1 Latent Class Regresson Statstcs for Psychosocal Research II: Structural Models December 4 and 6, 2006
2 Latent Class Regresson (LCR) What s t and when do we use t? Recall the standard latent class model from last term: Items measure dagnoses rather than underlyng scores Patterns of responses are thought to contan nformaton above and beyond aggregaton of responses The goal s clusterng ndvduals rather than response varables We add structural pece to model where covarates predct class membershp
3 Structural Equaton-type Depcton Measurement Pece x 1 y 1 x 2 η y 2 y 3 x 3 y 4 y 5 Structural Pece
4 When to use LCR Multple dscrete outcome varables bnary examples yes/no questons present/absent symptoms all measurng same latent construct We want to construct as outcome varable Responses to questons/tems measure underlyng states (.e. classes) wth error NOT approprate for counts or other way of groupng response patterns responses measure underlyng score wth error Note: Latent Varable s DISCRETE
5 Example: Depresson Is depresson contnuous or categorcal? Latent trat (IRT) assumes t s contnuous. Latent class model assumes t s dscrete Densty % class 1 80 class 2 15 class Depresson
6 Recall LC model M: number of latent classes K: number of symptoms p km : probablty of reportng symptom k gven latent class m π m : proporton of ndvduals n class m η : the true latent class of ndvdual, 1,,N m 1,,M; k 1,,K y 1, y 2,,y k : symptom presence/absence for ndvdual.
7 ECA wave 3 data (1993) N1126 n Baltmore Symptoms: weght/appette change sleep problems slow/ncreased movement loss of nterest/pleasure fatgue gult concentraton problems thoughts of death dysphora Covarates of nterest gender age martal status educaton ncome How are the above assocated wth depresson?
8 Assumptons Condtonal Independence: gven an ndvdual s depresson class, hs symptoms are ndependent P(y k, y j η ) P(y k η ) P(y j η ) Non-dfferental Measurement: gven an ndvdual s depresson class, covarates are not assocated wth symptoms P(y k x, η ) P(y k η )
9 Why LCR may be better than another analytc method LCR versus usng counts (e.g. number of symptoms) Pros: dstngushes meanngful patterns from trvally dfferent ones whch may be hard to dscern emprcally acknowledges measurement error precson and estmates of regresson coeffcents reflect measurement error Cons: may overdstngush prevalent patterns and mask dfferences n rare ones volaton of assumptons make nferences nvald
10 Why LCR may be better than another analytc method (contnued) Versus factor-type methods Pros: less severe assumptons (statstcally) easer to check assumptons Cons: lose statstcal power f construct s actually dmensonal (.e. contnuous) dentfablty harder to acheve (need bg sample) Practcally Pro: Allows for dsease/dsorder classfcaton whch s useful n a treatment vs. no treatment settng
11 Structural Equaton-type Depcton Measurement Pece x 1 x 2 x 3 y 1 β p What are y 2 the parameters that the arrows η y represent? 3 In other words, what are β and y 4 p n the LCR model? y 5 Structural Pece
12 Parameter Interpretaton Measurement Pece (p s) p km : probablty that an ndvdual from class m reports symptom k. η p kη y k Same as standard latent class model from last term
13 Parameter Interpretaton How do we relate η s and β s? In classc SEM, we have lnear model. What about when η s categorcal? What f η s bnary? x 1 x 2 x 3 β η
14 Parameter Interpretaton How do we relate η to x s? Consder smplest case: 2 classes or equvalently, β 1 and β 2 are log odds ratos x x P P ) ( 1 2) ( log β β β η η + + x x P P ) ( 2) ( log log β β β η η π π + +
15 p same as last term KxM p s π j P(η j) β Model Results Condtonal on x s No longer proporton of ndvduals n class Now, only can nterpret to mean probablty of class membershp gven covarates for ndvdual To get sze of class j, can sum of π j for all (M-1)*(H+1) β s where H number of covarates M-1: one class s reference class so all of ts β coeffcents are techncally zero H+1: for each class, there s one β for each covarate plus another for the ntercept.
16 Solvng for π j P(η j) π log π 2 1 log P( η P( η 2 x 1 x 1 1, x, x 2 2 ) ) β 0 + β x β x 2 2 π π P( η 2 x, x ) P( η 1 x, x ) e 1+ e 1+ β + β x + β x β + β x + β x β + β x + β x e
17 Parameter Interpretaton Example: e β1 2 and x 1 1 f female, 0 f male Women have twce the odds of beng n class 2 versus class 1 than men, holdng all else constant e β 1 P( η 2 x1 1, x2 c) P( η 1 x 1, x c) 1 2 P( η 2 x1 0, x2 c) P( η 1 x 1, x c) 1 2
18 More than two classes? Need more than one equaton Need to choose a reference class x x x x P x x P ), 1 ( ), 2 ( log β β β η η + + x x x x P x x P ), 1 ( ), 3 ( log β β β η η + + e e e e e β β β β β β OR for class 2 versus class 1 for females versus males OR for class 3 versus class 1 for females versus males OR for class 3 versus class 2 for females versus males /
19 Solvng for π j P(η ) π log π 2 1 log P( η 2) P( η 1) β 02 + β 12 x 1 + β 22 x 2 π 2 P( η 2) e 1+ e + β02 + β12x1+ β22x2 e 3 r 1 e β + β x + β β + β x + β x β + β x + β x β + β x + β x 0r 1r 1 2r 2 e x Where we assume that β01 β11 β21 0
20 Depresson Example: LCR coeffcents (log ORs) n 3 class model Class Class Class 3 vs 1 2 vs 1 3 vs 2 Log(age) -1.2* -1.5* 0.23 Female 0.85* 0.76* 0.09 Sngle Sep/wd/dv 0.86* 0.83* HS dploma * 0.51 * ndcates sgnfcant at the 0.10 level Note: class 1 s non-depressed, class 2 s mld, class 3 s severe
21 Depresson Example: ODDS RATIOS n 3 class model Class Class Class 3 vs 1 2 vs 1 3 vs 2 Log(age) 0.3* 0.22* 1.26 Female 2.34* 2.13* 1.09 Sngle Sep/wd/dv 2.36* 2.29* 0.99 HS dploma * 1.67 * ndcates sgnfcant at the 0.10 level Note: class 1 s non-depressed, class 2 s mld, class 3 s severe
22 Step 1: Model Buldng Get the measurement part rght! Ft standard latent class model frst. Use methods we dscussed last term to choose approprate model Step 2: add covarates one at a tme It s useful to perform smple regressons to see how each covarate s assocated wth latent varable before adjustng for others. Many of same ssues n lnear and logstc regresson (e.g. multcollnearty)
23 Estmaton Same caveats as last term Maxmum lkelhood: Iteratve fttng procedure. Packages Mplus Splus, R SAS Bayesan approach Computatonally ntensve WnBugs Splus, R SAS
24 Propertes of Estmates (β, p) If N s large, coeffcents are approxmately normal confdence ntervals and Z-tests are approprate. Nested models can be compared by usng chsquare test. But, recall problems of ch-square test when sample sze s large! And problems when the sample sze s small! Also can use AIC, BIC, etc. to compare nested AND non-nested models (e.g. s age as contnuous better than 3 age categores).
25 Specfcs Statstcally Standard LCM Lkelhood Latent Class Regresson Lkelhood ) ( , 4 4, 3 3, 2 2, 1 1 ) (1 ) ( ) ( k y k km M m K k y km m y Y y Y y Y y Y y Y y Y p p P P π ) ( , 4 4, 3 3, 2 2, 1 1 ) (1 ) ( ) ( ) ( k y k km M m K k y km m x y Y y Y y Y y Y y Y y Y p p x P P π M m x x m m m e e x 1 ) ( β β π where
26 Example: 3 class model coeffcent estmate se 95% confdence nterval b b b2age b3age b2sex b3sex p[1,1] p[1,2] p[1,3] p[2,1] p[2,2] p[2,3] etc..
27 Some Addtonal Concepts (1) η s a NOMINAL varable (2) Data Setup: Centerng covarates can help. Due to need to ntalze algorthm n ML. Due to prors on β s n Bayesan settng Wll be meanngful n model checkng, too. Need to choose startng values for model estmaton for regresson coeffcents n some ML packages. Ths s easer f they are centered. Not an ssue for Mplus: only need startng values for measurement part.
28 Choosng Values for Intalzaton A: Measurement model 1. Use results from standard latent class model B: Structural pece 1. choose all β s equal to 0 (wll work f there s a LOT of data and no ID problems) 2. a. Make a surrogate latent class (e.g. choose cutoffs based on number of symptoms) b. Perform mlogt on surrogate wth covarates c. Use log ORs as startng values
29 Choosng Values for Intalzaton 3. Use ML pseudo-class approach a. Usng pseudo-classes from standard LC model, treat class assgnment as fxed b. Regress class membershp on covarates (polytomous logstc regresson) c. Model buldng strategy -- gves ntal dea of whch covarates are assocated. d. Also, can use ths as a model checkng strategy post hoc 4. Use MCMC class assgnment approach: same as 3, but wth classes assgned usng MCMC model
30 Important Identfablty Issue Must run model more than once usng dfferent startng values to check dentfablty!
31 Model Checkng Very mportant step n LCR LCR can gve msleadng fndngs f measurement model assumptons are volated Two types of model checks: (1) model ft do y patterns behave as model would predct? (2) volaton of assumptons do y s relate to x s as expected?
32 ECA wave 3 data (1993) N1126 n Baltmore Symptoms: weght/appette change sleep problems slow/ncreased movement loss of nterest/pleasure fatgue gult concentraton problems thoughts of death dysphora Covarates of nterest gender age martal status educaton ncome How are the above assocated wth depresson?
33 Models Model A: log(age), gender, race Model B: log(age), gender, race, dploma
34 Do y patterns behave as model predcts? Compare observed pattern frequences to expected pattern frequences PFC plot How does addton of regresson change nterpretaton? Evaluatng ft of measurement pece Wll be same as n standard LC model unless..
35 o 2 class x 3 class 4 class
36 LCA x LCR-A o LCR-B
37 Does pattern frequency behave as predcted by covarates? Idea: focus on one tem at a tme Recall: M K yk (1 y 1 y1,..., YK yk x ) ( x ) pkm (1 pkm) m 1 k 1 P( Y π If nterested n tem r, gnore ( margnalze over ) other tems: k ) P( Y y x ) π ( x r r M m 1 ) p yk rm (1 p rm ) (1 y k )
38 Comparng Ftted to Observed
39 Categorcal Covarates Easer than contnuous (computatonally) Example Calculate: Predcted males wth gult Observed males wth gult Predcted females wth gult Observed females wth gult
40 Assume LC regresson model wth only gender Gender 0 f male, 1 f female Item of nterest f gult. Want fnd how many class 2 men we would expect to report gult based on the model P( gult and male and class 2) P( gult and class 2 male) P( male) P( gult male and class 2) P( class 2 male) P( male) P( gult class 2) P( class 2 male) P( male) p km β0 e 1+ e β 0 Pmale ( ) β0 e Expected ( gult and male and class 2) N pkm β Pmale ( ) 0 1+ e Calculate ths for each of the classes and sum up: Wll tell us the expected number of males reportng gult.
41 Falure n Ft Check Assumptons non-dfferental measurement condtonal ndependence Non-dfferental Measurement: P(y k x, η ) P(y k η ) In words, wthn a class, there s no assocaton between y s and x s. Check ths usng logstc regresson approach
42 Checkng Non-dfferental Measurement Assumpton For bnary covarates and for each class m and tem k consder P( yk 1 x 1, η m) / P( yk 0 x 1, η m) ORkmx P( y 1 x 0, η m) / P( y 0 x 0, η m) k If assumpton holds, ths OR wll be approxmately equal to 1. Why may ths get trcky? We don t KNOW class assgnments. Need a strategy for assgnng ndvduals to classes. k
43 Checkng NDM: Maxmum Lkelhood Approach (a) assgn ndvduals to pseudo-classes based on posteror probablty of class membershp recall posteror probablty based on observed pattern e.g. ndvdual wth 0.20, 0.05, 0.75 better chance of beng n class 3 not necessarly n class 3 (b) calculate OR s wthn classes. (c) repeat (a) and (b) at least a few tmes (d) compare OR s to 1.
44 Checkng NDM: Maxmum Lkelhood Approach What about contnuous covarates? Use same general dea, but estmate the logor wthn classes by logstc regresson Example: age
45 Checkng NDM: MCMC (Bayesan) approach At each teraton n Gbbs sampler, ndvduals are automatcally assgned to classes no need to manually assgn. At each teraton, smply calculate the OR s of nterest. Then, margnalze or average over all teratons. Results s posteror dstrbuton of OR
48 Checkng Condtonal Independence Assumpton In words, wthn a class, there s no assocaton between y k and y j, j k. Same approach Only dfference: OR jkm P( y P( y j j 1, y 1, y k k 1 η 0 η m) / m) / P( y P( y j j 0, 0, y y k k 1 η 0 η m) m) Stll use pseudo-class assgnment (ML) or class assgnment at each teraton (MCMC)
51 Identfablty (brefly) General Idea: dfferent parameters can lead to the same model ft 2 step rule: If (a) polytomous logstc regresson s ID ed (b) standard LCM s ID ed Then model s ID ed t-rule: need more data cells than parameters complcaton: contnuous covarates, but they usually don t make unid ed.
52 Utlty of Model Checkng May modfy nterpretaton to ncorporate lack of ft/volaton of assumpton May help elucdate a transformaton that that would be more approprate (e.g. log(age) versus age) May lead to beleve that LCR s not approprate.