More about linear mixed models

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1 Faculty of Health Sciences Contents More about linear mixed models Analysis of repeated measurements, NFA 2016 Julie Lyng Forman & Lene Theil Skovgaard Department of Biostatistics, University of Copenhagen Linear mixed models in general Case: a cross-over study Model asssumptions and diagnostics Missing data and methods for handling it Suggested reading FLW (2011) chapters 10, 18, / 68 Outline Specification of linear mixed models (LMMs) Mixed refers to a mixture of fixed and random effects. Linear mixed models in general Case: a cross-over study Model assumptions and how to check them Missing data Appendix: Other methods for handling missing data Systematic variation Effect of covariates (time, treatment, gender, age, etc.), described by population parameters. Random variation: Random effects described by subject specific parameters. Serial correlation Measurement error Note: Interactions between systematic and random effects are always random. 3 / 68 4 / 68

2 The covariance/correlation structure Linear model specification of LMMs Random effects: Model repeated outcomes on subjects/clusters as a linear function Serial correlation (correlated error terms): Y ij = X ij β + Z ij b i + ε ij of systematic effect(s) of covariate(s) X ij. random effect(s) of covariates Z ij. with possibly correlated residual error terms ε ij s. Measurement error (uncorrelated error terms): We see this if the same outcome is measured twice at the same time (e.g. duplicates or triplicates). We assume that the b i s and ε i s are independent and normally distributed with mean zero and covariance given by: The G-matrix: Var(b i ) = G. The R-matrix: Var(ε i ) = R. 5 / 68 6 / 68 SAS: PROC MIXED Nonidentifiability model: describes the mean value structure (i.e. covariates / fixed effects) random: describes the random effects repeated: describes the residual covariance. Very flexible modeling framework! Example: It is possible to model, e.g. longitudinal series of measurements (2 level model)... with repeated series on each subject and with different treatments along the way (3 level model)... where subjects belong to different clusters (4 level model). Warning: Make sure you understand your model! Modeling random effects together with a residual error covariance may result in unidentifiable covariance parameters, i.e. nonconvergence or strange results, unless done with some care. Example: Compound symmetry can be specified as either of: RANDOM id; RANDOM intercept / SUBJECT=id; REPEATED time / TYPE=CS SUBJECT=id; in case two of these lines are included in the same program, results may be incomprehensible. 7 / 68 8 / 68

3 Advice for making mixed model analyses First make a plan (perhaps with help from a statistician): Think about what results you want to obtain and how you want to present them. Think about an appropriate model (and its assumptions). Then do the analysis (perhaps with help from a statistician): Try to be calm and systematic. Don t just put data in your favourite statistical program and run (away). Make as many graphical displays as you can think of. When interpreting your output, check whether the results look plausible - and whether they in fact answer the relevant scientific questions. Outline Linear mixed models in general Case: a cross-over study Model assumptions and how to check them Missing data Appendix: Other methods for handling missing data Most likely you will have to do this a few times over again: Something unexpected? Think about it and revise the plan. 9 / / 68 Example: Cross-over study of headache Picture ignoring period effect and pairing Patients with chronic headache are randomized into two groups: Both groups receive LNMMA and placebo, on two different days, with a suitable wash-out period in-between Group 1 was treated first with placebo (period 1), and then with LNMMA (period 2) Group 2 was treated first with LNMMA (period 1), and then with placebo (period 2) Pain was measured subjectively on a VAS-scale (small is good), at baseline and at 30, 60, 90 and 120 minutes after treatment. Reference: Ashina et al (1999), Lancet, pp / / 68

4 Model building for complicated cross over study Analysis based on summary statistics Fixed effects: time, treat*time, period (e.g. due to weather conditions) possibly a carry-over effect: treat*(time)*period? note the baseline measurement at time=0 Suggested covariance structure: We expect observations from the same period (and patient) to be more strongly correlated when they are close in time. We also expect correlation between observations from different periods, but this is most likely not as strong. RANDOM patient; REPEATED time / TYPE=SP(POW)(time) SUBJECT=patient*period; Note the non-equidistant time points. Unfortunately, we do not have access to the full data with the two series of repeated measurements over time. The analysis was based on summary statistics: Difference between average follow-up measurements and baseline, Y 30 + Y 60 + Y Y 0 for each period, we have paired data. negativ values are good (pain decreases). 13 / / 68 Observations vs. period and treatment Averages Sequence groups: Group 1 (P+A), Group 2 (A+P) Period 1 Period 2 Mean (SD) of LNMMA (41.0) (65.0) Mean (SD) of Placebo (41.5) 3.3 (17.8) Pairs are obviously correlated. 15 / 68 Seemingly much larger treatment effect in period 2 -?? 16 / 68

5 PROC MIXED: test of carry-over effect Interpretation of the carry-over effect PROC MIXED DATA=ashina; CLASS patient treat period; MODEL effect=treat period treat*period / SOLUTION CL DDFM=KR; RANDOM intercept / SUBJECT=patient; RUN; Effect treat period Estimate StdError DF t Value Pr > t Intercept treat lmmma treat placebo period period treat*period lmmma treat*period lmmma treat*period placebo treat*period placebo Estimated effect of LNMMA vs placebo is in period 2 (reference) and =-8.2 in period 1. The carry-over effect is usually interpreted as an different effect of placebo when given after the active treatment. This could be explained as a psychological effect, in the sense that subjects expect something better (namely what they experienced in the previous period). Estimate 62.3 (95%CI : 25.4to149.9), i.e. nonsignificant. The carry-over effect (placebo following active) has a positive value, corresponding to a higher intensity of the headache. 17 / / 68 Traditional approach Conclusion on treatment effect First test the hypothesis H 0 : no carry-over effect: Two-sample T-test group 1 vs group 2. Outcome: sum of "effect" in the two periods. Expected mean difference: carry over effect. Main effects of treatment and period cancels out. If this is accepted, test H 1 : no treatment effect: Two-sample T-test, group 1 vs group 2. Outcome: difference of "effect", period 1 minus 2. Expected mean difference: two times treatment effect. Main effects of period cancels out. Same results as the mixed model analysis when there are no missing data. Method Estimate (95% CI) P-value Only period (-53.99,37.59) 0.71 Only period ( ,-11.55) 0.02 Both periods (no carry-over) (-68.70,-9.97) 0.01 Comments on the design: Use a paired design for higher power. Power calculation: t-test of treatment effect. Assumption of no carry-over is argued from using a suitably long wash-out period.... not too long, though, since then we would loose some of the power (correlation) from the pairing. 19 / / 68

6 Outline Case: Calcium supplements (from lecture 2) Linear mixed models in general Case: a cross-over study Model assumptions and how to check them Missing data Appendix: Other methods for handling missing data Bone mineral density in (g/cm 3 ) vs years since baseline 21 / / 68 Suggested models for BMD data How can we choose between models? Response profiles: Unstructured mean and unstructured covariance (only for balanced data, not actual time of observation) Think about it or otherwise: By direct visual assessment of fit Spaghettiplots and predicted profiles. Linear growth over time with either: Compound symmetry covariance pattern (same as random intercept for each girl) Autoregressive covariance pattern (or other similar) Random regression (i.e. random intercept and slope for each girl) By use of residual diagnostics Automatic model checks, using ods graphics More extensice model checks using output data sets By testing against more flexible alternatives Models for the mean tested by the usual F-tests in output. Covariance patterns evaluated by χ 2 -tests on 2 log L 23 / / 68

7 Assumptions in linear mixed models Observation Y ij from subject i measured at occation j described as Y ij = X ij β + Z ij b i + ε ij with covariate(s) X ij (fixed effects) and Z ij (random effects). Model assumptions are: Independence between individuals Linearity in covariates X ij s and Z ij s. Residuals ε i are multivariate normal ε i N (0, R i ) and the covariance pattern (R-matrix) is correctly specified. Random effects b i are (multivariate) normal b i N (0, G) and their (co)variance (G-matrix) is correctly specified. Random effects b i are independent of covariates X i s. Importance of assumptions Important: Independence between individuals (normally not an issue) Linearity Independence between X ij and b i Appropriateness of the covariance structure (may be circumvented with balanced complete data by using the empirical sandwich estimator, see lecture 2). Less important (especially when the sample size is large): Normality of residuals ε i Normality of random effects b i 25 / / 68 Model diagnostics in proc mixed Population predictions (fixed effects) ods graphics on; title1 random regression (model diagnostics) ; proc mixed data=calcium plots=all; class grp girl; model bmd = obstime grp*obstime / solution cl ddfm=kr outpm=fitpm outp=fitp residual vciry; random intercept obstime / type=un subject=girl g; run; Generates panels to check stability of variance and normal distribution of residuals. Creates two output data sets for further model checking. outpm: Population predictions and ordinary residuals. outp: Subject predictions and conditional residuals. Is the predicted population mean reasonable? 27 / / 68

8 Subject predictions (fixed + random effects) Goodness of fit, ordinary residuals Observed minus predicted population mean: Y ij X ij ˆβ Are the predicted subject means reasonable? 29 / / 68 Goodness of fit, conditional residuals Observed minus predicted subject mean: Y ij (X ij ˆβ + Zijˆbi ) Residuals in proc mixed Raw residuals: Observed - Predicted. No good for checking normality if variance changes with time. Pearson residuals: (Observed - Predicted) / SD. Same variance, but residuals on same subject are correlated. Studentized residuals: (Observed - Predicted) / SD. Similar to Pearson residuals (the same in large samples) but takes estimation uncertainty in predictions into account when estimating SD. Residuals on same subject are correlated. Scaled residuals: vciry-option in proc mixed. Scales residuals by the square root of their covariance-matrix. Independent and N (0, 1) if the model is correct. 31 / / 68

9 Goodness of fit, scaled residuals Additional model checks Plots of residuals against individual covariates: Do we see deviations from linearity? (next slide) title1 Ordinary residuals vs time ; proc sgplot data=fitpm; loess Y=Resid X=obstime / group=grp; refline 0; run; Is the variance similar in different groups? (not shown) Systematic deviation seen in the residual plot. title1 Scaled residuals vs groups ; proc sgplot data=fitpm; vbox ScaledResid / group=grp; run; 33 / / 68 Effect of time: Ordinary residuals (outpm) Effect of time: Conditional residuals (outp) 35 / / 68

10 Effect of time, scaled residuals (vciry + outpm) Influential observations Beware of single observations or subjects with a large influence on the estimates (mean or covariance parameters) e.g. due to: An unusual combination of covariates X i or Z i Large ordinary residuals (i.e. outliers) may occur naturally... or due to model misspecification. Loads of diagnostics are generated by proc mixed using e.g.: model bmd=obstime grp*obstime / ddfm=kr influence(iter=10); Some deviation from linearity seen in ordinary and scaled residuals, consistently for the two groups 37 / 68 Note: This takes a while to run because the model is re-fitted without each subject in turn. 38 / 68 Cooks distance Real trouble In some cases valid analysis can be more difficult or outright impossible to do. Call for statistical aid if: Your data set is very small. The (co)variance depend on the covariates. E.g. patients react more similarly to placebo than to treatment (or the other way around). One or more covariates depend on previous outcome values. E.g. dose adjustment. There are missing data. Tentative limit for "influential": 4 n = There are unmeasured confounders. 39 / / 68

11 Outline Missing data Linear mixed models in general Case: a cross-over study Model assumptions and how to check them Missing data Appendix: Other methods for handling missing data Most investigations are planned to be balanced but almost inevitable turn out to have missing values, or patients who drop-out for some reason... Just by coincidence (blood sample lost or ruined). The patient moved away (may be worrysome). The patient has recovered (worrying, i.e. carrying information). The patient is too ill to show up (very serious, i.e. carrying unretrievable information). 41 / / 68 Look-alike missing data When you have missing data The most important question is WHY. Missing data is not to be confused with: Non-existent data due to death. Censored data such as measurements above or below the limit of detection. In other case observed data may be replaced by missing values, In case of non-compliance or switch to rescue medication outcomes no longer reflect the treatment effect... Specialized statistical analyses are needed to handle these kinds of data. Investigate: Make separate spaghettiplots for completers and drop-outs. Make a table comparing the distribution of covariates and other characteristics between the drop-outs and the completers. If possible, ask the patients! Speculate: Think about what differences there might be between completers and drop-outs in terms of unmeasured outcomes and confounders. How could these affect the results of an analysis. 43 / / 68

12 Types of missingness Case: Missing data in calcium study MCAR Missing completely at random. Unrelated to the outcome and the covariates. MAR Missing at random. Missingness may depend on covariates and possible also on previously observed outcome values. NNAR Not missing at random. Missingness depends on the unobserved outcome either directly or indirectly by depending on an unmeasured confounder. Patterns of missingness Monotone Drops out and stays out. Intermittent Comes back later. 45 / / 68 Case: Missing data in calcium study Biased average curves Think about it... Likely causes of drop-out: Family moves away or too busy to participate (MCAR, unless related to unmeasured confounders). Parents learn at the visit that BMD is low and decides to withdraw because they think the girl is on placebo but needs treatment (MAR). The average curve is representative of the whole population when data is balanced and complete. When there are missing data (MAR or NMAR) it is biased. Note: Missing at random (MAR) has to be argued - it cannot be tested or otherwise verified from the data. Spaghettiplots are of course always ok. 47 / / 68

13 Example of missing mechanism MAR Low values are good (e.g. blood pressure): When the patient learns he is doing well, he might decide he no longer needs treatment. Example of missing mechanism NMAR Low values are bad (e.g. lung function): When the patient is sufficiently ill, he drops out of the labour market. Result: Average stays high, treatment effect is underestimated 49 / 68 Result: Average stays high, healthy worker effect. 50 / 68 Methods for handling missing data Likelihood inference Up-to-date methods valid under MAR-assumption: Likelihood inference (default in LMMs, e.g. proc mixed) Inverse probablility weighting (IPW) Multiple imputations Historical approaches that has been abandonned due to likely bias: Complete case analysis LOCF (or LVCF): Last observation (value) carried forward Predicted value (e.g. mean) imputation. BUT: The optimal method of handling missing data is of course to avoid it in the first place by choosing a missing-proof study design. 51 / 68 Perform the usual LMM analysis (e.g. in proc mixed where likelihood inference is default). Valid under MAR-assumption if the model is correct (all model assumptions must be fulfilled). Note: it is important that all covariates that missingness depend on are included in the model. Also it is important that observations are included - don t delete the incomplete cases! Properties: Efficient - makes optimal use of the available observations. Also applies to generalized linear mixed models for non-normal outcomes (lectures 5 and 6). Recommended whenever possible 52 / 68

14 NMAR: Not missing at random Paired T-test with missing data Tryptase level before and after operation, for 120 patients No statistical method can make up for data being NMAR. It has been suggested to perform sensitivity analyses over a range of plausible models for the missingness / unobserved data. However: All such models rely on unverifiable assumptions that can never be checked with the data. These analyses currently cannot be performed with standard statistical software. Red observations are the singletons: Reference: Garvey et al. (2010a,b) 53 / / 68 Missing values Complete case analysis What would happen if some patients were only measured once? We randomly delete 55 (23%) of the observations. time NObs N NMiss :before :after Total Note: Missing data are missing completely at random (MCAR). Paired t-test on complete pairs: The TTEST Procedure Difference: logafter - logbefore N Mean Std Dev Std Err Minimum Maximum Mean 95% CL Mean Std Dev 95% CL Std Dev DF t Value Pr > t BUT: only 70 patients have observations both before and after operation; only 140 of the 185 available observations are used. 55 / / 68

15 Two-sample t-test This uses all available observations... The GLM Procedure Dependent Variable: logtryptase Sum of Source DF Squares Mean Square F Value Pr > F Model Error Corrected Total R-Square Coeff Var Root MSE logtryptase Mean Standard Parameter Estimate Error t Value Pr > t Intercept B <.0001 time 1:before B time 2:after B... Parameter 95% Confidence Limits Intercept time 1:before time 2:after.. BUT: Correlation within pairs (about 0.94) is neglected! 57 / 68 Likelihood inference A linear mixed model uses all available observations and accounts for correlation within pairs. Number of Observations Number of Observations Read 240 Number of Observations Used 185 Number of Observations Not Used 55 Covariance Parameter Estimates Cov Parm Estimate patient Between Residual Within Solution for Fixed Effects Standard Effect time Estimate Error DF t Value Pr > t Alpha Intercept < time 1:before time 2:after Effect time Lower Upper Intercept time 1:before time 2:after.. 58 / 68 Estimated decrease after operation (on log-scale) When should we use what approach? Analyses on incomplete data compared to full data (benchmark). Method Effect N Confidence Interval P-value Mixed model ( , ) Paired T-test ( , ) Two-sample T-test ( , ) 0.21 All data ( , ) < Note: Very similar results from the mixed model and paired t-test. Paired T-test: Almost right when the correlation is strong, and only few observations are missing. Unpaired T-test: Almost right when the correlation is weak, and many observations are missing. Mixed effects model: Always right. But: This depends on what kind of missing we are dealing with. All variables and covariates that missingness depend on must be included in the model. T-tests are biased unless data is MCAR, while the mixed model can handle MAR. In case of NMAR data, nothing is rigth. 59 / / 68

16 Outline Complete case analysis Linear mixed models in general Case: a cross-over study Make an analysis including only those individuals who are observed at all available time points. Default choice for oldfashioned software (e.g. MANOVA). Valid under MCAR-assumption Model assumptions and how to check them Missing data Appendix: Other methods for handling missing data Consequences: Likely biased if there are specific reasons for the missingness. Inefficient because partial information from non-completers is lost. Not recommended 61 / / 68 Last observation carried forward (LOCF) LOCF in clinical trials If an individual has no observed value at time t, replace the missing value by the previously observed value. For drop-outs, all subsequent values will be identical. Consequences: The time effect is most likely biased. The natural variation is obscurred. Definitely not recommended The LOCF is the easiest imputation approach for missing data to be understood by the non-statisticians. However, the LOCF approach has been the target for criticisms from the statisticians for its lack of a sound statistical foundation and for its biases in either direction (i.e., it is not necessarily conservative). After the National Academies published its draft report "The prevention and treatment of missing data in clinical trials, using LOCF approach seemed to be out-dated and markedly out of step with modern statistical thinking. Reference: 63 / / 68

17 Predicted value imputation Replace the missing values with more or less qualified guesses of what they might have been, e.g. Time average imputation: Replace missing value with average over remaining subjects Model prediction imputation: Replace missing values with predicted values from regression models including previous observations and other covariates. Consequences: May be biased if the prediction model is not correct. Underestimates SEs because imputed values are treated as if they were the actual observations and because predictions are less variable than genuine observations. Not recommended 65 / 68 Multiple imputations Similar to predicted value imputation only random error terms are added to the predictions (and the model parameters). 1. This is repeated a number of times and analysis is performed on each imputed dataset. 2. Finally estimates are averaged and SEs are computed according to Rubin s rule. Properties Valid under the MAR-assumption if the imputation model is correct and includes all covariates that missingness depend on. Results are very similar to likelihood inference, but SEs tend to be slightly conservative in small samples. Recommended, but likelihood inference in LMMs is valid under 66 / 68 the same assumptions and easier to do in practice. Inverse probability weighting (IPW) Inverse probability weighting (IPW) Intuition: If patients with a certain characteristic are twice as likely to drop out than others, then those who actually remain in the study must count for two. This requires a model for drop out (usually logistic regression) including relevant covariates and previous outcomes on which missingness may depend. Model the outcome "Y ij is observed" (i.e. not dropping out). Save the predicted probabilities ˆπ ij from this model (the probability that the next outcome Y ij is observed). Use the inverse probabilities ˆπ ij 1 as weights when analysing the available data using generalized estimating equations (lectures 5 and 6). Properties: Valid under MAR-assumption if both the model for the missingness and the model for the observations are correct. In case of monotone missingness, it suffices that one of the two models is correct (estimates are doubly robust). Recommended for handling monotone missing data in the population average models for non-normal outcomes (lectures 6). 67 / / 68