Germán Rodríguez

Generalized Linear Models
Princeton University

We now turn our attention to models for *ordered* categorical
outcomes. Obviously the multinomial and sequential logit models
can be applied as well, but they make no explicit use of the fact
that the categories are ordered. The models considered here are
specifically designed for ordered data.

We will use data from 1681 residents of twelve areas in Copenhagen, classified in terms of the type of housing they have (tower blocks, apartments, atrium houses and terraced houses), their feeling of influence on apartment management (low, medium, high), their degree of contact with the neighbors (low, high), and their satisfaction with housing conditions (low, medium, high).

The data are available in the datasets page and can be read directly from there:

. use http://data.princeton.edu/wws509/datasets/copen, clear (Housing Conditions in Copenhagen)

We will treat satisfaction as the outcome and type of housing, feeling of influence and contact with the neighbors as categorical predictors.

It will be useful for comparison purposes to fit the saturated
multinomial logit model, where each of the 24 combinations of
housing type, influence and contact, has its own distribution.
The group code can easily be generated from the observation number,
and the easiest way to fit the model is to treat the code as a
*factor variable*. If you are running an earlier version of
Stata try the `xi:`

prefix.

. gen group = int((_n-1)/3)+1 . quietly mlogit satisfaction i.group [fw=n] . estimates store sat . di e(ll) -1715.7108

The log likelihood is -1715.7.

The next task is to fit the additive ordered logit model from Table 6.5 in the notes. We could use factor variables for simplicity, but will construct dummy variables instead

. gen apart = housing == 2 . gen atrium = housing == 3 . gen terrace = housing == 4 . local housing apart atrium terrace . gen influenceMed = influence == 2 . gen influenceHi = influence == 3 . local influence influenceMed influenceHi . gen contactHi = contact == 2

With that done, here's the additive model

. local housing apart atrium terrace . local influence influenceMed influenceHi . ologit satis `housing' `influence' contactHi [fw=n] Iteration 0: log likelihood = -1824.4388 Iteration 1: log likelihood = -1739.8163 Iteration 2: log likelihood = -1739.5747 Iteration 3: log likelihood = -1739.5746 Ordered logistic regression Number of obs = 1681 LR chi2(6) = 169.73 Prob > chi2 = 0.0000 Log likelihood = -1739.5746 Pseudo R2 = 0.0465 ------------------------------------------------------------------------------ satisfaction | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- apart | -.5723499 .119238 -4.80 0.000 -.8060521 -.3386477 atrium | -.3661863 .1551733 -2.36 0.018 -.6703205 -.0620522 terrace | -1.091015 .151486 -7.20 0.000 -1.387922 -.7941074 influenceMed | .5663937 .1046528 5.41 0.000 .361278 .7715093 influenceHi | 1.288819 .1271561 10.14 0.000 1.039597 1.53804 contactHi | .360284 .0955358 3.77 0.000 .1730372 .5475307 -------------+---------------------------------------------------------------- /cut1 | -.496135 .1248472 -.7408311 -.2514389 /cut2 | .6907081 .1254719 .4447876 .9366286 ------------------------------------------------------------------------------ . estimates store additive . lrtest . sat, force Likelihood-ratio test LR chi2(40) = 47.73 (Assumption: additive nested in sat) Prob > chi2 = 0.1874

The log-likelihood is -1739.6, so the deviance for this model
compared to the saturated multinomial model is 47.7 on 40 d.f.
This is a perfectly valid test because the models are nested,
but Stata is cautious and if you type `lrtest . sat`

it will complain that the
test involves different estimators: mlogit vs. ologit.
Fortunately we can insist with the `force`

option,
which is what I have done. Use cautiously!

The bottom line is that the deviance is not much more than one would expect when saving 40 parameters, so we have no evidence against the additive model. To be thorough, however, we will explore individual interactions just in case the deviance is concentrated on a few d.f.

The next step is to explore two-factor interactions. We can use factor variables to simplify our search:

. quietly ologit satis i.housing#i.influence i.contact [fw=n] . lrtest . sat, force Likelihood-ratio test LR chi2(34) = 25.22 (Assumption: . nested in sat) Prob > chi2 = 0.8623 . quietly ologit satis i.housing#i.contact i.influence [fw=n] . lrtest . sat, force Likelihood-ratio test LR chi2(37) = 39.06 (Assumption: . nested in sat) Prob > chi2 = 0.3773 . quietly ologit satis i.housing i.influence#i.contact [fw=n] . lrtest . sat, force Likelihood-ratio test LR chi2(38) = 47.52 (Assumption: . nested in sat) Prob > chi2 = 0.1385

The interaction between housing and influence reduces the deviance by about half at the expense of only six d.f., so it is worth a second look. The interaction between housing and contact makes a much smaller dent, and the interaction between influence and contact adds practically nothing. (we could have compared each of these models to the additive model, thus testing the interaction directly. We would get chi2 of 22.51 on 6 d.f., 8.67 on 3 d.f. and 0.21 on 2 d.f.)

Clearly the interaction to add is the first one, allowing the association between satisfaction with housing and a feeling of influence on management net of contact with neighbors to depend on the type of housing. To examine parameter estimates we build the dummy variables and refit the model:

. gen apartXinfMed = apart * influenceMed . gen apartXinfHi = apart * influenceHi . gen atriuXinfMed = atrium * influenceMed . gen atriuXinfHi = atrium * influenceHi . gen terrXinfMed = terrace * influenceMed . gen terrXinfHi = terrace * influenceHi . local housingXinf apartXinfMed-terrXinfHi . ologit satis `housing' `influence' `housingXinf' contactHi [fw=n] Iteration 0: log likelihood = -1824.4388 Iteration 1: log likelihood = -1728.6182 Iteration 2: log likelihood = -1728.3201 Iteration 3: log likelihood = -1728.32 Ordered logistic regression Number of obs = 1681 LR chi2(12) = 192.24 Prob > chi2 = 0.0000 Log likelihood = -1728.32 Pseudo R2 = 0.0527 ------------------------------------------------------------------------------ satisfaction | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- apart | -1.188494 .1972418 -6.03 0.000 -1.575081 -.8019072 atrium | -.6067061 .2445664 -2.48 0.013 -1.086047 -.1273647 terrace | -1.606231 .2409971 -6.66 0.000 -2.078576 -1.133885 influenceMed | -.1390175 .2125483 -0.65 0.513 -.5556044 .2775694 influenceHi | .8688638 .2743369 3.17 0.002 .3311733 1.406554 apartXinfMed | 1.080868 .2658489 4.07 0.000 .5598135 1.601922 apartXinfHi | .7197816 .3287309 2.19 0.029 .0754809 1.364082 atriuXinfMed | .65111 .3450048 1.89 0.059 -.0250869 1.327307 atriuXinfHi | -.1555515 .4104826 -0.38 0.705 -.9600826 .6489795 terrXinfMed | .8210056 .3306666 2.48 0.013 .172911 1.4691 terrXinfHi | .8446195 .4302698 1.96 0.050 .0013062 1.687933 contactHi | .372082 .0959868 3.88 0.000 .1839514 .5602126 -------------+---------------------------------------------------------------- /cut1 | -.8881686 .1671554 -1.215787 -.56055 /cut2 | .3126319 .1656627 -.012061 .6373249 ------------------------------------------------------------------------------ . lrtest . sat, force Likelihood-ratio test LR chi2(34) = 25.22 (Assumption: . nested in sat) Prob > chi2 = 0.8623 . lrtest additive . Likelihood-ratio test LR chi2(6) = 22.51 (Assumption: additive nested in .) Prob > chi2 = 0.0010

The model deviance of 25.2 on 34 d.f. is not significant. To test for the interaction effect we compare this model with the additive, obtaining a chi-squared statistic of 22.5 on six d.f., which is significant at the 0.001 level.

At this point one might consider adding a second interaction. The obvious choice is to allow the association between satisfaction and contact with neighbors to depend on the type of housing. This would reduce the deviance by 7.95 at the expense of three d.f., a gain that just makes the conventional 5% cutoff with a p-value of 0.047. In the interest of simplicity we will not pursue this addition.

The estimates indicate that respondents who have high contact with their neighbors are more satisfied than respondents with low contact who live in the same type of housing and have the same feeling of influence on management. The difference is estimated as 0.372 units in the underlying logistic scale. Dividing by the standard deviation of the (standard) logistic distribution we obtain

. display _b[contactHi]/(_pi/sqrt(3)) .20513955

So the difference in satisfaction between high and low contact with neighbors among respondents with the same housing and influence is 0.205 standard deviations.

Alternatively, we can exponentiate the coefficient:

. di exp(_b[contactHi]) 1.4507519

The odds of reporting high satisfaction (relative to medium or low), are 45% higher among respondents who have high contact with the neighbors than among tenants with low contact in the same type of housing and influence. The odds of reporting medium or high satisfaction (as opposed to low) are also 45% higher in this group.

Interpretation of the effects of housing type and influence requires taking into account the interaction effect. In the notes we describe differences by housing type among those who feel they have little influence in management, and the effects of influence in each type of housing.

Let us do something a bit different here, and focus on the joint effects of housing and influence, combining the main effects and interactions. The easiest way to do this is to refit the model omitting the main effects, which causes Stata to fold them into the interaction. We then plot them. I also divide the coefficients by π/√(3) to express them in standard deviation units.

. gen coef = . (72 missing values generated) . gen hc = . (72 missing values generated) . gen ic = . (72 missing values generated) . mata b = st_matrix("e(b)") . mata st_store(1::12, "coef", b[2::13]':/(pi()/sqrt(3))) . mata st_store(1::12, "hc", (1::4) # (1\1\1)) . mata st_store(1::12, "ic", (1\1\1\1) # (1::3)) . label values hc housing . label values ic lowmedhi . keep in 1/12 (60 observations deleted) . graph bar coef, over(ic, gap(0)) over(hc) asyvar /// > bar(1, color("221 238 255")) bar(2, color("128 170 230")) /// > bar(3, color("51 102 204")) ytitle(Satisfaction) /// > legend(ring(0) pos(7) cols(1) region(lwidth(none))) . graph export fig63.png, replace width(500) (file fig63.png written in PNG format)

Satisfaction with housing conditions is highest for residents of tower blocks who feel they have high influence, and lowest for residents of terraced houses with low influence. Satisfaction increases with influence in each type of housing, but the difference is largest for terraced houses and apartments than atrium houses and towers.

Another way to present the results is by focusing on the effects of influence within each type of housing or, alternatively, on the effects of housing type within each category of influence. All we need to do is substract the first row (or the first colum) from our predicted values:

. // I . gen coef_i = coef . replace coef_i = coef_i - coef_i[_n - ic + 1] (4 real changes made) . graph bar coef_i, over(ic, gap(0)) over(hc) asyvar /// > bar(1, color("221 238 255")) bar(2, color("128 170 230")) /// > bar(3, color("51 102 204")) ytitle(Satisfaction) /// > legend(ring(0) pos(7) cols(1) region(lwidth(none))) name(i, replace) . // H . gen coef_h = coef . replace coef_h = coef_h - coef_h[_n - 3*(hc-1)] (2 real changes made) . graph bar coef_h, over(hc, gap(0) /// > relabel(1 "Tower" 2 "Apart" 3 "Atrium" 4 "Terrace")) over(ic) asyvar /// > bar(1, color("221 238 255")) bar(2, color("157 193 238")) /// > bar(3, color("101 147 221")) bar(4, color("51 102 204")) /// > legend(rows(1) region(lwidth(none))) ytitle(Satisfaction) name(h, replace) . // both . graph combine h i, xsize(6) ysize(3) . graph export fig64.png, replace width(750) (file fig64.png written in PNG format)

On the left panel we see more clearly the differences by influence in each type of housing. As noted above having influence is good, particularly of you live in a terraced house or apartments. The right panel shows differences by type of housing within categories of influence. Tower residents are generally speaking more satisfied than residents of other types of housing, and the differences tend to be larger when influence is low.

Let us consider predicted probabilities.
Just as in multinomial logit models, the `predict`

command
computes predicted probabilities (the default) or logits.
With probabilities you need to specify an output variable for each
response category.
With logits you specify just one variable which stores the linear
predictor x'β, without the cutpoints.
Let us predict the probabilities for everyone

. predict pSatLow pSatMed pSatHigh (option pr assumed; predicted probabilities)

We'll look at these results for tower block dwellers with
little influence and with high and low contact with neighbors.
The first of these groups is, of course, the reference cell.
In the listing I add the condition `sat==1`

to
list the probabilities just once for each group:

. list contact pSatLow pSatMed pSatHigh /// > if housing==1 & influence==1 & sat==1, clean contact pSatLow pSatMed pSatHigh 1. low .2914879 .2860397 .4224724 4. high .2209308 .2642111 .5148581

We see that among tower tenants with low influence, those with high contact with their neighbors have a higher probability of high satisfaction and a lower probability of medium or low satisfaction, than those with low contact with the neighbors.

It is instructive to reproduce these calculations 'by hand'. For the reference cell all we need are the cutpoints. Remember that the model predicts cumulative probabilities, which is why we difference the results.

. scalar c1 = _b[/cut1] . scalar c2 = _b[/cut2] . di invlogit(c1), invlogit(c2)-invlogit(c1),1-invlogit(c2) .2914879 .28603966 .42247244

For the group with high contact we need to *subtract* the
corresponding coefficient from the cutpoints. The change of sign
is needed to convert coefficients from the latent variable to
the cumulative probability formulations (or from upper to
lower tails).

. scalar h1 = c1 - _b[contactHi] . scalar h2 = c2 - _b[contactHi] . di invlogit(h1), invlogit(h2)-invlogit(h1),1-invlogit(h2) .22093075 .26421111 .51485814

Results agree exactly with the outpout of the `predict`

command.

We now consider ordered probit models, starting with the additive model in Table 6.6:

. oprobit satis `housing' `influence' contactHi [fw=n] Iteration 0: log likelihood = -1824.4388 Iteration 1: log likelihood = -1739.9254 Iteration 2: log likelihood = -1739.8444 Iteration 3: log likelihood = -1739.8444 Ordered probit regression Number of obs = 1681 LR chi2(6) = 169.19 Prob > chi2 = 0.0000 Log likelihood = -1739.8444 Pseudo R2 = 0.0464 ------------------------------------------------------------------------------ satisfaction | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- apart | -.3475367 .0722909 -4.81 0.000 -.4892244 -.2058491 atrium | -.2178875 .0947661 -2.30 0.021 -.4036256 -.0321495 terrace | -.6641735 .0918 -7.24 0.000 -.8440983 -.4842487 influenceMed | .3464228 .0641371 5.40 0.000 .2207164 .4721291 influenceHi | .7829146 .0764262 10.24 0.000 .633122 .9327072 contactHi | .2223858 .0581227 3.83 0.000 .1084675 .3363042 -------------+---------------------------------------------------------------- /cut1 | -.2998279 .0761537 -.4490865 -.1505693 /cut2 | .4267208 .0764043 .2769711 .5764706 ------------------------------------------------------------------------------ . lrtest . sat, force Likelihood-ratio test LR chi2(40) = 48.27 (Assumption: . nested in sat) Prob > chi2 = 0.1734

The model has a log-likelihood of -1739.8, a little bit below that of the additive ordered logit. This is also reflected in the slightly higher deviance.

Next we add the housing by influence interaction

. oprobit satis `housing' `influence' `housingXinf' contactHi [fw=n] Iteration 0: log likelihood = -1824.4388 Iteration 1: log likelihood = -1728.7767 Iteration 2: log likelihood = -1728.6654 Iteration 3: log likelihood = -1728.6654 Ordered probit regression Number of obs = 1681 LR chi2(12) = 191.55 Prob > chi2 = 0.0000 Log likelihood = -1728.6654 Pseudo R2 = 0.0525 ------------------------------------------------------------------------------ satisfaction | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- apart | -.7280621 .1205029 -6.04 0.000 -.9642434 -.4918808 atrium | -.3720768 .1510259 -2.46 0.014 -.6680821 -.0760716 terrace | -.9789998 .1455862 -6.72 0.000 -1.264343 -.6936561 influenceMed | -.0863672 .130327 -0.66 0.508 -.3418033 .169069 influenceHi | .5164514 .1639345 3.15 0.002 .1951457 .8377571 apartXinfMed | .6600102 .1625787 4.06 0.000 .3413618 .9786586 apartXinfHi | .4479134 .1970667 2.27 0.023 .0616698 .834157 atriuXinfMed | .4108389 .2133778 1.93 0.054 -.007374 .8290517 atriuXinfHi | -.0779656 .2496472 -0.31 0.755 -.5672652 .4113339 terrXinfMed | .496378 .2016362 2.46 0.014 .1011783 .8915777 terrXinfHi | .5216698 .2587276 2.02 0.044 .0145731 1.028767 contactHi | .2284567 .0583151 3.92 0.000 .1141612 .3427522 -------------+---------------------------------------------------------------- /cut1 | -.5439821 .1023487 -.7445818 -.3433824 /cut2 | .189167 .1018442 -.0104438 .3887779 ------------------------------------------------------------------------------ . lrtest . sat, force Likelihood-ratio test LR chi2(34) = 25.91 (Assumption: . nested in sat) Prob > chi2 = 0.8387

We now have a log-likelihood of -1728.7 and a deviance of 25.9. which is almost indistinguishable from the corresponding ordered logit model.

The estimates indicate that tenants with high contact with the neighbors are 0.228 standard deviations higher in the latent satisfaction scale than tenants with low contact, who live in the same type of housing and have the same feeling of influence in management. Recall that the comparable logit estimate was 0.205.

The probabilities for the two groups compared earlier can be
computed using the `predict`

command or more
instructively 'by hand', using exactly the same code as before but
with the `normal()`

c.d.f. instead of the logistic c.d.f.
`invlogit()`

. scalar z1 = _b[/cut1] . scalar z2 = _b[/cut2] . di normal(z1), normal(z2)-normal(z1),1-normal(z2) .29322689 .28179216 .42498095 . scalar h1 = z1 - _b[contactHi] . scalar h2 = z2 - _b[contactHi] . di normal(h1), normal(h2)-normal(h1),1-normal(h2) .21992729 .26440244 .51567027

The main thing to note here is that the results are very close to the corresponding predictions based on the ordered logit model.

The third model mentioned in the lecture notes uses a complementary log-log link and has a proportional hazards interpretation. Unfortunately, this model can not be fit to ordered multinomial data using Stata. It is, of course, possible to fit c-log-log models to binary data, and proportional hazards models to survival data, as we will see in the next chapter.