# The Time Course of Grief Reactions to Spousal Loss: Evidence From a National Probability Sample

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6 GRIEF REACTIONS TO SPOUSAL LOSS 481 Table 1 Characteristics of the Widowed Sample Total (n 768) Men (n 149) Women (n 619) Characteristic M SD M SD M SD Average age (years) Average age when widowed (years) Education (number of years) Spouse s age at death (years) Total number of children % currently married Race White (%) Black (%) Other (%) Religion Protestant (%) Catholic (%) Jewish (%) Other/none (%) Time since death (years) Died suddenly (%) Number of years married included because it could plausibly be associated with both time since widowhood and the grief response, thereby leading to a bias in the relationship between the two. These adjusted means (also knows as least squares means) represent the predicted level of a given outcome for each time interval for individuals at the mean on the control variables, and they were computed using the LS- MEANS option in the general linear model (GLM) procedure available in SAS software (SAS Institute, 004). As we illustrate in the graphs presented below, although many of the process variables showed a negative exponential pattern with time, some showed what appeared to be a linear trend, and some showed no relationship with time. In our modeling of the functional form, we allowed for all three possibilities. The simplest or null model, shown in Figure 1(i), was one in which, after adjusting for the control variables (x i ) listed above, there was no relationship between time since widowhood (t) and a given process measure, labeled y. In formal terms, y a 0 t c i x i ε. (1) Each control variable (x i ) was mean centered, that is, the sample mean was subtracted from each person s score (resulting in a mean of 0 for all control variables). With the controls coded in this way, the intercept a is the mean of y, and the relationship between time (t) and y net of the controls is constrained to be 0. Each c i represents the unique linear relationship between control variable i and y. Finally, ε is a random variable that represents all other remaining influences on y. It is assumed that these (a) average to zero, (b) are uncorrelated with the other variables in the model, and (c) have the same variability over time and the covariates. Note that time t in this and the other two regression models discussed below is the original, continuous time scale, not the categorized version of time since widowhood used to estimate the adjusted means discussed above. The second or linear model, shown in Figure 1(ii), specified that after adjusting for the control variables, there was a linear relationship between time since widowhood and y, as follows: y a b t c i x i ε. () The intercept a is the value of y for a typical person who has just been widowed. The slope b is the expected difference in each recovery measure y associated with a one-year difference in time since widowhood. Here, each c i represents the unique linear relationship between control variable i and y, adjusting for time. As in the null model, ε is a random variable related to the unmeasured Figure 1. Characteristic forms of the null, linear, and negative exponential models.

7 48 CARNELLEY, WORTMAN, BOLGER, AND BURKE Table Goodness-of-Fit and Improvement-in-Fit Measures for Null, Linear, and Negative Exponential Models: Measures of Continued Involvement and Emotional Resolution Measure RMSE R adj R adj p Frequency of thoughts and memories Null model Linear model Negative exponential model Frequency of conversations Null model.5.14 Linear model Negative exponential model Positive affect Null model Linear model Negative exponential model Negative affect Null model Linear model Negative exponential model Frequency of anniversary reactions Null model Linear model Negative exponential model Length of anniversary reactions Null model Linear model Negative exponential model Intensity of anniversary reactions Null model Linear model Negative exponential model Note. RMSE root-mean-square error. influences on y, conforming to the same set of assumptions laid out above. The third or negative exponential model, illustrated in Figure 1(iii), specified that after adjusting for the control variables (x i ), there was a negative exponential relationship between time since widowhood (t) and y such that adjustment was relatively rapid at first but gradually slowed until an asymptote was reached. The model was as follows: y f d e s t c i x i ε, (3) in which y is the measure of recovery, f is the final or asymptotic level of y, d is the distance between the initial level and the final level of y, e is a mathematical constant representing the base of the natural logarithm, and t is time in years since widowhood. The final parameter, s, is known as the decay constant. It is a positive number related to the rate of adjustment, with larger numbers representing more rapid adjustment. In this specification, f d is the equivalent of a and a in the previous models, that is, a. Again, each c i represents the unique linear relationship between control variable i and y. Once again, ε corresponds to all other random, unmeasured influences on y. The same assumptions described for the null model also apply to this residual term in the negative exponential model. The estimation of all three models was accomplished using the NLIN procedure available with SAS software (SAS Institute, 004), although the null and linear models could have equivalently been estimated with several other SAS procedures (e.g., REG or GLM) or any software capable of least squares linear regression. To determine which of the three models was most appropriate for each measure, we computed adjusted goodness-of-fit indices for each model for each measure (shown in Tables and 3). Like many other methods for computing regression estimates, the NLIN procedure uses least squares estimation to arrive at these estimates (SAS Institute, 004). That is, the estimates produced are those that minimize the sum of squared residuals. Note that because the linear and negative exponential models are not nested models, they cannot be directly compared in terms of their fit. They can, however, be compared in terms of how much each improves the fit compared with the null model. A simple change in R statistic would not suffice for this purpose, though, because it does not take the relative degrees of freedom of the two models into account, namely, that one degree of freedom is lost when fitting the linear model, whereas two degrees of freedom are lost when fitting the negative exponential model. To account for the different degrees of freedom of the linear and negative expo- Unlike some other regression programs, the NLIN procedure requires the user to provide starting values for the parameter estimates. These starting values can be estimated from descriptive data and plots. An additional feature of NLIN is the ability to set bounds on the parameter estimates. In all negative exponential regression analyses presented here, f and f d from Equation 3 were constrained to be between 0 and 100, and s was constrained to be 0 or larger.

9 484 CARNELLEY, WORTMAN, BOLGER, AND BURKE Figure. Functional relationships between years since widowhood and measures of continuing involvement and emotional resolution, adjusted for sex, race, education, age of respondent; age of spouse and number of children at time of widowhood; whether the death was expected or not; whether the death was due to murder, accident, or suicide; and whether the respondent became remarried. Points indicate least squares means for 5-year intervals, and error bars represent standard errors of those means. improvement in fit, we show results for both in the tables. We also indicate the rate of change of each measure as a function of time. For those variables showing a linear relationship, this corresponds to b in Equation and represents the number of units on the dependent variable corresponding to a one-year difference in time since widowhood, adjusting for the control variables listed above. For example, knowing that the linear rate of change in frequency of thoughts about the deceased spouse is.89 units per year means that we would expect respondents widowed 10 years apart to differ on this variable by 8.9 units. For those variables showing a negative exponential relationship, the value listed in the table corresponds to s in Equation 3, although no such easy calculation of adjustment level is possible here because the rate of change depends on time itself. To determine how respondents were doing

10 GRIEF REACTIONS TO SPOUSAL LOSS 485 Figure 3. Functional relationships between years since widowhood and measures of meaning finding, adjusted for sex, race, education, age of respondent; age of spouse and number of children at time of widowhood; whether the death was expected or not; whether the death was due to murder, accident, or suicide; and whether the respondent became remarried. Points indicate least squares means for 5-year intervals, and error bars represent standard errors of those means. after a specified number of years, it is necessary to calculate the predicted levels of y using Equation 3. Tables 4 6 provide several additional pieces of information about the models we estimated. First, they provide the initial value or intercept predicted for each model. For all three models, this has the same interpretation, namely, the predicted level of the dependent variable for people at the point of widowhood, adjusting for the control variables. Second, for measures in which the negative exponential model provided the best fit, these tables provide what we call the 90% asymptotic level. This value represents the level of adjustment corresponding to 90% of the distance between the initial value and the asymptotic value. Finally, for those variables

11 486 CARNELLEY, WORTMAN, BOLGER, AND BURKE Figure 4. Functional relationship between years since widowhood and measures of personal growth, adjusted for sex, race, education, age of respondent; age of spouse and number of children at time of widowhood; whether the death was expected or not; whether the death was due to murder, accident, or suicide; and whether the respondent became remarried. Points indicate least squares means for 5-year intervals, and error bars represent standard errors of those means. in which there was a negative exponential relationship between time since widowhood and adjustment, we used Equation 3 to estimate the number of years it would take to reach 5%, 50%, 75%, and 90% of the distance to the asymptotes. These values provide a view of adjustment as a function of time for the negative exponential models that is easier to interpret than the s parameter of Equation 3. In those cases in which the relationship between the dependent variable and time since widowhood is linear, identifying a specific long-term recovery level is not possible because the rate of recovery does not change over time. Thus, for those variables in which the model was linear, we left the 5%, 50%, 75%, and 90% columns blank. A Note on the Figures Figures 4 show the best fitting model for each measure of recovery as well as the least squares means estimated for each 5-year interval since widowhood (described earlier). Although the total sample contained respondents widowed as little as a few months to as long as 64 years prior to the survey, small cell sizes Table 4 Linear and Negative Exponential Regressions Relating Indicators of Continuing Involvement and Emotional Resolution to the Duration of Time Since Widowhood Dependent variable & form of relationship Level of dependent variable Number of years to % of asymptotic level Speed of change SE Initial SE 90% asymptotic SE 5% 50% 75% 90% Frequency of memories Linear.89* *.0 Negative exponential * * a Frequency of conversations Negative exponential.06* * 3.4.8* Positive affect b Null 0 60.* 1.3 Negative affect b Negative exponential.18* * * Frequency of anniversary reactions Negative exponential.04* * * Duration of anniversary reactions c Null * 1. Intensity of anniversary reactions c Negative exponential.3* * * Note. For linear results, rates are in units per year. All analyses adjust for sex, race, education, age of respondent, age of spouse, and number of children at time of widowhood; whether the death was expected or not; whether the death was due to murder, accident, or suicide; and whether the respondent became remarried. a Projected value is beyond the range of the data. b Total n 697. c Total n 484. p.10. * p.05.

12 GRIEF REACTIONS TO SPOUSAL LOSS 487 Table 5 Linear and Negative Exponential Regressions Relating Indicators of Meaning Finding to the Duration of Time Since Widowhood Level of dependent variable Number of years to % of asymptotic level Dependent variable & form of relationship Speed of change SE Initial SE 90% asymptotic SE 5% 50% 75% 90% View death as meant to be Linear.60* *.7 Negative exponential.0* * * b b View death as something that just happens Linear.65* *.3 Negative Exponential.03* * * b Currently searching for meaning? a Negative exponential.1* * 6.5.* If searched, found meaning? a Null 0 5.9* 1.7 View death as senseless and unfair Linear.50* *.6 Negative exponential * * View of spouse as better off now Null * 1.5 Note. For linear results, rates are in units per year. All analyses adjust for sex, race, education, age of respondent, age of spouse, and number of children at time of widowhood; whether the death was expected or not; whether the death was due to murder, accident, or suicide; and whether the respondent became remarried. a Total n 313. b Projected value is beyond the range of the data. * p.05. limited the interpretability of the least squares means analysis to widows of 35 years or less. For this reason, Figures 4 depict only the first 35 years since widowhood, although all analyses made use of the entire widowed sample. Also, each of these figures has two ordinate axes, one showing the scale of the dependent measure and one showing the actual verbal labels the respondents were given. Continuing Involvement and Emotional Resolution We first present results for measures of continuing involvement with the lost loved one, such as the frequency of memories and conversations about one s spouse, and measures of emotional resolution, such as positive and negative feelings from thinking and talking about the loss and the frequency, duration, and intensity of feeling upset with reminders of the loss. Table summarizes the goodness-of-fit results, whereas Table 4 summarizes the fitted models. For frequency of thoughts and memories about the deceased spouse, both the linear and negative exponential models showed equal improvement in fit over the null model. For simplicity, we chose to graph only the negative exponential results ( R adj.0477, p.001), as shown in Figure (i). These results show a moderate decline in thoughts and memories as a function of time since widowhood. Respondents who were recently widowed reported having thoughts and memories of their loved one approximately two to three times per week (corresponding to 80.5 units). The 90% asymptotic value (34.5 units) corresponds to thoughts or memories about once per month, and the number of years widowed corresponding to this value is 71.3 years. This value is beyond the observed range of our data (0 64 years postwidowhood). Table 6 Linear and Negative Exponential Regressions Relating Perceptions of Personal Growth to the Duration of Time Since Widowhood Dependent variable & form of relationship Level of dependent variable Speed of change SE Initial SE Number of years to % of asymptotic level 90% asymptotic SE 5% 50% 75% 90% Increased self-confidence Linear.34* *.4 Negative exponential * * Stronger person as a result Null * 1.3 Note. Rates are in units per year. All analyses adjust for sex, race, education, age of respondent, age of spouse, and number of children at time of widowhood; whether the death was expected or not; whether the death was due to murder, accident, or suicide; and whether the respondent became remarried. * p.05.

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