Perform hypothesis testing
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1 Multivariate hypothesis tests for fixed effects Testing homogeneity of level-1 variances In the following sections, we use the model displayed in the figure below to illustrate the hypothesis tests. Partial output is shown below. Weighting Specification Weight Variable Weighting? Name Normalized? Level 1 no Level 2 no The outcome variable is MATHACH The model specified for the fixed effects was: Level-1 Level-2 Coefficients Predictors INTRCPT1, B0 INTRCPT2, G00 SECTOR, G01 MEANSES, G02 #* SES slope, B1 INTRCPT2, G10 SECTOR, G11 MEANSES, G12 '#' - The residual parameter variance for this level-1 coefficient has been set to zero. '*' - This level-1 predictor has been centered around its group mean. The model specified for the covariance components was: Sigma squared (constant across level-2 units) Tau dimensions INTRCPT1 227
2 hlm6 Summary of the model specified (in equation format) Level-1 Model Y = B0 + B1*(SES) + R Level-2 Model B0 = G00 + G01*(SECTOR) + G02*(MEANSES) + U0 B1 = G10 + G11*(SECTOR) + G12*(MEANSES) Note, the middle section of output has been deleted. We proceed directly to the final results page. Final estimation of fixed effects: Standard Approx. Fixed Effect Coefficient Error T-ratio d.f. P-value INTRCPT2, G SECTOR, G MEANSES, G INTRCPT2, G SECTOR, G MEANSES, G Final estimation of fixed effects (with robust standard errors) Standard Approx. Fixed Effect Coefficient Error T-ratio d.f. P-value INTRCPT2, G SECTOR, G MEANSES, G INTRCPT2, G SECTOR, G MEANSES, G Final estimation of variance components: Random Effect Standard Variance df Chi-square P-value Deviation Component INTRCPT1, U level-1, R Statistics for current covariance components model Deviance = Number of estimated parameters = 2 Multivariate hypothesis tests for fixed effects Consider the model displayed above, a user can test the following composite null hypothesis: where is the effect of SECTOR on the intercept and is the effect of sector on the SES slope. 228
3 Below is a procedure that illustrates a Windows execution of the hypothesis test. To pose a multivariate hypothesis test among the fixed effects, open the Other Settings menu and select the Estimation Settings option to open the Hypothesis Testing HLM2 dialog box. Click "1" to open the General Linear Hypothesis: Hypothesis 1 dialog box and to specify the first hypothesis (see the figure below for the contrast comparing the effect of SECTOR on the intercept and on the SES slope). Click OK. 229
4 hlm6 The HLM2 output associated with this test is as follows. Results of General Linear Hypothesis Testing Coefficients Contrast INTRCPT2, G SECTOR, G MEANSES, G INTRCPT2, G SECTOR, G MEANSES, G Chi-square statistic = Degrees of freedom = 2 P-value = The table above is a reminder of the multivariate contrast specified. The chi-square statistic and associated p-value indicate that it is highly unlikely that the observed estimates for G01 and G11 could have occurred under the specified null hypothesis. Testing homogeneity of level-1 variances HLM2 assumes homogeneity of residual variance at level 1. That is, it specifies a common within each of the J level-2 units. As an option, HLM2 tests the adequacy of this assumption. Consider again the same model shown above, to test homogeneity of level-1 variances, click the Test homogeneity of level-1 variance box. 230
5 The HLM2 output associated with this test is shown below. (For a further discussion of this test see Hierarchical Linear Models, pp ) Test of homogeneity of level-1 variance Chi-square statistic = Number of degrees of freedom = 159 P-value = These results indicate that there is variability among the (J = 160) level-2 units in terms of the residual within-school (i.e., level-1) variance. 231
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