Describing Change over Time: Adding Linear Trends

Transcription

1 Describing Change over Time: Adding Linear Trends Longitudinal Data Analysis Workshop Section 7 University of Georgia: Institute for Interdisciplinary Research in Education and Human Development Section 7: Describing (up to Linear) Change over Time

2 Covered this Section The big picture of modeling change Fixed and random effects models for up to linear change Section 7: Describing (up to Linear) Change over Time 2

3 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) Section 7: Describing (up to Linear) Change over Time 3

4 Its about time DECIDING UPON THE METRIC OF TIME Section 7: Describing (up to Linear) Change over Time 4

5 Deciding Upon the Metric of Time Summary across steps: The goal of creating statistical models of change is to describe the overall pattern of and predict individual differences in change over time These models employ an often unrecognized assumption that we know exactly what time should be. So the missing Step 2 is a pre cursor to every other step in longitudinal analysis, and involves 2 related concerns: What should time be? How should time be modeled when people differ in time? Section 7: Describing (up to Linear) Change over Time 5

6 Why Time Matters: A Level 1 with Level 2 Information The issue with longitudinal data is that variables collected at level 1 often contain level 2 information In longitudinal: time varying covariates In many studies, time may appear to be level 1, but may have level 2 information that we have to control for statistically Therefore, we must put the level 2 information into the statistical model This is typically accomplished in MLM by adding the mean of the cluster In longitudinal: adding the person mean across all time points For time, this may be a number of things Section 7: Describing (up to Linear) Change over Time 6

7 Considerations for Time First: What should time be? Which metric of time best matches the causal process thought to be responsible for observed change? Do alternative metrics of time for multiple processes create different pictures of change and individual differences therein? Relevant for aligning different persons onto same time trajectory, but this distinction is not relevant within persons Second: What do we do when people differ in time? How should time be modeled in accelerated designs? When people begin a study at different points in time, how should we distinguish effects of between person differences in time from effects of within person changes in time? Section 7: Describing (up to Linear) Change over Time 7

8 Example Data: Octogenarian (Twin) Study of Aging 173 persons (65% women) Measured up to 5 occasions over 8 years Known dates of birth and death Estimated dates of dementia diagnosis (91 Alz., 50 Vas., 32 Mixed) Baseline occasion time variability: 79 to 100 years of age (M = 84, SD = 3) 16 to 0 years from death (M = 6, SD = 4) 12 to 18 years from diagnosis (M = 0, SD = 5) Correlation Death Dementia Age Death.52 Cognition outcomes (each T scored): General: Mini Mental Status Exam Memory: Object Recall Spatial Reasoning: Block Design #Persons per #Occasions Section 7: Describing (up to Linear) Change over Time 8

9 Alternative Metrics of Time (and ICC) Chronological Age as Time (47% BP) Individual differences are organized around the mean level for a given distance from birth (84 years) and change with distance from birth Years to Death as Time (24% BP) Individual differences are organized around the mean level for a given distance from death ( 7 years) and change with distance from death Years to Dementia Diagnosis as Time (70% BP) Individual differences are organized around the mean level for a given distance from diagnosis (=0) and change with distance from diagnosis Years in Study as Time (0% BP) Individual differences are organized around the mean level for a given distance from baseline (=0) and change with distance from baseline Section 7: Describing (up to Linear) Change over Time 9

10 General Cognition: MMSE Section 7: Describing (up to Linear) Change over Time 10

11 What about just time as Time? Time in study models separate BP and WP effects Accelerated time (age, death ) model Grand mean centering Time in study model Person/group mean centering Time in study models can be made equivalent to models with accelerated time metric in their fixed effects, but not in their random effects For our running example, we will only consider time in study What we can show is that this also has 0% Between Person variation So only a level 1 variabel Section 7: Describing (up to Linear) Change over Time 11

12 Back to our Running Example 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) Section 7: Describing (up to Linear) Change over Time 12

13 DECIDING UPON A CENTERING POINT FOR TIME Section 7: Describing (up to Linear) Change over Time 13

14 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) Section 7: Describing (up to Linear) Change over Time 14

15 Deciding Upon a Centering Point for Time What we will see shortly is that depending on where we center time, a number of our parameters describing random effects may change r = +1 r = 0 r = 1 Which interceptslope correlation is the right one? Section 7: Describing (up to Linear) Change over Time 15

16 Step #4 ESTIMATE A SATURATED MEANS MODEL AND PLOT INDIVIDUAL TRAJECTORIES Section 7: Describing (up to Linear) Change over Time 16

17 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) Section 7: Describing (up to Linear) Change over Time 17

18 Example Data Individual Observed Trajectories (N = 101, n = 6) Number Match 3 Response Times by Session Section 7: Describing (up to Linear) Change over Time 18

19 The Big Picture of Longitudinal Data: Models for the Means (Fixed Effects) What kind of change occurs on average over time? Two baseline models for comparison: Empty model only a fixed intercept (predicts no change) Saturated model all occasion mean differences from time 0 (ANOVA model that uses # fixed effects= n) *** may not be possible in unbalanced time data Parsimony Empty Model: Predicts NO change over time 1 Fixed Effect In between options: polynomial slopes, piecewise slopes, nonlinear models Saturated Means: Reproduces mean at each occasion # Fixed Effects = # Occasions Good fit Section 7: Describing (up to Linear) Change over Time 19

20 Baseline Models for the Means Number Match 3 Mean Response Times by Session RT in msec 2,000 1,950 1,900 1,850 1,800 1,750 1,700 1,650 1,600 1,550 1,500 Saturated Means (ANOVA) Model = 6 parameters (1 mean per session) Empty Means Model = 1 fixed intercept (means predicted to be equal over sessions) Session Section 7: Describing (up to Linear) Change over Time 20

21 The Big Picture of Longitudinal Data: Models for the Variance (Random Effects) From a substantive perspective: Are there individual differences in change? Individual differences in the level of an outcome? At what time point are individual differences in outcome level important for your hypotheses (beginning, middle, end)? Individual differences in magnitude of change? Each aspect of change (e.g., linear change, quadratic change) can potentially exhibit individual differences (data permitting) From a statistical perspective: What kind of pattern do the variances and covariances exhibit over time? Do the variances increase or decrease over time? Are the covariances differentially related based on time? Section 7: Describing (up to Linear) Change over Time 21

22 The Big Picture of Longitudinal Data: Models for the Variance Compound Symmetry (CS) Unstructured (UN) Parsimony Univariate RM ANOVA Most useful model: likely somewhere in between! Good fit What is the pattern of variance and covariance over time? CS and UN are just two of the many, many options available within MLM, including random effects models (for change) and alternative covariance structure models (for fluctuation). Section 7: Describing (up to Linear) Change over Time 22

23 Baseline Models for the Variance Variance in Number Match 3 Response Times by Session Variance in RT in msec 325, , , , ,000 Unstructured Variance Model = 21 parameters (all vars. and covars.) Random Intercept Model = 2 parameters ( and ) (variances predicted to be equal over sessions) 200, , Session Section 7: Describing (up to Linear) Change over Time 23

24 Summary: Modeling Means and Variances We have two tasks in describing within person change: Choose a Model for the Means What kind of change in the outcome do we have on average? What kind and how many fixed effects do we need to predict that mean change as parsimoniously but accurately as possible? Choose a Model for the Variances What pattern do the variances and covariances of the outcome show over time because of individual differences in change? What kind and how many random effects do we need to predict that pattern as parsimoniously but accurately as possible? Section 7: Describing (up to Linear) Change over Time 24

25 Back to our Example 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) Section 7: Describing (up to Linear) Change over Time 25

26 Step #5 EVALUATE FIXED AND RANDOM EFFECTS OF TIME: FIXED AND RANDOM LINEAR TIME TRENDS Section 7: Describing (up to Linear) Change over Time 26

27 Back to our Example 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) Section 7: Describing (up to Linear) Change over Time 27

28 Using Polynomial Functions of Time to Examine Trends Predict mean change with polynomial fixed effects of time: Linear constant amount of change (up or down) Quadratic change in linear rate of change (acceleration/deceleration) Cubic change in acceleration/deceleration of linear rate of change (known in physics as jerk, surge, or jolt) Terms work together to describe curved trajectories Can have polynomial fixed time slopes UP TO: n 1* 3 occasions = 2nd order (time 2 )= Fixed Quadratic Time or less 4 occasions = 3rd order (time 3 ) = Fixed Cubic Time or less Interpretable polynomials past cubic are rarely seen in practice *This rule can be broken in unbalanced data (but cautiously) Section 7: Describing (up to Linear) Change over Time 28

29 Augmenting the Empty Means, Random Intercept Model with Time 2 questions about the possible effects of time: 1. Is there an effect of time on average? If the line describing the sample means not flat? Significant FIXED effect of time 2. Does the average effect of time vary across individuals? Does each individual need his or her own line? Significant RANDOM effect of time Section 7: Describing (up to Linear) Change over Time 29

30 Empty Means, Random Intercept Model 3 Total Parameters: Model for the Means (1): MLM Empty Model: Level 1: Level 2: Fixed Intercept Model for the Variance (2) Level 1 Variance of : Level 2 Variance of : Residual = time specific deviation from individual s predicted outcome Fixed Intercept =grand mean (because no predictors yet) Random Intercept = individual specific deviation from predicted intercept Composite equation/ Overall Model: Section 7: Describing (up to Linear) Change over Time 30

31 Fixed and Random Effects of Time (Note: The intercept is random in every figure) No Fixed, No Random Yes Fixed, No Random No Fixed, Yes Random Yes Fixed, Yes Random Section 7: Describing (up to Linear) Change over Time 31

32 Fixed Linear Time, Random Intercept Model Multilevel Model Residual = time specific deviation from individual s predicted outcome with estimated variance of Level 1 model: Fixed Intercept = predicted mean outcome at time 0 Level 2 model: Random Intercept = individual specific deviation from fixed intercept with estimated variance of Fixed Linear Time Slope = predicted mean rate of change per unit time Overall model: β 0i β 1i Because the effect of time is fixed, everyone is predicted to change at exactly the same rate. Section 7: Describing (up to Linear) Change over Time 32

33 Explained Variance from Fixed Linear Time Most common measure of effect size in MLM is Pseudo R 2 Is supposed to be variance accounted for by predictors Multiple variance components mean multiple possible values of pseudo R 2 (can be calculated per variance component or per model level) A fixed linear effect of time will reduce level 1 residual variance σ in R By how much is the residual variance σ reduced? If time varies between persons, then level 2 random intercept variance τ in G may also be reduced: But you are likely to see a (net) INCREASE in τ instead. Here s why: Section 7: Describing (up to Linear) Change over Time 33

34 Increases in Random Intercept Variance Level 2 random intercept variance τ will often increase as a consequence of reducing level 1 residual variance σ Observed level 2 τ is NOT just between person variance Also has a small part of within person variance (level 1 σ ), or: Observed = True + ( /n) As n occasions increases, bias of level 1 σ is minimized Likelihood based estimates of true τ use (σ /n) as correction factor: True = Observed ( /n) For example: observed level 2 τ =4.65, level 1 σ =7.06, n=4 True τ = 4.65 (7.60/4) = 2.88 in empty means model Add fixed linear time slope reduce σ from 7.06 to 2.17 (R 2 =.69) But now True τ = 4.65 (2.17/4) = 4.10 in fixed linear time model Section 7: Describing (up to Linear) Change over Time 34

35 Random Intercept Models Imply People differ from each other systematically in only ONE way in intercept ( ), which implies ONE kind of BP variance, which translates to ONE source of person dependency (covariance or correlation in the outcomes from the same person) If so, after controlling for BP intercept differences (by estimating the variance of as τ in the G matrix), the residuals (whose variance and covariance are estimated in the R matrix) should be uncorrelated with homogeneous variance across time, as shown: Level 2 G matrix: RANDOM TYPE=UN Level 1 R matrix: REPEATED TYPE=VC G and R matrices combine to create a total V matrix with CS pattern Section 7: Describing (up to Linear) Change over Time 35

36 Back to our Example 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) Section 7: Describing (up to Linear) Change over Time 36

37 Summary so far Regardless of what kind of model for the means you have Empty means = 1 fixed intercept that predicts no change Saturated means = 1 fixed intercept + n 1 fixed effects for mean differences that perfectly predict the means over time Is a description, not a model, and may not be possible with unbalanced time Fixed linear time = 1 fixed intercept, 1 fixed linear time slope that predicts linear average change across time Is a model that works with balanced or unbalanced time May cause an increase in the random intercept variance by explaining residual variance A random intercept model Predicts constant total variance and covariance over time Should be possible in balanced or unbalanced data Still has residual variance (always there via default R matrix TYPE=VC) Now we ll see what happens when adding other kinds of random effects, such as a random linear effect of time Section 7: Describing (up to Linear) Change over Time 37

38 Random Linear Time Model (6 total parameters) Multilevel Model Residual = time specific deviation from individual s predicted outcome with estimated variance of Level 1 model: Fixed Intercept = predicted mean outcome at time 0 Level 2 model: Fixed Linear Time Slope = predicted mean rate of change per unit time Random Intercept = individual specific deviation from fixed intercept with estimated variance of Random Linear Time Slope= individualspecific deviation from fixed linear time slope with estimated variance of Overall model: Also has an estimated covariance of random intercepts and slopes of β 0i β 1i Because the effect of time is fixed, everyone is predicted to change at exactly the same rate. Section 7: Describing (up to Linear) Change over Time 38

39 Random Linear Time Model y st = (γ 00 + U 0s )+ (γ 10 + U 1s )(Time st )+ e st Fixed Intercept Random Intercept Deviation Fixed Slope Random Slope Deviation error for person i at time t 6 Parameters: e st = -1 2 Fixed Effects: γ 00 Intercept, γ 10 Slope γ 00 =10 u 1s = +2 γ 10 = 6 2 Random Effects Variances: U 0s Intercept Variance U 1s Slope Variance Int Slope Covariance U 0s = -4 1 e st Residual Variance = Section 7: Describing (up to Linear) Change over Time 39

40 Unbalanced Time or Different Time Occasions Across Persons? No problem! Rounding time can lead to incorrect estimates! Code time as exactly as possible. This red predicted slope will probably be made steeper because it is based on less data, though the term for this is shrinkage. MLM uses each complete time point for each person Section 7: Describing (up to Linear) Change over Time 40

41 Back to our Example 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) Section 7: Describing (up to Linear) Change over Time 41

42 FURTHER EXAMINING HOW RANDOM EFFECTS OF TIME MODEL DEPENDENCIES IN MLMS Section 7: Describing (up to Linear) Change over Time 42

43 Summary: Sequential Models for Effects of Time Level 1: Level 2: Composite: Empty Means, Random Intercept Model: 3 parms = γ 00,, Level 1: Level 2: Fixed Linear Time, Random Intercept Model: 4 parms = γ 00, γ 10,, Composite: Level 1: Level 2: Random Linear Time Model: 6 parms = γ 00, γ 10,,,, (covariance of U 0s and U 1s ) Composite: Section 7: Describing (up to Linear) Change over Time 43

44 How MLM Handles Dependency Common description of the purpose of MLM is that it addresses or handles correlated (dependent) data But where does this correlation come from? 3 places (here, an example with health as an outcome): 1. Mean differences across persons Some people are just healthier than others (at every time point) This is what a random intercept is for 2. Differences in effects of predictors across persons Does time (or stress) affect health more in some persons than others? This is what random slopes are for 3. Non constant within person correlation for unknown reasons Occasions closer together may just be more related This is what ACS models are for Section 7: Describing (up to Linear) Change over Time 44

45 MLM Handles Dependency Where does each kind of person dependency go? Into a new random effects variance component: Level 1 Residual Variance Within Person Differences Original Residual Variance Level 1 Residual Variance Level 2 Random Intercept Variance Level 2 Random Linear Slope Variance Level 2 Random Intercept Variance Between Person Differences covariance Section 7: Describing (up to Linear) Change over Time 45

46 Variance Components By adding a random slope, we carve up our total variance into 3 components: BP (error) variance around intercept BP (error) variance around slope WP (error) residual variance These 2 components are 1 components of error variance in Univ. RM ANOVA But making components does NOT make error variance go away Level 1 (one source of) Within Person Variation: gets accounted for by time level predictors Residual Variance ( ) FIXED effects make variance go away (explain variance). RANDOM effects just make a new variance component Level 2 (two sources of) Between Person Variation: gets accounted for by personlevel predictors BP Int Variance ( ) covariance BP Slope Variance ( ) Section 7: Describing (up to Linear) Change over Time 46

47 Fixed vs. Random Effects of Persons Person dependency: via fixed effects in the model for the means or via random effects in the model for the variance? Individual intercept differences can be included as: N 1 person dummy code fixed main effects OR 1 random U0i Individual time slope differences can be included as: N 1*time person dummy code interactions OR 1 random U1i*timeti Either approach would appropriately control for dependency (fixed effects are used in some programs that control SEs for sampling) Two important advantages of random effects: Quantification: Direct measure of how much of the outcome variance is due to person differences (in intercept or in effects of predictors) Prediction: Person differences (main effects and effects of time) then become predictable quantities this can t happen using fixed effects Summary: Random effects give you predictable control of dependency Section 7: Describing (up to Linear) Change over Time 47

48 Quantification of Random Effects Variances Likelihood ratio tests tell us if a random effect is significant, but random effects variances are not likely to have inherent meaning e.g., I have a significant fixed linear time effect of γ10 = 1.72, so people increase by 1.72/time on average. I also have a significant random linear time slope variance of = 0.91, so people need their own slopes (people change differently). But how much is a variance of 0.91, really? 95% Random Effects Confidence Intervals can tell you Can be calculated for each effect that is random in your model Provide range around the fixed effect within which 95% of your sample is predicted to fall, based on your random effect variance: So although people improve on average, individual slopes are predicted to range from 0.15 to 3.59 (so some people may actually decline) Section 7: Describing (up to Linear) Change over Time 48

49 Random Linear Time Models Imply: People differ from each other systematically in TWO ways in intercept ( ) and slope ( ), which implies TWO kinds of BP variance, which translates to TWO sources of person dependency (covariance or correlation in the outcomes from the same person) If so, after controlling for both BP intercept and slope differences (by estimating the τ and τ variances in the G matrix), the residuals (whose variance and covariance are estimated in the R matrix) should be uncorrelated with homogeneous variance across time, as shown: Level 2 G matrix: RANDOM TYPE=UN Level 1 R matrix: REPEATED TYPE=VC G and R combine to create a total V matrix whose per person structure depends on the specific time occasions each person has (very flexible for unbalanced time) Section 7: Describing (up to Linear) Change over Time 49

50 Random Linear Time Model (6 total parameters: effect of time is now RANDOM) How the model predicts each element of the V matrix: Level 1: Level 2: Composite: Predicted Time Specific Variance: Section 7: Describing (up to Linear) Change over Time 50

51 Random Linear Time Model (6 total parameters: effect of time is now RANDOM) How the model predicts each element of the V matrix: Level 1: Level 2: Composite: Predicted Time Specific Covariances (Time A with Time B): Section 7: Describing (up to Linear) Change over Time 51

52 Random Linear Time Model (6 total parameters: effect of time is now RANDOM) Scalar mixed model equation per person: X i = n x k values of predictors with fixed effects, so can differ per person (k = 2: intercept, linear time) γ = k x 1 estimated fixed effects, so will be the same for all persons (γ 00 = intercept, γ 10 = linear time) Z i = n x u values of predictors with random effects, so can differ per person (u = 2: intercept, linear time) U i = u x 2 estimated individual random effects, so can differ per person E i = n x n time-specific residuals, so can differ per person Section 7: Describing (up to Linear) Change over Time 52

53 Random Linear Time Model (6 total parameters: effect of time is now RANDOM) Predicted total variances and covariances per person: Z i = n x u values of predictors with random effects, so can differ per person (u = 2: int., time slope) Z it = u x n values of predictors with random effects (just Z i transposed) G i = u x u estimated random effects variances and covariances, so will be the same for all persons (τ = int. var., τ = slope var.) R i = n x n time-specific residual variances and covariances, so will be same for all persons (here, just diagonal σ ) Section 7: Describing (up to Linear) Change over Time 53

54 Building V Across Persons: Random Linear Time Model V for two persons with unbalanced time observations: The giant combined V matrix across persons is how the multilevel or mixed model is actually estimated in SAS Known as block diagonal structure where predictions are given for each person, but 0 s are given for the elements that describe relationships between persons (because persons are supposed to be independent here!) Section 7: Describing (up to Linear) Change over Time 54

55 Building V Across Persons: Random Linear Time Model V for two persons also with different n per person: The block diagonal does not need to be the same size or contain the same time observations per person R matrix can also include non 0 covariance or differential residual variance across time (as in ACS models), although the models based on the idea of a lag won t work for unbalanced or unequal interval time Section 7: Describing (up to Linear) Change over Time 55

56 G, R, and V: The Take Home Point The partitioning of variance into specific components Level 2 = BP G matrix of random effects variances/covariances Level 1 = WP R matrix of residual variances/covariances G and R combine via Z to create V matrix of total variances/covariances Many flexible options that allows the variances and covariances to vary in a timedependent way that better matches the actual data Can allow variance and covariance due to other predictors, too Compound Symmetry (CS) Unstructured (UN) Parsimony Univariate RM ANOVA Random effects models use G and R to predict something in between! Multivariate RM ANOVA Good fit Section 7: Describing (up to Linear) Change over Time 56

57 Back to our Example 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) Section 7: Describing (up to Linear) Change over Time 57

58 Wrapping Up This section covered aspects of the investigation of the time trend in data What time should be What centering point should be used How to add fixed linear/random intercept and random linear effects of time Each of the models we added had very specific implications for our overall variance/covariance of our conditional observations (the V matrix) Covariance is the key to getting multivariate models right especially in longitudinal analyses Next we will further delve into time exploring higher order polynomials After that we can start to add other predictors to our analyses Longitudinal analyses are very similar to other MLMs, but with a very large design effect to control for time Section 7: Describing (up to Linear) Change over Time 58