W. Bernado. Johnson and Wales University.
A hospital administrator wishes to estimate the mean weight of babies born in her hospital discount 10mg prasugrel with visa. How large a sample of birth records should be taken if she wants a 99 percent confidence interval that is 1 pound wide? The director of the rabies control section in a city health department wishes to draw a sample from the department’s records of dog bites reported during the past year in order to estimate the mean age of persons bitten discount prasugrel 10mg otc. He wants a 95 percent confidence interval discount 10mg prasugrel with amex, he will be satisfied to let d ¼ 2:5, and from previous studies he estimates the population standard deviation to be about 15 years. A physician would like to know the mean fasting blood glucose value (milligrams per 100 ml) of patients seen in a diabetes clinic over the past 10 years. Determine the number of records the physician should examine in order to obtain a 90 percent confidence interval for m if the desired width of the interval is 6 units and a pilot sample yields a variance of 60. For multiple sclerosis patients we wish to estimate the mean age at which the disease was first diagnosed. We make use of the fact that one-half the desired interval, d, may be set equal to the product of the reliability coefficient and the standard error. Estimating p As we see, both formulas require knowledge of p, the proportion in the population possessing the characteristic of interest. Since this is the parameter we are trying to estimate, it, obviously, will be unknown. One solution to this problem is to take a pilot sample and compute an estimate to be used in place of p in the formula for n. Sometimes an investigator will have some notion of an upper bound for p that can be used in the formula. For example, if it is desired to estimate the proportion of some population who have a certain disability, we may feel that the true proportion cannot be greater than, say,. Since p ¼ :5 in the formula yields the maximum value of n, this procedure will give a large enough sample for the desired reliability and interval width. It may, however, be larger than needed and result in a more expensive sample than if a better estimate of p had been available. This procedure should be used only if one is unable to arrive at a better estimate of p. Solution: If the finite population correction can be ignored, we have 2 ð 1:96 :35 :65 n ¼ 2 ¼ 349:59 ð :05 The necessary sample size, then, is 350. An epidemiologist wishes to know what proportion of adults living in a large metropolitan area have subtype ayr hepatitis B virus. Determine the sample size that would be required to estimate the true proportion to within. In a similar metropolitan area the proportion of adults with the characteristic is reported to be. If data from another metropolitan area were not available and a pilot sample could not be drawn, what sample size would be required? A survey is planned to determine what proportion of the high-school students in a metropolitan school system have regularly smoked marijuana. If no estimate of p is available from previous studies, a pilot sample cannot be drawn, a confidence coefficient of. A hospital administrator wishes to know what proportion of discharged patients is unhappy with the care received during hospitalization. How large a sample should be drawn if we let d ¼ :05, the confidence coefficient is. A health planning agency wishes to know, for a certain geographic region, what proportion of patients admitted to hospitals for the treatment of trauma die in the hospital. We have discussed only one criterion of quality—unbiasedness—so let us see if the sample variance is an unbiased estimator of the population variance. To be unbiased, the average value of the sample variance over all possible samples must be equal to the population variance. To see if this condition holds for a particular situation, let us refer to the example of constructing a sampling distribution given in Section 5. It will be recalled that two measures of dispersion for this population were computed as follows: P 2 P 2 2 xi À m 2 xi À m s ¼ ¼ an d S ¼ ¼ 10 N N À 1 2 P 2 If we compute the sample variance s ¼ xi À x = n À 1 for each of the possible samples shown in Table 5. When we do this, we have P 2 ÀÁ2 s 0 þ 2 þÁÁÁþ2 þ 0 200 Es ¼ i ¼ ¼ ¼ 8 Nn 25 25 and we see, for example, that when sampling is with replacement EsðÞ2 s2, where s2 ¼ P 2 2 P 2 ð xi À x = n À 1 and s ¼ xi À m =N. Sampling Without Replacement If we consider the case where sampling is 2 without replacement, the expected value of s is obtained by taking the mean of all variances above (or below) the principal diagonal. These results are examples of general principles, as it can be shown that, in general, EsðÞ2 s2 when sampling is with replacement EsðÞ2 S2 when sampling is without replacement When N is large, N À 1 and N will be approximately equal and, consequently, s2 2 and S will be approximately equal. In passing, let us note that although s is an unbiased estimator of s2; s is not an unbiased estimator of s. Interval Estimation of a Population Variance With a point estimate available, it is logical to inquire about the construction of a confidence interval for a population variance. Whether we are successful in constructing a confidence interval for s2 will depend on our ability to find an appropriate sampling distribution. Heagerty, and Thomas Lumley, Biostatistics: A Methodology for the Health Sciences, 2nd Ed. If samples of size n are drawn from a normally distributed population, this quantity has a distribution known as the chi-square ðÞx2 distribution with n À 1 degrees of freedom. As we will say more about this distribution in chapter 12, we only say here that it is the distribution that the quantity n À 1 s2=s2 follows and that it is useful in finding confidence intervals for s2 when the assumption that the population is normally distributed holds true. The column headings give the values of x2 to the left of which lies a proportion of the total area under the curve equal to the subscript of x2. To obtain a 100 1 À a percent confidence interval for s2, we first obtain the 100 1 À a percent confidence interval for n À 1 s2=s2. To do this, we select the values of x2 from Appendix Table F in such a way that a=2 is to the left of the smaller value and a=2 is to the right of the larger value. In other words, the two values of x2 are selected in such a way that a is divided equally between the two tails of the distribution. First, let us divide each term by n À 1 s2 to get x2 x2 a=2 1 1ÀðÞa=2 2 < 2 < 2 ð n À 1 s s n À 1 s If we take the reciprocal of this expression, we have ð n À 1 s2 n À 1 s2 2 2 > s > 2 xa=2 x1ÀðÞa=2 Note that the direction of the inequalities changed when we took the reciprocals. Our 95 percent confidence interval for 1ÀðÞa=2 a=2 s2 is 639:763 2 639:763 < s < 14:449 1:237 16:512 < s2 < 192:868 The 95 percent confidence interval for s is 4:063 < s < 13:888 We are 95 percent confident that the parameters being estimated are within the specified limits, because we know that in the long run, in repeated sampling, 95 percent of intervals constructed as illustrated would include the respective parameters. First, the assumption of the normality of the population from which the sample is drawn is crucial, and results may be misleading if the assumption is ignored. Another difficulty with these intervals results from the fact that the estimator is not in the center of the confidence interval, as is the case with the confidence interval for m. The practical implication of this is that the method for the construction of confidence intervals for s2, which has just been described, does not yield the shortest possible confidence intervals.
The objective has been to provide enough of a “feel” for the subject so that the probabilistic aspects of statistical inference can be more readily understood and appreciated when this topic is presented later buy prasugrel 10mg. We defined probability as a number between 0 and 1 that measures the likelihood of the occurrence of some event 10mg prasugrel amex. Objective probability can be categorized further as classical or relative frequency probability prasugrel 10 mg for sale. After stating the three properties of probability, we defined and illustrated the calculation of the following kinds of probabilities: marginal, joint, and conditional. We also learned how to apply the addition and multiplication rules to find certain probabilities. We learned the meaning of independent, mutually exclusive, and complementary events. We learned the meaning of specificity, sensitivity, predictive value positive, and predictive value negative as applied to a screening test or disease symptom. Finally, we learned how to use Bayes’s theorem to calculate the probability that a subject has a disease, given that the subject has a positive screening test result (or has the symptom of interest). Define the following: (a) Probability (b) Objective probability (c) Subjective probability (d) Classical probability (e) The relative frequency concept of probability (f) Mutually exclusive events (g) Independence (h) Marginal probability (i) Joint probability (j) Conditional probability (k) The addition rule (l) The multiplication rule (m) Complementary events (n) False positive (o) False negative (p) Sensitivity (q) Specificity (r) Predictive value positive (s) Predictive value negative (t) Bayes’s theorem 2. The study used data from the Behavioral Risk Factor Surveillance System surveys of adults age 18 years or older conducted in 1999 and 2000. The table below reports the number of observations of Hispanic and non-Hispanic women who had received a mammogram in the past 2 years cross-classified with marital status. Wilson, “Breast and Cervical Cancer Screening Practices Among Hispanic and Non-Hispanic Women Residing Near the United States–Mexico Border, 1999–2000,” Family and Community Health, 26 (2003), 130–139. The table below shows the skill retention numbers in regard to overall competence as assessed by video ratings done by two video evaluators. The researchers classified subjects into four personality types: obsessiod, asthenic=low self-confident, asthenic=high self-confident, nervous=tense, and undeterminable. A certain county health department has received 25 applications for an opening that exists for a public health nurse. If a selection from among these 25 applicants is made at random, what is the probability that a person over 30 or a person with a master’s degree will be selected? Made a low score on the examination given that he or she graduated from a superior high school. For a variety of reasons, self-reported disease outcomes are frequently used without verification in epidemiologic research. They used the self-reported cancer data from a California Teachers Study and validated the cancer cases by using the California Cancer Registry data. The following table reports their findings for breast cancer: Cancer Reported (A) Cancer in Registry (B) Cancer Not in Registry Total Yes 2991 2244 5235 No 112 115849 115961 Total 3103 118093 121196 Source: Arti Parikh-Patel, Mark Allen, William E. Wright, and the California Teachers Study Steering Committee, “Validation of Self-reported Cancers in the California Teachers Study,” American Journal of Epidemiology, 157 (2003), 539–545. In a certain population the probability that a randomly selected subject will have been exposed to a certain allergen and experience a reaction to the allergen is. If a subject is selected at random from this population, what is the probability that he or she will have been exposed to the allergen? Suppose that 3 percent of the people in a population of adults have attempted suicide. It is also known that 20 percent of the population are living below the poverty level. In a certain population of women 4 percent have had breast cancer, 20 percent are smokers, and 3 percent are smokers and have had breast cancer. The probability that a person selected at random from a population will exhibit the classic symptom of a certain disease is. For a certain population we define the following events for mother’s age at time of giving birth: A ¼ under 20 years; B ¼ 20–24 years; C ¼ 25–29 years; D ¼ 30–44 years. For a certain population we define the following events with respect to plasma lipoprotein levels (mg=dl): A ¼ (10–15); B ¼ð! State in words the meaning of the following events: (a) A [ B (b) A B (c) A C (d) A [ C 20. Since they show all possible values of a random variable and the probabilities associated with these values, probability distributions may be summarized in ways that enable researchers to easily make objective deci- sions based on samples drawn from the populations that the distributions represent. This chapter introduces frequently used discrete and continuous probability distributions that are used in later chapters to make statistical inferences. We build on these concepts in the present chapter and explore ways of calculating the probability of an event under somewhat more complex conditions. In this chapter we shall see that the relationship between the values of a random variable and the probabilities of their occurrence may be summarized by means of a device called a probability distribution. A probability distribution may be expressed in the form of a table, graph, or formula. Knowledge of the probability distribution of a random variable provides the clinician and researcher with a powerful tool for summarizing and describing a set of data and for reaching conclusions about a population of data on the basis of a sample of data drawn from the population. The purpose of the study was to examine hunger rates of families with children in a local Head Start program in Athens, Ohio. In addition, participants were asked how many food assistance programs they had used in the last 12 months. We wish to construct the probability distribution of the discrete variable X, where X ¼ number of food assistance programs used by the study subjects. We compute the probabilities for these values by dividing their respective frequencies by the total, 297. These are not phenomena peculiar to this particular example, but are characteristics of all probability distributions of discrete variables. With its probability distribution available to us, we can make probability statements regarding the random variable X. Solution: To answer this question, we use the addition rule for mutually exclusive events. The cumulative probability distribution for the discrete variable whose probability distribution is given in Table 4. The length of each vertical line represents the same probability as that of the corresponding line in Figure 4. Solution: Since a family that used fewer than four programs used either one, two, or three programs, the answer is the cumulative probability for 3. Solution: To find the answer we make use of the concept of complementary probabili- ties. The set of families that used five or more programs is the complement of the set of families that used fewer than five (that is, four or fewer) programs.
Several studies have shown that of these patients prasugrel 10mg mastercard, up to 30% will require an additional surgery for recurrence of prolapse [3 quality prasugrel 10 mg,4] discount prasugrel 10 mg visa. Caucasian and Latina females have a fourfold to fivefold increase when compared to African- American females . McCall in 1957 using the culdoplasty technique that revealed the importance of this suspension at the time of a vaginal hysterectomy to prevent an enterocele and posthysterectomy vaginal vault prolapse . Now nearly 60 years after McCall described his technique, the same attachment points and surgical principles are used via laparoscopic approaches. Laparoscopic Approach for High Uterosacral Ligament Suspension Laparoscopic Port Placement Traditionally, three laparoscopic ports are placed in the abdomen. The first is the camera port placed in the umbilicus or up to 2 cm caudal to the umbilicus depending on the patient’s habitus. The second and third ports are 5 mm ports placed suprapubically and at the right paramedian  (Figures 87. The patient is transitioned into steep Trendelenburg to assist with bowel retraction and to better visualize the pelvic sidewalls. To further assist with visualization of the uterosacral ligaments, a probe is placed in the vagina to hold the ligaments on tension. This will assist the surgeon in tracing the ligaments back to their proximal origin . This is performed so that the pubocervical fascia and the rectovaginal fascia can be visualized. Next, a figure-eight suture is then placed approximately two- thirds cephalad to the proximal origin of the ligament and 1 cm caudal to the most anterior palpable margin of the sacrum . We recommend using a delayed absorbable or permanent suture cut to a length of 36–48 in. Additionally, when first performing this technique, it is recommended to use the extracorporeal approach with a closed knot pusher to secure knot placement, as this is the easiest method to both learn and teach. After knots are secured in place, the ureters should be inspected on each side, and an intraoperative cystoscopy should be performed. It is important that cystoscopy is performed prior to the removal of the laparoscopic ports, so that any ureteral compromise can be addressed prior to the completion of the case. For the main operative ports, a line is marked 16 cm cephalad to the inferior margin of the symphysis pubis. A left and right port are then placed 9 cm lateral to this mark once the abdomen has been insufflated. This is to prevent the robot arms from colliding with each other during the procedure. The third arm port is then placed on the left side, 4 cm superior and 3 cm lateral to the camera port. An assistant 5 mm diameter port is placed 5 cm lateral to the camera port  (Figure 87. Technique Once the patient is placed in steep Trendelenburg, the robot patient side cart is positioned either between the patient’s legs to align the center of the cart with the patient’s midline (central docking) or aligned alongside the patient for parallel docking. Identification of the uterosacral ligaments can be achieved by placing traction on the vaginal apex 1341 with a probe in the vagina when the uterus is not present or by using a uterine manipulator when the uterus is present. If the later method is indicated, after the uterus has been completely devitalized and prior to colpotomy, upward pressure is placed on the uterine manipulator to help with the identification of the uterosacral ligaments. The complete pelvic course of the ureter is then identified prior to placing sutures. At this level, the uterosacral ligaments are the furthest from the ureters as they are heading toward S3 and the ureters are clearing the pelvic brim. The distal end of the sutures are then passed through the pubocervical and rectovaginal fascia and then incorporated into the vaginal cuff. The vaginal cuff is then tied down after the completion of the cuff closure with polyglactin sutures. A third row of sutures can be used in the instance of elongated uterosacral ligaments. Again as stated in the laparoscopic section, an intraoperative cystoscopy should be performed prior to removal of the ports. The goal of both procedures is to restore normal anatomical support by suspending the apex of the vagina above the level of the ischial spines toward the level of the sacrum without causing any significant distortion to the vaginal axis . In most cases, removing the offending suture(s) will alleviate any obstruction without consequence as long as the surgeon finds the problem intraoperatively [14,15]. Other potential complications include bowel injury, pelvic abscess, dyspareunia, hemorrhage and in rare cases bladder injury, and exposure of permanent sutures into the vaginal lumen . Outcomes Although multiple studies and meta-analyses have been performed evaluating the long-term success of uterosacral ligament suspension when approached vaginally, there are few studies describing the long- term outcomes associated with the laparoscopic and robotic-assisted approach. However, in this study, the uterus was conserved in the laparoscopic approach . In this series, at 6 months follow-up, there was a 100% objective success rate [18,19]. Contrary to many previous studies, this further suggests that the laparoscopic approach is as effective as the traditional vaginal approach. The laparoscopic and robotic approach to performing uterosacral ligament suspension allows the surgeon to have a more global view to inspect the pelvic cavity. Other advantages include the ability to use pneumoperitoneum to access better surgical planes and the also the accuracy of suture placement to achieve an optimal result . The postoperative advantages of this approach are less blood loss, shorter hospital stays, decreased postoperative pain, and the ability to perform adhesiolysis when necessary to obtain a better anatomical result . Although the procedure has been modified through the years, the same principles of using multiple interrupted permanent sutures to attach mesh to the vagina and elevating this up to the anterior longitudinal ligament at the level of the sacrum are still important today. Lane also describes the importance of reapproximating the peritoneum over the synthetic material to avoid interaction of the graft with other pelvic structures. He further emphasized the importance of using mesh to replace the inadequately supported structures that contribute to prolapse as a disorder . Although the gold standard for treatment of apical prolapse is the abdominal sacrocolpopexy due to its effectiveness and availability around the world, many institutions have adopted the more minimally invasive approaches of laparoscopy and robotic-assisted techniques [17,23,24]. Nezhat in 1994 first introduced the laparoscopic sacrocolpopexy, showing its decrease in operative blood loss and, most importantly, patient recovery time while still producing high success rates . Although there is no strict definition of success for this procedure, we and many other authors define a successful procedure based on both the “clinical cure” and “objective anatomic cure” rates [12,29]. Laparoscopic Approach for Sacrocolpopexy Port Placement Traditionally, four laparoscopic ports are placed in the abdomen. A 10 mm suturing port is placed in the left paramedian region, and two additional 5 mm ports are placed. The first accessory port is placed suprapubically, and the second is placed in the right paramedian region  (Figures 87. Technique To assist with clearly visualizing the vaginal apex, a probe is placed in the vagina. Next, the peritoneum is dissected away from the vaginal apex anteriorly, exposing the full thickness of the vaginal wall.