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Orthostatic hypertension is sometimes defined based on an unusual change in blood pressure after changing position. Suppose we define a normal range for change in systolic blood pressure (SBP) based on change in SBP from the recumbent to the standing position in Table 2.16 that is between the upper and lower decile. What should the normal range be? 


Table 2.16 Effect of position on blood pressure

aSystolic blood pressure
bDiastolic blood pressure

For each variable (other than ID), obtain appropriate descriptive statistics (both numeric and graphic)?


Table 2.17: Format for FEV.DAT

The data in Table 2.17 are available for each child.

Use both numeric and graphic measures to assess the relationship of FEV to age, height, and smoking status. (Do this separately for boys and girls.)


Table 2.17: Format for FEV.DAT

The data in Table 2.17 are available for each child.

Compare the pattern of growth of FEV by age for boys and girls. Are there any similarities? Any differences?


Table 2.17: Format for FEV.DAT

The data in Table 2.17 are available for each child.

Compute appropriate descriptive statistics for each nutrient for both DR and FFQ, using both numeric and graphic measures? 

Table: 2.18 Format for VALID.DAT

Variable ________________________ Format or code
ID number............................................XXXXX.XX
Saturated fat—DR (g) .........................XXXXX.XX
Saturated fat—FFQ (g)........................XXXXX.XX
Total fat—DR (g)..................................XXXXX.XX
Total fat—FFQ (g)................................XXXXX.XX
Alcohol consumption—DR (oz)..........XXXXX.XX
Alcohol consumption—FFQ (oz) .......XXXXX.XX
Total calories—DR (K-cal).................XXXXXX.XX
Total calories—FFQ (K-cal) ..............XXXXXX.XX

Use descriptive statistics to relate nutrient intake for the DR and FFQ. Do you think the FFQ is a reasonably accurate approximation to the DR? Why or why not? 

Table: 2.18 Format for VALID.DAT

Variable ________________________ Format or code
ID number............................................XXXXX.XX
Saturated fat—DR (g) .........................XXXXX.XX
Saturated fat—FFQ (g)........................XXXXX.XX
Total fat—DR (g)..................................XXXXX.XX
Total fat—FFQ (g)................................XXXXX.XX
Alcohol consumption—DR (oz)..........XXXXX.XX
Alcohol consumption—FFQ (oz) .......XXXXX.XX
Total calories—DR (K-cal).................XXXXXX.XX
Total calories—FFQ (K-cal) ..............XXXXXX.XX

A frequently used method for quantifying dietary intake is in the form of quintiles. Compute quintiles for each nutrient and each method of recording, and relate the nutrient composition for DR and FFQ using the quintile scale. (That is, how does the quintile category based on DR relate to the quintile category based on FFQ for the same individual?) Do you get the same impression about the concordance between DR and FFQ using quintiles as in Problem 2.27, in which raw (ungrouped) nutrient intake is considered?

In nutritional epidemiology, it is customary to assess nutrient intake in relation to total caloric intake. One measure used to accomplish this is nutrient density, which is defined as 100% × (caloric intake of a nutrient/total caloric intake). For fat consumption, 1 g of fat is equivalent to 9 calories?


Refer to Problem 2.27,

Use descriptive statistics to relate nutrient intake for the DR and FFQ. Do you think the FFQ is a reasonably accurate approximation to the DR? Why or why not?


Table: 2.18 Format for VALID.DAT

Variable ________________________ Format or code
ID number............................................XXXXX.XX
Saturated fat—DR (g) .........................XXXXX.XX
Saturated fat—FFQ (g)........................XXXXX.XX
Total fat—DR (g)..................................XXXXX.XX
Total fat—FFQ (g)................................XXXXX.XX
Alcohol consumption—DR (oz)..........XXXXX.XX
Alcohol consumption—FFQ (oz) .......XXXXX.XX
Total calories—DR (K-cal).................XXXXXX.XX
Total calories—FFQ (K-cal) ..............XXXXXX.XX

Compute the nutrient density for total fat for the DR and FFQ, and obtain appropriate descriptive statistics for this variable. How do they compare?


Table: 2.18 Format for VALID.DAT

Variable ________________________ Format or code
ID number............................................XXXXX.XX
Saturated fat—DR (g) .........................XXXXX.XX
Saturated fat—FFQ (g)........................XXXXX.XX
Total fat—DR (g)..................................XXXXX.XX
Total fat—FFQ (g)................................XXXXX.XX
Alcohol consumption—DR (oz)..........XXXXX.XX
Alcohol consumption—FFQ (oz) .......XXXXX.XX
Total calories—DR (K-cal).................XXXXXX.XX
Total calories—FFQ (K-cal) ..............XXXXXX.XX

Relate the nutrient density for total fat for the DR versus the FFQ using the quintile approach in Problem 2.28. Is the concordance between total fat for DR and FFQ stronger, weaker, or the same when total fat is expressed in terms of nutrient density as opposed to raw nutrient?


Refer to Problem 2.28

frequently used method for quantifying dietary intake is in the form of quintiles. Compute quintiles for each nutrient and each method of recording, and relate the nutrient composition for DR and FFQ using the quintile scale. (That is, how does the quintile category based on DR relate to the
quintile category based on FFQ for the same individual?)
Do you get the same impression about the concordance between DR and FFQ using quintiles as in Problem 2.27, in which raw (ungrouped) nutrient intake is considered? In nutritional epidemiology, it is customary to assess nutrient intake in relation to total caloric intake. One measure used
to accomplish this is nutrient density, which is defined as 100% × (caloric intake of a nutrient/total caloric intake). For fat consumption, 1 g of fat is equivalent to 9 calories?


Table: 2.18 Format for VALID.DAT

Variable ________________________ Format or code
ID number............................................XXXXX.XX
Saturated fat—DR (g) .........................XXXXX.XX
Saturated fat—FFQ (g)........................XXXXX.XX
Total fat—DR (g)..................................XXXXX.XX
Total fat—FFQ (g)................................XXXXX.XX
Alcohol consumption—DR (oz)..........XXXXX.XX
Alcohol consumption—FFQ (oz) .......XXXXX.XX
Total calories—DR (K-cal).................XXXXXX.XX
Total calories—FFQ (K-cal) ..............XXXXXX.XX

Compare the exposed and control groups regarding age and gender, using appropriate numeric and graphic descriptive measures?

Compare the exposed and control groups regarding verbal and performance IQ, using appropriate numeric and graphic descriptive measures?

To assess the variability of the assay, the investigators need to compute the coefficient of variation. Compute the coefficient of variation (CV) for each subject by obtaining the mean and standard deviation over the 2 replicates for each subject?


Cardiovascular Disease
Activated-protein-C (APC) resistance is a serum marker that has been associated with thrombosis (the formation of blood clots often leading to heart attacks) among adults. A study assessed this risk factor among adolescents. To assess the reproducibility of the assay, a split-sample technique was used in which a blood sample was provided by 10 people; each sample was split into two aliquots (sub-samples), and each aliquot was assessed separately. Table 2.19 gives the results.

Table 2.19: APC resistance split-samples data

Compute the average CV over the 10 subjects as an overall measure of variability of the assay. In general, a CV of <10% is considered excellent, ≥10% and <20% is considered good, ≥20% and <30% is considered fair, and ≥30% is considered poor.

How would you characterize the reliability of the APC assay based on these criteria? 


Cardiovascular Disease
Activated-protein-C (APC) resistance is a serum marker that has been associated with thrombosis (the formation of blood clots often leading to heart attacks) among adults. A study assessed this risk factor among adolescents. To assess the reproducibility of the assay, a split-sample technique was used in which a blood sample was provided by 10 people; each sample was split into two aliquots (sub-samples), and each aliquot was assessed separately. Table 2.19 gives the results.

Table 2.19: APC resistance split-samples data

Compute appropriate descriptive statistics for I and Data set available U plants?


Microbiology
A study was conducted to demonstrate that soy beans inoculated with nitrogen-fixing bacteria yield more and grow adequately without expensive environmentally deleterious synthesized fertilizers. The trial was conducted under controlled conditions with uniform amounts of soil. The initial hypothesis was that inoculated plants would outperform their uninoculated counterparts. This assumption is based on the facts that plants need nitrogen to manufacture vital proteins and amino acids and that nitrogen-fixing bacteria would make more of this substance available to plants, increasing their size and yield. There were 8 inoculated plants (I) and 8 uninoculated plants (U). The plant yield as measured by pod weight for each plant is given in Table 2.20.


Table 2.20:Pod weight (g) from inoculated (I) and uninoculated (U) plants

The data for this problem were supplied by David Rosner.

Use graphic methods to compare the two groups?


Microbiology
A study was conducted to demonstrate that soy beans inoculated with nitrogen-fixing bacteria yield more and grow adequately without expensive environmentally deleterious synthesized fertilizers. The trial was conducted under controlled conditions with uniform amounts of soil. The initial hypothesis was that inoculated plants would outperform their uninoculated counterparts. This assumption is based on the facts that plants need nitrogen to manufacture vital proteins and amino acids and that nitrogen-fixing bacteria would make more of this substance available to plants, increasing their size and yield. There were 8 inoculated plants (I) and 8 uninoculated plants (U). The plant yield as measured by pod weight for each plant is given in Table 2.20.


Table 2.20:Pod weight (g) from inoculated (I) and uninoculated (U) plants

The data for this problem were supplied by David Rosner.

What is your overall impression concerning the pod weight in the two groups?


Microbiology
A study was conducted to demonstrate that soy beans inoculated with nitrogen-fixing bacteria yield more and grow adequately without expensive environmentally deleterious synthesized fertilizers. The trial was conducted under controlled conditions with uniform amounts of soil. The initial hypothesis was that inoculated plants would outperform their uninoculated counterparts. This assumption is based on the facts that plants need nitrogen to manufacture vital proteins and amino acids and that nitrogen-fixing bacteria would make more of this substance available to plants, increasing their size and yield. There were 8 inoculated plants (I) and 8 uninoculated plants (U). The plant yield as measured by pod weight for each plant is given in Table 2.20.


Table 2.20:Pod weight (g) from inoculated (I) and uninoculated (U) plants

The data for this problem were supplied by David Rosner.

1. For each pair of twins, compute the following for the lumbar spine:

A = BMD for the heavier-smoking twin − BMD for the lighter-smoking twin = x1 − x2

B = mean BMD for the twinship = (x1 + x2)/2

C = 100% × (A/B)

Derive appropriate descriptive statistics for C over the entire study population. 

2. Suppose we group the twin pairs according to the difference in tobacco use expressed in 10 pack-year groups (0–9.9 pack-years/10–19.9 pack-years/20–29.9 pack-years/30–39.9 pack-years/40+ pack-years). Compute appropriate descriptive statistics, and provide a scatter plot for C grouped by the difference in tobacco use in pack-years?

3. What impression do you have of the relationship between BMD and tobacco use based on Problem 2? 

Answer Problems 1–3 for BMD for the femoral neck?

Answer Problems 1–3 for BMD for the femoral shaft?

Provide a box plot of LVMI by blood pressure group?

What is the geometric mean of LVMI by blood pressure group?

What is the arithmetic mean of LVMI by blood pressure group?

Based on the box plot, does the arithmetic mean or the geometric mean provide a more appropriate measure of location for this type of data?

What does A1 ∪ A2 mean?

Consider a family with a mother, father, and two children. Let A1 = {mother has influenza}, A2 = {father has influenza}, A3 = {first child has influenza}, A4 = {second child has influenza}, B = {at least one child has influenza}, C = {at least one parent has influenza}, and D = {at least one person in the family has influenza}.

What does A1 ∩ A2 mean?

Consider a family with a mother, father, and two children. Let A1 = {mother has influenza}, A2 = {father has influenza}, A3 = {first child has influenza}, A4 = {second child has influenza}, B = {at least one child has influenza}, C = {at least one parent has influenza}, and D = {at least one person in the family has influenza}.

Are A3 and A4 mutually exclusive?

Consider a family with a mother, father, and two children. Let A1 = {mother has influenza}, A2 = {father has influenza}, A3 = {first child has influenza}, A4 = {second child has influenza}, B = {at least one child has influenza}, C = {at least one parent has influenza}, and D = {at least one person in the family has influenza}.

What does A3 ∪ B mean?

Consider a family with a mother, father, and two children. Let A1 = {mother has influenza}, A2 = {father has influenza}, A3 = {first child has influenza}, A4 = {second child has influenza}, B = {at least one child has influenza}, C = {at least one parent has influenza}, and D = {at least one person in the family has influenza}.

What does A3 ∩ B mean?

Consider a family with a mother, father, and two children. Let A1 = {mother has influenza}, A2 = {father has influenza}, A3 = {first child has influenza}, A4 = {second child has influenza}, B = {at least one child has influenza}, C = {at least one parent has influenza}, and D = {at least one person in the family has influenza}.

Express C in terms of A1, A2, A3, and A4?

Consider a family with a mother, father, and two children. Let A1 = {mother has influenza}, A2 = {father has influenza}, A3 = {first child has influenza}, A4 = {second child has influenza}, B = {at least one child has influenza}, C = {at least one parent has influenza}, and D = {at least one person in the family has influenza}.

Express D in terms of B and C?

Consider a family with a mother, father, and two children. Let A1 = {mother has influenza}, A2 = {father has influenza}, A3 = {first child has influenza}, A4 = {second child has influenza}, B = {at least one child has influenza}, C = {at least one parent has influenza}, and D = {at least one person in the family has influenza}.

What does A1 mean?

Consider a family with a mother, father, and two children. Let A1 = {mother has influenza}, A2 = {father has influenza}, A3 = {first child has influenza}, A4 = {second child has influenza}, B = {at least one child has influenza}, C = {at least one parent has influenza}, and D = {at least one person in the family has influenza}.

What does A2 mean?

Consider a family with a mother, father, and two children. Let A1 = {mother has influenza}, A2 = {father has influenza}, A3 = {first child has influenza}, A4 = {second child has influenza}, B = {at least one child has influenza}, C = {at least one parent has influenza}, and D = {at least one person in the family has influenza}.

Represent C in terms of A1, A2, A3, and A4?

Consider a family with a mother, father, and two children. Let A1 = {mother has influenza}, A2 = {father has influenza}, A3 = {first child has influenza}, A4 = {second child has influenza}, B = {at least one child has influenza}, C = {at least one parent has influenza}, and D = {at least one person in the family has influenza}.

Represent D in terms of B and C?

Consider a family with a mother, father, and two children. Let A1 = {mother has influenza}, A2 = {father has influenza}, A3 = {first child has influenza}, A4 = {second child has influenza}, B = {at least one child has influenza}, C = {at least one parent has influenza}, and D = {at least one person in the family has influenza}.

Are the events A1 = {mother has influenza} and A2 = {father has influenza} independent? 


Suppose an influenza epidemic strikes a city. In 10% of families the mother has influenza; in 10% of families the father has influenza; and in 2% of families both the mother and father have influenza.

What is the probability that at least one child will get influenza? 


Suppose an influenza epidemic strikes a city. In 10% of families the mother has influenza; in 10% of families the father has influenza; and in 2% of families both the mother and father have influenza.

Based on Problem 3.12, what is the conditional probability that the father has influenza given that the mother has influenza? 


Suppose an influenza epidemic strikes a city. In 10% of families the mother has influenza; in 10% of families the father has influenza; and in 2% of families both the mother and father have influenza.

Based on Problem 3.12, what is the conditional probability that the father has influenza given that the mother does not have influenza? 


Suppose an influenza epidemic strikes a city. In 10% of families the mother has influenza; in 10% of families the father has influenza; and in 2% of families both the mother and father have influenza.

What is the probability that all three of these individuals have Alzheimer’s disease? 


Mental Health
Estimates of the prevalence of Alzheimer’s disease have recently been provided by Pfeffer et al. [8]. The estimates are given in Table 3.5.

Table 3.5: Prevalence of Alzheimer’s disease (cases per 100 population)

Suppose an unrelated 77-year-old man, 76-year-old woman, and 82-year-old woman are selected from a community.

What is the probability that at least one of the women has Alzheimer’s disease? 


Mental Health
Estimates of the prevalence of Alzheimer’s disease have recently been provided by Pfeffer et al. [8]. The estimates are given in Table 3.5.

Table 3.5: Prevalence of Alzheimer’s disease (cases per 100 population)

Suppose an unrelated 77-year-old man, 76-year-old woman, and 82-year-old woman are selected from a community.

What is the probability that at least one of the three people has Alzheimer’s disease? 


Mental Health
Estimates of the prevalence of Alzheimer’s disease have recently been provided by Pfeffer et al. [8]. The estimates are given in Table 3.5.

Table 3.5: Prevalence of Alzheimer’s disease (cases per 100 population)

Suppose an unrelated 77-year-old man, 76-year-old woman, and 82-year-old woman are selected from a community.

What is the probability that exactly one of the three people has Alzheimer’s disease? 


Mental Health
Estimates of the prevalence of Alzheimer’s disease have recently been provided by Pfeffer et al. [8]. The estimates are given in Table 3.5.

Table 3.5: Prevalence of Alzheimer’s disease (cases per 100 population)

Suppose an unrelated 77-year-old man, 76-year-old woman, and 82-year-old woman are selected from a community.

Suppose we know one of the three people has Alzheimer’s disease, but we don’t know which one. What is the conditional probability that the affected person is a woman? 


Mental Health
Estimates of the prevalence of Alzheimer’s disease have recently been provided by Pfeffer et al. [8]. The estimates are given in Table 3.5.

Table 3.5: Prevalence of Alzheimer’s disease (cases per 100 population)

Suppose an unrelated 77-year-old man, 76-year-old woman, and 82-year-old woman are selected from a community.

Suppose we know two of the three people have Alzheimer’s disease. What is the conditional probability that they are both women? 


Mental Health
Estimates of the prevalence of Alzheimer’s disease have recently been provided by Pfeffer et al. [8]. The estimates are given in Table 3.5.

Table 3.5: Prevalence of Alzheimer’s disease (cases per 100 population)

Suppose an unrelated 77-year-old man, 76-year-old woman, and 82-year-old woman are selected from a community.

Suppose we know two of the three people have Alzheimer’s disease. What is the conditional probability that they are both younger than 80 years of age? 


Mental Health
Estimates of the prevalence of Alzheimer’s disease have recently been provided by Pfeffer et al. [8]. The estimates are given in Table 3.5.

Table 3.5: Prevalence of Alzheimer’s disease (cases per 100 population)

Suppose an unrelated 77-year-old man, 76-year-old woman, and 82-year-old woman are selected from a community.

What is the conditional probability that the man will be affected given that the woman is affected? How does this value compare with the prevalence in Table 3.5? Why should it be the same (or different)? 


Suppose the probability that both members of a married couple, each of whom is 75–79 years of age, will have Alzheimer’s disease is .0015.

Table 3.5: Prevalence of Alzheimer’s disease (cases per 100 population)

What is the conditional probability that the woman will be affected given that the man is affected? How does this value compare with the prevalence in Table 3.5? Why should it be the same (or different)? 


Suppose the probability that both members of a married couple, each of whom is 75–79 years of age, will have Alzheimer’s disease is .0015.

Table 3.5: Prevalence of Alzheimer’s disease (cases per 100 population)

What is the probability that at least one member of the couple is affected? 


Suppose the probability that both members of a married couple, each of whom is 75–79 years of age, will have Alzheimer’s disease is .0015.

Table 3.5: Prevalence of Alzheimer’s disease (cases per 100 population)

What is the expected overall prevalence of Alzheimer’s disease in the community if the prevalence estimates in Table 3.5 for specific age–gender groups hold? 


Suppose a study of Alzheimer’s disease is proposed in a retirement community with people 65+ years of age, where the age–gender distribution is as shown in Table 3.6.

Table 3.6: Age–gender distribution of retirement community

aPercentage of total population.

Assuming there are 1000 people 65+ years of age in the community, what is the expected number of cases of Alzheimer’s disease in the community? 


Suppose a study of Alzheimer’s disease is proposed in a retirement community with people 65+ years of age, where the age–gender distribution is as shown in Table 3.6.

Table 3.6: Age–gender distribution of retirement community

aPercentage of total population.

Suppose 3 children in a village ages 3, 5, and 7 are vaccinated with the QIV vaccine. What is the probability that at least one child among the 3 will get influenza? 


Infectious Disease
Commonly used vaccines for influenza are trivalent and contain only one type of influenza B virus. They may be ineffective against other types of influenza B virus. A randomized clinical trial was performed among children 3 to 8 years of age in 8 countries. Children received either a quadrivalent vaccine (QIV) that had more than one influenza B virus or a trivalent Hepatitis A vaccine (control) (Jain, et al., [9]. New England Journal of Medicine 2013: 369(26): 2481–2491). An attack rate (i.e.,% of children who developed influenza) starting 14 days after vaccination until the end of the study was computed for each vaccine group, stratified by age. The following data were reported:

What % of 3–4-year-old children in the village will get influenza? 


Suppose that 80% of 3–4-year-old children and 70% of 5–8-year-old children in a village are vaccinated with QIV vaccine. Also assume that children who are not vaccinated have twice the incidence of influenza as the control group in Table 3.7.

Table 3.7 Attack rate for influenza by age and treatment group

What % of 5–8-year-old children in the village will get influenza? 


Suppose that 80% of 3–4-year-old children and 70% of 5–8-year-old children in a village are vaccinated with QIV vaccine. Also assume that children who are not vaccinated have twice the incidence of influenza as the control group in Table 3.7.

Table 3.7 Attack rate for influenza by age and treatment group

Suppose we identify a 5–8-year-old child with influenza in the village but are uncertain whether the child was vaccinated. If we make the same assumptions as in Problems 3.29–3.30, then what is the probability that the child was vaccinated?


Suppose that 80% of 3–4-year-old children and 70% of 5–8-year-old children in a village are vaccinated with QIV vaccine. Also assume that children who are not vaccinated have twice the incidence of influenza as the control group in Table 3.7.

Table 3.7 Attack rate for influenza by age and treatment group

What is the probability that in a family with two children, both siblings are affected? 


Genetics
Suppose that a disease is inherited via a dominant mode of inheritance and that only one of the two parents is affected with the disease. The implications of this mode of inheritance are that the probability is 1 in 2 that any particular offspring will get the disease.

What is the probability that exactly one sibling is affected?


Genetics
Suppose that a disease is inherited via a dominant mode of inheritance and that only one of the two parents is affected with the disease. The implications of this mode of inheritance are that the probability is 1 in 2 that any particular offspring will get the disease.

What is the probability that neither sibling is affected? 


Genetics
Suppose that a disease is inherited via a dominant mode of inheritance and that only one of the two parents is affected with the disease. The implications of this mode of inheritance are that the probability is 1 in 2 that any particular offspring will get the disease.

Suppose the older child is affected. What is the probability that the younger child is affected? 


Genetics
Suppose that a disease is inherited via a dominant mode of inheritance and that only one of the two parents is affected with the disease. The implications of this mode of inheritance are that the probability is 1 in 2 that any particular offspring will get the disease.

If A, B are two events such that A = {older child is affected}, B = {younger child is affected}, then are the events A, B independent? 


Genetics
Suppose that a disease is inherited via a dominant mode of inheritance and that only one of the two parents is affected with the disease. The implications of this mode of inheritance are that the probability is 1 in 2 that any particular offspring will get the disease.

What is the probability that in a family with two children, both siblings are affected? 


Suppose that a disease is inherited via an autosomal recessive mode of inheritance. The implications of this mode of inheritance are that the children in a family each have a probability of 1 in 4 of inheriting the disease.

What is the probability that exactly one sibling is affected? 


Suppose that a disease is inherited via an autosomal recessive mode of inheritance. The implications of this mode of inheritance are that the children in a family each have a probability of 1 in 4 of inheriting the disease.

What is the probability that neither sibling is affected? 


Suppose that a disease is inherited via an autosomal recessive mode of inheritance. The implications of this mode of inheritance are that the children in a family each have a probability of 1 in 4 of inheriting the disease.

In a family with one male and one female sibling, what is the probability that both siblings are affected? 


Suppose that a disease is inherited via a sex-linked mode of inheritance. The implications of this mode of inheritance are that each male offspring has a 50% chance of inheriting the disease, whereas the female offspring have no chance of getting the disease.

What is the probability that exactly one sibling is affected?


Suppose that a disease is inherited via a sex-linked mode of inheritance. The implications of this mode of inheritance are that each male offspring has a 50% chance of inheriting the disease, whereas the female offspring have no chance of getting the disease.

What is the probability that neither sibling is affected?


Suppose that a disease is inherited via a sex-linked mode of inheritance. The implications of this mode of inheritance are that each male offspring has a 50% chance of inheriting the disease, whereas the female offspring have no chance of getting the disease.

Answer Problem 3.40 for families with two male siblings?


Suppose that a disease is inherited via a sex-linked mode of inheritance. The implications of this mode of inheritance are that each male offspring has a 50% chance of inheriting the disease, whereas the female offspring have no chance of getting the disease.

Answer Problem 3.41 for families with two male siblings?


Suppose that a disease is inherited via a sex-linked mode of inheritance. The implications of this mode of inheritance are that each male offspring has a 50% chance of inheriting the disease, whereas the female offspring have no chance of getting the disease.

Answer Problem 3.42 for families with two male siblings?


Suppose that a disease is inherited via a sex-linked mode of inheritance. The implications of this mode of inheritance are that each male offspring has a 50% chance of inheriting the disease, whereas the female offspring have no chance of getting the disease.

Assume that the dominant, recessive, and sex-linked modes of inheritance follow the probability laws given in Problems 3.32, 3.37, and 3.40 and that, without prior knowledge about the family in question, each mode of inheritance is equally likely. What is the posterior probability of each mode of inheritance in this family? 


Suppose that in a family with two male siblings, both siblings are affected with a genetically inherited disease. Suppose also that, although the genetic history of the family is unknown, only a dominant, recessive, or sex-linked mode of inheritance is possible.


Refer to Problem 3.32,

What is the probability that in a family with two children, both siblings are affected?

Refer to Problem 3.40,

In a family with one male and one female sibling, what is the probability that both siblings are affected?

Answer Problem 3.46 for a family with two male siblings in which only one sibling is affected?


Suppose that in a family with two male siblings, both siblings are affected with a genetically inherited disease. Suppose also that, although the genetic history of the family is unknown, only a dominant, recessive, or sex-linked mode of inheritance is possible.

Answer Problem 3.46 for a family with one male and one female sibling in which both siblings are affected?


Suppose that in a family with two male siblings, both siblings are affected with a genetically inherited disease. Suppose also that, although the genetic history of the family is unknown, only a dominant, recessive, or sex-linked mode of inheritance is possible.

Answer Problem 3.48 where only the male sibling is affected?


Suppose that in a family with two male siblings, both siblings are affected with a genetically inherited disease. Suppose also that, although the genetic history of the family is unknown, only a dominant, recessive, or sex-linked mode of inheritance is possible.

What is the probability of having a low birth-weight infant? 


Obstetrics
The following data are derived from the Monthly Vital Statistics Report (October 1999) issued by the National Center for Health Statistics [10]. These data are pertinent to livebirths only.


Suppose that infants are classified as low birthweight if they have a birthweight <2500 g and as normal birthweight if they have a birthweight ≥2500 g. Suppose that infants are also classified by length of gestation in the following five categories: <28 weeks, 28–31 weeks, 32–35 weeks, 36 weeks, and ≥37 weeks. Assume the probabilities of the different periods of gestation are as given in Table 3.8. Also assume that the probability of low birthweight is .949 given a gestation of <28 weeks, .702 given a gestation of 28–31 weeks, .434 given a gestation of 32–35 weeks, .201 given a gestation of 36 weeks, and .029 given a gestation of ≥37 weeks.


Table 3.8: Distribution of length of gestation
Length of gestation ____________ Probability
<28 weeks ........................................ .007
28–31 weeks ................................... .012
32–35 weeks ................................... .050
36 weeks ......................................... .037
≥37 weeks ....................................... .893

Show that the events {length of gestation ≤ 31 weeks} and {low birthweight} are not independent?


Obstetrics
The following data are derived from the Monthly Vital Statistics Report (October 1999) issued by the National Center for Health Statistics [10]. These data are pertinent to livebirths only.


Suppose that infants are classified as low birthweight if they have a birthweight <2500 g and as normal birthweight if they have a birthweight ≥2500 g. Suppose that infants are also classified by length of gestation in the following five categories: <28 weeks, 28–31 weeks, 32–35 weeks, 36 weeks, and ≥37 weeks. Assume the probabilities of the different periods of gestation are as given in Table 3.8. Also assume that the probability of low birthweight is .949 given a gestation of <28 weeks, .702 given a gestation of 28–31 weeks, .434 given a gestation of 32–35 weeks, .201 given a gestation of 36 weeks, and .029 given a gestation of ≥37 weeks.


Table 3.8: Distribution of length of gestation
Length of gestation ____________ Probability
<28 weeks ........................................ .007
28–31 weeks ................................... .012
32–35 weeks ................................... .050
36 weeks ......................................... .037
≥37 weeks ....................................... .893

What is the probability of having a length of gestation ≤ 36 weeks given that an infant is low birth-weight? 


Obstetrics
The following data are derived from the Monthly Vital Statistics Report (October 1999) issued by the National Center for Health Statistics [10]. These data are pertinent to livebirths only.


Suppose that infants are classified as low birthweight if they have a birthweight <2500 g and as normal birthweight if they have a birthweight ≥2500 g. Suppose that infants are also classified by length of gestation in the following five categories: <28 weeks, 28–31 weeks, 32–35 weeks, 36 weeks, and ≥37 weeks. Assume the probabilities of the different periods of gestation are as given in Table 3.8. Also assume that the probability of low birthweight is .949 given a gestation of <28 weeks, .702 given a gestation of 28–31 weeks, .434 given a gestation of 32–35 weeks, .201 given a gestation of 36 weeks, and .029 given a gestation of ≥37 weeks.


Table 3.8: Distribution of length of gestation
Length of gestation ____________ Probability
<28 weeks ........................................ .007
28–31 weeks ................................... .012
32–35 weeks ................................... .050
36 weeks ......................................... .037
≥37 weeks ....................................... .893

Suppose the smoking habits of the parents are independent and the probability that the mother is a current smoker is .4, whereas the probability that the father is a current smoker is .5. What is the probability that both the father and mother are current smokers? 


Pulmonary Disease
The familial aggregation of respiratory disease is a wellestablished clinical phenomenon. However, whether this aggregation is due to genetic or environmental factors or both is somewhat controversial. An investigator wishes to study a particular environmental factor, namely the relationship of cigarette-smoking habits in the parents to the presence or absence of asthma in their oldest child age 5 to 9 years living in the household (referred to below as their offspring). Suppose the investigator finds that (1) if both the mother and father are current smokers, then the probability of their offspring having asthma is .15; (2) if the mother is a current smoker and the father is not, then the probability of their offspring having asthma is .13; (3) if the father is a current smoker and the mother is not, then the probability of their offspring having asthma is .05; and (4) if neither parent is a current smoker, then the probability of their offspring having asthma is .04.

Consider the subgroup of families in which the mother is not a current smoker. What is the probability that the father is a current smoker among such families? How does this probability differ from that calculated in Problem 3.53? 


Pulmonary Disease
The familial aggregation of respiratory disease is a wellestablished clinical phenomenon. However, whether this aggregation is due to genetic or environmental factors or both is somewhat controversial. An investigator wishes to study a particular environmental factor, namely the relationship of cigarette-smoking habits in the parents to the presence or absence of asthma in their oldest child age 5 to 9 years living in the household (referred to below as their offspring). Suppose the investigator finds that (1) if both the mother and father are current smokers, then the probability of their offspring having asthma is .15; (2) if the mother is a current smoker and the father is not, then the probability of their offspring having asthma is .13; (3) if the father is a current smoker and the mother is not, then the probability of their offspring having asthma is .05; and (4) if neither parent is a current smoker, then the probability of their offspring having asthma is .04.

If the probability that the father is a current smoker is .5, what is the probability that the father is a current smoker and that the mother is not a current smoker? 


Suppose, alternatively, that if the father is a current smoker, then the probability that the mother is a current smoker is .6; whereas if the father is not a current smoker, then the probability that the mother is a current smoker is .2. Also assume that statements 1, 2, 3, and 4 above hold.

Are the current smoking habits of the father and the mother independent? Why or why not? 


Suppose, alternatively, that if the father is a current smoker, then the probability that the mother is a current smoker is .6; whereas if the father is not a current smoker, then the probability that the mother is a current smoker is .2. Also assume that statements 1, 2, 3, and 4 above hold.

Under the assumptions made in Problems 3.55 and 3.56, find the unconditional probability that the offspring will have asthma?


Suppose, alternatively, that if the father is a current smoker, then the probability that the mother is a current smoker is .6; whereas if the father is not a current smoker, then the probability that the mother is a current smoker is .2. Also assume that statements 1, 2, 3, and 4 above hold.


Refer to Problem 3.55 and 3.56,

3.55 If the probability that the father is a current smoker is .5, what is the probability that the father is a current smoker and that the mother is not a current smoker?

3.56 Are the current smoking habits of the father and the mother independent? Why or why not?

Suppose a child has asthma. What is the posterior probability that the father is a current smoker? 


Suppose, alternatively, that if the father is a current smoker, then the probability that the mother is a current smoker is .6; whereas if the father is not a current smoker, then the probability that the mother is a current smoker is .2. Also assume that statements 1, 2, 3, and 4 above hold.

What is the posterior probability that the mother is a current smoker if the child has asthma? 


Suppose, alternatively, that if the father is a current smoker, then the probability that the mother is a current smoker is .6; whereas if the father is not a current smoker, then the probability that the mother is a current smoker is .2. Also assume that statements 1, 2, 3, and 4 above hold.

Answer Problem 3.58 if the child does not have asthma?


Suppose, alternatively, that if the father is a current smoker, then the probability that the mother is a current smoker is .6; whereas if the father is not a current smoker, then the probability that the mother is a current smoker is .2. Also assume that statements 1, 2, 3, and 4 above hold.

Answer Problem 3.59 if the child does not have asthma?


Suppose, alternatively, that if the father is a current smoker, then the probability that the mother is a current smoker is .6; whereas if the father is not a current smoker, then the probability that the mother is a current smoker is .2. Also assume that statements 1, 2, 3, and 4 above hold.

Are the child’s asthma status and the father’s smoking status independent? Why or why not?


Suppose, alternatively, that if the father is a current smoker, then the probability that the mother is a current smoker is .6; whereas if the father is not a current smoker, then the probability that the mother is a current smoker is .2. Also assume that statements 1, 2, 3, and 4 above hold.

Are the child’s asthma status and the mother’s smoking status independent? Why or why not?


Suppose, alternatively, that if the father is a current smoker, then the probability that the mother is a current smoker is .6; whereas if the father is not a current smoker, then the probability that the mother is a current smoker is .2. Also assume that statements 1, 2, 3, and 4 above hold.


What is the sensitivity using this cut-point?


Genetics, Obstetrics
Precise quantification of smoking during pregnancy is difficult in retrospective studies. Routinely collected blood specimens from newborns for screening purposes may provide a low cost method to objectively measure maternal smoking close to the time of delivery. Serum cotinine is an important bio-marker of recent smoking. A study was performed comparing cotinine levels in dried blood spots in newborns with those
in umbilical cord blood (the gold standard) among 428 newborns in the California Genetic Screening Program (Yang et al. [11]). The lowest detection limit for dried blood spot cotinine was 3.1 ng/mL. The data in Table 3.9 were presented relating dried blood spot cotinine determinations to umbilical
cord blood cotinine determinations.

Table 3.9: Distribution of Cotinine Level in Dried Blood Spots from Newborns by Maternal Active Smoking Status* close to the time of delivery among 428 babies delivered in California, 2001–2003


*Maternal active smoking at the time of delivery was defined as cord blood levels of ≥10 ng/mL.

Suppose a cutoff of ≥ 5 ng/mL is proposed as a criterion for testing positive based on dried blood spot cotinine levels.

What is the specificity using this cut-point?


Genetics, Obstetrics
Precise quantification of smoking during pregnancy is difficult in retrospective studies. Routinely collected blood specimens from newborns for screening purposes may provide a low cost method to objectively measure maternal smoking close to the time of delivery. Serum cotinine is an important bio-marker of recent smoking. A study was performed comparing cotinine levels in dried blood spots in newborns with those
in umbilical cord blood (the gold standard) among 428 newborns in the California Genetic Screening Program (Yang et al. [11]). The lowest detection limit for dried blood spot cotinine was 3.1 ng/mL. The data in Table 3.9 were presented relating dried blood spot cotinine determinations to umbilical
cord blood cotinine determinations.

Table 3.9: Distribution of Cotinine Level in Dried Blood Spots from Newborns by Maternal Active Smoking Status* close to the time of delivery among 428 babies delivered in California, 2001–2003


*Maternal active smoking at the time of delivery was defined as cord blood levels of ≥10 ng/mL.

Suppose a cutoff of ≥ 5 ng/mL is proposed as a criterion for testing positive based on dried blood spot cotinine levels.

What is the probability that a mother smokes at the time of delivery if the dried blood specimen is ≥ 5 ng/mL? 


Suppose it is estimated based on a large sample of births in California that 20% of mothers smoke at the time of delivery.


Suppose the screening test for detecting whether a mother smokes at the time of pregnancy is based on a cutoff of ≥ 5 ng/mL using dried blood specimens from the newborn.

What is another name for this quantity?


Suppose it is estimated based on a large sample of births in California that 20% of mothers smoke at the time of delivery.


Suppose the screening test for detecting whether a mother smokes at the time of pregnancy is based on a cutoff of ≥ 5 ng/mL using dried blood specimens from the newborn.

What is the sensitivity of the test for light-smoking students (students who smoke ≤ 14 cigarettes per week)? 


Suppose the self-reports are completely accurate and are representative of the number of eighth-grade students who smoke in the general community. We are considering using an SCN level ≥ 100 μg/mL as a test criterion for identifying cigarette smokers. Regard a student as positive if he or she smokes one or more cigarettes per week.


Pulmonary Disease
Research into cigarette-smoking habits, smoking prevention, and cessation programs necessitates accurate measurement of smoking behavior. However, decreasing social acceptability of smoking appears to cause significant underreporting. Chemical markers for cigarette use can provide objective indicators of smoking behavior. One widely used noninvasive marker is the level of saliva thiocyanate (SCN). In a Minneapolis
school district, 1332 students in eighth grade (ages 12–14) participated in a study [12] whereby they

(1) Viewed a film illustrating how recent cigarette use could be readily detected from small samples of saliva

(2) Provided a personal sample of SCN

(3) Provided a self-report of the number of cigarettes smoked per week The results are given in Table 3.10.


Table 3.10: Relationship between SCN levels and self-reported cigarettes smoked per week

What is the sensitivity of the test for moderate-smoking students (students who smoke 15–44 cigarettes per week)? 


Suppose the self-reports are completely accurate and are representative of the number of eighth-grade students who smoke in the general community. We are considering using an SCN level ≥ 100 μg/mL as a test criterion for identifying cigarette smokers. Regard a student as positive if he or she smokes one or more cigarettes per week.


Pulmonary Disease
Research into cigarette-smoking habits, smoking prevention, and cessation programs necessitates accurate measurement of smoking behavior. However, decreasing social acceptability of smoking appears to cause significant underreporting. Chemical markers for cigarette use can provide objective indicators of smoking behavior. One widely used noninvasive marker is the level of saliva thiocyanate (SCN). In a Minneapolis
school district, 1332 students in eighth grade (ages 12–14) participated in a study [12] whereby they

(1) Viewed a film illustrating how recent cigarette use could be readily detected from small samples of saliva

(2) Provided a personal sample of SCN

(3) Provided a self-report of the number of cigarettes smoked per week The results are given in Table 3.10.


Table 3.10: Relationship between SCN levels and self-reported cigarettes smoked per week

What is the sensitivity of the test for heavy-smoking students (students who smoke ≥ 45 cigarettes per week)? 


Suppose the self-reports are completely accurate and are representative of the number of eighth-grade students who smoke in the general community. We are considering using an SCN level ≥ 100 μg/mL as a test criterion for identifying cigarette smokers. Regard a student as positive if he or she smokes one or more cigarettes per week.


Pulmonary Disease
Research into cigarette-smoking habits, smoking prevention, and cessation programs necessitates accurate measurement of smoking behavior. However, decreasing social acceptability of smoking appears to cause significant underreporting. Chemical markers for cigarette use can provide objective indicators of smoking behavior. One widely used noninvasive marker is the level of saliva thiocyanate (SCN). In a Minneapolis
school district, 1332 students in eighth grade (ages 12–14) participated in a study [12] whereby they

(1) Viewed a film illustrating how recent cigarette use could be readily detected from small samples of saliva

(2) Provided a personal sample of SCN

(3) Provided a self-report of the number of cigarettes smoked per week The results are given in Table 3.10.


Table 3.10: Relationship between SCN levels and self-reported cigarettes smoked per week

What is the specificity of the test?


Suppose the self-reports are completely accurate and are representative of the number of eighth-grade students who smoke in the general community. We are considering using an SCN level ≥ 100 μg/mL as a test criterion for identifying cigarette smokers. Regard a student as positive if he or she smokes one or more cigarettes per week.


Pulmonary Disease
Research into cigarette-smoking habits, smoking prevention, and cessation programs necessitates accurate measurement of smoking behavior. However, decreasing social acceptability of smoking appears to cause significant underreporting. Chemical markers for cigarette use can provide objective indicators of smoking behavior. One widely used noninvasive marker is the level of saliva thiocyanate (SCN). In a Minneapolis
school district, 1332 students in eighth grade (ages 12–14) participated in a study [12] whereby they

(1) Viewed a film illustrating how recent cigarette use could be readily detected from small samples of saliva

(2) Provided a personal sample of SCN

(3) Provided a self-report of the number of cigarettes smoked per week The results are given in Table 3.10.


Table 3.10: Relationship between SCN levels and self-reported cigarettes smoked per week

What is the PV+ of the test?


Suppose the self-reports are completely accurate and are representative of the number of eighth-grade students who smoke in the general community. We are considering using an SCN level ≥ 100 μg/mL as a test criterion for identifying cigarette smokers. Regard a student as positive if he or she smokes one or more cigarettes per week.


Pulmonary Disease
Research into cigarette-smoking habits, smoking prevention, and cessation programs necessitates accurate measurement of smoking behavior. However, decreasing social acceptability of smoking appears to cause significant underreporting. Chemical markers for cigarette use can provide objective indicators of smoking behavior. One widely used noninvasive marker is the level of saliva thiocyanate (SCN). In a Minneapolis
school district, 1332 students in eighth grade (ages 12–14) participated in a study [12] whereby they

(1) Viewed a film illustrating how recent cigarette use could be readily detected from small samples of saliva

(2) Provided a personal sample of SCN

(3) Provided a self-report of the number of cigarettes smoked per week The results are given in Table 3.10.


Table 3.10: Relationship between SCN levels and self-reported cigarettes smoked per week

What is the PV− of the test?


Suppose the self-reports are completely accurate and are representative of the number of eighth-grade students who smoke in the general community. We are considering using an SCN level ≥ 100 μg/mL as a test criterion for identifying cigarette smokers. Regard a student as positive if he or she smokes one or more cigarettes per week.


Pulmonary Disease
Research into cigarette-smoking habits, smoking prevention, and cessation programs necessitates accurate measurement of smoking behavior. However, decreasing social acceptability of smoking appears to cause significant underreporting. Chemical markers for cigarette use can provide objective indicators of smoking behavior. One widely used noninvasive marker is the level of saliva thiocyanate (SCN). In a Minneapolis
school district, 1332 students in eighth grade (ages 12–14) participated in a study [12] whereby they

(1) Viewed a film illustrating how recent cigarette use could be readily detected from small samples of saliva

(2) Provided a personal sample of SCN

(3) Provided a self-report of the number of cigarettes smoked per week The results are given in Table 3.10.


Table 3.10: Relationship between SCN levels and self-reported cigarettes smoked per week

Assuming the percentage of students with SCN ≥ 100 μg/mL in these two subgroups is the same as in those who truly report 1–4 and 5–14 cigarettes per week, compute the specificity under these assumptions?


Suppose we regard the self-reports of all students who report some cigarette consumption as valid but estimate that 20% of students who report no cigarette consumption actually smoke 1–4 cigarettes per week and an additional 10% smoke 5–14 cigarettes per week.

Compute the PV− under these altered assumptions. How does the true PV− using a screening criterion of SCN ≥ 100 μg/mL for identifying smokers compare with the PV− based on self-reports obtained in Problem 3.73? 


Suppose we regard the self-reports of all students who report some cigarette consumption as valid but estimate that 20% of students who report no cigarette consumption actually smoke 1–4 cigarettes per week and an additional 10% smoke 5–14 cigarettes per week.



If the manual measurements are regarded as the “true” measure of reactivity, then what is the sensitivity of automated DBP measurements? 


Hypertension
Laboratory measures of cardiovascular reactivity are receiving increasing attention. Much of the expanded interest is based on the belief that these measures, obtained under challenge from physical and psychological stressors, may yield a more biologically meaningful index of cardiovascular function than more traditional static measures. Typically, measurement of cardiovascular reactivity involves the use of an automated blood-pressure monitor to examine the changes in blood pressure before and after a stimulating experience (such as playing a video game). For this purpose, blood-pressure measurements were made with the Vita- Stat blood-pressure machine both before and after playing a video game. Similar measurements were obtained using manual methods for measuring blood pressure. A person was classified as a “reactor” if his or her DBP increased by 10 mm Hg or more after playing the game and as a nonreactor otherwise. The results are given in Table 3.11.


Table 3.11: Classification of cardiovascular reactivity using an automated and a manual sphygmomanometer

What is the specificity of automated DBP measurements?


Hypertension
Laboratory measures of cardiovascular reactivity are receiving increasing attention. Much of the expanded interest is based on the belief that these measures, obtained under challenge from physical and psychological stressors, may yield a more biologically meaningful index of cardiovascular function than more traditional static measures. Typically, measurement of cardiovascular reactivity involves the use of an automated blood-pressure monitor to examine the changes in blood pressure before and after a stimulating experience (such as playing a video game). For this purpose, blood-pressure measurements were made with the Vita- Stat blood-pressure machine both before and after playing a video game. Similar measurements were obtained using manual methods for measuring blood pressure. A person was classified as a “reactor” if his or her DBP increased by 10 mm Hg or more after playing the game and as a nonreactor otherwise. The results are given in Table 3.11.


Table 3.11: Classification of cardiovascular reactivity using an automated and a manual sphygmomanometer

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