8756-3282l92 $5.00 + .OO Copyright 0 1992 Pergamon Press Ltd.
Bone, 13, 243-247, (1992) printed in the USA. All rights reserved.
A Method for Estimating the Uncertainty of Future Bone Mass Y. F. HE, P. D. ROSS, J. W. DAVIS,
C. J. MACLEAN,
R. EPSTEIN
and R. D. WASNICH
Hawaii osteoporosis
Center, Honolulu, HI, USA; Department of Psychiatry, Virginia Commonwealth University, Richmond, VA, USA; Merck, Sharp and Dohme Research Laboratories, West Point, PA, USA; University of Maryland Medical School, Baltimore, MD, USA
Address for corresnondence and reorints: Dr. Philiu D. Ross. Euidemiology . Honolulu, HI 968i4, USA. ’
Abstract
Center, 932 Ward Avenue,
Suite 400,
dalities is not widespread (Cauley et al. 1990; Melton et al. 1990; Barlow et al. 1991). It appears that an objective measure of increased fracture risk, such as low bone mass, increases the willingness of women to initiate such treatment and become more active in their own preventive health care (Rubin & Cummings 1990). Numerous prospective studies have demonstrated that baseline bone mass measurements are strong predictors of the risk of various fractures (Wasnich et al. 1985; Hui et al. 1989; Ross et al. 1990a; Cummings et al. 1990). However, the period of observation in those studies was generally less than 5 to 10 years. Considerable differences in bone loss rate exist between individuals (Davis et al. 1989, 1991; Hui et al. 1990). Furthermore, measurements of current bone loss rate may not accurately reflect future loss rate (Hui et al. 1990). These factors will increase the uncertainty in future bone mass, which may impair the ability of single, baseline bone mass measurements to accurately predict fracture risk. We present here a mathematical model for estimating both the future level of bone mass and the uncertainty associated with this estimate. This model will find application in future cost/benefit analyses such as (1) improving estimates of the utility of single baseline bone mass measurements, and (2) determining the optimal frequency of bone mass measurements.
The development of statistical models for estimating fracture probability is a promising method for quantitatbtg and optimizing the clintcal utility of bone mass measurements. Earlier models have assmned that future bone mass could be predkted exactly and were, therefore, limited to analyses that assume tbe loss rate is known in advance. Since bone loss rates may vary over time and caMot be predicted accurately, we have developed a new model, based on empirical data, that estimates the degree of uncertainty associated with predicted bone mass. Without a bone mass measurement, the population mean must be assmned for an individual. For the calcaneus, the standard deviation of the population distribution is about 60 n&m’. By measuring bone mass, one can determine how close or far from the mean an individual’s true bone mass is, with a standard deviation (SD) of about 3 m&m”. Without a subsequent bone mass measurement, our model predicts that the uncertainty (standard deviation) in calcaneaf bone mass will increase approximately sixfold (relative to the reproducibility at the initial measurement) over a period of five years for women under age 60, from 3 mgkm’ to 19 mg/cm2. The five-year increase in uncertainty is approximately fourfold for women over age 60, from 3 to 13 m&m’. However, the muxrtainty in bone mass for an individual five years after the initial measurement is still only one third to one fifth that of the entire population, and can be reduced to the initial level by obtaining another measurement. Furthermore, the predicted (or measured) vabres are usually much better estimates of an individual’s true bone mass than simply assuming tbe population average. Thii model can be adapted to other types of bone mass measurements, and will find application in cost-effectiveness analyses of osteoporosis treatment and prophylaxis, including estimation of the optimal time interval between bone mass measurements. Key Words: Osteoporosis-Fracture prediction-Bone Longitudinal studies-Statistical models.
Section, Hawaii Osteoporosis
Methods When developing fracture prediction models, it is important to calculate the most likely value of the patient’s bone mass at each age in the future, since bone mass declines with age and exerts a strong influence on fracture risk. Although it is impossible to predict these values perfectly, a probability distribution can be constructed that defines the uncertainty associated with estimates of future bone mass. In this section we will first describe a function for predicting the most likely value of bone mass at a specified time in the future, and then explain how to construct a probability distribution for the predicted bone mass. Model parameters were estimated empirically from data acquired from a cohort study of bone loss and fractures (Yano et al. 1984; Davis et al. 1991). This cohort consists of 1098 Japanese-American women living on the island of Oahu, Hawaii (Heilbnm et al. 1985, 1991). The analyses here are limited to the 495 postmenopausal women who were not using estrogen, thiazide, or corticosteroids at any of the examinations. These women (mean age = 64; range = 45-81 years) have had serial bone mass measurements, beginning in February, 1981, at approximately one-
mass-
Introduction Although therapeutic regimens such as hormone replacement therapy are available to retard the progressive loss of bone which contributes to osteoporotic fractures, long-term use of these mo243
Y. F. He et al.: Future bone mass prediction model
244 to two-year intervals. A custom-made, single-photon (‘*‘I), rectilinear densitometer based on a design used by NASA was used for all measurements. Details of this procedure have been reported previously (Vogel et al. 1987). A maximum of six measurements (mean = 5.1) were obtained, with a mean followup of 5.3 years. Although any skeletal site could be used to model bone loss and predict fractures, we used the calcaneus here bccause the number of longitudinal data points was greater than for other sites, and also because we have had more experience using this site to develop previous fracture prediction models (Ross et al. 1987, 1988, 1990b). Predicting future bone mass Bone loss with age can be considered to be a stochastic process (i.e., involving variables at each moment of time) with a population mean of b(t) and standard deviation u(t), where t is age. Both u(t) and b(t) are deterministic functions: [a(
= var{x(t)}
and b(t) = E{x(t)}
where var = variance and E = the estimated value (mean) of the quantity enclosed by brackets. We can define the bone mass x of a person at age t as x(t) = u(t) . u(t) + b(t)
(1)
where u(t) is a stochastic process with a population mean of zero and variance of 1 .O. Simply put, the value of u(t) is a “Z-score,” and u(t) . u(t) describes the distance of the individual’s bone mass from the population mean. The correlation between values of u(t) at two points in time is described by the correlation function, R,,(s) = E{u(t) * u(t + s)}, where s is the age interval. The correlation over time (age) for bone mass of an individual is given by R,,(s)
= u(t) . u(t+s) . R,,(s)
(2)
Several studies with large data sets (Smith et al. 1975; Mazess et al. 1987; Ross et al. 1988) indicate that the population variance in bone mass is essentially constant with age, so that we assume u(t) = a, and R,,(s) = R,(s)lu*. For an individual with initial bone mass x&J at age to, the true bone mass at a future age, to + s, is equal to x(t,, + s) = a . u(t, + s) + b(t,, + s). Although the true future bone mass cannot be determined at present, it can be estimated as follows. Assuming that u(t) is normally distributed, E{u(t, + s))u(t,,)} = R,,(s) . u(t,J, and assuming a normal distribution of u(tO + s), conditional on u&J, with a mean of R,,(s) * u(t,,) and standard deviation of l-R,,(s), then the future bone mass, conditional on current bone mass, is given by E{x(t, + s)]x(&,)] = R,,(s)
. [x(&J - %)I
+ b(r, + s)
(3)
Although individuals differ with respect to loss rate, which would tend to increase the population variance over time, the observed population variance remains constant with age. Accordingly, an individual’s predicted future bone mass must converge toward the population mean over time. The rate of convergence is determined by the correlation of bone mass over time, as expressed in eqn (3). Although the constant population variance may be explained in part by the fact that those with the lowest bone mass tend to die earlier, this would not affect the results of the present study, which are based on survivors. Estimating uncertainty in predicted bone mass The uncertainty in a person’s
initial bone mass measurement, which has been reported for various densitometry techniques (Wasnich
x&J at age to, is defined by the short-term reproducibility,
et al. 1987; Peck et al. 1988; Gluer et al. 1990). The uncertainty (variance, var) in predicted values of bone mass can be calculated as: var{x(t,, + s))x(r,J} = u2 . (1 -R,,(s)) Empirical estimation of model parameters Decline of bone muss with age. A total of 2517 measurements were collected, with a maximum of six measurements per subject, among the 495 postmenopausal women who were not using thiazides, estrogens, or corticosteroids. These measurements were classified according to the woman’s age at the time of each measurement. The mean and variance were then calculated for each age group, in one-year increments. Since each subject may contribute to more than one age group, this analysis combines both cross-sectional and longitudinal data. An exponential model of the decline of mean bone mass with age was then constructed by linear regression of the log transform of the single year, age-specific means, weighted by the number of observations for each mean. The variance in bone mass was assumed constant for all ages (SD = 60.1 mg/cm*) and was estimated from the agestratified, total sample. Correlation with initial bone mass. First, each observed value was converted to a Z-score, calculated as the observed value minus the mean, divided by the standard deviation, using mean and standard deviation values predicted from the exponential model of age and bone mass, above. These Z-scores were then sorted into categories (O-1.0, 1.01-2.0, 2.01-3.0 years, etc.) according to the time interval between measurements, using all available prospective bone mass combinations for each individual (i.e., first and second, first and third, second and third time points, etc.). The correlation between the two bone mass measurements was then calculated separately for each time interval category. We hypothesized that the correlation over time might be better at older ages, if large fluctuations in bone mass occur soon after the menopause. To test the consistency of correlations for different age groups, the analyses were repeated after stratifying the data by age. To test the significance of the observed differences between correlation coefficients, we used the following method (Sachs 1982). The correlation coefficients for the two age groups, r, and r2, with sample sizes n1 and n2, are transformed using the R.A. Fisher transformation, x = (OS)ln[( 1 + r)/( 1 - r)]. The distribution of x is approximately normal, with SD, s, = ll(n-3)0.5. Then,
Ix1x
=
[l&l,
x21
- 3) + ll(n2 - 3)]“.5
If the hypothesis that r, = r2 is correct, X will have a normal distribution with mean = 0 and SD = 1 .O. If X was greater than 1.96, we rejected the null hypothesis, since p < 0.05.
Decline of bone mass with age The observed means of calcaneus bone mass by year of age are provided together with values predicted by the exponential model in Table I. The population mean bone mass at a given age (t) is described by the following equation, with SD = 60.1 mg/cm* at each age: mean (t) = eatb . ‘, where a = 6.576, b = -0.0138. The predicted values closely approximate the observed values at most ages. Although the observed values deviate
245
Y. F. He et al.: Future bone mass prediction model
Table I. Observed and predicted values of calcaneus bone mass
and above; therefore, the data in Table II are categorized by initial age less than 60, and 60 and above. Thus, the value of the correlation coefficient used in predicting future bone mass and variance will depend on the patient’s age. The standard deviation associated with the predicted bone mass for each time interval is also provided in Table II. These standard deviations were calculated from the correlation coefficients in Table II, using eqn (4). for ages 70
Age
n
Observed
Fkdicted
51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80
5 10 11 24 29 30 54 61 86 107 112 158 180 169 186 184 174 155 149 118 106 89 83 59 43 38 24 17 15 11
350 331 338 341 333 312 318 310 321 307 310 303 301 300 293 293 288 281 279 215 267 266 262 264 258 249 238 214 224 198
354 350 345 340 335 331 326 322 317 313 309 304 300 296 292 288 284 280 216 213 269 265 261 258 254 251 241 244 241 237
An example
illustrate how bone mass measurements classify individuals, consider the situation in Fig. 1. Without any measurement of bone mass, an individual would be assumed to have the population mean and variance for that age. By obtaining a single measurement, the person’s bone mass can be determined with a very small uncertainty, equal to the short-term reproducibility of the method, represented by the points labeled A and B for two hypothetical people. For the calcaneus, the reproducibility corresponds to a standard deviation of 3.0 mg/cm* (Ross et al. 1991). The model predicts that five years later, based on the average of previous observations for all people, these two individuals will have the probability distributions shown for the points labeled A’ and B’. The mean indicates the most likely value, and the standard deviation describes the uncertainty associated with that value. The predicted mean is slightly closer to the population mean, and the variance has increased considerably, relative to the earlier time point. Even so, the predicted value is a good estimate of the person’s true bone mass; without an initial measurement, we would have to assume the population average bone mass and variance. If the initial measurement were obtained before age 60, the predicted standard deviation will increase from 3.0 (at the initial age) to 19.3 mg/cm* over the next five years (point A’). For measurements made after age 60, the model predicts that the standard deviation will increase from 3.0 to 13.3 mg/cm* during the subsequent five years (point B’). To
Note: Bone mass units are mg/cm*.
slightly from predicted values at both ends, the sample sixes were very small at both ends. This exponential model of bone loss predicts a gradual slowing of bone loss with age, with an average loss rate of 4.87 mg/cm* per year at age 5 1, and 3.30 mg/cm* per year at age 79. Correlation of bone mass over time
Correlation coefficients for various categories of time intervals are provided in Table II. There is a clear, consistent trend of decreasing correlation as the time interval increases. In preliminary analyses categorized by both 5- and lO-year age groups (data not shown), the correlation coefficients for initial ages less than 60 were lower than those for older ages. The coefficients for ages 670 were not significantly different (p > .05) than those
Discussion As reported previously using only cross-sectional data (Ross et al. 1988) bone mass decreased consistently and progressively with age. Although this decrease could have been fit using a simple linear regression model, we elected to use an exponential model because it avoids the tendency to predict negative bone mass values at older ages, which might occur using a linear model. The deviation of the observed mean bone mass for the youngest and oldest ages from that predicted by the model could be due to (1) random variation caused by inadequate sample size (2) lack of fit of the model (perhaps a polynomial model would
Table II. Correlation of calcaneus bone mass with initial measurement by time interval and age category with corresponding predicted standard deviations Total Interval (Y) Cl.0 1 .Ol-2.0 2.01-3.0 3.01-4.0 4.01-5.0 5.01-6.0 6.01-7.0
Age
Bone, 13, 243-247, (1992) printed in the USA. All rights reserved.
A Method for Estimating the Uncertainty of Future Bone Mass Y. F. HE, P. D. ROSS, J. W. DAVIS,
C. J. MACLEAN,
R. EPSTEIN
and R. D. WASNICH
Hawaii osteoporosis
Center, Honolulu, HI, USA; Department of Psychiatry, Virginia Commonwealth University, Richmond, VA, USA; Merck, Sharp and Dohme Research Laboratories, West Point, PA, USA; University of Maryland Medical School, Baltimore, MD, USA
Address for corresnondence and reorints: Dr. Philiu D. Ross. Euidemiology . Honolulu, HI 968i4, USA. ’
Abstract
Center, 932 Ward Avenue,
Suite 400,
dalities is not widespread (Cauley et al. 1990; Melton et al. 1990; Barlow et al. 1991). It appears that an objective measure of increased fracture risk, such as low bone mass, increases the willingness of women to initiate such treatment and become more active in their own preventive health care (Rubin & Cummings 1990). Numerous prospective studies have demonstrated that baseline bone mass measurements are strong predictors of the risk of various fractures (Wasnich et al. 1985; Hui et al. 1989; Ross et al. 1990a; Cummings et al. 1990). However, the period of observation in those studies was generally less than 5 to 10 years. Considerable differences in bone loss rate exist between individuals (Davis et al. 1989, 1991; Hui et al. 1990). Furthermore, measurements of current bone loss rate may not accurately reflect future loss rate (Hui et al. 1990). These factors will increase the uncertainty in future bone mass, which may impair the ability of single, baseline bone mass measurements to accurately predict fracture risk. We present here a mathematical model for estimating both the future level of bone mass and the uncertainty associated with this estimate. This model will find application in future cost/benefit analyses such as (1) improving estimates of the utility of single baseline bone mass measurements, and (2) determining the optimal frequency of bone mass measurements.
The development of statistical models for estimating fracture probability is a promising method for quantitatbtg and optimizing the clintcal utility of bone mass measurements. Earlier models have assmned that future bone mass could be predkted exactly and were, therefore, limited to analyses that assume tbe loss rate is known in advance. Since bone loss rates may vary over time and caMot be predicted accurately, we have developed a new model, based on empirical data, that estimates the degree of uncertainty associated with predicted bone mass. Without a bone mass measurement, the population mean must be assmned for an individual. For the calcaneus, the standard deviation of the population distribution is about 60 n&m’. By measuring bone mass, one can determine how close or far from the mean an individual’s true bone mass is, with a standard deviation (SD) of about 3 m&m”. Without a subsequent bone mass measurement, our model predicts that the uncertainty (standard deviation) in calcaneaf bone mass will increase approximately sixfold (relative to the reproducibility at the initial measurement) over a period of five years for women under age 60, from 3 mgkm’ to 19 mg/cm2. The five-year increase in uncertainty is approximately fourfold for women over age 60, from 3 to 13 m&m’. However, the muxrtainty in bone mass for an individual five years after the initial measurement is still only one third to one fifth that of the entire population, and can be reduced to the initial level by obtaining another measurement. Furthermore, the predicted (or measured) vabres are usually much better estimates of an individual’s true bone mass than simply assuming tbe population average. Thii model can be adapted to other types of bone mass measurements, and will find application in cost-effectiveness analyses of osteoporosis treatment and prophylaxis, including estimation of the optimal time interval between bone mass measurements. Key Words: Osteoporosis-Fracture prediction-Bone Longitudinal studies-Statistical models.
Section, Hawaii Osteoporosis
Methods When developing fracture prediction models, it is important to calculate the most likely value of the patient’s bone mass at each age in the future, since bone mass declines with age and exerts a strong influence on fracture risk. Although it is impossible to predict these values perfectly, a probability distribution can be constructed that defines the uncertainty associated with estimates of future bone mass. In this section we will first describe a function for predicting the most likely value of bone mass at a specified time in the future, and then explain how to construct a probability distribution for the predicted bone mass. Model parameters were estimated empirically from data acquired from a cohort study of bone loss and fractures (Yano et al. 1984; Davis et al. 1991). This cohort consists of 1098 Japanese-American women living on the island of Oahu, Hawaii (Heilbnm et al. 1985, 1991). The analyses here are limited to the 495 postmenopausal women who were not using estrogen, thiazide, or corticosteroids at any of the examinations. These women (mean age = 64; range = 45-81 years) have had serial bone mass measurements, beginning in February, 1981, at approximately one-
mass-
Introduction Although therapeutic regimens such as hormone replacement therapy are available to retard the progressive loss of bone which contributes to osteoporotic fractures, long-term use of these mo243
Y. F. He et al.: Future bone mass prediction model
244 to two-year intervals. A custom-made, single-photon (‘*‘I), rectilinear densitometer based on a design used by NASA was used for all measurements. Details of this procedure have been reported previously (Vogel et al. 1987). A maximum of six measurements (mean = 5.1) were obtained, with a mean followup of 5.3 years. Although any skeletal site could be used to model bone loss and predict fractures, we used the calcaneus here bccause the number of longitudinal data points was greater than for other sites, and also because we have had more experience using this site to develop previous fracture prediction models (Ross et al. 1987, 1988, 1990b). Predicting future bone mass Bone loss with age can be considered to be a stochastic process (i.e., involving variables at each moment of time) with a population mean of b(t) and standard deviation u(t), where t is age. Both u(t) and b(t) are deterministic functions: [a(
= var{x(t)}
and b(t) = E{x(t)}
where var = variance and E = the estimated value (mean) of the quantity enclosed by brackets. We can define the bone mass x of a person at age t as x(t) = u(t) . u(t) + b(t)
(1)
where u(t) is a stochastic process with a population mean of zero and variance of 1 .O. Simply put, the value of u(t) is a “Z-score,” and u(t) . u(t) describes the distance of the individual’s bone mass from the population mean. The correlation between values of u(t) at two points in time is described by the correlation function, R,,(s) = E{u(t) * u(t + s)}, where s is the age interval. The correlation over time (age) for bone mass of an individual is given by R,,(s)
= u(t) . u(t+s) . R,,(s)
(2)
Several studies with large data sets (Smith et al. 1975; Mazess et al. 1987; Ross et al. 1988) indicate that the population variance in bone mass is essentially constant with age, so that we assume u(t) = a, and R,,(s) = R,(s)lu*. For an individual with initial bone mass x&J at age to, the true bone mass at a future age, to + s, is equal to x(t,, + s) = a . u(t, + s) + b(t,, + s). Although the true future bone mass cannot be determined at present, it can be estimated as follows. Assuming that u(t) is normally distributed, E{u(t, + s))u(t,,)} = R,,(s) . u(t,J, and assuming a normal distribution of u(tO + s), conditional on u&J, with a mean of R,,(s) * u(t,,) and standard deviation of l-R,,(s), then the future bone mass, conditional on current bone mass, is given by E{x(t, + s)]x(&,)] = R,,(s)
. [x(&J - %)I
+ b(r, + s)
(3)
Although individuals differ with respect to loss rate, which would tend to increase the population variance over time, the observed population variance remains constant with age. Accordingly, an individual’s predicted future bone mass must converge toward the population mean over time. The rate of convergence is determined by the correlation of bone mass over time, as expressed in eqn (3). Although the constant population variance may be explained in part by the fact that those with the lowest bone mass tend to die earlier, this would not affect the results of the present study, which are based on survivors. Estimating uncertainty in predicted bone mass The uncertainty in a person’s
initial bone mass measurement, which has been reported for various densitometry techniques (Wasnich
x&J at age to, is defined by the short-term reproducibility,
et al. 1987; Peck et al. 1988; Gluer et al. 1990). The uncertainty (variance, var) in predicted values of bone mass can be calculated as: var{x(t,, + s))x(r,J} = u2 . (1 -R,,(s)) Empirical estimation of model parameters Decline of bone muss with age. A total of 2517 measurements were collected, with a maximum of six measurements per subject, among the 495 postmenopausal women who were not using thiazides, estrogens, or corticosteroids. These measurements were classified according to the woman’s age at the time of each measurement. The mean and variance were then calculated for each age group, in one-year increments. Since each subject may contribute to more than one age group, this analysis combines both cross-sectional and longitudinal data. An exponential model of the decline of mean bone mass with age was then constructed by linear regression of the log transform of the single year, age-specific means, weighted by the number of observations for each mean. The variance in bone mass was assumed constant for all ages (SD = 60.1 mg/cm*) and was estimated from the agestratified, total sample. Correlation with initial bone mass. First, each observed value was converted to a Z-score, calculated as the observed value minus the mean, divided by the standard deviation, using mean and standard deviation values predicted from the exponential model of age and bone mass, above. These Z-scores were then sorted into categories (O-1.0, 1.01-2.0, 2.01-3.0 years, etc.) according to the time interval between measurements, using all available prospective bone mass combinations for each individual (i.e., first and second, first and third, second and third time points, etc.). The correlation between the two bone mass measurements was then calculated separately for each time interval category. We hypothesized that the correlation over time might be better at older ages, if large fluctuations in bone mass occur soon after the menopause. To test the consistency of correlations for different age groups, the analyses were repeated after stratifying the data by age. To test the significance of the observed differences between correlation coefficients, we used the following method (Sachs 1982). The correlation coefficients for the two age groups, r, and r2, with sample sizes n1 and n2, are transformed using the R.A. Fisher transformation, x = (OS)ln[( 1 + r)/( 1 - r)]. The distribution of x is approximately normal, with SD, s, = ll(n-3)0.5. Then,
Ix1x
=
[l&l,
x21
- 3) + ll(n2 - 3)]“.5
If the hypothesis that r, = r2 is correct, X will have a normal distribution with mean = 0 and SD = 1 .O. If X was greater than 1.96, we rejected the null hypothesis, since p < 0.05.
Decline of bone mass with age The observed means of calcaneus bone mass by year of age are provided together with values predicted by the exponential model in Table I. The population mean bone mass at a given age (t) is described by the following equation, with SD = 60.1 mg/cm* at each age: mean (t) = eatb . ‘, where a = 6.576, b = -0.0138. The predicted values closely approximate the observed values at most ages. Although the observed values deviate
245
Y. F. He et al.: Future bone mass prediction model
Table I. Observed and predicted values of calcaneus bone mass
and above; therefore, the data in Table II are categorized by initial age less than 60, and 60 and above. Thus, the value of the correlation coefficient used in predicting future bone mass and variance will depend on the patient’s age. The standard deviation associated with the predicted bone mass for each time interval is also provided in Table II. These standard deviations were calculated from the correlation coefficients in Table II, using eqn (4). for ages 70
Age
n
Observed
Fkdicted
51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80
5 10 11 24 29 30 54 61 86 107 112 158 180 169 186 184 174 155 149 118 106 89 83 59 43 38 24 17 15 11
350 331 338 341 333 312 318 310 321 307 310 303 301 300 293 293 288 281 279 215 267 266 262 264 258 249 238 214 224 198
354 350 345 340 335 331 326 322 317 313 309 304 300 296 292 288 284 280 216 213 269 265 261 258 254 251 241 244 241 237
An example
illustrate how bone mass measurements classify individuals, consider the situation in Fig. 1. Without any measurement of bone mass, an individual would be assumed to have the population mean and variance for that age. By obtaining a single measurement, the person’s bone mass can be determined with a very small uncertainty, equal to the short-term reproducibility of the method, represented by the points labeled A and B for two hypothetical people. For the calcaneus, the reproducibility corresponds to a standard deviation of 3.0 mg/cm* (Ross et al. 1991). The model predicts that five years later, based on the average of previous observations for all people, these two individuals will have the probability distributions shown for the points labeled A’ and B’. The mean indicates the most likely value, and the standard deviation describes the uncertainty associated with that value. The predicted mean is slightly closer to the population mean, and the variance has increased considerably, relative to the earlier time point. Even so, the predicted value is a good estimate of the person’s true bone mass; without an initial measurement, we would have to assume the population average bone mass and variance. If the initial measurement were obtained before age 60, the predicted standard deviation will increase from 3.0 (at the initial age) to 19.3 mg/cm* over the next five years (point A’). For measurements made after age 60, the model predicts that the standard deviation will increase from 3.0 to 13.3 mg/cm* during the subsequent five years (point B’). To
Note: Bone mass units are mg/cm*.
slightly from predicted values at both ends, the sample sixes were very small at both ends. This exponential model of bone loss predicts a gradual slowing of bone loss with age, with an average loss rate of 4.87 mg/cm* per year at age 5 1, and 3.30 mg/cm* per year at age 79. Correlation of bone mass over time
Correlation coefficients for various categories of time intervals are provided in Table II. There is a clear, consistent trend of decreasing correlation as the time interval increases. In preliminary analyses categorized by both 5- and lO-year age groups (data not shown), the correlation coefficients for initial ages less than 60 were lower than those for older ages. The coefficients for ages 670 were not significantly different (p > .05) than those
Discussion As reported previously using only cross-sectional data (Ross et al. 1988) bone mass decreased consistently and progressively with age. Although this decrease could have been fit using a simple linear regression model, we elected to use an exponential model because it avoids the tendency to predict negative bone mass values at older ages, which might occur using a linear model. The deviation of the observed mean bone mass for the youngest and oldest ages from that predicted by the model could be due to (1) random variation caused by inadequate sample size (2) lack of fit of the model (perhaps a polynomial model would
Table II. Correlation of calcaneus bone mass with initial measurement by time interval and age category with corresponding predicted standard deviations Total Interval (Y) Cl.0 1 .Ol-2.0 2.01-3.0 3.01-4.0 4.01-5.0 5.01-6.0 6.01-7.0
Age
