Two-level Bernoulli and Binomial models for Thailand data

Bernoulli model with logit link function
• The data
• Reading in the data
• Model building
• Specifying the distribution type
• Running the model
Binomial model with logit link function
• The data
• Reading in the data
• Model building
• Specifying the distribution type
• Running the model

Bernoulli model with logit link function

The data

Data are from a national survey of primary education in Thailand conducted in 1988, and yielding, for our analysis, complete data on 7516 sixth graders nested within 356 primary schools. Of interest is the probability that a child will repeat a grade during the primary years. The first few lines of the comma separated values (CSV) file thai_all_u.csv are shown below.

Inspection of the outcome variable REP1 shows that 86% of the students did not repeat a grade, and 14% did. Roughly half of students had pre-primary education. The mean SES of schools ranged between -0.77 and 1.49, with mean approximately equal to zero.

The model

It is hypothesized that the child’s gender, pre-primary experience and the school mean SES will be associated with the probability of repetition. Every level-1 record corresponds to a student, with a single binary outcome per student, so the model type is Bernoulli. The sampling distribution of the Bernoulli distribution is

and the variance is


Inspection of the outcome variable shows that 86% of the students did not repeat a grade, and 14% did. Roughly half of students had pre-primary education. The mean of schools ranged between -0.77 and 1.49, with mean approximately equal to zero.
Using the logit link function, the model we consider is:

The model for the means can be expressed as

The model we intend to fit can be formulated as


Reading in the data

To start a new analysis, select the New Analysis option on the landing page.

The Data page will open. As a first step, select the data file thai_all_u.csv and click Open to obtain the image shown below. Note that the data file may reside on a local hard drive, OneDrive, or Google Drive. A description of the variables contained in the data file are given below the image.

The following information is available:
• REP1 indicates whether a child repeated a grade (1 if yes, 0 if no).
• MALE = 1 if the student is male and 0 if female.
• If a student had pre-primary experience, PPED = 1. For students without pre-primary experience , PPED= 0.
• At the school level, the school mean socio-economic status is represented by the variable
The first step is to indicate the hierarchical structure by selecting ID variable(s). This being two-level data, we need only select SCHOOLID by checking the box associated with this variable in the ID 2 line of the second table. Next, we check the boxes for all other variables in the Variables line as we intend to use them all in the model.

Finally, we click update to prompt the program to automatically allocate the variables at the appropriate levels. When complete, the page updates to

Data file specification complete, we can now move to model building. Click on Models on the main menu bar to move on to the Models page.

Model building

Model building starts with selection of the outcome variable using the Set Response field. In this case, the outcome variable is REP1. Select the variable as shown below.

The page automatically updates to display an unconditional model.

The next step is to include predictors in the model. At level-1, we want to include the predictors MALE and PPED as uncentered predictors. Open the Level-1 Variables field, and while holding the Control button down, click on the two names. By default, these will be entered as individual uncentered predictors as shown in the image below.

Drag these variables into the level-1 equation before releasing the mouse button. The model is updated again, and the level-1 model specification is now complete.

At level-2, the mean socio-economic status of the school has to be added as a grand-mean centered variable. Open the Level-2 variables field and select the variable MSESC, taking care to click grand centered before dragging this variable into the intercept equation.

The screen updates to the model shown below. This is the model we intend to fit, so model building is now complete and we can move on to the Settings page, where we can specify the distribution type and link function. To do so, click on Settings at the top of the model page.


Specifying the distribution type and link function

The program suggests a distribution type based on its reading of the data. For these data, it suggests the use of a Bernoulli model. The default link function for the Bernoulli model is the logit link function. As these suggested settings match our intentions, we make no changes on the Settings page. Analysis specification is now complete, and all that remains is to run the model.


Running the model

To run the model, we move to the Run page, accessed by clicking on Run at the top of the window. When this page opens, only two options are available: saving the information on the model built in a MLCJSN file and running the syntax.

By default, the program will assign the same file name to the MLCJSN file as that of the data file read in, in this case thai_all_u. To save this analysis for potential reuse, we opt to save it using the Save option under the name thai_bern.mlcjsn before clicking Run Syntax to instruct the program to perform the analysis.

For this model, the following results were obtained. The model specifications and descriptive statistics are given first. Note that descriptive statistics are given for both categories of the outcome variable (rep11 and rep12).


This is followed by results and fit statistics for the model without any random effects.

In the next section, fit statistics at convergence are reported:

For models with non-continuous outcomes marginal (population average) or conditional (subject-specific or unit specific) estimates can be obtained.
• The population average estimates are estimates of the regression coefficient for a person-level predictor which assumes the baseline value is the same for all individuals (i.e., averages over all the individuals).
• In contrast, the unit specific estimates represent the expected value of the regression coefficient for a person-level predictor assuming the effects of predictors vary across individuals and allowing individuals to have a different baseline value (effects are conditioned on the intercept variance).
Typically, the population average results are reported as these results represent an average over the sample and are thus more appropriate for estimating the predicted population proportion.
The first set of results are the unit specific estimates.

The results for random effects in the model follow next. The program automatically also reports an intra-cluster correlation coefficient if there is no more than one random effect at each level of the hierarchy.

It is assumed that the level-1 error variance is equal to π²/₃ for the logit link function if the model is true (see, e.g., Hedeker & Gibbons (2006), p. 157). Using this approximation, the formulae for the standard ICCs can be adjusted.

This is followed by the second set of final results, namely the population average results.

The two sets of results are very similar apart from the estimate of the intercept coefficient. In the population average model the intercept represents the expected log-odds of repetition for a person with values of zero on the predictors (and therefore, for a female without pre-primary experience attending a school of average SES. In the unit specific results, the intercept is the expected log-odds of repetition rate for the same kind of student, but one who attends a school that not only has a mean SES of 0, but also has a random effect of zero (that is, a school with a “typical” repetition rate for the school of its type).
The estimated intercept in the population average model is -1.7501, which is the average logit. The estimated coefficients associated with gender is 0.4411, which indicates that the male respondents (MALE = 1) have a larger . The estimate for the indicator of pre-primary education shows that students with pre-primary education have a lower value.
To describe the ’s in a more accessible way to readers of reports, we need the link functions to transform them into probabilities.
First, we substitute the regression weights and obtain the function for

Recall that the mean value of MSESC over schools was zero. We can thus use the expression above to calculate

for the four groups of students formed by the cross-classification of gender by pre-primary education at a mean value of MSESC.

Group
Males with pre-primary education	-1.8255
Males without pre-primary education	-1.3090
Females with pre-primary education	-2.2666
Females without pre-primary education	-1.7482

Next, we transform the ’s into corresponding probabilities by using the logit link function:


Group
Males with pre-primary education	0.1388
Males without pre-primary education	0.2127
Females with pre-primary education	0.0939
Females without pre-primary education	0.1480

For students from a school with mean SES the probability of repeating a grade was highest for males without pre-primary education and lowest for females with pre-primary education.
In the table below, we consider the impact of the mean SES, by calculating the probabilities for the lowest and highest values of MSESC.
Males with pre-primary education

MSESC	Group
MSESC = -0.77	0.1628
	Males without pre-primary education	0.2458
	Females with pre-primary education	0.1112
	Females without pre-primary education	0.1736
MSESC = 1.49	Males with pre-primary education	0.1007
	Males without pre-primary education	0.1581
	Females with pre-primary education	0.0672
	Females without pre-primary education	0.1080

Students from schools with higher mean values were less likely to repeat a grade and females, regardless of their pre-primary education, were less likely than males to repeat a grade regardless of their school’s MSESC. This conclusion is also clear when we look at the odds ratios, where being male was associated with the highest odds ratio (1.5544).

Binomial model with logit link function

The data

We again use the data from a national survey of primary education in Thailand (see Raudenbush & Bhumirat, 1992, for details), conducted in 1988. We again focus on the probability that a child will repeat a grade during the primary years.
The first few lines of the data file GTHAI1.CSV are shown below. When the contents of this file is compared to that used for fitting the Bernoulli model, we see that in this case there seems to be fewer records associated with each SCHOOLID. A new variable, TRIAL, also appears in this file.

The variable TRIAL reports the number of observed repetitions within each of four cells, formed by the cross-tabulation of the two binary variables MALE and PPED. For each school, there may be a maximum of 4 lines of data, representing (MALE, PPED) = (0, 1), (1,1), (0, 0) and (1,0).
• REP1 indicates whether a grade was repeated (1 if yes, 0 if no).
• MALE indicates the gender of students within a cell, with 1 if the student is male and 0 if female.
• PPED = 1 if student in the cell had pre-primary experience. For students without pre-primary experience, PPED = 0.
• The number of trials is represented by the variable TRIAL.

The model

It is hypothesized that the child’s gender and pre-primary experience will be associated with the probability of repetition. Every level-1 record corresponds to a cell with students nested within it with the number of observed repetitions for the cell denoted by TRIAL.
The sampling distribution of the Binomial distribution is

and the variance is

The model we consider again uses the logit link function:

The model for the means can be expressed as

and in terms of our variables we can write it as shown below, also utilizing the variable TRIAL to indicate the number of repetitions per cell.


Reading the data

To start a new analysis, select the New Analysis option on the landing page.

The Data page will open. As a first step, select the data file gthai1.csv and click Open to obtain the image shown below.

Start by selecting the variable SCHOOLID as the variable denoting the hierarchical structure by checking its box in the ID 2 line. Next, select all other variables in a similar way in the Variables line of the second table as shown below.

Click Update to prompt the automatic allocation of variables to the different levels of the hierarchy.


Model building

With data specification complete, model building can begin. To start, click on Models at the top of the window. Following the same steps as for the Bernoulli model, set up the model shown below.


Specifying the distribution type and link function

The program suggests a distribution type based on its reading of the data. For these data, it suggests the use of a Poisson model. We need to make changes here, as we want to fit a Binomial model with logit link function to the data.

Change the Distribution of Outcome field to Binomial by clicking the radio button for the Binomial model in the Distribution of Outcome field. When updated, the Settings page shows that the default link function for this model is the logit link function. We also set the Number of Trials Variable by selecting the variable TRIAL from the drop-down menu associated with this option. Analysis specification is now complete, and all that remains is to run the model.


Running the model

We move to the Run page by clicking on Run at the top of the window. When this page opens, only two options are available: saving the information on the model built in a MLCJSN file and running the syntax.

By default, the program will assign the same file name to the MLCJSN file as that of the data file read in, in this case gthai1. To save this analysis for potential reuse, we opt to save it using the Save option under the name thai_bnml.mlcjsn before clicking Run Syntax to instruct the program to perform the analysis.

For this model, the following results were obtained. The model specifications and descriptive statistics are given first. Note that descriptive statistics are given for both categories of the outcome variable (rep11 and rep12).


This is followed by results and fit statistics for the model without any random effects.

The fit statistics at convergence follows, followed by the final unit specific results.

Information on the variance components is given next. The intracluster correlation coefficient is also provided.

Finally, final population average results are reported.

The two sets of results are very similar apart from the estimate of the intercept coefficient. The estimated intercept in the population average model is -1.7220, which is the average logit. The estimated coefficients associated with gender (MALE) is 0.4539, which indicates that the male respondents (MALE = 1) have a larger . The estimate for the indicator of pre-primary education (PPED) shows that students with pre-primary education have a lower value. Females with pre-primary education again have the lowest value.