As indicated in the previous unit, discrete-time survival analysis treats time, not as a continuous variable, but as being divided into discrete chunks or units. We will be able to analyze discrete time data using logistic or cloglog regression with indicator variables for each of the time periods. We will illustrate discrete-time survival analysis using the cancer.dta dataset.
Cancer Example
After reading in the dataset, we will describe the variables and list several variables for patient 5, 10 and 20.
use http://www.gseis.ucla.edu/courses/data/cancer
describe
Contains data from cancer.dta
obs: 48 Patient Survival in Drug Trial
vars: 7 2 Jan 1904 13:58
size: 1,248 (99.1% of memory free)
-------------------------------------------------------------------------------
storage display value
variable name type format label variable label
-------------------------------------------------------------------------------
id float %9.0g
studytime int %8.0g Months to death or end of exp.
died int %8.0g 1 if patient died
drug float %9.0g
age int %8.0g Patient's age at start of exp.
distime float %9.0g
censor float %9.0g
-------------------------------------------------------------------------------
tab distime
distime | Freq. Percent Cum.
------------+-----------------------------------
1 | 11 22.92 22.92
2 | 13 27.08 50.00
3 | 6 12.50 62.50
4 | 8 16.67 79.17
5 | 4 8.33 87.50
6 | 6 12.50 100.00
------------+-----------------------------------
Total | 48 100.00
univar age
-------------- Quantiles --------------
Variable n Mean S.D. Min .25 Mdn .75 Max
-------------------------------------------------------------------------------
age 48 55.88 5.66 47.00 50.50 56.00 60.00 67.00
-------------------------------------------------------------------------------
list distime drug age died censor if id==5
distime drug age died censor
5. 1 0 56 1 0
list distime drug age died censor if id==10
distime drug age died censor
10. 2 0 58 0 1
list distime drug age died censor if id==20
distime drug age died censor
20. 4 0 52 1 0Patient 5 (56 years old, did not receive a drug treatment) was observed for one time period, died. So, the observation for that patient was not censored. Patient 10 (58, no drug) was observed for two time periods did not die, i.e., observation was censored. Finally, patient 20 (52, no drug) was observed for four time periods, died (not censored).
In this dataset there is one observation for each patient. In order to do discrete-time survival analysis we need to have as many observations as there are time periods for each patient. For patients that die we need a response variable that is zero until the last time period when it is coded one. For patients that don't die the response variable will be zero for every observation.
A collection of Stata commands written by Alexis Dinno (Harvard School of Public Health) will help with the analysis. Here is how we can obtain the commands.
net from http://www.doyenne.com/stata net get dthaz
The command that we are interested in is prsnperd which creates the type of dataset that we want. prsnperd wants a variable that indicates whether the observation is censored or not which in our dataset is the variable censor. prsnperd creates the following variables: _period which is the time period, _Y which is the response variable and _d1 through _d6 which are the dummy coded time periods. Here is what it looks like.
prsnperd id distime censor
list id _period _Y if id==5
id _period _Y
5. 5 1 1
list id _period _Y if id==10
id _period _Y
11. 10 1 0
12. 10 2 0
list id _period _Y if id==20
id _period _Y
35. 20 1 0
36. 20 2 0
37. 20 3 0
38. 20 4 1Now we can actually do the discrete-time survival analysis using the logit command. The logit command estimates a proportional odds discrete-time survival model. We will run logit with and without the cluster and nocons options. The nocons options is used so that the dummy variables for all of the time periods can be included.
logit _Y _d1-_d6, cluster(id) nocons
Logit estimates Number of obs = 143
Wald chi2(6) = 39.22
Log pseudo-likelihood = -74.24676 Prob > chi2 = 0.0000
(standard errors adjusted for clustering on id)
------------------------------------------------------------------------------
| Robust
_Y | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
_d1 | -1.335001 .359167 -3.72 0.000 -2.038955 -.6310468
_d2 | -1.13498 .3872325 -2.93 0.003 -1.893942 -.3760181
_d3 | -1.609438 .5534596 -2.91 0.004 -2.694199 -.5246771
_d4 | -.9555114 .5318035 -1.80 0.072 -1.997827 .0868043
_d5 | -1.386294 .7989229 -1.74 0.083 -2.952155 .1795658
_d6 | -1.609438 1.106919 -1.45 0.146 -3.77896 .5600838
------------------------------------------------------------------------------
logit _Y _d1-_d6, nocons
Logit estimates Number of obs = 143
LR chi2(6) = .
Log likelihood = -74.24676 Prob > chi2 = .
------------------------------------------------------------------------------
_Y | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
_d1 | -1.335001 .3554076 -3.76 0.000 -2.031587 -.6384149
_d2 | -1.13498 .3831778 -2.96 0.003 -1.885995 -.3839652
_d3 | -1.609438 .5476933 -2.94 0.003 -2.682897 -.5359788
_d4 | -.9555114 .5262348 -1.82 0.069 -1.986913 .0758898
_d5 | -1.386294 .7905632 -1.75 0.080 -2.93577 .1631811
_d6 | -1.609438 1.095387 -1.47 0.142 -3.756356 .5374803
------------------------------------------------------------------------------
If we use the predict command we obtain the predicted probabilities for each of the time intervals. These probabilities are, in fact, the estimated hazard probabilities for each time interval.
predict p1 tablist _period p1 +---------------------------+ | _period p1 Freq | |---------------------------| | 1 .2083333 48 | | 2 .2432432 37 | | 3 .1666667 24 | | 4 .2777778 18 | | 5 .2 10 | | 6 .1666667 6 | +---------------------------+
Now we add the covariate drug to the model. Drug is a binary indicator of whether the patient received chemotherapy or not.
logit _Y drug _d1-_d6, nocons
Logit estimates Number of obs = 143
LR chi2(7) = .
Log likelihood = -61.192357 Prob > chi2 = .
------------------------------------------------------------------------------
_Y | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
drug | -2.553352 .5554143 -4.60 0.000 -3.641944 -1.46476
_d1 | -.305909 .4172929 -0.73 0.464 -1.123788 .5119701
_d2 | .246559 .5014684 0.49 0.623 -.7363011 1.229419
_d3 | .2249641 .7090498 0.32 0.751 -1.164748 1.614676
_d4 | 1.259138 .7556111 1.67 0.096 -.2218328 2.740108
_d5 | 1.167058 .9661703 1.21 0.227 -.7266011 3.060717
_d6 | .9439143 1.228204 0.77 0.442 -1.463321 3.35115
------------------------------------------------------------------------------
To get additional information we run the model with a constant.
logit _Y drug _d2-_d6
Logit estimates Number of obs = 143
LR chi2(6) = 27.14
Prob > chi2 = 0.0001
Log likelihood = -61.192357 Pseudo R2 = 0.1815
------------------------------------------------------------------------------
_Y | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
drug | -2.553352 .5554142 -4.60 0.000 -3.641944 -1.46476
_d2 | .552468 .606679 0.91 0.362 -.636601 1.741537
_d3 | .5308731 .7680236 0.69 0.489 -.9744256 2.036172
_d4 | 1.565047 .7879091 1.99 0.047 .0207732 3.10932
_d5 | 1.472967 .9836119 1.50 0.134 -.454877 3.400811
_d6 | 1.249823 1.241971 1.01 0.314 -1.184395 3.684042
_cons | -.305909 .4172929 -0.73 0.464 -1.123788 .51197
------------------------------------------------------------------------------
fitstat
Measures of Fit for logit of _Y
Log-Lik Intercept Only: -74.761 Log-Lik Full Model: -61.192
D(136): 122.385 LR(6): 27.138
Prob > LR: 0.000
McFadden's R2: 0.181 McFadden's Adj R2: 0.088
Maximum Likelihood R2: 0.173 Cragg & Uhler's R2: 0.267
McKelvey and Zavoina's R2: 0.271 Efron's R2: 0.208
Variance of y*: 4.512 Variance of error: 3.290
Count R2: 0.811 Adj Count R2: 0.129
AIC: 0.954 AIC*n: 136.385
BIC: -552.562 BIC': 2.639
Next we add in the second covariate, age, to the model.
logit _Y drug age _d1-_d6, nocons
Logit estimates Number of obs = 143
LR chi2(8) = .
Log likelihood = -55.65503 Prob > chi2 = .
------------------------------------------------------------------------------
_Y | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
drug | -3.024052 .6347086 -4.76 0.000 -4.268058 -1.780046
age | .1607128 .051414 3.13 0.002 .0599433 .2614823
_d1 | -9.309867 2.922645 -3.19 0.001 -15.03815 -3.581589
_d2 | -8.335442 2.780394 -3.00 0.003 -13.78491 -2.885969
_d3 | -8.326742 2.823744 -2.95 0.003 -13.86118 -2.792306
_d4 | -7.071942 2.734906 -2.59 0.010 -12.43226 -1.711624
_d5 | -7.19799 2.811519 -2.56 0.010 -12.70847 -1.687513
_d6 | -7.622593 2.988678 -2.55 0.011 -13.48029 -1.764892
------------------------------------------------------------------------------
logit, or
Logit estimates Number of obs = 143
LR chi2(8) = .
Log likelihood = -55.65503 Prob > chi2 = .
------------------------------------------------------------------------------
_Y | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
drug | .0486039 .0308493 -4.76 0.000 .014009 .1686304
age | 1.174348 .0603779 3.13 0.002 1.061776 1.298854
_d1 | .0000905 .0002646 -3.19 0.001 2.94e-07 .0278315
_d2 | .0002399 .0006669 -3.00 0.003 1.03e-06 .0558007
_d3 | .000242 .0006832 -2.95 0.003 9.55e-07 .0612797
_d4 | .0008486 .0023208 -2.59 0.010 3.99e-06 .1805723
_d5 | .0007481 .0021033 -2.56 0.010 3.03e-06 .184979
_d6 | .0004893 .0014623 -2.55 0.011 1.40e-06 .1712052
------------------------------------------------------------------------------Both drug and age are significant with the older patients more likely to die and those on drug therapy less likely. It is useful to look at the hazard function (and survival function) to ascertain the effects over time. The dthaz command (from Dinno) will produce a table with hazard and survival values for each time period. We will specify the function for drug=1 (drug therapy) and age=56 (the median age).
dthaz drug age, specify(1 56)
Discrete-Time Estimation of Conditional Hazard and Survival Probabilities
------------------------------------------------------------------------------
Time Parameterization: Fully Discrete
Additional predictors specified as:
drug = 1
age = 56
-----------------------------------------
Period p(Hazard) p(Survival)
(T_j) ^H(T_j) ^S(T_j)
-----------------------------------------
0 -- 1
1 0.0344 0.9656
2 0.0863 0.8822
3 0.0870 0.8055
4 0.2505 0.6037
5 0.2276 0.4663
6 0.1616 0.3910
-----------------------------------------
Logit Link (assumes proportional odds)Notice that the hazard maxes out at time period four and then declines. Now if we do the same for drug=0, we see a completely different pattern with much greater hazards.
dthaz drug age, specify(0 56)
Discrete-Time Estimation of Conditional Hazard and Survival Probabilities
------------------------------------------------------------------------------
Time Parameterization: Fully Discrete
Additional predictors specified as:
drug = 0
age = 56
-----------------------------------------
Period p(Hazard) p(Survival)
(T_j) ^H(T_j) ^S(T_j)
-----------------------------------------
0 -- 1
1 0.4231 0.5769
2 0.6603 0.1960
3 0.6622 0.0662
4 0.8730 0.0084
5 0.8584 0.0012
6 0.7986 0.0002
-----------------------------------------
Let's check the fit of this model. To do this we will rerun the model with a constant and then run the fitstat command. Notice that we have to drop one of the time dummies if we include the constant. Models without the constant but with all of the dummies and models with the constant dropping one dummy are equivalent they are merely different parameterizations of the same model.
logit _Y drug age _d1- _d5
Logit estimates Number of obs = 143
LR chi2(7) = 38.21
Prob > chi2 = 0.0000
Log likelihood = -55.65503 Pseudo R2 = 0.2556
------------------------------------------------------------------------------
_Y | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
drug | -3.024052 .6346992 -4.76 0.000 -4.26804 -1.780064
age | .1607128 .0514134 3.13 0.002 .0599444 .2614813
_d1 | -1.687274 1.312815 -1.29 0.199 -4.260344 .8857956
_d2 | -.7128491 1.273976 -0.56 0.576 -3.209797 1.784099
_d3 | -.7041496 1.317687 -0.53 0.593 -3.286768 1.878469
_d4 | .5506505 1.296978 0.42 0.671 -1.991381 3.092682
_d5 | .4246033 1.409744 0.30 0.763 -2.338444 3.187651
_cons | -7.622593 2.988652 -2.55 0.011 -13.48024 -1.764942
------------------------------------------------------------------------------
fitstat
Measures of Fit for logit of _Y
Log-Lik Intercept Only: -74.761 Log-Lik Full Model: -55.655
D(135): 111.310 LR(7): 38.213
Prob > LR: 0.000
McFadden's R2: 0.256 McFadden's Adj R2: 0.149
Maximum Likelihood R2: 0.234 Cragg & Uhler's R2: 0.362
McKelvey and Zavoina's R2: 0.410 Efron's R2: 0.271
Variance of y*: 5.579 Variance of error: 3.290
Count R2: 0.818 Adj Count R2: 0.161
AIC: 0.890 AIC*n: 127.310
BIC: -558.674 BIC': -3.473We can then compare the fit with a model that treats time as a linear variable.
logit _Y drug age _period
Logit estimates Number of obs = 143
LR chi2(3) = 35.96
Prob > chi2 = 0.0000
Log likelihood = -56.782549 Pseudo R2 = 0.2405
------------------------------------------------------------------------------
_Y | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
drug | -2.903144 .6025565 -4.82 0.000 -4.084133 -1.722154
age | .1480103 .0495297 2.99 0.003 .0509339 .2450867
_period | .4508214 .1868341 2.41 0.016 .0846332 .8170096
_cons | -8.849313 2.82147 -3.14 0.002 -14.37929 -3.319333
------------------------------------------------------------------------------
fitstat
Measures of Fit for logit of _Y
Log-Lik Intercept Only: -74.761 Log-Lik Full Model: -56.783
D(139): 113.565 LR(3): 35.958
Prob > LR: 0.000
McFadden's R2: 0.240 McFadden's Adj R2: 0.187
Maximum Likelihood R2: 0.222 Cragg & Uhler's R2: 0.343
McKelvey and Zavoina's R2: 0.378 Efron's R2: 0.262
Variance of y*: 5.290 Variance of error: 3.290
Count R2: 0.811 Adj Count R2: 0.129
AIC: 0.850 AIC*n: 121.565
BIC: -576.270 BIC': -21.069Based on the deviance (111.31 vs 113.565) and BIC (-558.674 vs -576.27) it appears that the model using indicator variables for time fits slightly better, although it uses four more degrees of freedom then the model with linear time.
Using a model that includes both _period and t-2 dummy coded time variables indicates that the dummy coded time does not account for significantly more variability than using linear time alone.
logit _Y drug age _d1-_d4 _period
Logit estimates Number of obs = 143
LR chi2(7) = 38.21
Prob > chi2 = 0.0000
Log likelihood = -55.65503 Pseudo R2 = 0.2556
------------------------------------------------------------------------------
_Y | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
drug | -3.024052 .6346992 -4.76 0.000 -4.26804 -1.780064
age | .1607128 .0514134 3.13 0.002 .0599444 .2614813
_d1 | -3.810291 6.209824 -0.61 0.539 -15.98132 8.36074
_d2 | -2.411262 4.800772 -0.50 0.615 -11.8206 6.998079
_d3 | -1.977959 3.441179 -0.57 0.565 -8.722546 4.766627
_d4 | -.2985561 2.099693 -0.14 0.887 -4.413879 3.816767
_period | -.4246033 1.409744 -0.30 0.763 -3.187651 2.338444
_cons | -5.074973 7.898617 -0.64 0.521 -20.55598 10.40603
------------------------------------------------------------------------------
test _d1 _d2 _d3 _d4
( 1) _d1 = 0
( 2) _d2 = 0
( 3) _d3 = 0
( 4) _d4 = 0
chi2( 4) = 2.15
Prob > chi2 = 0.7087
A Proportional Hazards ModelA discrete-time proportional hazards model can be estimated using the cloglog command.
use http://www.gseis.ucla.edu/courses/data/cancer, clear
(Patient Survival in Drug Trial)
prsnperd id distime censor
cloglog _Y drug age _d1-_d6, cluster(id) nocons
Complementary log-log regression Number of obs = 143
Zero outcomes = 112
Nonzero outcomes = 31
Wald chi2(8) = 62.45
Log likelihood = -55.701255 Prob > chi2 = 0.0000
(standard errors adjusted for clustering on id)
------------------------------------------------------------------------------
| Robust
_Y | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
drug | -2.460096 .4825704 -5.10 0.000 -3.405917 -1.514276
age | .1273736 .036167 3.52 0.000 .0564875 .1982597
_d1 | -7.770962 2.04366 -3.80 0.000 -11.77646 -3.765462
_d2 | -7.085621 2.036474 -3.48 0.000 -11.07704 -3.094204
_d3 | -7.077778 1.925394 -3.68 0.000 -10.85148 -3.304075
_d4 | -5.956626 1.963904 -3.03 0.002 -9.805807 -2.107444
_d5 | -6.079785 2.009739 -3.03 0.002 -10.0188 -2.140768
_d6 | -6.377616 2.162588 -2.95 0.003 -10.61621 -2.139022
------------------------------------------------------------------------------
glm _Y drug age _d1-_d6, fam(bin) link(clog) nocons
Generalized linear models No. of obs = 143
Optimization : ML: Newton-Raphson Residual df = 135
Scale param = 1
Deviance = 111.4025091 (1/df) Deviance = .8252038
Pearson = 132.6671546 (1/df) Pearson = .9827197
Variance function: V(u) = u*(1-u) [Bernoulli]
Link function : g(u) = ln(-ln(1-u)) [Complementary log-log]
Standard errors : OIM
Log likelihood = -55.70125456 AIC = .8909266
BIC = 71.69975209
------------------------------------------------------------------------------
_Y | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
drug | -2.460096 .4880544 -5.04 0.000 -3.416665 -1.503527
age | .1273736 .040619 3.14 0.002 .0477619 .2069853
_d1 | -7.770962 2.354981 -3.30 0.001 -12.38664 -3.155284
_d2 | -7.085621 2.266325 -3.13 0.002 -11.52754 -2.643705
_d3 | -7.077778 2.277786 -3.11 0.002 -11.54216 -2.613399
_d4 | -5.956626 2.250857 -2.65 0.008 -10.36822 -1.545026
_d5 | -6.079785 2.342926 -2.59 0.009 -10.67183 -1.487735
_d6 | -6.377616 2.474713 -2.58 0.010 -11.22796 -1.527268
------------------------------------------------------------------------------
glm, eform noheader
------------------------------------------------------------------------------
_Y | ExpB Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
drug | .0854267 .0416929 -5.04 0.000 .0328217 .2223446
age | 1.135841 .0461367 3.14 0.002 1.048921 1.229964
_d1 | .0004218 .0009933 -3.30 0.001 4.17e-06 .0426263
_d2 | .0008371 .001897 -3.13 0.002 9.85e-06 .0710974
_d3 | .0008436 .0019216 -3.11 0.002 9.71e-06 .073285
_d4 | .0025886 .0058266 -2.65 0.008 .0000314 .2133062
_d5 | .0022887 .0053622 -2.59 0.009 .0000232 .2258837
_d6 | .0016992 .004205 -2.58 0.010 .0000133 .2171281
------------------------------------------------------------------------------
There is also a program called pgmhaz (findit pgmhaz) that esitmates two
different discrete time proportional
hazards models, one of which incorporates a
gamma mixture distribution to summarize unobserved individual
heterogeneity (or "frailty").
pgmhaz drug age _d1- _d6, id(id) seq(_period) dead( _Y) nocons
(1) PGM hazard model without unobserved heterogeneity
Residual df = 135 No. of obs = 143
Pearson X2 = 132.6589 Deviance = 111.4025
Dispersion = .9826588 Dispersion = .8252038
Bernoulli distribution, cloglog link
------------------------------------------------------------------------------
_Y | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
drug | -2.460075 .4854314 -5.07 0.000 -3.411503 -1.508647
age | .1273721 .0395082 3.22 0.001 .0499374 .2048068
_d1 | -7.770891 2.310665 -3.36 0.001 -12.29971 -3.242069
_d2 | -7.085539 2.18251 -3.25 0.001 -11.36318 -2.807897
_d3 | -7.077696 2.209682 -3.20 0.001 -11.40859 -2.746799
_d4 | -5.956576 2.168435 -2.75 0.006 -10.20663 -1.706522
_d5 | -6.079717 2.260327 -2.69 0.007 -10.50988 -1.649557
_d6 | -6.377553 2.405179 -2.65 0.008 -11.09162 -1.663489
------------------------------------------------------------------------------
Log likelihood (-0.5*Deviance) = -55.701255
Cf. log likelihood for intercept-only model (Model 0) = -74.761305
Chi-squared statistic for Model (1) vs. Model (0) = 38.120101
Prob. > chi2(7) = 2.875e-06
(2) PGM hazard model with Gamma distributed unobserved heterogeneity
PGM hazard model with Gamma heterogeneity Number of obs = 143
Model chi2(8) = .
Prob > chi2 = .
Log Likelihood = -55.4864576
------------------------------------------------------------------------------
_Y | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
hazard |
drug | -2.96652 1.000103 -2.97 0.003 -4.926685 -1.006355
age | .1574424 .0688267 2.29 0.022 .0225445 .2923403
_d1 | -9.2916 3.703838 -2.51 0.012 -16.55099 -2.032211
_d2 | -8.306118 3.325512 -2.50 0.013 -14.824 -1.788235
_d3 | -8.129395 3.189794 -2.55 0.011 -14.38128 -1.877515
_d4 | -6.90202 3.039802 -2.27 0.023 -12.85992 -.9441166
_d5 | -6.970558 3.035035 -2.30 0.022 -12.91912 -1.021999
_d6 | -7.160581 3.054771 -2.34 0.019 -13.14782 -1.173341
-------------+----------------------------------------------------------------
ln_varg |
_cons | -.9778874 1.690064 -0.58 0.563 -4.290352 2.334577
------------------------------------------------------------------------------
Gamma variance, exp(ln_varg) = .37610483; Std. Err. = .63564123; z = .59169358
Likelihood ratio statistic for testing models (1) vs (2) = .42959389
Prob. test statistic > chi2(1) = .51218829
pgmhaz, eform
PGM hazard model with Gamma heterogeneity Number of obs = 143
Model chi2(8) = .
Prob > chi2 = .
Log Likelihood = -55.4864576
------------------------------------------------------------------------------
_Y | : Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
hazard |
drug | .0514821 .0514874 -2.97 0.003 .0072505 .3655489
age | 1.170513 .0805626 2.29 0.022 1.022801 1.339559
_d1 | .0000922 .0003415 -2.51 0.012 6.49e-08 .1310455
_d2 | .000247 .0008214 -2.50 0.013 3.65e-07 .1672551
_d3 | .0002947 .0009402 -2.55 0.011 5.68e-07 .1529698
_d4 | .0010058 .0030573 -2.27 0.023 2.60e-06 .3890231
_d5 | .0009391 .0028503 -2.30 0.022 2.45e-06 .3598749
_d6 | .0007766 .0023723 -2.34 0.019 1.95e-06 .3093318
-------------+----------------------------------------------------------------
ln_varg |
------------------------------------------------------------------------------
Gamma variance, exp(ln_varg) = .37610483; Std. Err. = .63564123; z = .59169358
Likelihood ratio statistic for testing models (1) vs (2) = .42959389
Prob. test statistic > chi2(1) = .51218829
Categorical Data Analysis Course
Phil Ender