# The application of logistic regression to pedestrian-walkway safety.

Executive Summary

The cost of walkway accidents--pedestrian slips and falls--is substantial. To reduce the incidence of slips, and subsequent falls, companies install and maintain 'slip-resistant' floor surfaces. Monitoring floor slipperiness using an instrument called a tribometer allows businesses to record quantitative information that allows them to apply quality-control protocols to maintenance procedures, in order to minimize fall-accident rates.

In regression analysis, you are given data that has (hypothetically) been generated by a mathematical function whose parameters are not known, and you must estimate those underlying parameters. Ordinary regression is used to predict a continuous outcome. Logistic Regression is used when the outcome is dichotomous: yes/no. Because slipping--or not slipping--is a dichotomous event, and because Logistic Regression is a mathematical model that can explicitly model dichotomous events, it has recently been utilized in walkway-safety analysis. For example, Logistic Regression has been used to describe the likelihood of falling, and to associate walkway and gait characteristics to the probability of falling. Researchers have used the characteristics of the flooring surface materials, contaminants, and shoe sole materials and textures as factors that might help to predict increased risk of falls or the ability to recover from a slip; they have also considered aspects of normal gait to see if these might be associated with falling. Logistic Regression has been used by the authors to characterize tribometric instruments used to evaluate walkway safety, as well as to evaluate a novel method for barefoot-friction metrology.

In summary. Logistic Regression provides a powerful tool for improved understanding of how the tribometer can be used accurately to assess walkway slip-resistance.

The Objective of this Paper

This paper will explore how Logistic Regression has recently been applied to walkway-safety prediction and tribometer characterization.

Introduction: The Magnitude of the Problem

The cost of walkway accidents (pedestrian slip-, trip-, fall-, and misstep-precipitated injuries) is huge, both to society and to business. Rice and MacKenzie reported that, for 1985, the economic cost of slip and fall injuries to society--direct, morbidity, and mortality costs taken together--was estimated to exceed 37 billion dollars (Rice, et al., 1989). They found that fall accidents were the second largest generator of unintentional, accidental-injury costs, and the largest generator of accidental mortality in the elderly. Englander, Hodson and Terragrossa (Hodson, et al., 1996, 733-746) projected these costs to the year 2020, taking into account demographic trends, i.e., the aging of the population. They estimated that the cost to the United States would be over 64 billion dollars in 1995, and over 85 billion in the year 2020. Leamon and Murphy investigated the cost of walkway accidents to business; their 1995 research paper was entitled, rather understatedly, "More than a Trivial Problem," with an estimated per-worker cost of falls ranging from $44 to $550, depending upon the industrial sector. Leamon (Leamon, et al., 1995). Buck and Coleman (Buck, et al., 1985, pp. 949-958), Proctor and Coleman (Proctor, et al., 1988, pp. 269-285), and Proctor (Proctor, 1993, 367-377), studied walkway accidents in workplaces in the United Kingdom. The cost to the U.K. was thought to exceed 150 million pounds annually (1982 data).

The collection and analysis of fall-related injury statistics within the United States is accomplished by a number of governmental and private organizations, viz., the Bureau of Labor Statistics, the National Electronic Injury Surveillance System (NEISS), and the National Safety Council. The Bureau of Labor Statistics (BLS) classifies falls in a hierarchal manner. Exhibit 1 provides only a portion of their extensive list. For example: the percentage of total injuries for 'falls on the same level' (item 1.3.0) averaged 64% from 1999 to 2001. The average number of injuries (1999-2001) for falls resulting in days away from work was approximately 300,000 (Yuon et. al, 2006, pp. 83-93). Slips precipitate the plurality, if not the majority (a), of walkway accidents (Manning, D.P. et al., 1988, pp. 121-130, and Bentley et al. 1998, pp. 1859-1872).

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The National Safety Council (2002) reported that the slips and falls are the leading cause of death in the workplace and the cause of more than 20% of disabling injuries. The National Safety Council has also estimated that costs associated with employee slip-and-fall accidents (compensation and medical) are approximately $70 billion/year (National Safety Council, 2002). Liberty Mutual Workplace Safety Index reported that costs associated with falls on the same level were $5.7 billion/year (Liberty Mutual Research Institute, 2002). Regardless of the source, it is clear that the number of slip accidents and the resulting direct costs to business are significant and, due to the aging of the population, shows no signs of abating.

To reduce the incidence of slips and falls, a number of Risk-management-based loss prevention techniques have been discussed in industry periodicals. For example banking industry literature suggests that banks can reduce the walkway-accident-related claims of their customers and employees by developing slip-and-fall prevention policy and by analyzing losses to gauge the effectiveness of implemented policies and programs. Among the measures suggested are the design (or re-design) and maintenance of the bank's property to reduce the potential for slips and falls, good housekeeping, periodic inspections, and employee education and training (Erikson, 2004).

Food-service companies, viz., supermarkets and convenience stores, hotels, motels, and restaurants, are subject to slip-precipitated accidents and injuries both from customers and from their employees (Kohr, 1991, and DiPilla, 2006). These companies, the industry literature suggests, can mandate (and supply) the footwear used by their employees, design and install 'slip-resistant' kitchen floor surfaces, and implement an integrated program that includes facility design, inspection and maintenance. Use of checklists that require employees to monitor flooring conditions periodically (sweep logs), both in the store and the parking lot, are common measures, as is the installation of slip-resistant floor surfaces in areas that are likely to become contaminated.

Monitoring floor slipperiness, by using a portable tribometer, (b) allows business owners to collect quantitative information that allows them to determine the flooring condition with respect to pedestrian safety. The use of decision-science tools--both to characterize the chance of a fall and to analyze tribometers' results--is a relatively recent development in walkway safety. Because slipping--or not slipping--is a dichotomous pair of events, a mathematical model that reflects that dichotomous situation is especially useful for analyzing walkway-safety situations. Similarly, a tribometer that has a slip/no-slip result can be effectively characterized by a mathematical model that can model a dichotomous-event set. Logistic regression is just such a technique, where a dichotomous dependent variable is predicted employing a continuous variable (or variables) and/or, using dummy-variable sets, (c) to employ dichotomous, trichotomous, etc., events as independent variables. In other words, we can use a set of continuous inputs (like friction) and/or discrete inputs (like wet/dry) to predict either the chance, i.e., the probability of, a fall, or the chance (the probability) that a tribometer will indicate a slip at a given setting.

Introduction: Regression

Regression is a term used to describe the process of fitting a line or curve to a dataset. Logistic regression is the term used to describe the process of fitting a logistic curve to a dataset. Let's gently look at each of these ideas in some detail.

Regression analysis is the statistical equivalent of the TV game show, Jeopardy. In that game show, an answer is given, and the contestant must respond with the question that would have generated that answer. In regression analysis, one is given data that could have been generated by some hypothetical but unknown mathematical function, and one must estimate the underlying parameters of the function that would best generate that set of data. In some situations, regression analysis is more complex than Jeopardy: in the TV game, the contestant gets to choose from a number of categories; in regression analysis the categories themselves may or may not be known.

Ordinary Regression

For example, take the following data, which represents the emergency response time for an ambulance as a function of the straight-line distance between the emergency vehicle to the destination.

It is easier to visualize such data if it is plotted in a graphical format. It is conventional to plot the variable that determines the relationship (called the independent variable, which here is the distance-as-the-crow-flies that the ambulance must travel) on the horizontal axis, and the determined variable (the dependent variable; here, the response time) on the vertical axis.

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The relationship between the variables can be determined by, first, hypothesizing what the underlying relationship should look like and, secondly, estimating the parameters of that relationship. In this example, we can hypothesize that there is a certain amount of time that it takes to receive the call from the dispatcher, followed by the time that it takes to travel to the destination site. The former time is approximately constant. The latter time is roughly proportional to the distance that the ambulance must travel; it is not exactly proportional because of the interplay between traffic, one-way streets, and the like. Based upon this information, we can hypothesize that the underlying form of the relationship looks like a straight line:

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The relationship is that of a straight line, where a is the y-intercept (the y value where the straight line crosses the x-axis, i.e., y where x = 0). And b is the slope of the line, the incremental time it takes to traverse each additional kilometer.

Exhibit 4: Relationship between variables of Dispatch Call and Ambulance Travel.

[y.sub.i] = a + [bx.sub.i]

where

[y.sub.i] is the response time in minutes for the ith call

a = the estimated time it takes to receive the call

b = the estimated typical speed that the ambulance travels, in km per minute

[x.sub.i] = the distance between the ambulance and the destination for the ith call

In high school, many of us studied this linear equation, estimating a and b by means of a clear-plastic straightedge, placing it so that about half of the data points were above the edge and half below, and "eyeballing" the slope and y-intercept. The problem with the eyeball method is that it cannot produce consistent results: what one person considers the best line may not be (and probably won't be) what a different person considers the best line.

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To achieve a method that will give consistent results, we turn to statistics. Without going into the minute details here, you can find the How-To in any solid Business-Statistics textbook. Suffice it to say (in true Jeopardy fashion) that the values of a and b that best give us the answers in the table above are: a = 0.9 minutes and b = 0.8 kilometers per minute (about 30 miles per hour). That is, assuming that the hypothetical underpinning, the equation [y.sub.i] = a + [bx.sub.i] reasonably represents the underlying process that generates response times as a function of distance between the ambulance and the destination, the best such equation would be

Response_Time (min) = 0.9 + 0.8 x Distance_Between_Ambulance_and_Destination (km)

Again, selecting the correct class of relationship is critical in making the analysis work. In Jeopardy, if you mistakenly think that you are in the Famous Ships category, and you are actually in the United States Presidents category, you are going to mistakenly answer Nimitz-Class Aircraft Carrier, rather than 34th president of the United States, when Eisenhower comes up.

Logistic Regression

There is a model: a class of relationships, where the answer is either: It Happened, or, It Did Not Happen. A brokerage house did or did not go belly up, an IPO was or was not successful. In the field of Walkway Safety, two very important questions have a Yes/No answer: (a) did a person slip or not (d), and (b) did a Variable Angle Tribometer indicate a slip. In those situations best modeled by a dichotomous result (again, a rather fancy way of saying a Yes/No result), the logistic-regression model is quite useful. In the simplest case, we try to use a mathematical model that will predict whether a specific event will or will not happen (expressed as the chance that the event will occur) as a function of some independent variable. In walkway-safety analysis, the first question is, what is the chance that person i will slip ([y.sub.i]), given that the friction between the shoe and the walkway is [x.sub.i]. Again, this is most simply modeled by what is known as a logistic curve. (e) The logistic curve is roughly s-shaped. (In the examples below, because the b parameter, analogous to the slope, is negative, the s-shape is sloped backwards.) The vertical limits of the curve are zero and one, just as the laws of probability restrict the chance of any event occurring to be between zero (no chance) and one (a sure thing).

For a simple logistic model, where the chance of an event occurring is a function of a single independent variable (again, analogous to the ambulance response-time model discussed above) there would be two parameters, which we can again call a and b. (f)

Parameter a is indeed analogous to the y-intercept, a, in the response-time model. The a parameter shifts the curve to the left and right, which changes the y-intercept.

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A change in the b parameter, on the other hand, as it does with the ambulance-response-time model, changes the slope of the logistics curve. The larger the absolute value of the b parameter, the steeper the transit of the logistics curve between its upper and lower extremities. I note in passing that, since both curves in the diagrams have a value of a of 1.0, both have the same y-intercept (0.73).

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Finally, the sign of the slope, b, defines the direction that the curve will traverse between its extremes. For positive b, the curve will go from zero to one as x increases. For negative b, the curve descends from one to zere as x increases.

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The Application of the Logistic Regression in Walkway Safety

Risk of falling is a natural place to apply the statistical analysis of logistic regression. As opposed to the ambulance problem above, the outcome we are measuring is not "time it takes to fall" (which would be on a continuous scale), but instead is "did the person fall?", a dichotomous (yes/no) variable. We know from experience that there is not a definitive "fall/ no fall" criteria for footing; one might cross an ice-covered sidewalk nine times without falling, and fall on the tenth crossing. (Or, perhaps, you would fall nine times out of ten.) The logistic regression allows us to describe the likelihood (probability) of falling, given a certain situation.

Logistic regression was first used (1989) in fall-factor identification in the context of analyzing what factors in observational studies were predictive of falls in rehabilitation and gerontological settings (Mion, et al., 1978, pp. 17-22, and Robbins, et al., 1989, pp. 1628-1633).

Application of Logistic Regression to Walkway Safety

Logistic regressions have been used to describe likelihood of falling, and to associate gait characteristics and probability of falling. In order to do this, we need to decide on which parameters we wish to use to predict whether someone will fall (Lord, et al., 1993, pp. 240-245). Many parameters could be chosen; consider the ice-covered sidewalk above. Extrinsic parameters (those associated with the environment) that would affect one's risk of falling might include type of footwear, walking speed, or temperature ("wet" ice is slipperier than "dry" ice), etc. In addition, you might have intrinsic characteristics (those associated with the person) to consider: age, level of fitness, presence of a health condition that affects balance, etc. Researchers have chosen different ways to characterize risk of falling.

Hanson et al., (Hanson, et al., 1999, pp. 1619-1633) and Burnfield and Powers (Burnfield, et al., 2006, pp. 982-995) used the available and required friction between the foot and the floor to characterize not only slips, but slips leading to falls. Hanson concentrated on the surface characteristics of the shoe and flooring surfaces. They chose a standard shoe with a PVC sole. Using a tribometer, they measured the Dynamic (that is, moving) Coefficient of Friction (DCOF) between this shoe and two flooring materials: vinyl composition tile (VCT; the standard industrial flooring tiles you see everywhere) and a low loop carpet. They then had subjects wearing these shoes walk down an adjustable ramp over these flooring surfaces, at angles of 0, 10 and 20 degrees. The Required Coefficient of Friction (RCOF) was directly measured with force sensors under the floor on the ramp.

In this simple model, if the RCOF is greater than the DCOF, then we would expect the subject to fall. Of course, in real life, this criterion is not completely black-and-white. However, Hanson did find that if the difference between DCOF and RCOF (DCOF - RCOF) was greater than 0.24, there was only a 5% probability of falling; if this difference was 0.08, this probability increased to 50%. Thus, using the logistic regression, a characterization of the available friction (DCOF) and required friction (RCOF) could be used to predict falls fairly accurately.

Burnfield and Powers (ibid.) also looked at the ability to predict slips and falls using the available and required friction. Their measurements were made with subjects walking on a level surface over a force sensor covered with vinyl tile, in both an uncontaminated and a contaminated state (the contaminant used was WD-40, a common spray lubricant). Using logistic regression, they evaluated the probability that a person would slip based on knowing both the available and required Coefficient of Friction; they could correctly classify individuals 89.5% of the time. They also developed a second regression model, based solely on the available COF. Note that this is a property of the shoe and the flooring, measured with the tribometer; there is no person walking when this is calculated. Using just the available COF, the regression model could correctly categorize fall/no-fall 78.9% of the time.

Researchers have also looked at predictors related to how a person is walking--are they walking slowly or quickly? Are they taking large or small steps? Brady et al. (Brady et al., 2000, pp. 803-808) used logistic regression to distinguish between slips leading to falls and slips from which one could recover. Predictors in her model included slip displacement (how far a person's foot slid along the ground), slip duration (how long, in seconds, they were slipping), the velocity of the slipping foot and the angle between the foot and the floor at initial heel strike. They looked at each of these parameters individually as a predictor of probability of falling, and also in combination (using a multivariate logistic regression). Using slip displacement alone, they correctly classified the probability of recovering from a slip approximately 70% of the time. Interestingly, they found that adding additional parameters to this model did not improve its accuracy (ibid.).

Moyer et al., (Moyer et al., 2006, pp. 329-343) looked at additional gait (walking) parameters when assessing "hazard risk" (the risk of a slip causing a fall). Parameters included in their model were the position and velocity of the heel at heel strike, gait speed, cadence, and step length. Moyer studied two populations of adults: a younger group and an older group (average ages 23.5 and 60.9 years respectively).

Using a multivariate logistic regression, they found that two variables, cadence and step length ratio (the step length divided by the leg length, to account for height of subjects) to be significant predictors of hazard risk. These parameters also interacted with one another: while both longer step length and faster cadence increased the hazard probability, as cadence increased, the hazard probability decreased, given a fixed step length; as step length increased, hazard probability increased given a fixed cadence. Interestingly, while older adults had shorter step lengths and lower cadences than did younger adults, the same relationship between increased step length and cadence and increased probability of falls was found in both populations.

In each of these studies, the logistic regression allowed the researcher to characterize a probability of a dichotomous event (fall) occurring, and to develop prediction equations which would allow one to evaluate the importance of various risk factors to the overall risk of falls.

Logistic Regression Used in Tribometer Characterization

Logistic regression has also been used to evaluate the instruments used to evaluate walkway safety, specifically, the Portable Inclinable Articulated Strut Tribometer (PIAST) class of instruments. Recall that a tribometer is a device that measures the friction at the interface between the floor (or any walking surface) and the shoe or sole material. The manner in which this interface is characterized by the PIAST tribometer is to repeatedly "drop" the tribometer "foot" onto the surface at increasing angles until a slip occurs; the incident angle at first slip describes the coefficient of friction. But as has already been described, even with the same materials as flooring and shoe, slipping is stochastic (that is, there is an element of randomness/probability involved; the process is non-deterministic).

In essence, wearing the same shoes on the same floor, sometimes you slip and sometimes you don't. Marpet, Medoff, and Besser used logistic regression to characterize this function for tribometers (Medoff, et al., 2007, p. 172, Besser, et al., 2007, pp. 182-183, Marpet, et al., 2007, Distribution on CD-ROM, and Medoff, et al., 2007, Distribution on CD-ROM) and to characterize a novel method for barefoot-friction metrology (Besser, et al., 2008, pp., 51-58). The logistic regression provides a powerful tool for improved understanding of how the tribometer can be used accurately to assess walkway slip-resistance.

The statistic itself has also been used as an evaluation tool for assessing biofidelity of walkway measurement techniques. Besser, et al., (ibid.) used logistic regression to evaluate three different instrumentation variations for in vivo characterization of barefoot walking. This area of study is of considerable importance, as many slips and falls take place in the bathroom, gymnasium or swimming pool deck, where the person is barefoot and the surfaces are wet. Such situations are problematic with the standard tribometer, as there is no good surrogate for human skin to use on the "test foot." Our device consisted of a standing frame and adjustable floor surface, and was used to assess validity of slip measurements with the person standing unconstrained, as well as standing and seated trials with the lower leg constrained to move vertically. We were looking at characterizing the differences between the three variations of the instrumentation. We varied the angle of the test surface (the independent variable) and tested repeatedly at a range of settings from the point where no slips were recorded to the point where slipping always occurred. Logistic regression was used to model the relationship between the test-surface angle and the chance of a slip.

The logistic regression parameters can give significant insight into which instrument variation does the best job. First, which instrumentation variation has the best repeatability (the same results on repeated testing; this is the same as saying that the logistic regression parameters a and b would be similar from test to test). Secondly, which instrumentation variation has the best discriminatory capability (which has the highest value of b, implying the logistic regression curve would be steep, providing a sharp delineation between the "fall" and "no fall" regions). Exhibit 11 is from Besser, et al. (ibid.):

The three variations were with the subject standing with the leg free to move (Unconstrained), standing with the leg constrained to move vertically move under subject power (Standing Unconstrained), and seated with the leg mechanically constrained to move vertically (Sitting Constrained). I The logistic regressions show that the (Unconstrained) (represented by solid lines in the graph) subjects had wildly differing Logistic Regression Curves, suggesting that this was not a viable test method. The two curves, representing two subjects, have widely differing a and b values; the curves neither look like each other nor are they located near each other. The Standing Constrained variation (represented by the dotted lines) showed poor discrimination (a low value of b). It was found that the Seated Constrained instrument variation (using a seated subject with the leg constrained to move vertically, and represented by the 'dash-dot' lines) provided optimal performance for assessing barefoot surface friction, having a high value of the b parameter and both the & and b parameters for different subjects each close to each other.

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Conclusion

Because the cost to both business and to society is significant, efforts to minimize fall accidents and their concomitant costs is desirable. Logistic regression is a natural candidate for mathematical modeling of fall accidents and, more recently, for the modeling of tribometers, that have a slip/no-slip output. By using logistic regression to model what contributes to fall accidents, and to model which tribometers best measure friction, costs to business and to society from this significant cause of accidents can be minimized.

References

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Marpet, M. I., H. P. Medoff, and M. P. Besser. 2007. Uncertainty Issues in Walkway-Safety Tribometric Testing: Isolating the Contribution of the Tribometric Instrument. 62nd Annual Meeting of the Society of Tribologists and Lubrication Engineers, Philadelphia, PA, June. Extended abstract in conference proceedings available on CD.

Medoff, H. P., M. P. Besser, and M. I. Marpet. 2007. The Characterization of the Slip-Test PIAST Tribometer by Characteristic Functions Based Upon Logistic Regression. 59th Annual Meeting of the American Academy of Forensic Sciences, San Antonio, TX, February. Abstract in conference proceedings, p. 172.

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Endnotes

(a) The reason that one cannot be more clear here is that falls are first categorized as occurring on one level or falling to a lower level. Many of the latter falls are precipitated by a slip, but the classification system in use does not, unfortunately, catch that information.

(b) A tribometer is a device for measuring the friction between a walkway (or a test surface) and a shoe or foot bottom (or a test foot).

(c) A dummy variable is the name that statisticians give to dichotomous variables.

(d) The criteria for a slip (Yes/No) is whether the friction that is required between the shoe and the walkway to prevent a slip is in fact available between the two: [f.sub.available] > [f.sub.required]

(e) Technically, the logistic curve is a representation of the natural logarithm of the odds of something occurring. If the chance of an event occurring is [p.sub.i], the odds of that event occurring is [[P.sub.i]/(1-[P.sub.i])]. The natural log of that ratio is called the logit. For example, if there is a 20% chance of an event occurring, the odds of it occurring area [[0.2]/(1-0.2)] = [[0.2]/[0.8]] =25% and the logit is-1.39.

Simply put, in logistic regression, rather than analyzing the relationship between x and y using simple linear regression, as we did when we analyzed the predictive relationship between travel distance and ambulance response time, we analyze, using regression, the predictive relationship between x and the logit(y). (f) a and b are in a different font, because they are different parameters than those in the ambulance response-time example.

Marcus Besser, Thomas Jefferson University

marcus.besser@jefferson.edu

Mark Marpet, The Peter J. Tobin College of Business, St. John's University

marpetm@stjohns.edu

Howard Medoff, Pennsylvania State University

hpm1@psu.edu

The cost of walkway accidents--pedestrian slips and falls--is substantial. To reduce the incidence of slips, and subsequent falls, companies install and maintain 'slip-resistant' floor surfaces. Monitoring floor slipperiness using an instrument called a tribometer allows businesses to record quantitative information that allows them to apply quality-control protocols to maintenance procedures, in order to minimize fall-accident rates.

In regression analysis, you are given data that has (hypothetically) been generated by a mathematical function whose parameters are not known, and you must estimate those underlying parameters. Ordinary regression is used to predict a continuous outcome. Logistic Regression is used when the outcome is dichotomous: yes/no. Because slipping--or not slipping--is a dichotomous event, and because Logistic Regression is a mathematical model that can explicitly model dichotomous events, it has recently been utilized in walkway-safety analysis. For example, Logistic Regression has been used to describe the likelihood of falling, and to associate walkway and gait characteristics to the probability of falling. Researchers have used the characteristics of the flooring surface materials, contaminants, and shoe sole materials and textures as factors that might help to predict increased risk of falls or the ability to recover from a slip; they have also considered aspects of normal gait to see if these might be associated with falling. Logistic Regression has been used by the authors to characterize tribometric instruments used to evaluate walkway safety, as well as to evaluate a novel method for barefoot-friction metrology.

In summary. Logistic Regression provides a powerful tool for improved understanding of how the tribometer can be used accurately to assess walkway slip-resistance.

The Objective of this Paper

This paper will explore how Logistic Regression has recently been applied to walkway-safety prediction and tribometer characterization.

Introduction: The Magnitude of the Problem

The cost of walkway accidents (pedestrian slip-, trip-, fall-, and misstep-precipitated injuries) is huge, both to society and to business. Rice and MacKenzie reported that, for 1985, the economic cost of slip and fall injuries to society--direct, morbidity, and mortality costs taken together--was estimated to exceed 37 billion dollars (Rice, et al., 1989). They found that fall accidents were the second largest generator of unintentional, accidental-injury costs, and the largest generator of accidental mortality in the elderly. Englander, Hodson and Terragrossa (Hodson, et al., 1996, 733-746) projected these costs to the year 2020, taking into account demographic trends, i.e., the aging of the population. They estimated that the cost to the United States would be over 64 billion dollars in 1995, and over 85 billion in the year 2020. Leamon and Murphy investigated the cost of walkway accidents to business; their 1995 research paper was entitled, rather understatedly, "More than a Trivial Problem," with an estimated per-worker cost of falls ranging from $44 to $550, depending upon the industrial sector. Leamon (Leamon, et al., 1995). Buck and Coleman (Buck, et al., 1985, pp. 949-958), Proctor and Coleman (Proctor, et al., 1988, pp. 269-285), and Proctor (Proctor, 1993, 367-377), studied walkway accidents in workplaces in the United Kingdom. The cost to the U.K. was thought to exceed 150 million pounds annually (1982 data).

The collection and analysis of fall-related injury statistics within the United States is accomplished by a number of governmental and private organizations, viz., the Bureau of Labor Statistics, the National Electronic Injury Surveillance System (NEISS), and the National Safety Council. The Bureau of Labor Statistics (BLS) classifies falls in a hierarchal manner. Exhibit 1 provides only a portion of their extensive list. For example: the percentage of total injuries for 'falls on the same level' (item 1.3.0) averaged 64% from 1999 to 2001. The average number of injuries (1999-2001) for falls resulting in days away from work was approximately 300,000 (Yuon et. al, 2006, pp. 83-93). Slips precipitate the plurality, if not the majority (a), of walkway accidents (Manning, D.P. et al., 1988, pp. 121-130, and Bentley et al. 1998, pp. 1859-1872).

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Exhibit 1. A classification of Falls as to Type, from the Bureau of Labor Statistics 1.0 Falls 1.0 Falls, unspecified 1.1. Falls to a lower level 1.1.0 Falls to a lower level, unspecified 1.1.1 Falls down stairs of steps 1.1.2 Falls from floor, dock, or ground level 1.1.2.0 Falls from floor, dock, or ground level, unspecified ... 1.2.0 Jump from a scaffold, platform, loading dock 1.2.1 Jump from structure, structural element, n.e.c. 1.2.2 Jump from nonmoving vehicle 1.2.9 Jump to lower level, n.e.c. 1.3 Falls on same level 1.3.0 Fall on same level, unspecified 1.3.1 Fall to floor, walkway, or other surface 1.3.2 Fall onto or against objects 1.3.9 Fall on same level, n.e.c. ... 1.9 Fall, n.e.c. n.e.c. means "not elsewhere classified" U.S. Dept of Labor, Bureau of Labor Statistics, 2007, pp. DE-4-DE5 This is just a small portion of the full list, which classifies thousands of items.

The National Safety Council (2002) reported that the slips and falls are the leading cause of death in the workplace and the cause of more than 20% of disabling injuries. The National Safety Council has also estimated that costs associated with employee slip-and-fall accidents (compensation and medical) are approximately $70 billion/year (National Safety Council, 2002). Liberty Mutual Workplace Safety Index reported that costs associated with falls on the same level were $5.7 billion/year (Liberty Mutual Research Institute, 2002). Regardless of the source, it is clear that the number of slip accidents and the resulting direct costs to business are significant and, due to the aging of the population, shows no signs of abating.

To reduce the incidence of slips and falls, a number of Risk-management-based loss prevention techniques have been discussed in industry periodicals. For example banking industry literature suggests that banks can reduce the walkway-accident-related claims of their customers and employees by developing slip-and-fall prevention policy and by analyzing losses to gauge the effectiveness of implemented policies and programs. Among the measures suggested are the design (or re-design) and maintenance of the bank's property to reduce the potential for slips and falls, good housekeeping, periodic inspections, and employee education and training (Erikson, 2004).

Food-service companies, viz., supermarkets and convenience stores, hotels, motels, and restaurants, are subject to slip-precipitated accidents and injuries both from customers and from their employees (Kohr, 1991, and DiPilla, 2006). These companies, the industry literature suggests, can mandate (and supply) the footwear used by their employees, design and install 'slip-resistant' kitchen floor surfaces, and implement an integrated program that includes facility design, inspection and maintenance. Use of checklists that require employees to monitor flooring conditions periodically (sweep logs), both in the store and the parking lot, are common measures, as is the installation of slip-resistant floor surfaces in areas that are likely to become contaminated.

Monitoring floor slipperiness, by using a portable tribometer, (b) allows business owners to collect quantitative information that allows them to determine the flooring condition with respect to pedestrian safety. The use of decision-science tools--both to characterize the chance of a fall and to analyze tribometers' results--is a relatively recent development in walkway safety. Because slipping--or not slipping--is a dichotomous pair of events, a mathematical model that reflects that dichotomous situation is especially useful for analyzing walkway-safety situations. Similarly, a tribometer that has a slip/no-slip result can be effectively characterized by a mathematical model that can model a dichotomous-event set. Logistic regression is just such a technique, where a dichotomous dependent variable is predicted employing a continuous variable (or variables) and/or, using dummy-variable sets, (c) to employ dichotomous, trichotomous, etc., events as independent variables. In other words, we can use a set of continuous inputs (like friction) and/or discrete inputs (like wet/dry) to predict either the chance, i.e., the probability of, a fall, or the chance (the probability) that a tribometer will indicate a slip at a given setting.

Introduction: Regression

Regression is a term used to describe the process of fitting a line or curve to a dataset. Logistic regression is the term used to describe the process of fitting a logistic curve to a dataset. Let's gently look at each of these ideas in some detail.

Regression analysis is the statistical equivalent of the TV game show, Jeopardy. In that game show, an answer is given, and the contestant must respond with the question that would have generated that answer. In regression analysis, one is given data that could have been generated by some hypothetical but unknown mathematical function, and one must estimate the underlying parameters of the function that would best generate that set of data. In some situations, regression analysis is more complex than Jeopardy: in the TV game, the contestant gets to choose from a number of categories; in regression analysis the categories themselves may or may not be known.

Ordinary Regression

For example, take the following data, which represents the emergency response time for an ambulance as a function of the straight-line distance between the emergency vehicle to the destination.

Exhibit 2a. An example of the type of data, in tabular form, that is amenable to analysis using regression analysis. Distance between ambulance (km) Response and destination (km) time (minutes) [x.sub.i] [y.sub.i] 0.5 1.3 1.0 1.8 1.2 1.7 1.6 2.1 1.8 2.4

It is easier to visualize such data if it is plotted in a graphical format. It is conventional to plot the variable that determines the relationship (called the independent variable, which here is the distance-as-the-crow-flies that the ambulance must travel) on the horizontal axis, and the determined variable (the dependent variable; here, the response time) on the vertical axis.

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The relationship between the variables can be determined by, first, hypothesizing what the underlying relationship should look like and, secondly, estimating the parameters of that relationship. In this example, we can hypothesize that there is a certain amount of time that it takes to receive the call from the dispatcher, followed by the time that it takes to travel to the destination site. The former time is approximately constant. The latter time is roughly proportional to the distance that the ambulance must travel; it is not exactly proportional because of the interplay between traffic, one-way streets, and the like. Based upon this information, we can hypothesize that the underlying form of the relationship looks like a straight line:

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The relationship is that of a straight line, where a is the y-intercept (the y value where the straight line crosses the x-axis, i.e., y where x = 0). And b is the slope of the line, the incremental time it takes to traverse each additional kilometer.

Exhibit 4: Relationship between variables of Dispatch Call and Ambulance Travel.

[y.sub.i] = a + [bx.sub.i]

where

[y.sub.i] is the response time in minutes for the ith call

a = the estimated time it takes to receive the call

b = the estimated typical speed that the ambulance travels, in km per minute

[x.sub.i] = the distance between the ambulance and the destination for the ith call

In high school, many of us studied this linear equation, estimating a and b by means of a clear-plastic straightedge, placing it so that about half of the data points were above the edge and half below, and "eyeballing" the slope and y-intercept. The problem with the eyeball method is that it cannot produce consistent results: what one person considers the best line may not be (and probably won't be) what a different person considers the best line.

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To achieve a method that will give consistent results, we turn to statistics. Without going into the minute details here, you can find the How-To in any solid Business-Statistics textbook. Suffice it to say (in true Jeopardy fashion) that the values of a and b that best give us the answers in the table above are: a = 0.9 minutes and b = 0.8 kilometers per minute (about 30 miles per hour). That is, assuming that the hypothetical underpinning, the equation [y.sub.i] = a + [bx.sub.i] reasonably represents the underlying process that generates response times as a function of distance between the ambulance and the destination, the best such equation would be

Response_Time (min) = 0.9 + 0.8 x Distance_Between_Ambulance_and_Destination (km)

Again, selecting the correct class of relationship is critical in making the analysis work. In Jeopardy, if you mistakenly think that you are in the Famous Ships category, and you are actually in the United States Presidents category, you are going to mistakenly answer Nimitz-Class Aircraft Carrier, rather than 34th president of the United States, when Eisenhower comes up.

Logistic Regression

There is a model: a class of relationships, where the answer is either: It Happened, or, It Did Not Happen. A brokerage house did or did not go belly up, an IPO was or was not successful. In the field of Walkway Safety, two very important questions have a Yes/No answer: (a) did a person slip or not (d), and (b) did a Variable Angle Tribometer indicate a slip. In those situations best modeled by a dichotomous result (again, a rather fancy way of saying a Yes/No result), the logistic-regression model is quite useful. In the simplest case, we try to use a mathematical model that will predict whether a specific event will or will not happen (expressed as the chance that the event will occur) as a function of some independent variable. In walkway-safety analysis, the first question is, what is the chance that person i will slip ([y.sub.i]), given that the friction between the shoe and the walkway is [x.sub.i]. Again, this is most simply modeled by what is known as a logistic curve. (e) The logistic curve is roughly s-shaped. (In the examples below, because the b parameter, analogous to the slope, is negative, the s-shape is sloped backwards.) The vertical limits of the curve are zero and one, just as the laws of probability restrict the chance of any event occurring to be between zero (no chance) and one (a sure thing).

For a simple logistic model, where the chance of an event occurring is a function of a single independent variable (again, analogous to the ambulance response-time model discussed above) there would be two parameters, which we can again call a and b. (f)

Parameter a is indeed analogous to the y-intercept, a, in the response-time model. The a parameter shifts the curve to the left and right, which changes the y-intercept.

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A change in the b parameter, on the other hand, as it does with the ambulance-response-time model, changes the slope of the logistics curve. The larger the absolute value of the b parameter, the steeper the transit of the logistics curve between its upper and lower extremities. I note in passing that, since both curves in the diagrams have a value of a of 1.0, both have the same y-intercept (0.73).

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Finally, the sign of the slope, b, defines the direction that the curve will traverse between its extremes. For positive b, the curve will go from zero to one as x increases. For negative b, the curve descends from one to zere as x increases.

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The Application of the Logistic Regression in Walkway Safety

Risk of falling is a natural place to apply the statistical analysis of logistic regression. As opposed to the ambulance problem above, the outcome we are measuring is not "time it takes to fall" (which would be on a continuous scale), but instead is "did the person fall?", a dichotomous (yes/no) variable. We know from experience that there is not a definitive "fall/ no fall" criteria for footing; one might cross an ice-covered sidewalk nine times without falling, and fall on the tenth crossing. (Or, perhaps, you would fall nine times out of ten.) The logistic regression allows us to describe the likelihood (probability) of falling, given a certain situation.

Logistic regression was first used (1989) in fall-factor identification in the context of analyzing what factors in observational studies were predictive of falls in rehabilitation and gerontological settings (Mion, et al., 1978, pp. 17-22, and Robbins, et al., 1989, pp. 1628-1633).

Application of Logistic Regression to Walkway Safety

Logistic regressions have been used to describe likelihood of falling, and to associate gait characteristics and probability of falling. In order to do this, we need to decide on which parameters we wish to use to predict whether someone will fall (Lord, et al., 1993, pp. 240-245). Many parameters could be chosen; consider the ice-covered sidewalk above. Extrinsic parameters (those associated with the environment) that would affect one's risk of falling might include type of footwear, walking speed, or temperature ("wet" ice is slipperier than "dry" ice), etc. In addition, you might have intrinsic characteristics (those associated with the person) to consider: age, level of fitness, presence of a health condition that affects balance, etc. Researchers have chosen different ways to characterize risk of falling.

Hanson et al., (Hanson, et al., 1999, pp. 1619-1633) and Burnfield and Powers (Burnfield, et al., 2006, pp. 982-995) used the available and required friction between the foot and the floor to characterize not only slips, but slips leading to falls. Hanson concentrated on the surface characteristics of the shoe and flooring surfaces. They chose a standard shoe with a PVC sole. Using a tribometer, they measured the Dynamic (that is, moving) Coefficient of Friction (DCOF) between this shoe and two flooring materials: vinyl composition tile (VCT; the standard industrial flooring tiles you see everywhere) and a low loop carpet. They then had subjects wearing these shoes walk down an adjustable ramp over these flooring surfaces, at angles of 0, 10 and 20 degrees. The Required Coefficient of Friction (RCOF) was directly measured with force sensors under the floor on the ramp.

In this simple model, if the RCOF is greater than the DCOF, then we would expect the subject to fall. Of course, in real life, this criterion is not completely black-and-white. However, Hanson did find that if the difference between DCOF and RCOF (DCOF - RCOF) was greater than 0.24, there was only a 5% probability of falling; if this difference was 0.08, this probability increased to 50%. Thus, using the logistic regression, a characterization of the available friction (DCOF) and required friction (RCOF) could be used to predict falls fairly accurately.

Burnfield and Powers (ibid.) also looked at the ability to predict slips and falls using the available and required friction. Their measurements were made with subjects walking on a level surface over a force sensor covered with vinyl tile, in both an uncontaminated and a contaminated state (the contaminant used was WD-40, a common spray lubricant). Using logistic regression, they evaluated the probability that a person would slip based on knowing both the available and required Coefficient of Friction; they could correctly classify individuals 89.5% of the time. They also developed a second regression model, based solely on the available COF. Note that this is a property of the shoe and the flooring, measured with the tribometer; there is no person walking when this is calculated. Using just the available COF, the regression model could correctly categorize fall/no-fall 78.9% of the time.

Researchers have also looked at predictors related to how a person is walking--are they walking slowly or quickly? Are they taking large or small steps? Brady et al. (Brady et al., 2000, pp. 803-808) used logistic regression to distinguish between slips leading to falls and slips from which one could recover. Predictors in her model included slip displacement (how far a person's foot slid along the ground), slip duration (how long, in seconds, they were slipping), the velocity of the slipping foot and the angle between the foot and the floor at initial heel strike. They looked at each of these parameters individually as a predictor of probability of falling, and also in combination (using a multivariate logistic regression). Using slip displacement alone, they correctly classified the probability of recovering from a slip approximately 70% of the time. Interestingly, they found that adding additional parameters to this model did not improve its accuracy (ibid.).

Moyer et al., (Moyer et al., 2006, pp. 329-343) looked at additional gait (walking) parameters when assessing "hazard risk" (the risk of a slip causing a fall). Parameters included in their model were the position and velocity of the heel at heel strike, gait speed, cadence, and step length. Moyer studied two populations of adults: a younger group and an older group (average ages 23.5 and 60.9 years respectively).

Using a multivariate logistic regression, they found that two variables, cadence and step length ratio (the step length divided by the leg length, to account for height of subjects) to be significant predictors of hazard risk. These parameters also interacted with one another: while both longer step length and faster cadence increased the hazard probability, as cadence increased, the hazard probability decreased, given a fixed step length; as step length increased, hazard probability increased given a fixed cadence. Interestingly, while older adults had shorter step lengths and lower cadences than did younger adults, the same relationship between increased step length and cadence and increased probability of falls was found in both populations.

In each of these studies, the logistic regression allowed the researcher to characterize a probability of a dichotomous event (fall) occurring, and to develop prediction equations which would allow one to evaluate the importance of various risk factors to the overall risk of falls.

Logistic Regression Used in Tribometer Characterization

Logistic regression has also been used to evaluate the instruments used to evaluate walkway safety, specifically, the Portable Inclinable Articulated Strut Tribometer (PIAST) class of instruments. Recall that a tribometer is a device that measures the friction at the interface between the floor (or any walking surface) and the shoe or sole material. The manner in which this interface is characterized by the PIAST tribometer is to repeatedly "drop" the tribometer "foot" onto the surface at increasing angles until a slip occurs; the incident angle at first slip describes the coefficient of friction. But as has already been described, even with the same materials as flooring and shoe, slipping is stochastic (that is, there is an element of randomness/probability involved; the process is non-deterministic).

In essence, wearing the same shoes on the same floor, sometimes you slip and sometimes you don't. Marpet, Medoff, and Besser used logistic regression to characterize this function for tribometers (Medoff, et al., 2007, p. 172, Besser, et al., 2007, pp. 182-183, Marpet, et al., 2007, Distribution on CD-ROM, and Medoff, et al., 2007, Distribution on CD-ROM) and to characterize a novel method for barefoot-friction metrology (Besser, et al., 2008, pp., 51-58). The logistic regression provides a powerful tool for improved understanding of how the tribometer can be used accurately to assess walkway slip-resistance.

The statistic itself has also been used as an evaluation tool for assessing biofidelity of walkway measurement techniques. Besser, et al., (ibid.) used logistic regression to evaluate three different instrumentation variations for in vivo characterization of barefoot walking. This area of study is of considerable importance, as many slips and falls take place in the bathroom, gymnasium or swimming pool deck, where the person is barefoot and the surfaces are wet. Such situations are problematic with the standard tribometer, as there is no good surrogate for human skin to use on the "test foot." Our device consisted of a standing frame and adjustable floor surface, and was used to assess validity of slip measurements with the person standing unconstrained, as well as standing and seated trials with the lower leg constrained to move vertically. We were looking at characterizing the differences between the three variations of the instrumentation. We varied the angle of the test surface (the independent variable) and tested repeatedly at a range of settings from the point where no slips were recorded to the point where slipping always occurred. Logistic regression was used to model the relationship between the test-surface angle and the chance of a slip.

The logistic regression parameters can give significant insight into which instrument variation does the best job. First, which instrumentation variation has the best repeatability (the same results on repeated testing; this is the same as saying that the logistic regression parameters a and b would be similar from test to test). Secondly, which instrumentation variation has the best discriminatory capability (which has the highest value of b, implying the logistic regression curve would be steep, providing a sharp delineation between the "fall" and "no fall" regions). Exhibit 11 is from Besser, et al. (ibid.):

The three variations were with the subject standing with the leg free to move (Unconstrained), standing with the leg constrained to move vertically move under subject power (Standing Unconstrained), and seated with the leg mechanically constrained to move vertically (Sitting Constrained). I The logistic regressions show that the (Unconstrained) (represented by solid lines in the graph) subjects had wildly differing Logistic Regression Curves, suggesting that this was not a viable test method. The two curves, representing two subjects, have widely differing a and b values; the curves neither look like each other nor are they located near each other. The Standing Constrained variation (represented by the dotted lines) showed poor discrimination (a low value of b). It was found that the Seated Constrained instrument variation (using a seated subject with the leg constrained to move vertically, and represented by the 'dash-dot' lines) provided optimal performance for assessing barefoot surface friction, having a high value of the b parameter and both the & and b parameters for different subjects each close to each other.

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Conclusion

Because the cost to both business and to society is significant, efforts to minimize fall accidents and their concomitant costs is desirable. Logistic regression is a natural candidate for mathematical modeling of fall accidents and, more recently, for the modeling of tribometers, that have a slip/no-slip output. By using logistic regression to model what contributes to fall accidents, and to model which tribometers best measure friction, costs to business and to society from this significant cause of accidents can be minimized.

References

Bentley, T. A. and R. A. Haslam. 1998. Slip and Fall Accidents Occurring During the Delivery of Mail. Ergonomics. 41:12: 1859-1872.

Besser, M. P., M. I. Marpet, and H. P. Medoff. 2007. Uncertainty Issues in Tribometric Testing: Isolating the Contribution of the Tribometer. 59th Annual Meeting of the American Academy of Forensic Sciences, San Antonio, TX, February. Abstract in conference proceedings, pp. 182-183.

Besser, M. P., M. I. Marpet, and H. P. Medof. 2008. Barefoot-Pedestrian Tribometry: In Vivo Method of Measurement of Available Friction Between the Human Heel and the Walkway. Industrial Health. Jan. 46(1): 51-58.

Brady, R. A., M. J. Pavol, T. M. Owings, and M. D. Grabiner. 2000. Foot Displacement but not Velocity Predicts the Outcome of a Slip Induced in Young Subjects While Walking. Journal of Biomechanics. Jul. 33(7): 803-808.

Buck, P. C. and V. P. Coleman. 1985. Slipping, Tripping, and Falling Accidents at Work: National Picture. Ergonomics 28(7): 949-958.

Burnfield, J. M., and C. M. Powers. 2006. Prediction of Slips: An Evaluation of Utilized Coefficient of Friction and Available Slip Resistance. Ergonomics. Aug. 1549(10): 982-95.

DiPilla, S. 2006. Managing Traffic Flow to Prevent Slips and Falls: Understanding the Traffic Patterns in Your Facility Can Reveal Opportunities to Reduce Your Exposure to Slip, Trip, and Fall Accidents. Food Management, November.

Englander, F., T. J. Hodson, and R. A. Terregrossa. 1996. Economic Dimensions of Slip and Fall Injuries. Journal of the Forensic Sciences. 41(5): 733-746.

Erickson, G. 2004. Protecting Against Slips and Falls. (St. Pauls's Traveler's) Community Banker, 2004.

Hanson J. P., M. S. Redfern, and M. Mazumdar. 1999. Predicting Slips and Falls Considering Required and Available Friction. Ergonomics. Dec. 42(12): 1619-1633.

Kohr, R. L. 1991. Accident Prevention for Hotels, Motels, and Restaurants. New York: Van Nostrand Reinhold.

Leamon, T. B. and P. L. Murphy. 1995. Occupational Slips and Falls:

More Than a Trivial Problem. Ergonomics 38: 487-498.

Liberty Mutual Research Institute for Safety. 2002. Annual Report of Science Activity.

Lord, S. R., J. A. Ward, P. Williams, and K. J. Anstey. 1993. An Epidemiological Study of Falls in Older Community-Dwelling Women: The Randwick Falls and Fractures Study. Australian Journal of Public Health. Sept. 17(3): 240-245.

Manning, D. P., et al. 1988. The incidence of Underfoot Accidents During 1985 in a Working Population of 10,000 Merseyside People. Journal of Occupational Accidents. 10, 121-130.

Marpet, M. I., H. P. Medoff, and M. P. Besser. 2007. Uncertainty Issues in Walkway-Safety Tribometric Testing: Isolating the Contribution of the Tribometric Instrument. 62nd Annual Meeting of the Society of Tribologists and Lubrication Engineers, Philadelphia, PA, June. Extended abstract in conference proceedings available on CD.

Medoff, H. P., M. P. Besser, and M. I. Marpet. 2007. The Characterization of the Slip-Test PIAST Tribometer by Characteristic Functions Based Upon Logistic Regression. 59th Annual Meeting of the American Academy of Forensic Sciences, San Antonio, TX, February. Abstract in conference proceedings, p. 172.

Medoff, H. P., M. I. Marpet, and M. P. Besser. 2007. The Characterization of Binary Output Walkway-Safety Tribometric Instruments (Slip Test Mark II and Mark III) by Characteristic Functions. 62nd Annual Meeting of the Society of Tribologists and Lubrication Engineers, Philadelphia, PA, June. Extended abstract in conference proceedings available on CD.

Mion, L. C, S. Gregor, M. Buettner, D. Chwirchak, O. Lee, and W. Paras. 1989. Falls in the Rehabilitation Setting: Incidence and Characteristics. Rehabilitation Nursing, 14 (1): 17-22.

Moyer, B. E., A. J. Chambers, M. S. Redfern, and R. Cham. 2006. Gait Parameters as Predictors of Slip Severity in Younger and Older Adults. Ergonomic, Mar. 1549(4): 329-343.

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Endnotes

(a) The reason that one cannot be more clear here is that falls are first categorized as occurring on one level or falling to a lower level. Many of the latter falls are precipitated by a slip, but the classification system in use does not, unfortunately, catch that information.

(b) A tribometer is a device for measuring the friction between a walkway (or a test surface) and a shoe or foot bottom (or a test foot).

(c) A dummy variable is the name that statisticians give to dichotomous variables.

(d) The criteria for a slip (Yes/No) is whether the friction that is required between the shoe and the walkway to prevent a slip is in fact available between the two: [f.sub.available] > [f.sub.required]

(e) Technically, the logistic curve is a representation of the natural logarithm of the odds of something occurring. If the chance of an event occurring is [p.sub.i], the odds of that event occurring is [[P.sub.i]/(1-[P.sub.i])]. The natural log of that ratio is called the logit. For example, if there is a 20% chance of an event occurring, the odds of it occurring area [[0.2]/(1-0.2)] = [[0.2]/[0.8]] =25% and the logit is-1.39.

Simply put, in logistic regression, rather than analyzing the relationship between x and y using simple linear regression, as we did when we analyzed the predictive relationship between travel distance and ambulance response time, we analyze, using regression, the predictive relationship between x and the logit(y). (f) a and b are in a different font, because they are different parameters than those in the ambulance response-time example.

Marcus Besser, Thomas Jefferson University

marcus.besser@jefferson.edu

Mark Marpet, The Peter J. Tobin College of Business, St. John's University

marpetm@stjohns.edu

Howard Medoff, Pennsylvania State University

hpm1@psu.edu

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Author: | Besser, Marcus; Marpet, Mark; Medoff, Howard |
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Publication: | Review of Business |

Geographic Code: | 4EUUK |

Date: | Mar 22, 2009 |

Words: | 5488 |

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