Documentation of Medicare Reimbursement Claims

The Pentad Home Health Agency provides home health services in five counties of an eastern state. The agency must be certain that its Medicare reimbursement claims are appropriately and correctly documented in order to ensure that Medicare will process these claims and issue benefits in a timely manner. All physician orders, including medications, home visits for physical therapy, home visits of skilled nursing staff, and any other orders for service, must be correctly documented on a Form CMS-485. Poorly or otherwise inadequately prepared documentation can lead to rejection or delay in processing of the claim for reimbursement by the Centers for Medicare and Medicaid Services (CMS).

Pentad serves about 800 clients in the five-county region. In order to assure themselves that all records are properly documented, the administration runs a chart audit of 1 in 10 charts each quarter. The audit seeks to determine (1) whether all orders indicated in the chart have been carried out and (2) if the orders have been correctly documented in the Form CMS-485. Orders that have not been carried out, or orders incorrectly documented, lead to follow-up training and intervention to address these issues and ensure that the orders and documentation are properly prepared going forward.

Historically, the chart audit has been done by selecting each tenth chart, commencing at the beginning or at the end of the chart list. Typically, the chart audit determines that the majority of charts, usually 85 to 95 percent, have been correctly documented. But there are occasionally areas, such as in skilled nursing care, where the percentage of correct documentation may fall below that level. When this happens, the administration initiates appropriate corrective action.

Sampling, Data Display, and Probability

One of the questions of the audit has been the selection of the sample. Because the list of clients changes relatively slowly, the selection of every tenth chart often results in the same charts being selected for audit from one quarter to the next. That being the case, a different strategy for chart selection is desirable. It has been suggested by statisticians that using a strictly random sample of the charts might be a better way to select them for quarterly review, as this selection would have a lesser likelihood of resulting in a review of the same charts from quarter to quarter. But how does one go about drawing a strictly random sample from any population? Or, for that matter, what does “strictly random” actually mean and why is it important beyond the likelihood that the same files may not be picked from quarter to quarter? These questions are addressed by statistics, specifically the statistics associated with sample selection and data collection.

Another question related to the audit concerns when to initiate corrective action. Suppose a sample of 1 in 10 records is drawn (for 800 clients that would be 80 records) and it is discovered that 20 of the records have been incorrectly documented. Twenty of 80 records incorrectly documented would mean that only 75 percent of the records were correctly documented. This would suggest that an intervention should be initiated to correct the documentation problem. But it was a sample of the 800 records that was examined, not the entire 800. Suppose that the 20 incorrectly documented records were, by the luck of the draw, so to speak, the only incorrectly documented records in the entire 800. That would mean that only 2.5 percent of the cases were incorrectly documented.

If the required corrective action were an expensive five-day workshop on correct documentation, the agency might not want to incur that expense when 97.5 percent of all cases are correctly documented. But how would the agency know from a sample what proportion of the total 800 cases might be incorrectly documented, and how would they know the likelihood that fewer than, say, 85 percent of all cases were correctly documented if 75 percent of a sample were correctly documented? This, again, is a subject of statistics.

Emergency Trauma Color Code

The emergency department (ED) of a university hospital was the site of difficulties arising from poor response time to serious trauma. Guidelines indicate that a trauma surgeon must attend for a certain level of trauma severity within 20 minutes and that other trauma, still severe but less so, should be attended by a trauma nurse within a comparable time. In general, it had been found that the response time for the ED in the university hospital was more or less the same for all levels of severity of trauma—too long for severe cases and often quicker than necessary, given competing priorities, for less severe cases.

The ED director knew that when a trauma case was en route to the hospital, the ambulance attendants called the ED to advise that an emergency was on its way. Part of the problem as perceived by the director of the ED was that the call-in did not differentiate the trauma according to severity. The ED director decided to institute a system whereby the ambulance attendants would assign a code red to the most severe trauma cases, a code yellow to less severe trauma cases, and no color code to the least severe trauma cases. The color code of the trauma would be made known to the ED as the patient was being transported to the facility. The intent of this coding was to ensure that the most severe traumas were attended within the 20-minute window. This in turn was expected to reduce the overall time from admission to the ED to discharge of the patient to the appropriate hospital department. (All trauma cases at the red or yellow level of severity are transferred from the ED to a hospital department.)

Descriptive Statistics, Confidence Limits, and Categorical Data

A major concern of the director of the ED was whether the new system actually reduced the overall time between admission to the ED, treatment of the patient in the ED, and discharge to the appropriate hospital department. The director of the ED had considerable information about each ED admission going back a period of several months before the implementation of the new color-coding system and six months of experience with the system after it was implemented. This information includes the precise time that each trauma patient was admitted to the ED and the time that the patient was discharged to the appropriate hospital department.

The information also includes the severity of the trauma at admission to the ED on a scale of 0 to 75, as well as information related to gender, age, and whether the admission occurred before or after the color-coding system was implemented. The ED director also has information about the color code assigned after the system was initiated that can generally be equated to the severity score assigned at admission to the ED. Trauma scoring 20 or more on the scale would be assigned code red; below 20, code yellow; and not on the scale, no color.

Inferential Statistics, Analysis of Variance, and Regression

The question the ED director wishes to address is how she can use her data to determine whether the color-coding system has reduced the time that trauma victims spend in the ED before being discharged to the appropriate hospital department. At the simplest level, this is a question that can be addressed by using a statistic called the t test for comparing two different groups. At a more complex level, the ED director can address the question of whether any difference in waiting time in the ED can be seen as at all related to changes in severity levels of patients before or after the color-coding scheme was introduced. She can also examine whether other changes in the nature of the patients who arrived as trauma victims before and after the introduction of the color-coding scheme might be the cause of possible differences in waiting time. These questions can be addressed by using regression analysis.

Two Caveats of Statistics: Establishing a Significant Difference and Causality

Two caveats regarding the use of statistics apply directly to this example. The first is that no statistical analysis may be needed at all if the waiting time after the initiation of the color-coding scheme is clearly shorter than the waiting time before. Suppose, for example, that the average waiting time before the color-coding scheme was three hours from admission to the ED to discharge to hospital department, and that after the initiation of the scheme, the average waiting time was 45 minutes. In this scenario, no statistical significance tests would be required to show that the color-coding scheme was associated with a clear improvement in waiting time. Furthermore, it is likely that the color-coding scheme would not only become a permanent part of the ED armamentarium of the university hospital but also be adopted widely by other hospitals.

However, suppose that after the initiation of the color-coding scheme the average waiting time in the ED was reduced from 3 hours to 2 hours and 50 minutes. A statistical test (probably the t test) would show whether 170 minutes of waiting was actually less, statistically, than 180 minutes. Although such a small difference may seem to have little practical significance, it may represent a statistically significant difference. In such a case, the administrator would have to decide whether to retain an intervention (the color-coding scheme) that had a statistical, but not a practical, effect.

The second caveat to the use of statistics is the importance of understanding that a statistical test cannot establish causality. It might be possible, statistically, to show that the color-coding scheme was associated with a statistical reduction in waiting time. But in the absence of a more rigorous study design, it is not possible to say that the color-coding scheme actually caused the reduction in waiting time. In a setting such as this, where measurements are taken before and after some intervention (in this case, the color-coding scheme), a number of factors other than the color-coding scheme might have accounted for an improvement in waiting time.

The very recognition of the problem and consequent concern by ED physicians and nurses may have had more effect on waiting time than the color-coding scheme itself. But this is not a question that statistics, per se, can resolve. Such questions may be resolved in whole or in part by the nature of a study design. A double-blind, random clinical trial, for example, is a very powerful design for resolving the question of causality. But, in general, statistical analysis alone cannot determine whether an observed result has occurred because of a particular intervention. All that statistical analysis can do is establish whether two events (in this case, the color-coding scheme and the improvement in waiting time) are or are not independent of each other. This notion of independence will come up many more times, and, in many ways, it is the focus of much of this book.

Length of Stay, Readmission Rates, and Cost per Case in a Hospital Alliance

The ever-increasing costs of providing hospital services have sparked a keen interest on the part of hospital administrators in practical mechanisms that can account for—and control or mitigate—those costs. The administrators for the Sea Coast Alliance, a system of eight hospitals, want to be able to use the previous case data to provide guidance on how to control costs. Because Sea Coast is associated with eight hospitals, it has a substantial volume of case data that the administrators believe can be useful in achieving their goal.

There are, in particular, three measures of hospital performance related to costs that need to be evaluated in this case: length of stay (LOS), readmission rates, and cost per case. One of the initial questions is whether there are real differences among the eight hospitals in these three cost-related measures of hospital performance. The question of what is a real difference is, of course, paramount. If the average LOS in one of Sea Coast's hospitals is five days for all hospital stays over the past year while the average LOS for another Sea Coast hospital is six days, is this a real difference? Given certain assumptions about what the average LOS for a year in these two hospitals represents, this is a question that can be answered with statistics.

Establishing a Statistical Difference between Two Groups Using t Tests and Analysis of Variance

If the interest is in comparing two hospitals with each other, the statistic that could be used would be a t test. In general, though, the real interest would be in deciding if there was any difference among all eight hospitals, taken simultaneously. This question can be examined in a couple different ways. One would be to use analysis of variance (ANOVA). Another would be to use multiple regression. If by using any of these statistical techniques it is determined that LOS is different from hospital to hospital, efforts could be directed toward determining whether lessons could be learned from the better performers about how to control costs that might be applied to the poorer performers. The same approach could be applied to understanding readmission rates and cost per case.

One particular focus of the Sea Coast administrators is diagnostic-related groupings (DRG) that have especially high costs. In addition to looking at the performance across the eight hospitals on high-cost DRGs, Sea Coast would like to be able to examine the question of whether individual physicians seemed to stand out in LOS, readmission rates, or cost per case. Identification of individual physicians who have unusually high LOS, readmission rates, or cost per case can allow Sea Coast to engage in selective educational efforts toward reduced costs. But an important question in looking at individual physician differences is whether what may appear to be unusually high values for LOS, readmission rates, or cost per case actually are unusual. Again, this question can be answered with statistics. In particular, predicted values for LOS, readmission rates, and cost per case can be determined by using regression analysis.

Establishing a Statistical Difference between Two Groups Using Regression

Regression can also be used to assess whether the differences that may exist across hospitals or across individual physicians could be attributed to differences in the mix of cases or patients whom the hospitals accept or the physicians see. Such differences may be attributable statistically to such characteristics of patients as sex, age, and payer, which may differ across the eight hospitals or the numerous physicians. There might also be differences across cases related to severity or diagnoses. If these were differentially distributed among hospitals or physicians, they could account for visible differences. All of these questions can be addressed (although not necessarily answered in full) by using multiple regression analysis.

At the Carteret Falls regional hospital, the emergency department has instituted a major change in how physicians are contracted to provide services and consequently how services are billed in the ED. Prior to January 1 of a recent year, emergency department physicians were employed by the regional hospital, and the hospital billed for their services. Beginning January 1, the physicians became private contractors working within the emergency department, essentially working on their own time and billing for that time directly. Bills are submitted to Medicare under five different coding levels that correspond to the level of reimbursement Medicare provides. The higher the coding level, the more Medicare actually reimburses for the service.

The practice manager for the physicians is concerned that she will begin to see the billing level creep upward as physicians begin billing for their own services. As the distinction between levels is frequently a matter of judgment, the practice manager is concerned that physicians may begin, even unconsciously, upgrading the level of the coding because it is directly tied to their reimbursement. The question the practice manager faces is how to decide if the physicians are upgrading their codes, consciously or not, after the initiation of the new system. If they are, the practice needs to take steps, either to ensure that the coding remains constant before and after the change in billing, or to have very good justification for CMS as to why it should be different.

Establishing a Difference Using Statistical Tests

The first problem is to determine if the billing levels have changed from before the change in billing to after the change. But it is simply not enough to say that there is a change, if one is seen to have occurred. It is critical to be able to say that this change is or is not a change that would have been expected, given the pattern of billing in the past. In other words, is any change seen as large enough to be deemed a real change and not just a chance occurrence? If a change has occurred, and if it is large enough to be viewed as a real change, then the second problem arises: determining whether anything in the nature of the ED cases before and after the billing change might account for the difference and thus be the explanation of the difference that will satisfy the Medicare administration.

Both of these problems can be examined by using statistics. In regard to the first problem, a difference between the distribution of billings across the five categories before and after the change in billing source can be assessed by using the chi-square statistic. Or, because the amount of a bill is constant within the five categories, it is also possible to compare the two groups, before and after, using the t test. The second problem, of whether any difference can be attributed to changed characteristics of the cases seen in the ED, can be assessed by using regression—when the cost of the bills before and after is the measure of change.

A Study of the Effectiveness of Breast Cancer Education

A resident at a local hospital has been asked by the senior physician to develop a pilot study on the effectiveness of two alternative approaches to breast cancer education, both aimed at women coming to a women's health center. The first alternative is the distribution of a brochure on breast cancer to the women when they arrive at the clinic. The second alternative is time specifically allocated during a clinic visit wherein the physician spends 5 to 10 minutes with each woman, giving direct information and answering questions on the same topics covered in the brochure.

The resident recognizes that a study can be designed in which one group of women would receive the brochure and a second group would participate in a session with a physician. She also believes that a questionnaire can be developed to measure the knowledge women have about breast cancer before the distribution of the brochure or session with the physician and after either event, to assess any difference in knowledge. She also is concerned about possibly needing a control group of women to determine whether either method of information dissemination is better than no intervention at all. And perhaps she is interested in whether the brochure and discussion with the physician together would be better than either alternative singly.

Although she has been asked to design a pilot study only, the student-resident wishes to be as careful and as thoughtful as possible in developing her study. She might consider a number of different alternatives. One would be a simple t test of the difference between a group of women who received the brochure and a group of women who participated in the sessions with a physician. The measurement for this comparison could be the knowledge assessment administered either after the distribution of the brochure or after the physician encounter.

Using Analysis of Variance versus t Tests

But the resident may very well be dissatisfied with the simple t test. One problem is that she wants to include a control group of women who received no intervention at all. She may also wish to include another group of women—those who received the brochure and spoke to a physician. Again, the effect of any intervention (or of none) could be measured using her previously developed knowledge assessment, administered after the fact. This assessment could be carried out using a one-way analysis of variance (ANOVA).

Again, however, the resident may not be entirely satisfied with either the t test or the one-way analysis of variance. She might wish to be sure that in her comparison she is not simply measuring a difference between women that existed prior to the receipt of the brochures or the physician sessions. To ensure this, she might wish to measure women's knowledge both before and after the interventions, at both times using her knowledge assessment questionnaire. This assessment could be carried out using a two-way analysis of variance.

Regardless of whether the resident decides to go with a t test, a one-way ANOVA, or a two-way ANOVA, one of the more important aspects of the study will be to randomly allocate women to the experimental or control group. When measuring knowledge only after the intervention, the resident will be able to ensure that prior knowledge is not responsible for any differences she might find only if she is certain that there is only a small chance that the groups of women receiving different interventions were not different to begin with. The only effective way to ensure this is through random assignment to the groups.

Calculating a Standard Hourly Rate for Health Care Personnel

In an article published in Healthcare Financial Management, Richard McDermott ([2001]) discusses the problem and importance of establishing standard hourly labor rates for employee reimbursement. He points out that many compensation systems have been worked out over a number of years by different human resources directors, each with his own compensation philosophy. As a result, these systems may fail to reflect market conditions and may be inconsistent in their treatment of differing categories of labor. McDermott suggests a regression approach to calculating labor rates that have both internal consistency and external validity.

The approach McDermott suggests for establishing labor rates is based on an example in which he provides data for 10 different positions. Each position is assigned a score from 0 to 5 based on the degree of complexity in the job in five separate categories, such as level of decision making, amount of planning required, educational requirements, and so on. He does not indicate specifically which five characteristics are employed in the example. The assigned scores in each category would have been developed through an examination of the requirements of the job by a compensation consultant after interviews with the incumbent of each position. Each of the 10 positions also includes an actual hourly wage.

Relating Variables via Regression Analysis: Some Issues

Regression analysis was used by McDermott to assess the relationship between each of the five characteristics of the job and the actual hourly compensation. The regression analysis indicates both the relationship between any one of the five characteristics (when all characteristics are considered simultaneously) and hourly compensation, and it provides a set of coefficients by which to translate assigned values on any set of characteristics into a predicted hourly compensation. This, then, becomes a relatively objective means of determining hourly compensation for a person in any position.

There are purely statistical problems in using this regression approach, at least as discussed by McDermott, to propose hourly compensation. Particularly, 10 observations (the jobs assessed) are rarely considered by statisticians to be an adequate number with which to assess the relationship between five predictor variables (the characteristics) and a sixth predicted variable (the hourly compensation). While there are no absolute rules for the number of observations needed relative to the number of variables assessed, it is often accepted that there should be at least three times as many observations as variables, and some statisticians suggest a ratio of as many as 10 observations to each variable.

A second problem with this approach to assigning hourly compensation is inherent in the fact that many jobs are essentially the same, with similar job titles and expectations. If such jobs are included in an analysis of the type discussed here, one of the basic premises of regression analysis, that there is no correlation between observations, will be violated. This can be overcome, in part, by the use of dummy variables.

Populations and Samples

Populations are those groups of entities about which there is an interest. Populations may be made up of people—for example, all citizens of the United States or all patients who have shown up or ever will show up at a specific emergency room clinic. Populations may be made up of organizations—for example, all hospitals in the United States, or all long-term care facilities in New York state. Populations may be made up of political entities—for example, all the countries in the world or all the counties in California. All the persons who might ever receive a particular type of assessment are a population, and all people who ever will have an magnetic resonance imaging (MRI) could be considered another population, or these two groups together could be considered a population.

In general, we are interested in characteristics of populations as opposed to characteristics of samples. We might wish to know the average cholesterol level of all persons age 55 or older (a population). We might wish to know the daily bed occupancy rate for hospitals in the United States (a population). Or we might wish to know the effect of a specific drug on cholesterol levels of some group of people (a population). If we knew these specific pieces of information, we would know a parameter. Parameters are information about populations. In general, except for some data collected by a complete census of the population (even most complete censuses are not complete), we do not know parameters. The best we can usually do is estimate parameters based on a subset of observations taken from populations.

Samples are subsets of populations. If a population of interest consists of all patients who have shown up or ever will show up at a specific emergency room clinic, a sample from that population could be all the patients who are there on a specific afternoon. If a population of interest consists of all long-term care facilities in New York state, a sample from that population might be all these facilities in Buffalo, Syracuse, and Albany. If a population of interest is all persons who have used or ever will use a cholesterol-reducing drug, a sample from that population might be all persons who received prescriptions for such a drug from a specific physician. An individual member of a sample might be referred to as an element of the sample or, more commonly, as an observation.

Information from samples can be used to make estimates of information about populations (parameters). When a specific value from a sample is used to make an estimate of the same value for a population, the sample value is known as a statistic. Statistics are to samples what parameters are to populations. If the parameter of interest is, for example, waiting time in emergency rooms, an estimate of that parameter could be the average waiting time for a small, carefully selected group of emergency rooms. The estimate would be a statistic. In general, we can know values of statistics but not parameters, even though we would wish to know the values of parameters.

Random and Nonrandom Samples

The samples, or subsets of a population, may be selected in a random manner or in a nonrandom manner. All patients in an emergency room on a specific afternoon would probably not constitute a random sample of all people who have used or will use an emergency room. All the hospitals in Buffalo, Syracuse, and Albany might be a random sample of all hospitals in New York state, but they probably would not be. All persons who received prescriptions for a cholesterol-reducing drug from a specific physician would, in general, not be a random sample of all persons who take such drugs. All of these examples would probably be considered nonrandom samples. Nonrandom samples may be drawn in many ways. In general, however, we are not interested in nonrandom samples. The study of statistics is based on and assumes the presence of random samples. This requires some discussion of what constitutes a random sample.

A random sample is a sample drawn in a manner whereby every member of the population has a known probability of being selected. At a minimum, this means that all members of the population must be identifiable. Frequently, there is a gap between the population of interest and the population from which the sample is actually drawn. For example, suppose a health department wished to draw a random sample of all families in its area of responsibility to determine what proportion believed that the health department was a possible source of any type of health services—prevention, treatment, advice—for members of the family.

The target population is all families in the area of responsibility. If we assume that this is a county health department, a random sample would assign a known probability of selection to each family in the county. In general, this would mean that each family in the county would have an equal probability of selection. If there were, for example, 30,000 families in the county, each one would have a probability of 1/30,000 of being selected as a member of the sample.

But, in general, it would be very difficult to be certain that every family in the county had exactly a 1/30,000 probability of being selected for the sample. The difficulty arises from the problem of devising an economically feasible mechanism of identifying and contacting every possible family in the county. For example, one relatively inexpensive way of collecting the information desired would be to contact a sample of families by telephone and ask them questions. But some families do not have phones, making their probability of selection into the sample not 1/30,000 but simply zero. Other families may have more than one phone, and if care is not taken to ensure that the family is not contacted twice, some families might have twice the chance (or more) of being selected into the sample. Still other families—especially in the present age of telemarketing—would decline to participate, which would make their probability of being included zero as well.

Other mechanisms of identifying all families in a county or other area have equal difficulties. Voter rolls list only registered voters. Tax rolls list only persons who pay taxes. Both of these rolls also contain single persons. A decision would have to be made about whether a single person constitutes a family. In summary, then, it is very often nearly impossible, or at least very expensive, to draw a truly random sample from a target population. What often happens instead is that the sample drawn is actually from a population very close to the target population but not the target population exactly. Instead of all the families in the county being the population from which the sample is drawn, the population may be all families with telephones.

The population from which the sample is actually drawn is known as the sampled population. Inferences from the sample to the population are always to the sampled population; one hopes that these inferences hold for the target population as well. Given that the population sampled may not be exactly the target population desired, there still needs to be a mechanism for assuring that each member of the population to be sampled has a known probability generally equal of being selected. There are lots of ways of assuring randomness in specific settings. Shuffling cards is a way of assuring that each person has an equal chance of getting the good cards and the bad cards during the deal—essentially a random distribution of the cards. Rolling dice is a way of ensuring a random distribution of the faces of a die. Flipping a coin is a way of ensuring the random appearance of a head or a tail.

But sampling from a population of all families served by a health department is more complicated. One workable mechanism might be to put every family's name on equal-sized slips of paper, put all the slips of paper in a box, shake the box, and without looking at the slips of paper, draw out the number of slips desired. But this approach, although it would produce a random sample of families, would be both cumbersome and time consuming. Excel provides several mechanisms that can be used to draw random samples.

There are basically four different types of random samples: systematic samples, simple random samples, stratified samples, and cluster samples.

Variables, Independent and Dependent

Throughout this book there are frequent references to the term “variable.” A variable is a characteristic of an observation or element of the sample that is assessed or measured. A value for a variable across all members of a sample (e.g., the average height of preadolescent teens) is typically referred to as a statistic. The comparable value for the population is a parameter. Most statistical activities are an attempt either to determine a value for a variable from a sample (and thus to be able to estimate the population value) or to determine whether there is a relationship between two or more variables.

In order to show a relationship between two or more variables, the variables must vary. That is to say that they must take on more than one value. Any characteristic of a population that does not vary is a constant. There can be no relationship between a constant and a variable. This is equivalent to saying that there can be no way of accounting for the value of any variable by referring to a constant. For example, if we wished to describe variations in adult-onset diabetes rates among persons with Hispanic surnames, it would be useless to employ Hispanic surname as an explanation, because it is constant for all these people. It cannot explain differences. But if we wished to describe differences in adult-onset diabetes among all the people living in, say, New Mexico, Hispanic surname or non-Hispanic surname might be a useful variable to employ.

Variables, Categorical and Numerical

Variables are typically classified as either of two types: categorical or numerical. Numerical variables are further classified as either discrete or continuous. These distinctions are important for the type of statistic that may effectively be employed with them.

Categorical variables are variables whose levels are distinguished simply by names. Hispanic and non-Hispanic surname is a two-level categorical variable that roughly distinguishes whether a person is of Hispanic ancestry. Sex is a two-level categorical variable that in general divides all persons into male or female. Categorical variables can take on more levels as well. Type of insurance coverage, for example, is a multilevel categorical variable that may take on the values Medicare, Medicaid, voluntary not-for-profit, for-profit, self-pay, and other. Other categorical variables may take on many levels.

Although a variable may be represented by a set of numbers, such a representation does not automatically mean that it is not a categorical variable. The ICD-9-CM code is a categorical variable, even though the codes are represented as numbers. The numbers simply classify diagnoses into numerical codes that have no actual numerical meaning. Another type of categorical variable that is assigned a number is the dummy variable. The dummy variable is a two-level categorical variable (e.g., sex) that is assigned a numerical value, usually the values 1 and 0. The value 1 might be assigned to female and 0 assigned to male, or vice versa. Although this type of variable remains a categorical variable, it can be treated as a numerical variable in some statistical applications that require numerical variables. However, a categorical variable with more than two levels, such as the ICD-9-CM code, can be treated as a numerical variable in analysis only by dividing the multilevel categorical variable into a number of two-level categorical variables that can be treated as dummy variables.

Numerical variables are, as the name implies, variables whose values are designated by numbers. But unlike ICD-9-CM codes, the numbers have some meaning relative to one another. At the very minimum, a numerical variable whose value is, for example, 23 is presumed to be larger than a numerical variable whose value is 17. Numerical variables may be measured on three scales: the ordinal scale, the interval scale, and the ratio scale.

Ordinal Scale

The ordinal scale is a scale in which the values assigned to the levels of a variable simply indicate that the levels are in order of magnitude. A common ordinal scale is the Likert scale, which requests a response to one of usually five alternatives: strongly agree, agree, undecided, disagree, or strongly disagree. These responses are then assigned values of 1 to 5, or 5 to 1, and treated as values of a numerical variable. Treating Likert scale responses like numerical variables assumes that the conceptual difference between strongly agree and agree is exactly the same, for example, as the conceptual difference between undecided and disagree. If that cannot be assumed, then ordered variables, such as Likert scale variables, even if assigned numerical values, should not be treated as numerical variables in analysis but must be treated as categorical variables.

Interval Scale

The interval scale is a scale in which the values assigned to the levels of a variable indicate the order of magnitude in equal intervals. The commonly employed measures of temperature, Fahrenheit and Celsius, are interval scales. For Celsius, for example, the value of 0 refers not to the complete absence of heat but simply to the temperature at which water freezes. One hundred on the Celsius scale refers to the temperature at which water boils at sea level. The distance between these two has been divided into 100 equal intervals. Because this is an interval scale measurement, it is accurate to say that the difference between 10 degrees Celsius and 15 degrees Celsius is the same as the distance between 20 degrees Celsius and 25 degrees Celsius. But it is not accurate to say that 20 degrees Celsius is twice as warm as 10 degrees Celsius.

Ratio Scale

The ratio scale is a scale in which the values assigned to the levels of a variable indicate both the order of magnitude and equal intervals but, in addition, assumes a real zero. The real zero represents the complete absence of the trait that is being measured. Temperature measured on the Kelvin scale has a real zero, which represents the complete absence of heat. At a more prosaic level, the number of patients in an emergency room is measured on a ratio scale. There can be zero patients in the emergency room, representing the complete absence of patients, or there can be any number of patients. Each new patient adds an equal increment to the number of patients in the emergency room. In general, any variable that is treated as numeric for statistical analysis is assumed to be measured on at least an interval scale and most commonly on a ratio scale.

Discrete and Continuous Numerical Variables

Discrete numerical variables are variables that can take on only whole number values. Discrete numerical variables are typically the result of the counting of things, persons, events, activities, and organizations. The number of persons in an emergency room is a discrete numerical variable. There must always be a whole number of persons in the room—for example, 23. There can never be 23.7 persons in an emergency room. The number of children born to an unmarried woman, the number of organizations that are accredited by a national accrediting body, the number of physicians on a hospital staff, the number of health departments in a state—these are all discrete numerical variables.

Continuous numerical variables are variables that can take on any value whatsoever. They can be whole numbers, such as 47, or they can be numbers to any number of decimal places, such as one-third (which is 0.33333 . . . and so on forever). The amount of time that a person spends in an emergency room is a continuous variable that can be stated to any level of precision (in terms of minutes, seconds, parts of seconds) that we have the ability and interest to measure. Body temperature, pulse rate, height, weight, and age are all continuous numerical variables. Measures that are created as the ratio of one variable to another, such as cost per hospital admission or cost per day or proportion of children fully immunized, are also continuous numerical variables.

Probabilities of occurrence of discrete or continuous numerical variables cannot be found in the same ways. In general, it is possible to find the exact probability of the occurrence of a discrete outcome based either on an a priori distribution or on empirical information. Probabilities of outcomes for continuous numerical variables, however, can only be approximated. Despite this, the distribution of continuous numerical variables has been extensively researched and in the form of the normal distribution: Particularly, it forms the basis of most statistical analyses for numerical variables, whether the variables are measured as discrete or continuous.

Chi-Square Test

The chi-square test can be used to assess the independence of two variables, both of which are categorical. Either of the two variables may take on two or more levels or categories, but the data itself are measured simply as named categories. For example, the chi-square can be used to determine whether coming to an emergency clinic for a true emergency or for a visit that is not an emergency (a two-level categorical variable) is independent of whether one comes during the day or during the night (another two-level categorical variable). Or a chi-square test could be used to determine whether the desire of women for an additional child (a two-level, yes–no variable) is independent of the number of children she already has in the three categories—for example, one, two or three, and four or more. The chi-square can be used on variables that take on larger numbers of values as well, but in practical terms, it is unusual to see a chi-square that involves variables having more than three or four levels.

t Test

The t test can be used to assess the independence of two variables. The t test assumes that one variable is a numerical variable measured in either discrete or continuous units and the other is a categorical variable taking on only two values. For example, the t test can be used to determine whether the score people receive on a test of knowledge about breast cancer measured on a 20-point scale (a numerical variable) is independent of whether those people were specifically and consciously exposed to knowledge about breast cancer or were not (a two-value categorical variable). Or a t test could be used to determine whether the cost of a hospital stay (a numerical variable) was independent of whether the patient was a member of an HMO or was not (a two-level categorical variable).

Analysis of Variance

Analysis of variance, or ANOVA, an extension of the t test, can be used to assess the independence of two or more variables. ANOVA assumes that one variable is a numerical variable measured in either discrete or continuous units and the others are categorical variables that may take on any number of values rather than only two. ANOVA, for example, could be used to assess not only whether a knowledge score about breast cancer was independent of exposure to knowledge about breast cancer but also whether the score might be independent of several different types of exposure. Exposure could be the reading of a brochure, a one-on-one discussion with a physician, both, or neither. Analysis of variance could also be used to determine whether the length of a hospital stay (a numerical variable) was independent of the hospital in which the stay took place over five separate hospitals (a categorical variable taking on five values).

Regression

Regression, a logical last stage in this progression, is a technique that can test the independence of two or more numerical variables measured in either discrete or continuous units. Regression may also include one or more categorical variables, any one of which can take on only two values (in which case, it is often referred to as analysis of covariance). Regression, then, could test the independence, for example, of the cost of a hospital stay (a numerical variable) and the length of a hospital stay (a second numerical variable) across an essentially unlimited number of hospitals. Or it could assess the independence of the dollar value of all hospital billings (a numerical variable) and the number of patients admitted (a second numerical variable) for a sample of for-profit and not-for-profit hospitals (a categorical variable taking on two values).

Logit

Logit, an extension of regression, can examine the independence of two or more variables where the dependent variable is a dichotomous categorical variable and the independent variable or variable set may be categorical (taking on only two values) or numerical—either discrete or continuous. Logit could be used, for example, to assess the independence of the outcome of an emergency surgical procedure (measured as successful or unsuccessful) and such variables as the degree of presurgery trauma, the length of time between the emergency and the surgical procedure, the age of the patient, and so on.