Cora J. M. Maas

Sufficient Sample Sizes for Multilevel Modeling

An important problem in multilevel modeling is what constitutes a sufficient sample size for accurate estimation. In multilevel analysis, the major restriction is often the higher-level sample size. In this paper, a simulation study is used to determine the influence of different sample sizes at the group level on the accuracy of the estimates (regression coefficients and variances) and their standard errors. In addition, the influence of other factors, such as the lowest-level sample size and different variance distributions between the levels (different intraclass correlations), is examined. The results show that only a small sample size at level two (meaning a sample of 50 or less) leads to biased estimates of the second-level standard errors. In all of the other simulated conditions the estimates of the regression coefficients, the variance components, and the standard errors are unbiased and accurate.

Meta-analysis of randomized response research.

This article discusses two meta-analyses on randomized response technique (RRT) studies, the first on 6 individual validation studies and the second on 32 comparative studies. The meta-analyses focus on the performance of RRTs compared to conventional question-and-answer methods. The authors use the percentage of incorrect answers as effect size for the individual validation studies and the standardized difference score (d-probit) as effect size for the comparative studies. Results indicate that compared to other methods, randomized response designs result in more valid data. For the individual validation studies, the mean percentage of incorrect answers for the RRT condition is .38; for the other conditions, it is .49. The more sensitive the topic under investigation, the higher the validity of RRT results. However, both meta-analyses have unexplained residual variances across studies, which indicates that RRTs are not completely under the control of the researcher.

The accuracy of multilevel structural equation modeling with pseudobalanced groups and small samples.

Hierarchical structured data cause problems in analysis, because the usual assumptions of independently and identically distributed variables are violated. Muthén (1989) described an estimation method for multilevel factor and path analysis with hierarchical data. This article assesses the robustness of the method with unequal groups, small sample sizes at both the individual and the group level, in the presence of a low or a high intraclass correlation (ICC). The within-groups part of the model poses no problems. The most important problem in the between-groups part of the model is the occurrence of inadmissible estimates, especially when group level sample size is small (50) while the intracluster correlation is low. This is partly compensated by using large group sizes. When an admissible solution is reached, the factor loadings are generally accurate. However, the residual variances are underestimated, and the standard errors are generally too small. Having more or larger groups or a higher ICC does not effectively compensate for this. Therefore, although the nominal alpha level is 5%, the operating alpha level is about 8% in all simulated conditions with unbalanced groups. The strongest factor is an inadequate sample size at the group level. Imbalance is only a problem for the overall fit test. For balanced data, the chi-square fit test is accurate. The size of the biases is comparable to the effect of moderate nonnormality in ordinary modeling, and in our view, the approximate solution remains a useful analysis tool, provided the group level sample size is at least 100.

The influence of violations of assumptions on multilevel parameter estimates and their standard errors

A crucial problem in the statistical analysis of hierarchically structured data is the dependence of the observations at the lower levels. Multilevel modeling programs account for this dependence and in recent years these programs have been widely accepted. One of the major assumptions of the tests of signi2cance usedin the multilevel programs is normality of the error d istributions involved. Simulations were used to assess how important this assumption is for the accuracy of multilevel parameter estimates andtheir stand arderrors. Simulations variedthe number of groups, the group size, andthe intraclass correlation, with the secondlevel resid ual errors following one of three non-normal distributions. In addition asymptotic maximum likelihood standard errors are comparedto robust (Huber/White) stand arderrors. The results show that non-normal residuals at the second level of the model have little or no e8ect on the parameter estimates. For the 2xedparameters, both the maximum likelihood -based stand arderrors andthe robust stand arderrors are accurate. For the parameters in the rand om part of the model, the maximum likelihood-based standard errors at the lowest level are accurate, while the robust standard errors are often overcorrected. The standard errors of the variances of the level-two random e8ects are highly inaccurate, although the robust errors do perform better than the maximum likelihooderrors. For goodaccuracy, robust stand arderrors needat least 100 groups. Thus, using robust standard errors as a diagnostic tool seems to be preferable to simply relying on them to solve the problem. c 2003 Elsevier B.V. All rights reserved.

Psychosocial Functioning of Spouses of Patients With Stroke From Initial Inpatient Rehabilitation to 3 Years Poststroke: Course and Relations With Coping Strategies

Background and Purpose— Few studies have focused on long-term changes in the caregiving experience after stroke. This study assessed changes in the psychosocial functioning of spouses (burden, depressive symptoms, harmony in the relationship between patient and spouse, and social relations) during the first 3 years after stroke and identified predictors of the course of spouses’ psychosocial functioning based on the characteristics of patients and spouses with special emphasis on coping style. Methods— We examined 211 couples shortly after the patient’s admission to a rehabilitation center, 197 2 months after discharge, 187 1 year poststroke, and 121 3 years poststroke. Burden was assessed using the Caregiver Strain Index, depressive symptoms with the Goldberg Depression Scale, harmony in the relationship with the Interactional Problem Solving Inventory, and social relations with the Social Support List. Multilevel regression analyses were performed. Results— A significant effect of time (P<0.01) was found for all 4 aspects of spouses’ psychosocial functioning. Although burden decreased, harmony in the relationship and social relations also decreased. The depression score showed a nonlinear pattern with an initial decrease but a long-term increase. All outcomes were significantly related to caregiver coping strategies. A total of 15% to 27% of the variance in psychosocial functioning could be explained. Conclusions— Follow-up of spouses of patients with stroke requires not only assessment of burden, but also other aspects of psychosocial functioning like harmony in the relationship, depression, and social relations, because our results show negative long-term consequences of stroke for these aspects of caregiver quality of life.

Cortisol secretion throughout the day, perceptions of the work environment, and negative affect

The effects of explanatory variables derived from a work stress model (the effort-reward imbalance model) on salivary cortisol were assessed. A multilevel analysis was used to distinguish the effects of single occasion and multiple occasion measurements of work stress and effect on cortisol. The single (or cross-sectional) factors include Effort-Reward Imbalance (ERI), need for control, negative affect, and other enduring factors (type of occupation, gender, and smoking). The multiple occasion measurements include momentary negative mood, Momentary Demand-Satisfaction Ratio (MD-SR), sleep quality, work load (workday versus day off), at work (versus not being at the workplace), and lunch. The effect of time of day on cortisol was controlled for before the effects of these variables were determined.Momentary negative mood but not trait negative affect was positively associated with ambulatory measured cortisol. The variables from the work stress model—effort, reward, need for control, and the multiple occasion measurements of demand and satisfaction—did not affect cortisol. As could be expected, time of day had an effect on cortisol, but a hypothesised interaction with momentary negative mood was not found. Additionally, the results show that the time course of cortisol differs between individuals and that the effect of sleep quality on cortisol can vary from person to person. This points to the necessity of continued efforts to single out sources of individual variability.The finding that variables derived from the effort-reward imbalance model are not related with cortisol does not support the hypothesis that ERI leads to short-term changes in cortisol, indicating no relation with hypothalamic-pituitary-adrenal (HPA) axis activity. On the other hand, the present results invite further qualification of negative affect as a potential determinant of HPA activity, at least, as far as can be deduced from cortisol measurements.

The Multilevel Approach to Repeated Measures for Complete and Incomplete Data

Repeated measurements often are analyzed by multivariate analysis of variance (MANOVA). An alternative approach is provided by multilevel analysis, also called the hierarchical linear model (HLM), which makes use of random coefficient models. This paper is a tutorial which indicates that the HLM can be specified in many different ways, corresponding to different sets of assumptions about the covariance matrix of the repeated measurements. The possible assumptions range from the very restrictive compound symmetry model to the unrestricted multivariate model. Thus, the HLM can be used to steer a useful middle road between the two traditional methods for analyzing repeated measurements. Another important advantage of the multilevel approach to analyzing repeated measures is the fact that it can be easily used also if the data are incomplete. Thus it provides a way to achieve a fully multivariate analysis of repeated measures with incomplete data.

Vagal cardiac control throughout the day: the relative importance of effort-reward imbalance and within-day measurements of mood, demand and satisfaction

The effects of variables derived from a work stress theory (the effort-reward imbalance theory) on the power in the high frequency (HF_HRV) band of heart rate (0.14-0.40 Hz) throughout a work day, were determined using multilevel analysis. Explanatory variables were analysed at two levels: at the lowest level (within-day level), the effects of positive mood, negative mood, demand, satisfaction, demand-satisfaction ratio, and time of day were assessed. At the highest level (the subject level), the effects of sleep quality, effort, reward, effort-reward imbalance, need for control, type of work (profession), negative affectivity, gender and smoking on HF_HRV were assessed. Need for control has a negative effect on HF_HRV after controlling for time of day effects, i.e. subjects with a high need for control have a lower vagal control of the heart. In the long run, these subjects may be considered to be at increased health risk, because they have less of the health protective effects of vagal tone. The interaction between effort-reward imbalance and time of day has a positive effect on HF_HRV, i.e. the cardiac vagal control of subjects with a high effort-reward imbalance increases as the day progresses. It is discussed that this probably reflects reduced effort allocation, ensuing from disengagement from the work demands.

Electronic momentary assessment in chronic pain. I: Psychological pain responses as predictors of pain intensity

Objectives and Methods:More than 7100 electronic diaries from 80 patients with chronic pain (mean: 89.3, range 30-115) entered multilevel analyses to establish the statistical prediction of disability by pain intensity and by psychological functioning (fear avoidance, cognitive, and spousal pain responses). We also tested the differences between prechronic, recently chronic, and persistently chronic pain in the prediction of disability (impaired physical and mental capacity, pain interference with activities, immobility due to pain). Results:Pain intensity explained 8% to 19% of the disability variance. Beyond this psychological functioning explained 7% to 16%: particularly fear-avoidance and cognitive pain responses predicted chronic pain disorder disability; spousal responses predicted immobility better than other aspects of disability. Immobility due to actual pain occurred infrequently. When it did, however, it was better predicted by avoidance behavior in the patient and by spousal discouragement of movement than by actual pain intensity. The prediction of immobility due to pain by, respectively, avoidance behavior and catastrophizing was better in chronic pain (>6 months) and that of physical impairment by catastrophizing better in persistently chronic pain (>12 months) than in pain of shorter duration. Discussion:The psychological prediction of chronic pain disorder disability was determined beyond that accounted for by pain intensity. Nonetheless, psychological functioning explained substantial variance in chronic pain disorder disability. The psychological prediction of immobility and physical impairment was stronger with longer pain duration. Patient characteristics and momentary states of disability-and in particular of immobility-should be carefully distinguished and accounted for in chronic pain disorder.

When a parent has a stroke - Clinical course and prediction of mood, behavior problems, and health status of their young children

Background and Purpose— The purpose of this research was to describe the clinical course of children’s functioning (depression, behavioral problems, and health status) during the first year after parental stroke and to determine which patient-, spouse-, or child-related factors at the start of inpatient rehabilitation can predict children’s functioning after parental stroke at 1-year poststroke. Methods— Interviews with 82 children (4 to 18 years of age) and their parents (n=55) shortly after admission to a rehabilitation center, 2 months after discharge from inpatient rehabilitation, and 1 year after stroke. Depression was assessed using the Children Depression Inventory, behavioral problems with the Child Behavior Check List, and health status with the Functional Status II. Potential predictors were gender and age (child), activities of daily living disability and communication ability (patient), and spouse’s depression and perception of the marital relationship. Results— At the start of the stroke patient’s rehabilitation, 54% of the children had ≥1 subclinical or clinical problems, which improved to 29% 1 year after stroke. Children’s functioning 1 year after stroke could best be predicted by their functioning at the start of rehabilitation. Spouse depression and perception of marital relationship were also significant predictors. A total of 28% to 58% of the variance in children’s functioning could be explained. Conclusions— Children’s functioning after parental stroke improved during the first year after stroke. Identifying children at risk for problems 1 year after stroke requires assessment of children’s functioning and the healthy spouse’s depressive symptoms and perception of the marital relationship at the start of rehabilitation. This demonstrates the need for a family-centered approach in stroke rehabilitation.