Longitudinal Data Analysis of Health Outcomes
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1 Longitudinal Data Analysis of Health Outcomes Longitudinal Data Analysis Workshop Running Example: Days 2 and 3 University of Georgia: Institute for Interdisciplinary Research in Education and Human Development Running Example
2 Example Data Data come from a study of aging adults On six occasions, participants were asked to report: A variety of health symptoms, which were tallied into a count (variables: nsymptoms1 nsymptoms6; ranging from 0 to 5) A five item survey assessing their negative affect the average of which was reported (variables: negmood1 negmood6) An indicator (dummy coded variable old) is given as to whether the participant is a young adult or an older adult Running Example 2
3 Philosophy of the Running Example In order to better demonstrate the issues involved in a longitudinal data analysis using multilevel models, we will make use of this running example I will switch back and forth between lecture on concepts and this example to better demonstrate the concepts at work Running Example 3
4 REPEATED MEASURES ANALYSIS Running Example 4
5 Repeated Measures Analysis The data could have been analyzed using a repeated measures analysis approach The analysis would not look at systematic change over time Only if change occurred Longitudinal analysis can help to predict change And allow it be moderate by other variables In general, the repeated measures analysis can be useful for determining if there was change But questions about nature of change are unanswered: Is there growth/decline? Is change accelerating/decelerating? Does change interact with other variables (person or time specific)? Each of these questions will next be answered by phrasing the analysis as a multilevel model Level 1: time Level 2: subject Running Example 5
6 The Repeated Measures Statistical Model Re tracing the analysis steps, we will consider our repeated measures analysis empty model: is the reported number of symptoms for subject at time is the (fixed) effect the average number of symptoms at time (there are T 1 of these as they multiply the indicator variables) Combination of intercept plus dummy effect of time represents average of symptoms at time is the error (level 1 residual) for subject at time Because we have 6 time points per person, there are six error terms With 6 error terms, we will use a 6 dimensional multivariate normal distribution to describe how they relate to each other 0, the set of residuals has unstructured error covariance matrix This is actually more analogous of a Multivariate ANOVA with the UN covariance matrix Running Example 6
7 SAS Syntax and Output Running Example 7
8 SAS Output Fixed Effects and Means Running Example 8
9 Post hoc Mean Comparisons We could use post hoc mean comparisons to see how change happened: Table is Bonferroni adjusted p values All times are different from time 1, but not from each other Time 1 Time 2 Time 3 Time 4 Time 5 Time 6 Time 1 Time Time Time Time 5 < Time 6 < Running Example 9
10 Plotting the Mean Number of Symptoms 2.25 Average Number of Symptoms Occasion Running Example 10
11 CALCULATING THE NATURE OF THE DEPENDENCY OF OBSERVATIONS WITHIN PERSON Running Example 11
12 Longitudinal Data Analysis: A 7 Step Program 1. Calculate nature of dependency (intraclass correlation or ICC) from an empty model 2. Decide on a metric of time to use 3. Decide on a centering point 4. Estimate a saturated means model and plot individual trajectories 5. If systematic change is present: evaluate fixed and random effects of time; otherwise evaluate alternative models for the variances 6. Consider possible alternative models for the residuals 7. (very seldom done) Evaluate remaining heterogeneity (not in workshop) Running Example 12
13 Initial Setup: Empty Model As with any multivariate analysis, the multilevel model begins with an empty model as a baseline: Level 1 model: Level 2 model: Overall model: This is the basic analysis model will be the grand mean (i.e., overall average here the number of symptoms) 0, where will be the overall grand variance This model is the independence model not useful for analysis purposes, but will describe how variances are decomposed in the multilevel model Running Example 13
14 SAS Syntax and Output R = 6 for 6 th subject Running Example 14
15 Next Step: Random Intercept Model After the baseline independence model, the next model to run is one with a random intercept per subject: Level 1 model: Level 2 model: Overall model: Where 0,, 0, The random intercept model will partition the overall variance into two parts: Variance at level 1 the matrix is Variance at level 2 the matrix (one term) The model assumes correlated observations The matrix is compound symmetric with a random intercept model Running Example 15
16 SAS Syntax and Output G matrix: R matrix: Running Example 16
17 Interpreting the Variance Components The level 1 error: Represents level 1 variation (how much symptoms vary within a subject) The level 2 error: Represents level 2 variation (how much subjects vary in their reported symptoms) The total variation is The intraclass correlation is the amount of level 2 variation over the total.5827 Represents the correlation of symptoms within a subject Also represents the proportion of variance that is between subjects Running Example 17
18 The Model Predicted Covariance Matrix The G matrix and the R matrix combine to produce the overall model predicted covariance matrix: As there are (up to) 6 observations per subject, the matrix will be size 6 by 6: Running Example 18
19 The Model Predicted Covariance Matrix in SAS Running Example 19
20 Model Comparison: Which Model Fits Best Before moving on to the time models, we must first decide which model fits the best We used REML but the fixed effects (the overall intercept) were the same So we can use a deviance test Baseline model 2LogL = Random Intercept model 2LogL = Deviance test: = Degrees of freedom = 1 P value: < P value using conservative naïve 1 distribution Conclusion: we reject the empty model as the random intercept variance is significantly different than zero Our end goal is to explain the between person variance and the within person variance Done by adding BP and WP predictors to our analysis Running Example 20
21 DECIDING UPON A METRIC OF TIME Running Example 21
22 Longitudinal Data Analysis: A 7 Step Program 1. Calculate nature of dependency (intraclass correlation or ICC) from an empty model 2. Decide on a metric of time to use 3. Decide on a centering point 4. Estimate a saturated means model and plot individual trajectories 5. If systematic change is present: evaluate fixed and random effects of time; otherwise evaluate alternative models for the variances 6. Consider possible alternative models for the residuals 7. (very seldom done) Evaluate remaining heterogeneity (not in workshop) Running Example 22
23 Deciding Upon A Metric of Time In our example data, we only have one option for time: We have which session the person attended (with session = 1, 2,, 6) Session is essentially time in study We can examine how the session variable looks between and within person The frequency distribution of session: Running Example 23
24 Examining BP versus WP Variance in Session Additionally, we can examine the proportion of variance BP by calculating the ICC for session: From our estimates: Here, the ICC = 0, so session is entirely a WP variable No need to control for average length in study Running Example 24
25 DECIDING UPON A CENTERING POINT OF TIME Running Example 25
26 Longitudinal Data Analysis: A 7 Step Program 1. Calculate nature of dependency (intraclass correlation or ICC) from an empty model 2. Decide on a metric of time to use 3. Decide on a centering point 4. Estimate a saturated means model and plot individual trajectories 5. If systematic change is present: evaluate fixed and random effects of time; otherwise evaluate alternative models for the variances 6. Consider possible alternative models for the residuals 7. (very seldom done) Evaluate remaining heterogeneity (not in workshop) Running Example 26
27 Deciding Upon A Centering Point for Time If we were to include session as our time variable, the intercept would not have any meaning as session started with the number 1 The number of reported symptoms at some undisclosed time before the study In order to providing meaning to the future parameters to appear in our analyses, we should consider centering time Our choice of centering point is arbitrary Some centering points may cause estimation difficulty (typically the beginning or end) We will choose to center at session #3: So we create a new variable called session3 Now empty model +time intercept will reflect # of symptoms at session #3 Running Example 27
28 ESTIMATE A SATURATED MEANS MODEL AND PLOT INDIVIDUAL TRAJECTORIES Running Example 28
29 Longitudinal Data Analysis: A 7 Step Program 1. Calculate nature of dependency (intraclass correlation or ICC) from an empty model 2. Decide on a metric of time to use 3. Decide on a centering point 4. Estimate a saturated means model and plot individual trajectories 5. If systematic change is present: evaluate fixed and random effects of time; otherwise evaluate alternative models for the variances 6. Consider possible alternative models for the residuals 7. (very seldom done) Evaluate remaining heterogeneity (not in workshop) Running Example 29
30 The Repeated Measures Statistical Model Re tracing the analysis steps, we will consider our repeated measures analysis empty model: is the reported number of symptoms for subject at time is the (fixed) effect the average number of symptoms at time (there are T 1 of these as they multiply the indicator variables) Combination of intercept plus dummy effect of time represents average of symptoms at time is the error (level 1 residual) for subject at time Because we have 6 time points per person, there are six error terms With 6 error terms, we will use a 6 dimensional multivariate normal distribution to describe how they relate to each other 0, the set of residuals has unstructured error covariance matrix This is actually more analogous of a Multivariate ANOVA with the UN covariance matrix Running Example 30
31 SAS Syntax and Output Running Example 31
32 SAS Output Fixed Effects and Means Running Example 32
33 Plotting the Mean Number of Symptoms 2.25 Average Number of Symptoms Do you see a trend in the means? Occasion Running Example 33
34 Plotting Individual Trajectories Do you see a trend in the individual trajectories? Running Example 34
35 EVALUATE FIXED AND RANDOM EFFECTS OF TIME: FIXED AND RANDOM LINEAR TIME TRENDS Running Example 35
36 Adding Time to the Analysis After determining the random intercept model fits better than the baseline model, we can now start to add time to our analysis The addition of time is a necessary next step in a longitudinal analysis You must first figure out which model for time fits the best before adding any other covariates The more time points you have, the more complicated the time model is likely to be We will discuss linear and quadratic time models here Many more options exist, though Running Example 36
37 Random Intercept, Fixed Linear Model of Time After the baseline independence model, the next model to run is one with a random intercept per subject: Level 1 model: 3 Level 2 models: Intercept: Slope: Overall model: 3 Where 0,, 0, We need to start with the fixed effect of time only As a baseline for comparison with the random time model Cannot use deviance tests in REML to compare with previous models new fixed effect ( ) Running Example 37
38 SAS Syntax and Output The p value for the slope tells us if the effect is needed that linear time is significant Running Example 38
39 Random Linear Model of Time After the baseline independence model, the next model to run is one with a random intercept per subject: Level 1 model: 3 Level 2 models: Intercept: Slope: Overall model: 3 Where 0,, 0, We now have three parameters in the G matrix the random intercept (variance of subject intercepts at time=0) the random linear slope (variance of subject slopes) the covariance of the random intercept and random slope Running Example 39
40 SAS Syntax and Output Running Example 40
41 Model Comparison Before inspecting parameters, we must first decide whether or not the random linear model is preferred to the random intercept, fixed linear model A deviance test of : 0 Note: the Chi square test will be overly conservative (variance is fixed to its boundary) Random intercept fixed linear model 2LogL = Random linear model 2LogL = Deviance test: = 27.6 Degrees of freedom = 2 P value: < Conclusion: we reject the random intercept, fixed time model Running Example 41
42 Model Parameters: Variance Components Variance Components: Running Example 42
43 Random Linear: Approximating V Matrix Compare the V matrix on the previous slide to that of the V (R) matrix from the RM ANOVA analysis The random linear approximates the UNSTRUCTURED V matrix using fewer (and more interpretable) parameters Approximations like this are common (see factor analysis) Running Example 43
44 Comparing Random Linear to RM ANOVA Note: Lines are from fixed effects as random effect do not provide prediction RM ANOVA Random Linear Running Example 44
45 Testing Quadratic Models of Change Although the visual ID from the graph showed a close approximation to the means using a random linear model, the next step is to look for higher order change Quadratic change This involves changing the fixed effects, so for REML we must start with the random linear, fixed quadratic model If the quadratic fixed effect is significant, then we can add a random quadratic term Running Example 45
46 Random Linear, Fixed Quadratic Model Level 1 model: 3 3 Level 2 models: Intercept: Linear Slope: Quadratic slope: Overall model: 3 3 Where 0, 3, 0, Running Example 46
47 SAS Syntax and Output We will call this significant Running Example 47
48 Random Quadratic Model Level 1 model: 3 3 Level 2 models: Intercept: Linear Slope: Quadratic slope: Overall model: 3 3 Where 0,, 0, Running Example 48
49 SAS Syntax and Output Running Example 49
50 Model Comparison Before inspecting parameters, we must first decide whether or not the random quadratic model is preferred to the random linear, fixed quadratic model A deviance test of : 0 Note: the Chi square test will be overly conservative (variance is fixed to its boundary) Random intercept fixed linear model 2LogL = Random linear model 2LogL = Deviance test: = 4.0 Degrees of freedom = 3 P value: Conclusion: we reject the random quadratic model We can now stop inspecting the trend across time Running Example 50
51 Comparing Models Note: Lines are from fixed effects as random effect do not provide prediction RM ANOVA Random Linear Random Linear, Fixed Quadratic Running Example 51
52 Notes About Time You cannot add a random component when a fixed component is not present No random slope without fixed slope The matrix should always be unstructured When using REML, you cannot do a deviance test to compare across models with different fixed effects The degrees of freedom method matters little for large sample sizes For small sample sizes, use DDFM = KENWARDROGER The number of level 1 and level 2 random coefficients cannot be more than the number of unique elements in the matrix 21 unique elements in our matrix In general, Nobs 1 random effect coefficients allowed Running Example 52
53 ADDING PREDICTORS: TIME INVARIANT (CONSTANT) COVARIATES Running Example 53
54 Adding Predictors in Longitudinal Models The nature of the predictor in longitudinal models plays a role in how it is added to an analysis Time invariant predictors All at level 2 (subject level) Time varying predictors Combination of effects (level 1 and level 2) The level of the predictor dictates which variance component it seeks to describe Level 2 describes level 2 variances ( matrix) Level 1 describes level 1 variances ( matrix) Although order entered may not ultimately matter, practice is to add level 2 predictors first (time invariant) Running Example 54
55 Time Invariant Predictor in Our Analysis The time invariant predictor in our analysis is the dummy coded age variable ( old ) If old = 0 subject is in not old group If old = 1 subject is in old group A more practical analysis would be to use the age of the subject directly Age must be a time varying predictor We do not have reported age, so old is all we have, which is time invariant In general, it is bad practice to categorize a continuous variable as it reduces variability that can be explained Running Example 55
56 Adding Old to Our model Our model is the random linear, fixed quadratic model: Level 1 model: 3 3 Level 2 models: Intercept: Linear Slope: Quadratic slope: Overall model: 3 Where 0,, 0, 3 Running Example 56
57 SAS Syntax and Output Running Example 57
58 Interpreting the Output: Fixed Effects The fixed effects with old were:.6228 (p = 0.002) Old group has lower intercept.1125 (p = ) Old group has higher (less negative) linear slope (p = ) Not significant Can remove from model The estimates of the new model without : Running Example 58
59 Plotting the Fixed Effects Old = 0 Old = Running Example 59
60 Inspecting the Random Effects Level 2 ( matrix): No OLD G. OLD G. Intercept Slope Intercept Slope Proportion reduction in generalized variance:.102 Proportion reduction of intercept variance:.076 Proportion reduction of slope variance:.049 OLD explains 10.2% of generalized level 2 variance A Pseudo Pseudo R 2 Level 1 ( matrix) negligible reduction (as it should be): No OLD: OLD: Running Example 60
61 ADDING PREDICTORS: TIME VARYING COVARIATES Running Example 61
62 Adding Time Varying Covariates Time varying covariates are variables that contain both level 1 and level 2 information Level 2 average level for a subject Level 1 time occasion fluctuation for each subject The key to using time varying covariates is that both portions must be put into the model Level 2 the subject mean (we will grand mean center this) Level 1 the subject mean centered predictor at each occasion Our time varying covariate is the variable NEGMOOD Representing the negative mood of a subject at each occasion Running Example 62
63 Mean Centering in SAS Running Example 63
64 Adding NEGMOOD to Our Model Level 1 model: (Note: is grand mean centered) 3 3 Level 2 models: Intercept: Linear Slope: Quadratic Slope: Negative Mood: Running Example 64
65 Overall Model Overall model: Main Effects y st = γ 00 + γ 10 (TIME st 3) + γ 30 (NM st ) + γ 01 OLD s + γ 02 + γ 20 (TIME st 3) 2 + γ 03 OLD s + γ 11 OLD s (TIME st 3) + γ 12 (TIME st 3) + γ 31 OLD s (NM st ) + γ 32 (NM st ) + γ 03 OLD s (TIME st 3) + γ 21 OLD s (TIME st 3) 2 + γ 22 (TIME st 3) 2 + γ 33 OLD s (NM st ) + γ 23 OLD s (TIME st 3) 2 + u 0s + u 1s (TIME st 3) + u 3s (NM st ) + e st Level 2 Random Effects 2 Way Interactions 3 Way Interactions 4 Way Interaction Running Example 65
66 SAS Syntax for Overall Model Running Example 66
67 Random NEGMOOD Slope? Before we do any additional analysis, we must first check to see if NEGMOOD has a random slope Model WITHOUT random NEGMOOD slope: 2LogL = Model WITH random NEGMOOD slope 2LogL = Deviance test (conservative): 3.3 (3 df) p = Conclusion: No NEGMOOD Random Slope Needed Running Example 67
68 Model Estimated Fixed Effects There are many fixed effects that are not significant Can remove, layer by layer (starting with 4 way interaction) Running Example 68
69 Removing Non Significant 4 Way Interaction Running Example 69
70 Removing Non Significant 3 Way Interactions Running Example 70
71 Removing Non Significant 2 Way Interactions Running Example 71
72 Removing Non Significant Main Effect of NEGMOOD New significant effects: increase in Number of Symptoms reported at each time point for every one unit increase in person Negative Mood mean (above grand mean) additional increase in Number of Symptoms reported at each time point for every one unit increase in person Negative Mood mean (above grand mean) for subjects in OLD = 1 group Running Example 72
73 Inspecting the Random Effects Level 2 ( matrix): OLD only G. OLD+NM G. Intercept Slope Intercept Slope Proportion reduction in generalized variance:.138 Proportion reduction of intercept variance:.118 Proportion reduction of slope variance:.001 NegMood explains additional 13.8% of generalized level 2 variance above OLD only A Pseudo Pseudo R 2 Level 1 ( matrix): no reduction (no level 1 predictors): OLD: OLD+NM: Running Example 73
74 Analysis Conclusions Number of reported symptoms generally decreased across study Quadratic model (time effects) Age group significantly related to number of reported symptoms Older group had overall lower reported (intercept) Older group declined slower (OLD*linear interaction) Average negative mood was significantly related to number of reported symptoms Overall increase in number of symptoms reported when negative mood is higher (main effect of negative mood) Age group had even higher increase in number of symptoms reported for higher average negative mood values (interaction of OLD and NEGMOOD) Occasion fluctuation in negative mood was not related to number of reported symptoms Findings are actually cross sectional in nature Running Example 74
75 Concluding Remarks This section was an (inadequately short) introduction to longitudinal modeling with the multilevel model approach You can take years of courses on the topic Big picture: using the same linear modeling framework, we can do powerful longitudinal analyses When conducting analyses, the order of analysis is: 1. The longitudinal trend (linear, quadratic, etc ) 2. The time invariant predictors 3. The time varying predictors Running Example 75
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