L5 – Repeated Measures and Longitudinal Analyses using MIXED1
Review of GLM Repeated / Intro to Longitudinal Analyses
The Chapter 5 data file, ch5hortest . . .
The hor stands for horizontal. We’ll see what that means shortly.
test1, test2, test3: Scores on identical achievement tests taken at 3 different times with approximately equal intervals between tests.
effective is a dichotomous variable equal to 1 if the teacher is perceived to be effective and 0 if not.
courses ???
female is a dichotomous variable equal to 1 if teacher is female?
ses is student SES as a Z-score.
ses_mean is mean SES of all students in a school.
courses_mean Mean of courses for a school.
Questions to ask (p. 142)
1. Is there change in average achievement across the 3 test periods?
This is the basic repeated measures question – change in means across measurement times.
2. If there is change, is it systematic as opposed to merely random?
Systematically increasing? Systematically decreasing?
3. It it’s systematic, what is the shape of the curve of change across time?
Linear?Quadratic?Cubic?
4. Is change related to teacher effectiveness or student ses?
Teacher effectiveness and ses are student characteristics – theyhave the potential to affect all of the scores a given student provides. That is, each student gives us a group of scores.
All of the scores of the student may be increased or decreased – an effect on the intercept.
Or, the change in scores across time may be affected by teacher effectiveness – an effect on the slope relating scores to time.
Teacher effectiveness and ses are Level 2 characteristics in the model.
Horizontal vs. Vertical arrangement
Traditional repeated measures analyses require that the repeated measures occupy different columns of the data editor. This arrangement will be called the horizontal arrangement. Here are the first few cases of the Ch 5 data arranged horizontally . . .
The multilevel R procedures and the SPSS MIXED procedure, on the other hand, requires thateach repeated measureoccupy a different rowof the data editor.
This will be called a vertical arrangement.
It’s also called person.period arrangement.
Here are the same data arranged vertically.
(Some other variables, e.g., time, have been added. More on those later.
Major points about going from horizontal to vertical . . .
1. Each time period is a row in the vertical arrangement.
2. Values which changed from time to time from column to column in the same row now change from row to row in the same column.
3. Values which were constant across time are copied from row to row.
The Ch5 Data : Descriptive statistics related to Questions 1 and 2
This table was created using the horizontal file.
Descriptive StatisticsN / Minimum / Maximum / Mean / Std. Deviation
test1 / 8670 / 24.35 / 99.99 / 48.6323 / 9.71254
test2 / 8670 / 26.64 / 99.99 / 53.1073 / 9.88757
test3 / 8670 / 25.29 / 99.98 / 57.0944 / 9.89402
Valid N(listwise) / 8670
Correlations
test1 / test2 / test3 / effective / ses
test1 / Pearson r / 1 / .324 / .327 / .159 / .018
Sig. (2-tailed) / .000 / .000 / .000 / .097
N / 8670 / 8670 / 8670 / 8670 / 8670
test2 / Pearson r / .324 / 1 / .479 / .232 / .007
Sig. (2-tailed) / .000 / .000 / .000 / .517
N / 8670 / 8670 / 8670 / 8670 / 8670
test3 / Pearson r / .327 / .479 / 1 / .490 / -.007
Sig. (2-tailed) / .000 / .000 / .000 / .540
N / 8670 / 8670 / 8670 / 8670 / 8670
effective / Pearson r / .159 / .232 / .490 / 1 / .000
Sig. (2-tailed) / .000 / .000 / .000 / .991
N / 8670 / 8670 / 8670 / 8670 / 8670
ses / Pearson r / .018 / .007 / -.007 / .000 / 1
Sig. (2-tailed) / .097 / .517 / .540 / .991
N / 8670 / 8670 / 8670 / 8670 / 8670
The test scores are positively correlated with each other.
Kids with high test1 scores also had high test2 scores and also had high test3 scores, for example.
I included the variables, effective, and ses, just to see whether or not test scores might be related to either.
I note that the test scores appear to be related to effective though not to ses. So kids with higher “effective” had higher test scores.
Changes in mean value across time.
Graphs of selected student learning curves using the vertical file. Later I’ll show how to create the vertical file.
Data -> Select CasesI selected the 1st 18 students
Graphs -> Legacy Dialogs -> Scatter/Dot -> Simple Scatter
Analyses of the horizontal file data using the GLM Repeated Measures procedure, p. 151
(These kinds of analyses should be vaguely familiar to you from last semester.)
These analyses require the horizontal file, ch5hortest.sav or heck2e_cht5growthdata_horizontal.sav
The data editor columns corresponding to the three times are specified to SPSS.
The output of the analysis using traditional repeated measures.
GLM test1 test2 test3
/WSFACTOR=time 3 Polynomial /MEASURE=test /METHOD=SSTYPE(3)
/PLOT=PROFILE(time) /EMMEANS=TABLES(time) /PRINT=DESCRIPTIVE ETASQ OPOWER
/CRITERIA=ALPHA(.05) /WSDESIGN=time.
General Linear Model
[DataSet1] G:\MdbT\P595C(Multilevel)\Multilevel and Longitudinal Modeling with IBM SPSS\Ch5Datasets&ModelSyntaxes\ch5hortest.sav
Within-Subjects FactorsMeasure:test
time / Dependent Variable
1 / test1
2 / test2
3 / test3
Descriptive Statistics
Mean / Std. Deviation / N
test1 / 48.6323 / 9.71254 / 8670
test2 / 53.1073 / 9.88757 / 8670
test3 / 57.0944 / 9.89402 / 8670
Multivariate Testsc
Effect / Value / F / Hypothesis df / Error df / Sig. / Partial Eta Squared / Noncent. Parameter / Observed Powerb
time / Pillai's Trace / .359 / 2424.124a / 2.000 / 8668.000 / .000 / .359 / 4848.248 / 1.000
Wilks' Lambda / .641 / 2424.124a / 2.000 / 8668.000 / .000 / .359 / 4848.248 / 1.000
Hotelling's Trace / .559 / 2424.124a / 2.000 / 8668.000 / .000 / .359 / 4848.248 / 1.000
Roy's Largest Root / .559 / 2424.124a / 2.000 / 8668.000 / .000 / .359 / 4848.248 / 1.000
a. Exact statistic
b. Computed using alpha = .05
c. Design: Intercept
Within Subjects Design: time
One factor – time – one test result.
The results of multivariate analyses say that there are significant differences in mean test score across time periods.
Recall that the data must “pass” Mauchly’s test if we wish to use the Sphericity Assumed test below.
Mauchly's Test of SphericitybMeasure:test
Within Subjects Effect / Mauchly's W / Approx. Chi-Square / df / Sig. / Epsilona
Greenhouse-Geisser / Huynh-Feldt / Lower-bound
time / .977 / 206.113 / 2 / .000 / .977 / .977 / .500
Tests the null hypothesis that the error covariance matrix of the orthonormalized transformed dependent variables is proportional to an identity matrix.
a. May be used to adjust the degrees of freedom for the averaged tests of significance. Corrected tests are displayed in the Tests of Within-Subjects Effects table.
b. Design: Intercept
Within Subjects Design: time
The data failed Mauchly’s test, so we’ll ignore the Sphericity Assumed line in the table below.
Tests of Within-Subjects Effects
Measure:test
Source / Type III Sum of Squares / df / Mean Square / F / Sig. / Partial Eta Squared / Noncent. Parameter / Observed Powera
time / Sphericity Assumed / 310760.337 / 2 / 155380.168 / 2581.645 / .000 / .229 / 5163.290 / 1.000
Greenhouse-Geisser / 310760.337 / 1.954 / 159031.307 / 2581.645 / .000 / .229 / 5044.748 / 1.000
Huynh-Feldt / 310760.337 / 1.955 / 158995.891 / 2581.645 / .000 / .229 / 5045.871 / 1.000
Lower-bound / 310760.337 / 1.000 / 310760.337 / 2581.645 / .000 / .229 / 2581.645 / 1.000
Error
(time) / Sphericity Assumed / 1043513.536 / 17338 / 60.186
Greenhouse-Geisser / 1043513.536 / 16939.944 / 61.601
Huynh-Feldt / 1043513.536 / 16943.717 / 61.587
Lower-bound / 1043513.536 / 8669.000 / 120.373
a. Computed using alpha = .05
A nice feature of the GLM Repeated Measures analysis procedure is its automatic test of the shape of the curve of means across time periods. That test assumes that the time periods are equally spaced, however, so don’t rely on it if they are not. As we’ll see, if we want such tests in MIXED, we’ll have to create them using polynomials (ugh). But analyzing the shape using multilevel techniques WILL allow the time periods to be unequally spaced. The test below tells us that the shape of the curve of means over the three time periods is not precisely linear, but curvilinear. Alas, it doesn’t tell us in this table, what the nature of the curve is – whether it is curved downward or curved upward.
Tests of Within-Subjects ContrastsMeasure:test
Source / time / Type III Sum of Squares / df / Mean Square / F / Sig. / Partial Eta Squared / Noncent. Parameter / Observed Powera
time / Linear / 310416.221 / 1 / 310416.221 / 4801.239 / .000 / .356 / 4801.239 / 1.000
Quadratic / 344.115 / 1 / 344.115 / 6.176 / .013 / .001 / 6.176 / .700
Error
(time) / Linear / 560479.947 / 8669 / 64.653
Quadratic / 483033.589 / 8669 / 55.720
a. Computed using alpha = .05
Be careful!! There is no between-subjects factor, but this table will always be displayed. It’s simply telling us that the overall mean of all the scores is significantly different from 0.
Tests of Between-Subjects Effects
Measure:test
Transformed Variable:Average
Source / Type III Sum of Squares / df / Mean Square / F / Sig. / Partial Eta Squared / Noncent. Parameter / Observed Powera
Intercept / 7.291E7 / 1 / 7.291E7 / 429850.489 / .000 / .980 / 429850.489 / 1.000
Error / 1470402.182 / 8669 / 169.616
a. Computed using alpha = .05
Estimated Marginal Means
These are means of the dependent variable within each group computed assuming that all covariates had the same value.
Since there are no covariates, these means will be the observed means.
timeMeasure:test
time / Mean / Std. Error / 95% Confidence Interval
Lower Bound / Upper Bound
1 / 48.632 / .104 / 48.428 / 48.837
2 / 53.107 / .106 / 52.899 / 53.315
3 / 57.094 / .106 / 56.886 / 57.303
Profile Plots
Traditional repeated analyses with between-subjects factors.
1) Teaching effectiveness:
0 = teacher not judged to be effective;
1 = teacher judged to be effective
2) Student SES as a between subjects covariate
Note that since we have a continuous covariate in the model, it makes sense to get the parameters of the equation corresponding to that covariate. That’s why I checked the “Parameter Estimates” box.
GLM test1 test2 test3 BY effective WITH ses
/WSFACTOR=time 3 Polynomial /MEASURE=test /METHOD=SSTYPE(3)
/PLOT=PROFILE(time time*effective) /EMMEANS=TABLES(time) WITH(ses=MEAN)
/EMMEANS=TABLES(effective*time) WITH(ses=MEAN)
/PRINT=DESCRIPTIVE ETASQ OPOWER PARAMETER /CRITERIA=ALPHA(.05)
/WSDESIGN=time /DESIGN=ses effective.
General Linear Model Output
[DataSet1] G:\MdbT\P595C(Multilevel)\Multilevel and Longitudinal Modeling with IBM SPSS\Ch5Datasets&ModelSyntaxes\ch5hortest.sav
Within-Subjects FactorsMeasure:test
time / Dependent Variable
1 / test1
2 / test2
3 / test3
Between-Subjects Factors
N
effective Teacher effectiveness / .00 / 3901
1.00 / 4769
Descriptive Statistics
effective Teacher effectiveness / Mean / Std. Deviation / N
test1 / .00 / 46.9255 / 12.14551 / 3901
1.00 / 50.0284 / 6.82068 / 4769
Total / 48.6323 / 9.71254 / 8670
test2 / .00 / 50.5716 / 12.39563 / 3901
1.00 / 55.1815 / 6.51977 / 4769
Total / 53.1073 / 9.88757 / 8670
test3 / .00 / 51.7330 / 10.02157 / 3901
1.00 / 61.4799 / 7.28562 / 4769
Total / 57.0944 / 9.89402 / 8670
Multivariate Testsc
Effect / Value / F / Hypothesis df / Error df / Sig. / Partial Eta Squared / Noncent. Parameter / Observed Powerb
time / Pillai's Trace / .360 / 2434.511a / 2.000 / 8666.000 / .000 / .360 / 4869.021 / 1.000
Wilks' Lambda / .640 / 2434.511a / 2.000 / 8666.000 / .000 / .360 / 4869.021 / 1.000
Hotelling's Trace / .562 / 2434.511a / 2.000 / 8666.000 / .000 / .360 / 4869.021 / 1.000
Roy's Largest Root / .562 / 2434.511a / 2.000 / 8666.000 / .000 / .360 / 4869.021 / 1.000
time * ses / Pillai's Trace / .001 / 2.196a / 2.000 / 8666.000 / .111 / .001 / 4.392 / .451
Wilks' Lambda / .999 / 2.196a / 2.000 / 8666.000 / .111 / .001 / 4.392 / .451
Hotelling's Trace / .001 / 2.196a / 2.000 / 8666.000 / .111 / .001 / 4.392 / .451
Roy's Largest Root / .001 / 2.196a / 2.000 / 8666.000 / .111 / .001 / 4.392 / .451
time * effective / Pillai's Trace / .104 / 502.407a / 2.000 / 8666.000 / .000 / .104 / 1004.814 / 1.000
Wilks' Lambda / .896 / 502.407a / 2.000 / 8666.000 / .000 / .104 / 1004.814 / 1.000
Hotelling's Trace / .116 / 502.407a / 2.000 / 8666.000 / .000 / .104 / 1004.814 / 1.000
Roy's Largest Root / .116 / 502.407a / 2.000 / 8666.000 / .000 / .104 / 1004.814 / 1.000
a. Exact statistic
b. Computed using alpha = .05
c. Design: Intercept + ses + effective
Within Subjects Design: time
Time: There were significant differences in mean test scores across the three time periods.
Time*ses: Since it’s notsignificant, it tells us that change across time was the same for low ses kids as it was for high ses kids.
Time*effective: Since it is significant, it tells us that the difference in mean test scores across times was different for kids with less effective teachers than it was for kids with more effective teachers. That is, effective moderates the Test~Time relationship.
The data have to pass Mauchly’s test of sphericity in order for us to be able to interpret the Sphericity Assumed line below.
Mauchly's Test of SphericitybMeasure:test
Within Subjects Effect / Mauchly's W / Approx. Chi-Square / df / Sig. / Epsilona
Greenhouse-Geisser / Huynh-Feldt / Lower-bound
time / .969 / 269.218 / 2 / .000 / .970 / .971 / .500
Tests the null hypothesis that the error covariance matrix of the orthonormalized transformed dependent variables is proportional to an identity matrix.
a. May be used to adjust the degrees of freedom for the averaged tests of significance. Corrected tests are displayed in the Tests of Within-Subjects Effects table.
b. Design: Intercept + ses + effective
Within Subjects Design: time
Sphericity did not hold, so we must ignore the “Sphericity Assumed” results on the next page.
(You must ignore them. Don’t mess with the God of Statistics!!)
Measure:test
Source / Type III Sum of Squares / df / Mean Square / F / Sig. / Partial Eta Squared / Noncent. Parameter / Observed Powera
time / Sphericity Assumed / 284416.583 / 2 / 142208.292 / 2486.938 / .000 / .223 / 4973.875 / 1.000
Greenhouse-Geisser / 284416.583 / 1.941 / 146558.224 / 2486.938 / .000 / .223 / 4826.248 / 1.000
Huynh-Feldt / 284416.583 / 1.942 / 146492.120 / 2486.938 / .000 / .223 / 4828.425 / 1.000
Lower-bound / 284416.583 / 1.000 / 284416.583 / 2486.938 / .000 / .223 / 2486.938 / 1.000
time * ses / Sphericity Assumed / 245.937 / 2 / 122.969 / 2.150 / .116 / .000 / 4.301 / .443
Greenhouse-Geisser / 245.937 / 1.941 / 126.730 / 2.150 / .118 / .000 / 4.173 / .436
Huynh-Feldt / 245.937 / 1.942 / 126.673 / 2.150 / .118 / .000 / 4.175 / .436
Lower-bound / 245.937 / 1.000 / 245.937 / 2.150 / .143 / .000 / 2.150 / .311
time * effective / Sphericity Assumed / 52072.332 / 2 / 26036.166 / 455.320 / .000 / .050 / 910.641 / 1.000
Greenhouse-Geisser / 52072.332 / 1.941 / 26832.572 / 455.320 / .000 / .050 / 883.612 / 1.000
Huynh-Feldt / 52072.332 / 1.942 / 26820.470 / 455.320 / .000 / .050 / 884.011 / 1.000
Lower-bound / 52072.332 / 1.000 / 52072.332 / 455.320 / .000 / .050 / 455.320 / 1.000
Error(time) / Sphericity Assumed / 991194.398 / 17334 / 57.182
Greenhouse-Geisser / 991194.398 / 16819.517 / 58.931
Huynh-Feldt / 991194.398 / 16827.107 / 58.905
Lower-bound / 991194.398 / 8667.000 / 114.364
a. Computed using alpha = .05
The results here are the same as found in the Multivariate Results table.
There is an effect of time;there is no time*ses interaction; and there is a time*effective interaction.
So mean test performance changes across time, and it changes in a different way for kids with effective teachers than it does for kids with uneffective teachers.
Measure:test
Source / time / Type III Sum of Squares / df / Mean Square / F / Sig. / Partial Eta Squared / Noncent. Parameter / Observed Powera
time / Linear / 283779.090 / 1 / 283779.090 / 4795.547 / .000 / .356 / 4795.547 / 1.000
Quadratic / 637.493 / 1 / 637.493 / 11.551 / .001 / .001 / 11.551 / .925
time * ses / Linear / 244.668 / 1 / 244.668 / 4.135 / .042 / .000 / 4.135 / .529
Quadratic / 1.270 / 1 / 1.270 / .023 / .879 / .000 / .023 / .053
time *effective / Linear / 47359.971 / 1 / 47359.971 / 800.330 / .000 / .085 / 800.330 / 1.000
Quadratic / 4712.361 / 1 / 4712.361 / 85.386 / .000 / .010 / 85.386 / 1.000
Error(time) / Linear / 512874.459 / 8667 / 59.176
Quadratic / 478319.939 / 8667 / 55.189
a. Computed using alpha = .05
time linear:The overall mean test scores changed in a linear fashion over time.
time quadratic: The shape of the overall curve of mean test scores to time was curved.
time*effective linear: The rate of change of the curves relating test scores to time were different for the two effectiveness levels.
time*effective quadratic: The shapes of the curves relating test scores to time were different for the two effectiveness levels.
Wow! These data are getting interesting.
Tests of Between-Subjects EffectsMeasure:test
Transformed Variable:Average
Source / Type III Sum of Squares / df / Mean Square / F / Sig. / Partial Eta Squared / Noncent. Parameter / Observed Powera
Intercept / 7.122E7 / 1 / 7.122E7 / 492920.714 / .000 / .983 / 492920.714 / 1.000
ses / 91.225 / 1 / 91.225 / .631 / .427 / .000 / .631 / .125
effective / 218038.615 / 1 / 218038.615 / 1509.048 / .000 / .148 / 1509.048 / 1.000
Error / 1252273.450 / 8667 / 144.488
a. Computed using alpha = .05
The dependent variable in these tests of Between-Subjects Effects is the mean across all levels of the repeated measures variable – the mean across the three time periods.
The mean scores across the 3 times were not related to student SES values (p = .427).
(Hmm- this is different from the result with the ch3 data.)
The average scores across the 3 times were related to teaching effectiveness.
High effective: Average of test1,test2,test3 is large
Low effective: Average of test1,test2,test3 is low
The following are reported because of the presence of a covariate, ses, and because I asked for Parameter Estimates.
At each time period, test scores were regressed onto SES. The relationship was NS for each regression.
Parameter EstimatesDependent Variable / Parameter / B / Std. Error / t / Sig. / 95% Confidence Interval / Partial Eta Squared / Noncent. Parameter / Observed Powera
Lower Bound / Upper Bound
test1 / Intercept / 50.020 / .139 / 360.028 / .000 / 49.748 / 50.293 / .937 / 360.028 / 1.000
ses / .221 / .132 / 1.681 / .093 / -.037 / .479 / .000 / 1.681 / .390
[effective=.00] / -3.103 / .207 / -14.990 / .000 / -3.509 / -2.697 / .025 / 14.990 / 1.000
[effective=1.00] / 0b / . / . / . / . / . / . / . / .
test2 / Intercept / 55.178 / .139 / 395.910 / .000 / 54.905 / 55.451 / .948 / 395.910 / 1.000
ses / .088 / .132 / .669 / .504 / -.171 / .347 / .000 / .669 / .103
[effective=.00] / -4.610 / .208 / -22.201 / .000 / -5.017 / -4.203 / .054 / 22.201 / 1.000
[effective=1.00] / 0b / . / . / . / . / . / . / . / .
test3 / Intercept / 61.483 / .125 / 491.978 / .000 / 61.238 / 61.728 / .965 / 491.978 / 1.000
ses / -.082 / .118 / -.696 / .486 / -.315 / .150 / .000 / .696 / .107
[effective=.00] / -9.747 / .186 / -52.348 / .000 / -10.112 / -9.382 / .240 / 52.348 / 1.000
[effective=1.00] / 0b / . / . / . / . / . / . / . / .
a. Computed using alpha = .05
b. This parameter is set to zero because it is redundant.
So, the average of all three test scores was not related to ses, as shown above in the Tests of Between-Subjects Effects.
This table adds to that by tellingusthat none of the individual test scores was related to ses – neither test1, nor test2, nor test2, as shown here.
Estimated Marginal Means
1. timeMeasure:test
time / Mean / Std. Error / 95% Confidence Interval
Lower Bound / Upper Bound
1 / 48.477a / .103 / 48.274 / 48.680
2 / 52.877a / .104 / 52.673 / 53.080
3 / 56.606a / .093 / 56.424 / 56.789
a. Covariates appearing in the model are evaluated at the following values: ses = .0370.
The estimated marginal means at each time are computed as if all participants had the same SES score- the average of all SES scores - .0370.
These could be referred to as mean test scores adjusted for ses.
2. Teacher effectiveness * timeMeasure:test
Teacher effectiveness / time / Mean / Std. Error / 95% Confidence Interval
Lower Bound / Upper Bound
.00 / 1 / 46.926a / .154 / 46.625 / 47.226
2 / 50.572a / .154 / 50.270 / 50.873
3 / 51.733a / .138 / 51.462 / 52.004
1.00 / 1 / 50.028a / .139 / 49.756 / 50.301
2 / 55.182a / .139 / 54.908 / 55.455
3 / 61.480a / .125 / 61.235 / 61.725
a. Covariates appearing in the model are evaluated at the following values: ses = .0370.
The estimated marginal means at each combination of effectiveness level and time period are computed as if all participants had the same SES score - .0370.
These could be referred to as test means at each combination of time period and effectiveness level adjusted for ses.
Since ses was not significant, these are virtually identical to the observed means reported several pages ago. They may not be in cases when the covariate is significantly related to the dependent variable.
Profile Plots
Overall plot across effective groups.
The graph below explains the slight downward bend in the overall curve.
Students with effective teachers actually gained momentum from the 2nd to the 3rd test.
Students with ineffective teachers were not able to continue upward at the same rate – they began falling behind even more.
Note that students with less effective teachers did more poorly from the beginning. Presumably, lack of effectiveness affected performance on the first test.
Obviously, these results could have huge policy implications.
Longitudinal Using MIXED I, 2E p. 191 ff Start on 2/14
We’ve analyzed these data using the traditional repeated measures techniques.
For those analyses, we used the file ch5hortest.sav .
When longitudinal (repeated measures) data are to be analyzed using multilevel methods with MIXED, the data must be arranged differently – in what is called a ppt or vertical file.
Here’s a screen shot of the Data Editor for the file we’ll be using now, ch5vertest.sav. . .or ch5growthdata-vertical.sav
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) All variables that applied to the person whose values did not change from one time period to the next, e.g., effective and ses, were copied downwards so they appear at each time period.
3) a variable called time, with values 0, 1, and 2 for each of the three successive lines of data has been added.
For longitudinal analyses,
Level 1 data:
The individual observations from a given person at different time periods are the Level 1 data
Characteristics that vary from one time to the next are Level 1 characteristics
Time is usually the Level 1 predictor.
Level 2 data:
People are the Level 2 entities – analogous to groups in cross-sectional analysis. Each person gives us a group of scores.
Characteristics of people – their ses, sex, etc, are Level 2 characteristics
Typically, these characteristics are assumed to affect the intercept of the relationships of observations to time and the slopes of the relationships of observations to time.