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**Questions From Yesterday**

Equation 2: r-to-z transform Equation is correct Comparable to other p-value estimates (z = r sqrt[n]) ANOVA will not be able to detect a group effect that has alternating + and – ICC Effect defined in terms of between and within group variability rather than being represented individually SPSS Advanced Models can be ordered at the VU Bookstore for $51

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**Hierarchical Linear Modeling (HLM)**

Theoretical introduction Introduction to HLM HLM equations HLM interpretation of your data sets Building an HLM model Demonstration of HLM software Personal experience with HLM tutorial

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**General Information and Terminology**

HLM can be used on data with many levels but we will only consider 2-level models The lowest level of analysis is Level 1 (L1), the second lowest is Level 2 (L2), and so on In group research, Level 1 corresponds to the individual level and Level 2 corresponds to the group level Your DV has to be at the lowest level

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**When Should You Use HLM? If you have mixed variables**

If you have different number of observations per group If you think a regression relationship varies by group Any time your data has multiple levels

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**What Does HLM Do? Fits a regression equation at the individual level**

Lets parameters of the regression equation vary by group membership Uses group-level variables to explain variation in the individual-level parameters Allows you to test for main effects and interactions within and between levels

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**The Level 1 Regression Equation**

Predicts the value of your DV from the values 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 number Since the coefficients B0, B1, and B2 change from group to group they have variability that we can try to explain

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Level 2 Equations Predict 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 parameter

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Combined Model We can substitute the L2 equations into the L1 equation to see the combined model Yij = G00 + G01 * W1j + u0j + (G10 + G11 * W1j + u1j) X1ij + (G20 + G21 * W1j + u2j) X2ij + rij Cannot estimate this using normal regression HLM estimates the random factors from the model with MLE and the fixed factors with LSE

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**Centering L1 regression equation:**

Yij = B0j + B1j * X1ij + B2j * X2ij + rij B0j tells us the value of Yij when X1ij = 0 and X2ij = 0 Interpretation 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 meaningful

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**Centering (continued)**

If you center the Xs on their group 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 group’s average value on the Xs You can also center an X on a meaningful fixed value

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Estimating the Model After you specify the L1 and L2 parameters you need to estimate your parameters We can examine the within and between group variability of L1 parameters to estimate the reliability of the analysis We examine estimates of L2 parameters to test theoretical factors

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**Interpreting Level 2 Intercept Parameters**

L2 intercept equation B0j = G00 + G01 * W1j + u0j G00 is the average intercept across groups If 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 mean u0 is the unaccounted group effect

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**Interpreting Level 2 Slope Parameters**

L2 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 effect

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**Building a HLM Model Start by fitting a random coefficient model**

All L1 variables included L2 equations only have intercept and error Examine the L2 output for each parameter If there is no random effect then parameter does not vary by group If there is no random effect and no intercept then the parameter is not needed in the model

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**Building a HLM Model (continued)**

Build the full intercepts- and slopes-as-outcomes model Use L2 predictor variables to explain variability in parameters with group effects Remove L2 predictors from equations where they are unable to explain a significant amount of variability

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