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Homework Assignment 2

For this assignment, you will need to use the TIMSS data provided on Canvas. You should be prepared to use HLM or any other software package the questions. For each question below, present the output relevant to the question asked. And, please make sure to answer all questions.

Descriptive Statistics

1. Create a table with the sample size, the mean, standard error and standard deviation for the following variables:

Student variables: bsmpv01 (dependent variable, 1^st plausible value in math)

Hbi (home background index)

Matatt (attitude toward math scale)

Class/School variables: clhbi (class/school mean home background index)

Clclimat (class/school climate composite)

Write a paragraph summarizing the descriptive statistics (means, standard deviations, sample sizes etc.) in the table.

The Cross-sectional One-way ANOVA or Unconditional Model:

2. Using the TIMSS data, estimate the unconditional model using bsmpv01 (math achievement) as the dependent variable. Present the statistical model as separate models and as a mixed model (i.e., all in one regression equation).

3. What is the value of the reliability of each group’s mean as an estimate of the true population mean? How was this calculated and what does the reliability estimate tell you? (HLM provides this output; not sure about STATA or R)

4. Fixed Effects:

a. Interpret the meaning of the fixed effect and its significance.

b. Calculate the 95% confidence interval around the grand mean, γ₀₀, to show the range of possible values for school means.

5. Random Effects:

a. What are the values of the random effects, σ² and τ₀₀?

b. Calculate the proportion of variance at the classroom/school level (i.e., the ICC).  How does it compare to the HSB data that we used in class?

c. Stating the statistical hypothesis and using the output, discuss whether τ₀₀ is statistically significantly different from 0. Are there significant differences among classrooms/schools’ mean achievement that warrants the inclusion of level-2 predictors in subsequent models?

Cross-sectional One-Way ANCOVA with Random Effects

6. Starting with the unconditional model where bsmpv01 is the outcome variable, estimate a conditional student-level model where you add the student level variables hbi and matatt at level-1 using grand mean centering. Do not allow the slopes associated with hbi and matatt to vary across the level-2 units. Present the statistical model as separate models and as a mixed model (i.e., all in one regression equation).

7. What is the reliability of the level-1 intercept? What does this tell you? (HLM provides this output; not sure about STATA or R)

8. Fixed Effects:

a. Interpret the meaning of the fixed effects (including the intercept), and stating the statistical hypotheses, test whether the student-level predictors, hbi and matatt, are significantly different from zero.

b. Calculate the 95% confidence interval around the conditional grand mean, γ₀₀, to show the range of possible values for classroom/school means after controlling for hbi and matatt. Is the range of possible classroom/school means different from unconditional range presented in question 4b?

9. Random Effects: (you will need to refer to the results from the unconditional model to interpret the random effects)

a. What are the values of the conditional random effects at level-1 (σ²) and the unconditional random effects at level-2 (τ₀₀)? With the addition of hbi and matatt, are the values of σ² and τ₀₀ approximately the same as you observed in question 5? Explain your answer.

b. Calculate the proportion of within-group/between-student variance in math achievement explained by hbi and matatt.

c. Calculate the proportion of the total variance in math achievement that was explained with the addition of hbi and matatt.

Remove hbi and matatt from the model.

Cross-sectional Means-as-outcomes model:

10. Starting with the unconditional model where bsmpv01 is the outcome variable, estimate a conditional classroom/school-level model where you enter clhbi (class/school mean HBI) and clclimat (class/school climate composite) uncentered into the level-2 model. Present the statistical model as separate models and as a mixed model (i.e., all in one regression equation).

11. What is the reliability of the level-1 intercept? What does this tell you? (HLM provides this output; not sure about STATA or R)

12. Fixed Effects:

a. Interpret the meaning of the fixed effects (including the intercept), and stating the statistical hypothesis, test whether the coefficients associated with the classroom-level predictors, clhbi and clclimat, are significantly different from zero.

b. Calculate the 95% confidence interval around the conditional grand mean, γ₀₀, to show the range of possible values for classroom/school means after controlling for clhbi and clclimat. Is the range of possible classroom/school means different from the unconditional range presented in question 4b, or from the range for the one-Way ANCOVA model with random effects in 8b?

13. Random Effects: (you will need to refer to the results from the unconditional model to interpret the random effects)

a. What are the values of the unconditional random effects at level-1 (σ²) and the conditional random effects at level-2 (τ₀₀)? With the addition of clhbi and clclimat, are the values of σ² and τ₀₀ approximately the same as you observed in question 5? Explain your answer.

b. Stating the statistical hypothesis and using the output, discuss whether the conditional τ₀₀ is statistically significantly different from 0. Is there significant variation in achievement among classrooms/schools after controlling for clhbi and clclimat that warrants the inclusion of additional level-2 predictors in subsequent models?

c. Calculate the proportion of between-class/school variance in math achievement that is explained by clhbi and clclimat.

d. Calculate the proportion of the total variance in math achievement that was explained with the addition of clhbi and clclimat.