Alfred DeMaris - Regression with Social Data, Modeling Continuous and Limited Response Variables

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pregnant, educational level attained, and so on. To eﬀectively model the inﬂuence of

these variables on the risk of event occurrence, we need to keep track of these

changes. On the other hand, causal inferences for such variables must be tendered

with caution, as these covariates are often subject to inﬂuence by the hazard process

itself. Yamaguchi (1991) presents a lucid discussion of the varieties of time-varying

covariates and their role in causal modeling. At any rate, the Cox model that incor-

porates time-varying covariates is

(t) ⫽ h

(t) exp(x

⬘

ββ

where the subscript t on the covariate vector indicates that one or more of the covari-

ates may change in value over time. To simplify the notation in later models, how-

ever, I omit this subscript, with the understanding that the covariate vector may

always include one or more time-varying covariates.

Time-varying covariates are readily speciﬁed in Cox regression software such as

STATA or SAS. This is done in PHREG, for example, via programming statements

after the initial speciﬁcation of the model. As Allison (1995) admonishes, however,

there is no way to check whether these covariate values are being implemented cor-

rectly. One advantage of the discrete-time approach discussed in Chapter 12 is that

the data can be scrutinized to ensure that time-varying covariates are coded correctly.

In the union disruption data there are two time-varying covariates of interest. One

pertains to the eventual marital status of couples who were cohabiting at time 1. Those

cohabiting unmarried in wave 1 who subsequently marry should experience greater

marital stability than those who remain unmarried. Marriage implies a greater com-

mitment to the permanence of the relationship and is more legally binding on the part-

ners. Therefore, the act of marrying, in and of itself, should reduce the risk of a breakup.

Of course, it is also likely that those with higher-quality relationships to begin with are

both more inclined to marry and less likely to break up. In this case the transition to

marriage may be aﬀected by the same process that aﬀects the hazard of disruption and

may not represent a distinct causal inﬂuence. In any case, it is desirable to distinguish

cohabitors who remained unmarried throughout the follow-up period from those who

married. Continuous cohabitation can be modeled via a time-invariant dummy. For

cohabitors who married, we can construct a time-varying dummy variable that takes the

value 0 until the month in which marriage occurs, at which point it changes to 1.

The other time-varying element of importance is the advent of a birth to the union.

Several couples experienced the birth of one or more children to the union during the

follow-up period. Having a child, or an additional child, should also mitigate against

union disruption. Couples are motivated to preserve their unions when children are

present due to the desirability of two-parent households for children’s development.

However, as in the case of the transition from cohabitation to marriage, fertility deci-

sions may also be inﬂuenced by the hazard of disruption. Couples with troubled rela-

tionships may postpone childbearing in the anticipation that the union might end.

Again, any causal role played by the event of a birth during the follow-up period

should be tentatively entertained. Nevertheless, experiencing a birth can be coded as

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a time-varying covariate that takes the value 0 until the date of the ﬁrst birth to the

union, at which point it changes to 1. In creating this variable, however, we need to

ensure that the variable is coded 1 only if the birth occurs prior to a disruption.

Models 4 and 5 in Table 11.2 add the two dummies for cohabiting status plus the

dummy for a union birth. In model 4 these dummies are treated as time invariant. In

other words, two dummies are created to represent the two types of cohabitation, and

a dummy is created to represent the advent of a birth to the union prior to censoring or

disruption. These dummies are treated as ﬁxed over the follow-up period. In model 5,

on the other hand, cohabiting to married and union birth are coded as time-varying

covariates, as described above. The diﬀerential treatment of these variables across

models is especially important for the cohabiting-status dummies. The method for cod-

ing the time-varying version of cohabiting to married necessarily lumps together

cohabitors who later marry with married couples, until these cohabitors make the tran-

sition to marriage, at which point they form a distinct group. Thus, the risk of disrup-

tion for cohabitors who remain unmarried may appear lower than it actually is, due to

the mixing of marrieds and cohabitors in the contrast category over time. Entering

cohabiting status as time-invariant in model 4 therefore provides a check on the robust-

ness of the cohabitation eﬀects.

The results suggest that the substantive conclusions would be unchanged regard-

less of speciﬁcation. According to either model, continuous cohabitation elevates the

risk of disruption, compared to being married. Cohabitors who transition to mar-

riage, on the other hand, experience no signiﬁcant inﬂation in the risk for disruption

compared to marrieds. Having a birth lowers the hazard of disruption compared to

having no birth. In particular, model 5 suggests that continuous cohabitation elevates

the risk for disruption by a factor of exp(1.54) ⫽ 4.665, or 367%, compared to being

married. Model 5 also suggests that there is about a 43% reduction in the hazard of

disruption upon the advent of a union birth.

Handling Nonproportional Effects

As indicated, the model in equation (11.11) assumes proportionality in the hazards.

When is this assumption violated? In truth, the model is already nonproportional as

soon as we include time-varying covariates (Allison, 1995; Collett, 1994). The reason

is that proportionality of hazards requires the eﬀect of each covariate to be constant

over time and also requires the number of units apart of any two people to be constant

over time. The latter is violated with time-varying covariates, since these will change

at diﬀerent rates for diﬀerent individuals, thereby changing the disparities in their

covariate values. Regardless, the primary violation of the proportional hazards assump-

tion occurs when eﬀects of covariates depend on time. Nonproportional eﬀects are

readily tested and modeled via cross-product terms in the Cox model. We simply add

the cross-product of the covariate in question, which can either be time-invariant or

time-varying, with survival time. Hence, if the model for the log-hazard is nonpropor-

tional in X

, it takes the form

ln[h

(t)] ⫽ λ

(t) ⫹ β

⫹

...

⫹ β

⫹ γX

T ⫹

...

⫹ β

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The eﬀect of X

on the log-hazard is then β

⫹ γT, which clearly varies over time. At

T ⫽ 0, or the inception of risk, the eﬀect is β

. At T ⫽ t, however, it is β

⫹ γt, and so

on. A general test for nonproportionality of covariate eﬀects would be conducted by

adding the block of interactions of all model covariates with time to the proportional

hazards model, and testing for a signiﬁcant improvement in ﬁt via the ∆χ

statistic.

Table 11.3 presents the results of estimating nonproportional eﬀects for two types

of covariates in the union-disruption model. The ﬁrst is alcohol or drug problem. A

reasonable hypothesis is that the deleterious impact on relationship stability of sub-

stance abuse problems recorded at a given point in the relationship dampens over

time. This can be due to the fact that couples become inured to such addictions on

the part of their partner. Or, this could be the result of attempts on the part of the sub-

stance abuser to seek treatment for the problem. The second factor investigated is

cohabiting status. As relationships endure over longer periods, the formalization of

unions via marriage may become less important to their stability, implying a declin-

ing eﬀect over time for the cohabitation dummies.

Model 1 examines the substance abuse eﬀect by adding an interaction term of the

form alcohol or drug problem * time. As expected, the eﬀect is signiﬁcant and nega-

tive. The impact of alcohol or drug problem on the log-hazard of union disruption is

seen to be .972 ⫺ .014 time. The value of .972 suggests that the hazard ratio for those

with vs. without substance abuse problems is exp(.972) ⫽ 2.643 at union inception.

After ﬁve years, however, the eﬀect on the log-hazard drops to .972 ⫺ .014(60) ⫽ .132,

REGRESSION MODELS IN SURVIVAL ANALYSIS 407

Table 11.3 Cox Models for Union Disruption Illustrating the Handling

of Nonproportional Effects of Covariates

Predictor Model 1 Model 2 Model 3

Model 4

Relationship duration (months) ⫺.022*** ⫺.024*** ⫺.029*** ⫺.025***

Female’s age at union ⫺.055*** ⫺.055*** ⫺.062*** ⫺.056***

Both in a ﬁrst union ⫺.618*** ⫺.611*** ⫺.728*** ⫺.639***

Alcohol or drug problem .972** .965** .348* .377*

Continuously cohabiting 1.540*** 2.230***

Cohabiting to married

.319 .296

Union birth

⫺.568*** ⫺.603*** ⫺.719*** ⫺.606***

Open disagreement .031* .031* .030* .031*

Conﬂict resolution style ⫺.188*** ⫺.184*** ⫺.236*** ⫺.185***

Alcohol or drug problem ⫻ time ⫺.014* ⫺.014

Continuously cohabiting ⫻ time ⫺.017**

Cohabiting to married ⫻ time ⫺.0003

Likelihood ratio χ

319.722*** 329.853*** 178.307*** 127.310***

Degrees of freedom 10 12 7 7

Note: n ⫽ 1230.

Unstratiﬁed model.

Model is stratiﬁed by marital status.

Time-varying covariates.

p ⫽ .051.

* p ⬍ .05. ** p ⬍ .01. *** p ⬍ .001.

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implying a hazard ratio of only 1.141. Model 2 adds the interaction of continuously

cohabiting * time and cohabiting to married * time to model 1. Testing the additional

terms as a block results in a ∆χ

⫽ 329.853 ⫺ 319.722 ⫽ 10.131, which, with 2 df,is

signiﬁcant at p ⫽ .0063. As predicted, both interaction terms are negative, although

only the ﬁrst is signiﬁcant. Hence, the eﬀect on the log-hazard of disruption for con-

tinuously cohabiting, compared to being married, is 2.23 at inception of the union,

suggesting a hazard ratio of exp(2.23) ⫽ 9.3. In other words, at the start of a union,

those in strictly cohabiting unions have about nine times the risk of disrupting

compared to marrieds. After ﬁve years, however, this risk ratio drops to exp[2.23 ⫺

.017(60)] ⫽ 3.353.

Stratified Models

In the event that one suspects nonproportionality in the eﬀect of a nominal covariate

and that covariate’s eﬀect per se is of tangential importance, it can also be modeled

as a stratiﬁying factor. Stratiﬁed models allow the baseline hazard to vary across

groups while constraining the eﬀects of covariates to be the same across groups. Why

does variation in the baseline hazard function across groups imply nonproportional-

ity of the grouping variable’s eﬀect? Consider a parametric model (a Gompertz model,

in particular) for the log-hazard, which is linear in time T, with an interaction eﬀect

of X

with time. Its form is

ln[h

(t)] ⫽ β

⫹ γ

T ⫹ β

⫹

...

⫹ β

⫹ γ

T ⫹

...

⫹ β

The eﬀect of time is then γ

⫹ γ

, which implies that the manner in which the

log-hazard varies with time depends on the value of X

. In the Cox model the para-

metric form of the relationship with time, rather than being linear, can take any

form, but the idea is the same. Interactions of covariates with time imply either that

the eﬀect of the covariate depends on time, as shown in the previous paragraph, or

that the relationship of the hazard with time depends on the covariate or grouping

factor.

In stratiﬁed models, each stratum, or value of the grouping factor, has a separate

baseline hazard function. Thus, the Cox model for each stratum, s, is (Hosmer and

Lemeshow, 1999)

(t) ⫽ h

(t) exp(x

⬘

ββ

where s ⫽ 1,2,...,S. As there is no subscript on

ββ

, the eﬀects of covariates on the

hazard are held to be the same for each stratum. The contribution to the partial like-

lihood for the sth stratum is (Hosmer and Lemeshow, 1999)

(

ββ

t,x) ⫽



i⫽1

exp(x

⬘

ββ

)

ᎏᎏ



ο∈R(t

(i))

exp(x

⬘

ββ

)

’

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where k

refers to the total number of events in the sth stratum and k ⫽ k

⫹ k

⫹

...

⫹

. The full stratiﬁed partial likelihood is (Hosmer and Lemeshow, 1999)

PL(

ββ

t,x) ⫽ 

s⫽1

(

ββ

t,x). (11.15)

The PL estimate of

ββ

is obtained by maximizing function (11.15) with respect to the

parameters.

A few comments about stratiﬁcation are in order. First, the stratifying variable is not

in the model, so one loses the ability to estimate the eﬀect of the stratifying variable

itself. This is not of particular concern if the stratifying variable is primarily a control

rather than a focus variable. Moreover, the eﬀects of all covariates whose values are

constant within each stratum are incorporated into the stratum-speciﬁc baseline hazard

function (Hosmer and Lemeshow, 1999). This implies that one cannot estimate the

eﬀects of any other characteristics that only vary across strata rather than within strata.

This can be seen as a plus, too, since these factors are all being held constant. That is,

any nuisance factors that vary across strata are being automatically controlled in the

estimation process. Stratiﬁcation also oﬀers an advantage over simply incorporating

into the model an interaction of the stratifying factor with time (Allison, 1995).

Including a term in the model of the form G * time, where G denotes the stratifying vari-

able, assumes a linear interaction with time, which may not be the correct form. With

stratiﬁcation, the functional form of the interaction with time is accounted for auto-

matically by allowing separate baseline hazard functions that remain unspeciﬁed.

Testing Parameter Invariance. A key assumption of stratiﬁed analysis is that the

parameter vector,

ββ

, is invariant over strata. If this is violated, it simply means that

covariates have diﬀerent eﬀects, depending on stratum, and the stratiﬁed model is

not warranted. It is a simple matter to test this assumption, however (Klein and

Moeschberger, 1997). One simply estimates the stratiﬁed Cox model and obtains

⫺2 ln L for it, denoted ⫺2lnL

.Then one estimates the Cox model for each stratum

separately, denoting ⫺2 ln L for each analysis as ⫺2lnL

, where j ⫽ 1, 2,..., S.

Minus twice the log-likelihood for the model that allows

ββ

to vary over strata is then

⫺2lnL

⫽



j⫽1

⫺2lnL

The test statistic for the null hypothesis of parameter invariance is

⫽⫺2lnL

⫺ (⫺2lnL

(11.16)

ββ

is invariant over strata, this statistic has the chi-squared distribution with K(S ⫺ 1)

degrees of freedom. If the test is signiﬁcant and invariance is rejected, one can always

just report the results of estimating the model separately for each stratum.

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Union Disruption Example. The last two models in Table 11.3 illustrate the results

of stratiﬁcation. Above, we saw that cohabiting status was seen to interact with time

(model 2). Therefore, I estimated a model including only the main eﬀects of covari-

ates (i.e., no interactions with time) as both unstratiﬁed (model 3) and stratiﬁed by

cohabiting status (model 4). The unstratiﬁed version, model 3, assumes the same

baseline hazard across cohabiting statuses, an assumption that has already been

rejected. The three strata for model 4 are marrieds, continuous cohabitors, and

cohabitors who subsequently married. Although substantive conclusions are not

dramatically altered with stratiﬁcation, there are noticeable reductions in the eﬀects

of being in a ﬁrst union, having a union birth,and the style of conﬂict resolution in

the stratiﬁed model compared to the unstratiﬁed one. Hazard ratios are interpreted

in the same manner as in unstratiﬁed models, so exp(⫺.606) ⫽ .546 is the ratio of

hazards for those with a union birth, versus others, and so on. That is, although the

baseline hazards diﬀer across strata, having a union birth is posited to magnify the

baseline hazard to the same degree in each stratum. Are the eﬀects of covariates

invariant across cohabiting-status groups? This was tested by performing the test in

equation (11.16). First, I estimated the model (model 3) for each cohabiting-status

group separately and summed ⫺2 ln L for the three analyses to arrive at 3660.921

for ⫺2lnL

. Minus twice the log-likelihood for model 4 is 3678.86. The test sta-

tistic is χ

⫽ 3678.86 ⫺ 3660.921⫽ 17.939 and has 7(3 ⫺ 1) ⫽ 14 degrees of

freedom. As p ⬎ .2, the result is nonsigniﬁcant, suggesting that the stratiﬁed model

is valid.

Assessing Model Fit

Discriminatory Power. As always, the degree to which a given model ﬁts the data

is of considerable interest. This can be assessed in terms of either the model’s dis-

criminatory power or its empirical consistency. As to the ﬁrst attribute, there is even

less agreement on which R

analog is best in survival analysis than in the techniques

covered in earlier chapters. An investigation by Schemper and Stare (1996) consid-

ered several possible measures of discriminatory power for survival models.

Nonetheless, the authors admonish that there is no uniformly superior measure, par-

ticularly since performance depends heavily on the degree of censoring. The meas-

ures most recommended by the authors tend to be computationally cumbersome.

However, a measure that appears to perform reasonably well under all conditions

and is simple to calculate is the generalized R

introduced in Chapter 7, a measure

also recommended by other authors (Allison, 1995; Hosmer and Lemeshow, 1999).

For the Cox model, the formula is

⫽ 1 ⫺ exp



ᎏ

⫺L

Rχ

ᎏ



As an example, discriminatory power for model 2 in Table 11.3 is

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⫽ 1 ⫺ exp



ᎏ

⫺3

853

ᎏ



⫽ .235.

As Schemper and Stare (1996) have found R

analogs in survival analysis to range

typically between .10 and .45, model 2 can be said to exhibit a moderate amount of

discriminatory power.

Empirical Consistency. A test for goodness of ﬁt, or empirical consistency, of the

Cox model has been recently developed by May and Hosmer (1998). It has been

designed in the same spirit as the Hosmer and Lemeshow (2000) goodness-of-ﬁt test

for logistic regression. The procedure is as follows. First, ﬁt the Cox model of inter-

est and obtain b, the estimate of the parameter vector. Second, calculate the esti-

mated risk score, r

, for each person as rˆ

⫽ x

⬘b. (The risk score is nothing more than

the linear predictor for each case.) Third, order the risk scores from lowest to high-

est and then group individuals according to deciles of risk. If there are too few cases

to create 10 groups, the authors suggest that a minimum number of groups should be

ﬁve. Fourth, create dummy variables representing the group each case falls into; for

10 groups, there should be 9 dummies, etc. Fifth, add the block of dummies to the

model of interest in a separate step and test whether the block is signiﬁcant using

∆χ

. If the block is not signiﬁcant, there is evidence that the model has an adequate

ﬁt to the data.

As an example, I performed the test on model 4 in Table 11.2 using deciles of risk

scores. This is not the ﬁnal model of interest, as it treats the time-varying covariates

as time-invariant. However, the calculation of risk scores with time-varying covari-

ates is problematic since people’s risk scores change over time when time-varying

covariates are involved. Model 4, on the other hand, facilitates the calculation of a

single risk score for each case. I calculated risk scores using the formula rˆ

⫽⫺.028

relationship duration ⫺ .074 female’s age at union ⫺

...

⫺ 1.457 union birth. Based

on a frequency distribution for the scores, I identiﬁed the deciles of risk and created

nine dummies to represent them in the model. The LRχ

values for the models with

and without the decile-of-risk dummies are 431.4674 and 417.3661, respectively.

The test statistic is therefore ∆χ

⫽ 431.4674 ⫺ 417.3661 ⫽ 14.1013, which, with nine

degrees of freedom, is not signiﬁcant ( p ⫽ .119). In this case, then, there is no evidence

to suggest that the model lacks empirical consistency.

EXERCISES

11.1 The following partial table refers to 1000 domestic violence oﬀenders fol-

lowed for one year after release from arrest. The event of interest (a “fail-

ure”) was the ﬁrst subsequent arrest for domestic violence recorded by

police. Subjects were considered censored if they were arrested for a

diﬀerent oﬀense during the follow-up period or if they had not been arrested

by the end of the study.

EXERCISES 411

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412 INTRODUCTION TO SURVIVAL ANALYSIS

Show the estimated survival and hazard functions for these data using the

life-table estimators.

11.2 The following data represent times to arrest or censoring for six of the

oﬀenders from Exercise 11.1, along with a dummy variable for whether the

oﬀender was employed at the time of arrest:

1 ⫽ uncensored; 0 ⫽ censored.

If the Cox estimate for the eﬀect of employment is ⫺1.073, give ⫺2 ln L for

the Cox model for these data with employment as the predictor.

11.3 For the data in Exercise 11.2, the standard error of the eﬀect of employment

is 1.235. Also, ⫺2 ln L for a model with no covariates is 10.386. Use both

LRχ

and Wald tests to test whether the eﬀect of employment is signiﬁcant.

Regardless of signiﬁcance, interpret the eﬀect of employment on the hazard

of rearrest for domestic violence.

11.4 Suppose that the number of weeks to rearrest for domestic violence after an

initial arrest has an exponential distribution with λ ⫽ .0035. Give:

(a) The probability of arrest within one year after the initial arrest.

(b) The probability of surviving one year without being arrested.

11.5 The survival function for the Weibull distribution is (Blossfeld et al., 1989)

S(t) ⫽ exp[(⫺λt)

], where λ and σ are positive constants. Show that this dis-

tribution implies that ln[⫺ln S(t)] is linear in the log of t.

Censoring

Subject Indicator

Weeks Employed

1 1 39 0

2 1 17 1

3 1 46 1

4 1 12 0

5 0 22 0

6 0 52 1

Time (weeks) Number Number

Interval Failed Censored

[0,10) 25 3

[10,20) 30 8

[20,30) 28 7

[30,40) 46 19

[40,⬁) 82 752

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11.6 Show that if time has an exponential density with λ⫽ 1, the log of time has a

Gumbell density. [Hint: Write the expression for F(t). Then let Y ⫽ ln t and let

G( y) be the distribution function for Y. Then note that G( y) ⫽ P(Y ⱕ y) ⫽

P(ln t ⱕ y) ⫽ P(t ⱕ e

). Now express P(t ⱕ e

) in terms of F(t). This gives you

G( y), and taking its ﬁrst derivative with respect to y gives the density of Y. This

is a well-known technique for ﬁnding density and distribution functions for a

function of some variable, where the variable has some known distribution.

See, e.g., Hoel et al. (1971).]

11.7 Based on model 1 in Table 11.2, give the predicted survival time for a cou-

ple together for 2 months at wave 1, in which the woman was 19 at the start

of the union, in which the man had been married before, in which both part-

ners had drinking problems, whose open disagreement score was 30, and

whose conﬂict resolution score was 3.

11.8 Suppose that there is a time-varying dummy, M, for marital status such that

M ⫽ 1 if married and 0 if unmarried. Continuing with the substantive exam-

ple from Exercise 11.1, say that we have two subjects with the following val-

ues for M at the times indicated:

Let the Cox model for the hazard of rearrest be ln h

(t) ⫽ λ

(t) ⫹ δM

(t),

where δ⫽0. Demonstrate, using these data points, that the ratio of hazards

for case 1 versus case 2 is nonproportional, due to the time-varying covari-

ate in the model.

Use the following information for Exercises 11.9 to 11.11. Yamaguchi (1991)

employed Cox regression to explore the impact of several factors on the hazard of

dropping out of college. The sample consisted of 265 students from the High

School & Beyond Survey who were high school seniors in 1980 and who

subsequently entered four-year colleges. The follow-up period was 1980–1984.

The event constituting a “failure” consisted of either dropping out of the initial

college of choice or transferring to a diﬀerent college. Those who graduated or

were still in the same school at the end of the study (1984) were considered

censored observations. Duration in college was measured in months. The

predictors used were:

Time M for M for

(weeks) Case 1 Case 2

10 0 1

15 0 0

30 1 1

42 1 0

EXERCISES 413

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SEX ⫽ dummy for being female.

GRD ⫽ self-reported high-school grades (1 ⫽ “mostly A’s”to8⫽ “mostly D’s”).

PRT ⫽ dummy for being a part-time student (“full-time student” is the contrast).

LAG ⫽ time lag (months) between high-school graduation and college entry.

MS ⫽ time-varying dummy for being married at time t.

EMP ⫽ time-varying dummy for being employed at time t.

11.9 The ﬁrst model investigated by Yamaguchi resulted in the following equation:

rˆ

⫽ .324 SEX ⫹ .285 GRD ⫹ 1.462 PRT ⫹ .127 LAG ⫺ .086 PRT * LAG.

LR χ

(5)

⫽ 49.57.

Interpret the results by interpreting all coeﬃcients. (All

eﬀects except for SEX are signiﬁcant at p ⬍ .05.)

11.10 Suppose that the baseline survivor function for the ﬁrst ﬁve months is 1,

.995, .991, .926, .914. Give the estimated survival function for a male full-

time student whose high-school grades were mostly B’s (GRD ⫽ 3), and

who waited 12 months before starting college.

11.11 The second model investigated produced the following equation:

⫽ .340 SEX ⫹ .289 GRD ⫹ 1.368 PRT ⫹ .125 LAG ⫺ .081 PRT * LAG

⫹ 1.255 MS ⫹ .512 EMP.

LR χ

(7)

⫽ 62.70.

(a) Test whether the effects of both time-varying covariates simultaneously

equal zero.

(b) Interpret the eﬀects of marriage and employment.

11.12 Teachman (2003) examined the inﬂuence on the hazard of ﬁrst union for-

mation (either marriage or cohabitation) of a number of demographic covari-

ates for a sample of 7477 women taken from the 1995 National Survey of

Family Growth. One model produced the following hazard ratios (net of

other covariates) for ever having lived before age 16 in various alternative

family forms, compared to continuous habitation with both biological par-

ents (the reference group):

Mother-only family: 1.10.

Step-parent family: 1.21.

Parent and cohabiting partner family: 1.26.

Two cohabiting parents family: 1.12.

Residual family type: 1.24.

Interpret the eﬀect of living in a family with two cohabiting parents. Give

the hazard ratios for living in a mother-only family versus living in each of

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