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Objectives. We used population-based data to evaluate whether caring for a child with health problems had implications for caregiver health after we controlled for relevant covariates.

Methods. We used data on 9401 children and their caregivers from a population-based Canadian study. We performed analyses to compare 3633 healthy children with 2485 children with health problems. Caregiver health outcomes included chronic conditions, activity limitations, self-reported general health, depressive symptoms, social support, family functioning, and marital satisfaction. Covariates included family (single-parent status, number of children, income adequacy), caregiver (gender, age, education, smoking status, biological relationship to child), and child (age, gender) characteristics.

Results. Logistic regression showed that caregivers of children with health problems had more than twice the odds of reporting chronic conditions, activity limitations, and elevated depressive symptoms, and had greater odds of reporting poorer general health than did caregivers of healthy children.

Conclusions. Caregivers of children with health problems had substantially greater odds of health problems than did caregivers of healthy children. The findings are consistent with the movement toward family-centered services recognizing the link between caregivers' health and health of the children for whom they care.

Caring for a child with health problems can entail greater than average time demands,1,2 medical costs,3,4 employment constraints,5,6 and childcare challenges.68 These demands may affect the health of caregivers, a notion supported by a variety of small-scale observational studies that have shown increased levels of stress, distress, emotional problems, and depression among caregivers of children with health problems.1,2,5,912

Whether these problems are caused by the additional demands of caring for children with health problems or by confounding variables is difficult to answer definitively. The literature reports the identification of a variety of factors purported to be associated with caregiver health, including contextual factors such as socioeconomic status1317; child factors such as level of disability,1,11,13,1821 presence of behavior problems,2225 and overall child adjustment26; and caregiver-related characteristics such as coping strategies11,22,27 and support from friends and family.15,17,28,29 In general, this work has been based on small clinic-based samples9,30 or specific child populations (e.g., cerebral palsy,5,25 attention-deficit/hyperactivity disorder31,32), and typically has been hampered by limited generalizability and a lack of careful, multivariate analysis. Furthermore, most studies have focused on caregivers' psychological health,1,2,5,912 although physical health effects may also exist among caregivers.5,19,25,33

One of the few studies to involve large-scale, population-based data compared the health of 468 caregivers of children with cerebral palsy to the health of a population-based sample of Canadian parents.5 The study showed that caregivers of children with cerebral palsy had poorer health on a variety of physical and psychological health measures. Furthermore, the data were consistent with a stress process model,5,25 which proposes that additional stresses associated with caring for a child with cerebral palsy directly contribute to poorer caregiver health. However, these findings were based on a specific subpopulation of caregivers and univariate comparisons that could not control for potentially important confounders such as variation in caregiver education, income, and other demographic factors.

We used population-based data to test the hypothesis that the health of caregivers of children with health problems would be significantly poorer than that of caregivers of healthy children, even after we controlled for relevant covariates. Our approach of using large-scale, population-based data representing a broad spectrum of childhood health problems34 makes 4 key contributions to the current literature. First, our use of population-based data rather than small-scale, clinic-based studies yielded results that are potentially generalizable to a wide group of caregivers caring for children with health problems. Second, our examination of children with and without health problems allowed us to examine caregiver health effects across a wide variety of caregiving situations. Third, consideration of physical health outcomes (in addition to more regularly studied psychological outcomes) increased our knowledge of the breadth of caregiver health issues. Finally, controlling for relevant covariates allowed us to rule out a number of alternative explanations for caregiver health effects.

Survey

Data for this study were drawn from the National Longitudinal Survey of Children and Youth (NLSCY), a long-term study of the physical and social development of Canadian children conducted jointly by Statistics Canada and Human Resources and Social Development Canada.35 The NLSCY identifies the person most knowledgeable about the child to provide information on the selected child, as well as information about herself or himself and the individual's spouse or partner (when appropriate). In 90% of cases, the person most knowledgeable about the child was the child's biological mother. Children were sampled from all areas of the country proportionate to the regional population. Exceptions were the northern territories (Yukon, Northwest Territories, Nunavut) and First Nations reserves, which were not included in the NLSCY sample. The first wave of interviews was conducted in 1994 and it has been repeated biennially since.

Sample

The study sample was drawn from the first NLSCY cohort of children aged 0 to 11 years who were interviewed in 1994 and 1995 (n = 22 831 children). The sample for our study was limited to children aged 4 to 11 years (n = 13 790) to ensure consistency of health outcome items, because items differed for younger children. In 1994 and 1995, the NLSCY interviewed multiple children per selected household, up to a maximum of 4 children. To eliminate repeated measures of the same caregiver, we selected 1 child per household. In households in which all children were healthy (3722 households, representing 4903 children) 1 child was randomly selected. In households in which all children were defined as having a health problem (4087 households, representing 5301 children) 1 child was randomly selected. In households in which there was a mix of healthy and non-healthy children (1592 households, representing 3586 children) we selected only the children who met our criteria for having a significant health problem. If more than 1 such child existed in the household, 1 was randomly selected. After these exclusions, our sample included 9401 child–caregiver pairs.

Categorizing Child Health Problems

Categorizing childhood health problems in the context of large-scale population-based surveys is a challenge, with no consensus on the best approach.36,37 In the past, chronic condition checklists have been widely used in surveys because of their simplicity.7,30 However, weaknesses of such an approach (e.g., exclusion of rare diseases, inconsistency of diagnostic labels, low reliability when categorizing children) have been noted.38 An alternative approach used increasingly in the United States involves grouping children with diverse conditions according to the consequences of their disorders (e.g., increased use of health care services, functional limitations) rather than categorically by clinical condition.38,39 Another approach commonly used in Canada and elsewhere considers chronic conditions along with functional limitations in a manner broadly consistent with the International Classification of Functioning Disability and Health.40,41 Arguments have been made for using both concepts in the context of childhood disability.33,42

A recent systematic review of measures and definitions of chronic health conditions in childhood43 identified at least 20 different definitions of chronic health conditions, which yielded prevalence estimates ranging from 0.22% to 44%. The included studies targeted different aspects of childhood health with a variety of categories, and prevalence estimates differed accordingly. Aaronson et al. argued for a clarification in terminology and increased transparency of methodology to facilitate an understanding of how these different aspects of childhood health are related. Furthermore, they proposed an initial framework in which chronic conditions comprised the most comprehensive category of childhood health problems, those with activity limitations forming a subset of this larger category, and those with high use of services constituting a subset of the first 2 categories.

In a recent article,44 we examined the relationship between different constructs related to childhood health problems (i.e., chronic conditions, activity limitations, functional problems, use of services) with the NLSCY. Our results suggested a different relationship between constructs than that suggested by the van der Lee et al. classification.37 Of those with activity limitations, 15% did not report a chronic condition; for those with elevated service use, more than 40% did not report a chronic condition or activity limitation. Rather than supporting a classification in which limitations and services use are subsets of chronic conditions, our findings suggested that each indicator identifies quite different subgroups of children and families.

In this study, we examined how caregiver health was related to poor child health broadly defined, at the population-based level. Although different candidate child health indicators yielded subgroups with very different characteristics, focusing instead on children identified by at least 2 of the 4 available indicators yielded a sample that (1) had considerably poorer health than those classified by 0 or 1 indicators, and (2) incorporated a wide range of childhood health problems and, therefore, caregiving situations.44 Consequently, we compared caregivers of healthy children—i.e., children who were not identified by any of the 4 key indicators—to caregivers of children with health problems, defined as those who were identified by 2 or more of the key indicators. We excluded from our analyses those individuals identified by 1 indicator alone, because this group is highly heterogeneous in terms of child factors, family composition, and child health status, and is likely to be less sensitive to caregiver group differences.44

Key Indicators

Activity limitation was based on an item that asked whether the child had any long-term conditions that limited participation at home, school, or at play, or in any other activity common for a child his or her age. A second question asked specifically about limitations caused by asthma. A positive response to either question indicated an activity limitation.

Functional problems were based on responses to the Health Utility Index (HUI).45 The HUI is a multiattribute system and scale used to classify an individual's functional abilities in 8 attributes: vision, hearing, speech, ambulation, dexterity, cognition, emotion, and pain. With an algorithm based on population preferences, a summary score is derived. An HUI score of 1.0 represents perfect health, whereas a score of 0.0 represents death; negative scores represent functional impairments that are considered worse than death by community citizens whose values were used to compute the utility values. According to a cutpoint proposed by Kopec et al.,46 children with a score of 0.81 or lower on the HUI are considered to have functional problems.

To measure chronic conditions, we used a checklist on which caregivers reported whether the child had been diagnosed by a health professional with any of the following conditions: asthma, allergies, bronchitis, cerebral palsy, epilepsy, heart condition or disease, kidney condition or disease, mental handicap, or “other.” For children 6 years and older, caregivers also reported whether the child had been diagnosed with a learning disability or any emotional, psychological, or nervous difficulties.

Elevated service use was determined using the NLSCY items related to 4 of the 5 concepts in the Children With Special Health Care Needs screener (CSHCN)47: (1) need or use of medicine prescribed by a doctor, other than vitamins; (2) need or use of more medical care, mental health, or educational services than is usual for most children of the same age; (3) special therapy, such as physical, occupational, or speech therapy; and (4) treatment or counseling for emotional, developmental, or behavioral problems. The fifth concept in the Children With Special Health Care Needs screener—limitations in ability to do the things most children of the same age can do—was omitted because it was conceptually similar to the activity limitation indicator. Additional information on the derivation of this indicator is provided elsewhere.44 Caregiver health outcomes, along with family, caregiver, and child covariates, are described in Table 1.

TABLE 1 Description of Study Variables for Comparison of Health Among Caregivers of Healthy Children Versus Caregivers of Children With Health Problems: National Longitudinal Survey of Children and Youth (NLSCY), Canada, 1994–1995

TABLE 1 Description of Study Variables for Comparison of Health Among Caregivers of Healthy Children Versus Caregivers of Children With Health Problems: National Longitudinal Survey of Children and Youth (NLSCY), Canada, 1994–1995

 Variable Description Caregiver health outcomes Chronic conditions Caregivers reported via checklist diagnosis of any chronic condition that had lasted, or was expected to last, 6 months or more. Checklist included food allergies, other allergies, asthma, arthritis or rheumatism, back problems excluding arthritis, high blood pressure, migraine headaches, chronic bronchitis or emphysema, sinusitis, diabetes, epilepsy, heart disease, cancer, stomach or intestinal ulcers, effects of stroke, urinary incontinence, Alzheimer disease or other dementia, cataracts, glaucoma, or any other long-term condition. Activity limitations Caregivers were asked whether a chronic health condition limited the kind or amount of activity they could do in 5 domains: at home, at school, at work, in caring for children, or in other activities. Those who reported being limited in 1 or more domains were said to have an activity limitation. General health Caregivers rated their general health on a 5-point scale, ranging from 1 (poor) to 5 (excellent). Caregivers' general health was dichotomized as being excellent/very good or good/fair/poor. Depressive symptoms Shortened version (12 items) of the Center for Epidemiological Studies Depression Scale48 was used to assess the occurrence and severity of caregiver depressive symptoms in the week prior to the survey. A 4-point rating scale was used: 0 = rarely or none of the time (less than 1 day in past week), 1 = some or a little of the time (1–2 days), (2) occasionally or a moderate amount of time (3–4 days), and (3) most or all of the time (5–7 days). The scale ranges from 0 to 36, with higher values indicating more—and more frequent—depressive symptoms. Continuous scores were classified according to reported symptom severity: 0–11 = minimal symptoms, 12–20 = somewhat elevated symptoms, and 21–36 = very elevated symptoms of depression.49 Family functioning Twelve-item General Functioning subscale of the McMaster Family Assessment Device was used to assess general family functioning.50 Caregivers reported degree to which each statement was true with a 4-point scale: 0 = strongly agree, 1 = agree, 2 = disagree, and 3 = strongly disagree. The scale ranged from 0 to 36, with higher scores indicating greater family dysfunction. Social support Level of social support caregivers received from friends, family, and others was assessed with a shortened version (6 items) of the Social Provisions Scale.51 Caregivers rated degree to which they agreed with statements of perceived support on a 4-point scale: 0 = strongly disagree, 1 = disagree, 2 = agree, and 3 = strongly agree. Scale ranged from 0 to 18, with higher scores indicating a greater perceived presence of social support. Marital satisfaction Caregiver's satisfaction with marital relationship was assessed with a single-item scale, ranging from 1 = completely dissatisfied to 11 = completely satisfied. Item was only asked of caregivers who reported being in a marital relationship, including common-law relationships. Family, caregiver, and child covariates Caregiver smoking status Single item was used to categorize caregiver smoking frequency at the time of interview: daily, occasionally, or not at all. Categorization method combined former smokers with lifetime nonsmokers. Caregiver educational attainment Three questions were asked regarding caregivers' educational attainment: whether graduated from high school, ever attended postsecondary education, and if so, what degree was earned. Highest level of education attained was categorized from the responses: less than secondary school, secondary school graduate, some postsecondary education, and graduation with a postsecondary degree or diploma (including trade, technical, or vocational colleges). Single- versus 2-parent families The NLSCY included a roster of individuals in the household and their relationship to the interviewed child. Children were described as living with 1 or 2 parents, including biological, adoptive, step– and foster parents. Number of children in household From NLSCY roster, which gave the number of children aged 0 to 17 years living in the household, including the interviewed child. Household income and income adequacy Estimate of the total household income before taxes and deductions also used to categorize the income adequacy of the household, which accounts for household size. The NLSCY provided 5 categories of income adequacy: (1) lowest adequacy, a household income less than Can $10 000 for 1 to 4 persons or less than Can$15 000 for 5 or more persons; (2) lower-middle adequacy, a household income of Can $10 000–$14 999 for 1 or 2 persons, Can $10 000–$19 999 for 3 or 4 persons, or Can $15 000–$29 999 for 5 or more persons; (3) middle adequacy, a household income of Can $15 000–$29 999 for 1 or 2 persons, Can $20 000–$39 999 for 3 or 4 persons, or Can $30 000–$59 999 for 5 or more persons; (4) upper-middle adequacy, a household income of Can $30 000–$59 999 for 1 or 2 persons, Can $40 000–$79 999 for 3 or 4 persons, or Can $60 000–$79 999 for 5 or more persons; and (5) highest adequacy, a household income of Can $60 000 or more for 1 or 2 persons, or Can$80 000 or more for 3 or more persons. For purposes of this study, the 2 lowest income adequacy groups were combined. Child age and gender Child's age and gender were examined as covariates related to caregiver health.
Analysis

We conducted descriptive comparisons between caregivers of children with health problems and healthy children by using the χ2 test (categorical) or t test (continuous) as appropriate. We used regression models to assess the independent impact of caring for a child with a health problem on caregiver health over and above the effect of covariates such as single versus 2-parent families, household size, income adequacy, and caregiver gender, age, educational attainment, and smoking status. To account for the complex survey design of the NLSCY, we weighted and estimated all analyses with the bootstrap procedure with SAS-callable SUDAAN version 9 (Research Triangle Institute, Research Triangle Park, NC).

Table 2 describes family, caregiver, and child characteristics for 3 groups: the total sample (N = 9401), the subgroup of children who were not categorized by any indicator of health problems (n = 3633, or 38.6% of the total sample), and the subgroup of children categorized as having health problems (n = 2495, or 26.5% of the total sample). The total sample of 9401 families had a mean income of Can $51 784; 82.0% were headed by 2 parents. Primary caregivers were most often women (90.0%), and usually the child's biological mother (88.0%). Caregivers typically had education levels beyond high school (65.0%), were nonsmokers (67.8%), and had a mean age of 35.8 years (standard error = 0.1). Of the 9401 children in the total sample, 6.6% had an activity limitation, 10.8% had functional problems, 37.4% had some kind of chronic health condition, and 43.0% had elevated service use. TABLE 2 Descriptive Analysis of Family, Caregiver, and Child Characteristics and Statistical Comparison of the 2 Subgroups: National Longitudinal Survey of Children and Youth, Canada, 1994–1995 TABLE 2 Descriptive Analysis of Family, Caregiver, and Child Characteristics and Statistical Comparison of the 2 Subgroups: National Longitudinal Survey of Children and Youth, Canada, 1994–1995  Total Sample (N = 9401), % or Mean (SE) Healthy Children (n = 3633), % or Mean (SE) Children With Health Problemsa (n = 2495), % or Mean (SE) Healthy Children Versus Children With Health Problems Characteristics Comparison Test P Family characteristics Family type Single parent 17.99 16.31 22.52 χ21 = 12.63 <.001 Two parents 82.01 83.69 77.48 Household income, Can$ $51 784 ($686) $51 353 ($965) $49 835 ($1 113) t = 1.05 .29 No. children in household 2.22 (0.02) 2.18 (0.03) 2.19 (0.03) t = 0.33 .74 Income adequacy categoryb Lowest or lower middle 17.53 17.25 19.27 χ23 = 1.77 .62 Middle 30.59 31.29 29.74 Upper middle 36.17 35.98 35.88 Highest 15.72 15.47 15.11 Caregiver characteristics Women 89.97 89.30 90.87 χ21 = 1.60 .21 Relationship to child Biological mother 87.97 87.13 88.25 χ22 = 3.04 .22 Biological father 9.49 10.35 8.56 Other 2.54 2.52 3.19 Age, y 35.80 (0.09) 36.12 (0.16) 35.44 (0.18) t = 2.68 .007 Educational attainment of the person most knowledgeable about the child Less than secondary school 16.48 15.95 17.89 χ23 = 5.75 .13 Secondary school graduate 18.57 19.59 16.87 Beyond high school 28.59 28.02 30.65 College or university degree (including trade) 36.36 36.45 34.59 Current smoking status Nonsmoker 67.83 70.84 63.35 χ22 = 20.10 <.001 Occasional 4.56 4.63 4.12 Daily 27.61 24.53 32.53 Child characteristics Age, y 7.50 (0.02) 7.43 (0.05) 7.72 (0.07) t = 2.98 .003 Boy 52.27 48.12 60.05 χ21 = 33.62 <.001 Has a chronic health condition 37.38 0 89.99 Has an activity limitation 6.56 0 23.59 Has functional problems 10.77 0 29.45 Has elevated use of health services 43.00 0 93.94

aOur definition of children with health problems only included those who had been classified by 2 or more indicators of health problems; children who met only 1 indicator (n = 3273) were excluded from the analyses.

bEstimate of the total household income before taxes and deductions. See Table 1 for categorical definitions.

The final column of Table 2 compares data concerning the children with health problems with those of healthy children on all demographic variables. Families of children with health problems were similar to families of healthy children on most sociodemographic characteristics, with exceptions that, compared with healthy children, children with health problems more often came from single-parent families (22.5% vs 16.3%; χ21 = 12.6; P < .001), had slightly younger caregivers (35.4 years vs 36.1 years; t = 2.7; P = .007), and were more likely to have a caregiver who was a daily smoker (32.5% vs 24.5%; χ22 = 20.1; P < .001).

Table 3 shows that caregivers of children with health problems more often had a chronic condition themselves (56.6% vs 34.5%; χ21 = 110.8; P < .001) and had more chronic conditions (mean number of chronic conditions: 2.0 vs 1.7; t = 5.4; P < .001) than did caregivers of healthy children. Caregivers of children with health problems also more often had an activity limitation (16.3% vs 7.3%; χ21 = 50.7; P < .001) and less often reported their health status as being excellent or very good (67.8% vs 77.5%; χ21 = 26.4; P < .001). Caregivers of children with health problems indicated more depressive symptoms than did caregivers of healthy children (mean scores: 5.9 vs 4.2 on a range of 0–36; t = 6.3; P < .001), but did not differ on measures of family functioning (mean scores: 8.3 vs 8.0 on a range of 0–36), social support (mean scores: 14.5 vs 14.5 on a range of 0–18), or marital satisfaction (mean scores: 9.4 vs 9.4 on a range of 1–11).

TABLE 3 Caregiver Health and Social Support Outcomes and Statistical Comparison of the 2 Subgroups: National Longitudinal Survey of Children and Youth, Canada, 1994–1995

TABLE 3 Caregiver Health and Social Support Outcomes and Statistical Comparison of the 2 Subgroups: National Longitudinal Survey of Children and Youth, Canada, 1994–1995

 Total Sample (N = 9401), % or Mean (SE) Healthy Children (n = 3633), % or Mean (SE) Children With Health Problems (n = 2495), % or Mean (SE) Healthy Children Versus Children With Health Problems Caregiver Outcome Comparison Test P At least 1 chronic condition 44.28 34.45 56.62 χ21 = 110.82 <.001 Number of chronic conditionsa 1.80 (0.03) 1.65 (0.05) 2.02 (0.05) t = 5.41 <.001 Limitation in at least 1 domain of activity 10.58 7.29 16.32 χ21 = 50.74 <.001 General health status excellent or very goodb 73.31 77.50 67.76 χ21 = 26.36 <.001 Depression scalec Mean score 4.86 (0.10) 4.23 (0.16) 5.88 (0.22) t = 6.25 <.001 Minimal score (0–11) 89.03 90.94 85.70 χ21 = 15.03 <.001 Somewhat elevated score (12–20) 8.17 7.03 9.70 χ21 = 5.89 .02 Very elevated score (21–36) 2.80 2.03 4.60 χ21 = 8.54 .004 Social support total scored 14.46 (0.05) 14.46 (0.08) 14.47 (0.11) t = 0.02 .98 Family functioning total scoree 8.06 (0.10) 7.99 (0.14) 8.25 (0.17) t = 1.20 .23 Marital satisfactionf 9.38 (0.04) 9.38 (0.06) 9.35 (0.08) t = 0.31 .76

aAmong those with at least 1 chronic condition.

bReference group is good/fair/poor general health.

cRange = 0–36; higher score indicates more symptoms of depression.

dRange = 0–18; higher score indicates better social support.

eRange = 0–36; higher score indicates greater family dysfunction.

fRange = 1–11; higher score indicates greater marital satisfaction; only asked of caregivers in a marriage relationship, including common law.

To examine the independent effect of a child's health status (i.e., healthy or with health problems) on his or her caregiver's health, we conducted a series of regressions on each of the 4 caregiver health outcomes (i.e., presence of chronic conditions, activity limitation, general health status, and depressive symptoms), with control for other family, caregiver, and child characteristics. As shown in Table 4, after we controlled for these characteristics, caregivers of children with health problems had more than twice the odds of reporting a chronic condition (odds ratio [OR] = 2.53; 95% confidence interval [CI] = 2.14, 3.00) or an activity limitation (OR = 2.42; 95% CI = 1.88, 3.12) and had greater odds of rating their overall health as relatively poor (OR = 1.64; 95% CI = 1.35, 1.99) than did caregivers of healthy children. Caregivers of children with health problems also had more than twice the odds of reporting very elevated depressive symptom scores (OR = 2.48; 95% CI = 1.40, 4.40). Caregiver groups did not differ on measures of family functioning, social support, or (for the subset of 2-parent families) marital satisfaction.

TABLE 4 The Effects of Child Health Status on Caregiver Health Outcomes After Control for Family, Child, and Caregiver Characteristics: National Longitudinal Survey of Children and Youth, Canada, 1994–1995

TABLE 4 The Effects of Child Health Status on Caregiver Health Outcomes After Control for Family, Child, and Caregiver Characteristics: National Longitudinal Survey of Children and Youth, Canada, 1994–1995

 Any Chronic Condition (n = 6062), OR (95% CI) Activity Limitation (n = 6063), OR (95% CI) General Health Statusa (n = 6063), OR (95% CI) Depression Score,b OR (95% CI) Caregiver of child with health problems (Ref = caregiver of healthy child) 2.53 (2.14, 3.00) 2.42 (1.88, 3.12) 1.64 (1.35, 1.99) 2.48 (1.40, 4.40) Child characteristics Male child (Ref = female child) 0.96 (0.80, 1.14) 0.86 (0.66, 1.12) 0.86 (0.71, 1.04) 0.50 (0.27, 0.92) Child's agec 0.99 (0.95, 1.03) 1.06 (0.99, 1.13) 0.99 (0.95, 1.04) 0.92 (0.81, 1.05) Family characteristics Single-parent family (Ref = 2 parents) 1.14 (0.89, 1.47) 1.56 (1.09, 2.21) 1.08 (0.83, 1.40) 2.20 (1.19, 4.09) Number of children in householdc 0.90 (0.81, 0.99) 0.88 (0.75, 1.03) 0.93 (0.83, 1.03) 0.96 (0.71, 1.31) Income adequacy categoryd (Ref = middle adequacy) Lowest or lower middle 1.52 (1.16, 1.98) 1.39 (0.95, 2.04) 1.45 (1.08, 1.94) 3.00 (1.40, 6.43) Upper middle 1.20 (0.97, 1.47) 0.72 (0.52, 1.00) 0.64 (0.50, 0.81) 0.83 (0.38, 1.83) Highest 1.29 (0.97, 1.71) 0.83 (0.52,1.34) 0.53 (0.37, 0.74) 0.64 (0.22, 1.84) Caregiver characteristics Men (Ref = women) 0.44 (0.32, 0.59) 0.92 (0.59, 1.43) 1.08 (0.76, 1.53) 0.98 (0.31, 3.05) Age 1.03 (1.01, 1.05) 1.01 (0.98, 1.04) 1.03 (1.01, 1.05) 1.05 (1.00, 1.11) Person most knowledgeable about the child had at least secondary school diploma (Ref = less than secondary) 1.09 (0.86, 1.37) 1.07 (0.78, 1.46) 0.53 (0.42, 0.68) 0.89 (0.47, 1.69) Caregiver's current smoking status (Ref = nonsmoker) Occasional 0.83 (0.52, 1.34) 0.96 (0.50, 1.84) 1.35 (0.85, 2.14) 2.78 (0.87, 8.80) Daily 1.10 (0.92, 1.33) 1.13 (0.86, 1.49) 1.84 (1.50, 2.25) 1.60 (0.85, 3.02)

Note. OR = odds ratio; CI = confidence interval.

aPredicting poorer health (poor/fair/good vs very good/excellent).

bVery elevated symptom scores.

cContinuous measure; baseline = 0.

dEstimate of the total household income before taxes and deductions. See Table 1 for categorical definitions.

The results of this study suggest that the health of caregivers of children with health problems is significantly poorer than the health of caregivers of healthy children. After we controlled for relevant family, caregiver, and child factors, caregivers of children with health problems had substantially greater odds of reporting chronic conditions, activity limitations, poor general health, and symptoms of depression than did caregivers of healthy children. This work contributes to the existing literature by having used large-scale, population-based data and a broad spectrum of child health problems to ensure generalizability to a wide range of caregiving situations; by broadening our knowledge of caregivers' physical health issues; and by isolating the independent effect of caring for a child with a health problem on caregiver health, after controlling for relevant covariates.

Note that in our effort to examine caregiver health effects at a broad, population-based level, we chose a categorization scheme that grouped together a heterogeneous sample of child health problems and caregiving situations. Our sample of children with health problems, therefore, included both children with mild conditions who nevertheless used a greater number of health services, and severely disabled children with complex disorders, functional problems, and activity limitations. Because we included caregivers of children with such a range of conditions in our sample, the observed caregiver health effects are all the more remarkable.

One logical next step of this work is to examine whether increased severity of child health problems is related to poorer caregiver health. Prior to this, however, the relationship between child health problem severity and caregiver burden needs to be clarified. Caregivers who feel burdened by caring for children with relatively mild conditions may fare more poorly than caregivers who feel less burdened by children with more-severe health problems; that is, it may be “perceived severity” rather than the actual severity of the child's health problem that affects caregiver health. Alternatively, it may be that the number and complexity of child health problems predict caregiver health better than the severity of a specific child health problem. Such issues are likely to be better addressed in the context of specific childhood conditions, when measures of disease severity have been validated, the prevalence of comorbidities is well understood, and specific caregiving challenges are known.

Given the cross-sectional nature of our study, the findings of an association between child and caregiver health do not unequivocally imply that the additional burdens associated with caring for a child with a health problem cause caregiver health problems. Although we were able to estimate the effect of caring for a child with a health problem after we controlled for important covariates, unmeasured variables (e.g., genetic predispositions) could account for the health differences between caregiver groups.

However, 2 intriguing initial results lead to consideration of some form of causal relationship. First, in a previous study5 we showed that the pattern of physical chronic conditions reported by caregivers of children with cerebral palsy was consistent with a stress process explanation. Conditions associated with physical or emotional stress (e.g., back problems, ulcers) were much more prevalent among these caregivers, whereas other conditions less obviously associated with stress (e.g., food allergies, bronchitis) showed much smaller effects. Second, we conducted exploratory analyses with the current data set to see whether group differences in caregiver health remained robust for caregivers who are not biologically related to their children. We reasoned that if both caregiver health and child health problems were caused primarily by a common genetic predisposition, an interaction between type of parent (i.e., biological, nonbiological) and caregiving situation (i.e., healthy, with health problems) on caregiver health would become apparent. No such interaction was evident, although small sample sizes precluded any strong conclusions. These admittedly preliminary data suggest that caregiver health may be affected by the additional stresses associated with caring for a child with a health problem.

Limitations

A number of limitations warrant consideration. First, although our sample of child–caregiver dyads was population based and included a wide range of child health problems and caregiving situations, it cannot be considered representative of the Canadian population of caregivers. Threats to representativeness included (1) our decision to exclude the 1-indicator-only group from these analyses; (2) the fact that the NLSCY was designed to obtain a nationally representative sample of children, not caregivers, and thus, the complex sampling strategies necessary to ensure national representativeness were not applied to the sample of caregivers; and (3) by limiting our sample to 1 child per caregiver and to a certain child age range, we may have selected a biased sample.

Indeed, higher rates of caregiver income, education, and 2-parent households compared with national norms52 suggest that such a bias may be present. Whether such a bias would substantially affect our findings is unknown, although, if anything, an affluence bias might suggest that the true caregiver health effects are underestimated in our study. Replication and related work in other countries (e.g., the United States) with different health care and social systems would allow assessment of the pervasiveness of this association between child and caregiver health, and perhaps identify interventions, such as family-centered services,22 that have the potential to improve caregiver health.

Second, interpretation of our results must be tempered by the knowledge that the data were collected cross-sectionally, yet the effects on caregiver health of stress associated with caring for a child with a health problem are likely to be complex, dynamic, and manifest themselves over time. Analysis of the health of the caregiver over time will be necessary to gain a better understanding of these processes. We plan to address this issue, making use of the longitudinal data collection schedule of the NLSCY.

Third, all indicators and outcomes were based on parent-reported measures. Parental reports of children's health conditions may not accurately reflect the child's health problems. Moreover, there is an issue of shared method variance; that is, parents who report poor health in their children may be more likely to report themselves in poor health. However, parent reports have been shown to be relatively reliable for severe health conditions.5356 Certainly, physician diagnoses and data linked from administrative sources would be a useful area for further study. For the present, our validated method for classifying children with health problems44 makes the best use of the self-reported data by combining several widely recognized indicators into a broad category of childhood health problems.

Finally, the definition of childhood illness used in our study included many nonspecific health problems and focused on the consequences of childhood health problems (e.g., use of health services) rather than specific child health conditions. This approach ensured that caregivers from very different caregiving situations would be grouped together in our analyses, potentially masking differences between specific caregiving situations. As described earlier, this practice is garnering support in the caregiver literature and is defensible given that policymakers must make their decisions based on broad categorizations. However, future work will compare subgroups of interest to examine relative effects on caregiver health; for example, children with externalizing behavior problems in addition to other health conditions may pose a particular caregiving challenge that might impact caregiver health.22,25,57

Conclusions

This study shows that after we controlled for differences in relevant family, child, and caregiver factors, caregivers of children with health problems (broadly defined) themselves have substantially greater odds of reporting chronic conditions, activity limitations, poorer health, and higher depressive symptoms than did caregivers of healthy children. Regardless of the causal pathways leading to poorer caregiver health, the results have implications for health providers and policymakers. Clinicians will be better prepared if they are aware of the increased risk for health problems that is associated with caring for children with health problems. Research examining family-centered services (comprehensive services that address child health and parental well-being together) could detail our understanding of the pathways by which such services improve care. Policy decisions that encourage the use of such family-centered services would underline the importance of healthy families to the care and support of children with health problems.

## Acknowledgments

This project was funded by the Canadian Institutes of Health Research (grant MOP-77620). J. C. Brehaut is a Career Scientist with the Ontario Ministry of Health and Long-Term Care. D. E. Kohen would like to acknowledge the support of the Social Sciences and Humanities Research Council Initiative for a New Economy. A. R. Miller is supported by an award from the Sunny Hill Foundation for Children. A. F. Klassen received salary support from the Canadian Institutes of Health Research and the Michael Smith Foundation for Health Research. P. L. Rosenbaum holds a Canada Research Chair in Childhood Disability.

We thank Alison Lott for her administrative assistance, and 3 anonymous reviewers for their detailed and thoughtful comments.

## Human Participation Protection

This work has been approved by the Ottawa Hospital Research Ethics Board.

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No related items

### ARTICLE CITATION

Jamie C. Brehaut, PhD, Dafna E. Kohen, PhD, Rochelle E. Garner, PhD, Anton R. Miller, MB, ChB, FRCPC, Lucyna M. Lach, PhD, MSW, Anne F. Klassen, DPhil, and Peter L. Rosenbaum, MDJamie C. Brehaut is with the Ottawa Health Research Institute, Ottawa, Ontario, and the Department of Epidemiology and Community Medicine at the University of Ottawa, Ottawa. Dafna E. Kohen is with the Information and Research Division at Statistics Canada, Ottawa, and the Department of Epidemiology and Community Medicine at the University of Ottawa, Ottawa. Rochelle E. Garner is with the Information and Research Division at Statistics Canada, Ottawa. Anton R. Miller is with the Department of Pediatrics, University of British Columbia, Vancouver, and the Centre for Community Child Health Research, Child and Family Research Institute, Vancouver. Lucyna M. Lach is with McGill University's School of Social Work, Montreal, Quebec. Anne F. Klassen is with the Department of Pediatrics, McMaster University, Hamilton, Ontario. Peter L. Rosenbaum is with the McMaster Child Health Research Institute, Hamilton. “Health Among Caregivers of Children With Health Problems: Findings From a Canadian Population-Based Study”, American Journal of Public Health 99, no. 7 (July 1, 2009): pp. 1254-1262.

https://doi.org/10.2105/AJPH.2007.129817

PMID: 19059861


text:
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# Homework Help: Missing component to get the impendance ohmic

1. Jul 19, 2011

### Lindsayyyy

1. The problem statement, all variables and given/known data

Hi, following graphic is given (attachment)

2. Relevant equations
-

3. The attempt at a solution

I tried it with the following equation:

$$\omega iL-\frac i {\omega C} + X =R$$

but I don't get to the solution, is this the right attempt? If yes, I'll try later on again, have to go to the university right now.

Thanks for the help

#### Attached Files:

• ###### Unbenannt.JPG
File size:
4.9 KB
Views:
105
2. Jul 19, 2011

### cupid.callin

what is the question?

3. Jul 19, 2011

### Lindsayyyy

ah sorry, I was in a hurry. I shall find the missing component so that the impendance/resistance of the series connection is just ohmic

4. Jul 19, 2011

### ehild

L and R are parallel. What is their resultant impedance?

ehild

5. Jul 19, 2011

### cupid.callin

Will that work?
the unknown resistance is also the part of LR circuit

6. Jul 19, 2011

### Lindsayyyy

The impendance is:

$$\frac 1 Z = \frac 1 {i\omega L} - \frac {\omega C} {i}$$

Do I have to solve it afterwards with the equation

$$R=X+Z$$

And find the X ?

7. Jul 19, 2011

### cupid.callin

what is C?

8. Jul 19, 2011

### ehild

That box in the figure is not a capacitor but a resistor R connected parallel to the inductor. And yes, Z+X has to be real.

ehild

9. Jul 19, 2011

### Lindsayyyy

Sorry everyone, I'm totally asleep at the switch, having exams in a few days and things are getting confused. I'll try it lateron, have so solve some other things first.

@ Cupid, I don't have any idea why I thought theres a capacity. maybe I should better look at the drawing :grumpy:

thanks everyone

10. Jul 19, 2011

### Lindsayyyy

I'm stuck, can't solve it.

I tried to calculate Z, my Z is the parallel connection between R and L

and I get for Z:

$$\frac 1 Z = \frac 1 R + \frac {1} {i \omega L}$$

after some conversions I get to:

$$Z= \frac {R\omega^{2}L+iR^{2}\omega L} {R^{2}+\omega L^{2}}$$

Now I want to get a pure ohmic resistance so

$$Z+X=R$$

but I just can't solve it. The task says the missing component is a "normal" one, so I guess it should be a capacity, inductor etc. Though I tried to take one of these to test if I get a solution, I don't get to an answer. Can anyone help me

11. Jul 19, 2011

### Staff: Mentor

Check your algebra. The ω in the term in the denominator should be squared, and the L in the real portion of the numerator should also be squared.

Once you've sorted that out, divide the expression into two parts: the real part and the imaginary part. It's the imaginary part that you want to "take care of" with the new impedance X.

12. Jul 19, 2011

### Lindsayyyy

Hope I haven't done any mistakes again:

$$X=R- \frac {R\omega^{2}L^{2}} {R^{2}+\omega^{2}L^{2}} - \frac {iR^{2}\omega L} {R^{2}+\omega^{2}L^{2}}$$

so in order to get a pure ohmic term my X has to be:

$$X= \frac {iR^{2}\omega L} {R^{2}+\omega^{2}L^{2}}$$

but what component is that? I don't get it :(

13. Jul 19, 2011

### Staff: Mentor

If you let Z be the impedance of the parallel R and L components, then you've found that

$$Z = \frac {R\omega^{2}L^{2}} {R^{2}+\omega^{2}L^{2}} + i\frac {R^{2}\omega L} {R^{2}+\omega^{2}L^{2}}$$

In order to cancel the imaginary term you need to put in series an impedance that is its negative, that is,

$$X = - i\frac {R^{2}\omega L} {R^{2}+\omega^{2}L^{2}}$$

Now, what sort of component has an impedance that yields a negative imaginary term?

14. Jul 19, 2011

### Lindsayyyy

I'd say capcity with:

$$R_{C}=-\frac i {\omega C}$$

if that answer is right, I still don't understand why to use this one

15. Jul 19, 2011

### Staff: Mentor

Yes, a capacitor gives a negative imaginary impedance.

If you set the above expression for a capacitor's impedance equal to the X expression then you can solve for C, the value that the capacitor must have in order to cancel out the imaginary term of the impedance in Z at a given frequency ω.

Last edited: Jul 19, 2011
16. Jul 19, 2011


text:
# Quantum operator derivation

1. Apr 14, 2013

### BruceW

Hi all! I was reviewing some basic quantum mechanics, and I was trying to 'derive' the equation
$$-i \hbar \frac{\partial}{\partial x} \psi_{(t,x)} = <x| \hat{P} | \psi >$$
using the commutator relation, and the form of the identity operator. OK, I know that the proper, mathematical way to prove the above equation, is by the Stone-von Neumann theorem. But anyway, I tried doing a sloppy, physicist's 'derivation', so that I could get a more intuitive sense for why it works. I am posting this partly so that it helps it stick in my brain, partly in case anyone has heard of any other interesting derivations and partly because I am not certain that all my steps are correct, so hopefully if I went wrong somewhere, maybe someone can kindly point it out. I do get to the right answer, but that doesn't mean all the steps were correct :)

Anyway, so assuming the commutator relation
$$\hat{X} \hat{P} - \hat{P} \hat{X} = i \hbar$$
And pre-multiply by an eigenvector of the X operator, and post-multiply by another (not necessarily the same) eigenvector of the X operator, we get:
$$<x| \hat{X} \hat{P} |x'> - <x| \hat{P} \hat{X} |x'> = i \hbar <x|x'>$$
For this derivation, I need to use this form of the identity operator:
$$\int dx \ |x><x|$$
(In this post, I am just considering 1d, and always assume an integral is over all space, unless explicitly stated otherwise). I'm also going to assume that:
$$<x|x'> = \delta (x-x')$$
Is the Dirac-delta function. And so, using this with the above equation, and using the fact that the X operator acts on its eigenvectors to give eigenvalues, we get:
$$(x-x') \ <x| \hat{P} |x'> = i \hbar \delta(x-x')$$
And from now, I will use the notation:
$$<x| \hat{P} |x'> = P_{(x,x')}$$
Which means our equation looks like:
$$(x-x') \ P_{(x,x')} = i \hbar \delta(x-x')$$
(I will refer to this as equation 1). And now, introducing the thing we are interested in:
$$<x| \hat{P} | \psi >$$
The first thing we can do, is to insert identity operators around the P operator, and use the notation:
$$<x'| \psi> = \psi_{(x')}$$
So that we get:
$$<x| \hat{P} | \psi > = \int dx' \ P_{(x,x')} \psi_{(x')}$$
Now, I'm going to define a new function, to put this equation into a useful form. The new function is defined by this equation:
$$\psi_{(x')} = (x-x') U_{(x,x')}$$
(Where the subscripts are variables which the function is dependent on). So now, using this substitution, we get:
$$<x| \hat{P} | \psi > = \int dx' \ P_{(x,x')} (x-x') U_{(x,x')}$$
Now, we can use equation 1, to get:
$$<x| \hat{P} | \psi > = \int dx' \ i \hbar \delta(x-x') U_{(x,x')}$$
And now, hooray, the Dirac-delta function does its thing, so then:
$$<x| \hat{P} | \psi > = i \hbar U_{(x,x)}$$
Notice that the function 'U' now depends on the two same variables, because of the integration with the Dirac-delta function. So now, we need to find what U(x,x) is. Right, so starting with our original definition, but I'm going to swap around the primes, just for convenience (it is still the same equation):
$$\psi_{(x)} = (x'-x) U_{(x',x)}$$
And now, again for convenience, I am going to stop writing the subscripts in the next few equations. Remember, psi and U still have the subscripts in the way I have just written them, and I should really write them out in full, but I think that would take up a lot of space. Now, doing a partial differentiation on both sides with respect to x', while keeping x constant, and keep in mind that psi does not vary when x is held constant, we get:
$$0 = U + (x'-x) \frac{\partial U}{\partial x'}$$
Also, doing a full derivative with respect to x on both sides gives:
$$\frac{d \psi}{dx} = ( \frac{dx'}{dx} - 1)U + (x'-x)( \frac{\partial U}{\partial x'} \frac{dx'}{dx} + \frac{\partial U}{\partial x} )$$
And now combining the last two equations gives:
$$\frac{d \psi}{dx} = -U + (x'-x) \frac{\partial U}{\partial x}$$
So now, writing it out properly with the subscripts:
$$U_{(x',x)} = (x'-x) \frac{\partial U_{(x',x)} }{\partial x} - \frac{d \psi_{(x)} }{dx}$$
So now, we see that:
$$U_{(x,x)} = - \frac{d \psi_{(x)} }{dx}$$
Which is awesome, because using this substitution for U(x,x) in the equation way back earlier, we have:
$$<x| \hat{P} | \psi > = -i \hbar \frac{d \psi}{dx}$$
Which is the result I was hoping for. Now, you'll notice that this is different to the standard equation (which is a partial differentiation, while keeping t constant). This is because I haven't introduced a time variable yet. It is not necessary for this derivation. And now, if I want to introduce a time variable, then the equation simply changes to:
$$<x| \hat{P} | \psi > = -i \hbar \frac{\partial \psi_{(t,x)}}{\partial x}$$
So x is being differentiated, while t is held constant. (which is the standard equation). Whew. So, what do you guys think? It is longer than I had hoped. Sorry about that. Hope you enjoyed reading it though.

Last edited: Apr 14, 2013
2. Apr 14, 2013

### Jano L.

Bruce, your procedure shows that you know the Dirac notation quite well. Unfortunately, this notation oversimplifies things and betrays students all the time. In the step

"Now, we can use equation 1, to get"

you substitute

$$P_{xx'} (x-x')$$

by

$$i\hbar \delta(x-x').$$

That would be OK if the function $U$ of $x'$ next to it was continuous in some neighborhood of the point $x$. But you define the function $U$ to have singularity around the point $x$ ; it behaves as $\frac{1}{x'-x}$. For such functions, the above substitution is not valid, because

$$\int dx' \delta(x-x') \frac{1}{x'-x}$$

is an invalid expression.

Before you try to circumvent this problem and find another derivation, consider how everything becomes much more clear and simple, if you turn the question around: given that the operator of momentum is $\hat p_x = i\hbar \frac{\partial}{\partial x}$, derive the action of $x\hat p_x - \hat p_x x$ on any function $\psi(x)$. No distributions, no kets, no problems :-)

3. Apr 14, 2013

### BruceW

You are right for sure that:
$$\delta(x-x') \frac{1}{x'-x}$$
Is a pretty messed up kind of expression. But I don't think it is necessarily invalid. I just did a google search, and found it being referred to as a 'derivative of the Dirac-delta function' whatever that means... I agree that I haven't shown that it works in a rigorous way. But rigor is definitely not something I was aiming for, hehe. It is probably the most mathematically questionable derivation ever! Well, maybe that's a bit of an exaggeration.

Also, yes, I totally agree that it is much easier to show that the commutation relation holds when the operator of momentum takes that form. Maybe I am a masochist? ;) was that too creepy? Is saying something like that against forum rules??!

4. Apr 15, 2013

### Jano L.

5. Apr 15, 2013

### vanhees71

The usual derivation uses the fact that
$$[\hat{x},\hat{p}]=\mathrm{i}$$
($\hbar=1$, one-dimensional motion) implies that the momentum operator is the generator of translations.

To prove this consider the generalized eigenvectors of the position operator $|x \rangle$. For the transformed vector we get
$$\hat{x} \exp(-\mathrm{i} \delta x \hat{p}) |x \rangle = \exp(-\mathrm{i} \delta x \hat{p}) \exp(\mathrm{i} \delta x \hat{p}) \hat{x} \exp(-\mathrm{i} \delta x \hat{p}) |x \rangle = \exp(-\mathrm{i} \delta x \hat{p}) [\hat{x}-\mathrm{i} \delta x [\hat{x},\hat{p}] + \mathcal{O}(\delta x^2)] |x \rangle.$$
Now due to the above assumed commutation relation we get
$$\hat{x} \exp(-\mathrm{i} \delta x \hat{p}) |x \rangle = (x+\delta x) \exp(-\mathrm{i} \delta x \hat{p}) |x \rangle.$$
This means that, up to a phase factor which we can choose conveniently to be 1, we have
$$\exp(-\mathrm{i} \delta x \hat{p} ) |x \rangle = |x+\delta x \rangle = |x \rangle -\mathrm{i} \delta x \hat{p} x \rangle + \mathcal{O}(\delta x^2).$$
From this you derive
$$\frac{\mathrm{d}}{\mathrm{d} x} |x \rangle =\lim_{\delta x \rightarrow 0} \frac{1}{\delta x} (|x + \delta x \rangle - |x \rangle)=-\mathrm{i} \hat{p} |x \rangle.$$
Taking the adjoint of this equation, gives
$$\frac{\mathrm{d}}{\mathrm{d} x} \langle x | = +\mathrm{i} \langle x | \hat{p}.$$
Multiplying from the right with an appropriate true Hilbert-space vector gives the position representation of the momentum operator to be
$$\langle x | \hat{p}| \psi \rangle:=\hat{p} \psi(x)=-\mathrm{i} \frac{\mathrm{d}}{\mathrm{d} x} \psi(x),$$
and that's the formula you wanted to prove.

6. Apr 15, 2013

### BruceW

ah, very nice. And much quicker derivation :) So we define momentum as the generator of translations, and use the commutation relation, and this gives the position representation of the momentum operator. Generators turn up a lot in more advanced quantum theory, right? I've only used them a few times, but I feel like they are a useful 'tool'.

@ Jano L. Wikipedia seem to define the derivative of the Dirac-delta in this way:
$$x \delta'_{(x)} = - \delta_{(x)}$$
http://en.wikipedia.org/wiki/Dirac_delta_function So using their definition,
$$P_{(x,x')} = -i \hbar \delta'_{(x-x')}$$
And since
$$<x| \hat{P} | \psi > = \int dx' P_{(x,x')} \psi_{(x')}$$
we can sub in the derivative of the Dirac-delta:
$$<x| \hat{P} | \psi > = \int dx' (-)i \hbar \delta'_{(x-x')} \psi_{(x')}$$
Right, so now we have a convolution of psi with the derivative of the Dirac-delta, and wikipedia say that:
$$\delta' \ast f = f'$$
(i.e. the convolution of the derivative of the Dirac-delta with a general function gives the derivative of that function). Therefore, we have:
$$<x| \hat{P} | \psi > = -i \hbar \delta' \ast \psi = -i \hbar \psi'$$
Which is the formula I wanted to prove, and done a bit faster this time, by making use of the properties of the derivative of the Dirac-delta, instead of doing it the longer way like I did the first time. (But still it is not as fast as vanhees' method).

7. Apr 16, 2013

### Jano L.

Yes, the above procedure makes sense, but I would not call it a proof. It is rather an exercise in correct handling of the Dirac notation.. You derived differentiation as integration over $\delta'$, irregular disitrbution. The difficulty is that if you wanted to give the above manipulations solid mathematical meaning, you would spend many pages on it and I am afraid that you would need to use much less intuitive knowledge than $\hat p_x = i\hbar \frac{\partial}{\partial x}$ in the process, namely theory of distributions, questions of function spaces etc. So, as an exercise in the Dirac notation, its OK, but as a proof which is supposed to explain the complicated in terms of the simple, it is useless.

8. Apr 16, 2013

### Jano L.

Another thing came to my mind. Back to your procedure, the distribution

$$P_{xx'} = C\delta(x-x') - i\hbar \delta'(x-x')$$

satisfies

$$(x-x')P_{xx'} = i\hbar \delta (x-x')$$

too. So you have yet to find the value of $C$...

9. Apr 16, 2013

### BruceW

ah darn, you're right. My second derivation doesn't work. Also I am not happy with my first derivation. Even if we can assume that psi can be re-written as:
$$\psi_{(x)} = (x'-x) U_{(x',x)}$$
(Which I am now thinking is a mistake, because I don't think we can generally do this... Unless someone knows of a theorem which says that you can do this if psi is a differentiable function?) Anyway, even if we assume this is possible, then I don't think that this implies that:
$$U_{(x,x)} = - \frac{d \psi_{(x)} }{dx}$$
I think I was just tricking myself by re-writing things. d'oh. So anyway, I think the second derivation is looking better than the first. Even though there is that C, which is mysteriously unknown. Maybe if we make some assumptions about the wavefunction, then the C problem goes away? I'm having a hard time thinking what this assumption might be..


text:
Errors and Warnings

There are three types of R error messages: errors, warnings and messages. These are ‘thrown’ with stop(), warning() and message() respectively.

stop("This isn't working.")
## Error: This isn't working.
warning("I'm not sure about this")
## Warning: I'm not sure about this
message("By the way...")
## By the way...

Some error messages are quite informative:

rnorm("hello")
## Warning: NAs introduced by coercion
## Error: invalid arguments

Some are less so.

f = function(lambda, dat) sum(dat*log(lambda) - lambda)
y = rpois(100, lambda=2)
newton(f, start=1, dat=y)
## Warning: NaNs produced
## Error: missing value where TRUE/FALSE needed

When you write code try to make your messages as specific and as helpful as possible!

Tracebacks

The most powerful way to find out what’s gone wrong is to ‘step in’ to the function at the point of failure.

options(error=recover)

When your function throws an error, you’ll be presented with a call stack, and you can select any environment to enter into. My advice is to start in the deepest environment containing code which you wrote yourself.

In the case below (we’re calling the code from the profiling page) you don’t have a choice: press ‘1’ to step inside your function.

> newton(f, start=1, dat=y)
Error in while (max(abs(FUN(x, ...))) > tol) { :
missing value where TRUE/FALSE needed
In log(lambda) : NaNs produced

Enter a frame number, or 0 to exit

1: newton(f, start = 1, dat = y)

Selection:

Warnings

A warning is usually an indication that something is wrong, even if it hasn’t generated a full blown error, so don’t ignore them! For debugging purposes it’s useful to set

options(warn=2)

which turns warnings straight into errors (the default is 0).

> newton(f, start=1, dat=y)
Error in log(lambda) : (converted from warning) NaNs produced

Enter a frame number, or 0 to exit

1: newton(f, start = 1, dat = y)
2: #4: FUN(x, ...)
3: #1: .signalSimpleWarning("NaNs produced", quote(log(lambda)))
4: withRestarts({
.Internal(.signalCondition(simpleWarning(msg, call), msg, call))
.Int
5: withOneRestart(expr, restarts[[1]])
6: doWithOneRestart(return(expr), restart)

Selection:

If you step into frame number 2 (which is the function f being called) we can check what the dat and lambda are:

Browse[1]> lambda
[,1]
[1,] -0.6477124

So f was called with inappropriate arguments. If we step out of this frame and back to frame 1…

Browse[2]> x
[,1]
[1,] -0.6477124

Now we see the problem: we stepped outside the valid range of our function.

Error Handling

In order to make newton() deal with this problem, we have to allow it to handle an error given to it by the function FUN.

## step()
##' Obtain one step of a Newton algorithm
##'
##' @param FUN    function taking single vector argument
##' @param x      current value of argument to \code{FUN}
##' @param eps    length of step size used for calculating numerical derivative
##'
##' @return numeric vector giving next \code{x} value
step = function(FUN, x, ..., eps = 1e-8) {
val = FUN(x, ...)
deriv = matrix(NA, length(val), length(val))

for (i in 1:length(val)) {
eps_vector = rep(0, length(val))
eps_vector[i] = eps

deriv[,i] = (FUN(x + eps_vector, ...) - FUN(x, ...)) / eps
}
out = -solve(deriv) %*% val
return(out)
}

## newton()
##' Run Newton algorithm to find root of function
##'
##' @param FUN    function taking single vector argument
##' @param start  starting value of argument to FUN
##' @param tol    maximum value of function for convergence
##' @param eps    length of step size used for calculating numerical derivative
##'
##' @return numeric vector of point achieved after convergence
##' of the Newton algorithm
newton = function(FUN, start, ..., tol = 1e-8, eps = 1e-8) {
x = start

val = tryCatch(FUN(x, ...),
error = function(e) stop("Invalid starting value for FUN")
)

while (max(abs(val)) > tol) {
move = step(FUN, x, ..., eps=eps)
x = x + move

val = tryCatch(FUN(x, ...),
error = function(e) stop("Moved outside valid range")
)

}

return(x)
}

newton(f, start=2, dat=y)
## Warning: NaNs produced
## Error: missing value where TRUE/FALSE needed

At this point you might notice that the function f I keep passing is the log-likelihood, but I should be giving the derivative of the log-likelihood. Hence:

Df = function(lambda, dat) sum(dat/lambda - 1)
newton(Df, start=1, dat=y)
##      [,1]
## [1,] 1.82

And now it works!

debug()

Sometimes the traceback doesn’t help, because the problem occurred before an error was thrown. If you call debug() on a function, you can ‘step through’ the function as it runs line-by-line, and see what it’s doing at each stage.

debug(lm)
x = rnorm(20); y = x+rnorm(20)
lm(y ~ x)

There are some special commands you can use: f will ‘finish’ the current loop; s allows you to step into the function calls in the next statement, again using the browser. Q finishes all executions and gets you back to the ordinary command line.

Run undebug(lm) (or equivalent) to stop this behaviour.

trace()

The trace() function provides a principled way of adding (for example) print statements to your code, without actually modifying it permanently. The most intuitive way to do this is with the edit=TRUE option, which gives you an editor containing a copy of your function. Any changes you make here will not alter your real code, so you can play around as you like. This version of the function will be run until you redeine your function, or call untrace().

trace(lm, edit=TRUE)
# try adding a print statement or two


text:
# Tag cloud: Wikis

Note: Many of our articles have direct quotes from sources you can cite, within the Wikipedia article! This article doesn't yet, but we're working on it! See more info or our list of citable articles.

# Encyclopedia

A tag cloud with terms related to Web 2.0
A Tag Cloud for Searching on Google

A tag cloud or word cloud (or weighted list in visual design) is a visual depiction of user-generated tags, or simply the word content of a site, typically used to describe the content of web sites. Tags are usually single words and are normally listed alphabetically, and the importance of a tag is shown with font size or color.[1] Thus, both finding a tag by alphabet and by popularity are possible. The tags are usually hyperlinks that lead to a collection of items that are associated with a tag.

## History

The first use of tag clouds on a high-profile website was on the photo sharing site Flickr, created by Flickr co-founder and interaction designer Stewart Butterfield.[2] That implementation was based[citation needed] on Jim Flanagan's Search Referral Zeitgeist,[3] a visualization of Web site referrers. Tag clouds have also been popularized by Del.icio.us and Technorati, among others. Flickr would later apologize to the web-development community in their five-word acceptance speech for the 2006 "Best Practices" Webby Award, where they simply stated "sorry about the tag clouds."[4]

The first published appearance of a tag cloud (or at least a weighted list) in the English language may have been as the "subconscious files" in Douglas Coupland's Microserfs (1995)[citation needed]; a German appearance occurred at least three years earlier.[5]

Prior to weighted list representation of tag clouds, paper maps had used the concept of weighted font size and font weights to represent relative size or importance of towns and cities. On 24 March 2009, CNN created what they claimed was the "largest word cloud in the free world" for that night's Anderson Cooper 360°. It was a word cloud of President Obama's address to the press earlier that day.[citation needed]

In recent years tag clouds gained even more popularity because of their role in search engine optimization of web pages. Properly implemented tag clouds make the website appear to search engine spiders more interlinked which tends to improve its search engine rank.[6]

## Types

A data cloud showing the population of each of the world's countries. Color visually separates the countries, font size indicates 2007 population.

There are three main types of tag cloud applications in social software, distinguished by their meaning rather than appearance.[citation needed] In the first type, there is a tag for the frequency of each item, whereas in the second type, there are global tag clouds where the frequencies are aggregated over all items and users. In the third type, the cloud contains categories, with size indicating number of subcategories.

In the first type, size represents the number of times that tag has been applied to a single item.[7] This is useful as a means of displaying metadata about an item that has been democratically 'voted' on and where precise results are not desired. Examples of such use include Last.fm (to indicate genres attributed to bands) and LibraryThing (to indicate tags attributed to a book).

In the second, more commonly used type,[citation needed] size represents the number of items to which a tag has been applied, as a presentation of each tag's popularity. Examples of this type of tag cloud are used on the image-hosting service Flickr, blog aggregator Technorati and on Google search results with DeeperWeb .

In the third type, tags are used as a categorization method for content items. Tags are represented in a cloud where larger tags represent the quantity of content items in that category.

More generally, the same visual technique can be used to display non-tag data[8], as in a word cloud or a data cloud.

## Visual appearance

A data cloud showing stock price movement. Color indicates positive or negative change, font size indicates percentage change.

Tag clouds are typically represented using inline HTML elements. The tags can appear in alphabetical order, in a random order, they can be sorted by weight, and so on. Most popular is a rectangular tag arrangement with alphabetical sorting in a sequential line-by-line layout. The decision for an optimal layout should be driven by the expected user goals.[9] Some prefer to cluster the tags semantically[10][11][12] so that similar tags will appear near each other. Heuristics can be used to reduce the size of the tag cloud whether or not the purpose is to cluster the tags.[11]

## Data clouds

A data cloud or cloud data is a data display which uses font size and/or color to indicate numerical values[13] It is similar to a tag cloud[14] but instead of word count, displays data such as population or stock market prices.

## Text clouds

A text cloud or word cloud is a visualization of word frequency in a given text as a weighted list.[15] The technique has recently been popularly used to visualize the topical content of political speeches.[16]

## Collocate clouds

Extending the principles of a text cloud, a collocate cloud provides a more focused view of a document or corpus. Instead of summarising an entire document, the collocate cloud examines the usage of a particular word. The resulting cloud contains the words which are often used in conjunction with the search word. These collocates are formatted to show frequency (as size) as well as collocational strength (as brightness). This provides interactive ways to browse and explore language.[17]

## Perception of tag clouds

Tag clouds have been subject of investigation in several usability studies. The following summary is based on an overview of research results given by Lohmann et al.:[9]

• Tag size: Large tags attract more user attention than small tags (effect influenced by further properties, e.g., number of characters, position, neighboring tags).
• Scanning: Users scan rather than read tag clouds.
• Centering: Tags in the middle of the cloud attract more user attention than tags near the borders (effect influenced by layout).
• Exploration: Tag clouds provide suboptimal support when searching for specific tags (if these have not a very large font size).

## Creation of a tag cloud

Tag cloud of the most frequently used tags at Flickr.

In principle, the font size of a tag in a tag cloud is determined by its incidence. For a word cloud of categories like weblogs, the frequency of use for example, corresponds to the number of weblog entries that are assigned to a category. For small frequencies it's sufficient to indicate directly for any number from one to a maximum font size. For larger values, a scaling should be made. In a linear normalization, the weight ti of a descriptor is mapped to a size scale of 1 through f, where tmin and tmax are specifying the range of available weights.

$s_i = \left \lceil \frac{f_{\mathrm{max}}\cdot(t_i - t_{\mathrm{min}})}{t_{\mathrm{max}}-t_{\mathrm{min}}} \right \rceil$ for ti > tmin; else si = 1

• si: display fontsize
• fmax: max. fontsize
• ti: count
• tmin: min. count
• tmax: max. count

Since the number of indexed items per descriptor is usually distributed according to a power law [18], for larger ranges of values, a logarithmic representation makes sense [19].

Implementations of tag clouds also include text parsing and filtering out unhelpful tags such as common words, numbers, and punctuation.

## References

1. ^ Martin Halvey and Mark T. Keane, An Assessment of Tag Presentation Techniques, poster presentation at WWW 2007, 2007
2. ^ Paul Bausch, Jim Bumgardner (2006). "Make a Flickr-Style Tag Cloud". Flickr Hacks. O'Reilly Press. ISBN 0596102453.
3. ^ A copy of Jim Flanagan's Search Referral Zeitgeist was available at archive.org but has since been blocked. In the comments of a blog entry, a user identified as Steve Minutillo attribute the idea to Jim Flanagan, stating that Flanagan's site had such displays in 2002.
4. ^ http://www.webbyawards.com/press/archived-speeches.php#2006
5. ^ Gilles Deleuze, Felix Guattari (1992). Tausend Plateaus. Kapitalismus und Schizophrenie. ISBN 3883960942.
6. ^ Article: Free tag cloud generator script for PHP web pages Retrieved 2009-11-17
7. ^ Bielenberg, K. and Zacher, M., Groups in Social Software: Utilizing Tagging to Integrate Individual Contexts for Social Navigation, Masters Thesis submitted to the Program of Digital Media, Universität Bremen (2006)
8. ^ Kamel Aouiche, Daniel Lemire, Robert Godin, Collaborative OLAP with Tag Clouds: Web 2.0 OLAP Formalism and Experimental Evaluation, WEBIST 2008, 2008.
9. ^ a b Lohmann, S., Ziegler, J., Tetzlaff, L. Comparison of Tag Cloud Layouts: Task-Related Performance and Visual Exploration, T. Gross et al. (Eds.): INTERACT 2009, Part I, LNCS 5726, pp. 392–404, 2009.
10. ^ Hassan-Montero, Y., Herrero-Solana, V. Improving Tag-Clouds as Visual Information Retrieval Interfaces
11. ^ a b Owen Kaser and Daniel Lemire, Tag-Cloud Drawing: Algorithms for Cloud Visualization, Tagging and Metadata for Social Information Organization (WWW 2007), 2007
12. ^ Salonen, J. 2007. Self-organising map based tag clouds - Creating spatially meaningful representations of tagging data. Proceedings of the 1st OPAALS conference, 26-27 November 2007, Rome, Italy.
13. ^ Apel, Warren. "ManyEyes Visualization and Commentary: World Population Data Cloud.". Retrieved 2007-08-26.
14. ^ Wattenberg, Martin. "ManyEyes Visualization: Ad cloud". Retrieved 2007-03-12.
15. ^ Lamantia, Joe. "Text Clouds: A New Form of Tag Cloud?". Retrieved 2008-09-11.
16. ^ Mehta, Chirag. "US Presidential Speeches Tag Cloud". Retrieved 2008-09-11.
17. ^ "Collocate cloud". Retrieved 2008-12-05.
18. ^ Jakob Voss: Collaborative thesaurus tagging the Wikipedia way. April 2006 [1]
19. ^ Kentbyte: Tag Cloud Font Distribution Algorithm. June 2005 [2]


text:
# Program: Mass Drug Administration to Control Schistosomiasis

The content that appears below was created several years ago. This content is likely to be no longer fully accurate, both with respect to the research it presents and with respect to what it implies about our views and positions.

# In a nutshell

• The Problem: Schistosomiasis infection can cause anemia (and other chronic conditions) and in serious cases, death. More information on schistosomiasis available here.
• The Program: Mass administration of praziquantel (PZQ). (This program is often combined with administration of drugs targeting soil-transmitted helminths - more on the combination program here.)
• Track record: There's strong evidence from repeated trials that praziquantel effectively reduces prevalence of schistosomiasis infection.
• Cost-effectiveness: Estimates place this intervention among the most cost-effective.
• Bottom line: This is a proven, inexpensive method for improving lives in the developing world.

## Basics of the program

### What is the program? What problem does it target?

Schistosomiasis is a parasite infection that can cause chronic malnutrition, pain, anemia, and in some cases, death. It is further discussed here. The World Health Organization recommends treatment with praziquantel at least three times during childhood to cure non-severe morbidity and prevent the development of severe symptoms later in life.1

### What are the components required to implement this program - how does it work?

This program requires praziquantel and a means of administering the drug as efficiently as possible.
• Drugs: Praziquantel is donated by Merck to high disease burdened areas and thus freely available.2
• Distribution mechanism: The Disease Control Priorities Report states that schools provide a strong infrastructure for administration and that teachers can be trained to deliver drugs safely.3 In the absence of training, mobile teams can come in to implement drug administration, but this method can cost significantly more.4

## Program track record

### Micro evidence: Has this program been rigorously evaluated and shown to work?

A Cochrane review of 24 randomized and quasi-randomized controlled trials included 6,315 participants in interventions to treat urinary schistosomiasis. The review concluded that "praziquantel and metrifonate are effective treatments for urinary schistosomiasis and have few adverse events. Metrifonate requires multiple administrations and is therefore operationally less convenient in community-based control programmes."5 A Cochrane review of 13 randomized or pseudo-randomized trials treating intestinal schistosomiasis concluded that "oxamniquine and praziquantel both appear to be effective for treatment of S. mansoni, although lower doses of oxamniquine (less than 30 mg/kg) may not be as effective in some areas."6

### Macro evidence: Has this program played a role in large-scale success stories?

We have identified no such success stories. More on the general idea of "macro evidence" here.

## Recommendations and concerns

### Do expert reviews of the comparative merits of interventions endorse this one?

See this page for sources consulted. Experts endorse this intervention as part of the deworming intervention, which includes albendazole for soil-transmitted helminths (more on this combination program here).
• The Copenhagen Consensus ranks deworming and other nutrition programs at school as its 6th most promising program.7
• Jamison 2008 does not endorse this program8
• The Disease Control Priorities report states, "Until new technologies become available, anthelmintic chemotherapy for school-age children remains the most practical and substantive means to control ... schistosome infections in the developing world."9

### What versions of the intervention are best?

Three versions of this intervention are (1) population intervention, where everyone in a region is treated; (2) targeted intervention, where certain demographic subgroups are treated; and (3) selective intervention, where individuals selected by diagnosis or suspicion of infection are treated.10 Population deworming (treatment for all people in an area of high infection) is recommended by the World Health Organization as its first choice.11 However, treating school-aged children as a targeted population group is the version that has been most closely evaluated.12 It also may be the most cost-effective because school-aged children typically have the highest rate of helminth infection and reinfection, and schools offer an infrastructure for delivery.13


text:
## 6c.19 f

Acidity $K_{a}$
Basicity $K_{b}$
The Conjugate Seesaw $K_{a}\times K_{b}=K_{w}$

Fdonovan 3D
Posts: 101
Joined: Sat Aug 17, 2019 12:16 am

### 6c.19 f

In the answer key, it says H2CO3 is the stronger acid because C is more electronegative. However, I thought that when comparing acids in which the only difference is the central atom, you compare the bond strengths for elements in the same group. Since Ge and C come from the same group (not the same period), shouldn't the justification come from comparing the bond strengths, not the electronegativity?

Ethan Lam 4A
Posts: 69
Joined: Thu Jul 11, 2019 12:17 am

### Re: 6c.19 f

Electronegativity and bond strengths both play a factor in determining acid strength. Since it is more electronegative, it becomes more polar which makes it easier to dissociate.

Ruby Richter 2L
Posts: 103
Joined: Thu Jul 25, 2019 12:17 am

### Re: 6c.19 f

So, in this case, the difference in electronegativity would dominate over the difference in bond strength?


text:
# Year 10 Maths Find the length and width that will maximize the area of rectangle

#### liang123993

##### New member

The question is in the image. Working out with every step would be much appreciated.

#### Greg

##### Perseverance
Staff member
Here's a start:

Let $W$ be width, $L$ be length an $A$ be the desired area. Then,

$$\displaystyle 5W+2L=550$$

$$\displaystyle LW=A$$

Can you make any progress from there?

#### Greg

##### Perseverance
Staff member
Here's a start:

Let $W$ be width, $L$ be length an $A$ be the desired area. Then,

$$\displaystyle 5W+2L=550$$

$$\displaystyle LW=A$$

Can you make any progress from there?
$$\displaystyle W=\frac AL$$

$$\displaystyle \frac{5A}{L}+2L=550$$

$$\displaystyle 5A+2L^2=550L$$

$$\displaystyle A=110L-\frac{2L^2}{5}$$

$A$ has a maximum at the vertex of this inverted parabola, so $L=\frac{275}{2}$. Finding $A$ and $W$ from here should be straightforward.


text:
2 added 191 characters in body

Here is an outline of such a procedure.

Checking that a vertex $u$ is a top one can be done by solving a linear program, as follows: write $p$ in baricentric coordinates, i.e. $p=p_x=\sum_{v\in V} x_v v$, $x_v\geq 0$ for any $v\in V$, and $\sum_{v\in V} x_v=1$ (I denote by $V$ the set of vertices of the simplex). To check $u$ is top is equivalent to checking that there exists $x=(x_{v_1},\dots,x_{v_{|V|}})$ satisfying $x\geq 0$, $\sum_{v\in V} x_v=1$, $u\geq p_x$, and $x_u<1$.

So you solve the linear program $$\min x_u \text{ subject to x\geq 0, \sum_{v\in V} x_v=1, u\geq p_x},$$ and it can be done in polynomial time. If the value of the objective is strictly less than 1 then $u$ is top.

Testing for $u$ to be bottom is essentially the same, just replace the inequalities $u\geq p_x$ by $u\leq p_x$.

Finally, do this for each vertex. Needless to say, if your simplex is given by inequalities rather than by vertices, it still can be done in polynomial time, as you can enumerate the vertices of a simplex in polynomial time.

1

Here is an outline of such a procedure.

Checking that a vertex $u$ is a top one can be done by solving a linear program, as follows: write $p$ in baricentric coordinates, i.e. $p=p_x=\sum_{v\in V} x_v v$, $x_v\geq 0$ for any $v\in V$, and $\sum_{v\in V} x_v=1$ (I denote by $V$ the set of vertices of the simplex). To check $u$ is top is equivalent to checking that there exists $x=(x_{v_1},\dots,x_{v_{|V|}})$ satisfying $x\geq 0$, $\sum_{v\in V} x_v=1$, $u\geq p_x$, and $x_u<1$.

So you solve the linear program $$\min x_u \text{ subject to x\geq 0, \sum_{v\in V} x_v=1, u\geq p_x},$$ and it can be done in polynomial time. If the value of the objective is strictly less than 1 then $u$ is top.

Testing for $u$ to be bottom is essentially the same, just replace the inequalities $u\geq p_x$ by $u\leq p_x$.

Finally, do this for each vertex.


text:
# For the synchronous sequential circuit shown below, the output Z is zero for the initial conditions $${Q_A}{Q_B}{Q_C} = Q_A'Q_B'Q_C' = 100$$.The minimum number of clock cycles after which the output Z would again become zero is _______.

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## Answer (Detailed Solution Below) 6

Free
CT 1: Ratio and Proportion
4927
10 Questions 16 Marks 30 Mins

## Detailed Solution

DA = QC, DB = QA, DC = QB

$$D_A' = \bar Q_C',{\rm{\;}}D_B' = \bar Q_A',{\rm{\;}}D_C' = \bar Q_B'$$

$$Z = \left( {{Q_A} \oplus Q_A'} \right) + \left( {{Q_B} \oplus Q_B'} \right) + \left( {{Q_C} \oplus Q_C'} \right)\;$$

Z becomes zero when all the inputs of OR gates are zero

$$\Rightarrow {Q_A} \oplus Q_A' + {Q_B} \oplus Q_B' + {Q_C} \oplus Q_C' = 0$$

$$\Rightarrow {Q_A} = Q_A',{Q_B} = Q_B',{Q_C} = Q_C'$$

 Clock DA QA DB QB DC QC DA’ QA’ DB’ QB’ DC’ QC’ Z Initial 1 0 0 1 0 0 1 0 0 1 1 0 0 1 1 1 1 0 0 1 2 0 0 0 0 1 1 1 1 1 1 1 1 1 3 1 1 0 0 0 0 0 0 1 1 1 1 1 4 0 0 1 1 0 0 0 0 0 0 1 1 1 5 0 0 0 0 1 1 0 0 0 0 0 0 1 6 1 1 0 0 0 0 1 1 0 0 0 0 0

At 6th clock pulse

$${Q_A}{Q_B}{Q_C} = Q_A'Q_B'Q_C' = 100$$