Statistical Studies in Populations
In order for a study to be performed, the population must be defined. When one studies the extent of diabetes mellitus amongst the Hispanic population of Los Angeles, does that mean both males and females, adults only, persons who have emigrated from Latin America, persons who live just in the city of Los Angeles, etc? One must be very careful about defining the population to be studied.
Since it is not practical to perform tests or measures on all members of a population, then one must obtain a sample of that population. There are methods available to randomize the sampling of the population. The closer the measurements are to the "real" or true value for a population, the more unbiased the study. Precision in a study simply refers to how repeatable it is. The larger the sample, the more precise the study.
Example of a study:
You are conducting a health screening program in a community. You obtain a series of findings for a set of persons attending the program. This study includes adult men and women between the ages of 20 and 76 on a particular day in a particular community. This population is not narrowly defined. The results are as follows:
|Patient||BP (systolic)||BP (diastolic)||Glucose||Height||Weight
| 11||140 ||95||86||171||79
Distribution and Central Tendency:
A measure of the probability of a distribution of values is known as the central tendency and can be simply calculated as a mean, median, or mode:
Mean: This is the sum of a list of numbers, divided by the total number of numbers in the list. It is also called the arithmetic mean.
Median: This is the middle value in a list and is the smallest number such that at least half the numbers in the list are no greater than it. If the list has an odd number of entries, the median is the middle entry in the list after sorting the list into increasing order. If the list has an even number of entries, the median is equal to the sum of the two middle (after sorting) numbers divided by two.
Mode: For lists, the mode is the most common (frequent) value. A list can have more than one mode.
The range of values gives an indication of distribution of values and is just the highest value minus the lowest value.
In the above set of patients:
Measurement of Variability
Variability occurs in a set of values. Variability is the amount of distribution of values away from central tendency. Measures of the deviation of values from the central tendency can include the variance and the standard deviation.
Variance is the average of the squared deviations from the arithmetic mean. The standard deviation is a square root from variance value. The standard deviation is a measure of the variability of values around the mean and is meant to be used with values that are normally distributed (e.g., follow a normal curve). The standard normal curve is a bell-shaped curve. Non
normal (skewed) data can sometimes be transformed to give a graph of normal shape by performing some mathematical transformation (such as using the variable's logarithm, square root, or reciprocal). Some data, however, cannot be transformed into a smooth pattern. The data for height and weight are "positively" skewed because such measures do not approach zero.
Skewed distributions have a median that lies to the left or right of the mean. A measurement of the amount of skew can be given by the formula:
skew = 3(mean - median)/SD
In the above distribution of glucose values, the mean of 105 is slightly greater than the median of 101, so the skew is +0.5, or very slightly skewed to the right.
For most bell-shaped curves, 68% of the values fall within 1 standard deviation of the mean, 95% within 2 SD's, and 97.7% within 3 SD's. For most laboratory tests, the "normal range" is defined as values falling within 2 SD's of the mean. This is sometimes called the "95% confidence limits". In general, a "significant" P value of <0.05 corresponds to a 95% confidence limit. It is not possible to know the exact population mean, because we cannot perform measurements on everyone, but we can take a sample (preferably large) of persons to try and estimate the population mean.
For bigger numbers for a set of values, the standard deviation is bigger, but does this imply that the values are more variable than for a set of values with a smaller mean? The coefficient of variation can be calculated to determine this variability when comparing two sets of data with different means. The CV is calculated as the SD divided by the mean and multiplied by 100.
Another measure is the "standard error of the mean" or just standard error (SE) which is calculated as the standard deviation of each set of values divided by the square root of the number of the observations in the sample.
Confidence Limits and the t Test
The 95% confidence limits are typically 2 SD's from the mean for a large sample size, typically over 60 values. For smaller sample sizes, such as the one above, there is more likely to be variation from the mean. For analyzing the variance and estimating the standard deviation for a small sample, the "student" or "t" test is done. In such a test, the number of "degrees of freedom" is calculated, which is the sample size minus one, or 14 for the above group. One then uses a table of pre-calculated values for different confidence limits for different degrees of freedom. In the table, for 14 degrees of freedom at 0.05 probability, the value is 2.145. Thus, the 95% confidence limits would be 2.145 SD's from the mean, or slightly more than the 2 SD's for a larger group.
The "t" test is a "two-tailed" test because the "tail" of the distribution on each side of the mean is analyzed. For many laboratory measurements or clinical trials, one would want a two
tailed test because the value or the outcome could be either above or below the mean.
Note that the above set of patients has a mean, 105 mg/dL, and a SD, 22 mg/dL, which are much larger than for a typical "normal" population in which the mean is usually 90 mg/dL and the SD 10 mg/dL. Thus, the typical "normal range" for glucose is given as 70 to 110 mg/dL.
What is the likelihood that the populations are, indeed, different, and our population is abnormal compared to the "normal" population from which the normal range was calculated? The difference in means is 15 mg/dL, and the standard error of the mean for our population is 5.7 mg/dL. Dividing the former by the latter gives a "z" value of 2.63, which is more than 2 SD's, and therefore beyond the 95% confidence limits, so our sample study group is different from the normal population. This is a "one sample t test" because it measures the difference of sample mean from the population mean.
A t test comparing the difference in the means of two samples can also be calculated with a more complex formula. A "paired t test" can be performed using matched sets of data from a study group and a control group, for example.
A "Chi-square" test can be done to compare sets of observations, classically arranged in a "2 X 2" table, as in the comparison of compliance with two different treatment plans (two columns for compliance and non-compliance; two rows for treatment A and for treatment B). There can be more columns and rows, but the math gets more complex. A comparison is made of observed and expected values as follows:
Chi-square = sum of (observed - expected)2/expected
The degrees of freedom are calculated as: df = (rows -1)(columns - 1)
Thus, for a study comparing compliance with running and swimming as exercise regimens for weight maintenance, we might get the following data:
|Exercise||Complied||Did not comply||Total|| %Compliance
The overall compliance rate is 34.6%, so for the null hypothesis to be true, then 34.6% of each group would be expected to comply. Thus, the expected number for each group is given in parentheses, as follows:
|Running||15 (21)|| 46 (40)|| 61
|Swimming||29 (23)|| 37 (43)|| 66
Chi-square tests are not reliable for small numbers (for a total less than 40 and an expected number in a row less than 5).
Correlation and Regression
An association between data can be determined by a correlation coefficient. This can be done if the relationship is linear. It is often the case that a scatter plot of data comparing two measurements is done. For the patients above, one can plot the relationship of weight to glucose, as follows:
Looking at the plot suggests that the glucose is higher for persons who have a greater weight, but what is the correlation coefficient?
The correlation coefficient is measured on a scale that varies from + 1 through 0 to - 1. Complete correlation between two variables is expressed by either + 1 or -1. When one variable increases as the other increases the correlation is positive; when one decreases as the other increases it is negative. Complete absence of correlation is represented by 0.
The formula is a bit complex:
r = sum of paired (x)(y) - (n)(mean of x)(mean of y) / (n-1)(SD of x)(SD of y)
A t test can be done to determine the significance of this r value for the number of paired data items, in this case 15.
t = r (square root of (n-2)/1 - r2)
When the data on the x axis change as a function of data on the y axis, then there is a relationship known historically as "regression" and the "regression line" on a scatter plot is the line drawn through the dots that defines the amount of correlation. A line sloping at 45 degrees, with dots closer together, indicates better correlation, while a flat line indicates no correlation. Remember: correlation is NOT causation!
Covariance is a measure of how two data sets vary with respect to each other. Analysis of variance, or ANOVA, is the term given to the method of analysing data from two or more groups.
All statistical tests are either parametric (assuming the data were sampled from a particular type of distribution, such as a normal distribution) or non-parametric (no assumption of type of distribution is made). In general, parametric tests are better than non-parametric tests. Non-parametric tests generate a rank order of values and ignore the absolute differences between values. The statistical significance is more difficult to show with non-parametric tests.
Other Types of Distributions
In some clinical studies, results are recorded simply as positive or negative, with no gradation or quantification. Did the colon cancer therapy work or not? Data from such studies form what is called a "binomial" distribution. 95% confidence limits can be set for such a distribution, similar to a normal distribution.
The "Poisson" distribution is used to describe discrete quantitative data such as numbers of events when the size of the sample population is large but the probability of an event is small, though the number of events is moderate. An example is the number of deaths from a particular type of cancer in a community on a particular day.
"Null" hypothesis, type I error and type II error:
In comparing two sets of values from population groups, one can make the assumption that they will be the same. This is called the "null hypothesis". For most statistical studies the goal is to show that the null hypothesis is unlikely, so a difference which is greater than the limits set, and which we therefore regard as "significant", will make the null hypothesis unlikely.
To reject the null hypothesis when it is true is to make what is known as a type I error, or "alpha" error (a false positive). The level at which a result is declared significant is known as the type I error rate, often denoted by alpha.
If the null hypothesis is not rejected when there is a real difference between the groups, then this is known as a type II error, or "beta" error (a false negative).
More about studies:
In order to conduct a study, subjects must be selected. Selection is the process by which a sample is recruited from a population. If the sample selected is truly random then observations from this population will be expected to yield a sample that is representative of the entire population.
However, recruitment is often not random. Instead, studies often rely on volunteers, and this is "self-selection" which is a non-random recruitment. This could constrain the population and hinder the ability to generalize any study findings.
Assignment is the process by which a sample is further divided into experimental and control groups. A random assignment of the sample will be expected to generate groups that are similar and, hence, any difference between them will be due to the experimental conditions. Any non-random assignment will limit a study's ability to control for non-experimental characteristics.
Thus, randomization of a self-selected (volunteer) sample will generally produce similar experimental and control groups and yield results close to those of a truly random sample selection process.
Internal validity refers to the validity of the sample observed in a study and the conditions under which the data were gathered for that particular study.
External validity refers to the validity of generalizing the sample data from the study to the population as a whole.
Types of Studies
The best type of study is the one that is performed prospectively. A retrospective study starts with diseased subjects and then examines for possible causes. The best example of a prospective study with patients is the randomised controlled trial in which the subjects with a disease are randomised to one of two (or more) treatments, one of which may be a control treatment. Randomization insures that treatment groups will be balanced with both known and unknown prognostic factors. The treatments tested in the study should be concurrent, with the tested and control treatments given over the same time frame. The subjects need to be compliant with the treatment in order to have valid outcomes.
A parallel group design is one in which treatment and control are allocated to different subjects in the study. One set of subjects typically receives a "placebo" treatment that appears identical to the study treatment, but without a real effect (such as an inert compound instead of a real drug).
The best study is double blinded so that neither the investigator nor the subjects in the study are aware of which treatment is given to which subject. This is easiest to do for drugs.
A crossover study measures the effects of two or more sequential treatments given to the same set of subjects. In such as study each subject acts as his or her own control, reducing the requirement for more subjects to serve as a control group. However, there may be a carry over effect from the first treatment to affect outcomes for the second treatment.
A cohort study follows initially disease free subjects over a period of time. During that time, some subjects are exposed to risk factors, such as alcohol consumption, and the outcomes are measured. The cohort may be defined as persons born in a particular year (making them all the same age), persons who lived in a particular community, or persons who worked in a particular place The outcome may be a particular disease state (cirrhosis) or death. The prevalence of the disease studied is known or can be estimated from the data. Cohort studies need large numbers of subjects studied for long periods of time to be valid. Here is an example of such a study examining the risk for developing diabetes mellitus with a risk factor of cheeseburger consumption in a cohort of persons born in 1940:
|Subject type||Subjects with diabetes mellitus in follow-up||Subjects without diabetes mellitus in follow-up||Total
|Subjects eating cheeseburgers||125 (a)|| 575 (b)|| 700 (a + b)
|Subjects not eating cheeseburgers||62 (c)|| 638 (d)|| 700 (c + d)
The risk for developing the disease for those exposed is: a / a + b
The risk for develolping the disease for those not exposed is: c / c + d
The "relative risk" or RR is the ratio of these: RR = a(a + b) / c(c + d)
In a "case control" study, one defines a population with a particular disease and then finds a suitable control group without the disease. One then compares the two using a particular risk factor for the disease. The cases and the controls may be matched for variables such as age, sex, and race, but they may be unmatched. Here is an example of an unmatched case control study comparing the risk for esophageal adenocarcinoma in persons with GERD:
|Disease state||Subjects with adenocarcinoma||Subjects without adenocarcinoma
|GERD present||15 (a)||985 (b)
|GERD not present||2 (c)||998 (d)
Unlike the cohort study, a relative risk is not used in a case control study, where the prevalence of the disease is not known, and the apparent prevalence is based upon the ratio of sample cases to controls. Instead, an "odds ratio" is computed as follows:
Odds Ratio = a X d / b X c
An odds ratio can give a reasonable estimate of the relative risk when the proportion of subjects with the disease is small.
A "cross-sectional" study includes subjects without reference to their history of exposure or to their disease. Instead, the cross-sectional study analyzes the cases prevalent at the time of the study. Such studies may be based upon methods of sampling of a population, such as questionnaires, but there are problems with true random sampling.
Prevalence and Incidence and Rate:
The prevalence of a disease is the proportion of a population that are cases at a point in time. For example, the prevalence of systemic lupus erythematosus may stated to be 7 per 1000. The group measured can be defined more narrowly, as: the prevalence of diabetes mellitus in women ages 50 to 60 is 7%.
The incidence of a disease is the rate at which new cases occur in a population during a specified period. For example, the incidence of influenza in the year 2001 is 20 per 1000 per year.
In general, prevalence is used to track diseases that are more chronic, while incidence is used for more acute conditions, such as infectious diseases, with a shorter course.
The incidence contributes to the prevalence. For example, if there were 12 women in 1000 diagnosed with breast cancer last year (the incidence) and there were 39 in the same population already diagnosed and known to be living with breast cancer, a tumor registry would then report the prevalence as 0.051.
A rate is the number of events per unit of population over a particular time span. An example of this is mortality. Mortality is the incidence of death from a disease. The crude mortality rate for a given year is stated as:
Crude Mortality Rate = (Number of deaths in a year / mid-year population) X 1000
An "age-specific" mortality rate can be given as:
ASMR = (Number of deaths in a specific age group / mid-year population of that group) X 1000
This is a quantitative measure of uncertainty. For a coin flip, the probability of either heads or tails is 0.5. Probability is given a fraction between 0 and 1. A probability of 0 means that the event cannot occur; a probability of 1 means that an event will always happen. The probability of an event is the ratio of the number of outcomes that constitute the event to the total number of possible outcomes:
P(event) = (number of event outcomes ) / (number of total outcomes).
In case of tossing coin, the probability of getting heads is: P(head) = 1/2 = 0.5.
Patients in randomised trials are not a random sample from the population of people with the disease in question but are a highly selected set of patients who were available and were willing to participate. However, it is possible to randomize this set into treated and untreated groups so that any differences in outcomes between the two treatment groups are due solely to differences in the treatment to be studied.
Laboratory Testing Principles
Results fall into four categories:
True positives (TP) Persons who really have the disease and test positive
False negatives (FN) Persons who really have the disease but test negative
True negatives (TN) Persons who do not have the disease and test negative
False positives (FP) Persons who do not have the disease but test positive
The usefulness of a laboratory test can be measured by:
Diagnostic Sensitivity: how well can the test detect persons who really have the disease?
Sensitivity = true positives ÷ (true positives + false negatives)
Diagnostic Specificity: how well can the test exclude persons without the disease?
Specificity = true negatives ÷ (true negatives + false positives)
Example: In a given population, 1000 persons are tested for the presence of a particular disease. Of these, 80 are found to test positive. However, only 40 of these are found on subsequent confirmatory testing to really have the disease. Furthermore, follow-up of the original group of patients reveals that there were 10 people who really had the disease, but were missed by the initial screening test. Calculate the diagnostic sensitivity and specificity for the original screening test:
Sensitivity = 40 true positives ÷ (40 true positives + 10 false negatives)
Specificity = 910 true negatives ÷ (910 true negatives + 40 false positives)
So what does a positive or negative test really mean? This can be measured by positive and negative predictive values (PV):
PV of a positive test = true positives ÷ (true positives + false positives)
PV of a negative test = true negatives ÷ (true negatives + false negatives)
In the example of the screening test above:
PV(+) = 40 true positives ÷ (40 true positives + 40 false positives)]
PV(-) = 910 true negatives ÷ (910 true negatives + 10 false negatives)
Predictive values have a lot to do with the prevalence of the disease, or the number of persons in the population who actually have the disease (incidence of disease is only the new cases that are reported). In the above example, the prevalence of the disease was 6%, which is quite high. Few diseases have that high a prevalence in a population.
The prevalence of most diseases is low. Thus positive predictive value, even for a good test with a sensitivity of 95%, can be poor when there are few persons with the disease, and most of the positives will be false positives.
As an example, the best test in the laboratory is the HIV antibody test, which has a sensitivity of 99.9% and a specificity of 99.7%. In a given population (such as in rural areas) where the prevalence of the disease being tested is around 1:10,000 the predictive value of a positive test will be quite low. Of course, the test is still useful, but it is a screening test, and a repeat assay and an additional confirmatory test (Western blot) are needed to find the true positives.
The following chart indicates the performance of testing based upon prevalence:
|Prevalence of Disease (%)
||Predictive Value of a Positive Test (%)
- The generalist, primary care physician is the initial person who sees many patients and who has to deal with the problem of ordering and interpreting screening tests.
- Why can't you have both 100% sensitivity and 100% specificity? The ranges of test values in a population typically have some overlap for persons with and without the disease:
- You can obtain maximum sensitivity at point A, but only at the expense of generating many more false positives that require additional workup to exclude.
- You can obtain maximum specificity at point B, but only at the expense of generating many more false negatives and miss patients with the disease.
- You can improve predictive value by first narrowing down the population to be tested with standard history and physical exam (e.g., don't order superfluous lab tests). Example: you can progressively improve you chances of getting a meaningful result for a prostate specific antigen test if you order it on: men (this should be obvious), older men, older men with a palpable nodule.
- Performance Characteristics of Laboratory Testing
- Accuracy: How well does the test measure what is really there? Agreement of the test results with the patient's condition is the best measure of accuracy.
- Example: clinical diagnosis of acute appendicitis is about 90-95% accurate
- Question: How accurate is the standard history and physical exam?
Question: What is the "gold standard" by which you measure accuracy? Is it the word of your attending physician? A consultant? A laboratory test result? Autopsy? (Note: a courtroom decision on a medical matter may not be based upon scientific principles, but nonetheless can modify how we practice.)
Example: you perform a physical examination on a newborn and determine that the baby has slanting epicanthal folds, bilateral transverse palmar creases, and an absent distal flexion crease on the fifth digits of both hands. You suspect Down syndrome. The "gold standard" is cytogenetic analysis of baby's cells, which reveals a 47, XY, +21 karyotype.
- Precision: How reproducible is the test under the same conditions? The laboratory tries to assure reproducibility by the use of control specimens with each run of patient specimens. The instruments have a routine maintenance and check procedure performed as well.
- You can be precise but not accurate by making the same error consistently.
Example: you may be using improper technique to measure blood pressure, but you will keep getting the same result, which is different from what the nurse (who is positioning the cuff properly and listening appropriately) records.
- Accuracy and precision can apply to written and verbal communcations. Lack of understanding or failure to properly record observations can have an impact.
- Example: the patient's primary physician palpates a "lump" on the left side of the neck of his 17 year old patient. An imaging study is performed, and the lesion is a 3 cm well-circumscribed cyst in the soft tissue of the left lateral neck. The lesion is recorded as consistent with a "brachial cleft cyst". The surgeon's operative report records removal of a "brachial cleft cyst". The surgical specimen is sent to pathology, and the final reported diagnosis is "brachial cleft cyst". The resulting medical record is quite precise, but totally innaccurate, because everyone has made the classic freshman anatomy mistake of confusing the terms "brachial", "branchial", and "bronchial". (It is a branchial cleft cyst.)
- CV: Coefficient of variation. Just how variable are the test results. This depends upon the test methodology, the instrument being used, and the range of results. (The CV is calculated by dividing the standard deviation by the mean.)
- Sodium (Na) of 138 mmol/L is probably between 137.5 and 138.5 mmol/L
- Hgb of 10.0 g/dL is probably between 9.8 and 10.2 g/dL
- Glucose of 800 mg/dL is probably between 770 and 830 mg/dL
- Thus, a change in values from one day to the next generally has to be 10% or more to be of major significance. Just specimen handling, processing, and instrument variation can account for some changes. Running a test in duplicate will show this.
- Bear in mind that there is also "physiologic variation" in patients that is dictated by factors such as the degree of hydration, diet, and exercise.
Example: an elderly person admitted with an apparently normal hemoglobin, but an elevated urea nitrogen and glucose, may be dehydrated, and upon administering fluids will be found to have anemia, but normal renal function, and the glucose was slightly high because she just ate.
- If you rely on specific "numbers" for decision points, you may run into trouble.
Example: it is late afternoon and the physician checks lab values for tests ordered on his patients. He notes that his elderly patient has a hemoglobin of 9.9 g/dL, whereas the value was 10.1 g/dL early in the morning that same day. The physician's "set point" for ordering a transfusion is 10 g/dL, even though this is not a recognized practice standard. Using such criteria, an unnecessary transfusion, subjecting the patient to potential complications, would be given. But the two values could have come from either the morning or afternoon specimens run in duplicate!
- What is "normal"?
- The laboratory sets "normal" ranges for laboratory tests based upon population studies. A test may have a single normal range, or there may be different normal ranges based upon age, sex, race, or other factors. Sometimes, more history is needed for interpretation (such as with maternal serum alpha-fetoprotein in pregnancy, which is dependent upon the gestational age--the later in gestation, the more AFP is present normally) so that is why this information needs to be provided. Otherwise, you may have an uninterpretable result.
- Standard "normal" ranges for tests with numeric values are based upon use of a bell shaped curve. "Normal" is defined as those test values that fall within 2 standard deviations of the mean, which includes 95% of all results. The standard deviation is just a measure of dispersion.
Thus, there is a 1 in 20 chance that an "abnormal" test may really be normal. If you perform 20 or more independent tests (which is not uncommon on patients admitted to hospital), then there is a greater than 50% likelihood that one or more tests will be "abnormal" just from statistical variation. If you keep ordering more tests just to track these down, you can go on for a long time and spend a lot of money.
However, size counts! The farther out of range the test result is, the more likely that the result reflects real disease.
- GUIDING RULE: It is better to treat the patient than the numbers.
- What are the accuracy, precision, and preditive values for clinical assessment?
- Very little may sometimes be done in regard to quality assessment of clinical activities, such as history taking and physical examination. The following story illustrates this point:
- In 1888, Nellie Bly (Elizabeth Cochrane) was a reporter for the New York World, the premier tabloid of its era. She was one of the first true investigative reporters, although a lot of what she did was publicity stunts to sell newspapers (such as her most famous stunt, "Around the world in 80 days" which was made in 72 days, 6 hours, and 11 minutes). One of her stunts that served a useful purpose was an exposé of the New York mental health care system, which consisted of asylums where the mentally ill were placed. She acted the part of an insane woman and allowed herself to be committed to Blackwell's Island, New York City's most notorious insane asylum. She then wrote an exposé of the mistreatment of patients that got the attention of reformers and readers alike, shown in the front pages of the New York World, and that got the asylum closed down. She described the asylum as "
a human rat-trap. It is easy to get into the place, but once you are there, it is impossible to get out." In fact, the editor of the newspaper had to get the police to extricate her 10 days later from the asylum. The diagnostic tools and criteria employed were so poor that the staff could not, or would not, determine who was really mentally ill and who wasn't.
- Medical Necessity
- When you order tests or procedures, you must document the medical necessity for the order (i.e., you must justify what you are doing). Failure to do so will result in the charges for the test or procedure being denied (i.e., you or the institution for which you work will not get paid).
- If you order tests based upon misinterpretation of findings from previous testing, the problem is compounded.
- Every test ordered must have a reason. Charges for tests which have documentation that indicates they were done as "standing orders" or as "routine" will be flatly denied.
- Tests may be appropriate depending upon the time course of a workup for disease. Primary physicians may appropriately order screening tests. However, if a urologist were to order a "screening prostate specific antigen test" then the charge would be denied.
- COCHRANE'S APHORISM: Before ordering a test, decide what you will do if the test is 1) positive, or 2) negative. If both answers are the same, do not order the test.
Answers to Questions:
1. What is the mean for the glucose?
Answer: 105 mg/dL
2. What is the median weight?
Answer: 79 kg
3. What is the mode for height?
Answer: 170 cm
4. What is the range for systolic blood pressure?
Answer: 70 mm Hg
5. What is the standard deviation for glucose values?
Answer: 22 mg/dL
6. What is the CV for systolic B.P.? For diastolic?
Answer: The SD for systolic is 18 and for diastolic is 11, which appear considerably different, but the CV for systolic is 13.8 and for diastolic is 12.6, much closer, indicating that the two sets are not that much different in terms of variability.
7. What is the standard error (SE) for glucose in the above patients?
Answer: 5.7 mg/dL
8. What is the Chi-square for this study and what is the significance?
Answer: Computing the Chi-square gives a value of 1.7 + 0.9 + 1.6 + 0.8 = 4
This is more than the value of 3.841 given in a table of chi square values for 1 degree of freedom for a probability of < 0.05, so this difference between the groups is significant.
9. In the above case, what is the correlation coefficient?
Answer: r = 0.88
10. In this case, what is t and what does it mean?
Answer: t = 6.6
For 13 degrees of freedom, 6.6 is much larger than the value of 2.16 for a 0.05 probability, and larger than the value of 4.22 for a 0.001 probability. Thus, this correlation is significant.
11. What is the RR is this study for developing diabetes mellitus from eating cheeseburgers?
Answer: RR = 2
Of course, in reality there are multiple factors at work, and the risk for any one event is determined by many risks.
12. What is the odds ratio for development of esophageal adenocarcinoma when GERD is present in this study?
Answer: Odds Ratio = 7.6