Covariate Effects Via Cox Models
We have just seen evidence that the hazard for natural death is lower than that for the other two exit types, but that it increases more rapidly with time. On the other hand, the hazard functions for constitutional and nonconstitutional exits are nearly indistinguishable (despite the apparent differences in Output 6.2). We now look at whether the effects of covariates are the same or different across event types by fitting a Cox model to each type. The SAS program for doing this is as follows:
proc phreg data=leaders;
model years*lost(0)=manner start military age conflict
loginc growth pop land literacy / ties=efron;
strata region;
run;
proc phreg data=leaders;
model years*lost(0,1,2)=manner start military age conflict
loginc growth pop land literacy / ties=efron;
strata region;
run;
proc phreg data=leaders;
model years*lost(0,1,3)=manner start military age conflict
loginc growth pop land literacy / ties=efron;
strata region;
run;
proc phreg data=leaders;
model years*lost(0,2,3)=manner start military age conflict
loginc growth pop land literacy / ties=efron;
strata region;
run;The first model treats all event types the same. I’ve included that model for comparison, as well as for some hypothesis tests that we’ll look at shortly. The second model focuses on type 3 by treating types 1 and 2 as censoring. That’s followed by a models for type 2, treating types 1 and 3 as censoring, and a model for type 1, treating types 2 and 3 as censoring. Notice that I have stratified by region. Alternatively, I could have created three dummy variables to represent the four regions and included them in the model. However, stratification is easier to specify and less restrictive (it allows for interactions between region and time).
In Output 6.4 we see what happens when no distinction is made among the different kinds of exits. Four variables are statistically significant by conventional criteria. The strongest effect is age at entry into power, with each additional year of age associated with a 2.3 percent increase in the risk of leaving power. Recall from Chapter 5, moreover, that if age at origin is a covariate in a Cox model, its coefficient may actually be interpreted as the effect of age as a time-dependent covariate. We also see that
leaders who attained power by nonconstitutional means (MANNER=1) have a 47 percent greater risk of leaving power
leaders in countries with higher per capita GNP (LOGINC) have a lower risk of exit
the risk of exit declined by about 2 percent per year (START) since 1960.
The coefficient for the logarithm of GNP (–.1822) needs a little explanation. If both the hazard and a covariate in a regression model are logged, we can interpret the coefficient (without transformation) as the percent change in the hazard for a 1-percent increase in the covariate (in its original metric). Thus, we conclude that a 1-percent increase in per capita GNP yields a .18 percent decrease in the risk of exit.
Output 6.4. PROC PHREG Analysis for Exits by Any Means
Testing Global Null Hypothesis: BETA=0
Without With
Criterion Covariates Covariates Model Chi-Square
-2 LOG L 2596.320 2561.077 35.243 with 10 DF (p=0.0001)
Score . . 34.829 with 10 DF (p=0.0001)
Wald . . 34.628 with 10 DF (p=0.0001)
Analysis of Maximum Likelihood Estimate
Parameter Standard Wald Pr > Risk
Variable DF Estimate Error Chi-Square Chi-Square Ratio
MANNER 1 0.382797 0.15520 6.08355 0.0136 1.466
AGE 1 0.022743 0.00556 16.73312 0.0001 1.023
START 1 -0.017531 0.00814 4.63251 0.0314 0.983
MILITARY 1 -0.231220 0.16342 2.00180 0.1571 0.794
CONFLICT 1 0.127089 0.13131 0.93667 0.3331 1.136
LOGINC 1 -0.182206 0.08259 4.86652 0.0274 0.833
GROWTH 1 -0.002041 0.02138 0.00912 0.9239 0.998
LAND 1 0.000010945 0.0000480 0.05194 0.8197 1.000
POP 1 -0.000063245 0.0006373 0.00985 0.9210 1.000
LITERACY 1 0.000702 0.00321 0.04785 0.8268 1.001 |
The picture changes somewhat when we focus on the hazard for nonconstitutional exits in Output 6.5. Leaders who acquired power by nonconstitutional means are 2.5 times as likely as other leaders to exit by nonconstitutional means. Income also has a stronger effect: a 1-percent increase in per capita GNP now yields a .43 percent decrease in the risk of nonconstitutional exit. Age is no longer important, but we still see a reduction in the hazard over time since 1960. In addition, we find that leaders in countries with ethnic conflict have a 66 percent greater chance of nonconstitutional exit.
Constitutional exits show a different pattern (Output 6.6). Manner of acquiring power is not important, nor is income or conflict. Age shows up again, however, and we also see an effect of literacy. Each one-percentage-point increase in the literacy rate is associated with a 1.4 percent increase in the risk of a constitutional exit.
Output 6.5. PROC PHREG Analysis for Nonconstitutional Exits
Parameter Standard Wald Pr > Risk Variable DF Estimate Error Chi-Square Chi-Square Ratio MANNER 1 0.928161 0.22047 17.72317 0.0001 2.530 AGE 1 0.008973 0.00845 1.12869 0.2881 1.009 START 1 -0.033498 0.01221 7.53257 0.0061 0.967 MILITARY 1 -0.413794 0.22757 3.30625 0.0690 0.661 CONFLICT 1 0.504451 0.20335 6.15408 0.0131 1.656 LOGINC 1 -0.433399 0.14135 9.40072 0.0022 0.648 GROWTH 1 -0.048583 0.03112 2.43758 0.1185 0.953 LAND 1 0.000020113 0.0000802 0.06286 0.8020 1.000 POP 1 -0.001027 0.00154 0.44253 0.5059 0.999 LITERACY 1 -0.005693 0.00452 1.58425 0.2081 0.994 |
Output 6.6. PROC PHREG Analysis for Constitutional Exits
Parameter Standard Wald Pr > Risk Variable DF Estimate Error Chi-Square Chi-Square Ratio MANNER 1 -0.308283 0.25750 1.43333 0.2312 0.735 AGE 1 0.023977 0.00862 7.73043 0.0054 1.024 START 1 0.002392 0.01201 0.03969 0.8421 1.002 MILITARY 1 -0.012040 0.26053 0.00214 0.9631 0.988 CONFLICT 1 -0.029853 0.20372 0.02147 0.8835 0.971 LOGINC 1 -0.130681 0.11939 1.19812 0.2737 0.877 GROWTH 1 0.033905 0.03453 0.96398 0.3262 1.034 LAND 1 -0.000026111 0.0000703 0.13792 0.7104 1.000 POP 1 0.000412 0.0008461 0.23656 0.6267 1.000 LITERACY 1 0.013688 0.00561 5.95030 0.0147 1.014 |
For deaths from natural causes (Output 6.7), it is not terribly surprising that age is the only significant variable. Each one-year increase in age is associated with an 8.3 percent increase in the hazard of death.
Output 6.7. Deaths from Natural Causes
Parameter Standard Wald Pr > Risk Variable DF Estimate Error Chi-Square Chi-Square Ratio MANNER 1 0.302690 0.70202 0.18591 0.6663 1.353 AGE 1 0.079380 0.02038 15.17644 0.0001 1.083 START 1 -0.058506 0.03530 2.74749 0.0974 0.943 MILITARY 1 -0.299045 0.78251 0.14605 0.7023 0.742 CONFLICT 1 -0.553633 0.50851 1.18536 0.2763 0.575 LOGINC 1 0.194861 0.28309 0.47381 0.4912 1.215 GROWTH 1 0.092281 0.08538 1.16806 0.2798 1.097 LAND 1 0.000034665 0.0001797 0.03720 0.8471 1.000 POP 1 0.000956 0.00220 0.18847 0.6642 1.001 LITERACY 1 -0.012390 0.01361 0.82909 0.3625 0.988 |
We see, then, that the coefficients can differ greatly across different event types. But perhaps these differences are merely the result of random variation. What we need is a test of the null hypothesis that βj = β for all j, where βj is the vector of coefficients for event type j. A test statistic is readily constructed from output given by PROC PHREG. For each model, PROC PHREG reports –2 × log-likelihood (for the model with covariates). For the four models we just estimated, the values are
| All types combined | 2561.08 |
| Nonconstitutional | 1202.04 |
| Constitutional | 1139.41 |
| Natural Death | 155.38 |
To construct the test, we sum the values for the three specific death types, yielding a total of 2496.83. We then subtract that from the value for all types combined, for a difference of 64.25. This is a likelihood-ratio chi-square statistic for the null hypothesis. How many degrees of freedom? Well, when we estimated separate models for the three event types, we got a total of 30 coefficients. When we collapsed them all together, we only estimated 10. The difference of 20 is the degrees of freedom. Since the chi-square statistic is significant at well beyond the .01 level, we may reject the hypothesis that the coefficients are all equal across event types.
That result is not terribly surprising because deaths from natural causes are unlikely to have the same determinants as constitutional and nonconstitutional exits. A more interesting question is whether the covariates for the latter two event types have identical coefficients. We can obtain a test statistic for that null hypothesis by estimating a model that combines constitutional and nonconstitutional exits and that treats natural deaths as censoring. For that model, –2 × log-likelihood=2392. If we sum the values for the two separate models (given above), we get 2341.45. The difference of 50.55 has 10 degrees of freedom (the difference in the number of estimated parameters) for a p-value less than .0001. So again, we may conclude that different models are required for constitutional and nonconstitutional exits. (This test is only valid when using the Breslow method for handling ties or when there are no tied event times. In other cases, a more complicated procedure is necessary).
If we had concluded, instead, that corresponding coefficients were equal across all event types, a natural next step would be to test whether they were all equal to 0. No special computations are needed for that test, however. Just estimate the model without distinguishing among the event types, and examine the global statistics for the null hypothesis that all coefficients are equal to 0. In Output 6.4, all three chi-square statistics (Wald, score, and likelihood ratio) have values of about 35 with 10 d.f., giving strong evidence against that null hypothesis. Note that this is a conditional test, the condition being that the coefficients are equal. To test the hypothesis that corresponding coefficients are equal and that they are equal to 0, simply add this conditional chi-square statistic to the chi-square statistic for testing equality, and then add the degrees of freedom as well. Thus, for this example, the likelihood-ratio statistic is 35.23+64.25=99.48 with 40 d.f.
We can also construct test statistics for hypotheses about coefficients for specific covariates, using only the coefficients and their standard errors (Lagakos 1978). The coefficient for CONFLICT for nonconstitutional exits was .5044 with a standard error of .2034. By contrast, the coefficient for CONFLICT for constitutional exits was –.02985 with a standard error of .2037. The first coefficient is significantly different from 0; the second is not. But is there a significant difference between them? A 1-degree-of-freedom Wald chi-square statistic for testing the null hypothesis that β1 = β2 is easily calculated by the following formula:
Equation 6.8
where b1 is the estimate of β1 and s.e.(.) means estimated standard error. For our particular question, we have
Because that does not exceed the .05 criterion, we have insufficient statistical justification for concluding that the two coefficients are different.
This particular statistic, in one form or another, is widely used to test for differences in parameter estimates across two independent groups. Here, however, we do not have independent groups because the same set of 472 leaders is used to estimate models for both constitutional and nonconstitutional exits. This suggests that we may need a covariance term in the denominator of the statistic. In fact, we do not. What justifies this statistic is the fact that the likelihood function factors into a distinct likelihood for each event type. It follows that the parameter estimates for each event type are asymptotically independent of the parameter estimates for all other event types. This only holds for mutually exclusive event types, however. We cannot use this statistic to test for the difference between a coefficient for constitutional exits and the corresponding coefficient for all types of exits.
Because we found no reason to reject the hypothesis that the two coefficients for CONFLICT are equal, we may want to go further and test whether they are 0. This is easily accomplished by taking the reported chi-square statistics for CONFLICT, for both constitutional and nonconstitutional exits, and summing them: .021+6.15=6.27 with 2 d.f. This is just barely significant at the .05 level.
The tests for a single covariate can be generalized to more than two event types. For a given covariate, let bj be its coefficient for event type j; let sj2 be the squared, estimated standard error of bj, and let Xj2=bj2/sj2 be the reported Wald chi-square statistic for testing that βj=0. To test the hypothesis that all the coefficients for the chosen covariate are 0, we sum the Wald chi-square statistics
Equation 6.9
which has degrees of freedom equal to the number of event types. To test the hypothesis that all coefficients are equal to each other, we calculate
Equation 6.10
which has degrees of freedom equal to one less than the number of event types.