Estimating Discrete-time Survival Models as. Structural Equation Models

Transcription

1 Estimating Discrete-time Survival Moels as Structural Equation Moels Shawn Baulry Kenneth A. Bollen August 13, 2008 Abstract Survival analysis is a common tool in the analysis of the timing of events. Survival moels have been applie to a vast number of areas of sociological research. In this paper we show how to embe survival moels in structural equation moels (SEMs) while taking avantage of aitional features possible in a SEM approach. Using empirical examples of time to promotion an timing of intercourse, we show that we can replicate the more traitional survival moel results, but more importantly, we show how to control for measurement error an permit meiating variables. By oing so we provie aitional options useful to sociologists who use survival moels. 1

2 1 Introuction Following their introuction into the iscipline in the late 1970s, survival moels have become a common tool in the analysis of the timing an occurence of events of sociological signi cance. Sociologists have rawn on survival moels, or event-history moels, to further our unerstaning of social processes, incluing the i usion of protests (e.g., Anrews an Biggs 2006), marriage formation (e.g., Sweeney 2002), the passage of laws (e.g., Behrens, Uggen, an Manza 2003), an organizational emography (e.g., Sørensen 2004). As the use of survival moels has sprea throughout sociology the moels have been extene to aress various epartures from the stanar moel. For instance, Allison (1984) covers how to hanle multiple types of events an repeate events, Guo (1993) emonstrates how to aress left-truncate ata, an Allison (2005) reviews xe-e ects survival moels. Until recently, however, it has not been known how to accomoate measurement error in the covariates an how to simultaneously moel systems of equations allowing for the e ects of meiators. Accommoating measurement error is important since ignoring such error will bias our estimates of the impact of covariates an will mislea researchers. Simultaneous equations permits a fuller unerstaning of a variable s irect, inirect, an total e ects. Survival analysis as currently practice reveals only the irect e ect which can i er from its inirect e ects. Recent work in the area of eucational statistics has emonstrate that survival moels can be treate as a speci c case in the more general class of structural equation moels (Asparouhov, Masyn, an Muthén 2006; Muthén 2

3 an Masyn 2005; Masyn 2003). In this article, we provie an overview of this work with a focus on iscrete-time survival moels for a sociological auience. These prior works o not consier multiequation moels an meiating variables. Given the avantages of oing so, we also inclue a iscussion of such moels in our presentation. Following a brief iscussion of iscrete-time survival moels, we illustrate how these moels t into a structural equation moeling framework. Next, using Allison s (1995) ata on the careers of biochemists, we emonstrate how to estimate several basic moels an in the process ocument that one obtains the same results as the more conventional approach to estimating these moels. We conclue with an extene empirical example in which we illustrate the avantages of this approach in accounting for measurement error an allowing for meiating an inirect e ects. 2 Discrete-Time Survival Moels The two primary types of survival moels are continuous-time an iscretetime. Continuous-time moels are typically employe when the exact timing of an event is known (e.g., the ay an year of eath). Discrete-time moels, in contrast, are generally use when only the interval in which an event occurs is known or the event itself occurs in iscrete intervals (e.g., the year in which responents marry). In practice, the istinction between whether an event is measure in continuous-time or iscrete-time is arbitrary (in a technical sense, all events of sociological interest can be consiere as measure in iscrete-time) 3

4 an the choice between continuous-time an iscrete-time moels epens on other factors, such as the existence of ties in the ata (i.e., iniviuals who experience the event of interest at the same time point), whether a speci c parametric function of time is appropriate, an computational consierations. As many sociological applications involve the use of iscrete-time moels, we focus on this class of moels in this work; however, it is also possible to estimate continuous-time survival moels in a structural equation moeling framework (see Asparouhov et al. 2006). Survival ata consists of at least two pieces of information for each case regaring an event: (1) the time until an event occurs or the case is no longer observe an (2) an inicator of whether the case experience the event. A case that oes not experience the event while uner observation is sai to be censore as the time to event is unknown. Allison (1995) escribes three types of censoring: Type I, Type II, an ranom (see also Singer an Willett (2003) for a helpful iscussion). Type I censoring occurs when the censoring time is xe by esign an all observations share the same censoring time (e.g., the en of a stuy). Type II censoring occurs when cases are no longer observe after a preetermine number of events. Ranom censoring refers to the situation when a case is no longer observe for any reason not uner the control of the investigator (e.g., sample attrition in a longituinal stuy). These cases can be problematic for the analysis. If the reason for censoring, conitional on the covariates in the moel, is relate to the event of interest, then the censoring is consiere informative an parameter estimates can be baly biase (Allison 4

5 1995). 1 For example, in an analysis of the uration of spells of unemployment, the people who are not able to be followe over time (i.e., censore) may be more likely to experience longer spells of unemployment. Unfortunately, it is not possible to test whether ranom censoring is informative or non-informative (but, see Allison (1995) for one approach to assessing the sensitivity of results to the potential presence of informative censoring). In the following evelopment of the moels we assume censoring is non-informative; if researchers have reason to believe otherwise, as usual, results shoul be interprete with caution. 2.1 Survival an Hazar Functions In the iscrete-time framework with non-repeate events there are two probabilities of primary interest: the hazar probability an the survival probability. The hazar probability refers to the probability of a case experiencing an event in a time interval given that the case has not experience the event in any previous time interval. Formally, let T be a iscrete ranom variable, i an inex for cases, an j an inex for time perios. Then T takes on values T i which inicate the time perio j when case i experiences an event. The hazar probability expresse as a function of time is given as the conitional probability ensity function h(t ij ) = Pr[T i = jjt i j]: (1) 1 As we will iscuss below, the concepts of informative an non-informative censoring closely mirror i erent types of missing ata. 5

6 The conitional nature of the hazar probability is important to keep in min. A case can only experience an event in time perio j if an only if the case has not alreay experience the event. Such cases that are eligible to experience an event are sai to be in the risk set. The survival probability is e ne as the probability of surviving, or not experiencing an event, beyon a given time perio j. We can express the survival probability as S(t ij ) = Pr[T i > j]: (2) As one woul expect, the hazar an survival probability are closely relate. In a iscrete-time setting the survival probability can be consiere the probability of not experiencing the event in any prior time perio. For any single time perio, the probability of not experiencing the event, given the case is in the risk set, is simply one minus the hazar probability. Taking the prouct, the survival probably can also be written as jy S(t ij ) = (1 h(t ik )) : (3) k=1 Using this formula, it is easy to calculate the estimate survival probability for any given time point using the estimates of the hazar probabilities (ue to censoring, however, it is not possible to calculate the hazar probabilities from the survival probabilities). 6

7 2.2 Maximum Likelihoo Estimation The metho of maximum likelihoo is the most common approach to obtain estimates of the hazar probabilities for the population. For iscrete-time survival moels, the likelihoo function expresses the probability of observing the pattern of the occurrences of the event in the ata. The pattern of events arises from arraying the ata in a case-perio format such that each case contributes as many observations as time perios they are in the risk set. In this format an inicator variable ( ij ) enotes for each time perio whether a case experiences an event. As note in (1), the hazar captures the probability that case i experiences an event in time perio j conitional on being in the risk set. Therefore if case i experiences the event in time perio j the case contributes h(t ij ) to the likelihoo function. Conversely, if case i oes not experience the event in time perio j, the the case contributes 1 h(t ij ) to the likelihoo function. Taking the prouct over all cases (inexe by N) an all time perios cases remain in the risk set (inexe by J i ), we can write the likelihoo function as NY YJ i L = h(tij ) ij (1 h(t ij )) 1 ij : (4) i=1 j=1 As we will emonstrate below, we erive an ientical likelihoo function estimating these moels as structural equation moels. In most research sociologists are intereste in estimating the relationship between a set of covariates an the hazar probabilities. One of the avantages of survival moels is that covariates are not force to be static, they may vary 7

8 over time. In the iscrete-time setting the logit link is the most wiely use an the resulting moel can easily incorporate both time invariant an time-varying covariates. Let X an X j respectively be matrices of time invariant an timevarying covariates, an vectors of coe cients, an a vector of intercepts for each time perio. Then we can write the logit of the hazar probability as h(tij ) log = + X + X j : (5) 1 h(t ij ) In this form of the moel, the e ects of both the time invariant an the timevarying variables are constraine to be the same for each time perio. This is often referre to as the proportional hazar os property. In many instances, it is not reasonable to assume that the e ects of the covariates will remain constant over time. For example, one might imagine that the e ect of selfesteem on the hazar of rst intercourse coul change as iniviuals grow oler. Fortunately, it is easy to relax the proportional hazar os property an test it; one simply as interactions between the covariates an the time perio inicators containe in the vector of intercepts. 2.3 Structural Equation Moels In this section, we provie a brief overview of structural equation moels (SEMs) before turning to how we can estimate iscrete-time survival moels in a structural equation moeling framework. SEMs are a general class of statistical moels consisting of multiequation systems that represent relationships between 8

9 latent an observe variables. Many moels commonly employe by sociologists can be estimate as SEMs. For instance, general linear moels, factor analysis, an simultaneous equations are all special cases of SEMs (Bollen 1989). Structural equation moels have two primary components: the latent variable moel an the measurement moel. The latent variable moel is i = + B i + i + i ; (6) where i is a vector of latent enogenous variables, is a vector of intercept terms, B is a matrix of coe cients containing the e ects of the latent enogenous variables on each other, i is a vector of latent exogenous variables, is a coe cient matrix containing the e ects of the latent exogenous variables on the latent enogenous variables, an i is a vector of isturbances. The i subscript inexes the ith case in the sample. In this moel we assume that E( i ) = 0; COV ( 0 i; i ) = 0, an that (I B) is invertible. Exogenous variables are etermine outsie the system of equations, while enogenous variables are in uence by other variables in the system. The measurement moel links the latent to the observe variables (inicators). The following two equations capture this relationship: y i = y + y i + " i (7) x i = x + x i + i ; where y i an x i are respective vectors of inicators of the latent variables in 9

10 respectively i an i, y an x are respective vectors of intercept terms, y an x are respective matrices of factor loaings, an " i an i are vectors of isturbances. We assume that E(" i ) = E( i ) = 0 an that " i ; i ; i ; an i are all uncorrelate. SEMs have been generalize to account for categorical epenent variables. Following Long (1997), a limite-epenent variable moel can be written as y i = X i + " i ; (8) where y i is an unobserve continuous variable relate to the observe categorical variable y i. The precise relationship epens on the nature of the categorical variable. For a ichotomous variable, the relationship can be written as 8 9 >< 1 if yi y i = > >= : (9) >: 0 if yi >; This equation represents a threshol moel. When the unerlying continuous variable excees some threshol, a 1 is observe, otherwise a 0 is observe. In orer for the moel to be ienti e, one must assume a istribution for the isturbance " i. In practice, the efault assumption for SEMs is that the isturbance follows a stanar normal istribution, in which case we have a probit regression moel. For our purposes, however, it will be convenient to assume that the isturbance follows a stanarize logistic istribution with a mean of 0 an a variance of 2 =3, in which case we have a logistic regression 10

11 moel. 3 Discrete-time Survival Moels as SEMs To incorporate iscrete-time survival moels into a SEM framework we nee a way to estimate the hazar probabilities for each time interval. Returning to our e nition of a hazar probability as the conitional probability of a case experiencing an event in a time interval given the case is in the risk set suggests one approach. Imagine for each time perio that each case in the risk set has an unerlying latent propensity to experience an event an if a threshol is exceee then the case experience the event. In this set-up, if we assume a stanarize logistic istribution for the isturbance, then the hazar probability for the jth time perio is relate to the threshol for the jth time perio j as h(j) = exp[ j ] : (10) We can then estimate the hazar probabilities for all of the time points simultaneously through a system of logistic moels. We illustrate this in Figure 1, where the circles represent the unerlying latent propensities to experience an event for each time perio, the rectangles represent observe event inicators for each time perio, an the jagge lines represent the nonlinear relationships between the latent propensities an the observe inicators. Figure 1 about here 11

12 Estimating the hazar probabilities using this approach requires a wie format for the ata rather than the long, or case-perio, format neee for stanar iscrete-time survival moels (see Figure 2). To prepare the ata, the analyst nees to create event inicators for every time perio in the ata. For each case, the event inicators can take three values: i not experience event in time perio j >< >= E ij = 1 experience event in time perio j : (11) >: : not in risk set in time perio j >; Once a case experiences an event or is censore, all of the subsequent event inicators shoul be set to missing. This ensures that only cases in the risk set are inclue in the estimation of the hazar probability for each time interval. Estimating the system of logistic moels with missing ata inclue relies on the assumption that the ata are missing at ranom (Little an Rubin 2002), which in this setting correspons to the assumption of noninformative censoring. With the ata in wie format, time invariant covariates simply occupy a column for each, but time-varying covariates nee to be prepare in a similar fashion to the event inicators (see Figure 2 for an illustration). Figure 2 about here 3.1 Maximum Likelihoo Estimation Given the organization of the ata, we take a i erent approach than use above in eriving the likelihoo function in the SEM framework. Because the 12

13 ata is in wie format an the cases not in the risk set for any time interval are treate as missing on the respective event inicators, we note that the conitional probability of experiencing an event in each time interval is given by h(t ij ) = Pr[E ij = 1] (12) an, conversely, the conitional probability of not experiencing an event is given by 1 h(t ij ) = Pr[E ij = 0]: (13) Letting i be an inicator for whether a case experiences an event while uner observation, we can write the iniviual likelihoo function as JY i 1 L i = Pr[E ij = 1] i j=1 Pr[E ij = 0]: (14) Taking the prouct over all of the cases gives us the likelihoo function for the sample, an with a little rearrangement an substiting in (12) an (13) we obtain the same likelihoo function that we erive above (4); L = = 2 3 NY JY i 1 4Pr[E ij = 1] i Pr[E ij = 0] 5 i=1 j=1 NY YJ i h(tij ) ij (1 h(t ij )) 1 ij : n=1 j=1 The equality follows ue to the fact that for the rst J i 1 terms ij = 0 an for the nal Jth term 1 ij = 0. 13

14 With an approach for estimating the hazar probabilities in place, we can exten our moel to inclue time invariant an time-varying covariates. In Figure 3, we illustrate a moel with a single time invariant covariate (x 1 ) an a single time-varying covariate (x 2j ). In this moel, we inclue a composite () that captures the time invariant in uences on the hazar probability. In contrast to an enogenous latent variable, this composite oes not have a isturbance associate with it, rather it is completely etermine by the time invariant covariate. The composite oes not vary over time, so the e ects of the timevarying covariate are represente irectly as paths from the covariate to the latent propensities for the respective time points. Note, however, that in this moel the e ects of the time-varying covariate are constraine to be equal across all of the time intervals. This correspons with the proportional hazar os property an, as note above, can be relaxe an teste. Figure 3 about here More generally, the moel illustrate in Figure 3 can be represente by the following system of equations: h(j) log 1 h(j) = j + + X j (15) = X: X an X j are respectively matrices of time invariant an time-varying covariates, an an are vectors of coe cients. As oppose to the vector of intercepts in (5), we have a vector of threshols ( j ). In aition, we see that 14

15 the log os of the hazar is a function of, which in turn is etermine by X. In orer to relax the proportional hazar os property, we can simply free the constraint that the parameter estimates are equal across the time points. For time-varying covariates, this is easily re ecte in our moel by aing a subscript j to. Since the composite oes not vary over time, in orer to relax this constraint for time invariant covariates they neee to be ae to the log hazar os function. The system allowing for all coe cient estimates to vary over time can be written with the single matrix equation, h(j) log 1 h(j) = j + X j + X j j : (16) 4 Careers of Biochemists To illustrate how to estimate a few basic iscrete-time survival moels in a structural equation moeling framework we raw on a ataset on the careers of biochemists provie as an example in Allison (1995). All of our structural equation moels are estimate using Mplus (Version 5.1) (Muthén an Muthén 2007) an our comparison moels are estimate using Stata (Version 10.0). We provie the Mplus coe for all of our moels in the Appenix. The ata for our examples consists of the years to promotion for 301 assistant professors of biochemistry (see Table 1 for escriptive statistics). The biochemists careers were followe for up to 10 years after they were hire. Of the 301 professors, 72 percent were promote uring the 10 year perio of ob- 15

16 servation. With this ata we are able to examine the relationship between three time-invariant covariates an one time-vary covariate an the hazar of promotion. For time-invariant covariates, we have a measure of the selectivity of the unergrauate institution the iniviuals attene, whether the iniviual earne his or her Ph.D. from a meical school, an the prestige of the Ph.D. granting institution. For our time-varying covariate, we have a cumulative count of the number of article publishe by each iniviual for each year. Table 1 about here We rst estimate a baseline moel, illustrate in Figure 4 panel A, that just inclues the hazar rate for each time perio (see Table 2). Because this moel oes not inclue any covariates, the estimates for the hazar rates shoul be equal to the proportion of the sample promote for that time perio. This is, in fact, the case, which can be seen using the following formula relating the threshol estimates to the hazar rate an comparing the results with the escriptive statistics in Table 1: h(j) = exp( j ) : (17) For example, the estimate hazar probability for promotion in year 5 base 1 on the moel is 1+exp(1:09) = 0:25, which matches the proportion of the sample promote in year 5 (see Table 1). We also note that the parameter estimates obtaine from estimating the survival moel as a SEM are within a hunreth of a ecimal point of those obtaine from estimating the moel using the stanar 16

17 logistic regression approach. The tau estimates are relate to the intercept an parameter estimates for the time perio inicators by j = + j, (18) so, for example, we see that the negative of the threshol estimate for year 3, 2:78, equals the intercept plus the estimate of the coe cient for the year 3 inicator variable, 5:70 + 2:92 = 2:78. Finally, we also note that we obtain the same log likelihoo with both approaches. Figure 4 about here Table 2 about here Aing covariates to the moel (see Figure 4 panel B), we n that both unergrauate selectivity an the cumulative number of articles publishe are signi cantly positively associate with the hazar of promotion. As with our baseline moel, we n that there are essentially no i erences in the parameter estimates an the log-likelihoos between the two approaches. We also note that the stanar errors for the estimates of the covariates are ientical. With this example we have little theoretical reason to expect the e ects of the covariates may vary over time; however, it is generally a goo iea to test whether the proportional hazar os property shoul be relaxe. We illustrate how to o this by relaxing the constraint of equal parameter estimates for the e ect of unergrauate selectivity (see Figure 4 panel C). We see some inication that the e ect of unergrauate selectivity on the hazar of promotion varies 17

18 over time (see Table 3), but it oes not seem to follow a meaningful pattern an for most years the e ect is not statistically signi cant. Furthermore, a likelihoo ratio test (see Table 4) inicates that the moel relaxing the proportional hazar os property oes t the ata signi cantly better than a moel maintaining the property. We also note that once again the parameter estimates we obtain using a SEM framework match those using a stanar estimation proceure. 5 Extene Empirical Example The last section illustrate that SEMs can incorporate the traitional iscretetime survival moel an replicate results obtaine with more traitional approaches. For our nal empirical example we consier a moel that illustrates some of the avantages of estimating iscrete-time survival moels as structural equation moels. Research on the transition to rst intercourse among aolescents has foun religiosity an attitues regaring sex are important preictors of initiation (Meier 2003). In theoretical moels of this process, religiosity is posite to have both a irect e ect on the probability of having sex an an inirect e ect through its relationship to attitues about sex. There are two avantages to estimating such a moel as a structural equation moel. First, religiosity an attitues about sex are concepts for which we have several inicators. Treating these concepts as latent variables allows us to account for measurement error an to get a sense of how well the inicators capture the theoretical concepts. Ignoring the measurement error woul create biase coef- 18

19 cient estimates. Secon, given that our theoretical moel inclues both irect an inirect e ects, SEMs provie a convenient way of exploring these relationships. 5.1 Time to First Intercourse Data For our analysis we use ata from the Waves I an III of the National Longituinal Stuy of Aolescent Health (A Health). A Health began as nationally representative sample of aolescents in graes 7 through 12 in 1994/95 (Harris et al. 2003). Responents were followe for two aitional waves of in-home interviews, with Wave III collecte in We etermine the age of rst intercourse base on responents self-reports at Wave III. There is some concern about the accuracy of retrospective reports of age of rst sex; however, Upchurch et al. (2002) foun in an analysis of A Health ata through Wave 2, inconsistencies in reporting appeare to be ranom. In orer to ensure that our measures of religiosity an attitues regaring sex are temporally prior to initiation of sexual activity, we rop any cases in which the responent reporte having sex for the rst time prior to the Wave I interview. 2 Finally, we rop a little less than 10 percent of the cases missing ata on mother s eucation. These leaves us with an analysis sample of 9,914 responents. Table 5 about here 2 Dropping these cases raises the possiblity of selection bias, which we o not aress in this example (see Meier (2003) an Bearman an Brückner (2001) for approaches to accounting for selection bias or testing the robustness of the results). 19

20 Of 9,914 responents, 18 percent i not initiate sexual activity between Wave 1 an Wave 3 of the stuy an so are treate as censore observations (see Table 5). The age of rst intercourse in our sample ranges from 12 to 25. We only observe iniviuals for roughly seven years (the time between Wave 1 an Wave 3), so in creating our event inicators we have missing ata at both ens of the age range. This shoul not be confuse with left-censoring, but rather simply re ects the fact that responents enter the ata at Wave 1 at i erent ages. Following Meier (2003), we consier four Likert-scale measures of religiosity: (1) In the past 12 months, how often i you atten religious services?, (2) In the past 12 months, how often i you atten such youth activities?, (3) How often o you pray?, an (4) How important is religion to you?. For the purpose of comparison, we construct a religiosity scale as the average of the non-missing items ( = 0.85). Also following Meier (2003), we consier six attitues about sex. 3 These items were only aske of responents at least 15 years ol an not marrie. Three of the attitues express negative sentiments: If you ha sexual intercourse, (1) your partner woul lose respect for you, (2) afterwar, you woul feel guilty, an (3) it woul upset [name of mother]. Three others express positive sentiments: if you ha sexual intercourse, (1) your friens woul respect you more, (2) it woul make you more attractive to women/men, an (3) you woul feel less lonely. Base on these items, we 3 Meier also uses a seventh item: if you ha sexual intercourse, it woul give you a great eal of physical pleasure. This item stoo out in our measurement moels an so we chose not to inclue it. [Probably nee a better/more precise justi cation for excluing this item.] 20

21 create two scales: (1) negative sentiments about sex ( = 0.67) an (2) positive sentiments about sex ( = 0.70). Research on the transition to intercourse has foun that the probability of initiation i ers by gener, race/ethnicity, an mother s eucation (Meier 2003, Bearman an Brückner (2001), Day (1992)). We inclue these backgroun characteristics to account for the known i erences. As our main purpose in this example is to explore the relationship between religiosity, attitues regaring sex, an the initiation of sexual activity, this is not intene to be an exhaustive list of backgroun characteristics that may be relate to the transition to rst intercourse. In our rst set of moels we only use cases with complete ata on the covariates. This involves ropping roughly 4,000 cases missing ata on the measures of attitues regaring sex ue to the legitimate skip pattern note above. Given the nature of the skip pattern an that we inclue the age of the responent at Wave 1 as a covariate, one can make the case that these ata are missing at ranom (MAR). Therefore, as a comparison moel, we take avantage of the same metho of hanling missing ata that we are using for the hazar component of the moel to estimate the parameters using the full analysis sample. 5.2 Analysis of Time to First Intercourse Measurement Moels Our rst task is to evelop the measurement moels for our latent covariates religiosity an attitues regaring sex. For religiosity we consier two mea- 21

22 surement moels. Our rst moel involves a single latent imension with four inicators, our secon is a moel with two latent imensions each with two inicators (see Figure 5, Panels A an B). A moel with a single latent imension allowing for no measurement error an constraining the loaings to equal one is equivalent to the stanar practice of creating a single scale from the four measures. We n that the moel with a single latent imension has a poor t with the ata accoring to a number of measures (see Table 6). In particular, the moel 2 is signi cant, the RMSEA is well over 0.05, an the BIC is large an positive. As a justi cation for our secon moel, we felt that the concept of religiosity may consist of two imensions: a imension relate to involvement in the religious community an a more personal imension. For each of these imensions we have two inicators. To ientify the moel, we scale the latent variable associate with involvement to our measure of attening services an we scale the latent variable associate with personal beliefs to the measure of the importance of religion. This moel ts the ata quite well. We n a non-sign cant moel 2, an RMSEA of 0.01, an a negative BIC. Figures 5 here Turning to the parameter estimates, we n both of the free factor loaings are signi cant in the expecte irection of e ect. We also n that three out of four of the measures have reliabilities between 0.7 an 0.85, but one, youth group attenance, has a more moest 0.42 reliability. Finally, we note that the estimate correlation between religious involvement an personal beliefs is As we woul expect, this correlation is high, but these o appear to be 22

23 two istinct imensions. Given the strong overall t an the sensible parameter estimates, we aopt this moel with two latent imensions of religiosity in the following analyses. Table 6 about here We consier a similar set of moels for attitues regaring sex (see Figure 6, Panels A, B, an C). We n that a one imensional moel oes not t the ata. We n that a two imensional moel incluing latent variables for positive an negative attitues relate to sex has a substantially better t with the ata, but there are still some inications that the moel oes not aequately t the ata (see Table 6). The 2 remains signi cant, the RMSEA is greater than 0.05, an the BIC is positive. We examine a thir moel that introuces a correlation among the error terms for the two measures involving respect (one s partner an one s friens). We n that this moel has a signi cantly better t than the previous moel, but the t is still not ieal for the same reasons as the secon moel. Figure 6 about here Looking at the parameter estimates, we n that the free factor loaings are all signi cant an in the expecte irection. 4 In terms of the reliabilities of the items, we n more variation than we i for the religiosity items. Three of the items have quite low reliabilities aroun 0.25: partner woul lose respect, it woul upset your mother, an friens woul respect you more. Two items 4 For ease of interpretation, we coe all of the attitue items such that high values woul be theoretically associate with an increase likelihoo of initiating sexual activity. 23

24 have reliabilities aroun 0.55: it woul make you more attractive an you woul feel less lonely. Finally, one item, you woul feel guilty, has a high reliability of The two imensions of sexual attitues o appear to capture fairly istinct imensions as they have a relatively low correlation of We also n that a signi cant negative association between the error terms for the two items measuring respect. This appears to be ue to the fact that the original woring of the items ha opposite valences, which were renere the same by reverse coing the secon item. Although the overall t an the reliabilities of some of the items inicate that we o not have as strong a moel for attitues regaring sex as we o for religiosity, we o not see any other theoreticallymotivate improvements to the moel. So, in the following analyses, we use our thir moel Structural Moels As a basis for comparison, we rst estimate a moel in which we use only observe variables that is, we use scales for religiosity, positive, an negative attitues regaring sex (see Figure 7, Panel A). This correspons to the typical sociological application where measurement error in covariates is ignore. In our secon moel, we replace the scales with the measurement moels we evelope for the two imensions of religiosity an positive an negative attitues relate to sex (Figure 7, Panel B). Finally, in our thir moel, we buil in the theoretical relationships among religiosity an attitues relate to sex (Figure 7, Panel C). Figure 7 about here 24

25 In our rst moel, consistent with past research we n that religiosity is signi cantly negatively associate with the hazar of rst intercourse (see Table 7). We also n that positive an negative attitues regaring sex are signi - cantly relate to the hazar of rst intercourse in the expecte irections (we reverse coe the negative attitues such that the expecte e ect is positive). When we account for measurement error in religiosity an attitues relate to sex a i erent pattern emerges. Although both imensions of sexual attitues remain signi cantly associate with the hazar of rst intercourse, neither imension of religiosity remains signi cant. This suggests that failing to account for measurement error may lea one to erroneously conclue that religiosity has a irect e ect on the initiation of sexual activity among aolescents. This is not to say that religiosity has no e ect. In moel three we examine the e ects of religiosity on attitues regaring sex. We n that both imensions of religiosity have a signi cant association with negative sexual attitues an the imension of religiosity relate to one s beliefs has a signi cant association with positive attitues concerning sex. In aition, both imensions of attitues regaring sex remain signi cant preictors of the hazar of rst intercourse. Therefore, as one woul anticipate from our theoretical moel, religiosity has an inirect e ect on the hazar of rst intercourse through it s e ect on attitues relate to sex. Table 7 about here In our nal moel, we reestimate our thir moel using all of the available cases. This increases our sample size by roughly 4,000 cases, almost entirely ue 25

26 to the missing ata for the sexual attitues items. This approach also allows us to examine cases initiate sexual activity as early as age 12. Although the precise parameter estimates i er, we obtain substantively the same pattern of results as in our complete case analysis. Table 8 about here 5.3 Summary We chose this example to emonstrate some of the bene ts of estimating iscretetime survival moels as structural equation moels. In this case, we see that accounting for measurement error in one of the key concepts is consequential for the substantive interpretation of the results. Without aressing measurement error in the religiosity scale, one might incorrectly conclue that religiosity has a irect e ect on the age of rst intercourse. 6 Conclusion In this paper we have emonstrate how researchers can estimate iscrete-time survival moels in a structural equation moeling framework. In our rst set of analyses, we ocumente that the parameter estimates from SEM iscrete-time survival moels match those obtaine from the stanar approach to estimating iscrete-time survival moels. In our secon set of analyses, we illustrate some of the potential bene ts estimating iscrete-time survival moels in a SEM framework. In particular, we provie an example in which accounting 26

27 for measurement error in the covariates ha an appreciable e ect on the results. Furthermore, we iscusse another bene t of the SEM approach, the ability to explore meiational relationships an inirect e ects. Given the potential presence of measurement error an inirect relationships in many areas of sociological an emographic research as well as the utility of survival analysis, the ability to estimate these moels as SEMs may prove wiely bene cial. 7 References Allison, Paul D Event History Analysis: Regression for Longituinal Event Data. Newbury Park, CA: Sage Publications Inc. Allison, Paul D Survival Analysis Using the SAS System: A Practical Guie. Cary, NC: SAS Institute Inc. Allison, Paul D Fixe E ects Regression Methos for Longituinal Data Using SAS. Cary, NC: SAS Institute Inc. Anrews, Kenneth T. an Michael Biggs The Dynamics of Protest Di usion: Movement Organizations, Social Networks, an News Meia in the 1960 Sit-Ins. American Sociological Review 71: Asparouhov, Tihomir, Katherine Masyn, an Bengt Muthén Continuous Time Survival in Latent Variable Moels. Proceeings of the Joint Statistical Meeting in Seattle, August Bearman, Peter an Hannah Brückner Promising the Future: Virginity Pleges an First Intercourse. American Journal of Sociology 106:

28 912. Behrens, Angela, Christopher Uggen, an Je Manza Ballot Manipulation an the Menace of Negro Domination : Racial Threat an Felon Disenfranchisement in the Unite States, American Journal of Sociology 109: Bollen, Kenneth A Structural Equations with Latent Variables. New York: Wiley. Day, Ranall The Transition to First Intercourse among Racially an Culturally Diverse Youth. Journal of Marriage an the Family 54: Guo, Guang Event-History Analysis for Left-Truncate Data. Sociological Methoology 23: Harris, Kathy M., F. Florey, J. Tabor, Peter Bearman, J. Jones, an J. R. Ury The National Longituinal Stuy of Aolescent Health: Research Design. Available online at Little, Roerick J. A. an Donal B. Rubin Statistical Analysis with Missing Data. (2n e.). New York: Wiley. Long, J. Scott Regression Moels for Categorical an Limite Depenent Variables. Thousan Oaks, CA: Sage. Masyn, Katherine Discrete-time Survival Mixture Analysis for Single an Recurrent Events Using Latent Variables. Unpublishe octoral issertation, University of California, Los Angeles. Meier, Ann M Aolescents Transition to First Intercourse, Religiosity, an Attitues about Sex. Social Forces 81:

29 Muthén, Bengt an Katherine Masyn Discrete-Time Survival Mixture Analysis. Journal of Eucational an Behavioral Statistics 30: Muthén, Bengt an Lina Muthén Mplus (Version 5.1) [computer software]. Los Angeles: Muthén an Muthén. Singer, Juith D. an John B. Willett Applie Longituinal Data Analysis: Moeling Change an Event Occurrence. New York: Oxfor University Press. Sørensen, Jesper B The Organizational Demography of Racial Employment Segregation. American Journal of Sociology 110: Sweeney, Megan M Two Decaes of Family Change: The Shifting Economic Founations of Marriage. American Sociological Review 67: Upchurch, Dawn M., Lee A. Lillar, Carol S. Aneshensel, an Nicole Fang Lee Inconsistencies in Reporting the Occurrence an Timing of First Intercourse among Aolescents. The Journal of Sex Research 39:

30 8 Tables an Figures Table 1: Descriptive Statistics N Mean SD N Mean SD Ever Promote Promote y Tot Articles y Promote y Tot Articles y Promote y Tot Articles y Promote y Tot Articles y Promote y Tot Articles y Promote y Tot Articles y Promote y Tot Articles y Promote y Tot Articles y Promote y Tot Articles y Promote y Tot Articles y Unergra Select PhD Me School PhD Prestige

31 Table 2: Comparison of Parameter Estimates Baseline Moel Moel with Covariates Mplus Stata Mplus Stata Est SE Est SE Est SE Est SE 1 / int / y / y / y / y / y / y / y / y / y Un Select PhD Me PhD Prest Cumul. Art N LL

32 LL Table 3: Comparison of Moels Relaxing Proportional Hazar Mplus Stata Est SE Est SE 1 / int / y / y / y / y / y / y / y / y / y PhD Me PhD Prest Cumul. Art Un. Sel. y1 / Un. Sel Un. Sel. y2 / Un. Sel. x y Un. Sel. y3 / Un. Sel. x y Un. Sel. y4 / Un. Sel. x y Un. Sel. y5 / Un. Sel. x y Un. Sel. y6 / Un. Sel. x y Un. Sel. y7 / Un. Sel. x y Un. Sel. y8 / Un. Sel. x y Un. Sel. y9 / Un. Sel. x y Un. Sel. y10 / Un. Sel. x y N

33 Table 4: Likelihoo Ratio Tests Moel relaxing proportional hazar constraint Moel with proportional hazar constraint Log Likelihoo -2*(Moel 2 LL - Moel 1 LL) f 9 Chi-square test

34 Table 5a: Descriptive Statistics N Mean SD Ever ha sex Age 12 rst sex Age 13 rst sex Age 14 rst sex Age 15 rst sex Age 16 rst sex Age 17 rst sex Age 18 rst sex Age 19 rst sex Age 20 rst sex Age 21 rst sex Age 22 rst sex Age 23 rst sex Age 24 rst sex Age 25 rst sex

35 Table 5b: Descriptive Statistics N Mean SD Age at wave Female Black Hispanic Asian Other race Mother s eucation How often atten services How often atten yth grp How often pray How important religion Religiosity scale If you ha sex: Partner woul lose respect You woul feel guilty Upset your mother Friens respect you more Make you feel more attractive Feel less lonely Neg Sex Attitues scale Pos Sex Attitues scale

36 Table 6: Moel Fit Statistics for Measurement Moels N Chi-Sq f sig CFI TLI RMSEA BIC Religiosity: 1 latent var latent vars Attitues: 1 latent var latent vars lat vars, cor err

37 Table 7: Parameter Estimates Using Complete Case Data (N=5980) Moel 1 Moel 2 Moel 3 Neg Sex Pos Sex Age First Sex Est SE Est SE Est SE Est SE Est SE Rel Scale Lat Atten Lat Belief Neg Sex Scale Lat Neg Sex Pos Sex Scale Lat Pos Sex Notes SEs are robust stanar errors. All moels control for gener, race/ethnicity, an mother s eucation. 37

38 Table 8: Parameter Estimates Using Full Sample (N=9914) Moel 3 Neg. Sex Att. Pos. Sex Att. Age First Sex Est SE Est SE Est SE Latent attenance Latent beliefs Latent negative sex att Latent positive sex att Notes: SEs are robust stanar errors. All moels control for gener, race/ethnicity, an mother s eucation. Figure 1: Discrete Time Survival Moel Represente as a Structural Equation Moel event t1* event t2* event tj* τ 1 τ 2 τ j event t1 event t2 event tj 38

39 Figure 3: Discrete Time Survival Moel with Covariates X 1 β 1 η X 2j κ 2 κ2 1 1 event t1* event tj* τ 1 τ j event t1 event tj 39

40 Figure 4: Moels for Careers of Biochemists Panel A: Baseline Moel η 1 1 year1* year10* τ 1 τ 10 year1 year10 Figure 4: Moels for Careers of Biochemists Panel B: Moel with Covariates me school Ph.D. prestige unergra selectivity articles y1 β 1 β 2 β 3 κ 1 η 1 1 year1* year10* τ 1 τ 10 articles y10 κ 1 year1 year10 40

41 Figure 4: Moels for Careers of Biochemists Panel C: Moel Relaxing Proportional Hazar Os Constraint for Unergrauate Selectivity me school Ph.D. prestige β 2 β 1 η 1 1 β 3,yr10 unergra selectivity β 3,yr1 year1* year10* κ 1 articles y1 τ 1 τ 10 articles y10 κ 1 year1 year10 Figure 5: Religiosity Moels Panel A Panel B Religiosity Atten Beliefs services ythgrp pray import services ythgrp pray import

42 Figure 6: Sex Attitues Moels Panel A Panel B Sex Att Pos Sex Att Neg Sex Att Q1 Q2 Q3 Q4 Q5 Q6 Q1 Q2 Q3 Q4 Q5 Q Figure 6: Sex Attitues Moels Panel C Pos Sex Att Neg Sex Att Q1 Q2 Q3 Q4 Q5 Q

43 Figure 7: Moels for Time to First Sex Panel A: Moel with Covariates (No Latent Variables) X Religiosity Pos Sex Att η 1 1 Neg Sex Att Age 15 Age 25 Age 15 Age 25 Figure 7: Moels for Time to First Sex Panel B: Moel with Covariates (Latent Variables) X Religiosity Sex Att η 1 1 Age 15 Age 25 Age 15 Age 25 43

44 Figure 7: Moels for Time to First Sex Panel C: Moel with Covariates (Latent Variables) X Religiosity Sex Att η 1 1 Age 15 Age 25 Age 15 Age 25 44