Nominal-logit model for McKinney data

• The data
• Reading in the data
• Model building
• Specifying the distribution type
• Running the model

The data

• ID is the subject ID (362 subjects in total).
• HOUSING represents the housing status at the time of interview: 0 = street, 1 = community, 2 = independent.
• SECTION8 indicates the Section 8 group, with 1 representing those using Section 8 certifications, and 0 those without.
• TIME1 to TIME3 are three dummy variables for time effects, and denote whether a classification was at baseline, or at the 6-, 12- or 24-month follow-up. If at the 6 months follow-up, TIME1 = 1 and TIME2 = TIME3 =0; if at 12 months, TIME2 = 1 while TIME1 = TIME3 = 0; and at the 24-month follow-up TIME3 = 1 and TIME1 = TIME2 = 0. With this coding scheme, the baseline serves as the reference group of classification. The coding structure is shown in the table below.
• Three Section 8 by time interaction terms follow: SECT8T1 is the product of SECTION8 and TIME1, and SECT8T2 and SECTION8 the products of SECTION8 and TIME2 and TIME3 respectively.
• NOSECT8 indicates the non-Section 8 group, with 0 = no, and 1 = yes.
• TIME represents the linear time contrast. At baseline, TIME = 0, at 6 months, TIME = 1, at 12 months TIME = 2, and at 24 months TIME = 3.
• SEC8TIME is the product of SECTION8 and TIME.

I = independent housing). These logits are given below.

Logits across time by group

Time point
Group	Status	Baseline	6 months	12 months	24 months
Control	Community vs. Street	-0.49	1.13	1.88	0.130
	Independent vs. Street	-1.66	0.24	1.31	1.22
Time point
Section 8	Status	Baseline	6 months	12 months	24 months
Control	Community vs. Street	-0.07	1.10	0.19	0.64
	Independent vs. Street	-1.12	1.91	1.0	1.69
Difference	Status	Baseline	6 months	12 months	24 months
Control	Community vs. Street	0.42	-0.03	-1.69	-0.66
	Independent vs. Street	0.54	1.67	0.49	0.47

The logits clearly show the increase in community and independent housing, relative to street housing, at all follow-up time points (6, 12, and 24 months). In terms of group-related differences, these appear most pronounced at 6 months for independent housing and 12 months for community housing. While examination of these logits is instructive, the subsequent analyses will more rigorously assess the degree to which these logits vary by time and group.

In this example, one random subject effect (i.e., a random subject intercept) is assumed and the repeated housing status classifications is modeled in terms of the dummy-coded time effects (6, 12, and 24 month follow-ups compared to baseline), a group effect (section 8 versus control), and group by time interaction terms.

The model

The nominal model is part of a family of models based on the multinomial distribution, which is a generalization of the Binomial distribution. It is commonly used in to describe the probability of the outcome of n independent trials each of which leads to a success for one of c categories, with each category having a given fixed probability of success. A nominal variable has categories that cannot be ordered.

where C represents the number of categories of the outcome variable.

For the nominal model, the logistic link function is specified.

In the current example, the outcome variable HOUSING is coded 0, 1, and 2. Therefore

As we wish to use the “street” category as reference category, we select the first category as reference category and will thus be considering the two logits

For c = 2 (community housing)

and for c = 3

.

The level-2 model is

Reading in the data

To construct this model, we select New Analysis on the landing page.

thereby opening the Data page. Select the CSV file and click Open to obtain the page below.

The first step is to select an ID variable. In this case, the variable ID defines the hierarchical structure, and we select it by checking the box associated with this variable in the ID 2 line. To select predictors, the boxes in the Variables line must be checked. We opt to select them all

before clicking Update. Once Update is clicked the program automatically determines the appropriate levels of all selected variables.

With data specification now complete, model building can begin. Click on Models at the top of the page to move to the Models page.

Building the model

The first step is specifying the response variable. Select HOUSING from the drop-down list associated with the Set Response field. An unconditional model is now displayed.

The TIME variables are selected from the Level-1 Variables field by holding down the Control button while clicking on the names. Next, drag them all into the level-1 equation before releasing the mouse button. The following level-1 model appears:

Add the three SECTION8 variables (SECT8T1, SECT8T2 and SECT8T3) in the same way to the level-1 equation. Then add the variable SECTION8 to the intercept equation at level-2 to obtain the final model

Specification of outcome type and associated options

Move to the Settings page by clicking on Settings at the top of the page. When the page opens, we see that the programs has correctly suggested a nominal model as appropriate for these data. The default values of reference category and link function for the nominal distribution are displayed.

As these settings are exactly what we intend to use, we need not make additional changes here. We can proceed to run the analysis.

Running the model

Run the model by going to the Run page by clicking on the Run link at the top of the page. The Run page opens, and two options are available:

One can Save the model information to a syntax file with file extension MLCJSN for potential reuse. By default, the default file name assigned corresponds to the file name of the data file. Using this option is optional, however, and one can simply click the Run Syntax option to start the analysis.

Portions of the output file are shown below. The first part of the output file gives a description of the model specifications. This is followed by descriptive statistics for all the variables included in the model. We note that the most frequent response was in category 2, i.e. "Independent" on the nominal outcome HOUSING, while 23% indicated that they were living on the street (HOUSING = 0, the reference category here represented by the indicator variable HOUSING3). The variable HOUSING1 represents the respondents living in community housing.

Note that the order of the indicator variables for HOUSING is the opposite of the original coding of the variable: as we selected the first category as reference category, the program automatically recodes it to make the selected reference category the last category in the working data set.

Results for the model without any random effects are given next. The first eight values are those for the intercept, SECTION8, TIME1, …, SECT8T3 for response code 1 vs. code 0, the last eight are for the same predictors, but for response code 2 vs. response code 0.

The results obtained with adaptive quadrature estimation are given next.

Comparing the log-likelihood value from this analysis to one where there are no random effects clearly supports inclusion of the random subject effect (likelihood ratio ). Expressed as intraclass correlations and for community versus street and independent versus street, respectively. Thus, the subject influence is much more pronounced in terms of distinguishing independent versus street living, relative to community versus street living. This is borne out by contrasting models with separate versus a common random-effect variance across the two category contrasts (not shown) which yields a highly significant likelihood ratio favoring the model with separate variance terms.

The unit specific results follow.

In terms of significance of the fixed effects, the time effects are observed to be highly significant. With the inclusion of the Time by Section 8 interaction terms, the time effects reflect comparisons between time points for the control group (i.e., SECTION8 coded as 0). Thus, subjects in the control group show increased use of both independent and community housing relative to street housing at all three follow-ups, as compared to baseline. Similarly, due to the inclusion of the interaction terms, the Section 8 effect is the group difference at baseline (i.e., when all time effects are 0). Using a .05 cutoff, there is no statistical evidence of group differences at baseline. Turning to the interaction terms, these indicate how the two groups differ in terms of comparisons between time points. Compared to controls, the increase in community versus street housing is less pronounced for section 8 subjects at 12 months (the estimate for SECT8T2 equals –1.92 in terms of the logit), but not statistically different at 6 months (SECT8T1) and only marginally different at 24 months (SECT8T3). Conversely, as compared to controls, the increase in independent versus street housing (response code 2 vs. code 0) is more pronounced for section 8 subjects at 6 months (the estimate equals 2.00 in terms of the logit), but not statistically different at 12 or 24 months.

In terms of community versus street housing (i.e., response code 1 versus 0), there is an increase across time for the control group relative to the Section 8 group. As the statistical test indicated, these groups differ most at 12 months. For the independent versus street housing comparison (i.e., response code 2 versus 0) there is a beneficial effect of Section 8 certificates at 6 months. Thereafter, the non-significant interaction terms indicate that the control group catches up to some degree. Considering these results of the mixed-effects analysis, it is seen that both groups reduce the degree of street housing but do so in somewhat different ways. The control group subjects are shifted more towards community housing, whereas Section 8 subjects are more quickly shifted towards independent housing.

This differential effect of Section 8 certificates over time is completely missed if one simply analyzes the outcome variable as a binary indicator of street versus non-street housing (i.e., collapsing community and independent housing categories). In this case (not shown), none of the section 8 by time interaction terms are observed to be statistically significant. Thus, analysis of the three-category nominal outcome is important in uncovering the beneficial effect of Section 8 certificates.

The output also includes the intracluster correlation coefficient for both categories, and population average results, as shown below.

References

Hedeker, D. & Gibbons, R.D. (2006). Longitudinal Data Analysis. Wiley: New York.

Hedeker, D. & Gibbons, R.D. (1997). Application of random-effects pattern-mixture models for missing data in longitudinal studies. Psychological Methods, 2(1): 64 – 78.

Hough, R.L., Harmon, S., Tarke, H., Yamashiro, S., Quinlivan, R., Landau-Cox, P., Hurlburt, M.S., Wood, P.A., Milone, R., Renker, V., Crowell, A. & Morris, E., (1997). Supported independent housing: Implementation issues and solutions in the San Diego McKinney Homeless Demonstration Research Project. In W.R. Breakey & J.W. Thompson (eds.), Mentally Ill and Homeless: Special Programs for Special Needs, pp. 95-117. New York: Harwood Academic Publishers.

Hurlburt, M.S., Wood, P.A., & Hough, R.L. (1996). Providing independent housing for the homeless mentally ill: a novel approach to evaluating long-term longitudinal housing patterns, Journal of Community Psychology, 24, 291-310.