3nd meeting: Multilevel modeling: introducing level 1 (individual) and level 2 (contextual) variables + interactions Subjects for today:  Intra Class Correlation once more  Including individual (level 1) variables  Including contextual (level 2) variables  (cross level) interactions

Last meeting: Intra Class Correlation: a fraction, it tells us what the relative share of level 2 variance is in the total variance: ICC = level 2 / level 2 + level 1 variance ICC is always between 0 (only level 1 variance, no clustering) and 1 (only level 2 variance) We test with a Chi-square test: difference in -2 loglikelihood between model with both level 1 and level 2 variance and model with level 2 variance set to zero. 2

Multilevel null model: ICC = / ( ) =.23 3

Test whether ICC is significant or whether level 2 variance is significant different from zero we perform a Chi-square test: -2 loglikelihood from 1 level model - -2 loglikelihood from 2 level model: – = 132 with 1 df. Which is highly significant. This test is superior to Z-test in SPSS because the latter assumes normal distribution. Note: we test one sided because outcome is always zero or higher Model with one level only: 4

Testing in MLwin with Chi-square (more info in document about testing, see “statistical testing in Mlwin.pdf” Type cprob 10 1 and press [Enter] Note that must be divided by 2 voor level 2 variance testing 5

So we have taken one step in the whole procedure: we tested whether there is significant level 2 variance. In the example with schools we indeed have a level 2 variance. The next step is to add level 1 variables to the equation: Y = b0j + b1 * level 1 variable + eij + uoj Why? First, we might have hypotheses about level 1 variables, for instance we assume that white students perform best on a math test compared to other students. Second, we might expect that level 1 variables can explain level 2 variance. How is that? 6

Suppose So we have had one step in the whole procedure: we tested whether there is level 2 variance. In the example with schools we indeed have a level 2 variance The next step is to add level 1 variables to the equation: Y = b0 + b1 * level 1 variable + eij + uoj Why? First, we might have hypotheses about level 1 variables, for instance we assume that white students perform better on a math test. Second, we might expect that level 1 variables can explain level 2 variance. How is that? 7

Well, suppose being white indeed increases the chances to have a higher grade at a math test. Let us further assume that the fraction whites across schools varies. So assume school 1 has 80% whites and school 2 has 40% whites and we have a positive effect for being white on a math test. This must result in a lower average grade for school 2 compared to school 1, all other things being equal. This is called a compositional effect: the mere fact that whites perform better on a math test, school 1 (with 80% whites) will perform better than school 2 (with 40% whites). This eventually means that part of the difference between the school’s average is a result of the race composition. As a consequence introducing the individual level variable ‘race’ will REDUCE level 2 variance!! 8

Including the dummy ‘white’ reduces level 2 variance while it was in the null model Level 1 variance is also reduced (was ) Note that -2 loglikelihood dropped to was Being white increases Grade by 4.7! 9

Testing the effect of being white: 2 options: One is traditional t-test: b / se = t with df = n – p (where n=sample, p=number of level 1 predictors: / = with 519 – 1 = 518. Data manipulation  Command interface and type: tprob Second option: take the -2 loglikelihood difference from model with and without level 1 predictor and use Chi-square test: = with df = 1 (in linear models it is equal to t-test because square root taken from = 3.946) So type cprob Take half of that because directional hypothesis! 10

Of course one will introduce more level 1 variables. First, to test hypotheses, second to find a (partly) explanation for level 2 variance (we now use the full dataset: LOGSCHOOL.SAV: 415 schools, >8000 pupils. After we included all relevant level 1 predictors we move on to step number 2: introduce level 2 variables: here the % whites (variable name: race2_mean) 415 schools, t-test: 7.555/.8 = 9!! With df = n2 – 1 – 1=

You see there is a difference between a compositional effect from being white and the level 2 effect of %whites. The first effect is just due to individual behavior that taken together influences the school’s average on a math test. The second effect tells us that individuals perform better when a school has a high percentage of whites. Now let us finally include public versus private school (students in public schools do worse: estimate ). 415 schools, df=415 –

Now we have done 3 things: 1) estimate ICC, 2) included level 1 variables, 3) include level 2 variables. But now…. We might think that the effect of homework varies across schools: 13

Let us get back to the model with no random effect of homework. So effect of homework is constant across schools: 14

There are two extra parameters: 1) the variance of the slope for Homework (not all with the same slope angle, even -! 2) a co-variance between slope and intercept (high on intercept, lower slope) Effect of homework set random across schools: 15

Ok, so we know that the effect of homework depends on what school you are in: in some schools the effect is low, while in others it is high. WHY is that? One explanation: it is so because in private schools quality of staff is higher so making homework is less important. This would imply that the effect of homework is stronger when school is public: Effect of homework when school = private equals.914. When public the effect doubles = = 1.831!! 16