# What can the intercepts from polr be used to?

Hi,
I estimated in total 10 ordered logit regressiona using the polr function and I have 10 variables in each where most of them have many categories. So, I do not think it make sense to determine each proability. However, the intercepts itself are not so much useable (as far I understooth).
Do you have any suggestions/recommendation for what I can do? What are the options using R?
The polr is from the MASS package, just to let you know.

Run the examples from `help(polr)`, shown below in the `reprex`. To better understand them, read through the rest of the help page carefully. If still uncertain read one or both of

Agresti, A. (2002) Categorical Data. Second edition. Wiley.
Venables, W. N. and Ripley, B. D. (2002) Modern Applied Statistics with S. Fourth edition. Springer.

or a text with similar coverage. If those are not helpful, it is probably necessary to get some formal instruction. We should never use features of `R` that we don't fully understand.

``````library(MASS)
options(contrasts = c("contr.treatment", "contr.poly"))
house.plr <- polr(Sat ~ Infl + Type + Cont, weights = Freq, data = housing)
house.plr
#> Call:
#> polr(formula = Sat ~ Infl + Type + Cont, data = housing, weights = Freq)
#>
#> Coefficients:
#>    InflMedium      InflHigh TypeApartment    TypeAtrium   TypeTerrace
#>     0.5663937     1.2888191    -0.5723501    -0.3661866    -1.0910149
#>      ContHigh
#>     0.3602841
#>
#> Intercepts:
#>  Low|Medium Medium|High
#>  -0.4961353   0.6907083
#>
#> Residual Deviance: 3479.149
#> AIC: 3495.149
summary(house.plr, digits = 3)
#>
#> Re-fitting to get Hessian
#> Call:
#> polr(formula = Sat ~ Infl + Type + Cont, data = housing, weights = Freq)
#>
#> Coefficients:
#>                Value Std. Error t value
#> InflMedium     0.566     0.1047    5.41
#> InflHigh       1.289     0.1272   10.14
#> TypeApartment -0.572     0.1192   -4.80
#> TypeAtrium    -0.366     0.1552   -2.36
#> TypeTerrace   -1.091     0.1515   -7.20
#> ContHigh       0.360     0.0955    3.77
#>
#> Intercepts:
#>             Value  Std. Error t value
#> Low|Medium  -0.496  0.125     -3.974
#> Medium|High  0.691  0.125      5.505
#>
#> Residual Deviance: 3479.149
#> AIC: 3495.149
## slightly worse fit from
summary(update(house.plr, method = "probit", Hess = TRUE), digits = 3)
#> Call:
#> polr(formula = Sat ~ Infl + Type + Cont, data = housing, weights = Freq,
#>     Hess = TRUE, method = "probit")
#>
#> Coefficients:
#>                Value Std. Error t value
#> InflMedium     0.346     0.0641    5.40
#> InflHigh       0.783     0.0764   10.24
#> TypeApartment -0.348     0.0723   -4.81
#> TypeAtrium    -0.218     0.0948   -2.30
#> TypeTerrace   -0.664     0.0918   -7.24
#> ContHigh       0.222     0.0581    3.83
#>
#> Intercepts:
#>             Value  Std. Error t value
#> Low|Medium  -0.300  0.076     -3.937
#> Medium|High  0.427  0.076      5.585
#>
#> Residual Deviance: 3479.689
#> AIC: 3495.689
## although it is not really appropriate, can fit
summary(update(house.plr, method = "loglog", Hess = TRUE), digits = 3)
#> Call:
#> polr(formula = Sat ~ Infl + Type + Cont, data = housing, weights = Freq,
#>     Hess = TRUE, method = "loglog")
#>
#> Coefficients:
#>                Value Std. Error t value
#> InflMedium     0.367     0.0727    5.05
#> InflHigh       0.790     0.0806    9.81
#> TypeApartment -0.349     0.0757   -4.61
#> TypeAtrium    -0.196     0.0988   -1.98
#> TypeTerrace   -0.698     0.1043   -6.69
#> ContHigh       0.268     0.0636    4.21
#>
#> Intercepts:
#>             Value  Std. Error t value
#> Low|Medium   0.086  0.083      1.038
#> Medium|High  0.892  0.087     10.223
#>
#> Residual Deviance: 3491.41
#> AIC: 3507.41
summary(update(house.plr, method = "cloglog", Hess = TRUE), digits = 3)
#> Call:
#> polr(formula = Sat ~ Infl + Type + Cont, data = housing, weights = Freq,
#>     Hess = TRUE, method = "cloglog")
#>
#> Coefficients:
#>                Value Std. Error t value
#> InflMedium     0.382     0.0703    5.44
#> InflHigh       0.915     0.0926    9.89
#> TypeApartment -0.407     0.0861   -4.73
#> TypeAtrium    -0.281     0.1111   -2.52
#> TypeTerrace   -0.742     0.1013   -7.33
#> ContHigh       0.209     0.0651    3.21
#>
#> Intercepts:
#>             Value  Std. Error t value
#> Low|Medium  -0.796  0.090     -8.881
#> Medium|High  0.055  0.086      0.647
#>
#> Residual Deviance: 3484.053
#> AIC: 3500.053

predict(house.plr, housing, type = "p")
#>          Low    Medium      High
#> 1  0.3784493 0.2876752 0.3338755
#> 2  0.3784493 0.2876752 0.3338755
#> 3  0.3784493 0.2876752 0.3338755
#> 4  0.2568264 0.2742122 0.4689613
#> 5  0.2568264 0.2742122 0.4689613
#> 6  0.2568264 0.2742122 0.4689613
#> 7  0.1436924 0.2110836 0.6452240
#> 8  0.1436924 0.2110836 0.6452240
#> 9  0.1436924 0.2110836 0.6452240
#> 10 0.5190445 0.2605077 0.2204478
#> 11 0.5190445 0.2605077 0.2204478
#> 12 0.5190445 0.2605077 0.2204478
#> 13 0.3798514 0.2875965 0.3325521
#> 14 0.3798514 0.2875965 0.3325521
#> 15 0.3798514 0.2875965 0.3325521
#> 16 0.2292406 0.2643196 0.5064398
#> 17 0.2292406 0.2643196 0.5064398
#> 18 0.2292406 0.2643196 0.5064398
#> 19 0.4675584 0.2745383 0.2579033
#> 20 0.4675584 0.2745383 0.2579033
#> 21 0.4675584 0.2745383 0.2579033
#> 22 0.3326236 0.2876008 0.3797755
#> 23 0.3326236 0.2876008 0.3797755
#> 24 0.3326236 0.2876008 0.3797755
#> 25 0.1948548 0.2474226 0.5577225
#> 26 0.1948548 0.2474226 0.5577225
#> 27 0.1948548 0.2474226 0.5577225
#> 28 0.6444840 0.2114256 0.1440905
#> 29 0.6444840 0.2114256 0.1440905
#> 30 0.6444840 0.2114256 0.1440905
#> 31 0.5071210 0.2641196 0.2287594
#> 32 0.5071210 0.2641196 0.2287594
#> 33 0.5071210 0.2641196 0.2287594
#> 34 0.3331573 0.2876330 0.3792097
#> 35 0.3331573 0.2876330 0.3792097
#> 36 0.3331573 0.2876330 0.3792097
#> 37 0.2980880 0.2837746 0.4181374
#> 38 0.2980880 0.2837746 0.4181374
#> 39 0.2980880 0.2837746 0.4181374
#> 40 0.1942209 0.2470589 0.5587202
#> 41 0.1942209 0.2470589 0.5587202
#> 42 0.1942209 0.2470589 0.5587202
#> 43 0.1047770 0.1724227 0.7228003
#> 44 0.1047770 0.1724227 0.7228003
#> 45 0.1047770 0.1724227 0.7228003
#> 46 0.4294564 0.2820629 0.2884807
#> 47 0.4294564 0.2820629 0.2884807
#> 48 0.4294564 0.2820629 0.2884807
#> 49 0.2993357 0.2839753 0.4166890
#> 50 0.2993357 0.2839753 0.4166890
#> 51 0.2993357 0.2839753 0.4166890
#> 52 0.1718050 0.2328648 0.5953302
#> 53 0.1718050 0.2328648 0.5953302
#> 54 0.1718050 0.2328648 0.5953302
#> 55 0.3798387 0.2875972 0.3325641
#> 56 0.3798387 0.2875972 0.3325641
#> 57 0.3798387 0.2875972 0.3325641
#> 58 0.2579546 0.2745537 0.4674917
#> 59 0.2579546 0.2745537 0.4674917
#> 60 0.2579546 0.2745537 0.4674917
#> 61 0.1444202 0.2117081 0.6438717
#> 62 0.1444202 0.2117081 0.6438717
#> 63 0.1444202 0.2117081 0.6438717
#> 64 0.5583813 0.2471826 0.1944361
#> 65 0.5583813 0.2471826 0.1944361
#> 66 0.5583813 0.2471826 0.1944361
#> 67 0.4178031 0.2838213 0.2983756
#> 68 0.4178031 0.2838213 0.2983756
#> 69 0.4178031 0.2838213 0.2983756
#> 70 0.2584149 0.2746916 0.4668935
#> 71 0.2584149 0.2746916 0.4668935
#> 72 0.2584149 0.2746916 0.4668935
#>
#> Model:
#> Sat ~ Infl + Type + Cont
#>           Df    AIC     LRT   Pr(Chi)
#> <none>       3495.1
#> Infl:Type  6 3484.6 22.5093 0.0009786 ***
#> Infl:Cont  2 3498.9  0.2090 0.9007957
#> Type:Cont  3 3492.5  8.6662 0.0340752 *
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
house.plr2 <- stepAIC(house.plr, ~.^2)
#> Start:  AIC=3495.15
#> Sat ~ Infl + Type + Cont
#>
#>             Df    AIC
#> + Infl:Type  6 3484.6
#> + Type:Cont  3 3492.5
#> <none>         3495.1
#> + Infl:Cont  2 3498.9
#> - Cont       1 3507.5
#> - Type       3 3545.1
#> - Infl       2 3599.4
#>
#> Step:  AIC=3484.64
#> Sat ~ Infl + Type + Cont + Infl:Type
#>
#>             Df    AIC
#> + Type:Cont  3 3482.7
#> <none>         3484.6
#> + Infl:Cont  2 3488.5
#> - Infl:Type  6 3495.1
#> - Cont       1 3497.8
#>
#> Step:  AIC=3482.69
#> Sat ~ Infl + Type + Cont + Infl:Type + Type:Cont
#>
#>             Df    AIC
#> <none>         3482.7
#> - Type:Cont  3 3484.6
#> + Infl:Cont  2 3486.6
#> - Infl:Type  6 3492.5
house.plr2\$anova
#> Stepwise Model Path
#> Analysis of Deviance Table
#>
#> Initial Model:
#> Sat ~ Infl + Type + Cont
#>
#> Final Model:
#> Sat ~ Infl + Type + Cont + Infl:Type + Type:Cont
#>
#>
#>          Step Df  Deviance Resid. Df Resid. Dev      AIC
#> 1                               1673   3479.149 3495.149
#> 2 + Infl:Type  6 22.509347      1667   3456.640 3484.640
#> 3 + Type:Cont  3  7.945029      1664   3448.695 3482.695
anova(house.plr, house.plr2)
#> Likelihood ratio tests of ordinal regression models
#>
#> Response: Sat
#>                                        Model Resid. df Resid. Dev   Test    Df
#> 1                         Infl + Type + Cont      1673   3479.149
#> 2 Infl + Type + Cont + Infl:Type + Type:Cont      1664   3448.695 1 vs 2     9
#>   LR stat.      Pr(Chi)
#> 1
#> 2 30.45438 0.0003670555

house.plr <- update(house.plr, Hess=TRUE)
pr <- profile(house.plr)
confint(pr)
#>                    2.5 %      97.5 %
#> InflMedium     0.3616415  0.77195375
#> InflHigh       1.0409701  1.53958138
#> TypeApartment -0.8069590 -0.33940432
#> TypeAtrium    -0.6705862 -0.06204495
#> TypeTerrace   -1.3893863 -0.79533958
#> ContHigh       0.1733589  0.54792854
plot(pr)
``````

``````pairs(pr)
``````

Created on 2022-11-17 by the reprex package (v2.0.1)

Thanks, I will have a look at these books. Can you explain what these two plots shows/indicate?

Sorry, that I was to early for asking before I do my own reaserch.. as far I understooth the second plot is actually a scatterplot.. then it can be used to detect outliers, right? However the second plot I still do not know how to interpret.. the same for the first plot.

That's right, an x-vector against a y-vector, derived from the `pr` object, which is of `profile.polr`, which is a very complex object. I can't tell just from looking at it which two vectors are being plotted and wouldn't use it until I understood.

``````library(MASS)
options(contrasts = c("contr.treatment", "contr.poly"))
house.plr <- polr(Sat ~ Infl + Type + Cont, weights = Freq, data = housing)
house.plr <- update(house.plr, Hess=TRUE)
pr <- profile(house.plr)
str(pr)
#> List of 6
#>  \$ InflMedium   :'data.frame':   12 obs. of  2 variables:
#>   ..\$ z       : num [1:12] -2.582 -2.064 -1.548 -1.031 -0.515 ...
#>   ..\$ par.vals: num [1:12, 1:6] 0.297 0.351 0.405 0.459 0.512 ...
#>   .. ..- attr(*, "dimnames")=List of 2
#>   .. .. ..\$ : NULL
#>   .. .. ..\$ : chr [1:6] "InflMedium" "InflHigh" "TypeApartment" "TypeAtrium" ...
#>  \$ InflHigh     :'data.frame':   12 obs. of  2 variables:
#>   ..\$ z       : num [1:12] -2.594 -2.073 -1.552 -1.033 -0.516 ...
#>   ..\$ par.vals: num [1:12, 1:6] 0.447 0.471 0.495 0.519 0.543 ...
#>   .. ..- attr(*, "dimnames")=List of 2
#>   .. .. ..\$ : NULL
#>   .. .. ..\$ : chr [1:6] "InflMedium" "InflHigh" "TypeApartment" "TypeAtrium" ...
#>  \$ TypeApartment:'data.frame':   12 obs. of  2 variables:
#>   ..\$ z       : num [1:12] -3.07 -2.56 -2.05 -1.54 -1.03 ...
#>   ..\$ par.vals: num [1:12, 1:6] 0.572 0.57 0.569 0.568 0.568 ...
#>   .. ..- attr(*, "dimnames")=List of 2
#>   .. .. ..\$ : NULL
#>   .. .. ..\$ : chr [1:6] "InflMedium" "InflHigh" "TypeApartment" "TypeAtrium" ...
#>  \$ TypeAtrium   :'data.frame':   13 obs. of  2 variables:
#>   ..\$ z       : num [1:13] -3.09 -2.57 -2.06 -1.54 -1.03 ...
#>   ..\$ par.vals: num [1:13, 1:6] 0.562 0.562 0.563 0.563 0.564 ...
#>   .. ..- attr(*, "dimnames")=List of 2
#>   .. .. ..\$ : NULL
#>   .. .. ..\$ : chr [1:6] "InflMedium" "InflHigh" "TypeApartment" "TypeAtrium" ...
#>  \$ TypeTerrace  :'data.frame':   12 obs. of  2 variables:
#>   ..\$ z       : num [1:12] -3.07 -2.56 -2.05 -1.54 -1.03 ...
#>   ..\$ par.vals: num [1:12, 1:6] 0.566 0.566 0.566 0.566 0.566 ...
#>   .. ..- attr(*, "dimnames")=List of 2
#>   .. .. ..\$ : NULL
#>   .. .. ..\$ : chr [1:6] "InflMedium" "InflHigh" "TypeApartment" "TypeAtrium" ...
#>  \$ ContHigh     :'data.frame':   12 obs. of  2 variables:
#>   ..\$ z       : num [1:12] -2.581 -2.064 -1.548 -1.031 -0.515 ...
#>   ..\$ par.vals: num [1:12, 1:6] 0.554 0.556 0.558 0.561 0.564 ...
#>   .. ..- attr(*, "dimnames")=List of 2
#>   .. .. ..\$ : NULL
#>   .. .. ..\$ : chr [1:6] "InflMedium" "InflHigh" "TypeApartment" "TypeAtrium" ...
#>  - attr(*, "original.fit")=List of 19
#>   ..\$ coefficients : Named num [1:6] 0.566 1.289 -0.572 -0.366 -1.091 ...
#>   .. ..- attr(*, "names")= chr [1:6] "InflMedium" "InflHigh" "TypeApartment" "TypeAtrium" ...
#>   ..\$ zeta         : Named num [1:2] -0.496 0.691
#>   .. ..- attr(*, "names")= chr [1:2] "Low|Medium" "Medium|High"
#>   ..\$ deviance     : num 3479
#>   ..\$ fitted.values: num [1:72, 1:3] 0.378 0.378 0.378 0.257 0.257 ...
#>   .. ..- attr(*, "dimnames")=List of 2
#>   .. .. ..\$ : chr [1:72] "1" "2" "3" "4" ...
#>   .. .. ..\$ : chr [1:3] "Low" "Medium" "High"
#>   ..\$ lev          : chr [1:3] "Low" "Medium" "High"
#>   ..\$ terms        :Classes 'terms', 'formula'  language Sat ~ Infl + Type + Cont
#>   .. .. ..- attr(*, "variables")= language list(Sat, Infl, Type, Cont)
#>   .. .. ..- attr(*, "factors")= int [1:4, 1:3] 0 1 0 0 0 0 1 0 0 0 ...
#>   .. .. .. ..- attr(*, "dimnames")=List of 2
#>   .. .. .. .. ..\$ : chr [1:4] "Sat" "Infl" "Type" "Cont"
#>   .. .. .. .. ..\$ : chr [1:3] "Infl" "Type" "Cont"
#>   .. .. ..- attr(*, "term.labels")= chr [1:3] "Infl" "Type" "Cont"
#>   .. .. ..- attr(*, "order")= int [1:3] 1 1 1
#>   .. .. ..- attr(*, "intercept")= int 1
#>   .. .. ..- attr(*, "response")= int 1
#>   .. .. ..- attr(*, ".Environment")=<environment: R_GlobalEnv>
#>   .. .. ..- attr(*, "predvars")= language list(Sat, Infl, Type, Cont)
#>   .. .. ..- attr(*, "dataClasses")= Named chr [1:5] "ordered" "factor" "factor" "factor" ...
#>   .. .. .. ..- attr(*, "names")= chr [1:5] "Sat" "Infl" "Type" "Cont" ...
#>   ..\$ df.residual  : int 1673
#>   ..\$ edf          : int 8
#>   ..\$ n            : int 1681
#>   ..\$ nobs         : int 1681
#>   ..\$ call         : language polr(formula = Sat ~ Infl + Type + Cont, data = housing, weights = Freq,      Hess = TRUE)
#>   ..\$ method       : chr "logistic"
#>   ..\$ convergence  : int 0
#>   ..\$ niter        : Named int [1:2] 51 13
#>   .. ..- attr(*, "names")= chr [1:2] "f.evals.function" "g.evals.gradient"
#>   ..\$ lp           : Named num [1:72] 0 0 0 0.566 0.566 ...
#>   .. ..- attr(*, "names")= chr [1:72] "1" "2" "3" "4" ...
#>   ..\$ Hessian      : num [1:8, 1:8] 190.5 0 86.4 25.8 31.4 ...
#>   .. ..- attr(*, "dimnames")=List of 2
#>   .. .. ..\$ : chr [1:8] "InflMedium" "InflHigh" "TypeApartment" "TypeAtrium" ...
#>   .. .. ..\$ : chr [1:8] "InflMedium" "InflHigh" "TypeApartment" "TypeAtrium" ...
#>   ..\$ model        :'data.frame':    72 obs. of  5 variables:
#>   .. ..\$ Sat      : Ord.factor w/ 3 levels "Low"<"Medium"<..: 1 2 3 1 2 3 1 2 3 1 ...
#>   .. ..\$ Infl     : Factor w/ 3 levels "Low","Medium",..: 1 1 1 2 2 2 3 3 3 1 ...
#>   .. ..\$ Type     : Factor w/ 4 levels "Tower","Apartment",..: 1 1 1 1 1 1 1 1 1 2 ...
#>   .. ..\$ Cont     : Factor w/ 2 levels "Low","High": 1 1 1 1 1 1 1 1 1 1 ...
#>   .. ..\$ (weights): int [1:72] 21 21 28 34 22 36 10 11 36 61 ...
#>   .. ..- attr(*, "terms")=Classes 'terms', 'formula'  language Sat ~ Infl + Type + Cont
#>   .. .. .. ..- attr(*, "variables")= language list(Sat, Infl, Type, Cont)
#>   .. .. .. ..- attr(*, "factors")= int [1:4, 1:3] 0 1 0 0 0 0 1 0 0 0 ...
#>   .. .. .. .. ..- attr(*, "dimnames")=List of 2
#>   .. .. .. .. .. ..\$ : chr [1:4] "Sat" "Infl" "Type" "Cont"
#>   .. .. .. .. .. ..\$ : chr [1:3] "Infl" "Type" "Cont"
#>   .. .. .. ..- attr(*, "term.labels")= chr [1:3] "Infl" "Type" "Cont"
#>   .. .. .. ..- attr(*, "order")= int [1:3] 1 1 1
#>   .. .. .. ..- attr(*, "intercept")= int 1
#>   .. .. .. ..- attr(*, "response")= int 1
#>   .. .. .. ..- attr(*, ".Environment")=<environment: R_GlobalEnv>
#>   .. .. .. ..- attr(*, "predvars")= language list(Sat, Infl, Type, Cont)
#>   .. .. .. ..- attr(*, "dataClasses")= Named chr [1:5] "ordered" "factor" "factor" "factor" ...
#>   .. .. .. .. ..- attr(*, "names")= chr [1:5] "Sat" "Infl" "Type" "Cont" ...
#>   ..\$ contrasts    :List of 3
#>   .. ..\$ Infl: chr "contr.treatment"
#>   .. ..\$ Type: chr "contr.treatment"
#>   .. ..\$ Cont: chr "contr.treatment"
#>   ..\$ xlevels      :List of 3
#>   .. ..\$ Infl: chr [1:3] "Low" "Medium" "High"
#>   .. ..\$ Type: chr [1:4] "Tower" "Apartment" "Atrium" "Terrace"
#>   .. ..\$ Cont: chr [1:2] "Low" "High"
#>   ..- attr(*, "class")= chr "polr"
#>  - attr(*, "summary")=List of 21
#>   ..\$ coefficients : num [1:8, 1:3] 0.566 1.289 -0.572 -0.366 -1.091 ...
#>   .. ..- attr(*, "dimnames")=List of 2
#>   .. .. ..\$ : chr [1:8] "InflMedium" "InflHigh" "TypeApartment" "TypeAtrium" ...
#>   .. .. ..\$ : chr [1:3] "Value" "Std. Error" "t value"
#>   ..\$ zeta         : Named num [1:2] -0.496 0.691
#>   .. ..- attr(*, "names")= chr [1:2] "Low|Medium" "Medium|High"
#>   ..\$ deviance     : num 3479
#>   ..\$ fitted.values: num [1:72, 1:3] 0.378 0.378 0.378 0.257 0.257 ...
#>   .. ..- attr(*, "dimnames")=List of 2
#>   .. .. ..\$ : chr [1:72] "1" "2" "3" "4" ...
#>   .. .. ..\$ : chr [1:3] "Low" "Medium" "High"
#>   ..\$ lev          : chr [1:3] "Low" "Medium" "High"
#>   ..\$ terms        :Classes 'terms', 'formula'  language Sat ~ Infl + Type + Cont
#>   .. .. ..- attr(*, "variables")= language list(Sat, Infl, Type, Cont)
#>   .. .. ..- attr(*, "factors")= int [1:4, 1:3] 0 1 0 0 0 0 1 0 0 0 ...
#>   .. .. .. ..- attr(*, "dimnames")=List of 2
#>   .. .. .. .. ..\$ : chr [1:4] "Sat" "Infl" "Type" "Cont"
#>   .. .. .. .. ..\$ : chr [1:3] "Infl" "Type" "Cont"
#>   .. .. ..- attr(*, "term.labels")= chr [1:3] "Infl" "Type" "Cont"
#>   .. .. ..- attr(*, "order")= int [1:3] 1 1 1
#>   .. .. ..- attr(*, "intercept")= int 1
#>   .. .. ..- attr(*, "response")= int 1
#>   .. .. ..- attr(*, ".Environment")=<environment: R_GlobalEnv>
#>   .. .. ..- attr(*, "predvars")= language list(Sat, Infl, Type, Cont)
#>   .. .. ..- attr(*, "dataClasses")= Named chr [1:5] "ordered" "factor" "factor" "factor" ...
#>   .. .. .. ..- attr(*, "names")= chr [1:5] "Sat" "Infl" "Type" "Cont" ...
#>   ..\$ df.residual  : int 1673
#>   ..\$ edf          : int 8
#>   ..\$ n            : int 1681
#>   ..\$ nobs         : int 1681
#>   ..\$ call         : language polr(formula = Sat ~ Infl + Type + Cont, data = housing, weights = Freq,      Hess = TRUE)
#>   ..\$ method       : chr "logistic"
#>   ..\$ convergence  : int 0
#>   ..\$ niter        : Named int [1:2] 51 13
#>   .. ..- attr(*, "names")= chr [1:2] "f.evals.function" "g.evals.gradient"
#>   ..\$ lp           : Named num [1:72] 0 0 0 0.566 0.566 ...
#>   .. ..- attr(*, "names")= chr [1:72] "1" "2" "3" "4" ...
#>   ..\$ Hessian      : num [1:8, 1:8] 190.5 0 86.4 25.8 31.4 ...
#>   .. ..- attr(*, "dimnames")=List of 2
#>   .. .. ..\$ : chr [1:8] "InflMedium" "InflHigh" "TypeApartment" "TypeAtrium" ...
#>   .. .. ..\$ : chr [1:8] "InflMedium" "InflHigh" "TypeApartment" "TypeAtrium" ...
#>   ..\$ model        :'data.frame':    72 obs. of  5 variables:
#>   .. ..\$ Sat      : Ord.factor w/ 3 levels "Low"<"Medium"<..: 1 2 3 1 2 3 1 2 3 1 ...
#>   .. ..\$ Infl     : Factor w/ 3 levels "Low","Medium",..: 1 1 1 2 2 2 3 3 3 1 ...
#>   .. ..\$ Type     : Factor w/ 4 levels "Tower","Apartment",..: 1 1 1 1 1 1 1 1 1 2 ...
#>   .. ..\$ Cont     : Factor w/ 2 levels "Low","High": 1 1 1 1 1 1 1 1 1 1 ...
#>   .. ..\$ (weights): int [1:72] 21 21 28 34 22 36 10 11 36 61 ...
#>   .. ..- attr(*, "terms")=Classes 'terms', 'formula'  language Sat ~ Infl + Type + Cont
#>   .. .. .. ..- attr(*, "variables")= language list(Sat, Infl, Type, Cont)
#>   .. .. .. ..- attr(*, "factors")= int [1:4, 1:3] 0 1 0 0 0 0 1 0 0 0 ...
#>   .. .. .. .. ..- attr(*, "dimnames")=List of 2
#>   .. .. .. .. .. ..\$ : chr [1:4] "Sat" "Infl" "Type" "Cont"
#>   .. .. .. .. .. ..\$ : chr [1:3] "Infl" "Type" "Cont"
#>   .. .. .. ..- attr(*, "term.labels")= chr [1:3] "Infl" "Type" "Cont"
#>   .. .. .. ..- attr(*, "order")= int [1:3] 1 1 1
#>   .. .. .. ..- attr(*, "intercept")= int 1
#>   .. .. .. ..- attr(*, "response")= int 1
#>   .. .. .. ..- attr(*, ".Environment")=<environment: R_GlobalEnv>
#>   .. .. .. ..- attr(*, "predvars")= language list(Sat, Infl, Type, Cont)
#>   .. .. .. ..- attr(*, "dataClasses")= Named chr [1:5] "ordered" "factor" "factor" "factor" ...
#>   .. .. .. .. ..- attr(*, "names")= chr [1:5] "Sat" "Infl" "Type" "Cont" ...
#>   ..\$ contrasts    :List of 3
#>   .. ..\$ Infl: chr "contr.treatment"
#>   .. ..\$ Type: chr "contr.treatment"
#>   .. ..\$ Cont: chr "contr.treatment"
#>   ..\$ xlevels      :List of 3
#>   .. ..\$ Infl: chr [1:3] "Low" "Medium" "High"
#>   .. ..\$ Type: chr [1:4] "Tower" "Apartment" "Atrium" "Terrace"
#>   .. ..\$ Cont: chr [1:2] "Low" "High"
#>   ..\$ pc           : int 6
#>   ..\$ digits       : num 4
#>   ..- attr(*, "class")= chr "summary.polr"
#>  - attr(*, "class")= chr [1:2] "profile.polr" "profile"
``````

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