Using PROC ARIMA in Forecasting the Demand and Utilization of Inpatient Hospital Services

Transcription

1 Using PROC ARIMA in Forecasting the Demand and Utilization of Inpatient Hospital Services John J. Hisnanick* U.S. Department of Health and Human Services Introduction In the course of any normal day, we are, either directly or indirectly, involved with univariate time series data. That is, by either working with such data or becomill.g part of a time-series through our economic and social behavior. A time-series can be defined as a set of numbers that measures the status of some ongoing process or activity overtime. Generally, one can think of a time-series as being composed into three components: a trend component, a seasonal component and anirregular, or random component. These components can be related in either an additive or a multiplicative fashion. In working with such data, one often wishes to describe the series regarding it's component parts and predict future observations for the series. Under SAS/ETS, there exists.the procedure ARIMA, which facilities the description and prediction of a univariate time-series data set. The focus of this study was to define an appropriate model specification for the forecasting of a monthly times-series, assess the issue of seasonality in the series, and evaluate the forecast model's effectiveness as a planning tool. Using an established taxonomy of diagnostic disease categories and the national hospital discharge database of the Indian Health Service (IHS) for the period , a forecasting model for inpatient hospitalizations was undertaken. This investigation represents an attempt at forecasting the demand for hospital services with a particular emphasis on the potential burden specific health conditions may impose on this health care delivery system. The Forecasting Process The forecasting process can be thought of as being composed of six, separate steps. These have been defined as; (1) define the forecasting problem, (2) collect and prepare the data, (3) select and apply a forecasting method, (4) review and adjust the preliminary forecasts, (5) track the forecast accuracy and (6) update the forecasts and the forecasting system [Hoff(1983)]. The following is an overview of each component part and how SAS software facilitated each step of the process. The Forecasting Problem: The forecasting problem was developed based upon the assumption that chronic health conditions due to alcoholism and alcohol abuse is a Significant health problem among the population served by the IHS. Further, this problem has been hypothesized as having imposed a serious strain on existing resources. The forecasting problem was formulated as "What will be the future impact on direct care inpatient services from alcoholism and alcohol abuse". The problem formulation was greatly facilitated given that there does exist a taxonomy of diagnostic disease categories which can capture inpatient hospitalizations related to chronic health conditions associated with alcoholism and alcohol abuse, Table 1. Therefore, using this taxonomy, could a monthly univariate time-series for hospitalizations be generated with a structure that could becharacterized and described. Collecting and Preparing the Data: The data base used in the construction of the timeseries was the national hospital discharge data base for the IHS. This data base is maintained in a SAS format on magnetic tapes and because of the number of records per file is most easily accessed and manipulated within a mainframe environment. Each record within the data base contains both administrative and demographic information related to a hospitalization of en individual. In addition, each event can have up to six possible discharge diagnoses recorded. Therefore, the generation of the time-series involved the aggregation of daily event specific information into a monthly format. Inclusion within the monthly aggregation was limited to those individuals fourteen years of age or older, with any mention of a diagnoses of one of the conditions listed in the taxonomy from Table 1. The construction of the monthly time-series was greatly facilitated by the useofproc FREQ procedure upon a SAS dataset sorted by year of discharge and then month of discharge. Thegenerated monthly time-seriescovered the years , accounting for 108 observations in the series. Figure 1 is a graphical representation of the respective monthly time-series for hospitalizations associated wi th alcohol related diagnoses (ARD). 383

2 Select and Apply a Forecasting Method: A univariate time-series represents a collection of data with a specific temporal ordering and is assumed to be generated by a stochastic process with a structure that can be characterized and described accordingly. The series can be thought of being com posed of three distinct components; a trend component, a seasonal component and a random or irregular component. The trend component represents the long-run movement of the series. The seasonal component reflects a possible seasonal pattern that repeats itself over some time period. And the irregular or random component reflects the nonsystematic movements that occur in the series. Therefore, the selection and application of an appropriate forecasting methodology can be quite involved, but is greatly facilitated by the SAS/ETS software. One major question that needs to be answered prior to the selecting of an appropriate forecasting model is to determine whether the generated time-series is stationary. Stationarity is an important characteristic of stochastic processes and implies that the time-series is invariant with respect to time and wanders more or less uniformly about some fixed level. Assessing stationarity can be done by looking at the autocorrelation function of the time-series. The autocorrelation function is a statistical measure that indicates how the time-series is related to itself over time. More precisely, autocorrelation has a possible range covering the interval (-1, +1) and measures how strongly the time-series values, a specific number of periods apart, are correlated to each other. If the values of the autocorrelation function do not fall off quickly, (or within the prescribed range of statistical insignificance), as the number of lags increases, this indicates that the series is nonstationary. In the case that a nonstationary series is encountered, it can be made stationary by differencing, one or more times. Differencing is the process of transforming a nonstationary series into a stationary one by taking the incremental difference one or more times for the series observations. The number of times that a series is differenced in order to achieve stationarity is called the order of homogeniety of the series. To assess whether a more stationary series could be obtained, further differencing could be done on a series. If the successive differencing does notresult in any significant qualitative change in the decline of the autocorrelation function, the appropriate order of homogeneity for the series is the level of differencing where no further qualitative changes occurred in the autocorrelation function. Figure 2 reflects the values of the autocorrelation function associated with the times-series of monthly admissions for ARD hospitalizations. The values estimated for the autocorrelation function do not fall off quickly, but do dampen towards zero. Therefore, it does appear that the time-series is stationary, but could be made more stationary by an appropriate transformation of the data. Therefore, the series was transformed using the natural logrithmic transformation in SAS, (Le., In_x=log(x)). The transformed series is presented in figure 3, and the series was smoothed considerably. Figure 4 presents the autocorrelation function for the transformed series. The autocorrelation function dampens quickly to zero, with the exception that significant "spikes" occur at the first and twelveth lags. This behavior of the autocorrelation implies that there is possible seasonality occuring in the data. While the autocorrelation function provides a way to evaluate the stationarity of a series, it is also helpful in addressing other modeling issues. More specifically, the question of seasonality within the data, as well as the number of disturbance terrns in the moving average (MA) portion and the number oflags in the autoregressive portion of an integrated model. On the issue of seasonality, the data points in the series should show some degree of correlation with corresponding data points. While this was not readily apparent in the time-series,figure 3, it showed up in the graphoftheautocorrelationfunctionforthetransformed series, figure 4. There were observed regular peaks in the autocorrelation function. The way to remove annual cycles in a monthly time-series is to take a twelve month difference ofthe data. Then the observed autocorrelation function will represent a "deseasonalized" series. Figure 5 represents the autocorrelation function for the deseasonalized time-series and this series was used in the estimation of the appropriate forecasting model. In regard to helping in the model specification, the autocorrelation function and the partial autocorrelation function can assist in selecting the appropriate ARIMA parameters. ARIMA is the acronym for anautoregressiveintegrated moving average specification. The specification of an appropriate ARIMA model has been termed more of an art, than a science, [Pindyck and Rubinfeld, (1976)]. The determination of the appropriate MA portion of the model can be done by evaluating the behavior of the autocorrelation function. In general, the autocorrelation function for the MA(q) process has "q" positive and negative values and is then zero for lag values greater than "q". From figure 5, the order of "q" was determined to be one. For the AR component of the model, the autocorrelation function patterns are somewhat more complicated than those for the MA component. For the AR component, it is clear that the current values of the 384

3 series, Xt, are related to past values of the series, Xt-p. However, as the lag values increase back in time, the level of association with current values of the series becomes weaker until for lagged values greater than "p", the level of association is zero. This behavior is reflected through the autocorrelation function in that it will dampen as the number of lags increases. This observed dampening can take the form of an exponential rate of decrease, a constant rate of decrease or an alternating pattern of decrease due to the sign change (positive to negative) of the AR parameters. The partial autocorrelation function tends to be more appropriate in evaluating the AR portion of the model. There using the partial autocorrelation function associated with figure 5, the appropriate order of the AR component for the model was determined to be one. In summary, for the series, the natural logarithmic transformation smoothed the data considerably and seasonally adjusting it aided in providing a suitable behavior in the autocorrelation function. This in turn help in resolving the appropriate order for the MA and AR components. Therefore, the model chosen for forecast the behavior of ARD hospitalizations was an ARIMA(1,0,1)12 X (1,0,1). That is, a seasonal adjusted model consisting of one moving average component and one autoregressive component. Table 2 summarizes the respective parameter estimates associated with the proposed ARIMA specification. Testing The Model Specification A question of concern regarding any model specification centers around whether what is being proposed is an adequate model specification. In dealing with an ARIMA specification, one can test the adequacy of the model by observing the sample autocorrelation function of the models residuals. These residuals, under the assumption of an adequate model specification, should be nearly uncorrelated with each other. A very convenient test to check for model adequacy is the Box-Pierce test [Box and Pierce, (1970». This test assumes that for large displacements or lags, the residual autocorrelations areuncorrelated,normallydistributed random variables with a mean of zero and a variance of lit; T being the number of observations in the series. The test statistic proposed by Box and Pierce was termed the "R" statistic. It is composed of the first "K" residual autocorrelations of the series. This test statistic is approximately distributed as a chi-squared distribution with (K-p-q) degrees of freedom. The "R" test statistic to assess model adequacy is computed as follows; R = T' L Tk 2 for k=l,...,k where "T" represents the number of observations in the series, and rk represents the residual autocorrelation computed for the Kth observation in the series. Some texts also refer to this statistic which tests for model adequacy as the Q-statistic. The procedure ARIMA under SAS/ETS provides output labeled as the "autocorrelation check of residuals". The output provides an estimate of the R-statistic at lag intervals of 6, 12, 18 and 24. In addition, it also reports the probability level at which the test statistic would be significant. Box and Pierce showed that for large displacements, (i.e., k > 5), the residual autocorrelations are uncorrelated, normally distributed random variables with a mean of zero and a variance of lit. Therefore, assuming a 95% confidence level, one can conclude that the residual autocorrelation from the model specification, as a whole, are not statistically significant. Table 3 summarized the values that lead to this conclusion. It is important to remember that the R statistic only gives an indication of an inadequate model and does not provide any insight into how one might improve on the model specification. However, it appears that the proposed ARIMA specification is adequate for modeling ARD hospitalizations. Track The Forecast Accuracy Using the parameter estimates presented in Table 2, for the seasonaily adjusted ARIMA(l,O,l), a forecast for the transformed series was done. A twelve period ahead forecast, the forecast standard error and the estimated upper and lower limits for a 95% confidence interval are reported in Table 4. This output is greatly facilitated under PROC ARIMA using the FORECAST statement. However, in order to get meaningful estimates for program planning purposes it is necessary to re-transform the estimated forecast values. Once again, this is greatly facilitated under PROC ARIMA and the FORECAST statement, in that one can direct the forecasted output to a new SAS dataset. This dataset can then be re-transformed. The results presented in Table 5 are the retransformed twelve period ahead forecast, with the upper and lower limits on the 95% confidence interval. In addition, figure 6 is a graphical representation of the series and the forecast values, with the appropriate upper and lower limits. For program planning this type of methodology for addressing the future demand of hospital services is quite appropria teo While the method is considered naive from a modeling prospective, other independent variables which would effect or influence increased hospitalizations due to ARD conditions are not readily available, and in some cases are not measureable. The 385

4 proposed ARIMA methods have a proven record in the study and tracting of such phenomena as gross national product, corn yield and livestock production. Further, given the relative ease that the SAS/ETS software provides in model specification, estimation and forecasting it seems natural to use in health care administration. The last step in the forecasting process concerns updating the forecasts and the forecast system. This can be done once new data is available for the series. It is important to remember that forecast modeling is a dynamicactivity. It israrelyaone-time activity and as time moves on, the assumptions, requirements and constraints that were initially developed will probably change. References * The opinions expressed in this paper are those of the author and do not reflect those of the U.S. Dept. of Health and Human Services or the IHS. Special thanks is extended to Lisa Preston for preparation of the final manuscipt. Box, G.E.P and D.A. Pierce,"Distribution of Residual Autocorrelations in Autoregress-Integrated Moving Average Time Series", Journal of the American Statistical Association. Vol. 65, December Pincyck, R.S. and D.L. Rubinfeld, Econometric Models And Economic Forecasts. New York, NY, McGraw-Hill Book Co., Hoff, J.e., A Practical Guide to Box-Jenkins Forecasting. Belmont, CA, Lifetime Learning Publications, Wadsworth, Inc., 1983 Kendall, M.G., The Advanced TheOlYOf Statistics. Vol. IT, New York, NY, Hafner Publishing Co., SAS Institute Inc., SAS/ETS User's Guide. Version 6. First Edition, Cary, NC: SAS Institute Inc., Table 1 Chronic Alcohol Related Diagnoses ICD-9-CM Diagnostic Categories and Codes For Chronic Conditions Often Associated With Alcohol Abuse Alcoholic Psychoses Alcohol Dependence Syndrome Alcohol Abuse Alcohol Polyneuropathy Alcohol Cardiomyopathy Alcohol Gastritis Alcohol Fatty Liver Acute Alcoholic Hepatitis Alcoholic Cirrhosis of Liver Alcoholic Liver Damage, Unspec. Chronic Hepatitis Cirrhosis of Liver w / 0 Mention of Alcohol Biliary Cirrhosis Other Chronic Non-alcoholic Liver Disease Unspec. Chronic Liver Disease w / 0 Mention of Alcohol Portal Hypertension Hepatorenal Syndrome Other Disorders of Liver Chronic Pancreatitis Cyst and Pseudocyst of Pancreas Gastrointestinal Hemorrhage ICD-9-CM Code Table 2 Parameter Estimates For Seasonally Adjusted ARIMA(1,O,l) Specification (No Mean Term Is Included In The Model) Autoregressive Components (B**I) -O (B**12) Moving Average Components -O (B**1 ) -O.99998(B**12) Table 3 Result of The Autocorrelation Check of Residuals Numer of Lags R-Statistic D.F. Probability

5 Table 4 Table 5 Forecast Estimates, 95% Upper and Lower Forecast Estimates, 95% Upper and Lower Confidence Limits And Actual Values Confidence Limits And Actual Values For The Ten Observation and Twelve Periods For The Ten Observation and Twelve Periods Ahead For The Transformed Series Ahead For The Un-Transformed Series Obs. Forecast Lower 95% Upper 95% Actual Obs. Forecast Lower 95% Upper 95% Actual

6 Figure 1 Monthly Series Covering 1980~ ~~ ' _ D 60 Months Figure 2 AC Function For Monthly Series 12 r , Lag Periods 388

7 Figure 3 I Transformed Monthly Series, 1980~ , ~ a 8' ' l'.;:j ~ Months Figure 4 Transformed AC Function 1.2, , ~ o ] U I I I I I I I I I I I I I I I I I I I I I I I I I o Lagged Periods

8 Figure 5 Seasonally Adjusted AC Function o ] u 0.6'" ", " ~ _.. ~~~~,. IL :1===:]==== o Lagged Periods Figure 6 Forecast Of Transformed Series ,., --:..,. 7~ _~_=-~~~~~~..,.~ ~ ,.-. Upper 95% Series For Lower 95% Mooths

9 Figure 7 Forecast Of Re-Transformed Series 10c0~ ~ Upper 95% Series For ~ Lower 95% _._-_._-_ Months 391