# Drinking patterns and drinking behaviors: theoretical models of risky acts.

Successful models linking drinking to subsequent injury require the complementary development of theories of drinking patterns and theories of drinking behaviors. Theories of drinking patterns characterize the primary dimensions of drinking that augment risks for injury, and theories of drinking behaviors characterize activities in which individuals participate during and after drinking and that expose drinkers to risk. This paper presents a theoretical approach to understanding these two aspects of risk for alcohol-involved injuries. In developing this approach, traditional methods of characterizing drinking patterns are reviewed, and their applicability to the prediction of drinking risks is evaluated (i.e., measures of total consumption, quantity/frequency measures, and measures of heavy drinking). The current approach provides a means of representing drinking risks in terms of three drinking measures--frequency, average drinks per occasion, and variance--and provides a mathematical representation of how injury risks can be predicted from drinking patterns.As a complement to the work on drinking patterns, the rather limited literature regarding drinking behaviors is reviewed with the objective of indicating the major components of human activities around alcohol. A skeletal framework for theoretically conceptualizing drinking behaviors is constructed around the idea that drinkers tend to maximize beverage quality and the amenity value of drinking locations within economic and time-energy budget constraints and outcome risks. That is, drinkers prefer to drink higher-quality beverages at locations where some form of entertainment is available (i.e., bars and restaurants). Yet drinkers must minimize costs in terms of time, money, and the possible risks associated with drinking.

A lot of things happen to people who drink. Among them are the pleasures of intoxication, the relaxation that comes from the sedating effects of alcohol, and the social camaraderie of drinking groups. Also among them are some rather unfortunate consequences: alcohol-related car crashes caused by impaired driving performance (Perrine et al., 1989), injuries arising from impaired judgment in using pedestrian walkways at busy intersections (Solnick and Hemenway, 1994; Holubowycz, 1995), increased risks of suicide and other forms of aggression related to the effects of alcohol on the neurochemical and perceptual-attentive bases of behavior (summarized in Giancola and Zeichner, 1995), and a number of other social consequences such as domestic disputes, lost work, and, not surprisingly, the occasional hangover affecting job performance (Clark and Hilton, 1991). The involvement of a particular individual in these outcomes depends, of course, on how much and how often he or she drinks alcohol. It also depends upon his or her participation in certain activities (i.e., driving, walking, latently aggressive interactions, and work). All of these outcomes require participation in certain patterns of drinking. All of them require context. One must drive and drink in order to drink and drive.

It is obvious that drinking patterns and drinking contexts, along with their interactions, form the basis for individual drinking problems. One must drive a car after drinking to be driving while under the influence of alcohol. At the individual level the behaviors of interest are well understood. Less well understood are the associations between drinking patterns and drinking behaviors aggregated over time. Here we have a different theoretical problem. In this case, patterns of alcohol use (e.g., drinking frequencies) and changes in routine human activities (e.g., the use of public transportation) may or may not alter the rates at which drinking driving occurs. The effects of drinking patterns and routine human activities on drinking and driving are conditional upon their linkage in time and space. Thus one may increase frequencies of drinking at home, leaving likelihoods of drinking and driving unaffected. For this reason, drinking driving and all other acute alcohol-related problems may be viewed as behavioral outcomes stochastically related to drinking and routine activity patterns, conditionally a temporal function of both. The empirical problem for the alcohol researcher is to capture enough of the stochastic variance in drinking patterns and drinking contexts over time to be able to predict "harmful" drinking outcomes.(1) The theoretical problem, of course, is more complex.

If one's only concern were with the prediction of drinking outcomes (e.g., rates of alcohol-involved crashes), one could simply enumerate all contexts in which drinking events occur (e.g., drinking at home, bars, restaurants, friends' homes, etc.), enumerate the extent of drinking in each of these contexts (e.g., conditional measures of frequency and quantity of drinking in each location), and determine the likely probability of driving after being in each of these locations. Indulging in the naivete of risk factor models, this would lead to a summary statement of the factors related to "harmful" drinking at "risky" locations in which drinking and driving was likely. This would not, however, be a theoretical statement of how these relationships came to be, but rather a static description of the relationships themselves. A theory of human behaviors related to drinking and driving would state how drinking and routine activity patterns come to be shaped by environmental contexts (i.e., genetic, physical, psychological, and social). The theorist would be concerned with the manifolds of human action, not human action itself. That is, the theorist would be interested in specifying the ways in which behaviors change in response to altered environmental inputs, not the facts of such changes. Such a theory would provide reasonable predictions of probabilities of drinking and driving and, more importantly, provide predictions regarding expected changes in the stochastic relationships between drinking, driving, and drinking and driving as environments change. Such a theory would be as applicable in Australia as in Africa, as useful in the US as in Great Britain.

This paper seeks to clarify a few of the relationships between drinking patterns and routine activities related to alcohol use that may help develop the basis for such a theory. It is hoped that these considerations will point the way to more thorough theory construction and indicate some of the areas in which additional empirical work is needed. In order to move quickly through the development of the models on which the arguments of the current paper are based, historical references are kept to a minimum. Additional references can be found in any of a large number of reviews (e.g., Clark and Hilton, 1991) and in a number of summaries of the relevant issues developed by the authors (Gruenewald, 1991; Gruenewald et al., 1990, 1993, 1995; Treno et al., 1993; Gruenewald and Nephew, 1994). The arguments presented are mathematical of necessity. An effort has been made to simplify the presentation; however, this was not possible without some sacrifice of important content.

Drinking patterns

Among all drinkers, time between drinking episodes, frequencies of drinking, and amounts consumed per drinking occasion all vary substantially over time. On any day, one either does or does not drink, and when one drinks, one consumes a given amount. Thus drinking patterns can be crudely separated into processes that define the onset of drinking events and processes that in turn define the continuation of drinking once drinking has begun (Gruenewald et al., 1990; Duffy and Alanko, 1992; Lemmens, Tan and Knibbe, 1992). These separable processes are considered individually and then combined for a discussion of a global model of drinking patterns.

Drinking episodes

Typically, contemporary researchers report rates of drinking episodes episodes using a single number: the frequency of drinking over time. Expressed interest in the temporal distribution of drinking episodes, or in the marginal distribution of drinking frequencies, is uncommon. Thus when one reviews studies of drinking episodes, one finds a peculiar disjunction between epidemiological and clinical observation. On the one hand, epidemiological studies assert the importance of measuring drinking frequencies (dating back at least to Strauss and Bacon, 1953), have advanced these observations to the study of drinking frequencies at various drinking levels (e.g., "moderate" and "heavy" drinking, Knupfer, 1966), and have entertained the measurement of variation in drinking frequencies over time (Cahalan et al., 1969). On the other hand, clinical observations assert the importance of temporal dependencies between drinking episodes (i.e., Jellinek's, 1960, delta alcoholics), have characterized "binge" drinkers as those who show an inability to abstain after each drinking episode, and consider observed drinking frequencies a byproduct of these temporal dynamics (Kahler et al., 1995). Although a few researchers have attempted to simulate the temporal dynamics of drinking episodes (e.g., Duffy and Alanko, 1992) and empirically explore time series of drinking data (Mundt et al., 1995), these studies have been limited in scope. Duffy and Alanko (1992) assume independence between drinking episodes. Mundt et al. (1995) suggest that temporal patterns of drinking may differ between alcohol dependent and non-dependent groups.

In order to provide some feel for the temporal dynamics of drinking episodes, Figure la shows a time series of drinking data from a recent study of drinking patterns (Epstein et al., 1995: the "hand classified" drinker). This figure shows quantities of drinks consumed per day over 122 days. As the figure indicates, drinking episodes were relatively frequent for this drinker, and the temporal pattern of drinking was punctuated with strong weekly periodicities. These periodicities are graphically emphasized in a reduction of the data to drinking episode format (Figure lb): a series of zeros and ones reflecting daily abstinence and drinking over time. This figure also shows that drinking episodes clump together in seven-day groups and suggests that there are changes in rates of drinking episodes over time. The suggested nonstationary and periodic patterns are confirmed in a logistic regression of the binary drinking episode data on linear and quadratic time trend and weekly dummy variables (G2 = 33.29, df = 7, p < .001). The suggested clumping of drinking events is confirmed in a significant autoregressive effect relating prior drinking episodes to current probabilities of drinking (G2 = 19.23, df = 1, p < .001) using the endogenous autoregressive model

(1) [y.sup.*][sub.t] = [X.sub.t]b + [ry.sub.t-1] + [epsilon]

where [y.sup.*][sub.t] represents the probability of initiating a drinking episode at time t (in log odds), t is the time index, [X.sub.t]b represents the exogenous effects of time, r is the autoregressive coefficient (also expressed in log odds), [y.sub.t-1] is the lagged drinking episode state, and [epsilon] iS an independently distributed error term.(2) Put simply, for this subject the probability of a drinking episode on each day depended on the date and day of the week, [X.sub.t]b, and whether or not the drinker had consumed alcohol the day before, [ry.sub.t-1]. Importantly, this drinker tended to persist in his or her drinking episodes with r > 0; drinking tomorrow was made more likely by drinking today (equivalent to Jellinek's (1960) definition of the inability to abstain characteristic of binge drinkers). Bootstrap confidence intervals for r were 1.485 [less than or equal to] r [less than or equal to] 2.012; as a consequence of drinking today, the odds of drinking tomorrow were increased by a factor between 4.41 and 7.48. Thus, at least for this drinker, current drinking probabilities are nonstationary and are dependent upon prior drinking events.(3)

The periodic nature of drinking episodes is brought out more strongly through a spectral analysis of the data presented in Figure 1c. Spectral analysis represents an alternative approach to autoregressive models of time series data (equation 1) by shifting decompositions of the series from the time to the frequency domain. Instead of addressing the dependencies in observations over time, spectral analysis focuses upon the periodicities to be found in time-ordered data. The two approaches to analyses of time series data can be shown to be equivalent (Box and Jenkins, 1976), but they differently inform the user about the characteristics of the processes that underlay the data (Shiavi, 1991). In this figure, the frequency decomposition of the ordering of drinking events reveals two primary features: first, there is a strong frequency component related to weekly drinking, indicated by the arrow in the figure; second, there is a dominance of low-frequency spectral components in the series. The former indicates important periodicities in frequencies of drinking. The latter supports the autoregressive effect observed in the previous analysis; positive autoregressive processes produce a predominance of low-frequency spectral components in Fourier analyses of time series data (Bloomfield, 1976).(4)

Providing some contrast to this drinking pattern, Figure 1d presents a spectral analysis of drinking episode data from a second drinker (Epstein et al., 1995: the "episodic" drinker). The power spectrum of drinking episodes for this drinker again shows a strong weekly periodicity (arrowed). However, although this drinker also exhibited strong autoregressive effects ([G.sup.2] = 7.12, df = 1, p < .001), r was substantially less than zero. Bootstrap confidence intervals for r were -1.150 [less than or equal to] r [less than or equal to] -.544; as a consequence of drinking today, the odds of drinking tomorrow were decreased by a factor between .32 and .58. That is, the probability of drinking on the current day was suppressed if drinking had occurred on the previous day. For this reason, the spectral analysis of this subject's data shows an abundance of high-frequency components. Episodic drinking is characterized by occasional "heavy" drinking followed by desistance.

As these examples make clear, there is little reason to believe that drinking episodes are either stationary in time or independent of one another, supporting clinical observation. Rather, drinking episodes are both time dependent (see also Lemmens and Knibbe, 1993) and endogenous autoregressive. That is, the occurrence of one drinking event may enhance or depress the likelihood of another. Thus, as the point we wish to make here, not all drinking frequencies are alike. Reports of the average drinking frequency for a person or a group of persons misses a key facet of alcohol use: the temporal patterning of drinking events. Obviously, if the periodic use of alcohol exhibited by both these cases is focused on intervals (e.g., weekends) during which certain activities are more likely than others (e.g., driving after drinking), the binge drinker is more likely than the episodic drinker to be exposed to risk.(5) The binge drinker is more likely to repeat his or her drinking event over the weekend and the episodic drinker is less likely.

Continued drinking

The consumption of alcohol within drinking episodes has been characterized by measures derived from two different perspectives on drinking patterns. When it comes to measuring drinks consumed per drinking episode, the most straight forward procedure is to ask subjects how much they "typically" drink on each drinking occasion (Strauss and Bacon, 1953). The strength of this approach, like the strength of the direct approach to evaluating drinking frequencies, is that a single number can be taken as representing this aspect of drinking patterns. The weakness is that the approach fails to recognize any temporal variation in this measure. An alternative to this procedure, popularized by the work of Cahalan et al. (1969), is to measure frequencies of consumption at graded quantities; the number of times one has had four or more drinks, eight or more drinks, and so on. This approach emphasizes the risks of "heavy" drinking (e.g., at eight or more drinks, Knupfer, 1966). This approach is also the basic source of the array of "heavy" drinking indices in the literature (with published criteria covering the astounding range of less than one to eight or more drinks per day (Abel and Kruger, 1995; Gruenewald et al., 1990)).

It is important to note that the introduction of measures of "heavy" drinking to the research literature emphasized the essential need to obtain some measure of the long tail of the daily drinking distribution. For, it was argued, it is essential to know to what extent individual drinkers go beyond average levels of consumption to drink at levels at which greater social and personal risks would be entailed (Cahalan et al., 1969). In one respect this assertion is incontrovertible; in another, dubious. It is incontrovertible in the sense that drinking events in the extreme tails of the daily drinking distribution (e.g., the number of times one drinks 12 or more drinks) will be associated with all manner of problematic outcomes (e.g., difficulties driving, walking, etc.). So for this reason alone measuring "heavy" drinking is important. It is dubious in the sense that it is difficult, at best, to ascertain what constitutes an average level of drinking against which to compare the "heavy" drinking of an individual. What is "heavy" for one is not "heavy" for another--due, for example, to differences in body mass (Mirand and Welte, 1994), and, given a degree of tolerance to alcohol, different drinkers (e.g., "light" versus "heavy") have different experiences of "heavy" drinking.

Although it is rarely directly stated, the concept of continued drinking is not alien to the literature on drinking patterns (reviewed in Gruenewald and Nephew, 1994). The term refers, simply enough, to the likelihood of continued drinking beyond some specified drinking level. All measures of "heavy" drinking are of this type. The differences between the general study of continued drinking patterns and individual studies of "heavy" drinking are that the former studies are theoretically motivated and a priori agnostic with respect to the notion of "heavy" drinking itself. The theoretical motivations for general studies of continued drinking patterns are (1) the construction of a model of the drinking process adequate to capture all possible measures of drinking patterns; (2) a description of the drinking process that generally reflects the forces underlying drinking within any given drinking episode; and (3) the design of a theory of drinking patterns adequate to characterize drinking-related risks (Gruenewald and Nephew, 1991). The agnosticism of this approach to the notion of "heavy" drinking arises from the demonstration that "heavy" drinking criteria are arbitrarily aligned with drinking risks; depending on circumstances, these criteria may be good for one purpose and bad for another.

In outline, the theoretical approach states that on any drinking occasion (a) the probability of continued drinking is a nonincreasing function over continued drinks; (b) this function can be used to characterize both continued drinking and individual drinking probabilities; and (c) the distribution of drinking probabilities has a mean and a variance reflecting average drinking quantities and the variance in drinking levels over drinking occasions. Underlying the conceptualization of the model is an implicit decision process in which individuals choose to continue drinking until a target level is reached and the likelihood of continued drinking goes into decline. This model is briefly outlined here.

Figure 2a presents the continued drinking function constructed from the data presented in Figure la. Along the x-axis of the figure is the continued drink index, x = 1, 2, 3, . . .; the first continued drink is the second drink consumed on any drinking occasion. Along the y-axis of the figure is the probability of consuming the xth continued drink calculated across drinking occasions. The dashed line connecting the black circles identifies the empirical continued drinking rates calculated directly from the data. The empirical estimates were obtained by summing across all drinking days to determine the frequencies with which the individual had 1, 2, 3, . . . or more drinks ([f.sub.1], [f.sub.2], [f.sub.3], . . .) and calculating the continued probabilities [f.sub.2]/[f.sub.1] for the first continued drink, [f.sub.3]/[f.sub.1] for the second continued drink, and so on. The empirical continued drinking function is, as desired, nonincreasing and, in fact, decreases toward zero over 24 continued drinks. The solid curve fit to these data is an approximating function developed by Gruenewald and Nephew (1994) to fit continued drinking self-reports from survey data:

(2) p(x) = [ax.sub.b] / (1 + [ax.sub.b]) :b<0.

The parameters of the model reflect the initial probability of continued drinking, a/(1 + a), and the slope of the continued drinking probability function, b. Greater values of b indicate greater difficulties desisting from continued consumption (loss of control, Gruenewald, 1991). As shown, the log-logistic model roughly approximates the empirical continued drinking function.(6) To interpret the empirical data, note that this drinker consumed eight or more continued drinks (nine or more drinks) on 40% of his or her drinking occasions (the model approximates this figure at a low 20%).

The function describing the probability of continued drinking, p(x), provides the basis for obtaining the drinking probability function, g(x); the probability that x-and-only-x continued drinks are consumed on any occasion. The continued drinking function can be used to construct g(x) through the recursive equation

(3) g(x) = p(x) - p(x - 1)

iteratively applied to continued probabilities, x = 1, 2, 3, . . . n. Noting that the probability of having two or more drinks is given by p(l), then the probability of having one-and-only-one drink (continued drink x = 0) is 1 - p(1), or 1 - a/(1+a) from equation 2. The probability of having two-and-only-two drinks is given by g(1), three-and-only-three drinks by g(2), and so on. This function is shown as the unimodal distribution toward the bottom left of Figure 2a.

The mean and variance of the drinking probability function, g(x), are given by

(4) dpo = [sigma] (x+1)g(x)

and

(5) [s.sub.2] = [sigma]((x+1) - dpo)[sub.2]g(x)_

summed over all values of continued drink index x = 0, 1, 2, . . . n. As shown in Figure 2a, the drinking probability function, g(x), has a number of valuable properties. It is unimodal, has an estimable mean (average drinks per occasion, dpo), and when multiplied by frequencies of having one or more drinks ([f.sub.1]) can be used to estimate total numbers of drinks consumed across all levels of continued drinking. This latter feature is of particular importance because it allows estimation of drinking probabilities from survey data at unobserved levels, interpolating some points and extrapolating to others (i.e., those not covered by direct questions). As a validation of the consistency of these methods with those previously used in the literature, Gruenewald and Nephew (1994) showed that estimates of modal drinking(7) and average drinks per occasion obtained from the model reflected levels of "typical" drinking obtained from other self-reports. Thus self-reported "typical" quantities consumed appear to reflect some compromise between the mode and the mean of the drinking probability distribution. As one would expect from this observation, the authors showed that the resulting estimates from the standard quantity--frequency questions overestimated low levels of consumption and underestimated high levels of consumption.(8)

As shown in Figure 2b, the variance of the underlying drinking probability function is very important to the characterization of drinking. For the "hand classified" drinker used in this example, the obtained mean was dpo = 6.4 and the obtained variance was [s.sup.2] = 10.7. At the same average quantity consumed per occasion, halving the variance reduces the tails of the distribution and doubling the variance inflates the tails of the distribution.(9) (The y-axis is displayed in probability coordinates to emphasize this fact.)

The continued drinking function, p(x), and the drinking probability function, g(x), can both be used to express the underlying forces that form drinking patterns within drinking occasions. As the basic theoretical model suggests, there are unobserved forces that underlie both the persistence and the desistance of drinking within any drinking occasion (Gruenewald, 1991). These may be revealed by estimating the probability of having each continued drink net of the probability that one has consumed up to that drinking level (equal to the probability of drinking at that level and beyond):

(6) h(x) = g(x)/p(x)

As in(x) increases, the likelihoods of having the next drink increase over drinks (expressing the dominance of forces that encourage persistence).(10) As h(x) decreases, the likelihoods of having the next drink decrease over drinks (expressing the dominance of forces that encourage desistance). Figure 2c shows a plot of this function over continued drinks for the "hand classified" drinker. The peak of this function occurs at 6.3 continued drinks (or 7.3 drinks on any occasion).

The arbitrariness of "heavy" drinking criteria

The agnosticism of the model with regard to "heavy" drinking is motivated by considerations of the drinking probability function, g(x). "Heavy" drinking is usually defined by a single criterion applied to the right tail of this function. Presumably, beyond this point there is something unique that elevates risk for harmful events. However, t4e arbitrariness of these criteria is revealed when one considers that all drinking beyond the "heavy" criterion is somehow equivalent between drinkers. For example, it is assumed that the impact of drinking beyond the criterion is equivalent between respondents who rarely drink beyond this level and, in the current instance, the "hand classified" respondent who drinks beyond this level 45% of the time (Figure 2a, continued drink index = 7). A reasonable alternative to this approach might be to determine "heavy" drinking by a criterion based on properties of the drinking probability function itself (e.g., dpo and [s.sup.2]): assessing "heavy" drinking as any drinking beyond one standard deviation above the mean. Of course, this might relativize the definition of "heavy" drinking too much; one individual might drink "heavily" whenever drinking beyond one drink. The "hand classified" drinker would drink "heavily" whenever he or she consumed beyond nine. The virtue of "heavy" drinking criteria, that they identify drinking levels at which most consumers are quite seriously impaired, would be lost.(11)

A preliminary formulation of drinking patterns

As suggested by the previous discussions, the current approach characterizes drinking patterns along several continuous dimensions of a continuous drinking space. Binge drinking fades into episodic drinking as the autoregressive properties of drinking episodes change character. High and low frequencies, quantities and variances of drinking differ only by degree. Thus, in the crudest terms, one may drink with some drinking frequency, f, some average drinks per occasion, dpo, and some variance, [s.sup.2]. If one does not drink at all, abstention can be represented by the triplet {0, 0, 0}. If one drinks with some frequency but drinks only one drink on every drinking occasion, this pattern can be represented by the triplet {f, 1, 0}. These rudimentary aspects of drinking patterns describe a three-dimensional state space in which individual loci represent individual drinking patterns. Distributions of individuals within this drinking space represent population distributions of drinking patterns among different consumers. Movements of individuals through this space represent drinking patterns as they evolve over the life course. Although this triplet needs supplementation by parameters characterizing other aspects of consumption (e.g., the autoregressive parameter, r, equation 1), the three parameters are sufficient to capture much, if not all, of the flavor of prior efforts at establishing drinker and drinking types.(12) Thus the current model incorporates Strauss and Bacon's (1953) formulation of drinking in terms of frequencies and quantities of use, explicates the ambiguities of "heavy" drinking, and provides a means of operationalizing Jellinek's (1960) characterization of inability-to-abstain.(13)

Risk and incapacitation

Although theoretically informative, the development of this analysis of consumption patterns would have little practical utility without application to alcohol-related problems. In this regard the model finds its best use in the analysis of alcohol-related acute injuries. While it is beyond dispute that risks of involvement in acute injury increase with the use of alcohol, applications of the model in this context show how drinking patterns may best be measured to predict injury outcomes.(14)

Incapacitation and risk

It is more than evident that alcohol consumption incapacitates a variety of physical and psychological processes. As suggested by the rather extensive experimental literature (reviewed in Kruger, 1995), it is reasonable to assert that the use of alcohol will eventually incapacitate every behavior. Over a number of drinks on any occasion there will be a decreased ability to perform any task, and the probability of failure at each task will increase to 1.00. Thus behaviors are incapacitated over drinks, with some simple behaviors (e.g., elementary reaction times) incapacitated more cleanly than more complex ones (e.g., driving a car; Moskowitz et al., 1985). It is also more than evident that different drinkers drink to different extents on different drinking occasions and so differentially expose themselves to the incapacitating effects of alcohol. Thus drinking patterns can be expected to interact with the incapacitating effects of alcohol to produce behavior failure.

A demonstration of the link between drinking patterns, incapacitation and probabilities of behavior failure is presented in Figure 3a. In this figure, the drinking probability function, g(x), for the "hand classified" drinker from Figure 2a is plotted along with a theoretical incapacitation function, i(x).(15) As presented, probabilities of incapacitation increase slowly at relatively low levels of consumption (five through seven continued drinks), accelerate, and then decelerate again at high levels of consumption (thirteen through fifteen continued drinks). Looking at the drinking probability function, although some proportion of the consumer's drinking takes place when levels of incapacitation are low (e.g., continued drinks five through seven), some drinking also takes place when levels of incapacitation are high (e.g., continued drinks ten through twelve). At these levels the risks of behavior failure are relatively high.

Noting that the unobserved incapacitation function is independent of the drinking probability distribution, the risk of the behavior failure at each continued drinking level is given by the cross-product of the two probabilities:

(7) r(x) = g(x) i(x)

Summing these probabilities across continued drinks provides the (also unobserved) cumulative risk function, R(x). As also shown in Figure 3a, this function asymptotes at some value between 0 and 1, depending on the relative positions and shapes of both i(x) and g(x). With g(x) all the way to the right of the incapacitation function, the target behavior will be incapacitated on most drinking occasions and the cumulative risk will approximate 1.00. With g(x) all the way to the left of the incapacitation function, the target behavior will rarely be incapacitated on any drinking occasion, and the cumulative risk will approximate 0.00. At asymptote, cumulative risk represents the probability of behavior failure over drinking occasions for an individual drinker, taking into account his or her drinking pattern and unique incapacitation function.(16) Thus the estimate of cumulative risk represents the proportion of expected drinking events in which behavior failure will be realized. Importantly, the measure assumes that the drinker always participates in the target act after every drinking occasion (e.g., always drives after drinking). Thus, at large values of x, R(x) provides an upper bound assessment of risk defined as the cumulative probability of target behavior failure (summarized as the cumulative risk, R).

Figures 3b and 3c demonstrate the importance of differences in drinking patterns to the risks of behavior failure. Figure 3b presents the effects of differences in location of the drinking probability function on cumulative risk. With dpo = 3.7, the leftward function presents a cumulative risk estimate of .036. With dpo = 6.4 (the "hand classified" respondent), the center function presents a cumulative risk estimate of .089. With a dpo of 9.4, the rightward function presents a cumulative risk estimate of .274. (A factor of 8 separates the low and high dpo estimates.) Figure 3c presents the effects of differences in the variance of the drinking probability function on cumulative risk. The halved variance function presents a cumulative risk estimate of .058. The standard ("hand classified") variance function presents a cumulative risk estimate of .089. The doubled variance function presents a cumulative risk estimate of .115. (A factor of 2 separates the low and high variance estimates.) Clearly, greater variance in drinking quantities can be associated with greater risks of incapacitation. Figure 3d presents an alternative scenario, of great qualitative importance, in which a rightward function displays a drinking pattern with reduced and enhanced variance. Significantly, in this situation increased variance in drinking patterns reduces risks of incapacitation. Thus the effects of variance on risks of incapacitation are determined by the location of the drinking probability function relative to the incapacitation function. Although it is likely that most configurations will lead to a positive relationship between drinking variance and risk, in some configurations (i.e., a leftward incapacitation function relative to a rightward drinking probability distribution) increased variance may be related to reduced risk.

Participation in risky events

Theoretical niceties aside, one might be wondering at this point what good is an unobservable theoretical estimate of cumulative risk? Moreover, what good is a measure of risk that requires that the individual drinker participate in the target behavior on every drinking occasion? Obviously, not very much good at all. On the other hand, if one knows the relative proportion of occasions on which a drinker participates in the target behavior contingent upon drinking, one knows a very great deal indeed. For then it is possible to transform unobserved cumulative risks into statements about the probability of involvement in the target event (e.g., drunken driving). One can also provide statements about the expected form of relationships between drinking patterns and the expected problematic outcome. Under the assumption that drinkers participate in the target behavior (e.g., driving) on a constant proportion of all drinking occasions, specific expectations about the best means of relating drinking patterns to probabilities of behavior failure (e.g., drunken driving) can be developed.(17)

Drinking patterns and drinking risks

One agenda of the current paper is to make a clear theoretical statement of scientifically plausible relationships between measures of drinking patterns and drinking problems. From the perspective of the current model, using total consumption to predict harmful outcomes is rather like using an object's volume to determine whether it can fit through a hole. Although not irrelevant to the situation, it is not to the point either. One would rather know the height, width and depth of the object, just as one would like to know the f, dpo and [s.sup.2] of drinking. It is the evaluation of the relationships of these measures of drinking patterns to drinking problems that will provide a rational scientific basis for evaluating drinking risks. Just as the total volume of an object will not reveal anything regarding its figure, the total volume of drinking will not reveal anything regarding the dimensions of consumption. Nor will the total volume of drinking inform studies of drinking risks (e.g., Norstrom, 1995; Rehm and Sempos, 1995; Room et al., 1995). To the degree that the separate dimensions of consumption separately affect drinking risks, measures of total drinking may be unrelated or perfectly related to probabilities of harmful outcomes. Total drinking may be equivalent between two subpopulations, yet have entirely different relationships to measures of harm.

In the absence of additional information regarding stochastic patterns of drinking, the three dimensions of consumption, f, dpo, [s.sup.2], are assumed to characterize an individual's drinking pattern over fixed periods of time. As shown above, with measures of these values and some assumptions about the form of the underlying incapacitation function, the likelihood that a particular drinking pattern will lead to a particular problem outcome can be computed. Of course, as also noted above, it must be assumed that the conditional probability of involvement in the potentially harmful activity (e.g., driving) after drinking remains constant over drinking patterns and drinking events themselves. With these assumptions in place, it is possible to make several qualitative predictions regarding the relationships between these measures of drinking patterns and any harmful outcome for alcohol consumers:

Risk levels:

1. Controlling for relative frequencies of consumption and variance in drinking quantities, probability of involvement in a harmful outcome will be an increasing decelerating function of dpo (Figures 3a and 3b).

2. Controlling for relative frequencies of consumption and average drinking levels, probability of involvement in a harmful outcome will be an increasing decelerating function of variance (Figure 3c).

3. Controlling for relative frequencies of consumption, at the highest drinking levels probability of involvement in a harmful outcome will be a decreasing function of variance (Figure 3d).

The relationship between frequencies of alcohol use and alcohol problems, of course, is itself a contingent and somewhat problematic one. Thus at low average quantities consumed (i.e., low dpos), it is expected that increases in frequencies of use will have negligible effects on probabilities of harmful outcomes. At higher quantities, it is expected that increases in frequencies of use will have greater effects on these probabilities.(18) In addition, it is to be expected that unless the individual participates in the target event to some degree (e.g., driving a car after drinking), behavior failure will never occur. Again, one must drive after drinking in order to drink and drive.

In order to capture these effects in a simple model of the relationships between frequencies of use and probabilities of behavior failure (e.g., drunken driving), a number of simplifying assumptions must be put in place. First, it is necessary to assume that dpo and [s.sup.2] remain stable over time. Thus the conditional probability of a harmful event on any given drinking occasion may be summarized by the cumulative estimate of risk per drinking occasion, R. Second, it must be assumed once again that the probabilities of involvement in the target behavior are constant over drinking occasions. That is, all other things being equal, environmental circumstances remain much the same from episode to episode of drinking. In the case of drunken driving these would include where one usually drinks, whether one drove to the place of consumption, whether one is inclined to drive home after drinking, and so on. These are the environmental circumstances that are assumed to be constant over occasions and are naively represented here by the "environmental" probability [P.sub.e] Keeping these conditions in mind, and noting that the observed relative frequency of drinking can be represented by [f.sub.r] = [f.sub.1]/T (where T is the time base of observation, 0 [less than or equal to] [f.sub.r] [less than or equal to] 1), the probability of observing a failure over k episodes (e.g., days on which drinking was possible) can be given by

(8) P[[f.sub.r], [P.sub.e], R, k] = 1 - [1 - [f.sub.r][P.sub.e]R][sup.k]

(the complement of the probability of obtaining k successes): a cumulative distribution function for a geometrically distributed discrete random variable (Ross, 1993).(19) Importantly, p[[f.sub.r], [P.sub.e], R(x), k] will be a positive decelerating function over frequencies of alcohol use.(20)

An examination of equation 8 shows that as the likelihood of a harmful event contingent on drinking patterns decreases, R [right arrow] 0, the probability of observing a failure also decreases, and, incidentally, the slope relating relative frequencies of use to probabilities of problem outcomes decreases. Similarly, as [P.sub.e] [right arrow 0] , probabilities of observing a failure decrease, and the slope of the frequency function decreases. Thus two statements regarding relative frequencies of drinking (and thereby exposures to risk) can be made:

Risk exposure:

1. Controlling for average drinking levels, variances in quantities consumed, and environmental circumstances that engage individuals in potentially harmful activities, probability of involvement in a harmful outcome will be an increasing decelerating function of frequencies of use.

2. Controlling for variances in quantities consumed and environmental circumstances that engage individuals in potentially harmful activities, probability of involvement in a harmful outcome will be related to the interaction of average drinking levels and frequencies of alcohol use.

As the model proposed here suggests, then, the relationships between drinking patterns and harmful events is expected to be complexly nonlinear, although having an expected general form, and subject to the largely unknown relationships between drinking patterns and the environmental circumstances in which drinkers find themselves when consuming alcohol (e.g., choices of places to drink). It is the strong expectation from these modeling activities that analyses of the relationships between drinking patterns and drinking problems that incorporate these relationships will outperform those based on single measures of total consumption or frequencies of use, dual measures of quantities and frequencies consumed, and the triplet f, dpo, [s.sup.2] itself. Thus well-developed models of the relationships between drinking patterns and drinking problems will provide better predictions of problem outcomes.

Routine activities and the optimization of drinking behaviors

To this point, the theoretical arguments developed in this paper have shown how measures of drinking patterns may be related to alcohol-related outcomes through a model of the incapacitating effects of alcohol. As should be obvious from the various caveats mentioned above, the major missing part of this theoretical puzzle is the role played by individual routine activities in establishing the contexts in which drinkers are exposed to harm. Quite simply, this issue has been assumed away. It has been assumed that the likelihoods of participation in routine activities that expose drinkers to risk are constant. This assumption is no doubt unrealistic.

Rightfully, the study of those behaviors in which individuals engage while drinking should complement studies of drinking patterns and risks. Thus one would expect that a rudimentary theory of how people become engaged in activities in which injury events occur would be a developed part of the literature. Surprisingly, considerations of the mechanisms of consumer choice related to locations of drinking have not been the subject of much discussion in the literature on alcohol problems. There has been a recent spate of articles that have sought to explicate the ways in which consumers find themselves in situations in which drinking is a potentially risky activity--for example, at bars and restaurants where one is expected to drive home (Single and Wortley, 1993: Casswell et al., 1993; Stockwell et al., 1993). These have focused, however, on descriptions of the patterns of choice, not on models of the choice process. Thus they show that there is substantial variation in consumers' choice of drinking locations across demographic variables (e.g., gender, age, education, income, and marital status). Drinking at bars appears greatest among young, low income, unmarried males; consumption at restaurants is greatest among older, more highly educated females with higher incomes (Single and Wortley, 1993). Self-reported alcohol problems vary as a function of choices of drinking environments (hotels, taverns, and clubs) and typical quantities consumed (Casswell et al., 1993). Violent incidents (e.g., arguments or fights) appear most likely to occur among young male heavy drinkers who drink in licensed premises (e.g., bars and restaurants; Stockwell et al., 1993).

Exhibiting the facts of all this variation in routine activities associated with drinking does not explain, of course, how individual drinkers go about making the drinking choices they do. In order to extend these descriptions to a far more generalizable predictive model of drinking behaviors, one must conceptualize the ways in which the desire to drink (e.g., as represented in drinking patterns) combines with the social and economic capital of drinkers to produce drinking behaviors. The beginnings of one such approach have been developed by Gruenewald et al. (1995). They suggest that net of their individual drinking patterns, individual drinkers seek to maximize both the quality of beverages consumed and the amenity value of drinking places. Simply, drinkers want to buy the "best" beverages (treated by the authors as the most expensive per unit pure ethanol) at the "best" environments (treated by the authors as any on-premises drinking location). Constrained by their incomes and time-energy budgets (represented by marital and family commitments), the attempt to maximize quality and amenity values shapes drinkers' preferences for beverage types and drinking`locations. Thus the authors show that increased income is related to greater utilization of restaurants and to purchases of beverages of greater quality. Increased time-energy budgets are related to greater consumption at all on-premises establishments. Importantly, the results of the study also show that greater frequencies of use of alcohol are related to reductions of consumption at on-premises establishments. That is, the greater the amount of disposable income likely to be used for the consumption of alcohol, the lower the amenity value attached to its consumption.

As the authors suggest, it appears that individual consumers attempt to optimize their drinking behaviors with respect to financial and personal constraints that restrict drinking. Thus in important respects drinking choices are the product of environmental constraints. As consumers attempt to maximize their satisfaction from drinking, they balance all the various aspects of drinking behavior that affect their lives (positive and negative) and optimize the result to produce the greatest satisfaction.(21) To the extent that consumers' optimization behaviors are short-sighted (e.g., they do not adequately account for risks associated with drinking alcohol), or to which the benefits of drinking outweigh the risks of getting drunk (e.g., the social capital of drinking groups), consumers are likely to place themselves at risk for some harmful outcome. Thus the general issues regarding involvement in risky behaviors come to abut specific issues of drinking and drinking problems (Turrisi and Jaccard, 1991, 1992).

At this juncture several important problems confront researchers exploring this area. These include (l) the need to more precisely specify the domains of routine activities required to characterize potential involvement in alcohol-related problems; (2) the construction of a model representing participation in these routine activities that incorporates the environmental constraints that shape these activities; (3) the development of a dynamic model of drinking activities that provides predictive power for explaining involvement in alcohol-related injury events; and (4) the theoretical definition of how routine drinking activities are (or are not) coupled to drinking patterns and problematic outcomes. Future work in these areas will no doubt enhance our ability to predict alcohol-related harmful events and, one hopes, to develop preventive interventions suitable to the needs of alcohol consumers.

Notes

(1.) A number of terms referring to specific drinking patterns are placed in quotes. These terms are received from the literature. The definitions of a number of these are made explicitly clear in the context of the current model. (2.) Heckman's (1979) endogenous autoregressive "state dependent" model was fit to the data using Grether and Maddala's (1982) method. Here [y*.sub.t], refers to the unobserved probability of initiating a drinking episode in time t, and [y.sub.t-1] refers to the observed (zero or one) drinking state at time t-1. (3.) Bias-corrected accelerated confidence intervals were defined using 2,000 resampled moving blocks bootstraps of order 1 (Efron and Tibshirani, 1993). It is significant at this point to note that this observation suggests that Duffy and Alanko's (1992) assertion that an exponential waiting time process could be used to model time between drinking events is not true for all drinkers. The assumptions of independent and stationary increments necessary for the model are both violated here. In an examination of the frequency distribution of drinking, Gruenewald and Nephew (1994) suggest that these assumptions may be violated for most drinkers. (4.) The spectral decomposition is based upon the demeaned binary series filtered on the first and second spectral components to eliminate trend effects. No spectral windows were applied to the series. (5.) The quotes are dropped deliberately. Binge drinkers are those drinkers whose sequences of drinking episodes are positive autoregressive. Episodic drinkers are those drinkers whose sequences of drinking episodes are negative autoregressive. (6.) Recognizing the importance of continued work in the area, there is obviously room for improvement here. (7.) Literally, the mode of the drinking probability function. (8.) This observation suggests, then, that estimates of total consumption from standard quantity-frequency questions will be substantially downward biased. Indeed, Gruenewald and Nephew (1994) show that model-based estimates of total consumption exceeded quantity-frequency-based estimates by 28%. (9.) It is also to be expected that drinkers with different dpo's may have the same [s.sup.2], showing the theoretical independence of these parameters. However, in analyses of drinking populations the form of the drinking probability distribution (censored at zero) will induce a positive correlation between dpo and [s.sup.2]. Gruenewald and Nephew (1994) note that with this consideration in mind, the only unbiased asymptotically consistent estimators of drinking patterns are to be obtained through applications of censored regression models to alcohol consumption data. The results of ordinary least squares regression models will always be biased toward zero. Because the drinking probability distribution is censored at zero, it is also to be expected that "heavy" (e.g., high dpo) drinkers will have greater variances in drinking quantities and lower apparent reliability of drinking reports over time (Skog and Duckert, 1993). The upshot of this observation is that as a stable estimator of the underlying drinking probability function, the reliability of [s.sup.2] may be very high, but the reliability of any particular continued drinking frequency or quantity may be quite low. Thus the model makes specific predictions regarding empirically assessed reliabilities of drinking measures and may be used to calibrate sampling frames from which to obtain stable estimates of drinking patterns. (10.) These probability values are commonly referred to as hazards (Lawless, 1982). In life course mortality studies, hazards reflect the force of mortality (i.e., the likelihood that one will die given that one has lived to a certain age). In the current context, as hazards increase or decrease the proportionate pressures on continued drinking increase and decrease. Although little considered to this point, this approach suggests the possibility of separate monitoring of these forces in future studies of consumption patterns. (11.) For similar reasons it is erroneous to identify an individual as a "heavy" drinker based on "typical" quantities consumed (e.g., "typically" drinking three or more drinks per occasion). The issue again is to consider with what distribution of drinking the assessment of "heavy" is to be compared. (12.) It is an assertion of the model that all typological characterizations of drinking patterns describe loci within the drinking space. For example, the "frequent heavy drinker" popularized by Cahalan et al. (1969) appears somewhere in the high frequency, high quantity and/or high variance quadrant of the space. It is a deficiency of Cahalan et al.'s (1969) typology that the location of this "type" cannot be more accurately ascertained. The model similarly reveals the ambiguities of terms such as "controlled" drinking, the application of which implies "control" over some aspect of drinking (f, dpo, [s.sup.2], r, b, etc.). (13.) This is a controversial statement. The model argues for a dispositional definition of binge drinking revealed in the persistence of drinking episodes over time. Therefore whether or not an individual drinker feels he or she suffers from inability to abstain is immaterial to his or her categorization as a binge drinker. Although subjective definitions of inability to abstain may be material as an indication of the enlightenment of the drinker, they do not necessarily constitute accurate introspections regarding drinking patterns per se. (14.) The term "best" is used here from a theoretical point of view. That is, based on theory it is expected that the various measures of alcohol use will have specific qualitative relationships to problem outcomes. It is to be hoped, of course, that specification of these qualitative relationships will provide measures that are also best from some empirical point of view. (15.) For simplicity it is assumed that i(x) is a logistic function (i.e., i(x) = exp(d + fx)/(l+exp(d+fx)); d = -10, f = 1). One could imagine i(x) as a step function representing, among other things, a 'heavy" drinking criterion beyond which it is assumed that the target behavior is fully incapacitated. This is, in fact, what the use of "heavy" drinking criteria suggests. (16.) This is an important caveat. For example, those drinkers with high total body water can be expected to have incapacitation functions right-shifted relative to those with low total body water, explaining the differential effects of drinking upon blood alcohol contents for men versus women. (17.) The assumption of constant rates of participation in target activities contingent upon drinking suggests, for example, that the choice to drive after drinking is unaffected by the amount that the individual drinker consumes. Although this may appear a surprising suggestion, there is no evidence in the literature to suggest that drinkers do in fact alter driving patterns in response to quantities consumed per occasion. There are suggestions that perceived risks may affect likelihoods of drinking and driving (Turrisi and Jaccard, 1991, 1992) and that these are related to average quantities consumed across occasions, but there is no evidence suggesting that individuals factor these risks into an account of how much they consume per occasion and then act on these risks accordingly. (18.) This is because total risks are a product of f and dpo. R, a function of dpo, is multiplied by f to obtain cumulative exposure to risk over drinking occasions. (19.) This is, of course, a vastly oversimplified model of the observed failure process, but one that does well to outline the kinds of effects expected from any mapping of probabilities of harmful outcomes to these measures of drinking patterns. (20.) Although the demonstration is far too complicated to present, the nonlinearities noted here and above obtain after transformation to probability coordinates. Therefore the expected effects remain nonlinear in Logits and Probits. This is an important point for analysis. (21.) Thus this model is aligned with economic models in which consumers' choices are assumed to be the result of the weighing of rational expectations. It does, however, extend the model to contexts in which social capital is relevant (Coleman, 1990).

[Figures 1a to 3a ILLUSTRATION OMITTED]

References

Abel, E.L. and Kruger, M.L. Hon v. Stroh Brewery Company: What do we mean by "moderate" and "heavy" drinking? Alcoholism: Clinical and Experimental Research, l9, 1024-1031, 1995.

Bloomfield, P. Fourier Analysis of Time Series: An Introduction. New York: John Wiley, 1976.

Box, G.E.P. and Jenkins, G.M. Time Series Analysis Forecasting and Control. Oakland, CA: Holden-Day, 1976.

Cahalan, D.I., Cisin, I. and Crossley, H. American Drinking Practices: A National Study of Drinking Behavior. New Brunswick, NJ: Rutgers Center of Alcohol Studies, 1969.

Casswell, S., Fang Zhang, J. and Wyllie, A. The importance of amount and location of drinking for the experience of alcohol-related problems. Addiction, 88, 1527-1534, 1993.

Clark, W.B. and Hilton, M.E. Alcohol in America: Drinking Practices and Problems. Albany, NY: State University of New York Press, 1991.

Coleman, J.S. The Foundations of Social Theory. Cambridge: The Belknap Press, 1990.

Duffy, J.C. and Alanko, T. Self-reported consumption measures in sample surveys: A simulation study of alcohol consumption. Journal of Official Statistics, 8, 327-350, 1992.

Efron, B. and Tibshirani, R.J. An Introduction to the Boofstrap. New York: Chapman and Hall, 1993.

Epstein, E.E., Kahler, C.W., McCrady, B.S., Lewis, K.D. and Lewis, S. An empirical classification of drinking patterns among alcoholics: Binge, episodic, sporadic, and steady. Addictive Behaviors. 20. 2341, 1995.

Giancola, P.R. and Zeichner, A. An investigation of gender differences in alcohol-related aggression. Journal of Studies on Alcohol, 56, 573579, 1995.

Grether, D.M. and Maddala, G.S. A time series model with qualitative variables. In M. Deistler, E. Furst, and G. Schwodiauer (eds.), Games, Economic Dynamics, and Time Series Analysis. Wien-Wurzburg: Physica-Verlag, 1982.

Gruenewald, P.J. Loss of control drinking among first offender drunk drivers. Alcoholism: Clinical and Experimental Research, 15, 634639, 1991.

Gruenewald, P.J., Stewart, K. and Klitzuer, M. Alcohol use and the appearance of alcohol problems among first offender drunk drivers. British Journal of Addiction, 85, 107-117, 1990.

Gruenewald, P.J., Millar, A. and Treno, A.J. Alcohol availability and the ecology of drinking. Alcohol Health and Research World, 17, 39-45, 1993.

Gruenewald, P.J. and Nephew, T.M. Consumption and death: The dynamics of alcohol consumption patterns. Paper presented at the annual meeting of the Research Society on Alcoholism, Marco Island, Florida, June 9-13, 1991.

Gruenewald, P.J. and Nephew, T.M. Drinking in California: Theoretical and empirical analyses of alcohol consumption patterns. Addiction, 89, 707-723, 1994.

Gruenewald, P.J., Treno, A.J., Nephew, T.M. and Ponicki, W.R. Routine activities and alcohol use: Constraints on outlet utilization. Alcoholism: Clinical and Experimental Research, 19, 44-53, 1995

Heckman, J. Statistical models for discrete panel data. Center for Mathematical Studies in Business and Economics, Report No. 7902, University of Chicago, 1979.

Holubowycz, O.T. Age, sex, and blood alcohol concentration of killed and injured pedestrians. Accident Analysis and Prevention, 27, 417-422, 1995.

Jellinek, E.M. The Disease Concept of Alcoholism. New Haven, CT: College and University Press, 1960.

Kahler, C.W., Epstein, E.E. and McCrady, B.S. Loss of control and inability to abstain: The measurement of and the relationship between two constructs in male alcoholics. Addiction, 90, 1025-1036, 1995.

Knupfer, G. Some methodological problems in tie epidemiology of alcoholic beverage usage: Definition of amount of intake. American Journal of Public Health, 2, 232-242, 1966.

Kruger, H.P. A behavioral model of low alcohol effects. In H.-P. Kruger, R. Kohnen and M.W. Perrine (eds.), Low Alcohol Effects-A Challenge for Science. NIAAA Research Monograph. Washington, DC: U.S. Government Printing Office; in press, 1995.

Lawless, J.F. Statistical Models and Methods for Lifetime Data. New York: John Wiley, 1982.

Lemmens, P., Tan, E.S. and Knibbe, R.A. Measuring quantity and frequency of drinking in a general population survey: A comparison of five indices. Journal of Studies on Alcohol, 53, 476-486, 1992.

Lemmens, P. and Knibbe, R.A. Seasonal variation in survey and sales estimates of alcohol consumption. Journal of Studies on Alcohol, 54, 157-163, 1993.

Mirand, A.L. and Welte, J.W. Total body water adjustment of mean alcohol intakes. Journal of Substance Abuse, 6, 419-425, 1994.

Moskowitz, H., Burns, M. and Williams, A.F. Skills performance at low blood alcohol levels. Journal of Studies on Alcohol, 46, 482-485, 1985.

Mundt, J.C., Searles, J.S., Perrine, M.W. and Helzer, J.E. Cycles of alcohol dependence: Frequency-domain analyses of daily drinking logs for matched alcohol-dependent and nondependent subjects. Journal of Studies on Alcohol, 56, 491-499, 1995.

Norstrom, T. Prevention strategies and alcohol policy. Addiction, 90, 515524, 1995.

Perrine, M.W., Peck, R.C. and Fell, J.C. Epidemiologic perspectives on drunk driving. In Surgeon General's Workshop on Drunk Driving: Background Papers. Bethesda, MD: U.S. Department of Health and Human Services, 35-76, 1989.

Rehm, J. and Sempos, C.T. Alcohol consumption and all-cause mortality. Addiction, 90, 471-480, 1995.

Room, R., Bondy, S.J. and Ferris, J. The risk of harm to oneself from drinking, Canada 1989. Addiction, 90, 499-513, 1995.

Ross, S.M. Introduction to Probability Models, 5th Edition. New York: Academic Press, 1993.

Shiavi, R. Introduction to Applied Statistical Signal Analysis. Homewood, IL: Aksen Associates, 1991.

Single, E. and Wortley, S. Drinking in various settings as it relates to demographic variables and level of consumption: Findings from a national survey in Canada. Journal of Studies on Alcohol, 54, 590599, 1993.

Skog, O.-J. and Duckert, F. The development of alcoholics' and heavy drinkers' consumption: A longitudinal study. Journal of Studies on Alcohol, 54, 178-188, 1993.

Solnick, S.J. and Hemenway, D. Hit the bottle and run: The role of alcohol in hit-and-run pedestrian fatalities. Journal of Studies on Alcohol, 55, 679-684, 1994.

Stockwell, R., Lang, E. and Rydon, P. High risk drinking settings: The association of serving and promotional practices with harmful drinking. Addiction, 88, 1519- 1526, 1993.

Strauss, R. and Bacon, S. Drinking in America. New Haven: Yale University Press, 1953.

Treno, A.J., Nephew, T.M., Ponicki, W.R. and Gruenewald, P.J. Alcohol beverage price spectra: Opportunities for substitution. Alcoholism: Clinical and Experimental Research, 17, 675-680, 1993.

Turrisi, R. and Jaccard, J. Judgment processes relevant to drunk driving. Journal of Applied Social Psychology, 21, 89- 118, 1991.

Turrisi, R. and Jaccard, J. Cognitive and attitudinal factors in the analysis of alternatives to drunk driving. Journal of Studies on Alcohol, 53, 405-414, 1992.

Paul J. Gruenewald is a senior research scientist at the Prevention Research Center (2150 Shattuck Ave., Suite 900, Berkeley, CA 94709), where he has worked and published on drinking-driver research and on the effects of alcohol availability. Andrew J. Treno is a senior research associate and Patrick R. Mitchell a research associate at the Prevention Research Center.

AUTHORS' NOTE: This paper was presented at the conference "Social and Health Effects of Different Drinking Patterns, " sponsored by the Alcohol Research Foundation, Toronto, Canada, November 13-17, 1995. Research for and preparation of this paper were supported by the National Institute on Alcohol Abuse and Alcoholism Research Center grant number AA06282.