EdPsy/Psych/Stat 587 Spring 2019 C.J. Anderson R Homework 4 Answer Key 1. (10 points) For models (n) – (v) that you fit to the TIMSS data in computer lab 2, write out the equation of the model in each of the following ways: (n) modeln ← lmer(science ∼ 1 + grpCmath + third + boy + hoursTV + hourscomputergames + grpMmath + (1 | idschool), lab2, REML=FALSE) Hierarchical model : Level 1 : (science) ij = β 0j + β 1j (grpCmath) ij + β 2j (third) ij + β 3j (boy) ij +β 4j (hoursTV) ij + β 5j (hourscomputergames) ij + R ij where R ij ∼N (0,σ 2 ) and independent. Level 2 : β 0j = γ 00 + γ 01 (grpMmath) j + U 0j β 1j = γ 10 β 2j = γ 20 β 3j = γ 30 β 4j = γ 40 β 5j = γ 5j where U 0j ∼N (0,τ 2 0 ) i.i.d and independent of R ij . Linear mixed model : (science) ij = γ 00 + γ 10 (grpCmath) ij + γ 20 (third) ij + γ 30 (boy) ij +γ 40 (hoursTV) ij + γ 5j (hourscomputergames) ij + γ 01 (grpMmath) j +U 0j + R ij Marginal model : (science) ij ∼N (μ ij , (τ 2 0 + σ 2 )) where μ ij = γ 00 + γ 10 (grpCmath) ij + γ 20 (third) ij + γ 30 (boy) ij +γ 40 (hoursTV) ij + γ 5j (hourscomputergames) ij + γ 01 (grpMmath) j 1
Transcript
EdPsy/Psych/Stat 587Spring 2019C.J. Anderson
R Homework 4Answer Key
1. (10 points) For models (n) – (v) that you fit to the TIMSS data in computer lab 2,write out the equation of the model in each of the following ways:
Note: re-scale version is the same model. From here on out I willuse (xgrpMmath)ij and (xgrpCmath)ij to indicate these were re-scaledto standard deviations equal 1.
(v) modelv ← lmer(science ∼ 1 + xgrpCmath + third + boy + hoursTV +hourscomputergames + xgrpMmath + short2 + short3 +xgrpCmath*xgrpMmath +(1 + xgrpCmath| idschool),Hierarchical model :
4. (5 points) For some of the models, there was a warning about different scales. We’lllook at model (u). in particular the warning message was that ”Warning message:
Some predictor variables are on very different scales: consider
rescaling’’ first appears in model (u). In model (u) a cross-level
interaction between grpCmath and grpMmath was added.”
(a) The standard deviation for this interaction is much larger than those for allothers; namely,
It seems that adding the cross-level interaction made the difference (modelswithout it we did not get this warning).
(b) After re-scaling the variables we find
• The exact same values of fit statistics (i.e., AIC, BIC, loglik, deviance)
• The exact same values for σ2
• The exact same t-statistics and p-values
• Difference values of γ’s and their standard errors for gCmath and gMmath.
• Difference values for τ 21 and τ 20 (and τ01).
Note the γs for gCmath and gMmath are now with respect a 1 standard deviationhigher. For example, for a 1 standard deviation higher of hours watching TV expectscience scores to be 0.1175 points lower.
5. (5 points) Where there any problems in estimating the new models or problems inthe solutions found by R? If yes, what was/were the error messages?
Yes. Some models yielded the message
1: In checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv, :
unable to evaluate scaled gradient
2: In checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv, :
Model failed to converge: degenerate Hessian with 1 negative eigenvalues
I will be delving into why this model may not be converging (maybe not adding tinynumber of diagonal to Hessian??)
Because there was convergence was not achieved, we also get the error message (fromlmerTest),
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Error in calculation of the Satterthwaite’s approximation. The output
of lme4 package is returned summary from lme4 is returned some computational
error has occurred in lmerTest
and at the very end
convergence code: 0
unable to evaluate scaled gradient
Model failed to converge: degenerate Hessian with 1 negative eigenvalues
and for model (x),
fixed-effect model matrix is rank deficient so dropping 1 column / coefficient
6. (5 points) Does it appear that you need a random slope for grpCmath, hours tv,and/or hours computer games? Explain your reasoning.
Things that could be said:
• Among the models that are the same except for which variable has a randomslope, model with a random slope for grpCmath fits the best (can look at−2LnLike or AIC.
• It seems to depend on what is in the model. Each variables can be random butwhat’s in the fixed part does matter.
• The following figures (at the end of the answer key) show that only grpCmathappears to have different slopes over groups than the other possible explanatory(micro) variables (not expected, but helpful to understand what’s going on).
• Anything that makes sense.
7. The variable grpMmath appears to be a useful predictors of the slope forgrpCmath (See models (q), (u), (v), (w) and (x)). The cross-level interactionsbetween grpCmath & type of community and between grpCmath & don’t appear“significant” in model (x).
8. (0 points) My favorite model ....from using SAS..... . .
Note that I found in model (v), which is my favorite model of those that we fit, thatγ04, which was the fixed effect parameter for (suburb)j was not significantly differentfrom 0, so I recoded the data as
(isolate)j =
{1 for isolated school0 otherwise
(rural)j =
{1 for isolated school0 otherwise
If a school was urban or suburban, then (isolate)j = (rural)j = 0.
Summary/Interpretation: We find higher science scores for boys in the 4th grade whohave higher math scores relative to their peers, and those who don’t watch a lot of TV. Also,we find higher science scores in schools that have high average math scores. Furthermore,students in schools have the higher science scores in isolated locations, followed by rural.There don’t appear to be differences between students in schools in urban & sub-urbanlocations. Those with the lowest scores are from urban and suburban locations.
We can give a more detailed explanation and put the parameter estimates into the models.For example,
• Science scores for boys are 1.21 points higher than those for girls.
• 3rd graders have science scores that are −.88 points lower then 4th graders (or scoresfor 4th graders are 0.88 points higher).
• Science scores tend to be -.14 of a point lower for every hour of TV watched duringthe week.
• Differences exist between schools.
– Interpretation of random intercept: Overall, students have science scores that are
∗ 0.90 points higher at schools where average math scores are 1 point higher.
∗ 4.00 points higher at isolated schools than at urban or sub-urban schools.
∗ 0.98 points higher at rural schools than at urban or sub-urban schools.
∗ The variance of the intercept over schools is 3.43 (standard deviation of1.85)— this is how much unexplained variation there is between schools interms of the overall level of science scores.
– Interpretation of random slope: For a 1 unit increase in a student’s math scoresrelative to their peers, the expected change in the student’s science score is3.27− 0.02(grpMmath)j . If the student goes to a school with an average mathscore that’s 1 unit higher, then the change in science score would be3.27− .02 = 3.25. In appears that if a student has a higher math score relative to
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their peers but that the school overall has lower average math scores, we expectthis student to have higher science scores.
This is an example where the micro and macro effects are in opposite directions.If we use student math score (raw) or overall mean centered we get the slope formath equal to (3.2725− 2.3751(grpMmath)j)mathij , which shows this oppositemicro/macro effect even more clearly. This seems to me counter-intuitive— anysuggestions or explanations?The unexplained variance between schools in terms of the effect of student mathscores on science scores is .002 (i.e., standard deviation of the slopes is prettysmall).
In the figures at the end of this answer key are plots of data. The plots of science scoresversus potential micro level predictors suggest that we need a random intercept. The“flatness” of the lines in the plots for gender, hour TV and hours computer games look likethere may not be effects for these or if there are they are relatively small. The figure forscience scores versus math scores indicate that this may be a stronger effect (not positiveslope) and need a random slope.
On the last two pages, are plots of a sample of individual schools by math scores with bestfit regression line for that school drawn in. It looks like linear regression OK (withinschools). Can also see various differences between schools.
These plots and re-coding data are things that we’ll cover in computer labs to come. . .
