Using Hierarchical Linear Models to Measure Growth. Measurement Incorporated Hierarchical Linear Models Workshop

Transcription

1 Using Hierarchical Linear Models to Measure Growth Measurement Incorporated Hierarchical Linear Models Workshop

2 Chapter 6 Motivation Linear Growth Curves Quadratic Growth Curves Some other Growth Curves Centering HLM/Repeated Measures/SEM Power 2

3 Motivation HLM is useful when we have observations (Level-1) nested within a Level-2 variable. Think students nested within classes. Useful because HLM: Explains dependencies. Computes relationship within each group. 3

4 Motivation These models can easily apply to repeated measures: Individual Change/Growth Models. We have repeated measures observed within a student. Assuming that we use appropriate measures, HLM can provide an integrated approach for studying the structure and indicators of individual growth. 4

5 Motivation So now we will have data like Level-2 e Analysis ayss (e.g. Character level) Because this is HLM, we can also have additional covariates at each level Level-1 Analysis (e.g. Repeated Measures of Time) 5

6 Motivation In this context we can use time as a level-1 covariate. This allows us to also look at change across time. Or even growth rates at a specific time. 6

7 Objective After this chapter you should be able to: Fit a basic (2-level) HLM to repeated measures data. Linear growth curves. Quadratic growth curves. Understand their basic interpretation. Understand the importance of centering and its effects. 7

8 Objective You will also: Understand the difference between HLM and other approaches. Have a better feel for power considerations. 8

9 The Model For the model we will have a different notation. Which is consistent with the notation introduced for the 3- level analyses. We will have observations of examinee i at time point t for some variable Y ti : Observations are nested within examinee. 9

10 The Model For the repeated measures growth curve we will have a is usually time points Y = π + π a + π a + L+ π a + ti 0i 1i ti 2i 2 ti Pi P ti e ti For the examinee level π pi = β p0 + Q p q= 1 β pq X qi + r All of the assumptions still apply pi These are individual characteristics 10

11 Brief Examples Students who have taken several tests (which are on the same scale). An athlete training for the olympics, has repeated measures of the time it takes on some event. Individuals who are in some kind of counseling with repeated observations of behavior. 11

12 Linear Growth Models We begin with a basic linear growth model. In this case, the rate of change is constant across time. While a linear model may not be the true model it works well. For short intervals of time. Few observations. 12

13 Linear Growth Models Time Repeated measures Y a + ti = π + π 0i 1i ti e ti Error What is this? What is this? π π oi 1i = β = β Q p q= 1 Q p q = 1 β β oq 1q X X qi qi + r + r 1i oi Level-2 and errors (we let them correlate) 13

14 Linear Growth Models Using this model we can: Estimate a mean growth slope (curve). Determine the reliability of status and change. Estimate relationship between initial status and rate of change. Provide some general descriptive statistics. Model relations of person-level variables to status and growth rate. 14

15 Working Example (p. 164) Study of children s growth during preschool and early elementary grades. Outcome measure is IRT examinee parameter value from a test of natural science knowledge. 143 examinees in a Head Start program. 15

16 Data So our data looks like 16

17 Working Example First let s assume that we want to: Estimate the average increase in ability per unit of time for the participants. Determine the reliability of status and change. Estimate relationship between initial status and rate of change. Estimate the variability across intercepts and rates. 17

18 Random Coefficient Model The first model that we deal with is the random coefficient model How would we know the average increase? Y a + ti = π + π 0i 1i ti e ti How would we know the variability? How would we know the reliability? πoi = β 00 + r 0i π 1 = β 10 + r i r 1i How would we know the relationship b/t starting stress and rate? What do these parameters mean for our problem? 18

19 HLM Program 19

20 HLM Model Parameters The following parameters are estimated: Fixed intercept: β 00 Fixed slope: β 10 Variance of random intercept: τ 00 Variance of random slope: τ 11 Covariance of random intercept and slope: τ 01 Level-1 error variance: σ 2 The following slides dissect the HLM output for each of these terms. 20

21 Fixed Effects Estimates Fixed intercept: β 00 Fixed slope: β 11 21

22 Variance Component Estimates Variance of random intercept: τ 00 Level-1 error variance: σ 2 Variance of random slope: τ 11 22

23 Random Coefficient Model So we see that this model is no different from what we have seen in the past. We make a basic assumption that dependencies in our observations are due to the second level of analysis (people). 23

24 Intercepts/Slopes as Outcomes Now we will use the model Same within person level model Y = π + π a + ti = 0i 1i ti e ti Now our intercept and slope are modeled as a function of what they are wearing π oi π = β W 00 + β01( X i ) + r0 i W 1 i = β 10 + β 11 ( X i ) + r 1 i So just think about what we are saying here 24

25 Intercepts/Slopes as Outcomes After looking at the results we have seen how a basic growth model could be useful. In addition, I think they have a really nice interpretation. BUT,,growth and change are not always linear. Learning Training Testing 25

26 A Quadratic Growth Curve For that reason we can expand our model (as was shown in the general case). In expanding the model we allow for the growth rate to change across time. Do you remember what parameter was used to indicate our growth rate in the linear growth model? 26

27 A Quadratic Growth Curve So to expand our model to the quadratic we will use = π + π ( ) + π ( ) 2 + ti 0i 1i ti 2i ti ti Y a L a L e Expected score at time L Q p + X + π 0 = β00 β0 i q= 1 q qi r 0i The growth rate at time L Q p + + π1 = β10 β1 i q= 1 q X qi r1 i Characterizes the acceleration over time π 2i = β20 + β2q qi + r Q p q= 1 X 2i 27

28 A Quadratic Growth Curve Intercept gives level at time L of the study. The second coefficient gives the spontaneous growth rate at time L. The third coefficient gives the acceleration at which the change is occurring. This means that the growth rate actually depends on time. 28

29 A Quadratic Growth Curve To determine the growth rate at any given time we use the first derivative of our level-1 model: Y = π + ti 2 0 i + π1 i ( ati L) + π2i ( ati L) e ti Level-1 Model Growth Rate=1 st derivative with respect to time GrowthRate = π 1 i + 2π 2 i ( ati L) 29

30 Example So now we go to the example that is nearly the same as is given in the book. We have verbal scores that are recorded at different ages. Level-1 variables: Verbal ability. Age (we will actually use age-12). Level-2: Study. Sex. MomSpeak (or the log of this variable). 30

31 Data Set 31

32 Quadratic Growth Curve The nice thing about this example is that it gives us a feel about how to complete a basic analysis with a quadratic growth curve. I will follow the example and we can go over the results. We will also discuss the interpretation of these variables. 32

33 Quadratic Growth Curve So the procedure is: Look at the data (ask: do we need quadratic?). Fit random coefficients data. Simplify (Do we need certain coefficients?) Fixed versus random effects. Intercepts versus not. Associations versus not between random effects. Fit second level model. 33

34 Graph Data Look at the data. In HLM this is also fairly easy. File Graph Data line plots, scatter plots. Put time variable on X-axis. Choose line plot. 34

35 Graph Data Result Trend seems to indicate some type of curved learning taking place. VOCAB AGE12 35

36 Fit Level-1 Analysis Here we need to determine our first level equation. We have already determined that we need a Quadratic so our basic model will be a random coefficients model. 36

37 The Model = π + π ( ) + π ( ) 2 + ti 0i 1i ti 2i ti ti Y a L a L e Again, we want to think about what these parameters mean. Also, What does it mean if we find that the level-2 error terms are correlated π π π 0 i = β 00 + r 0i 1 i = β 10 + r 1i 2 i = β 20 + r 2i i 37

38 HLM Model Parameters The following parameters are estimated: Fixed intercept: β 00 Fixed linear slope: β 10 Fixed quadratic slope: β 20 Variance of random intercept: τ 00 Variance of random linear slope: τ 11 Variance of random quadratic slope: τ 11 Covariance of random intercept and linear slope: τ 01 Covariance of random intercept and quadratic slope: τ 02 Covariance of random linear slope and random quadratic slope: τ 12 Level-1 error variance: σ 2 38

39 Simplification Things to look at: Fixed Coefficients. Random Coefficients. Associations. 39

40 HLM Input 40

41 Variance Component Estimates Level-1 error variance: σ 2 Random Effect Correlation Matrix Random Effect Covariance Matrix 41

42 Fixed Effect Estimates Fixed intercept: β 00 Fixed linear slope: β 10 Fixed quadratic slope: β 20 42

43 Other Growth Curves We can always fit more terms: We could have a polynomial with up to P terms as long as we have (P+1) observations with in several observations. We can always transform our dependent d variable. ibl The average level-1 equation can look different from the model at level-1. 43

44 Complex Level-1 Errors Also, if need be, we may be interested in: Specifying level-1 variance as a function of person variables. Specifying level-1 variance as a function of time variables. We can also allow the error terms within a person to correlate (AR(1)). 44

45 Piecewise Linear Growth Models While a quadratic growth curve may work just fine to describe the data, there is a alternative. There may be instances when you do not like assuming that the growth rate is constantly changing. This is also useful when we want to compare growth rates between two different periods. As an alternative, we can fit a nonlinear trend with two lines. 45

46 Piecewise Linear Growth Models To do this we get to define a few new variables. Depending on how we define the variables we will have two different interpretations. To do this I first present the table on page

47 Recoding These are for two different slopes This is for the incremental change in the slope between the two periods 47

48 Models Once we have our new variables we use the model: Y = π + π a + π a + r ti 0i 1i 1ti 2i 2ti ij Of course, we can model each level-1 coefficient as a function of level-2 coefficients. How are these parameters interpreted? 48

49 Time-Varying Covariates One other thing to notice is that by modeling responses using an HLM we can easily have time varying covariates. For example: Weather conditions. Amount of sleep on the night prior to a test. These could also be fixed or random effects. 49

50 Centering Next we see how centering can have an effect on growth curves. Specifically, we will look at: Centering in linear growth curves. Centering in quadratic growth curves. Bias of studying time-varying predictors. Variance growth parameters. 50

51 Centering (Linear Growth) As you may expect centering is all the same here. How we interpret the intercept is determined by where we center. Here, we are more likely to center based on a theoretically interesting point. 51

52 Centering (Quadratic Growth) Now in a quad growth curve things are going to be slightly different. Remember that the intercept π 0i is equal to that point at which our variable equals 0. Also π 1i is the instantaneous growth rate at that same time. 52

53 Centering (Quadratic Growth) This means that if we use the start time at 0 then: The intercept is the starting value. π 1i is the growth rate at the initial time individuals start. There could be little information at that point. This may also cause high correlations between these two values. 53

54 Centering (Quadratic Growth) Centering at the middle Intercept is ability at the middle of study. π 1i is the growth rate at the middle. This is also the average growth rate. More information about growth rate. Minimizes correlation between random effects. 54

55 Bias for Time-Varying Covariates In addition, Time-varying covariates may have biased slopes because of compositional effects (covered in detail in Chapter 5). The aggregate of our variable may mean something different than the individual observations. Think about thing like a variable has a job. 55

56 Variance of Growth Parms Centering can have an effect on variance, because of the change in interpretation of each parameter. If have equal time points across people centering will not change variance. If thinking of other covariates or time points that vary across people then it will matter. Grand mean will shrink standard errors. Group mean will remove effect of the covariate differences. 56

57 Growth Model Extensions 57 Measurement Incorporated HLM Workshop March 13-14, 2008

58 HLM, MRM, and SEM Now we provide a discussion about the comparisons of three different methods to deal with repeated measures: Hierarchical Linear Models Multivariate-Repeated Measures Structural Equation Models 58

59 HLM We begin by the concept of HLM. Repeated measures are conceptualized as a person s trajectory that changes as a function of person specific variables. Level-2 variables describe some of the variation across people. 59

60 MRM In MRM the repeated measures are treated as a response vector from a person. We define our groups: These can be defined by a combination of main effects and interactions. Then we have to define the variation across time. Then we compare across groups: This is just like a MANOVA with contrasts or profile analysis. 60

61 Limitations of MRM For the most part MRM requires that all time periods have been observed for all people: p This is not required by HLM. MRM does not extend to a situations ti where the people (level-2) are nested inside of a level-3. HLM does this naturally. 61

62 SEM In general, there are a number of growth curves that can be phrased as a structural equation model. In these cases: The SEM measurement part is the level-1 analysis. The latent variables are the growth parameters. The structural part then corresponds to the betweenperson part of the model. 62

63 SEM One major advantage of SEM is that: It can provide numerous correlation structures at either level. You will have any number of fit indices that generally are not supplied for HLM. 63

64 SEM The disadvantage is that: It requires that each examinee has the same spacing between time points. If we can conceptualize a situation where this would occur and we simply have missing data we can still use SEM. So there can be some variation in time points. It requires that level-1 predictors have the same distributions across all participants in the same subgroup. 64

65 HLM and Covariance Structures So just quickly What happens whenever we do not assume that our observations within a level-2 unit are independent? While something like this would be easy in SEM In HLM you would need to do something with a MHLM (which I will not talk about now) 65

66 Just a little Background Notice that HLM already models dependency among our observations (to take this a step farther we are thinking about how HLM will model the variance and covariance structure of our data) Think of a model with a random intercept (homogeneous variance): Uses 2 parameters. Think of a model with a random intercept and slope (homogeneous variance): Uses 4 parameters. Think of a model with a random intercept and slope (heterogeneous variances) Uses more parameters depends on the number of time points 66

67 Just a little Background There are times when this is not enough. Or the parameterization is not right. Think in terms of what happens if within a level-2 unit the observations have a certain correlation structure. 67

68 HLM with level-1 Cov Structures Probably the most common is an autoregressive correlation matrix. This assumes that an observed value at time t actually depends on the value at time (t-1). In the end this means that the correlation between observations given group is something other than 0. And can be characterized with a single parameter. 68

69 HLM and Covariance So what we can see is that HLM is generally thought as having independent errors within the level-2 units. We are able to incorporate some structure. Not as well as SEM, so there is a trade off. SEM has fit indices and more options with structure. HLM handles missing data better in certain situations. 69

70 One Other Point Again we get back to predicting future slopes for a person: OLS Works fine in situations where the coefficients are highly reliable. EB Will give better estimates in data with a lot of noise. We have shared information. 70

71 Power Considerations The same hold as before: That is, if we can, add more people. If we can t, add more observations. With observations we can either: Lengthen the study. Add more time points within a study. Software such as Optimal Design or PINT could be used to add in Power computations 71

72 Chapter 6 Summary Linear and quadratic growth curves. These are the same as any other 2-level HLM. Also, talked about interpretation on the parameters. 72

73 Chapter 6 Summary Piecewise linear growth models: Said by recoding we can explore the difference between two time periods. Time-Varying Covariates Can always add in time varying covariates. 73

74 Chapter 6 Summary Centering Effects Has the same effects as before. Covered effects of center on quadratic models. Comparison of HLM to MRM and SEM Models differ only slightly. SEM has added advantage of modeling association. HLM has added advantage of being flexible with time points. 74

75 Chapter 6 Summary Future Prediction: In predicting future growth curves we can use OLS if reliable. Otherwise use EB for reduced standard errors. Power: Same basic idea as before only now we had to consider length of study. 75