QuestionsFromYesterday

1、Questions From YesterdayEquation 2: r-to-z transformEquation is correctComparable to other p-value estimates (z = r sqrtn)ANOVA will not be able to detect a group effect that has alternating + and ICCEffect defined in terms of between and within group variability rather than being represented indivi

2、duallySPSS Advanced Models can be ordered at the VU Bookstore for $511Hierarchical Linear Modeling (HLM)Theoretical introductionIntroduction to HLMHLM equationsHLM interpretation of your data setsBuilding an HLM modelDemonstration of HLM softwarePersonal experience with HLM tutorial2General Informat

3、ion and TerminologyHLM can be used on data with many levels but we will only consider 2-level modelsThe lowest level of analysis is Level 1 (L1), the second lowest is Level 2 (L2), and so onIn group research, Level 1 corresponds to the individual level and Level 2 corresponds to the group levelYour

4、DV has to be at the lowest level3When Should You Use HLM?If you have mixed variablesIf you have different number of observations per groupIf you think a regression relationship varies by groupAny time your data has multiple levels4What Does HLM Do?Fits a regression equation at the individual levelLe

5、ts parameters of the regression equation vary by group membershipUses group-level variables to explain variation in the individual-level parametersAllows you to test for main effects and interactions within and between levels5The Level 1 Regression EquationPredicts the value of your DV from the valu

6、es of your L1 IVs (example uses 2)Equation has the general form of Yij = B0j + B1j * X1ij + B2j * X2ij + rij“i” refers to the person number and “j” refers to the group numberSince the coefficients B0, B1, and B2 change from group to group they have variability that we can try to explain6Level 2 Equa

7、tionsPredict the value of the L1 parameters using values of your L2 IVs (example uses 1)Sample equations:B0j = G00 + G01 * W1j + u0j B1j = G10 + G11 * W1j + u1j B2j = G20 + G21 * W1j + u2j You will have a separate equation for each parameter7Combined ModelWe can substitute the L2 equations into the

8、L1 equation to see the combined modelYij = G00 + G01 * W1j + u0j + (G10 + G11 * W1j + u1j) X1ij + (G20 + G21 * W1j + u2j) X2ij + rijCannot estimate this using normal regressionHLM estimates the random factors from the model with MLE and the fixed factors with LSE8CenteringL1 regression equation: Yij

9、 = B0j + B1j * X1ij + B2j * X2ij + rijB0j tells us the value of Yij when X1ij = 0 and X2ij = 0Interpretation of B0j therefore depends on the scale of X1ij and X2ij “Centering” refers to subtracting a value from an X to make the 0 point meaningful9Centering (continued)If you center the Xs on their gr

10、oup mean (GPM) then B0 represents the group mean on Yij If you center the Xs on the grand mean (GRM) then B0 represents the group mean on Yij adjusted for the groups average value on the XsYou can also center an X on a meaningful fixed value10Estimating the ModelAfter you specify the L1 and L2 param

11、eters you need to estimate your parametersWe can examine the within and between group variability of L1 parameters to estimate the reliability of the analysisWe examine estimates of L2 parameters to test theoretical factors11Interpreting Level 2 Intercept ParametersL2 intercept equationB0j = G00 + G

12、01 * W1j + u0j G00 is the average intercept across groupsIf Xs are GPM centered, G01 is the relationship between W1 and the group mean (main effect of W1)If Xs are GRM centered, G01 is the relationship between W1 and the adjusted group meanu0 is the unaccounted group effect12Interpreting Level 2 Slo

13、pe ParametersL2 slope equation B1j = G10 + G11 * W1j + u1j G10 is the average slope (main effect of X)G11 is relationship between W1 and the slope (interaction between X and W)u1 is the unaccounted group effect13Building a HLM ModelStart by fitting a random coefficient modelAll L1 variables includedL2 equations only have intercept and errorExamine the L2 output for each parameterIf there is no random effect then parameter does not vary by groupIf there is no random effect and no intercept then the parameter is not needed in the model14Building a HLM Model (continued)Build the full intercep