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SC968: Panel Data Methods for Sociologists Random coefficients models

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Overview Random coefficients models Continuous data Binary data Growth curves

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Random coefficients models Also known as Multilevel models MLwiN http://www.cmm.bristol.ac.uk/ Hierarchical models HLM http://www.ssicentral.com/ Mixed models Stata

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Random coefficients models for continuous outcomes

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Random coefficients models We started off with OLS models that pooled data across many waves of panel data We separated the between and within variance with fixed effects and between effects models Then we allowed intercepts to vary for each individual using random effects models We can also allow the coefficients for the independent variables to vary for each individual These models are called random coefficients or random slopes models

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Example of a random coefficients model Schools’ mean maths scores and student socioeconomic status (SES) Students at level 1 nested within schools at level 2 Using a random coefficients model we can estimate Overall mean maths score How SES relates to individual maths scores Within school variability in maths scores Between school variability in mean maths scores Between school variability in the relationship between SES and individual maths scores

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Another example Children’s emotional problems Suppose we have problems measured each year from 2000-2007 for pupils in a junior school Want to know if a school policy implemented in 2004 reduces problems Emotional problems for each year at level-1 and pupils at level-2 Using a random coefficients model we can examine Levels of emotional problems, averaged over years 2000-2007 Within pupil variability in emotional problems Between pupil variability in emotional problems Whether the intervention reduced emotional problems Whether the intervention had different effects for different children What pupil characteristics made the intervention more or less successful

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Possible combinations of slopes and intercepts with panel data Constant slopes Constant intercept The OLS model

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Possible combinations of slopes and intercepts with panel data The random effects model Constant slopes Varying intercepts

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Possible combinations of slopes and intercepts with panel data Varying slopes Constant intercept Unlikely to occur

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Possible combinations of slopes and intercepts with panel data Varying slopes Varying intercepts Random coefficients model - separate regression for each individual

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Random coefficients model for continuous data Fixed coefficients Random coefficients Residual

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Random coefficients model for continuous data Fixed intercept Fixed slope Random intercept Random slope Random error

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Partitioning unexplained variance in a random coefficients model Total variance at each level Variance explained by predictors Remaining unexplained variance Variancedue to random intercept Remaining unexplained variance Variancedue to random slopes Remaining unexplained variance

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Steps in multi-level modelling (Hox,1995) 1. Compute variance for the baseline/null/unconditional model which includes only the intercept. 2. Compute variance for the model with level-1 independent variables included and the variance components of the slopes constrained to zero (that is, a fixed coefficients model). 3. Use a chi-square difference test to see if the fixed coefficients model has a significantly better fit than the baseline model. If it does, then proceed to investigate random coefficients. At this stage can drop non-significant level-1 independents from the model.

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Steps in multi-level modelling (Hox,1995) 4. Identify which level-1 regression coefficients have significant variance across level-2 groups. Compute -2LL for the model with the variance components of the level-1 coefficients constrained to zero only for the coefficients which do not have significant variance across level-2 groups. 5. Add level-2 independent variables, determining which improve model fit. Drop variables which do not improve model fit. 6. Add cross-level interactions between explanatory level-2 variables and level-1 independent variables that had random coefficients (in step 3). Drop interactions which do not improve model fit.

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Worked example Random 20% sample from BHPS Waves 1 - 15 Ages 21 to 59 Outcome: GHQ likert scores Explanatory variable: household income last month (logged)

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Random coefficients model example where y ij = GHQ score for subject i, j = 1,…, J x ij = logged household income in month to wave j β 1 = mean slope b i = subject-specific random deviation from mean slope u i = subject-specific random intercept

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Linear random coefficients model Stata output

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Random slopes Fixed effect Estimates covariance between all random effects Least restrictive model. xtmixed hlghq1 lnfihhmn || pid: lnfihhmn, mle cov(unstr) variance Mixed-effects ML regression Number of obs = 18541 Group variable: pid Number of groups = 2508 Obs per group: min = 1 avg = 7.4 max = 15 Wald chi2(1) = 28.11 Log likelihood = -55286.004 Prob > chi2 = 0.0000 ------------------------------------------------------------------------------ hlghq1 | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- lnfihhmn | -.4015666.075741 -5.30 0.000 -.5500162 -.253117 _cons | 14.40864.5917387 24.35 0.000 13.24885 15.56843 ------------------------------------------------------------------------------ Random-effects Parameters | Estimate Std. Err. [95% Conf. Interval] -----------------------------+------------------------------------------------ pid: Unstructured | var(lnfihhmn) | 2.073304.3231129 1.527594 2.813961 var(_cons) | 144.5579 19.37199 111.1664 187.9793 cov(lnfihhmn,_cons) | -16.63265 2.488823 -21.51065 -11.75465 -----------------------------+------------------------------------------------ var(Residual) | 18.10746.2092684 17.70191 18.5223 ------------------------------------------------------------------------------ LR test vs. linear regression: chi2(3) = 5099.77 Prob > chi2 = 0.0000

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. xtmixed hlghq1 lnfihhmn || pid: lnfihhmn, mle cov(unstr) variance Mixed-effects ML regression Number of obs = 18541 Group variable: pid Number of groups = 2508 Obs per group: min = 1 avg = 7.4 max = 15 Wald chi2(1) = 28.11 Log likelihood = -55286.004 Prob > chi2 = 0.0000 ------------------------------------------------------------------------------ hlghq1 | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- lnfihhmn | -.4015666.075741 -5.30 0.000 -.5500162 -.253117 _cons | 14.40864.5917387 24.35 0.000 13.24885 15.56843 ------------------------------------------------------------------------------ Random-effects Parameters | Estimate Std. Err. [95% Conf. Interval] -----------------------------+------------------------------------------------ pid: Unstructured | var(lnfihhmn) | 2.073304.3231129 1.527594 2.813961 var(_cons) | 144.5579 19.37199 111.1664 187.9793 cov(lnfihhmn,_cons) | -16.63265 2.488823 -21.51065 -11.75465 -----------------------------+------------------------------------------------ var(Residual) | 18.10746.2092684 17.70191 18.5223 ------------------------------------------------------------------------------ LR test vs. linear regression: chi2(3) = 5099.77 Prob > chi2 = 0.0000 Fixed intercept Fixed coefficient

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Covariation between random intercept and random slope Random intercept Random slope. xtmixed hlghq1 lnfihhmn || pid: lnfihhmn, mle cov(unstr) variance Mixed-effects ML regression Number of obs = 18541 Group variable: pid Number of groups = 2508 Obs per group: min = 1 avg = 7.4 max = 15 Wald chi2(1) = 28.11 Log likelihood = -55286.004 Prob > chi2 = 0.0000 ------------------------------------------------------------------------------ hlghq1 | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- lnfihhmn | -.4015666.075741 -5.30 0.000 -.5500162 -.253117 _cons | 14.40864.5917387 24.35 0.000 13.24885 15.56843 ------------------------------------------------------------------------------ Random-effects Parameters | Estimate Std. Err. [95% Conf. Interval] -----------------------------+------------------------------------------------ pid: Unstructured | var(lnfihhmn) | 2.073304.3231129 1.527594 2.813961 var(_cons) | 144.5579 19.37199 111.1664 187.9793 cov(lnfihhmn,_cons) | -16.63265 2.488823 -21.51065 -11.75465 -----------------------------+------------------------------------------------ var(Residual) | 18.10746.2092684 17.70191 18.5223 ------------------------------------------------------------------------------ LR test vs. linear regression: chi2(3) = 5099.77 Prob > chi2 = 0.0000

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Post estimation predictions Stata output

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Post estimation predictions – random coefficients. predict re_slope re_int, reffects +----------------------------------+ | pid re_int re_slope | |----------------------------------| 1. | 10019057 -4.833731.4040526 | 4. | 10028005 -5.241494.3758182 | 16. | 10042571 -1.442705.1409662 | 17. | 10051538 -2.836288.305106 | 35. | 10059377 5.487209 -.5189736 | +----------------------------------+

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Predicted individual regression lines. gen intercept= _b[_cons] + re_int. gen slope = _b[lnfihhmn] + re_slope +---------------------------------+ | pid intercept slope | |---------------------------------| | 10019057 9.574909.002486 | | 10028005 9.167147 -.0257484 | | 10042571 12.96594 -.2606004 | | 10051538 11.57235 -.0964606 | | 10059377 19.89585 -.9205403 | +---------------------------------+

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Partitioning unexplained variance in a random coefficients model Total variance at each level Variance explained by predictors Remaining unexplained variance Variancedue to random intercept Remaining unexplained variance Variancedue to random slopes Remaining unexplained variance

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Calculating the variance partition coefficient Random intercepts model Between variance Total variance i.e. Between + Within

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Calculating the variance partition coefficient Random slopes model At the intercept, = 0 So at the intercept, the VPC for the random slopes model reduces to the same as the random intercepts model

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Variance partition coefficient for our example Tentative interpretation: least variability in GHQ for those on average incomes

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Random coefficients models Categorical outcomes

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Random coefficients model for binary data where β k is the mean coefficient or fixed effect of covariate k b ik is a subject-specific random deviation from mean coefficient u i is a subject-specific random intercept with mean zero

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Worked example Random 20% sample 15 waves of BHPS Ages 21 to 59 Outcome: GHQ binary scores (psychological morbidity cases: hlghq2 > 2) Explanatory variable: employment status (jbstat recoded to employed/unemployed/olf)

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Logistic random coefficients example where y ij = binary GHQ score for subject i, j = 1,…, J x ij = employment status in wave j β 1 = mean slope b i = subject-specific random deviation from mean slope u i = subject-specific random intercept

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Logistic random coefficients model Stata output

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. xtmelogit ghq unemp olf || pid: unemp olf,variance cov(unstr)

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No constant term with odds ratios No random residual Because logit model

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Random coefficients models for development over time

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Growth curve models Models change over time as a continuous trajectory Suitable for research questions such as What is the trajectory for the population? Are there distinct trajectories for each respondent? If individuals have distinct trajectories, what variables predict these individual trajectories?

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Linear growth curve model Individual growth curves t = 0 at baseline and 1,2,3 ….,T in successive waves Mean population growth curve

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Worked example Random 20% sample from BHPS Waves 1 - 15 All respondents over 16 years Outcome: self-rated health (hlstat) 5 point Likert scale with higher scores indicating poorer health Linear growth function

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Slope (change in health over time) Intercept (mean health at baseline)

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Individual differences in baseline health Individual differences in health change

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Adding time invariant covariates

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Interacting gender with time

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Adding time varying covariates

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Beyond linear change Polynomial trajectories Quadratic or cubic trajectories Piecewise linear trajectories Exponential trajectories

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Non linear growth curves

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Piecewise growth Research questions Children’s reading before and after summer holiday Grip strength before and after a stroke Change in well-being before and after retirement Household income before and after birth of first child What sort of growth function expected? Jointed piecewise growth with single function Disjointed piecewise growth with single function Disjointed piecewise growth with two functions Jointed piecewise growth with two functions

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Specifying time Metrics of time Wave of assessment Chronological age Time before/after an event Individually varying values of time

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Age-period-cohort A cohort is defined by their age in a particular period Impossible to separate age, period and cohort effects But any pair of the 3 factors are independent If wave is our metric of time, then usual to control for age at baseline (i.e. cohort) If age is our metric of time, then can control for period (i.e. wave) effects using dummy variables

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Accelerated panel designs Respondents of varying age sampled at same time-point then followed for several years Period and age effects not completely confounded as in a birth cohort design Assumption, after controlling for period growth curves for each cohort overlap and form smooth curve

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Checking the assumption Stratify sample into a number of cohorts Estimate growth curve model for each strata Plot growth curve

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Example of an accelerated panel design Random 20% sample from BHPS Waves 1 - 15 All respondents over 16 years Outcome: self-rated health (hlstat) Quadratic growth by age Controlling for period effects (wave)

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When are random coefficients not necessary? When number of units at level 2 is small Use dummy variables with OLS and the cluster option When correlation within subjects is small Using OLS and the cluster option will give similar estimates When want to correct for correlation between observations but not interested in variation between and within subjects Use fixed effects or random effects model

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Finally…. Random coefficients models can take a very long time to estimate Can speed things up by collapsing data and using frequency weights My personal recommendation is to use MLwiN Excellent online training material Easier to build up model step-by-step

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