Features exercises, with answers to alternate questions, enabling use as a course text. Written at an elementary mathematical level so that readers with high school mathematics will find the content accessible.
Graduate students studying genetic epidemiology, researchers and practitioners from genetics, epidemiology, biology, medical research and statistics will find this an invaluable introduction to statistics. Thetext includes an extensive discussion of measurement issues inepidemiology, especially confounding.
Maximum likelihood,Mantel-Haenszel, and weighted least squares methods are presentedfor the analysis of closed cohort and case-control data. Kaplan-Meier and Poisson methods are described for the analysis ofcensored survival data. A justification for using odds ratiomethods in case-control studies is provided.
Standardization ofrates is discussed and the construction of ordinary, multipledecrement and cause-deleted life tables is outlined. Sample sizeformulas are given for a range of epidemiologic study designs. Thetext ends with a brief overview of logistic and Cox regression. Other highlights include: Many worked examples based on actual data Discussion of exact methods Recommendations for preferred methods Extensive appendices and references Biostatistical Methods in Epidemiology provides anexcellent introduction to the subject for students, while alsoserving as a comprehensive reference for epidemiologists and otherhealth professionals.
For more information, visit www. The book also provides analysis of statistical software outcomes and their interpretations, includes guidance for critical evaluation of published scientific reports, and provides technical aspects for decision making and research communication.
Basic Epidemiology and Biostatistics provides information that will help public health, health care, and biomedical researchers in planning of their research, its execution and in-depth analysis of the data, and presenting the output from statistical testing. Offers an amalgamation of epidemiology and biostatistics principles Presents a selection of optimum research methodology Provides guidance for the interpretation of data for statistical and clinical significance.
Methods in Social Epidemiology Author : J. Covering the theory, models, and methods used to measure and analyze these phenomena, this book serves as both an introduction to the field and a practical manual for data collection and analysis.
American journal of epidemiology. Cadernos de saude publica. Power for tests of interaction: effect of raising the Type I error rate.
Analysis of contingency tables based on generalised median polish with power transformations and non-additive models. Health Inf. View 1 excerpt, cites methods. Prevention Science. Simulation of complex survival distributions. Epidemiology for the nuclear medicine technologist. Journal of nuclear medicine technology. Related Papers.
Abstract Topics Citations Related Papers. By clicking accept or continuing to use the site, you agree to the terms outlined in our Privacy Policy , Terms of Service , and Dataset License. The other cell frequencies only slightly increase the variance. Such a p-value indicates that the elevated value of the observed odds ratio comparing the odds of breast cancer between women who served and did not serve in Vietnam is plausibly due to random variation.
The Pearson chi-square test statistic is applied to compare the four observed values denoted oi j to four values generated as if no association exists expected values, denoted ei j. The four expected values represented by ei j are theoretical cell frequencies calculated as if risk factor and disease are exactly independent details follow in Chapter 2. For the breast cancer data, the four expected values are given in Table 1—4.
Thus, the odds ratio is exactly 1. In a statistical context, the term expected value has a technical meaning. The parallel statistical test based on the logarithm of the odds ratio is not the same as the Table 1—4. An odds ratio of 1. When Debbie read about folic acid in a March of Dimes brochure, she was astonished when she learned about the role of folic acid in preventing NTDs.
I immediately began taking folic acid and telling every women I could about it. However, the odds ratio, as the name indicates, is a ratio measure of association. It measures association in terms of a multiplicative scale and behaves in an asymmetric way on an additive scale.
Odds ratios between zero and 1 indicate a decrease in risk, and odds ratios greater than 1 1. Thus, an odds ratio of 0. Comparisons on a ratio scale are necessarily made in terms of ratios. The result of a statistical test of association, for example, is unchanged same p-value. It should be noted that the logarithms of values measured on a ratio scale produce values that are directly comparable on an additive scale.
An odds ratio of 0. Thus, the value log 0. The transformation of an odds Two Measures of Risk 11 Table 1—5. Table 1—7. The results from a survey of primary care physicians conducted by Schulman and colleagues were published in The New England Journal of Medicine The collected data showed a P correct white 0.
We should not have allowed the odds ratio in the abstract. At the beginning of the 18th century, Sir Thomas Bayes b. P B Less historic, but important to the study of disease and other binary variables, the odds ratio has the same value whether the association under study is measured by comparing the likelihoods of the risk factor between individuals with and Two Measures of Risk 15 without the disease or is measured by comparing the likelihoods of the disease between individuals with and without the risk factor.
Prospectively and crosssectionally collected data require the comparison of the likelihood of the disease among individuals with and without the risk factor.
The key to assessing an association using these two kinds of data is the comparison of the probability of the risk factor among those with the disease P F D to those without disease P F D , or the comparison of the probability of the disease among those with the risk factor P D F to those without the risk factor P D F.
That is, the test statistics produce exactly the same result. Another notable property of the odds ratio is that it is always further from 1.
In this sense, the relative risk is conservative, because any association measured by relative risk is more extreme when measured by an odds ratio. These concepts are properties of summary values calculated from data and are not properties of the sampled population. This extremely important distinction is frequently not made clear. The concepts are: interaction, confounding, and independence. The example binary variables are discussed in terms of a disease D, a risk factor F, and a third variable labeled C.
This introductory description is expanded in subsequent chapters. When the odds ratios calculated from each of two tables differ, the question arises: Do the estimated values differ by chance alone or does a systematic difference exist? The failure of a measure of association to be the same within each subtable is an example of an interaction Table 1—8A. An interaction, in simple terms, is the failure of the values measured by a summary statistic to be the same under different conditions.
The existence of an interaction is a central element in addressing the issues involved in combining estimates. When odds ratios systematically differ, a single summary odds ratio is usually not meaningful and, in fact, such a single value can be misleading an example follows.
However, when the odds ratios measuring association differ only by chance alone, Table 1—8A. The odds ratios between the risk factor F and disease D at the two levels of the variable C no interaction and a single odds ratio from the combined table. A single odds ratio is not only a meaningful summary but provides a useful, simpler, and more precise measure of the association between the risk factor and disease. When a single odds ratio is a useful summary no interaction , the second issue becomes the choice of how the summary value is calculated.
The choice is between a summary value calculated by combining the two odds ratios or by combining the data into a single table and then calculating a summary odds ratio. When the variable C is unrelated to the disease or the risk factor. Table 1—8C illustrates the case where all three odds ratios equal 4. However, important properties are illustrated.
Namely, the presence or absence of an interaction determines 18 Statistical Tools for Epidemiologic Research Table 1—8C. That is, the third variable is ignored. The topic of creating accurate summary values in applied situations is continued in detail in Chapter 2 and beyond. A new treatment denoted F appears more successful than the usual treatment denoted F for two kinds of cancer patients, but when the data are combined into a single table, the new treatment appears less effective.
Therefore, combining the data and calculating a single summary value produces a result that has no useful interpretation. The point is, when Two Measures of Risk 19 Table 1—9. The United States cancer mortality rate among individuals 60—64 years old was One gets a sense of the change in risk by comparing two such rates, but the answers to a number of questions are not apparent. Some examples are: Why report the number of deaths per person-years?
How do these rates differ from probabilities? What is their relationship to the mean survival time? The following is a less than casual but short of rigorous description of the statistical origins and properties of an average rate. To simplify the terminology, rates from this point on are referred to as mortality rates, but the description applies to disease incidence rates, as well as to many other kinds of average rates.
For a mortality rate, the mean number of deaths divided by the mean survival time experienced by those individuals at risk forms the average rate. The mean number of deaths is also the proportion of deaths.
A proportion is another name for a mean value calculated from zeros and ones. In the language of baseball, this mean value is called a batting average but elsewhere it is usually referred to as a proportion or an estimated probability Chapter 3. The mean survival time, similar to any mean value, is the total time at risk experienced by the n individuals who accumulated the relevant survival time divided by n.
An average rate can be viewed as the reciprocal of a mean value. R Two Measures of Risk 21 A more realistic case occurs when a group of individuals is observed for a period of time and, at the end of that time, not all at-risk persons have died. A single estimate requires that the value estimated be a single value.
Thus, when a single rate is estimated from data collected over a period of time, it is implicitly assumed that the risk described by this single value is constant over the same time period or at least approximately constant.
When a mortality rate is constant, then the mean survival time is also constant. R d Naturally, as the rate of death increases, the mean survival time decreases and vice versa. The estimated mean survival time is the total survival time observed for all n at-risk individuals divided by d number of deaths and not n number of observed individuals for the following reason. For these kinds of survival data, the exact time of death is not usually available.
However, a generally accurate but approximate average mortality rate can be calculated when the length of the age or time interval considered is not large. The total time alive at-risk is made up of two distinct contributions. Furthermore, a rate ratio and a ratio of probabilities are also essentially equal when applied to the same time interval. However, the approximate geometry is simple and is displayed in Figure 1—2. The key to a geometric description of a rate is the proportion of surviving individuals measured at two points in time.
The approximate mean person-years again consists of two parts, namely the mean years lived by those who survived the entire interval a rectangle and the mean years lived by those who died during the interval a triangle. The approximate and exact areas approximate and exact mean survival times describing the mortality pattern of a hypothetical population.
More succinctly, the mean time survived is 30 0. However, for the example, the exact rate generated from the survival probabilities Figure 1—2 is death per 10, person-years at risk for any interval constant mortality rate.
That is, a rectangle plus a triangle more accurately estimate the mean person-years at risk area under the curve. For the example, the approximate average rate is deaths per 10, person-years for a year interval, and the rate is deaths per 10, person-years for a 5-year interval. For left-handers, it was All individuals sampled are dead. Comparing the frequency of deaths among individuals with different exposures or risk factors is called a proportionate mortality study.
A fundamental property of proportionate mortality studies in general is that it is not possible to determine from the data whether 26 Statistical Tools for Epidemiologic Research the exposure increased the risk in one group or decreased the risk in another or both.
For example, the increased life-time observed in right-handed individuals could be due to a decrease in their risk, or to an increase in the risk of left-handed individuals, or both.
In addition, a frequent problem associated with interpreting proportionate mortality data is that the results are confused with an assessment of the risk of death, as illustrated by the Coren and Halpren report. Rates cannot be calculated from proportionate mortality data. The individuals who did not die are not included in the collected data. Unexposed n—x? Total n? Two Measures of Risk 27 In terms of the right- and left-handed mortality data, the frequencies of surviving left- and right-handed individuals are not known.
It is likely that the frequency of right-handedness is increased among the older individuals sampled, causing an apparent longer lifetime. Early in the 20th century, being lefthanded was frequently treated as a kind of disability and many naturally lefthanded children were trained to be right-handed. As the century progressed, this practice became less frequent.
By the end of the 20th century, therefore, the frequency of right-handedness was relative higher among older individuals or conversely, relative lower among younger individuals. The absence of the numbers of surviving right- and lefthanded individuals makes it impossible to compare the risk of death between these two groups. More simply, to estimate a rate requires an estimate of the personyears at risk accumulated by those who die, as well as by those who survive.
Similarly, to estimate the probability of death requires the number of deaths, as well as the number of those who did not die. In terms of analytic approaches, the two points of view are: 1. Regression analysis: What is the relationship between a k-level risk variable and a binary outcome? Two-sample analysis: Does the mean value of the k-level risk variable differ between two sampled groups? Both approaches are distribution-free, in the sense that knowledge or assumptions about the distribution that generates the sampled data is not required.
For simplicity, more than three cups consumed per day are considered as three, incurring a slight bias. Second, the kind of relationship between the k-level numeric variable and the binary outcome is explored How does pancreatic cancer risk change as the amount of coffee consumed increases? The symbol ni j represents the number of observations falling into both the ith row and the jth column; namely, the count in the i, j th -cell. The marginal frequencies are the sums of the columns or the rows and are represented by n.
In symbols, the marginal frequencies are n. For example, the total number of controls among the study subjects Table 2—1 is the marginal frequency of row 2 or n2. When categorical variables X and Y are unrelated, the cell probabilities are completely determined by the marginal probabilities Table 2—4. The expected number of observations in the i, j th -cell is nqi p j when, to repeat, X and Y are statistically independent. Because the underlying cell probabilities pi j are theoretical quantities population parameters , these values are almost always estimated from the collected data.
These expected values are calculated for each cell in the table and are compared to the observed values, typically using a chi-square test statistic to assess the conjecture that the categorical variables X and Y are unrelated. The marginal frequencies remain the same as those in the original table Table 2. A consequence of statistical independence is that a table is unnecessary.
For example, in column 1, the ratio is Thus, the chi-square statistic consists of the sum of eight comparisons, one for each cell in the table.
A small p-value, say in the neighborhood of 0. A chi-square statistic summarizes this variation. Notice that the chi-square statistic is a measure of the variability among a series of estimated values. The test statistic X 2 , therefore, provides an assessment p-value of the likelihood that this variability is due entirely to chance. The purpose of calculating a chi-square statistic is frequently to identify nonrandom variation among a series of estimated values. It is not a coincidence that the chi-square value to evaluate independence and the chi-square value to evaluate homogeneity are identical.
A little algebra shows that these two apparently different assessments produce identical summary values. One such opportunity arises when the k-level categorical variables are numeric. Proportion of cases of pancreatic cancer by consumption of 0, 1, 2 and 3 or more cups of coffee per day circles.
The ratio of the squared estimated slope to its 36 Statistical Tools for Epidemiologic Research estimated variance has an approximate chi-square distribution with one degree of freedom when the column variable X is unrelated to the conditional probabilities Pj. The chi-square statistic, XL2 , measures the extent to which an estimated line 2 , summarizes the deviates from a horizontal line. The test-statistic XNL a chi-square distribution when the variation is strictly random.
For an assessment of linear trend to be useful, the trend in the probabilities Pj associated with levels of X must be at least approximately linear. This moderately large chi-square value small pvalue suggests a likely failure of a straight line as an accurate summary of the relationship between pancreatic cancer risk and coffee consumption.
The extent to which a summary value fails, an analysis based on that summary equally fails. A potential threshold-like effect of coffee consumption is visible in Figure 2—1. That is, the risk associated with coffee drinkers appears to differ from non—coffee drinkers, but does not appear to increase in a consistent linear pattern.
Failure to reject the 38 Statistical Tools for Epidemiologic Research hypothesis of statistical independence does not mean that X and Y are unrelated. Y based on an estimated straight line. The analysis of the coffee consumption and pancreatic cancer data shows visual as well as statistical evidence of a nonlinear relationship, possibly a threshold response. Threshold responses, like the one illustrated, are prone to bias, particularly bias caused by other related variables.
For example, perhaps coffee drinkers smoke more cigarettes than non—coffee drinkers and, therefore, at least part of the observed case—control response might be due to an unobserved relationship with smoking.
A natural measure of association between a binary variableY and a numeric variable X is the difference between the two mean values x1 and x2. For the pancreatic cancer data, the mean level of coffee consumption among the n1. The primary question becomes: Is the observed mean increase likely due to random variation, or does it provide evidence of an underlying systematic increase in coffee consumption among the cases?
For example, each estimate is likely close to zero when the variables X and Y are unrelated. So far, the choice of the numeric values for X has not been an issue. This choice is, nevertheless, important and careful thought should be given to an appropriate scale for the variable X. In yet other situations, natural units arise, such as the number of cups of coffee consumed per day. The choice of scale is essentially nonstatistical and is primarily determined by subject-matter considerations.
Proportion of cases of childhood cancer for exposure to 0, 1, 2, 3, 4, and 5 or more x-rays during pregnancy. A test of linear trend, as pointed out, has meaning only when the data are effectively summarized by a straight line. A striking feature of these x-ray data is the apparent straight-line dose response. The partitioned chi-square values are summarized in Table 2—8. The strong and linear response is convincing evidence of a substantial and consistently increasing risk associated with each additional prenatal x-ray and the subsequent development of cancer in the children of the x-rayed mothers.
This observation 60 years ago brought a halt to using x-rays as a routine prenatal diagnostic tool. Table 2—8. Similar to the odds ratio summary, the differences in proportions of low-birth-weight infants between smokers and nonsmokers can be combined into a single summary difference and statistically assessed. Using a weighted average, Tabular Data 51 Table 2— Because the summary mean value d is a single value, the quantity estimated must also be a single value.
When the di -values randomly differ, the estimated summary mean value d is both precise and unbiased. The variability among the estimated differences is again key to the decision to combine or not combine the k differences between proportions of low-birth-weight infants into a single summary estimate interaction? The summary odds ratio produces an apparently useful single summary of low-birth-weight risk associated with smoking.
The summary difference in proportions does not. A question not addressed is: Do the four groups under study systematically differ in their risk—outcome relationships? The question addressed is: Can the four measures of association be combined to create a single unbiased measure of the risk—outcome relationship? The second answered question refers to the issues concerning the statistical analysis. The answer to the second question, as illustrated, depends on the choice of the measure of association.
The difference between these two summary odds ratios directly measures the extent of confounding Chapter 1. Tabular Data 53 Table 2— The difference between the two summary odds ratios, 2. The degree of confounding depends on how it is measured. This chapter outlines two techniques key to statistical estimation in general: namely, maximum likelihood estimation and the derivation of the properties of statistical functions.
These somewhat theoretical topics are not critical to understanding the application of statistical methods, but provide valuable insight into the origins of parameter estimates and the variances of their distributions. The complexity of this technique lies in the technical application and not its underlying principle.
Maximum likelihood estimation is conceptually simple. A small example introduces the fundamental logic at the heart of this process. Furthermore, three tacks are tossed, and one lands point up and the other two land point down. When two values are proposed as an estimate of the parameter p, it is not hard to decide the most likely to have produced the observed result one up and two down and, therefore, is the better estimate of the unknown probability p.
Maximum likelihood estimation is a natural extension of the same logic. It is again the value most consistent with the observed data. For the thumb tack example, this value is 0. No other choice of p makes the observed data one up and two down more likely. Expanding this example continues to illustrate the logic of the maximum likelihood estimation process.
That is, the data are up, up, down, up, down, down,. The expression labeled L is called the likelihood function.
0コメント