Categorical and Zero Inflated Growth Models

Transcription

1 Categorical and Zero Inflated Growth Models Alan C. Acock* Summer, 2009 *Alan C. Acock, Department of Human Development and Family Sciences, Oregon State University, Corvallis OR This was supported in part by 1R01DA13474, The Positive Action Program: Outcomes and Mediators, A Randomized Trial in Hawaii and R305L CFDA U.S. Department of Education; Positive Action for Social and Character Development. Randomized trial in Chicago, Brian Flay, PI. ; and R215S CFDA, Uintah Character Education Randomized Trial, U.S. Department of Education.

2 Topics to Be Covered Predicting Rare Events Binary Growth Curves Count Growth curves Zero-Inflated Poisson Growth Curves Latent Class Zero-Inflated Poisson Models A detailed presentation of the ideas is available at Alan C. Acock 2

3 Predicting Rare Events Physical conflict in romantic relationships Frequency of depressive symptoms Frequency of Parent-Child Conflict Frequency of risky sex last month Alan C. Acock 3

4 Poisson with too many zeros

5 Binary: Does Behavior Occur Structural zeros behavior is not in behavioral repertoire Do not smoke marijuana Didn t smoke last month Chance zeros Behavior part of repertoire, just not last month No fight with spouse last week, but... Alan C. Acock 5

6 Count Component Two-part model Equation for zero vs. not zero Equation for those not zero. Zeros are missing values Zero-inflated model includes both those who are structural zeros and chance zeros Alan C. Acock 6

7 Trajectory of the Probability of Behavior Alan C. Acock 7

8 Trajectory of the Count of behavior Alan C. Acock 8

9 Why Aren t Both Lines Straight? We use a linear model of the growth curve We predict the log of the expected count We predict log odds for the binary component Alan C. Acock 9

10 Why Aren t Both Lines Straight? For the count we are predicting Expected = ln(λ) = α + βt i T i (0, 1, 2,...) is the time period α is the intercept or initial value β is the slope or rate of growth Alan C. Acock 10

11 Why Aren t Both Lines Straight? Expected log odds or expected log count, are linear Expected probability or expected count, are not linear

12 Time Invariant Covariates Time invariant covariates are constants over the duration of study May influence growth in the binary and count components May influence initial level of binary and count components Different effects a major focus Alan C. Acock 11

13 Time Invariant Covariates Mother s education might influence likelihood of being structurally zero Mother s education might be negatively related to the rate of growth Alan C. Acock 12

14 Time Varying Covariates Time Varying Covariates variables that can change across waves Peer pressure may increase each year between 12 and 18 The peer pressure each wave can directly influence drug usage that year Alan C. Acock 13

15 Estimating a Binary Growth Curve

16 Example of Binary Component Brian Flay has a study in Hawaii evaluating the Positive Action Program in Grades 1-4 Key outcome reducing negative responses to behaviors that Positive Action promotes Gender is a time invariant covariate boys higher initially but to have just as strong a negative slope Alan C. Acock 15

17 Binary Model Alan C. Acock 16

18 Predicting a Threshold Thresholds Y* Where, u = 1 if Y* >! or u = o if Y* "! u Alan C. Acock 17

19 Binary Growth Curve Program Title: workshop binary growth.inp Data: File is workshop_growth.dat ; Variables: Names are idnum s1flbadc s2flbadc s3flbadc s4flbadc male s1flbadd s2flbadd s3flbadd s4flbadd s1flbadm s2flbadm s3flbadm s4flbadm c3 c4 s3techer room ; Usevariables are male s1flbadd s2flbadd s3flbadd s4flbadd c3 c4 ; Categorical are s1flbadd s2flbadd s3flbadd s4flbadd ; Missing are all (-9999) ; Analysis: Estimator = ML ; Alan C. Acock 18

20 Binary Growth Curve Program Part 2 Model: alpha beta s1flbadd@0 s2flbadd@1 s3flbadd@2 s4flbadd@3 ; alpha on male ; beta on male ; s3flbadd on c3 ; s4flbadd on c4 ; Output: Patterns sampstat standardized tech8; Alan C. Acock 19

21 Sample Proportions and Model Fit S1FLBADD Category Category S2FLBADD Category Category S3FLBADD Category Category S4FLBADD Category Category Loglikelihood H0 Value Information Criteria Number of Free Parameters 9 Akaike (AIC) Bayesian (BIC) Sample Size Adjusted BIC Proportion of Negative Responses drops each year Alan C. Acock 20

22 Model Estimates Estimates S.E. Est./S.E. Std StdYX ALPHA ON MALE BETA ON MALE S3FLBADD ON C S4FLBADD ON C BETA WITH ALPHA Intercepts ALPHA BETA Alan C. Acock 21

23 Gender Effects Unstandardized effect of male on the intercept, α, is. 548, z = 2.98, p <.01 Standardized Beta weight is.232 Partially standardized (standardized on latent variable only) is.464 Path to slope is not significant, B =.03, partially standardized path is.08 However effective the program is at reducing negative feelings, it is about as effective for boys as for girls Alan C. Acock 22

24 Implementation Effects Wave 3 Unstandardized effect of implementation for the Binary Component has a B = -.23, z = -2.71, p <.05-- Exponentiated odds ratio is e -.23 =.79 Wave 4 the unstandardized effect of implementation for the Binary Component has a B = -.64, z = , p < Exponentiated odds ratio is e -.64 =.53 Alan C. Acock 23

25 MODEL RESULTS (cont.) Thresholds Estimates S.E. Est./S.E. Std StdYX S1FLBADD$ S2FLBADD$ S3FLBADD$ S4FLBADD$ Residual Variances ALPHA BETA LOGISTIC REGRESSION ODDS RATIO RESULTS S3FLBADD ON C S4FLBADD ON C Alan C. Acock 24

26 Thresholds & Graphs Mplus does not graph estimated probabilities when there are covariates because variances depend on the covariate level We cannot estimate initial probability using threshold value. If no covariates, we would exponentiate the threshold. In Stata display exp(-.714) yields.49. Alan C. Acock 25

27 Thresholds & Graphs If you want a series of graphs (e.g., boy/low intervention both wave 3 and wave 4), you need to treat each combination as a separate group Each group would have no covariates; just be a subset of children. Results might not be consistent with the model using all of the data

28 Estimating a Count

29 Count Component Mplus uses a Poisson Distribution for estimating counts The Poisson distribution is a single parameter distribution with λ = M = σ 2 Without adjusting for the excess of zeros, the σ 2 is often greater than the M Alan C. Acock 27

30 Count Component Alan C. Acock 28

31 Count Program Part 1 Title: workshop count growth fixed effects.inp Data: File is workshop_growth.dat ; Variable: Names are idnum s1flbadc s2flbadc s3flbadc s4flbadc male s1flbadd s2flbadd s3flbadd s4flbadd s1flbadm s2flbadm s3flbadm s4flbadm c3 c4 s3techer room ; Usevariables are s1flbadc s2flbadc s3flbadc s4flbadc ; Missing are all (-9999) ; Count are s1flbadc s2flbadc s3flbadc s4flbadc ; Alan C. Acock 29

32 Count Program Part 2 Model: alpha beta s1flbadc@0 s2flbadc@1 s3flbadc@2 s4flbadc@3 ; alpha@0 ;!fixes var. of intercept at 0 beta@0 ;!fixes var.of slope at 0 Output: residual tech1 tech4 tech8; Plot: Type = Plot3 ; Series = s1flbadc s2flbadc s3flbadc s4flbadc(*) ; Alan C. Acock 30

33 Fixing Variances Fixing the variance of the intercept and slope makes this a fixed effects model Fixing the variance of the slope only makes it a random intercept model Not fixing them makes it a random intercept & random slope model This takes a very long time to run

34 Count Model Output- MODEL RESULTS Estimates S.E. S.E./Est. Means ALPHA BETA Variances ALPHA BETA Alan C. Acock 31

35 Interpreting the Est. Intercept We fixed the residual variances at zero The mean intercept is.56, z = , p <. 001 We can exponentiate this when there are no covariates to get the expected count at the intercept, e.56 = 1.75 Alan C. Acock 32

36 Interpreting the Est. Slope The mean slope is -.64, z = , p <.001. With no covariates we use exponentiation to obtain the expected count for each wave Expected count (wave1) = e α e β 0 = 1.75 Expected count (wave2) = e α e β 1 =.92 Expected count (wave3) = e α e β 2 =.48 Alan C. Acock 33

37 Sample and Estimated Count Alan C. Acock 34

38 Putting the Binary and Count Growth Curves Together

39 Two-Part Model Alan C. Acock 37

40 Putting the Binary and Two-Part Solution First part models binary outcome as we did here with binary data Second part deletes all people who have a count of zero at any wave. This leaves only children who have a count of at least 1 for every wave Second part estimated using a Poisson Model Alan C. Acock 38

41 Putting the Binary and Count Models Together Zero-Inflated Growth Curve Model estimates growth curve for structural zeros and for the count simultaneously Binary component includes all observations Count component includes all observations but is modeling only those zeros that are explainable by a random Poisson process Alan C. Acock 39

42 Zero-Inflated Poisson Regression Alan C. Acock 40

43 Here are 5 cases with counts s1flbadc s2flbadc s3flbadc s4flbadc Alan C. Acock 41

44 Here are there Binary Scores s1flbadd s2flbadd s3flbadd s4flbadd Alan C. Acock 42

45 ZIP Model With No Covarites Alan C. Acock 43

46 ZIP Model With No Alan C. Acock 44

47 Interpreting Inflation Model β_i under the Model Results, B = 2.353, z = 7.149, p <.001. The threshold for the zero-inflated part of the model is shown under the label of Intercepts. For each wave the threshold is , z = 3.45, p <.05. This large negative value will be confusing, unless we remember that the outcome for the inflated part of the model is predicting always zero. We are not predicting one.

48 Interpreting Inflation The more negative the threshold value the smaller the likelihood of being in the always zero class at the start. (display exp(-6.756).001.) Logistic regression usually is predicting the presence of an outcome, but now we are predicting its absence.

49 ZIP Model With No Covariates Alan C. Acock 46

50 ZIP Model With No Covariates Alan C. Acock 47

51 Binary Part: Probability of Inflation Alan C. Acock 48

52 Interpreting the Count Part Alan C. Acock 49

53 Count Part: Expected Count Alan C. Acock 50

54 ZIP Model with Covariates Alan C. Acock 51

55 ZIP Model with Covariates Covariates Effects Alan C. Acock 52

56 ZIP Model with Covariates Alan C. Acock 53

57 ZIP Model with Covariates Alan C. Acock 54

58 ZIP Model with Covariates: Intercept and Slope Alan C. Acock 55

59 Latent Class Growth Analysis Using Zero- Inflated Poisson Model

60 LCGA Poisson Models We use mixture models A single population may have two subpopulations, i.e., our Implementation variable is a class variable Usually assumes class membership explains differences in trajectory, thus a fixed effects model Alan C. Acock 57

61 Latent Profile Analysis for Variable Overall Two Classes Item Means First Second Class Class Stickers for PA Word of the week You put notes in icu box Teacher read ICU notes about you Teacher read your ICU notes Tokens for meeting goals PA Assembly activities Assembly Balloon for PA Whole school PA Days/wk taught PA N 1,550 1, Alan C. Acock 58

62 Applied To Count We can use a Latent Class Analysis in combination with a count growth model to See if there are several classes Classes are distinct from each other Members of a class share a homogeneous growth trajectory Alan C. Acock 59

63 Applied To Zero-Inflated Growth Sometimes referred to as Case or Person Centered rather than Variable Centered Subgroups of children rather than of variables Has advantages in ease of interpretation of results No w/n group variance of intercepts or slope assumes each subgroup is Alan C. Acock 60

64 LCGA Using Count Model, No Serves as a baseline for multi-class solutions Add Mixture to Analysis: section because we are doing a mixture model Add %Overall% to Model: section Later, we will add commands so each class can have differences Alan C. Acock 61

65 LCGA Count Model Program Part 1 Title: LCA zip poisson model NO covariates c1.inp Latent Class Growth Analysis for a count outcome using a ZIP Model with no covariates and just one class Data: File is workshop_growth.dat ; Variable: Names are idnum s1flbadc s2flbadc s3flbadc s4flbadc male s1flbadd s2flbadd s3flbadd s4flbadd s1flbadm s2flbadm s3flbadm s4flbadm c3 c4 s3techer room ; Usevariables are s1flbadc s2flbadc s3flbadc s4flbadc ; Missing are all (-9999) ; Classes = c(1) ;! this says there is a single class Alan C. Acock 62

66 LCGA Count Model Program Part 2 Analysis: Type = Mixture ; Starts 20 2 ; Model: %Overall% Alpha Beta s1flbadc@0 s2flbadc@1 s3flbadc@2 s4flbadc@3 ; Output: residual tech1 tech11 ;!tech11 gives you the Lo, Mendell, Rubin test Plot: Type = Plot3 ;! Series = s1flbadc s2flbadc s3flbadc s4flbadc(*) ; Alan C. Acock 63

67 LCGA Using ZIP Model, No Thestarts 20 2 ; Covariates Two Classes generates 20 starting values, does an initial estimation on each of these, then does full iterations on 2 best initial solutions. Best two did not converge with 150 starts for 4 classes We change classes = c(1) to = c(2) The following table compares 1 to 3 classes Alan C. Acock 64

68 LCGA Using ZIP Model, No Covariates Two Classes We will focus on the 2 class solution There is a normative class (902 children) and a deviant class with just 85 The biggest improvement in fit is from 1 to 2 classes

69 Comparison of 1 to 4 Classes 1 Class 2 Classes 3 Classes 4 Classes Free Parameters AIC BIC Sample Adjusted BIC Entropy Lo, Mendell, Rubin na 2 v 1 Value = p < v 2 Value = p <.01 4 v 3 Value = P <.05 N for each class C1 = 2927 C1=2651 C2=276 C1=235 C2=81 C3=2611 C1 = 19 C2 = 2559 C3 = 188 C4 = 161 Alan C. Acock 65

70 Two Class Solution CLASSIFICATION QUALITY Entropy CLASSIFICATION OF INDIVIDUALS BASED ON THEIR MOST LIKELY LATENT CLASS MEMBERSHIP Class Counts and Proportions Latent Classes Alan C. Acock 66

71 Two Class solution Parameter Estimate Class 1 N = 2651 Class 2 N=276 Mean α.379*** 1.395*** Mean β -.821*** -.395*** Count at time Count at time Count at time Count at time Display exp(alpha)*exp(beta*0,1,2,3) Alan C. Acock 67

72 Two Class solution Count Part Alan C. Acock 68

73 Interpretation This solution has the deviant group start with a higher initial count (α) and showing a significant improvement, decline in count, (β) The Normative group starts with a lower a lower count and drops less rapidly, possibly because there is a floor effect on the count. Alan C. Acock 69

74 Interpretation With a three class solution we got similar results with a normative group and a deviant group as the two main groups. However, the third class (only 85 kids) start with a low count and actually increase the count significantly over time. These 85 kids would be ideal for a qualitative sample because the program definitely fails Alan C. Acock 70

75 Adding Covariates We can add covariates These can be free across classes or constrained across classes The simple interpretation and graphs no longer work Alan C. Acock 71

76 Program for Freeing Constraint Model: %Overall% Alpha Beta ; Alpha_i Beta_i s1flbadc#1@0 s2flbadc#1@1 s3flbadc#1@2 s4flbadc#1@3 ; Alpha on male ; Beta on male ; S3flbadc on c3 ; S4flbadc on c4 ; %c#2% [s1flbadc#1 s2flbadc#1 s3flbadc#1 s4flbadc#1](1) ; [Beta_i] ; Alan C. Acock 72

77 Next Steps If you find distinct classes of participants who have different growth trajectories you can save the class of each participant. This is shown in the detailed document You can then compare the classes on whatever variable you think might be important in explaining the differentiation, e.g., parental support for program This will generate a new set of important covariates for subsequent research Alan C. Acock 73

78 Next Steps An introduction to growth curves and a detailed presentation of the ideas we ve discussed is available at Alan C. Acock 74

79 Summary Three Models Available from Mplus Traditional growth modeling where There is a common expectation for the trajectory for a sample Parameter estimates will have variances across individuals around the common expected trajectory (random effects) Covariates may explain some of this variance Alan C. Acock 75

80 Summary Three Models Latent Class Growth Models where We expect distinct classes that have different trajectories Class membership explains all of the variance in the parameters. Classes are homogeneous with respect to their growth curves (fixed effects) Alan C. Acock 76

81 Summary Three Models Mixture Models extending Latent Class Growth Models where We expect distinguishable classes that each have a different common trajectory Residual variance not explained by class membership are allowed (random effects) Covariates may explain some of this residual variance Alan C. Acock 77

82 Summary Mplus offers many features that are especially useful for longitudinal studies of individuals and families Many outcomes for family members are best studied using longitudinal data to identify growth trajectories Studies of growth trajectories can utilize time invariant, time variant, and distal outcomes Alan C. Acock 78

83 Summary Some outcomes for family members are successes or failures and the binary growth curves are useful for modeling these processes Some outcomes for family members are counts of how often some behavior or outcome occurs Alan C. Acock 79