Outline. Selection Bias in Multilevel Models. The selection problem. The selection problem. Scope of analysis. The selection problem

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1 International Workshop on tatistical Latent Variable Models in Health ciences erugia, 6-8 eptember 6 Outline election Bias in Multilevel Models Leonardo Grilli Carla Rampichini grilli@ds.unifi.it carla@ds.unifi.it Department of tatistics University of Florence L. Grilli & C. Rampichini - erugia 6 1 Aim: understanding the consequences of sample selection in multilevel linear models selection mechanisms in multilevel models the bivariate random intercept linear model consequences of selection theoretical results (in some special instances) simulation study (in more complex cases) future research L. Grilli & C. Rampichini - erugia 6 ample selection arises when an outcome Y ( = principal) is observed conditionally on another variable, e.g. Y > (incidental truncation) election is present in many settings, e.g. wage can be observed only for employed people roblems arise if the selection mechanism depends on unobserved variables correlated with the errors terms Consequences of selection and remedies are well established in standard (single-level) models and in random effects models for panel/longitudinal data (Vella, 1998) Applications in multilevel cross-section settings are rare (Borgoni & Billari, ; Bellio & Gori, 3; Grilli & Rampichini, 4) No systematic study on sample selection in multilevel models L. Grilli & C. Rampichini - erugia 6 3 L. Grilli & C. Rampichini - erugia 6 4 cope of analysis ample selection in a multilevel model is more complex than in a single-level model: the selection process can act at different hierarchical levels, giving rise to a wide variety of patterns the variance-covariance structure is often of primary interest, so it must be carefully assessed how it is affected by selection the selection process modifies the hierarchical structure (number of clusters and cluster sizes), a feature that is relevant in the estimation phase (estimation algorithms, asymptotic approximations, power of the tests) We consider sample selection in a two-level random intercept linear model Our analysis is quite general in several respects: the selection mechanism is driven by unobserved factors (errors) at both hierarchical levels the errors determining the selection are distinct from the errors determining the outcome (though they are allowed to be the same) the missingness pattern is arbitrary the analysis concerns the effect of selection on the properties of the model, rather than on specific estimators L. Grilli & C. Rampichini - erugia 6 5 L. Grilli & C. Rampichini - erugia 6 6 1

2 Model election mechanism BIVARIATE: each equation is two-level random intercept linear Y = z θ u e Y u e i i i i = zi θ i = 1,,, J clusters (level units) i = 1,,, n elementary (level 1) units Unbalanced hierarchy election equation rincipal equation Cluster-level covariates are allowed Usually the two equations have many covariates in common e iid iid i u ~ N, ~ N e i,, u The distributional assumption of Normality is not essential for the general discussion on selection bias, but it is used to derive the analytical results later shown L. Grilli & C. Rampichini - erugia 6 7 Y i observed Y > It operates at the elementary level (= it causes the missingness of level 1 units) It modifies the hierarchical structure of the data (cluster sizes and possibly also number of clusters) It depends on both covariances (level 1) and (level ) and it is ignorable when they are both null Within a given cluster the pattern of missingness can be of any kind (drop-out is ust a special case) i L. Grilli & C. Rampichini - erugia 6 8 election mechanism Consequences of selection Yi observed Yi > wi > zi θ w = u e Composite error of the election eq. i i { A = w > z θ } { w z θ } i i i i iy : iy i > : i Units with observed Y Units with unobserved Y Truncation event of cluster After selection = conditional on truncation on the composite errors Now consider a cluster with observed Y on the first unit (i=1) A1 w1 1 { } = > z θ Truncation event of unit 1 of cluster L. Grilli & C. Rampichini - erugia 6 9 When the selection mechanism is not ignorable it is of interest to determine the biases arising when fitting the rincipal equation alone Let us consider the first unit (i=1) of cluster, assuming it is observed Y = z θ u e independence is among clusters, but not within clusters The relevant conditioning is not on A 1 (truncation event of unit 1), but on A (truncation events of all units of the cluster) L. Grilli & C. Rampichini - erugia 6 1 Key quantities ( 1, A ) = z1θ ( 1, A ) EY u u Ee u ( 1 ) = z1θ ( u ) E( e1 ) E Y E A V Y1 = V( u ) V( e1 ) cov( u, e1 ) Marginal var. Due to the conditioning on A, the means and variances after selection depend on some features of the cluster: (1) the cluster size n Conditional mean Marginal mean Marginal w.r.t. the random effects () the missingness pattern (one out of n-1 e.g. it is not irrelevant if ) unit i= is observed or not (3) all the covariates of the election equation for all the level 1 units of the cluster L. Grilli & C. Rampichini - erugia 6 11 lopes In linear mixed models marginal slope = conditional slope Equality may break down after selection marginal slope and conditional slope must be treated separately ML and REML are based on marginal distribution they estimate the marginal slope L. Grilli & C. Rampichini - erugia 6 1

3 Marginal slope Marginal variance EY Eu Ee 1 1 = θk. zk1 zk1 zk1 lope lope after sel. before sel. level bias level 1 bias The two components of bias add up, they may have same signs or opposite signs (and even cancel out) The bias is null if covariate z k is not in the election equation, since A does not contain z k (but if covariate z k is correlated with others the estimable slope may be biased anyway) The effect of a covariate varies from unit to unit: The estimable slope is an average ossible to end with an incorrect specification with random slopes L. Grilli & C. Rampichini - erugia 6 13 V Y1 = 1 1 Vu ( ) Ve ( ) covu (, e ) After selection the errors may be no longer homoscedastic, nor independent the variance component structure breaks down: Level errors u may be correlated with level 1 errors e i Level 1 errors of different units may be correlated roblems: tandard estimators are inefficient and yield incorrect std errors ICC from mis-specified model ignoring selection may be above or below true ICC risk of over- or under-stating the role of clustering ICC: Intraclass Correlation Coefficient (between-cluster variance on total variance) L. Grilli & C. Rampichini - erugia 6 14 Research aims earch configurations of model parameters such that some of the potential selection biases are not in effect (e.g. the cluster level variance is unbiased, ) for any unit, it is enough to condition on its own truncation event (i.e. conditioning on A reduces to conditioning on A 1 ) earch analytical expressions of bias Tools: standard theory of Normal variates some recent results from the UN distribution (Unified kew-normal: Arellano-Valle & Azzalini, 6) Take a multivariate Normal and truncate on a subset of variables the other variables are UN distributed, e.g. u, e1 UN Three cases where selection causes biases, but things are not so bad Case 1 Case Case 3 election eq. cluster var > > Level cov. Level 1 cov. Reduction to oneelement truncation A1 yes no no Bias on slope Ee ( 1 1)/ zk1 Eu ( )/ zk1 Ee ( 1 )/ zk1 e1 u yes yes yes ei ei yes yes no Bias on level 1 v. downward no downward Bias on level v. no downward upward Bias on ICC upward downward upward L. Grilli & C. Rampichini - erugia 6 15 Analytical expressions of bias imulation design We exploit some general formulae in Johnson & Kotz (197) and Tallis (1961) We derive expressions in two cases: election eq. not mixed (so conditioning on A reduces to conditioning on A 1 ) well-known expressions of Heckman (1979) based on the inverse Mills ratio Balanced hierarchy with clusters of size In the general case expressions are too complex, e.g. for a balanced hierarchy with clusters of size n there are n-1 expressions, one for each missingness pattern expressions involve Normal distribution functions of dimension n Y = α β x β x γ v u e Y x x v u e i 1 1i i i i = α 1i β3 3i γ i level 1 covariate entering both and equation-specific level 1 covariates level covariate entering both and Covariates are independent (and generated once) Level 1 covariates only vary within clusters (i.e. identical cluster means) Hierarchical structure: 1 clusters of 5 units each True values: intercepts = ( missingness rate 5%) slopes = 1 variances = 1 covariances and in [-1,1] step.5 (a grid with 81 cells) L. Grilli & C. Rampichini - erugia 6 17 L. Grilli & C. Rampichini - erugia

4 Estimates of level 1 variance Estimates of level variance \ MC means on 1 runs \ MC means on 1 runs L. Grilli & C. Rampichini - erugia 6 19 L. Grilli & C. Rampichini - erugia 6 Future work on sample selection Questions and further material Understanding Linear mixed models with random slopes Non-linear mixed models, e.g. logit Other selection mechanisms, e.g. cluster-based selection Diagnostic tools olutions (two-equation models, instrumental variables, sensitivity analysis) grilli@ds.unifi.it Web: Thanks for your attention! L. Grilli & C. Rampichini - erugia 6 1 L. Grilli & C. Rampichini - erugia 6 References Estimates of slope of level covariate γ Arellano-Valle, R. B. and A. Azzalini (6) On the unification of families of skew-normal distributions. candinavian J. tat. Bellio, R. and E. Gori (3) Impact evaluation of ob training programmes: election bias in multilevel models. Journal of Applied tatistics 3, Borgoni, R. and F. C. Billari () A multilevel sample selection probit model with an application to contraceptive use. roc. XLI meeting Italian tatistical oc. Grilli, L. and C. Rampichini (4) A polytomous response multilevel model with a non ignorable selection mechanism. roceedings of the 19th IWM. Firenze Heckman, J. (1979) ample selection bias as a specificaton error. Econometrica 47, Johnson, N. L. and. Kotz (197) Distributions in tatistics: Continuous Multivariate Distributions. New York: Wiley & ons. Tallis, G. M. (1961) The moment generating function of the truncated multinormal distribution. Journal of the Royal tatistical ociety, B 3, 3 9. Vella, F. (1998) Estimating models with sample selection bias: A survey. Journal of Human Resources 33, \ MC means on 1 runs L. Grilli & C. Rampichini - erugia 6 3 L. Grilli & C. Rampichini - erugia 6 4 4

5 Estimates of slope of level 1 covariate MC mean percentage bias on 1 replications for different data structures (J= 1, n =, 5, 1, 5) \ This covariate has only within-cluster variation In general z = z ( z z ) i Between variation i Within variation MC means on 1 runs L. Grilli & C. Rampichini - erugia 6 5 parameter γ γ γ n L. Grilli & C. Rampichini - erugia 6 6 elementary level cov = cluster level covariance = E( e u, A ) 1 E ( u ) E( e ) 1 Var( u e ) 1 E( u ) Var( u ) lope biased due to correlation at level Marginal conditional Errors at different levels are independent Errors at level 1 e i are independent ICC under-estimated E( e1 ) E ( e1 ) Var( e ) 1 lope biased due to correlation at level 1 Marginal = conditional Errors at different levels are independent Errors at level 1 e i are not independent, except when the election eq. is not mixed ( = ) ICC over-estimated if the election eq. is not mixed The conditioning on A reduces to conditioning on A 1 only when the election eq. is not mixed ( = ) 5