# Analyzing/modelling discrete numerical responses and determining associations based on categorical co-variates

I need and would be grateful for some advice regarding the analysis of some data that has been gathered and coded by a third party. The data is part of a project regarding performance in the health care sector and the person who has the data would like to perform some sort of analysis on it.

There are 4 variables in the data set as follows: 1. hours: Hours worked (per week) from 1 – 37.5 (mostly numeric data but if a person works more than 37.5 hours worked this is coded as 37.5+). 2. location: 11 locations in the sample (categorical). 3. experience: Level of experience in years: 1-5, 6-9,… (categorical – but ordinal) 4. number_activities: Number of activities performed: 1 – 8 (integer). 0 is a valid response but the data sample does not contain any 0 values for number_activities.

There are 65 “independent” observations on these variables.

# Data User’s Question:

Specifically, the data user has said he is interested in whether there is an association between the 3 variables hours, location, experience, based on the values in number_activities. The data user is not really interested in what the relationship is (at the moment at any rate) but just wants to know if one exists, and in particular if we can perform a hypothesis test for this.

From my initial investigation I do not think this is as easy an analysis as the data user assumes.

The data user, who is not a statistician, but who has performed statistical analyses in the past, specifically chi-squared tests for goodness-of-fit and for independence in categorical tables, thinks that something like this may be possible with the current data.

The data user has suggested converting hours to a categorical variable (e.g. (0 - 7.5], (7.5 - 15],…), which I think may be necessary regardless of the analysis being performed, and using some form of chi-squared test approach.

Specifically he thinks:

1) we could build a contingency table based on the levels of the three variables (hours, location, experience) using the totals of number_activities as the frequency counts for each of the levels of the three variables;

or

2) analyse a number of 2x2 tables. For example we could build a number of 2 x 2 tables based on the levels of a third variable (partial tables) and try to perform a “goodness-of-fit” test for each of these multiple tables, assuming some null hypothesis regarding the distributions of number_activities at the different levels.

My opinion on this is as follows: For option 1) from my understanding of contingency tables the approach is incorrect – we would be using the wrong counts, totals of number_activities for each level of the variables is not the correct count for this type of analysis – we would need instead to use the counts of the observations that occur at the different levels, and this won’t tell us anything about the relationships based on number_activities. Further, given the number of levels in the variables compared to the 65 observations there would likely be a very large number of zero entries in the table.

For option 2) I’m not sure how this would work in practice. For the partial tables it would need a number of hypotheses to be formulated and there would be multiple tables with corresponding multiple hypotheses issues etc.

My initial opinion is that the data user would be better off using a modelling approach to the data with hours, location, experience as the explanatory variables and number_activities as the response/dependent variable. To answer the question of whether there is a relationship between the variables hours, location, experience we would need to include an interaction term in the model.

Specifically I could do with some advice as follows:

1) Is there any value in the approach suggested by the data user? What mean is there a single test one could do to test for association based on number_activities that I am clearly not aware of.

2) Is the modelling approach the best way forward here?

3) If the modelling approach is a good way forward then the models I’ve been considering are:

a. binomial – treat the number of activities as the number of successes in 8 independent trials – then for each observation number_activities then represents the of number of “successes”

b. Poisson – the number_activities is a count, but it is restricted to 0 – 8, so would this be a problem?

c. ordinal regression: ordered logistic or ordered probit models

4) Is there another approach – possibly simpler that I have just missed entirely?

It is a while since I've done this type of work and any advice/help would be most appreciated.

• Why not start much simpler and look for correlations? If the user is just wanting to look for associations, I think a natural easy place to start is by examining associations via pairwise correlations. Since you are limited in the number of observations you have, you're probably better off using simpler analyses rather than building models. I don't think a carefully selected $2\times 2$ tables approach is terrible if the goal at this stage is to detect associations. I think stratified $2\times 2$ tables might be helpful too. – StatsStudent Jan 23 at 16:58
• @StatsStudent Thanks for your input. Yes we will start with this type of approach and see what happens. Maybe there won't be more that we can easily say at this stage. – Ray Jan 24 at 13:33