var(g0i) [subject = id] UN (2,1) .354646 1.053148 .337 .736 -1.709485 2.418777 Cov(g0i,g1i)

UN (2,2) 7.526024 .964305 7.805 .000 5.854660 9.674523 Var(g1i)

Why were there no covariance terms specified for the etis? Because the matrix requested to be estimated was the diagonal matrix. Random Effect Covariance Structure (G)a

Intercept | id time | id Intercept | id 30.749900 .354646 time | id .354646 7.526024

Unstructured a. Dependent Variable: test. L8 19 Effect of changing the above Diagonal model of Level 1 variance/covariance matrix to an Unstructured model of Level 1 variances and covariances.

UNstructured specified.

Since the text reported they had failure to converge with the UN option, I multiplied the number of iterations by 10, from 100 to 1000. (See text at bottom of p. 171)

The algorithm still failed to converge. Key portions of the output (not shown in the text) follow L8 20 2 Yti = B00 + g0i + B10*timeti + g1i*timeti + B20 *time ti + eti (Rep=UN; Ran=UN)

Covariance Parameters

Estimates of Covariance Parametersb

Repeated Measures UN (1,1) 36.166756 1.215165 29.763 .000 33.861802 38.628607 Var(e1i)

UN (2,1) -18.430220 .975324 -18.897 .000 -20.341820 -16.518620 Cov(e2i,e1i)

UN (2,2) 43.517629 1.430301 30.425 .000 40.802675 46.413232 Var(e2i)

UN (3,1) -9.443867 .978212 -9.654 .000 -11.361127 -7.526607 Cov(e3i,e1i)

UN (3,2) -12.092249 1.075670 -11.242 .000 -14.200523 -9.983975 Cov(e2i,e2i) UN (3,3) 20.878378 1.307690 15.966 .000 18.466426 23.605362 Var(e3i) Intercept + time [subject UN (1,1) 50.330233a .000000 . . . . Var(g0i) = id] UN (2,1) -3.005638a .000000 . . . . Cov(g0i,g1i) UN (2,2) 8.087527a .000000 . . . . Var(g1i) a. This covariance parameter is redundant. The test statistic and confidence interval cannot be computed. b. Dependent Variable: test.

Although the program estimated the Level 1 variances and covariances, it was unable to estimate the standard errors of the Level 2 random parameters – variance of the intercept, variance of the time slope, and the covariance between them.

The Level 1 V/C matrix is estimated as

Time 1 Time 2 Time 3 Time 1 36.2 -18.4 -9.4 Time 2 43.5 -12.1 Time 3 20.9 L8 21 Finally, effect of changing the model of the Level 1 variances and covariances, (p. 172) to an autoregressive model. L8 22

2 Model is Yti = B00 + g0i + B10*timeti + g1i*timeti + B20 *time ti + eti (Rep=AR(1); Ran=Diag)

MIXED test WITH time quadtime /CRITERIA=CIN(95) MXITER(1000) MXSTEP(10) SCORING(1) SINGULAR(0.000000000001) HCONVERGE(0, ABSOLUTE) LCONVERGE(0, ABSOLUTE) PCONVERGE(0.000001, ABSOLUTE) /FIXED=time quadtime | SSTYPE(3) /METHOD=REML /PRINT=G SOLUTION TESTCOV /RANDOM=INTERCEPT time | SUBJECT(id) COVTYPE(DIAG) /REPEATED=time | SUBJECT(id) COVTYPE(AR1).

Mixed Model Analysis

[DataSet1] G:\MdbO\html\myweb\PSY5950C\ch5vertest.sav Model Dimensiona

Number of Covariance Number of Subject Number of Levels Structure Parameters Variables Subjects Fixed Effects Intercept 1 1 time 1 1 quadtime 1 1 Random Effects Intercept + time 2 Diagonal 2 id Repeated Effects time 3 First-Order 2 id 8670 Autoregressive Total 8 7

Information Criteriaa -2 Restricted Log Likelihood 189274.041 Akaike's Information Criterion 189282.041 (AIC) Hurvich and Tsai's Criterion 189282.042 (AICC) Bozdogan's Criterion (CAIC) 189318.705 Schwarz's Bayesian Criterion 189314.705 (BIC)

The information criteria are displayed in smaller- is-better forms. a. Dependent Variable: test. L8 23 2 Model is Yti = B00 + g0i + B10*timeti + g1i*timeti + B20 *time ti + eti (Rep=AR(1); Ran=Diag) Type III Tests of Fixed Effectsa

a. Dependent Variable: test. Covariance Parameters Estimates of Covariance Parametersa

Repeated Measures AR1 diagonal 59.644473 1.349246 44.206 .000 57.057767 62.348447 Var(eij) AR1 rho .056248 .018970 2.965 .003 .019007 .093333 Rho

Intercept + time [subject Var: Intercept 32.142176 1.171808 27.430 .000 29.925610 34.522922 Var(g0i)

= id] Var: time 2.885435 .462885 6.234 .000 2.106980 3.951502 Var(g1i)

The Level 1 V/C matrix is estimated as Time 1 Time 2 Time 3 Time 1 59.6 3.34 .187 1 .056 .003 Time 2 59.6 3.34 1 .056 Time 3 59.6 1 (3.34 = 59.6*.056) (.187=59.6*.0562)

Random Effect Covariance Structure (G)a Intercept | id time | id Intercept | id 32.142176 0 time | id 0 2.885435 Diagonal a. Dependent Variable: test. Note that the estimates of the Fixed parameters were not much affected by different assumptions concerning the variances and covariances of the Level 1 observations and the different assumptions concerning the covariance of the Level 2 random parameters – the random intercept and random time slope. The significance of the variance of the intercept parameter indicates that there may be factors affecting the heights of the Level 1 scatterplots. The significance of the variance of the linear slope parameter indicates that there may be factors affecting the slopes of the Level 1 scatterplots.