8.2 - Baseline-Category Logit Model

Goal:

Give a simultaneous representation (summary) of the odds of being in one category relative to being in a designated category, called the baseline category, for all pairs of categories. This is an extension of binary logistic regression model, where we will consider \(r − 1\) non-redundant logits.

Variables:

Suppose that a response variable \(Y\),

\(y_i=(y_{i1},y_{i2},\ldots,y_{ir})^T \),

has a multinomial distribution with index \(n_i=\sum_{j=1}^r y_{ij}\) and parameter

\(\pi_i=(\pi_{i1},\pi_{i2},\ldots,\pi_{ir})^T \).

A set of explanatory variables \(x_i = (x_{i1}, x_{i2}, \ldots, x_{ip})\) can be discrete, continuous, or a combination of both. When the explanatory/predictor variables are all categorical, the baseline category logit model has an equivalent loglinear model.

Model:

When the response categories \(1, 2,\ldots, r\) are unordered, the most popular way to relate \(\pi_i\) to covariates is through a set of \(r − 1\) baseline-category logits. Taking \(j^*\) as the baseline category, the model is

\(\log\left(\dfrac{\pi_{ij}}{\pi_{ij^\ast}}\right)=x_i^T \beta_j,\qquad j \neq j^\ast\)

Note here that \(x_i\), which has length \(p\), represents the vector of terms needed to include all the predictors and any interactions of interest. We may also take the intercept to represent a predictor \(x_{i1}=1\), so that may be included with this notation as well. The coefficient vector is then

\(\beta_j=(\beta_{1j},\beta_{2j},\ldots,\beta_{pj})^T,\qquad j \neq j^\ast\)

We can choose the baseline or let the software program do it automatically.

Questions: How many logits of the model are there for the study of the effects of the radiation exposure to mortality? Can you write these models down if the baseline level is "being alive"?

Interpretation of Parameter Estimates

The \(k^{th}\) element of \(\beta_j\) can be interpreted as the increase in log-odds of falling into category \(j\) versus category \(j^*\) resulting from a one-unit increase in the \(k^{th}\) predictor term, holding the other terms constant.

Removing the \(k^{th}\) term from the model is equivalent to simultaneously setting \(r − 1\) coefficients to zero.

Any of the categories can be chosen to be the baseline. The model will fit equally well, achieving the same likelihood and producing the same fitted values. Only the values and interpretation of the coefficients will change. However to develop a sensible model a 'natural' baseline is chosen. If the response is ordinal, usually the highest or the lowest category in the ordinal scale is chosen.

To calculate the \(\pi_{ij}\) values, we have for the non-baseline categories (\(j \ne j^*\))

\(\pi_{ij}=\dfrac{\exp(x_i^T \beta_j)}{1+\sum_{k \neq j^\ast}\exp(x_i^T \beta_k)}\)

And for the baseline-category, the probability is

\(\pi_{ij^\ast}=\dfrac{1}{1+\sum_{k \neq j^\ast}\exp(x_i^T \beta_k)}\)

Note that as expected, we have \(\sum_{j=1}^r\pi_{ij}=1\).

Goodness of Fit

If the estimated expected counts \(\hat{\mu}_{ij}=n_i \hat{\pi}_{ij}\) are large enough, we can test the fit of our model versus a saturated model that estimates \(\pi\) independently for \(i = 1,\ldots, N\). The deviance for comparing this model to a saturated one is (if data are grouped)

\(G^2=2\sum\limits_{i=1}^N \sum\limits_{j=1}^r y_{ij}\log\dfrac{y_{ij}}{\hat{\mu}_{ij}}\)

The saturated model has \(N(r − 1)\) free parameters, and the current model (i.e., the model we are fitting) has \(p(r − 1)\), where \(p\) is the length of \(x_i\), so the degrees of freedom are

\(df=(N-p)(r-1)\)

The corresponding Pearson statistic is

\(X^2=\sum\limits_{i=1}^N \sum\limits_{j=1}^r r^2_{ij}\)

where

\(r_{ij}=\dfrac{y_{ij}-\hat{\mu}_{ij}}{\sqrt{\hat{\mu}_{ij}}}\)

is the Pearson residual.

Under the null hypothesis that the current model is correct, both statistics are approximately distributed as \(\chi^2_{df}\) provided that

no more than 20% of the \(\hat{\mu}_{ij}\)'s are below 5.0, and
none are below 1.0.

In practice, this is often not satisfied, so there may be no way to assess the overall fit of the model. However, we may still apply a \(\chi^2\) approximation to \(G^2\) when comparing nested models, provided that \((N − p)(r − 1)\) is large relative to the difference in the number of parameters between them.

Overdispersion

As before, overdispersion means that the actual covariance matrix of \(Y_i\) exceeds that specified by the multinomial model,

\(V(Y_i)=n_i \left[\text{Diag}(\pi_i)-\pi_i \pi_i^T\right]\)

The expression above utilizes the format for a multivariate dispersion matrix. \(\pi_i\) denotes the vector of probabilities and \(\pi_{i}^{T}\) denotes the transpose of the vector. It is simply another way of writing the dispersion matrix instead of writing it element-wise.

It is reasonable to consider overdispersion if

the data are grouped (\(n_i\)'s are greater than 1),
\(x_i\) already contains all covariates worth considering, and
the overall \(X^2\) is substantially larger than its degrees of freedom \((N − p)(r − 1)\).

In this situation, it may be worthwhile to introduce a scale parameter \(\sigma^2\), so that

\(V(Y_i)=n_i \sigma^2 \left[\text{Diag}(\pi_i)-\pi_i \pi_i^T\right]\)

Recall that the usual estimate for \(\sigma^2\) is obtained from the maximal model (not saturated) by dividing the Pearson chi-square statistic with its degrees of freedom:

\(\hat{\sigma}^2=\dfrac{X^2}{(N-p)(r-1)}\)

which is approximately unbiased if \((N − p)(r − 1)\) is large. Introducing a scale parameter does not alter the estimate of \(\beta\) (which then becomes a quasi-likelihood estimate), but it does alter our estimate of the variability of \(\hat{\beta}\). If we estimate a scale parameter, we should

multiply the estimated ML covariance matrix for \(\hat{\beta}\) by \(\hat{\sigma}^2\) (SAS does this automatically; in R, it depends which function we use);
divide the usual Pearson residuals by \(\hat{\sigma}^2\); and
divide the usual \(X^2\) and \(G^2\) statistics by \(\hat{\sigma}^2\) (SAS reports these as scaled statistics; in R, it depends which function we use.).

These adjustments will have little practical effect unless the estimated scale parameter is substantially greater than 1.0 (say, 1.2 or higher).

8.2.1 - Example: Housing Satisfaction in SAS

Example: Housing Data (using SAS)

In this section, we will fit the baseline-category logit model to the data below via PROC LOGISTIC with LINK=GLOGIT in SAS. The model can be also fitted by using PROC GENMOD (see the reference link provided at the top page of this lesson).

The data in "housing.dat" gives information on housing conditions from a survey in Copenhagen. Variables are satisfaction (low, medium, high), perceived influence on management (low, medium, high), type of housing (tower, atrium, apartment, terrace), contact with other residents (low, high), and the frequency count of individuals in that row.

When the data are grouped, as they are in this example, SAS expects the response categories \(1, 2, \ldots , r\) to appear in a single column of the dataset, with another column containing the frequency or count as they are in this case. The lines that have a frequency of zero are not actually used in the modeling, because they contribute nothing to the log-likelihood; as such, we can omit them if we wish.

Let's describe these data by a baseline-category model, with Satisfaction as the outcome (baseline chosen as "medium") and other variables are predictors. Note that with this model, we treat the satisfaction levels as nominal; we'll revisit this data set with satisfaction treated as ordinal in the next section.

We let \(\pi_1\) = probability of low satisfaction, \(\pi_2\) = probability of high satisfaction, and \(\pi_3\) = probability of medium satisfaction so that the equations are

\(\log\left(\dfrac{\pi_j}{\pi_3}\right)=\beta_{0j}+\beta_{1j}x_1+\beta_{2j}x_2+\cdots\)

for \(j = 1,2\). The \(x\)'s represent the predictor terms, including any interactions we wish to fit as well. Note that these predictor terms do not depend on \(j\). Also, treating the predictors as nominal requires

two indicators for perceived influence on management
three indicators for type of housing
one indicator for contact with other residents

With first-order terms and all two-way interactions, this gives us a total of \((3-1)(1+6+11)=36\) parameters (1 intercept, 6 first-order terms, and 11 interaction terms for each of the two non-baseline satisfaction response categories)!

Specifying the Model

Note! In the model statement, we need to tell SAS about the existence of a count or frequency variable; otherwise, SAS will assume that the data are ungrouped, with each line representing a single observation. We also need to specify which of the categories is the baseline, e.g., model sat (ref="Medium") sets medium satisfaction as the baseline. The link function is glogit, for generalized logit.

To get fit statistics, we include the options aggregate and scale=none. To adjust for overdispersion, we would use scale=pearson.

data housing;
input Sat $ Infl $ Type $ Cont $ Freq;
cards;
     Low    Low     Tower  Low   21
  Medium    Low     Tower  Low   21
    High    Low     Tower  Low   28
     Low Medium     Tower  Low   34
  Medium Medium     Tower  Low   22
    High Medium     Tower  Low   36
     Low   High     Tower  Low   10
  Medium   High     Tower  Low   11
    High   High     Tower  Low   36
    Low    Low Apartment  Low   61
 Medium    Low Apartment  Low   23
   High    Low Apartment  Low   17
    Low Medium Apartment  Low   43
 Medium Medium Apartment  Low   35
   High Medium Apartment  Low   40
    Low   High Apartment  Low   26
 Medium   High Apartment  Low   18
   High   High Apartment  Low   54
    Low    Low    Atrium  Low   13
 Medium    Low    Atrium  Low    9
   High    Low    Atrium  Low   10
    Low Medium    Atrium  Low    8
 Medium Medium    Atrium  Low    8
   High Medium    Atrium  Low   12
    Low   High    Atrium  Low    6
 Medium   High    Atrium  Low    7
   High   High    Atrium  Low    9
    Low    Low   Terrace  Low   18
 Medium    Low   Terrace  Low    6
   High    Low   Terrace  Low    7
    Low Medium   Terrace  Low   15
 Medium Medium   Terrace  Low   13
   High Medium   Terrace  Low   13
    Low   High   Terrace  Low    7
 Medium   High   Terrace  Low    5
   High   High   Terrace  Low   11
    Low    Low     Tower High   14
 Medium    Low     Tower High   19
   High    Low     Tower High   37
    Low Medium     Tower High   17
 Medium Medium     Tower High   23
   High Medium     Tower High   40
    Low   High     Tower High    3
 Medium   High     Tower High    5
   High   High     Tower High   23
    Low    Low Apartment High   78
 Medium    Low Apartment High   46
   High    Low Apartment High   43
    Low Medium Apartment High   48
 Medium Medium Apartment High   45
   High Medium Apartment High   86
    Low   High Apartment High   15
 Medium   High Apartment High   25
   High   High Apartment High   62
    Low    Low    Atrium High   20
 Medium    Low    Atrium High   23
   High    Low    Atrium High   20
    Low Medium    Atrium High   10
 Medium Medium    Atrium High   22
   High Medium    Atrium High   24
    Low   High    Atrium High    7
 Medium   High    Atrium High   10
   High   High    Atrium High   21
    Low    Low   Terrace High   57
 Medium    Low   Terrace High   23
   High    Low   Terrace High   13
    Low Medium   Terrace High   31
 Medium Medium   Terrace High   21
   High Medium   Terrace High   13
    Low   High   Terrace High    5
 Medium   High   Terrace High    6
   High   High   Terrace High   13
; run;

* two-way interactions;
proc logistic data=housing;
freq freq;
class infl (ref="Low") type cont / param=ref;
model sat (ref="Medium") =  infl type cont 
 infl*type infl*cont type*cont /link=glogit 
 aggregate=(infl type cont) scale=none;
run;

* additive model;
proc logistic data=housing;
freq freq;
class infl (ref="Low") type cont / param=ref;
model sat (ref="Medium") =  infl type cont /link=glogit 
 aggregate=(infl type cont) scale=none;
run;

* type and cont only;
proc logistic data=housing;
freq freq;
class type cont / param=ref;
model sat (ref="Medium") = type cont /link=glogit 
 aggregate=(infl type cont) scale=none;
run;

* infl and cont only;
proc logistic data=housing;
freq freq;
class infl (ref="Low") cont / param=ref;
model sat (ref="Medium") = infl cont /link=glogit 
 aggregate=(infl type cont) scale=none;
run;

* infl and type only;
proc logistic data=housing;
freq freq;
class infl (ref="Low") type / param=ref;
model sat (ref="Medium") = infl type /link=glogit 
 aggregate=(infl type cont) scale=none;
run;

* intercept only;
proc logistic data=housing;
freq freq;
class / param=ref;
model sat (ref="Medium") = /link=glogit 
 aggregate=(infl type cont) scale=none;
run;

Output

Here is the output pertaining to the "response profile" that gives us three levels of the satisfaction response and clarifying the baseline as "Medium". For example, there are 668 individuals who responded with "High" satisfaction.


Response Profile
Ordered Value	Sat	Total Frequency
1	High	668
2	Low	567
3	Medium	446

Here is the output pertaining to the goodness of fit.


Deviance and Pearson Goodness-of-Fit Statistics
Criterion	Value	DF	Value/DF	Pr > ChiSq
Deviance	5.9443	12	0.4954	0.9189
Pearson	5.9670	12	0.4973	0.9177

Number of unique profiles: 24

There are \(N = 24\) profiles (unique combinations of the predictors) in this dataset. Recall that this part of the output tests the fit of the current model versus the saturated model. The saturated model, which fits a separate multinomial distribution to each profile, has \(24\cdot2 = 48\) parameters. Compared with the current model, which has 36 parameters to include up to two-way interactions only, this gives 12 degrees of freedom for the test.

Stop and Think!

Does this model fit well?

The model fits well because both the residual deviance and Pearson test statistics are small, relative to their chi-square distributions; both p-values are large and indicate very little evidence of lack of fit.

Output pertaining to the significance of the predictors:


Joint Tests
Effect	DF	Wald Chi-Square	Pr > ChiSq
Infl	4	13.2983	0.0099
Type	6	25.3474	0.0003
Cont	2	5.7270	0.0571
Infl*Type	12	20.8975	0.0519
Infl*Cont	4	0.8772	0.9278
Type*Cont	6	9.4081	0.1519

Note:Under full-rank parameterizations, Type 3 effect tests are replaced by joint tests. The joint test for an effect is a test that all the parameters associated with that effect are zero. Such joint tests might not be equivalent to Type 3 effect tests under GLM parameterization.

The first section (global null hypothesis) tests the fit of the current model against the null or intercept-only model. The null model has two parameters (one-intercept for each non-baseline equation). Therefore, the comparison has \(36 - 2 = 34\) degrees of freedom. This test is highly significant, indicating that at least one of the predictors has an effect on satisfaction.

The next section (Type III analysis of effects) shows the change in fit resulting from discarding any one of the predictors—influence, type, or contact—while keeping the others in the model. For example, the test for influence is equivalent to setting its two indicator coefficients equal to zero in each of the two logit equations; so the test for significance of influence has \(2\cdot2=4\) degrees of freedom. Judging from these tests, we see that

Perceived influence on management depends on the type of housing to a degree (interaction), but it's borderline insignificant, given that other terms are already in the model. Other interaction terms are relatively weak.
Both influence and housing type are strongly related to satisfaction, while contact is borderline insignificant.

This suggests that we may get a more efficient model by removing one or more interaction terms. Below are the goodness of fit statistics for the additive model (no interaction terms). This seems to fit reasonably well and has considerably fewer parameters: \((3-1)(1+6)=14\). We'll retain this additive model and interpret the parameter estimates next.

Interpreting Parameter Estimates

Let's look at the estimated coefficients of the current model that contains only the main effects:


Analysis of Maximum Likelihood Estimates
Parameter		Sat	DF	Estimate	Standard Error	Wald Chi-Square	Pr > ChiSq
Intercept		High	1	0.2805	0.1662	2.8473	0.0915
Intercept		Low	1	0.4192	0.1729	5.8768	0.0153
Infl	High	High	1	0.9477	0.1681	31.8016	<.0001
Infl	High	Low	1	-0.6649	0.1863	12.7338	0.0004
Infl	Medium	High	1	0.2885	0.1448	3.9704	0.0463
Infl	Medium	Low	1	-0.4464	0.1416	9.9443	0.0016
Type	Apartmen	High	1	-0.2999	0.1563	3.6834	0.0550
Type	Apartmen	Low	1	0.4357	0.1725	6.3769	0.0116
Type	Atrium	High	1	-0.5393	0.1996	7.3033	0.0069
Type	Atrium	Low	1	-0.1314	0.2231	0.3467	0.5560
Type	Terrace	High	1	-0.7458	0.2105	12.5494	0.0004
Type	Terrace	Low	1	0.6666	0.2063	10.4445	0.0012
Cont	High	High	1	0.1210	0.1293	0.8752	0.3495
Cont	High	Low	1	-0.3609	0.1324	7.4284	0.0064

The SAS System

The LOGISTIC Procedure


Model Information
Data Set	WORK.HOUSING
Response Variable	Sat
Number of Response Levels	3
Frequency Variable	Freq
Model	generalized logit
Optimization Technique	Newton-Raphson


Odds Ratio Estimates
Effect	Sat	Point Estimate	95% Wald Confidence Limits
Infl High vs Low	High	2.580	1.856	3.586
Infl High vs Low	Low	0.514	0.357	0.741
Infl Medium vs Low	High	1.334	1.005	1.772
Infl Medium vs Low	Low	0.640	0.485	0.845
Type Apartmen vs Tower	High	0.741	0.545	1.006
Type Apartmen vs Tower	Low	1.546	1.102	2.168
Type Atrium vs Tower	High	0.583	0.394	0.862
Type Atrium vs Tower	Low	0.877	0.566	1.358
Type Terrace vs Tower	High	0.474	0.314	0.717
Type Terrace vs Tower	Low	1.948	1.300	2.918
Cont High vs Low	High	1.129	0.876	1.454
Cont High vs Low	Low	0.697	0.538	0.904

How do we interpret them? Recall that there are two logit equations involved: 1) the log-odds of low satisfaction versus medium and 2) the log-odds of high satisfaction versus medium. This baseline reference is in addition to any baselines involved in the predictors. For the two indicators used for perceived influence, the baseline corresponds to "low". So, from the output, the estimated coefficient for high influence is interpreted as follows: for those with high perceived influence on management, the odds of high satisfaction (versus medium) is estimated to be \(e^{.9477}=2.58\) times as high as that for those with low influence. As with other glms we've dealt with to this point, we may change the baselines arbitrarily, which changes the interpretations of the numerical values reported but not the significance of the predictor (all levels together) or the fit of the model as a whole.

The intercepts give the estimated log-odds for the baseline groups. For example, the estimated log-odds of high satisfaction (versus medium) is \(e^{.2805}=1.32\) for those with low perceived influence, housing type tower, and low contact with other residents.

Although the above output shows all three predictor main effects are significantly related to satisfaction, we may consider the same model selection techniques here as we used earlier for binary logistic regression.

Model Selection

In the main effects model, the one we fitted above, the Wald statistics reported indicate that each of the three predictors is highly significant, given the others. We can also address this by comparing deviances, full model versus reduced model; we just have to be careful of the number of parameters since each effect we test applies to two (in general, \(r-1\)) logit models. The current (additive) model has deviance \(G^2=38.6622\) with 34 degrees of freedom versus the saturated model as we noted above. If we remove the influence indicators, the deviance increases to \(G^2=147.7797\) with 38 degrees of freedom, and the LRT statistic for comparing these two models would be

\(147.7797 - 38.6622=109.1175\)

which can be compared against a chi-square distribution with \(38-34=4\) degrees of freedom (p-value approximately 0). So, the effect of influence is considered (highly) significant. Similarly, we can compare any two models in this way, provided one is a special case of the other. Repeating the model-fitting for various sets of predictors, we obtain the following analysis-of-deviance table:

Model	\(G^2\)	df
Saturated	0.00	0
Infl + Type + Cont	38.6622	34
Type + Cont	147.7797	38
Infl + Cont	100.8891	40
Infl + Type	54.7219	36
...	...	...
Null	217.4560	46

We can refine the model by removing terms if doing so does not significantly reduce the fit.

Finally, an adjustment for overdispersion could also be implemented as it was with binary logistic regression. The assumption is that the multinomial variance assumed in the model is off by a scale parameter, which we can estimate from the Pearson goodness of fit test. For the additive model, this would be

\(\dfrac{X^2}{df}=\dfrac{38.9104}{34}=1.144\)

Since this is so close to 1, we would not opt to include such an adjustment in this case, but if we did, the effect would be to multiply all variance and standard error estimates by \(1.144\) and \(\sqrt{1.144}\), respectively.

8.2.2 - Example: Housing Satisfaction in R

Example: Housing Data (using R)

For R, there are a number of different packages and functions we can use. For example, vglm() from VGAM package, or multinom() from nnet package, or mlogit() from globaltest package from BIOCONDUCTOR; see the links at the first page of these lecture notes and the later examples. We will focus on vglm().

The vglm function expects the response categories \(1,2,\ldots,r\) to appear in separate columns, with rows corresponding to predictor combinations, much like the glm function. When the data are stacked as they are in this example, we can use the "spread" function to separate them. The lines that have a frequency of zero are not actually used in the modeling, because they contribute nothing to the log-likelihood; as such, we can omit them if we wish.

library(MASS)
library(tidyr)
data(housing)
housing = spread(housing, Sat, Freq)

The result is below. Note also that there are 24 rows corresponding to the unique combinations of the predictors.

> housing
     Infl      Type Cont Low Medium High
1     Low     Tower  Low  21     21   28
2     Low     Tower High  14     19   37
3     Low Apartment  Low  61     23   17
4     Low Apartment High  78     46   43
5     Low    Atrium  Low  13      9   10
6     Low    Atrium High  20     23   20
7     Low   Terrace  Low  18      6    7
8     Low   Terrace High  57     23   13
9  Medium     Tower  Low  34     22   36
10 Medium     Tower High  17     23   40
11 Medium Apartment  Low  43     35   40
12 Medium Apartment High  48     45   86
13 Medium    Atrium  Low   8      8   12
14 Medium    Atrium High  10     22   24
15 Medium   Terrace  Low  15     13   13
16 Medium   Terrace High  31     21   13
17   High     Tower  Low  10     11   36
18   High     Tower High   3      5   23
19   High Apartment  Low  26     18   54
20   High Apartment High  15     25   62
21   High    Atrium  Low   6      7    9
22   High    Atrium High   7     10   21
23   High   Terrace  Low   7      5   11
24   High   Terrace High   5      6   13

We let \(\pi_1\) = probability of low satisfaction, \(\pi_2\) = probability of high satisfaction, and \(\pi_3\) = probability of medium satisfaction so that the equations are

\(\log\left(\dfrac{\pi_j}{\pi_3}\right)=\beta_{0j}+\beta_{1j}x_1+\beta_{2j}x_2+\cdots\)

two indicators for perceived influence on management
three indicators for type of housing
one indicator for contact with other residents

With first-order terms and all two-way interactions, this gives us a total of \((3-1)(1+6+11)=36\) parameters (1 intercept, 6 first-order terms, and 11 i

Specifying the Model

The syntax of the vglm function is very similar to that of the glm function, but note that the last of the response categories is taken as the baseline by default. If we want to use "Medium" as the baseline, we need to place that last. Also, note the family is "multinomial".

library(tidyr)
library(VGAM)
library(MASS)

data(housing)
# see also https://rdrr.io/cran/MASS/man/housing.html
# baseline category logit model for nominal response

# unstacking the satisfaction response
housing = spread(housing, Sat, Freq)

# default baseline in R is last category
# all two-way interactions
fit.full = vglm(cbind(Low,High,Medium) ~ (Infl+Type+Cont)^2, family=multinomial, data=housing)
summary(fit.full)

# deviance test for lack of fit
g2 = deviance(fit.full)
df = df.residual(fit.full)
1 - pchisq(g2, df)

# pearson test for lack of fit
e = residuals(fit.full, type='pearson')
x2 = sum(e^2)
1 - pchisq(x2, df)

# significance tests based on LR statistics
anova(fit.full, type=3, test="LR")

# additive model
fit.add = vglm(cbind(Low,High,Medium) ~ Infl+Type+Cont, family=multinomial, data=housing)
summary(fit.add)
deviance(fit.add)
exp(coef(fit.add))
exp(confint(fit.add))
anova(fit.add, type=3, test="LR")
e.add = residuals(fit.add, type='pearson')
x2.add = sum(e.add^2)
df.add = df.residual(fit.add)
1 - pchisq(x2.add, df.add)

# type and cont only
fit.tc = vglm(cbind(Low,High,Medium) ~ Type+Cont, family=multinomial, data=housing)
deviance(fit.tc)

# infl and cont only
fit.ic = vglm(cbind(Low,High,Medium) ~ Infl+Cont, family=multinomial, data=housing)
deviance(fit.ic)

# infl and type only
fit.it = vglm(cbind(Low,High,Medium) ~ Infl+Type, family=multinomial, data=housing)
deviance(fit.it)

# intercept only
fit0 = vglm(cbind(Low,High,Medium) ~ 1, family=multinomial, data=housing)
deviance(fit0)

# stepwise algorithm using AIC
step4vglm(fit.full, direction='backward')

With \(N = 24\) profiles (unique combinations of the predictors) for each of the two baseline logit equations, the saturated model has \(24\cdot2 = 48\) parameters. Compared with the current model, which has 36 parameters to include up to two-way interactions only, this gives 12 degrees of freedom for the goodness of fit test.

> # deviance test for lack of fit
g2 = deviance(fit.full)
df = df.residual(fit.full)
1 - pchisq(g2, df)

# pearson test for lack of fit
e = residuals(fit.full, type='pearson')
x2 = sum(e^2)
1 - pchisq(x2, df)

Stop and Think!

Does this model fit well?

Output pertaining to the significance of the predictors:

> anova(fit.full, type=3, test="LR")

          Df Deviance Resid. Df Resid. Dev  Pr(>Chi)    
Infl       4  13.9044        16     19.849 0.0076066 ** 
Type       6  26.2749        18     32.219 0.0001979 ***
Cont       2   5.8487        14     11.793 0.0536990 .  
Infl:Type 12  21.0581        24     27.002 0.0495363 *  
Infl:Cont  4   0.8841        16      6.828 0.9268361    
Type:Cont  6   9.2898        18     15.234 0.1579255

These results are based on the likelihood-ratio tests and are very similar to SAS's Wald Type 3 statistics. They show the change in fit resulting from discarding any one of the predictors—influence, type, or contact—while keeping the others in the model. For example, the test for influence is equivalent to setting its two indicator coefficients equal to zero in each of the two logit equations; so the test for significance of influence has \(2\cdot2=4\) degrees of freedom. Judging from these tests, we see that

Perceived influence on management depends on the type of housing to a degree (interaction), but it's only borderline significant, given that other terms are already in the model. Other interaction terms are relatively weak.
Both influence and housing type are strongly related to satisfaction, while contact is borderline insignificant.

Interpreting Parameter Estimates

Let's look at the estimated coefficients of the current model that contains only the main effects:

 >summary(fit.add)

Call:
vglm(formula = cbind(Low, High, Medium) ~ Infl + Type + Cont, 
    family = multinomial, data = housing)

Coefficients: 
                Estimate Std. Error z value Pr(>|z|)    
(Intercept):1     0.4192     0.1729   2.424 0.015342 *  
(Intercept):2     0.2805     0.1662   1.687 0.091525 .  
InflMedium:1     -0.4464     0.1416  -3.153 0.001613 ** 
InflMedium:2      0.2885     0.1448   1.993 0.046306 *  
InflHigh:1       -0.6649     0.1863  -3.568 0.000359 ***
InflHigh:2        0.9477     0.1681   5.639 1.71e-08 ***
TypeApartment:1   0.4357     0.1725   2.525 0.011562 *  
TypeApartment:2  -0.2999     0.1563  -1.919 0.054955 .  
TypeAtrium:1     -0.1314     0.2231  -0.589 0.555980    
TypeAtrium:2     -0.5393     0.1996  -2.702 0.006883 ** 
TypeTerrace:1     0.6666     0.2063   3.232 0.001230 ** 
TypeTerrace:2    -0.7458     0.2105  -3.543 0.000396 ***
ContHigh:1       -0.3609     0.1324  -2.726 0.006420 ** 
ContHigh:2        0.1210     0.1293   0.936 0.349522    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Names of linear predictors: log(mu[,1]/mu[,3]), log(mu[,2]/mu[,3])
Residual deviance: 38.6622 on 34 degrees of freedom
Log-likelihood: -118.8993 on 34 degrees of freedom

How do we interpret this? If we jump to toward the lower part of the output, we see that that there are two logit models (i.e., linear predictors) being fitted and that this is a baseline logit model with the last category as the baseline level; this is "Medium" since we specified the response as cbind(Low,High,Medium) in the function call.

Names of linear predictors: log(mu[,1]/mu[,3]), log(mu[,2]/mu[,3])

The two logit equations predict the log-odds of

low satisfaction versus medium (1 vs 3)
high satisfaction versus medium (2 vs 3)

This baseline reference for the response is in addition to any baselines involved in the predictors. For the two indicators used for perceived influence, the baseline corresponds to "low". So, from the output, the estimated coefficient for high influence is interpreted as follows: for those with high perceived influence on management, the odds of high satisfaction (versus medium) is estimated to be \(e^{.9477}=2.58\) times as high as that for those with low influence. As with other glms we've dealt with to this point, we may change the baselines arbitrarily, which changes the interpretations of the numerical values reported but not the significance of the predictor (all levels together) or the fit of the model as a whole.

The intercepts give the estimated log-odds for the baseline groups. For example, the estimated log-odds of high satisfaction (versus medium) is \(e^{.2805}=1.32\) for birds versus fish in this group is −2.4633 for those with low perceived influence, housing type tower, and low contact with other residents.

To get odds ratios, we exponentiate the fitted model coefficients. Note the (rounded) value 2.58 for InflHigh : 2 corresponds to the odds ratio we found above.

> exp(coef(fit.add))
  (Intercept):1   (Intercept):2 
      1.5207882       1.3237730 
   InflMedium:1    InflMedium:2 
      0.6399304       1.3343807 
     InflHigh:1      InflHigh:2 
      0.5143068       2.5797584 
TypeApartment:1 TypeApartment:2 
      1.5460274       0.7408604 
   TypeAtrium:1    TypeAtrium:2 
      0.8768930       0.5831281 
  TypeTerrace:1   TypeTerrace:2 
      1.9475467       0.4743750 
     ContHigh:1      ContHigh:2 
      0.6970822       1.1285968

And to add 95% confidence limits, we can run

> exp(confint(fit.add))
                    2.5 %    97.5 %
(Intercept):1   1.0835940 2.1343757
(Intercept):2   0.9557058 1.8335924
InflMedium:1    0.4848852 0.8445523
InflMedium:2    1.0047350 1.7721807
InflHigh:1      0.3569534 0.7410253
InflHigh:2      1.8558064 3.5861247
TypeApartment:1 1.1024451 2.1680907
TypeApartment:2 0.5453913 1.0063859
TypeAtrium:1    0.5662886 1.3578612
TypeAtrium:2    0.3943523 0.8622705
TypeTerrace:1   1.2999442 2.9177698
TypeTerrace:2   0.3140002 0.7166607
ContHigh:1      0.5377582 0.9036100
ContHigh:2      0.8759248 1.4541555

The likelihood-ratio tests for main effects for the additive model can be found with the anova function:

> anova(fit.add, type=3, test="LR")

     Df Deviance Resid. Df Resid. Dev  Pr(>Chi)    
Infl  4  109.117        38    147.780 < 2.2e-16 ***
Type  6   62.227        40    100.889 1.586e-11 ***
Cont  2   16.060        36     54.722 0.0003256 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Model Selection

In the main effects model, the one we fitted above, the LR statistics reported indicate that each of the three predictors is highly significant, given the others. For example, the current (additive) model has deviance \(G^2=38.6622\) with 34 degrees of freedom versus the saturated model as we noted above. If we remove the influence indicators, the deviance increases to \(G^2=147.7797\) with 38 degrees of freedom, and the LRT statistic for comparing these two models would be

\(147.7797 - 38.6622=109.1175\)

Model	\(G^2\)	df
Saturated	0.00	0
Infl + Type + Cont	38.6622	34
Type + Cont	147.7797	38
Infl + Cont	100.8891	40
Infl + Type	54.7219	36
...	...	...
Null	217.4560	46

We can refine the model by removing terms if doing so does not significantly reduce the fit.

To compare models that are not of the form full versus reduced, we can use an information criterion, such as AIC. For example, to compare the models with only Type and Cont against the model with only Infl and Cont, the AIC values would be 366.9161 and 316.0256, respectively, in favor of the latter (lower AIC). Note that these differ from the values reported by SAS because of how the software packages differently handle constants in the likelihoods; their ordering is the same, however.

\(\dfrac{X^2}{df}=\dfrac{38.9104}{34}=1.144\)

> e.add = residuals(fit.add, type='pearson')
> x2.add = sum(e.add^2)
> df.add = df.residual(fit.add)
> x2.add/df.add
[1] 1.144424

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