Ordered Logit and Probit Models with PROC LOGIST and PROC PROBIT

This page is part of the documentation of the course in Generalized Linear Models offered by Robert Hanneman of the Department of Sociology at the University of California, Riverside. Your comments and suggestions are welcome. You can reach me as: rhannema@wizard.ucr.edu

This page has several parts:

- The problem
- SAS code
- Baseline 1: Regression by OLS on a linear contrast of Y
- Baseline 2: Multinomial logistic model (Y treated as nominal)
- PROC LOGISTIC
- Complementary log/log link
- Logit link
- Normit link
- PROC PROBIT
- Normal link
- Logistic link
- Gompertz link

The Problem

- left the program without completion of the master's degree
- in progress toward the master's degree
- completed the master's and left the program
- in progress toward the PhD
- completed the PhD

Information on a number of predictors is obtained for each of 117 students (missing data on some variables causes the loss of more than 20, however). We will examine the predictive power of scores on the Graduate Record Examination Verbal (GREV) and mathematical (GREM) sections, the student's gender (treated as a dummy variable with males serving as the reference group), United States citizenship (with non-citizens serving as the referenee group in a dummy code), and the year that the student entered the program. This last variable serves primarily as a control for students who have not yet completed their careers due to recency. This effect is confounded with secular trends in the success probabilities of students in the program, in general.

Leaving aside the control variable, the research hypotheses hope to identify significant positive partial effects of both verbal skills (above those expected from math skills and other factors) and mathematical skills on attainment. The research hypothesis hopes to identify a lack of any effect of gender and U.S. citizenship on the probabilities of attainment, once "ability" has been taken into account.

It is pretty obvious that this exercise should not be taken very seriously because of probably incompleteness of the model, and lack of good theoretical guidance about what kind of model to fit. In part because of this lack of specification of the problem substantively, there are a large number of plausible ways to approach the data analysis.

One idea is to think about the five outcomes as representing points along a linear scale (or, of course, one could impose some other weighting scheme, e.g. 1, 1.5, 5, 7.5, 10). One could then suppose that the degree of attainment was a function the X vector. The choice of a linking function is not obvious, nor is the nature of the sampling distribution underlying the conditional central tendency of the scale. One could, however, attack the problem by fitting a range of models with differing assumptions about the distribution of Y (normal, poisson, gamma), the interval distances in the weights assigned to levels of Y, and the link connecting X to Y (direct, logistic, cumulative normal, etc.). We don't like this approach much. But, to illustrate a very crude example of it, we have coded the dependent variable 1-5, assumed that it's sampling distribution is normal, and that the link with X is direct. Then, we apply OLS with SAS proc REG to estimate parameters. A more elegant approach would be to use SAS PROC GENMOD to fit with ML for different distributions of Y and link functions. Since we don't like this approach, we won't go to the trouble to do so. The OLS linear regression model is reported below as a baseline, but is not recommended as an approach to this problem.

A second approach goes to the opposite extreme, and ignores completely the ordering of the levels of the dependent variable. A simple multi-nomial model could be fit to the five outcomes, treating them as simple alternatives. With this approach, we could constrain the effects of X on the probability of each level of Y to be identical; a more logical approach is to allow the effects of X to differ across the outcomes. SAS PROC CATMOD can be used to fit this model, using generalized logits. Since this model ignores the ordering in the data, however, it too must be treated as a baseline, and is not recommended as an approach to the problem.

A third approach conceptualizes the dependent variable as a set of sequential stages, such that each subsequent outcome depends on the attainment of the prior one. Alternatively, one might think of this as a series of events, with people being "at risk" for an event only if the prior event in the sequence has occurred. This seems a more logical approach, as the process generating the data clearly has this character: one cannot complete the PhD without first completing the Master's. But, the situation here is made quite messy by including students who have not yet completed their degrees. One might think of these as sequential stages as well. For example, one must pass through the "event" of "in progress toward the Master's" as a conditon to attaining the "event" "earned the Master's." Thinking about the problem this way leads us to see the problem as a set of equations, each predicting the probability of a a "success" at each stage among those at risk for that event. That is, we would estimate a number of equations. If we were using the logistic formulation, we might look at the log odds of being in progress toward the master's or any further attainment, relative to having dropped out without the master's; the log odds of earning the masters (or more) relative to dropping out or still being in progress toward the master's, etc. For more developed examples of estimating nested or sequential probability models, see Liao, 1994).

A fourth approach is to treat the five stages as ordered categories with different overall response probabilities, but having the same responses to differences in X. That is, higher GREV scores might be hypothesized to increase the probability of falling in the next highest category of the dependent variable. This model (implemented in a number of variations below) analyzes the cumulative probability distribution of Y as we move from low to high on the ordered categories. The "distances" between the categories are adjusted by a series of intercepts, and the effects of X on falling in the next highest category of Y, relative to all prior categories is assumed to be homogeneous across all levels of Y. This last assumption (essentially the assumption of the homogeneity of association) or "proportionality" of effects of X is tested with a diagnostic chi-square statistic. PROC LOGISTIC and PROC PROBIT in SAS can be used to fit these proportional effects models and to test the proportionality assumptions. As an alternative, we might not wish to impose the constraint of proportional effects, but still analyze the cumulative logits (or normits, or gompits). This approach, in principle, can be accomplished in PROC CATMOD by specifying that the cumulative logits rather than the default generalized logits are to be analyzed in the multinomial model (I was not, however, able to get an example of this to run). Again, Liao, 1994 (chapter 5) provides a discussion of the proportional effects ordered logit and probit models.

A final word, before turning to the output. It is probably true that none of the approaches discussed above are ideal. The outcome categories are clearly dependent on one another, and are generally sequential -- but not purely so do to the inclusion of students in progress. The inclusion of recency of admission as a predictor is an attempt "after the fact" to deal with this messyness. It is not necessarily an efficacious approach. In the original analysis of these data, two efforts were made to overcome the problem. In one, the information on length of time to event was used, and event-history regression techniques were used to deal with the censoring of outcomes. This, however, changes the research question somewhat. In the other approach, we ran two simple logistic models, following the sequential outcome logic described above. In one variant, only students with completed careers were used; in the other variant, we made the assumption that students in progress toward a particular degree would attain it. The results of the various approaches, while hardly identical, were sufficiently similar to be convincing about the main patterns of effects (which were, in fact, somewhat surprising, and did not support the research hypotheses).

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SAS code

proc format ; value afmt 1='noma' 2='maip' 3='maonly' 4='phdip' 5='phd' ; data grads; input id sex race enterage uscit priorma priorsoc priorgpa ucrug toefl grev grem enteryr macen phdcen ninter nfcomp nfspec gpa grea matime phdtime tmatime tphdtime mai maii phdi phdii attain; format attain afmt. ; cards; 8 1 1 30 1 1 1 3.42 0 . 700 770 92 1 1 0 . . . 580 1 . 1 . . 1 . . 2 1 1 1 22 1 0 1 3.42 0 . 520 620 92 1 1 0 . . . 660 1 . 1 . . 1 . . 2 116 2 3 53 1 1 1 4. 0 . 540 410 78 0 0 1 0 1 3.61 320 13 21 . . 1 1 1 1 5 117 1 1 38 1 1 0 3.7 0 . 650 660 87 1 1 0 . . 3.17 610 . . 4 . 0 0 . . 1 ; data job ; set grads; options linesize=80 compress=yes nocenter; gender=0; if sex=1 then gender=0; else gender=1; racein=0; if race>1 and uscit=1 then racein=1; else racein=0; if race>1 and uscit=0 then raceout=1; else raceout=0; gre=grev+grem+grea; proc sort ; by attain ; proc reg ; model attain=gender grev grem enteryr uscit ; proc catmod data=job ; direct grev grem ; model attain=gender grev grem uscit; proc logistic order=data ; model attain=gender grev grem enteryr uscit/ link=cloglog ; proc logistic order=data ; model attain=gender grev grem enteryr uscit / link=logit ; proc logistic order=data ; model attain=gender grev grem enteryr uscit/ link=normit ; proc probit order=data ; class attain ; model attain=gender grev grem enteryr uscit / d=normal; proc probit order=data ; class attain ; model attain=gender grev grem enteryr uscit / d=logistic; proc probit order=data ; class attain ; model attain=gender grev grem enteryr uscit / d=gompertz; run;

The data are then sorted from low to high on the dependent variable, ATTAIN. In the various models that follow, we order SAS to order the categories of the dependent variable according to the order in which they are read from the data set (ORDER=DATA). These steps are one way (there are others) to make sure that SAS PROC LOGISTIC and PROC PROBIT are actually ordering the categories of the dependent variable for analysis as we intended.

The various logistic models analyze the cumulative log odds of categories relative to the last level of the variable (that is, attained the PhD). Three alternative link functions are examined, with the second (logit) being the "conventional" ordered logistic model and the "normit" being the cumulative normal link. The various probit models analyze the cumulative normal probability function from "no MA" to "no MA or MA in progress" etc. Again, three alternative linking functions are examined: the first d=normal is the normal "probit" model; the second model imposes a logistic link, and the third a gomperz link.

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Baseline 1: Regression by OLS on a linear Y

Dependent Variable: ATTAIN Analysis of Variance Sum of Mean Source DF Squares Square F Value Prob>F Model 5 105.85684 21.17137 15.882 0.0001 Error 90 119.97650 1.33307 C Total 95 225.83333 Root MSE 1.15459 R-square 0.4687 Dep Mean 2.95833 Adj R-sq 0.4392 C.V. 39.02831 Parameter Estimates Parameter Standard T for H0: Variable DF Estimate Error Parameter=0 Prob > |T| INTERCEP 1 23.622594 2.56256668 9.218 0.0001 GENDER 1 0.222260 0.24749930 0.898 0.3716 GREV 1 -0.002833 0.00123905 -2.287 0.0246 GREM 1 0.000254 0.00141177 0.180 0.8576 ENTERYR 1 -0.225816 0.02819551 -8.009 0.0001 USCIT 1 0.138575 0.42538609 0.326 0.7454

In the former regard, these results suggest a direct relationship between the X vector and the five stages of the graduate career, where these stages are treated as interval. The r-square is statistically significant, and suggests about 45% of the variance accounted for. The precision of the regression (RMSE and CV) appear fairly reasonable, though hardly impressive. An examination of residuals, however, would have shown very severe problems, and the sums of squares, in particular are quite suspect.

Even given all of that, we form the (very tentative) impression that perhaps female gender is, if anything, an advantage; as is U.S. citizenship. The affects, however, look quite weak. More recent admissions to the program have not had as high a level of attainment. This clearly reflects the nature of the data (with censored observations included) and may or may not reflect any secular tendency in attaiment probabilities. Math ability that is higher or lower than would be expected from the other independent variables appears to have no effect on attainment; unusual verbal ability appears to have the perverse effect of retarding attainment.

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CATMOD PROCEDURE Response: ATTAIN Response Levels (R)= 5 Weight Variable: None Populations (S)= 95 Data Set: JOB Total Frequency (N)= 96 Frequency Missing: 21 Observations (Obs)= 96 POPULATION PROFILES Sample Sample GENDER USCIT GREV GREM Size ------------------------------------------- 1 0 0 260 700 1 2 0 0 400 780 1 3 0 0 430 620 1 . . 93 1 1 700 520 1 94 1 1 730 600 1 95 1 1 740 540 1 RESPONSE PROFILES Response ATTAIN ---------------- 1 noma 2 maip 3 maonly 4 phdip 5 phd MAXIMUM-LIKELIHOOD ANALYSIS Sub -2 Log Convergence Parameter Estimates Iteration Iteration Likelihood Criterion 1 2 3 ------------------------------------------------------------------------------ 0 0 309.01208 1.0000 0 0 0 1 0 278.85294 0.0976 -4.3961 -5.4860 -4.5222 5 0 276.07723 9.412E-10 -3.7968 -4.5767 -5.7760 Parameter Estimates Iteration 4 5 6 7 8 9 --------------------------------------------------------------------------- 0 0 0 0 0 0 0 1 -3.8980 0.0900 -0.4065 -0.0122 -0.3984 0.008763 5 -3.3262 0.0971 -0.3227 0.3275 -0.4092 0.008562 Parameter Estimates Iteration 10 11 12 13 14 15 --------------------------------------------------------------------------- 0 0 0 0 0 0 0 1 0.008040 0.007623 0.002578 -0.000062 0.002586 0.000663 5 0.007488 0.0112 0.001638 -0.000647 0.001796 -0.000438 Parameter Estimates Iteration 16 17 18 19 20 -------------------------------------------------------------------- 0 0 0 0 0 0 1 0.003844 0.4723 0.3534 1.0175 0.0401 5 0.003799 0.5620 0.4500 1.6499 -0.005137 MAXIMUM-LIKELIHOOD ANALYSIS-OF-VARIANCE TABLE Source DF Chi-Square Prob -------------------------------------------------- INTERCEPT 4 5.01 0.2859 GENDER 4 4.27 0.3702 GREV 4 9.97 0.0409 GREM 4 1.53 0.8205 USCIT 4 6.37 0.1729 LIKELIHOOD RATIO 360 273.30 0.9998 ANALYSIS OF MAXIMUM-LIKELIHOOD ESTIMATES Standard Chi- Effect Parameter Estimate Error Square Prob ---------------------------------------------------------------- INTERCEPT 1 -3.7968 2.3860 2.53 0.1115 2 -4.5767 2.3686 3.73 0.0533 3 -5.7760 3.4117 2.87 0.0905 4 -3.3262 2.5562 1.69 0.1932 GENDER 5 0.0971 0.3160 0.09 0.7586 6 -0.3227 0.3156 1.05 0.3065 7 0.3275 0.4795 0.47 0.4947 8 -0.4092 0.3447 1.41 0.2351 GREV 9 0.00856 0.00354 5.86 0.0155 10 0.00749 0.00340 4.85 0.0276 11 0.0112 0.00468 5.70 0.0169 12 0.00164 0.00365 0.20 0.6537 GREM 13 -0.00065 0.00377 0.03 0.8638 14 0.00180 0.00368 0.24 0.6253 15 -0.00044 0.00501 0.01 0.9304 16 0.00380 0.00403 0.89 0.3464 USCIT 17 0.5620 0.6520 0.74 0.3887 18 0.4500 0.6113 0.54 0.4617 19 1.6499 0.7312 5.09 0.0240 20 -0.00514 0.6678 0.00 0.9939

To interpret the coefficients, we must remember that the logits here are formed by the frequencies of the first outcome (dropped out without MA) to the last (completed PhD); the second outcome (in progress toward the MA) to the last (completed PhD), etc. We see some evidence of non-proportional effects. The coefficients for gender differ across the four logits in sign, but none are significant. Mathematics test scores also display differences in sign, with none significant. Being a U.S. citizen appears to increase the odds of being in progress toward the PhD compared to having attained the degree, but does not matter otherwise (I interpret this effect as one due to high non-resident tuition charges forcing non-citizens onto the job market more quickly than citizens). Finally, we note that the perverse partial effect of GRE verbal scores is moderately homogeneous across the logits (although it is not significant on the distinction between being in progress toward the PhD versus having achieved it).

The multinomial model is also only a baseline model, in that it does not take the ordering of the categories of the dependent variable into account. We turn now to a number of variations on the proportional effects model for cumulative logits -- which are explicitly ordinal.

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Response Variable: ATTAIN Response Levels: 5 Number of Observations: 96 Link Function: Complementary log-log Response Profile Ordered Value ATTAIN Count 1 noma 22 2 maip 25 3 maonly 8 4 phdip 17 5 phd 24 Score Test for the Equal Slopes Assumption Chi-Square = 229.8320 with 15 DF (p=0.0001) Model Fitting Information and Testing Global Null Hypothesis BETA=0 Intercept Intercept and Criterion Only Covariates Chi-Square for Covariates AIC 305.258 245.000 . SC 315.516 268.080 . -2 LOG L 297.258 227.000 70.258 with 5 DF (p=0.0001) Score . . 55.394 with 5 DF (p=0.0001) Analysis of Maximum Likelihood Estimates Parameter Standard Wald Pr > Standardized Variable DF Estimate Error Chi-Square Chi-Square Estimate INTERCP1 1 -27.4623 3.8850 49.9669 0.0001 . INTERCP2 1 -26.2261 3.8467 46.4836 0.0001 . INTERCP3 1 -25.7676 3.8231 45.4279 0.0001 . INTERCP4 1 -24.7244 3.7326 43.8749 0.0001 . GENDER 1 -0.3047 0.2629 1.3425 0.2466 -0.117275 GREV 1 0.00339 0.00135 6.2714 0.0123 0.280696 GREM 1 -0.00038 0.00148 0.0667 0.7962 -0.028804 ENTERYR 1 0.2805 0.0416 45.4039 0.0001 0.940938 USCIT 1 -0.2842 0.4301 0.4366 0.5088 -0.073674 Association of Predicted Probabilities and Observed Responses Concordant = 76.2% Somers' D = 0.527 Discordant = 23.4% Gamma = 0.530 Tied = 0.4% Tau-a = 0.415 (3589 pairs) c = 0.764

A test is performed to examine the viability of the proportionality or equal slopes assumption. The null hypothesis is that of proportionality, and we see that we can be confident in rejecting that idea that effects are proportional. That is, we really ought to stop at this point, and not move on to the parameters. But, as this is an exercise, we will continue.

To interpret the parameters, one must know what logits are being examined. Here, the logits are composed by comparing the odds of the first category to the last; the first and second to the last; the first, second, and third to the last; etc. That is, cumulative logits. A "common sense" interpretation of parameters then is that parameters show the effects of a unit of X on the log odds of falling below a certain score versus achieving a certain score.

The multiple intercept terms are not normally directly interpreted, although one could use them to create the equation describing each cumulative logit, and to calculate the response probabilities. The slope coefficients for the independent variables describe effects on the log odds of failing to achieve a given point in the attainment continuum. Gender (female) reduces the odds of "failure" or, conversely, increases the odds of success, albeit not in a statistically significant way. Similarly, mathematical test scores and U.S. citizenship appear to have (non-significant) effects in reducing the odds of failure to attain. More recent entrants to the program have higher odds of falling lower on the attainment scale; and the perverse partial effect of verbal ability again surfaces in this analysis.

Most frequently, we would go no further than noting the directions and significance of effects. If we did wish to go further, several approaches are possible. A fairly reasonable thing to do is to calculate the expected cumulative logits at the sample mean values of the independent variables (or for a baseline "type" of case) and contrast the difference in the expected cumulative logits as each X score is varied by one unit (e.g. when we consider males instead of female, holding GREV, GREM, USCIT, ENTERYR constant at sample means or a baseline). The differences between the calculated logits are an "elasticity" of a sort, and can be compared across the variables to get a sense of the relative magnitudes of effects.

In this model, we specified that the link function was to be the complementary log-log formulation. Several other possible links might be examined, including the more common logistic and cumulative normal links.

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Link Function: Logit Score Test for the Proportional Odds Assumption Chi-Square = 174.9428 with 15 DF (p=0.0001) Model Fitting Information and Testing Global Null Hypothesis BETA=0 Intercept Intercept and Criterion Only Covariates Chi-Square for Covariates AIC 305.258 262.634 . SC 315.516 285.713 . -2 LOG L 297.258 244.634 52.625 with 5 DF (p=0.0001) Score . . 41.430 with 5 DF (p=0.0001) Analysis of Maximum Likelihood Estimates Parameter Standard Wald Pr > Standardized Odds Variable DF Estimate Error Chi-Square Chi-Square Estimate Ratio INTERCP1 1 -32.9038 5.3282 38.1359 0.0001 . . INTERCP2 1 -31.3751 5.2477 35.7459 0.0001 . . INTERCP3 1 -30.7907 5.2124 34.8956 0.0001 . . INTERCP4 1 -29.3892 5.1183 32.9703 0.0001 . . GENDER 1 -0.4927 0.4080 1.4588 0.2271 -0.134123 0.611 GREV 1 0.00471 0.00211 4.9874 0.0255 0.275859 1.005 GREM 1 -0.00068 0.00232 0.0853 0.7702 -0.036258 0.999 ENTERYR 1 0.3387 0.0570 35.2937 0.0001 0.803180 1.403 USCIT 1 -0.3279 0.6933 0.2236 0.6363 -0.060097 0.720 Association of Predicted Probabilities and Observed Responses Concordant = 76.6% Somers' D = 0.535 Discordant = 23.2% Gamma = 0.536 Tied = 0.2% Tau-a = 0.421 (3589 pairs) c = 0.767

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Link Function: Normit Score Test for the Equal Slopes Assumption Chi-Square = 101.2743 with 15 DF (p=0.0001) Model Fitting Information and Testing Global Null Hypothesis BETA=0 Intercept Intercept and Criterion Only Covariates Chi-Square for Covariates AIC 305.258 263.665 . SC 315.516 286.744 . -2 LOG L 297.258 245.665 51.594 with 5 DF (p=0.0001) Score . . 39.060 with 5 DF (p=0.0001) Analysis of Maximum Likelihood Estimates Parameter Standard Wald Pr > Standardized Variable DF Estimate Error Chi-Square Chi-Square Estimate INTERCP1 1 -19.6980 2.9166 45.6143 0.0001 . INTERCP2 1 -18.8136 2.8807 42.6523 0.0001 . INTERCP3 1 -18.4728 2.8663 41.5345 0.0001 . INTERCP4 1 -17.6519 2.8265 39.0014 0.0001 . GENDER 1 -0.2451 0.2396 1.0469 0.3062 -0.121013 GREV 1 0.00277 0.00123 5.0567 0.0245 0.294168 GREM 1 -0.00033 0.00137 0.0573 0.8108 -0.031761 ENTERYR 1 0.2030 0.0313 42.1107 0.0001 0.873398 USCIT 1 -0.1450 0.4090 0.1257 0.7230 -0.048203 Association of Predicted Probabilities and Observed Responses Concordant = 76.6% Somers' D = 0.534 Discordant = 23.2% Gamma = 0.535 Tied = 0.2% Tau-a = 0.420 (3589 pairs) c = 0.767

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PROC PROBIT: Normal link

Class Level Information Class Levels Values ATTAIN 5 noma maip maonly phdip phd Number of observations used = 96 Data Set =WORK.JOB Dependent Variable=ATTAIN Weighted Frequency Counts for the Ordered Response Categories Level Count noma 22 maip 25 maonly 8 phdip 17 phd 24 Observations with Missing Values= 21 Log Likelihood for NORMAL -122.8322946 Variable DF Estimate Std Err ChiSquare Pr>Chi Label/Value INTERCPT 1 -19.698018 3.011221 42.79171 0.0001 Intercept GENDER 1 -0.2451083 0.240532 1.038413 0.3082 GREV 1 0.00277196 0.001228 5.093648 0.0240 GREM 1 -0.0003277 0.00139 0.055579 0.8136 ENTERYR 1 0.20303478 0.0324 39.26857 0.0001 USCIT 1 -0.1449933 0.403835 0.12891 0.7196 INTER.2 1 0.88437332 0.154668 maip INTER.3 1 1.22515554 0.179669 maonly INTER.4 1 2.04603752 0.236945 phdip

It is clear at a glance that the main patterns of this model are the same as the logistics examined above. Interpretation of the parameters beyond direction, size, and significance is again probably best done by calculating elasticities. The only difference between caluculating elasticities from the logit rather than the probit model is that one calculates the expected probits for a given X (say, the baseline case where all variables are at sample means), and then consults a table of Z scores to determine the probability of each outcome (that is, the probability of not getting an MA versus getting a PhD; the probability of not getting an MA or having one in progress versus getting a PhD, etc.).

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Class Level Information Class Levels Values ATTAIN 5 noma maip maonly phdip phd Weighted Frequency Counts for the Ordered Response Categories Level Count noma 22 maip 25 maonly 8 phdip 17 phd 24 Log Likelihood for LOGISTIC -122.316893 Probit Procedure Variable DF Estimate Std Err ChiSquare Pr>Chi Label/Value INTERCPT 1 -32.904096 5.313054 38.35407 0.0001 Intercept GENDER 1 -0.4927412 0.410173 1.443127 0.2296 GREV 1 0.00471487 0.002056 5.261113 0.0218 GREM 1 -0.0006785 0.002419 0.078669 0.7791 ENTERYR 1 0.33865984 0.056532 35.88773 0.0001 USCIT 1 -0.327885 0.676406 0.234978 0.6279 INTER.2 1 1.52865774 0.276664 maip INTER.3 1 2.11305694 0.324375 maonly INTER.4 1 3.51462307 0.439274 phdip

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Log Likelihood for GOMPERTZ -113.500196 Variable DF Estimate Std Err ChiSquare Pr>Chi Label/Value INTERCPT 1 -27.462548 3.980509 47.59972 0.0001 Intercept GENDER 1 -0.3046536 0.262168 1.350376 0.2452 GREV 1 0.00339238 0.001404 5.834209 0.0157 GREM 1 -0.0003812 0.001514 0.063352 0.8013 ENTERYR 1 0.28054145 0.043464 41.66055 0.0001 USCIT 1 -0.2842309 0.423537 0.45036 0.5022 INTER.2 1 1.23624474 0.223105 maip INTER.3 1 1.69472946 0.256916 maonly INTER.4 1 2.73800891 0.336051 phdip

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